PAR
Christophe NICOLET
Ingnieur mcanicien diplm EPF
de nationalit suisse et orginaire de Cottens (FR)
Lausanne, EPFL
2007
Remerciements
Il me tient `a coeur, ici au debut de ce document, de remercier toutes les personnes qui
ont contribues de pr`es ou de loin `a la realisation de ce travail de th`ese.
Ce travail naurait pas ete possible sans le soutien financier des fonds de recherche de
la CTI, et du PSEL, ainsi que celui des exploitants EOS, BKW, EEF, SIG, EDF, de Electricite Suisse, mais egalement de celui des constructeurs VOITHSiemens et ALSTOM.
Je commence par exprimer tout ma gratitude `a mes deux directeurs de th`ese, le
Professeur Francois Avellan, directeur du Laboratoire de Machines Hydrauliques, et le
Professeur JeanJacques Simond, directeur du Laboratoire de Machines Electriques, pour
mavoir propose et encourage `a me lancer dans cette enrichissante aventure quest le
travail de th`ese. Durant cette periode, jai beaucoup apprecie la liberte de manoeuvre
et la confiance dont jai pu beneficier ainsi que de latmosph`ere stimulante et dynamique
quils ont cree et maintenu au cours du projet.
Mes remerciements vont egalement aux membres du jury, Professeur Paul Xirouchakis,
President du Jury, Dr. Erik Bollaert de lEPFL, Dr. Jiri Koutnik de VOITHSiemens
et Professeur Yoshinobu Tsujimoto de lUniversite de Osaka, pour le temps consacre `a
lecture de ce manuscrit ainsi que pour toutes les remarques et discussions constructives
lors de la defense privee.
Je souhaite egalement remercier les differents responsables du LMH, le Professeur
JeanLouis Kueny, pour sa bonne humeur et son enthousiasme, Jean Prenat pour ses
encouragements, et le Dr. Mohammed Farhat pour son aide lors de la realisation des
mesures.
Dans le cadre des projets industriels auxquels jai pu participer, jai beaucoup apprecie
la collaboration avec le Dr. JeanJacques Herou et Mme Nadine PajeanWassong de EDF,
le Dr. Johann G
ulich de Sulzer Pumps, et de M. Jordi Rossell de Meditecnic. Je remercie
tout particuli`erement le Dr. Jiri Koutnik et ses coll`egues de VOITHSiemens avec qui ca
a vraiment ete un plaisir de collaborer.
Je souhaite remercier le personnel du groupe GEM, et du BE, pour leur aide lors
de la realisation dessais, de dessins, pour leurs divers coups de mains et aussi pour
lambiance de travail agreable `a laquelle ils contribuent : merci `a Pierre Barmaverain,
Philippe Faucherre, Alain Renaud, Pierre Dutoit, HenriPascal Mombelli, Georges Crittin et `a Philippe Cerrutti. Toutes les experiences et autres manips nauraient bien s
ur
pas ete possibles sans le soutien de lequipe de latelier, coordonnee par Louis Bezencon,
que je remercie aussi pour ses bons tuyaux oenologiques, JeanDaniel Niederhauser, Raymond Fazan, Maxime Raton, Jerome Gruaz et Christian Sierro. Je tiens aussi bien s
ur
`a remercier Isabelle Stoudmann, Maria Anitua, Anne Ecabert et Shadije Avdulahi pour
leur aide au quotidien et leur contribution `a latmosph`ere du laboratoire.
Je souhaite bien s
ur remercier mes compagnons de laventure SIMSEN, je pense en
particulier `a Philippe Allenbach pour toutes les heures passees `a debugger le programme,
heureusement parfois recompensees par une mousse salvatrice, mais aussi pour son enthousiasme dans le projet. Je remercie egalement le Dr. Alain Spain, p`ere de SIMSEN,
qui a su me faire profiter de son experience et qui a toujours ete disponible pour nous
apporter son aide. Je remercie le Dr. Mai Tuxuan pour ses explications sur les machines
electriques et bien dautres domaines, sans oublier le Dr. Basile Kawkabani, Stefan Keller,
et Antoine Beguin avec qui la collaboration a ete agreable.
Je souhaite remercier les etudiants qui ont contribue `a ce travail au travers de leur
travaux de diplomes, merci `a Bob Greiveldinger, pour son aide pour la modelisation
des reseaux, `a Nicolas Ruchonnet, pour son aide dans les developpements rotorstator,
Yves Vaillant pour son aide avec la Kaplan, et Cecile Picollet pour son aide dans les
manipulations de caracteristiques.
Une mention toute speciale `a mon Silverback de compagnon dinfortune dans cette
aventure que represente une th`ese, le regrette Dr. Alexandre Perrig, qui maura toujours
impressionne par sa rigueur avec luimeme, mais aussi par ses capacites redactionnelles
dont il na jamais hesite `a me faire profiter, et avec qui les discussions scientifiques et
gastronomiques ont ete tr`es sympathiques. Merci pour la superbe ambiance tant au
travail que lors dexcursions valaisannes, gruy`eriennes ou meme nepalaises!
Je ne saurais oublier tous mes coll`egues et amis, anciens ou actuels, du laboratoire avec
qui les jeudis de langoisse et les terrasses de StSulpices resteront dexcellents moments,
comme dirait un certain Bob, cetait YES PUR, donc un grand merci `a Tino, Youss,
Sonia, Lluis, Silvia, Philippe, Ali, Olivier Braun, Nicolas, Fred, Javi, Stefan, Cecile,
Pierre, Faical, Bernd, PierreYves, Lavinia, Gabi, Georgetta et Monica, mais aussi Jorge,
Gino, Couty, Alex, Sebastiano, et scusi `a tous ceux que jaurais pu oublier...
Je remercie egalement mes amis et mes colocs avec qui jai pu partage cette aventure
Lausannoise, merci `a Win, Val, Guillaume, Ivano, Eric, Brigitte, Orane, Oli, Andreanne,
Lio, Nat, Reynald, Arrigo, Eli, Zoe, Christine, et tous les autres.
Enfin, je tiens `a remercier du fond du coeur mes parents Mutti et Padre et ma soeur
Anne, qui mont toujours encourages et mont soutenu tout au long de mon parcours !
R
esum
e
La production hydroelectrique representait en 1999 le 19% de la production mondiale
delectricite et il est `a prevoir quelle aie une forte augmentation dici `a 2030. Les turbines Francis ont un role majeur `a jouer dans cette production de par leur gamme etendue
dapplication. La deregulation du marche de lelectricite conduit `a une exploitation des
turbines hors de leurs conditions optimales de fonctionnement et engendre des sequences
darrets et de demarrages frequentes. De ce fait, les turbines Francis sont sujettes `a
des sollicitations relatives `a des phenom`enes transitoires et periodiques dont la nature
propagative est preponderante. Ce travail est une contribution `a la modelisation hydroacoustique des centrales hydroelectriques equipees de turbines Francis.
La premi`ere partie de ce travail presente la modelisation du comportement dynamique
et letude des phenom`enes transitoires survenant dans les centrales hydroelectriques.
Ainsi, un mod`ele monodimensionnel dune conduite est etabli `a partir des equations fondamentales de conservation de la quantite de mouvement et de continuite. Lutilisation
de schemas numeriques appropries permet de representer une conduite par un circuit
electrique equivalent en T. La precision de ce mod`ele numerique est evaluee dans le domaine frequentiel par comparaison `a une solution analytique.
Cette approche de modelisation est ensuite etendue `a dautres composants hydrauliques
tels que : vanne, cheminee dequilibre, reservoir dair, developpement de cavitation, etc.
Ensuite, la modelisation des turbines Francis, Pelton et Kaplan est presentee. Cette
modelisation est basee sur lutilisation de caracteristiques statiques des machines hydrauliques. Les mod`eles des composants hydrauliques ont ete implementes dans un logiciel
de simulation dinstallation electrique nomme SIMSEN. Apr`es la validation des mod`eles
des composants hydrauliques, une etude des regimes transitoires dans les centrales hydroelectriques est realisee. Il est montre en particulier, que la modelisation seule de la
partie hydraulique ou electrique de la centrale est suffisante pour le dimensionnement de
linstallation, alors que loptimisation des regulateurs de vitesse des turbines requiert une
modelisation de la centrale dans son ensemble.
La deuxi`eme partie de ce travail porte sur la modelisation et lanalyse de phenom`enes
de resonance ou dinstabilite survenant dans les centrales hydrauliques equipees de turbines Francis. La revue des sources dexcitations inherentes `a lexploitation des turbines
Francis indique que les fluctuations de pression dues au diffuseur et aux effets rotorstator
sont preponderantes. Etant donne que la modelisation des fluctuations de pression dans le
diffuseur survenant `a charge partielle est bien etablie, laccent est mis sur la modelisation
des phenom`enes tels que les fluctuations de pression de haut de charge partielle, de rotorstator ainsi que sur les instabilites de fortes charges.
Trois etudes hydroacoustiques sont realisees. Dabord, les phenom`enes de fluctuations de haut de charge partielle identifies experimentalement sur un stand dessai sont
Mots clefs:
Abstract
Hydropower represented in 1999 19% of the world electricity production and the absolute
production is expected to grow considerably during the next 30 years. Francis turbines
play a major role in the hydroelectric production due to their extended range of application. Due to the deregulated energy market, hydroelectric power plants are increasingly
subjecting to off design operation, startup and shutdown and new control strategies.
Consequently, the operation of Francis turbine power plants leads to transients phenomena, risk of resonance or instabilities. The understanding of these propagation phenomena
is therefore paramount. This work is a contribution to the hydroacoustic modelling of
Francis turbine power plants for the investigation of the aforementioned problematic.
The first part of the document presents the modelling of the dynamic behavior and the
transient analysis of hydroelectric power plants. Therefore, the onedimensional model
of an elementary pipe is derived from the governing equations, i.e. momentum and
continuity equations. The use of appropriate numerical schemes leads to a discrete model
of the pipe consisting of a Tshaped equivalent electrical circuit. The accuracy in the
frequency domain of the discrete model of the pipe is determined by comparison with the
analytical solution of the governing equations.
The modelling approach is extended to hydraulic components such as valve, surge
tanks, surge shaft, air vessels, cavitation development, etc. Then, the modelling of the
Francis, Pelton and Kaplan turbines for transient analysis purposes is presented. This
modelling is based on the use of the static characteristic of the turbines. The hydraulic
components models are implemented in the EPFL software SIMSEN developed for the
simulation of electrical installations. After validation of the hydraulic models, transient
phenomena in hydroelectric power plants are investigated. It appears that standard separate studies of either the hydraulic or of the electrical part are valid only for design
purposes, while full hydroelectric models are necessary for the optimization of turbine
speed governors.
The second part of the document deals with the modelling and analysis of possible
resonance or operating instabilities in Francis turbine power plants. The review of the
excitation sources inherent to Francis turbine operations indicates that the draft tube
and the rotorstator interaction pressure fluctuations are of the major concern. As the
modelling of part load pressure fluctuations induced by the cavitating vortex rope that
develops in the draft tube at low frequencies is well established, the focus is put on
higher frequency phenomena such as higher part load pressure fluctuations and rotorstator interactions or full load instabilities.
Three hydroacoustic investigations are performed. (i) Pressure fluctuations identified
experimentally at higher part load on a reduced scale model Francis turbine are investigated by means of hydroacoustic simulations and high speed flow visualizations. The
resonance of the test rig due to the vortex rope excitation is pointed out by the simulation while the special motion and shape of the cavitating vortex rope at the resonance
frequency is highlighted by the visualization. A description of the possible excitation
mechanisms is proposed. (ii) A pressure and power surge measured on a 4 400 MW
pumpedstorage plant operating at full load is investigated. The modelling of the entire
system, including the hydraulic circuit, the rotating inertias and the electrical installation
provides an explanation of the phenomenon and the related conditions of apparition. A
nonlinear model of the full load vortex rope is established and qualitatively validated.
(iii) The rotorstator interactions (RSI) are studied in the case of a reduced scale pumpturbine model. An original modelling approach of this phenomenon based on the flow
distribution between the stationnary and the rotating part is presented. The model provides the RSI pressure fluctuation patterns in the vaneless gap and enables to predict
standing waves in the spiral case and adduction pipe.
The proposed onedimensional modelling approach enables the simulation, analysis
and optimization of the dynamic behavior of hydroelectric power plants. The approach
has proven its capability of simulating properly both transient and periodic phenomena.
Such investigations can be undertaken at early stages of a project to assess the possible
dynamic problems and to select appropriate solutions ensuring the safest and optimal
operation of the facility.
Keywords:
Hydraulic systems, Francis turbine, transient simulation, hydroacoustic modelling, dynamic behavior, electrical analogy.
Contents
Introduction
1 Introduction
1.1 Hydropower: Facts and Issues . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Increase of Hydropower Production . . . . . . . . . . . . . . . . .
1.2.1 Increase of the Capacity . . . . . . . . . . . . . . . . . . . . . .
1.2.2 OverEquipment for Improving Network Stability . . . . . . . .
1.2.3 The Hydropower Challenge . . . . . . . . . . . . . . . . . . . .
1.3 Francis Turbine in the Context of Hydropower . . . . . . . . . . . . . .
1.3.1 The Francis Turbine . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Operating Stability of Francis Turbine Units . . . . . . . . . . .
1.3.3 Transient Behavior of Francis Turbine Units . . . . . . . . . . .
1.4 The Role of Numerical Simulation in Improving Hydropower Operation
1.5 Description of the Present Work . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Problematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Structure of the Document . . . . . . . . . . . . . . . . . . . . .
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2 Fundamental Equations
2.1 General . . . . . . . .
2.2 Momentum Equation .
2.3 Continuity Equation .
2.4 Simplified Equations .
2.5 Resolution Methods . .
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CONTENTS
3.3
3.4
3.5
Resolution of the Set of Hyperbolic Partial Differential Equation for Discrete System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Numerical Resolution of the Hyperbolic Partial Differential Equation set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Equivalent Scheme Representation . . . . . . . . . . . . . . . . .
3.3.3 Free Oscillation Analysis: Discrete System . . . . . . . . . . . . .
Comparison of Continuous and Discrete Simulation Model . . . . . . . .
3.4.1 Truncation Error . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Comparison of Hydroacoustic Models . . . . . . . . . . . . . . . .
3.4.3 Frequency Confidence Threshold of the Model . . . . . . . . . . .
Summary of the Approach . . . . . . . . . . . . . . . . . . . . . . . . . .
Hydraulic Systems
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6.3
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iv
CONTENTS
9.2.2
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9.2.4
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9.2.8
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CONTENTS
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Appendices
A Numerical Integration Methods
A.1 Integration Methods . . . . . . . . .
A.1.1 Explicit RungeKutta Method
A.1.2 Implicit RungeKutta Method
A.2 Comparison of the Methods . . . . .
A.2.1 Waterhammer Phenomenon .
A.2.2 Surge Phenomenon . . . . . .
A.2.3 Van der Pol Equation . . . . .
A.3 Stability Analysis of RK Methods . .
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283
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References
297
Index
309
List of Publications
311
Curriculum Vitae
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vi
CONTENTS
Notations
Nomenclature
Latin characters
a
dx
dt
f
g
h
lc
n
n
p
p
t
x
y
z
A
C
D
E
E
Ec
Er
H
K
N
R
T
p
Wave speed a = E/ (for infinite fluid)
Spatial discretization step
Temporal discretization step
Frequency f = 1/T
Terrestrial acceleration g ' 9.81 m/s2
Piezometric head h = z + p/(g)
Characteristic length
Polytropic coefficient
Rotational frequency
Static pressure
Fluctuating pressure
Time
Abscissa
Guide vane opening (GVO)
Elevation
Pipe area
Absolute flow velocity
Diameter
Massic energy E = gH = p/ + gz + C 2 /2
Bulk modulus
Young modulus
Massic energy loss
Head
Local loss coefficient
Rotational speed
Radius R = D/2
Torque
[m/s]
[m]
[s]
[Hz]
[m/s2 ]
[m/s2 ]
[m]
[]
[Hz]
[Pa]
[Pa]
[s]
[m]
[]
[m]
[m2 ]
[m/s]
[m]
[J/Kg]
[Pa]
[Pa]
[J/kg]
[m]
[]
[rpm]
[m]
[Nm]
viii
NOTATIONS
Period
[s]
Peripheral velocity
Volume
[m/s2 ]
Acoustic impedance
[m/s2 ]
[m/s]
[m3 ]
Greek symbols
Flow angle
[rad]
Blade angle
[rad]
Wave number = 2/ = /a
[1/m]
[1/m]
[]
Adiabatic coefficient
[]
[m]
[]
[Pa s]
[m2 /s]
Density
Stress
[Pa]
Shear stress
[Pa]
Phase
[rad]
Pulsation = 2f
[kg/m3 ]
[s]
[rad/s]
[]
Re
Reynolds number Re = C lc /
[]
St
Strouhal number St = C /f lc
[]
Efficiency = T /(QE)
[]
[]
Thoma number = (p
[]
[]
[]
[J/Kg]
NOTATIONS
ix
[s/m2 ]
Rd
[s/m2 ]
Rve
[s/m2 ]
L
L
[m2 ]
[m2 ]
[s2 /m2 ]
[s2 /m2 ]
Stay vanes
Guide vanes
1e 1i
1e
1i
NOTATIONS
Subscript
BEP
n
i
r
t
Introduction
Chapter 1
Introduction
1.1
The demand for electricity is constantly increasing because of the demographic growth
and social level increase of developing countries. Worldwide projections for the period
20032030 predict that the electricity consumption will more than double from 14781
TWh/year to 30116 TWh/year [50]. Figure 1.1 shows the evolution of the sources of
electricity generation from 2003 until 2030. To cope with this need, a gain in efficiency in
all domains, i.e. production, transport, consumption, but also an increase of renewable
energies capacity are required in order to refrain the development of solutions generating greenhouse gases. The development of the related infrastructure and technologies
minimizing social and environmental impacts represents a huge challenge for mankind.
400
Coal
Natural gas
Renewable
Nuclear
Oil
Btu
300
200
100
0
1990
2000
2010
Year
2020
2030
2040
Figure 1.1: Projection of the electricity generation by fuel type for 2003, 2015 and 2030
in Btu (British thermal units) [50].
The contribution of renewable energy in the future is expected to grow mainly due to
the increase of large hydropower capability. In 1999, the hydropower production covered
EPFL  Laboratoire de Machines Hydrauliques
CHAPTER 1. INTRODUCTION
19% of the world electricity needs with a total installed capacity estimated around 692
GW [156], [144], [78]. The geographical distribution of the total production capacity is
illustrated in figure 1.2 with about 31% of the installed capacity in Europe, 25% in Asia,
23% in North America, 15% in South America and the remaining 6% are shared between
Africa, Oceania and the Middle East. However, the 2633 TWh of hydroelectricity produced in 1999 represents only 33.2% of the economically exploitable resources and 18.3%
of the technically exploitable capability. Therefore, hydropower has still a high potential
for growth in the 21st century. As illustrated in figure 1.3, the regions with the highest
potential are Asia, Africa, South America but also in Europe. In addition, hydropower
presents the advantage of avoiding emissions of gases in spite of others environmental
impacts on the fauna, flora and sediments. The social impact are on the one hand detrimental because of the population displacements and land transformation but on the other
hand positive as hydropower offers the possibility to mitigate flood, enabling better fluvial
navigation and irrigation and providing employment. Moreover, the drawbacks related to
hydropower production can be mitigated by taking appropriate counter measures at the
early stages of the projects.
Middle East
0.6044
Oceania Africa
1.911
2.913
North America
23.12
Europe
30.96
South America
15.35
Asia
25.14
Figure 1.2: Distribution of hydropower capacity (for the year 1999); 100% being 692 GW
[144].
1.2
1.2.1
As illustrated in figure 1.1, the contribution of hydropower is expected to grow considerably in the next 30 years, and 553 GW of renewable production capability increase are
predicted, corresponding to an annual rate of 1.9% [156]. Regarding the development of
the hydroelectric production, it can be decomposed in 4 main areas:
EPFL  Laboratoire de Machines Hydrauliques
6000
[TWh/year]
4000
Oceania
Middle East
Europe
Asia
South America
North America
Africa
2000
Figure 1.3: Distribution of the hydropower technically and economically exploitable capability and generation (for the year 1999) [144].
1.2.2
Another aspect of development of the hydroelectric market is the increasing need for power
plants able to stabilize the global power network by allowing quick set point changes in
terms of both active and reactive power in generating and in motor mode. Indeed, the
increase of renewable energy source contribution such as wind power, whose availability
cannot be ensured, will represent a source of disturbances for power networks that are
nowadays considered to be stable. For example in Europe, where thermal power plants
still represent the major contribution, the impact of sudden changes of wind production
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CHAPTER 1. INTRODUCTION
1.2.3
Modern hydropower has to face new challenges related to completely different exploitation
strategies leading to an increase of the solicitation of the entire machine. Thus, hydraulic
machines are increasingly subject to offdesign operation, startup and shutdown sequences,
quick set point changes, etc. To be competitive on the energy market, the specific power
and efficiency of hydro units, which are already high, are constantly increased in order to
meet economical issues but lead to higher loading of the structure of the turbines.
Consequently, manufacturers, consultants and electric utilities of hydro power plants
need integrating new technologies and methodologies for improving dynamic performances,
ensuring the safety and increasing the competitiveness of hydroelectric power plants. This
requires developing appropriate experimental and numerical tools and methods for a better understanding and thus a more accurate prediction of the micro and macroscale
behavior of hydroelectric power plants.
1.3
The hydraulic power Ph results from the product of the mass flow M = Q and of the
specific energy E = g H and is therefore given by:
Ph = Q E
(1.1)
The role of a hydraulic turbine is to convert the hydraulic power into mechanical power
Pm = T with the highest hydraulic efficiency h which is given for a turbine by:
h =
Pm
T
=
Ph
QE
(1.2)
According to the hydrology and the exploitation strategy of a given hydraulic project,
a goal discharge Qplant and a goal specific energy Eplant are determined for the site. Then,
depending on the number of machines and the selection of the synchronous rotational
speed N = (fnetwork 60)/(pairepoles), the type of turbine can be chosen between the
standard hydraulic turbines which are: (i) Pelton turbines, (ii) Francis turbines, (iii)
Kaplan turbines, (iv) bulbe turbine, (v) or propellers.
Figure 1.4 shows the domain of application of the different types of turbines as function
of the nominal net head Hn and the nominal discharge Q of the machine. Typically,
EPFL  Laboratoire de Machines Hydrauliques
for high head, medium head and low head, Pelton, Francis and Kaplan turbines are
respectively chosen. However, for the intermediate range of head, 2 types of turbines are
in competition. Then, the final selection of a type of turbine results from an iteration
process aiming to maximize the produced energy on a standard year of exploitation taking
into account maintenance, civil work and flexibility of operation issues. Because of its
wide application range, the Francis turbine is often selected. Table 1.1 summarizes the
worldwide percentage sales of turbines of each type for the period 19972001. It arises
that 56% of the turbines are of the Francis type with an additional 5% of pumpturbines.
Table 1.1: Distribution of sales of the different types of turbines during the period 1997
to 2001 [124].
Francis Pelton Kaplan Bulbes Pumpturbines
%
%
%
%
%
56
15
15
9
5
Once the nominal discharge Qn , specific energy En and rotational pulsation n are
EPFL  Laboratoire de Machines Hydrauliques
CHAPTER 1. INTRODUCTION
(Qn /)1/2
(2 En )3/4
(1.3)
All turbines having the same specific speed are geometrically similar.
1.3.1
General
The francis turbine is made of 5 main components as illustrated by figure 1.5:
the spiral case: converts axial momentum into angular momentum and distributes
uniformly the flow into the stay vanes;
the stay vanes: are fixed blades having the structural role to close the force loop
of the pressurized spiral case ;
the guide vanes: are mobile blades allowing controlling the flow rate through the
turbine;
the runner: converts the angular momentum of the flow into mechanical momentum by deviating the flow from the inlet to the outlet so that the flow has no more
angular momentum at the outlet, the reaction force acting on the blade inducing
the mechanical torque;
the diffuser or draft tube: has the role to convert the kinetic energy of the flow
into potential energy and therefore enables increasing the efficiency of the turbine
by reducing the pressure level at the runner outlet.
I
Spiral case
Stay vanes
Guide vanes
Runner
Diffuser
Figure 1.6 shows an example of turbine prototype and figure 1.7 shows typical geometries of Francis turbine runner for different specific speed .
EPFL  Laboratoire de Machines Hydrauliques
Figure 1.6: Example of prototype turbine; Three Gorges (Sanxia) 26700 MW (Courtesy
of VoithSiemens Hydro).
Velocity Triangles
The absolute velocity C at any point of the turbine can be decomposed as the sum of
the peripheral velocity of the turbine U and of the relative velocity W and is therefore
given by:
C = U +W
(1.4)
The velocity triangle at the inlet and outlet of a Francis turbine runner for the optimum
velocity at the outlet C 1 , is usually almost axial. The influence of the discharge on the
velocity triangle at the runner outlet is illustrated in figure 1.9. It can be seen that for
discharge below the optimum discharge, the flow at the runner outlet is animated with
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10
CHAPTER 1. INTRODUCTION
= 0.235
= 0.273
0,20
0,16
0,69
0,62
0,75
= 0.380
= 0.323
1,00
1,00
1,00
0,29
0,56
1,00
0,11
1,00
0,24
1,00
= 0.209
= 0.184
0,13
= 0.165
0,09
0,07
= 0.146
1,00
1,00
0,91
0,83
= 0.533
1,15
= 0.634
0,35
0,34
1,00
1,09
1,00
1,15
= 0.792
0,36
= 0.444
0,32
1,00
1,00
1,15
1,00
1,15
Figure 1.7: Francis turbine runner geometry as function of the specific speed [10].
d
(
r m C) = T
dt
(1.5)
Et = U 1 C1 U 1 C1
(1.6)
The scalar value of the transformed energy is given by:
Et = U1 Cu1 U1 Cu1
(1.7)
T
Q
(1.8)
11
w
a1
U1
b1
C1
C1
a1
W1
U1
b1
W1
Figure 1.8: Velocity triangles at inlet and outlet of the runner blade.
Q<Qn
Cu1 U
1
C
C1 m1
W1
Q=Qn
C1
a1
Q>Qn
Cu1
U1
b1
W1
Cm1 C1
U1
W1
Figure 1.9: Influence of the discharge on the circumferential component of the absolute
velocity.
Et
E
(1.9)
The analysis of equation 1.7 shows that the energy transferred from the fluid to the
runner depends on the change of circumferential velocity Cu through the runner, i.e. the
deviation of the flow induced by the blades. It points out that it is interesting to minimize
the outlet Cu in order to reduces losses by residual kinetic energy.
Turbine Performances
The evaluation of the performances of a turbine on its whole operating range requires, according to equation 1.2, measuring the discharge Q, the specific energy E, the rotational
EPFL  Laboratoire de Machines Hydrauliques
12
CHAPTER 1. INTRODUCTION
speed N and the mechanical torque T for different guide vanes openings. The representation of the efficiency as function of the operating conditions [Q, E, N, T ] requires to
suppress one operating parameter to enable a 3 dimensional representation. For performances purposes, it is convenient to use dimensionless coefficients where the rotational
speed N is eliminated, leading to the expression of 2 coefficients:
the discharge coefficient: =
Q
R3
1e
2E
2 R2
1e
(1.10)
Then, for a given turbine, the hydraulic efficiency h can be evaluated on the complete
operating range of the turbine and represented as a function of the discharge and energy
coefficients: h = h (, ), providing the efficiency hill chart of the machine. However, it
is convenient to rate the 2 coefficients by their values at the Best Efficiency Point of the
machine (BEP), i.e. h = h (/BEP , /BEP ) as illustrated in figure 1.10.
2
/ BEP = 0,89
BEP
0,92
0,93
0,94
0,95
0,96
0,97
0,87
0,98
0,
6
0,99
0,995
1
VO
/G
VO
BE
P
7
0,
0,4
0,5
0,8 0,9
1,1
1,2
1,3
1,5 /
BEP
Cavitation
The velocity magnitudes in a turbine are very high and can easily reach 40 m/s. According to Bernoullis law, low pressure arise in areas of high velocities. Then, depending
on the local mean static pressure, the water can vaporize if the pressure drops below the
EPFL  Laboratoire de Machines Hydrauliques
13
vaporization pressure pv . This phenomenon, referred as cavitation, corresponds to vaporization at constant temperature due to pressure decrease. Consequently, the cavitation
is most likely to appear in the low pressure sides of a hydraulic machine. To qualify the
pressure level inside the turbine, the Net Positive Suction Energy (NPSE) is introduced
and refers to the downstream conditions of the turbine (I) as follows:
N P SE =
pI
1
pv
+ g (ZI Zref ) + CI2
(1.11)
Where pv is the vaporization pressure and Zref is a reference elevation, for example the
center line of the guide vanes.
The dimensionless cavitation number also known as the Thoma number is then introduced:
=
N P SE
E
(1.12)
Low cavitation numbers indicate high risks of cavitation. The undesirable effects of cavitation are the risk of erosion, noise and mechanical vibrations but also flow distortions
and efficiency drop [56].
At off design operating conditions, the outlet velocity triangles features a circumferential velocity component. At part load operation, the swirl flow induced in the draft tube
may lead to flow instabilities resulting in the apparition of a helicoidal vortex precessing
in the draft tube with a frequency of about 0.2 to 0.4 times the rotational frequency n
[108], [79]. This vortex, called the vortex rope, is visible if the tailrace water level is sufficiently low, i.e. for a low cavitation number. Depending on the cavitation number and
the rotational speed, a pressure surge may occurs, resulting into unacceptable pressure
and output power fluctuations amplitudes due to resonance between pressure excitation
induced by the vortex rope and the natural frequency of the hydraulic circuit [109].
1.3.2
14
CHAPTER 1. INTRODUCTION
(3) interblade cavitation vortices: can induce high cycle fatigue breaks [47], [100];
(5) part load vortex rope: can be associated with pressure source excitation that
may lead to resonance with the hydraulic system [40];
(6) full load vortex rope: can lead to selfexcited pressure fluctuations [85], [97];
(7) outlet edge bubble cavitation: can lead to erosion of cavitation [73].
Regarding the operating stability of Francis turbine prototype, the pressure fluctuations measurements are of major concerns. The pressure fluctuations measured at different
draft tube wall locations during the model tests can be presented in a waterfall diagram
as a function of the rated frequency and of the rated discharge as illustrated in figure
1.12. This diagram highlights pressure amplitudes of 3 different types:
(2) part load pressure fluctuations;
(3) upper part load pressure fluctuations;
(5) full load pressure fluctuations.
No pressure pulsations are measured close to the best efficiency operating point (4) in
figure 1.11 where the velocity triangle at the outlet leads to almost purely axial flow.
Part Load Draft Tube Vortex Rope
Francis turbine power plants operating at part load may present instabilities in terms of
pressure, discharge, rotational speed and torque. These phenomena are strongly linked to
the swirl flow structure at the runner outlet inducing a vortex core precession in the draft
tube [108]. This leads to hydrodynamic instabilities [79]. The decrease of the tailrace
pressure level makes the vortex core visible as a gaseous vortex rope. The volume of the
gaseous vortex rope is dependent on the cavitation number and affects the parameters
characterizing the hydroacoustic behavior of the entire power plant. As a result, eigen
frequencies of the hydraulic system decrease with the cavitation number. Interaction
between excitation sources like vortex rope precession and system eigen frequencies may
result in resonance effect and induce a so called draft tube surge and electrical power
swing [58]. Consequences on pressure fluctuations and power oscillations were observed
in the framework of many prototypes projects [126], [31], [52], [151], [58], [83] and [93].
Full Load Draft Tube Vortex Rope
Full load operation of Francis turbine creates a circumferential component of the outflow
velocity inducing a swirl flow rotating in the opposite direction of the runner. For some
operating points, the resulting axisymmetric vortex rope developing in the draft tube is
known to start breathing [81]. These specific operating conditions may lead on prototype
to severe self excited pressure fluctuations [85], [97], [93].
EPFL  Laboratoire de Machines Hydrauliques
15
2
9
/ BEP = 0,8
BEP
0,87
0,92
0,93
0,94
0,95
0,96
0,97
0,98
0,99
0,995
0,
=
P
BE
VO
/G
VO
0,
0,4
0,8 0,9
0,5
1,1
1,2
1,3
1,5 /
Rope Free
BEP
16
CHAPTER 1. INTRODUCTION
Figure 1.12: Waterfall diagram of pressure pulsations at the turbine draft tube cone at
constant [81].
RotorStator Interactions
Interactions between rotating parts and stationary parts of a Francis turbine result in
pressure fluctuations that propagate in the entire machine [37]. The combination of
these pressure waves may result in resonance effects and induces unacceptable pressure
fluctuations jeopardizing the safety of the whole power plant [70], [113], [57] and [42].
Mitigation Measures
The mitigation of resonance problems requires acting either on the excitation source
or on the system parameters in order to detune the excitation from the systems eigen
frequencies. The main solutions to achieve the mitigation are:
draft tube fins, used to induce swirl flow distortions and then modify pressure
source frequencies or amplitudes [64], [15];
cylinder in the draft tube or extensions of the runner cone, also used to
induce swirl flow distortions and then modify the pressure source frequencies or
amplitudes [64];
air injection, to modify the hydroacoustic parameters of the turbine in order to
detune the systems eigen frequencies from the excitation sources[64], [119], [118];
mechanical dampers act like Frahm dampers in order to absorb energy of resonance [4], [117];
active control of the pressure fluctuations, using a complex control strategy
based on a rotating valve or a mechanical piston in order to inject pressure fluctuations in the draft tube with the same amplitude as the source but in phase opposition
[21], [19];
EPFL  Laboratoire de Machines Hydrauliques
17
water jet control located in the center of the runner cone to modify the swirl
momentum ratio and eliminate pressure source [140].
However, the success of one of the above mitigating solutions is never ensured and
often has detrimental effects on the turbine efficiency. In addition, some of them are
technically complex solutions and difficult to set up.
Regarding the full load surge, injection of air appeared to be successful in some cases
[97], [8]. Use of fins is also sometimes beneficial.
The resonances resulting from rotorstator problems can usually be solved by modifying the rotorstator arrangement, i.e. the number of guide vanes or runner blades. It is
also possible to mitigate pressure excitations by changing the rotorstator interface geometry by increasing the guide vanerunner blade gap, or using blades with skew angles. If
resonance occurs with specific mechanical parts of the turbine, their structural characteristics can be modified in order to change their eigen frequencies, and again detune the
resonance.
1.3.3
General
In the deregulated electricity market, hydropower plants are more and more solicited in
order to adapt the production to the demand in energy. Consequently, the power plants
are victim of their availability and are subject to an increasing number of startup and
shutdown sequences. During operation, shortcircuits resulting from the failure of power
lines may trigger an emergency shutdown of the power plant. In addition, hydropower
plants are constantly modernized to increase their flexibility by taking advantage of new
control strategies or installing new technologies such as variable speed solutions. Such
events are parts of the todays normal operation of hydropower plants whose solicitations
changes according to technologies and energy market issues.
Transients: the Safety Issues
Transient phenomena result from a change in the operating conditions of a system. In
the case of hydroelectric power plants, transient phenomena can be caused by: (i) unit
shutdown or startup, (ii) change in operating set point, (iii) load rejection or acceptance,
(iv) emergency shutdown and (v) electrical faults such as earth fault, shortcircuit, out of
phase synchronization, and so on.
All the above listed events induce changes of discharge, pressure, rotational speed,
voltage, current and so on in the entire power plant. The impacts of these changes on
the safety of the power plants should be assessed at the early stage of any hydroelectric
project in order to be able to select the critical dimensions of the system with appropriate margins. Therefore transient analysis should be performed accounting for the entire
system; i.e. the whole adduction system, the hydraulic machines, the mechanical inertias, the electrical machines, the controllers, the emergency systems, etc, as illustrated in
figure 1.13. The transient analysis aims to determine: the pipe wall thickness, the surge
tank diameters, the coupling shaft diameters, appropriate emergency procedure, control
parameters, conductors diameters, etc.
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18
CHAPTER 1. INTRODUCTION
19
the Power System Stabilizer (PSS) acts also on the excitation voltage but has the
rotational speed and power as set points.
In addition, for variable speed solutions, a supervisor governor is necessary to select
appropriate rotational speed set points and enabling rotational speed changes. The determination of the parameters sets of these regulators represents a challenging task that
should account for the system layout and exploitation conditions.
In the control strategy of power networks, 3 different levels are distinguished: the
primary, secondary and tertiary levels. The related regulation times and devices are
summarized in table 1.2.
Level
Primary
Secondary
Tertiary
1.4
As illustrated in the previous sections, the design, operation and regulation of hydroelectric power plant require the ability to predict the dynamic behavior of the power plant
taking into account various aspects of the exploitation of the installation. Therefore,
mathematical models able to represent the dynamic behavior of hydropower plants with
high fidelity are necessary. The complexity of the model used for the simulation should
be adapted according to the issues.
Since most of the aforementioned issues regard optimization or parametrization, small
computation times are required. In addition, many of the investigations require a multiphysics model of the power plant, comprising: (i) the entire hydraulic circuit, (ii) the
mechanical inertias, (iii) the electrical installation, and (iv) the regulation systems.
For these applications, onedimensional models offer the best compromise in terms
of computational effort and accuracy. As propagation phenomena globally dominate in
the dynamic behavior of the entire hydraulic circuit, hydroacoustic models are the most
appropriate.
1.5
1.5.1
Hydropower, and more precisely Francis turbines are paramount in the context of electricity production. During the exploitation of Francis turbine power plants, the installation
EPFL  Laboratoire de Machines Hydrauliques
20
CHAPTER 1. INTRODUCTION
is subject to transients phenomena and to a risk of resonance or instabilities. The prediction of these phenomena is crucial for ensuring the safety of the power plant and of
the population. Numerical simulation of the dynamic behavior of the whole installation
enables these predictions. However, classical approaches features 2 main drawbacks:
the focus of the modelling is usually put either on the hydraulic or on the electrical
side, using too much simplified models for the other side;
most of the stability/resonance analysis are based on linearized models, however accurate prediction of amplitudes requires taking into account system nonlinearities.
Issue complexity
Power network
Transient phenomena
xxx
xx
xx
xx
xx
xx
xx
16
xx
xx
xx
xx
12
PSS optimisation
xx
xx
xx
xx
xx
xx
xx
15
Periodic phenomena
Issue
PSS
xx
Voltage regulator
Mechanical Inertias
xx
Turbine
xxx
x: simplified model
Electrical installation
Hydraulic circuit
xxx xxx
12
xxx xxx
12
xxx xxx
Model influence
18 16
xx
xx
xx
6
12 10
10
Classical modelling approaches are sufficient when safety margins is large and when exploitation configurations are simple. However, economical issues tend to increase specific
EPFL  Laboratoire de Machines Hydrauliques
21
power and to lead to completely different exploitation strategies, thus requiring appropriate simulation tools. These tools should enable multiphysics approaches and encompass
the system nonlinearities to improve the accuracy of the prediction of the solicitations
undergone by the facility.
1.5.2
Methodology
The proposed approach is based on the development of an appropriate tool for the simulation of the dynamic behavior of an entire hydropower plant and on the improvement of
models for resonance and instability phenomena assessment. Thus, the present work can
be decomposed as follows:
(1) setup of the models of hydraulic components for the simulation of their dynamic
behavior;
(2) validation of the proposed models;
(3) implementation of the models in an existing software application developed for
the simulation of the dynamic behavior of electrical installations;
(4) investigation of the hydroelectric transients;
(5) development and validation of new models of Francis turbine for the simulation
of vortex rope induced resonance or instabilities and for rotorstator resonance;
(6) development/proposition of system parameters and source excitations identification methods.
1.5.3
This document splits in two main parts. The first part is devoted to the establishment
of hydroacoustic models of hydraulic components, their validation and the simulation of
hydroelectric transients. The second part focuses on the modelling of possible resonance
and instabilities in hydroelectric power plants.
Part I: Hydroacoustic Modelling of Hydraulic Circuits
The fundamental equations used for the hydroacoustic modelling are introduced in chapter 2. In chapter 3, the different solution methods applicable to the set of equation derived
in chapter 2 are presented and compared. The use of the solution methods is illustrated in
chapter 4 by their application for the characterization of the hydroacoustic behavior of a
hydraulic pipe. The modelling approach selected is extended to other standard hydraulic
components in chapter 5. Quasistatic models of hydraulic turbines are also presented.
An analytical analysis of the simple hydraulic systems based on equivalent models developed previously are presented in chapter 6. Such methods provide some stability criteria
of the hydraulic installation for classical transients problems.
In chapter 7, after the validation of the hydraulic components models, hydroelectric
transients are investigated. Consequences of classical electrical faults on hydraulic installations are studied and the stability of turbine speed governor of a simplified hydroelectric
EPFL  Laboratoire de Machines Hydrauliques
22
CHAPTER 1. INTRODUCTION
power plant is assessed. The stability of turbine speed governor is then investigated for a
realistic case of hydroelectric power plant operating in an islanded power network.
Part II: Hydroacoustic Modelling of Pressure Fluctuations in Francis Turbine
Chapter 8 is a literature review of the possible problems of pressure pulsations in Francis
turbine and their possible interactions with the circuits. Existing models of vortex rope
and and rotorstator interactions are presented. Chapter 9 focuses on a peculiar problem
often encountered on scale model but never on prototype: the upper part load pressure
pulsations. The problem is investigated experimentally and numerically for a given operating point. Then, the results of flow visualizations and the influence of the operating
parameters are presented.
The modelling of full load pressure pulsations induced by the vortex rope is illustrated
in chapter 10 by an investigation of an overload surge event occurring on a prototype.
The modelling of the phenomenon is presented. The hydroacoustic rotorstator interactions are simulated for the case of a scale model of pumpturbine in chapter 11. A new
modelling approach is proposed. The problematic of the transposition from model tests to
prototype is presented in chapter 12 and proposition of a methodology for the resonance
risk assessment is made.
Part I
Hydroacoustic Modelling of
Hydraulic Circuits
Chapter 2
Fundamental Equations
2.1
General
A mathematical model based on mass and momentum conservation can properly describe
the dynamic behavior of a pipe filled with water. Hydraulic installations feature longitudinal dimensions greater than transversal dimensions, thus justifying a onedimensional
approach based on the following assumption:
the flow is normal to the crosssections A;
the pressure p, the flow velocity C and the density are uniform in a crosssection
A.
2.2
Momentum Equation
The momentum equation is applied to the control volume, dashedline in figure 2.1, of
length dx. The momentum equation expresses the balance of the forces acting on this
fluid volume, the momentum flux through the surfaces and the rate of change of the
momentum in the volume itself. The integral form of the momentum equation applied to
a volume of fluid is given by:
Z
Z
C n dV + C C
u
n dA = F
(2.1)
t
V
The momentum equation along the xaxis neglecting axial displacement of the pipe u,
and considering gravity, pressure and friction forces is expressed as:
Adx
DC
(pA)
p dx A
= pA [pA +
dx] + (p +
)
dx o Ddx gAdx sin() (2.2)
Dt
x
x 2 x
With:
A : pipe crosssection [m2 ];
: density [Kg/m3 ];
C : flow velocity [m/s];
: shear stress [N/m2 ];
EPFL  Laboratoire de Machines Hydrauliques
26
Piezometric line
hZ = p/(g)
dx
oD
pA +
xdx
(pA)/
pA
/xdx
/2)A
/xdx
x
p
+
(p
dx
Z
gAdx
control volume
Datum
Figure 2.1: Momentum equation applied to the control volume of length dx.
p
DC
+ o D + gA sin() + A
=0
x
Dt
(2.3)
o =
C 2
8
(2.4)
(2.5)
(2.6)
2.3
27
Continuity Equation
The mass balance in the control volume of length dx, in figure 2.2, can be expressed as
follow:
Z
Z
dM
=
dV + C u
n dA = 0
(2.7)
dt
t
V
Developing all the terms and neglecting the axial displacement of the pipe u, yields
to:
(AC)
(Adx)
= AC AC +
dx
t
x
(2.8)
Then:
(AC)
(A)
=
t
x
(2.9)
Piezometric line
hZ
u
 u) +
A(C
/xdx
C  u))
(A(
u)
A(C
dx
Z
control volume
Datum
Figure 2.2: Continuity equation applied to pipe control volume of length dx.
Expressing the partial derivative gives:
1
1 A
1 C
1
1 A
+
+
+
+
=0
x A x
C x
C t AC t
(2.10)
Introducing:
D
=
+C
Dt
t
x
and
DA
A
A
=
+C
Dt
t
x
(2.11)
(2.12)
28
Assuming barotropic behavior of the fluid, i.e. = (p), and introducing the fluid
bulk modulus Ewater yields to:
d
dp = Ewater
(2.13)
Then:
1 d
1 dp
=
(2.14)
dt
Ewater dt
The traction strain of the pipe wall can be expressed as follows:
dR
= Ec
(2.15)
R
With the change of pipe cross section:
dA
dR
2R2 d
= 2R
=
dt
dt
Ec dt
The strain in the pipe wall is deduced from figure 2.3:
pD
d
D dp
=
gives
=
2e
dt
2e dt
With:
e = pipe wall thickness [m]
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
C p
p
+
+C
=0
x
t
x
(2.21)
2.4
29
Simplified Equations
1 p C
C
C C
+
+C
+ g sin() +
=0
x
t
x
2D
(2.22)
C p
p
2
a
+
+C
=0
x
t
x
In hydraulics, it is useful to use the discharge Q and the piezometric head h as state
variables instead of the flow velocity C and pressure p. The discharge and the piezometric
head are defined as:
h=Z+
p
g
(2.23)
Q=C A
(2.24)
With Z the elevation [m]. The piezometric head is the pressure given in meters
of water column, mWC, above a given datum. Injecting equations 2.23 and 2.24 in
equation 2.22, assuming no vertical displacements of the pipe z/t
= 0 and noticing
that z/x = sin(), gives:
h
1 Q
Q
Q Q
x + gA t + C x + 2gDA2 = 0
(2.25)
h
h
a2 Q
+C
=0
+
t
x
gA x
Hydroacoustic phenomena are characterized by a high wave speed a(a = 1430 m/s at
20C) and low flow velocities (C = 10 m/s), thus the convective terms C/x related to
the transport phenomena can be neglected with respect to the propagative terms /t.
This simplification leads to the following set of of partial derivative equations:
h
1 Q Q Q
+
+
=0
x gA t
2gDA2
(2.26)
h
a2 Q
+
=0
t
gA x
2.5
Resolution Methods
Q
t
h
t
gA
+ a2
0
gA

{z
}
0
Q
x
h
x
=
QQ
2DA
0
[A]
(2.27)
30
The eigen values of this set of equations are the roots of the following characteristic
equation:
det ([A] [I]) = 0
(2.28)
(2.29)
Since the eigen values of the equation system 2.27 are real, it corresponds to a system
of hyperbolic partial differential equations. This type of equation is related to propagative
problems that can be solved with various methods:
arithmetic method of Allievi (1925) [2];
graphical method of SchnyderBergeron (1950) [14];
method of characteristics (MOC) [138], [141], [36];
transfer matrix method [138], [55];
impedance method [138].
All these methods enable analyzing the dynamic behavior of hydraulic system.
Chapter 3
Resolution Methods of the Set of
Hyperbolic Partial Differential
Equations
3.1
Electrical Analogy
The solution of a system of hyperbolic partial differential equations such as the set of
equations 2.26, was at first inspired by the methods developed in the field of telecommunication [116], [87]. The resolution of the propagation of electrical waves in conductors
is based on an equivalent scheme representation providing a high level of abstraction and
having a rigorous formalism. The study of electrical wave propagation in conductors
leads to the establishment of the set of equations expressed as follows, referred to as the
telegraphists equation:
U
i
+ L0e + Re0 i = 0
x
t
(3.1)
1
U
i
+ 0
=0
t
Ce x
Where:
i: electrical current [A]
U : electrical potential [V ]
Re0 : lineic electrical resistance [/m]
L0e : lineic electrical inductance [H/m]
Ce0 : lineic electrical capacitance [F/m]
The analogy between equation set 2.26 modelling the propagation of pressure waves in
hydraulic systems and the equation set 3.1 modelling the propagation of voltage waves in
conductors allows identifying a lineic hydraulic resistance R0 , a lineic hydraulic inductance
L0 and a lineic hydraulic inductance C 0 . The equation set 2.26 can be rewritten as:
Q
h
+ L0
+ R0 (Q)Q = 0
x
t
(3.2)
h + 1 Q = 0
t
C 0 x
EPFL  Laboratoire de Machines Hydrauliques
32
gA
[m];
a2
1
[s2 /m3 ];
gA
Q
[s/m3 ].
2gDA2
Hydraulic and electrical systems are both characterized by an extensive state variable,
i.e. discharge Q and current i, and by a potential state variable, i.e. piezometric head
h and voltage U . The electrical analogy permits to apply the mathematical formalism
developed initially for electrical purposes to hydroacoustic problems and to use powerful concepts such as equivalent scheme, impedance or transfer matrix. Two modelling
approaches of hydraulic system are distinguished:
to consider the system as a system with distributed parameters, or as a continuous
system;
to consider the system as a system with lumped parameters, or as a discrete system.
Analytical solutions can be derived for continuous systems, but require the linearization of the system and thus restricts the study to small perturbations. Analytical solutions are not possible for discrete systems of large dimension and thus numerical methods
should be used, but enables to take into account system nonlinearities. Nevertheless, this
modelling approach introduces approximation errors that have to be quantified.
3.2
Resolution of the Set of Hyperbolic Partial Differential Equations for Continuous System
Pressure wave propagation in hydraulic systems can be modelled using continuity and
momentum equations. The resulting set of hyperbolic partial differential equations can
be written as:
Q
h
+ L0
+ R0 Q = 0
x
t
(3.3)
h + 1 Q = 0
t
C 0 x
The set of equations 3.3 can be rewritten using the separation of variables method
assuming a sinusoidal variation of the piezometric head h(x, t) and of the discharge Q(x, t)
defined as complex function:
(
h(x, t) = h(x) est
(3.4)
Q(x, t) = Q(x) est
Where the constant s is the complex frequency also referred to as Laplace variable.
The complex frequency is composed of an imaginary part and of a real part:
s=+j
(3.5)
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33
For simplicity, the complex frequency s is written as s in the next section of this
document. The set of equations 3.3 is rewritten as follows:
2
h
2 = 2 h(x)
x
(3.6)
2
= Q(x)
x2
Where is the complex wave number:
2 = C 0 s (L0 s + R0 )
3.2.1
(3.7)
(3.8)
The constants C1 and C2 are to be determined from the boundary conditions. Thus
equation 3.6 for the piezometric head admits as solution both a progressive wave hp (x)
and a retrograde wave hr (x), defined from the boundary condition at x = 0:
hp (x) = hp (0) ex
(3.9)
hr (x) = hr (0) ex
(3.10)
Rewriting the equations set 3.3 assuming a sinusoidal evolution of the piezometric
head and of the discharge yields to:
h
= (L0 s + R0 ) Q
x
(3.11)
Q
= C0 s h
x
(3.12)
Equations 3.11 and 3.12 provide the solution of the discharge equation of equation 3.3
admitting as solution a progressive discharge wave Qp (x) and a retrograde discharge wave
Qr (x) expressed as:
Qp (x) =
()
hp (x)
hp (x)
1
h
(x)
=
p
(L0 s + R0 )
x
(L0 s + R0 )
Zc
Qr (x) =
(L0
h (x)
1
h (x)
hr (x) = r
r
= 0
0
0
(L s + R )
Zc
s+R)
x
(3.13)
(3.14)
34
The ratio between the piezometric head and the discharge variations is the characteristic impedance of the pipe Z c defined as:
r
(L0 s + R0 )
Zc =
(3.15)
C0 s
The solution to the equation system 3.6 is the sum of the progressive and the retrograde
waves yielding to:
h(x) = hp (x) + hr (x)
Q(x) =
(3.16)
hp (x) + hr (x)
Zc
(3.17)
Combining equations 3.9 and 3.10 with the equations 3.16 and 3.17 established for
x = 0, gives:
hp (x) =
h(0) + Z c Q(0) x
e
2
(3.18)
hr (x) =
h(0) Z c Q(0) x
e
2
(3.19)
h(0) + Z c Q(0)
2
(3.20)
C2 =
h(0) Z c Q(0)
2
(3.21)
Reformulating the equations 3.16 and 3.17 with equations 3.18 and 3.19, gives:
Z1 sinh( l)
cosh( l)
Q(l)
Q(0)
c
(3.23)
The transfer matrix offers the possibility to determine the fluctuations of piezometric
head and discharge at the end of a pipe resulting from the excitation at the other end.
For a system made of pipes in series, it is possible to compute the global transfer matrix
of the system by performing the matricial product of the transfer matrices of all the pipes
in series:
Y
[M tot ] =
[M i ]
(3.24)
1kn
3.2.2
35
Neglecting dissipation, (R0 = 0) the complex number of waves of equation 3.7 becomes:
s
gA
1
=
(3.25)
= s C 0 L0 = s
2
a
gA
a
Similarly, the characteristic complex impedance Z c of equation 3.15 becomes a scalar
value:
r
L0
a
(3.26)
Zc =
=
0
C
gA
Without damping, the complex frequency s is simplified as s = j. In addition,
noticing that cosh() = cos(j ) and sinh() = j sin(j ), the transfer matrix of
equation 3.23 for a frictionless pipe becomes:
l
cos( l
)
j
sinh(
)
h(l)
h(0)
c
a
a
=
(3.27)
j Z1c sinh( l
)
cosh( l
)
Q(l)
Q(0)
a
a
3.2.3
The spatial partial derivative of the piezometric head combined with the equation of
discharge of the equation system 3.3 gives:
2
2Q
Q
0
0 Q
=
C
+ C 0 R0
2
2
x
t
t
(3.28)
The time partial derivative of the piezometric head combined with the equation of
discharge of the equation system 3.3 gives:
2
2h
0
0 h
0
0 h
=
C
+
C
x2
t2
t
(3.29)
The above system expressed for a frictionless system, i.e. R0 = 0, yields to the wave
equation:
2Q
2Q
2 = a2
t
x2
(3.30)
2
h
2h
=a 2
t2
x
DAlembert has derived the general solution of this set of equations for the piezometric
head back in 1747:
h(x, t) = Fp (a t x) + Gr (a t + x) = Fp (t x/a) + Gr (t + x/a)
(3.31)
The function Fp (tx/a) is a progressive wave whose shape is fixed, and is propagating
at the wave speed a towards positive x values and Gr (t + x/a) is a retrograde wave
propagating at the same wave speed towards negative x values.
EPFL  Laboratoire de Machines Hydrauliques
36
(3.32)
A progressive wave reflected at an open end becomes a retrograde wave with the same
shape but opposite sign as presented in figure 3.1.
h=0
h+
Fp
Gr
x+
h
= (L0 s + R0 ) Q = (L0 s + 0) 0
x
(3.33)
= Fp (a t) + Gr (a t)
x
Fp (a t) = Gr (a t)
(3.34)
A progressive wave reflected at a dead end becomes a retrograde wave with same shape
and same sign as presented in figure 3.2.
Wave Reflection at a Junction
A progressive wave propagating in a pipe with change of hydroacoustic parameters in
the longitudinal axis is subject to wave reflection. A change of hydroacoustic nature can
EPFL  Laboratoire de Machines Hydrauliques
37
Q=0
Fp
Gr
h+
x+
h(x)
Q(x)
(3.35)
Expressing the impedance of the second pipe using the sum of incident and reflected
waves of pipe 1, and introducing equations 3.13 and 3.14 gives:
Z c2 =
ht
h + hr
= i
=
Qt
Qi + Qr
1
Z c1
hi + hr
(hi hr )
(3.36)
As the reflected wave is retrograde, its sign is negative, after rearranging it becomes:
Z c2 = Z c1
hi + hr
hi hr
(3.37)
From equation 3.37, one can express the ratio between the incident and the reflected
waves as:
hr
Z Z c1
= c2
(3.38)
hi
Z c2 + Z c1
In addition, the piezometric head at the junction is identical for both pipes, thus:
hi + hr = ht
(3.39)
The ratio between the incident and transmitted waves for the piezometric head is
therefore given by:
ht
2 Z c2
=
hi
Z c2 + Z c1
For the cases of open and dead end pipes, one get:
EPFL  Laboratoire de Machines Hydrauliques
(3.40)
38
Zc1
Zc2
hi
hr
ht
xo
x+
Figure 3.3: Wave reflection of incident wave i at a junction of 2 pipes having different
characteristic impedance Z c1 amd Z c2 .
hr
hi
0Z c1
0+Z c1
=
hr
hi
= 1
Z c1
+Z c1
Z c1 Z c2
Z c1 + Z c2
(3.41)
In addition the discharge at the junction is identical for both pipes, thus:
Qi + Qr = Qt
(3.42)
Then the ratio between the incident and the transmitted wave is given by:
Qt
Qi
2 Z c1
Z c1 + Z c2
(3.43)
Qr
Qi
Qr
Qi
Z c1 0
Z c1 +0
=1
Z c1
Z c1 +
These results agree well with the previous results obtained from dAlembert equation.
Paradox of the Wave Reflection Consecutive to a Waterhammer
The case of a pressure wave induced by the sudden closure of a valve downstream of a
pipe, as presented in figure 3.4, inducing a waterhammer in the pipe can be first analyzed
using the dAlembert solution. It is important to notice that the solution of the pressure
wave given by equation 3.31 for a given time t and a given location x, is the sum of all
the incident, transmitted and reflected waves.
The pressure wave of amplitude h generated by the downstream valve closure is
reflected by the upstream tank with a negative sign because of the open end boundary
condition. In turn the piezometric head in the pipe is the sum of the incident and the
reflected waves as the amplitude of both waves is identical and therefore, their sum is
EPFL  Laboratoire de Machines Hydrauliques
39
zero. As a result, when the reflected wave reaches the downstream valve, the piezometric
head in the pipe recovers its initial value, i.e. ho . At this moment, the reflected wave is
reflected again with the same sign, as the closed valve represents a dead end boundary
condition. The sum of the 3 waves propagating in the pipe, 1 incident wave with positive
amplitude h and 2 reflected waves with negative amplitudes h gives a piezometric
head below ho equal to h = ho h.
The paradox lies to the fact that observing the time evolution of the piezometric head
in the pipe gives the impression that the incident wave is reflected at the downstream
valve with the same sign, which is not the case. If the piezometric head recovers the value
ho , this is because the sum of incident and reflected wave is zero. Similarly, the reflection
at the downstream valve gives the impression that the wave is reflected with opposite
sign which is again not the case. If the piezometric head reaches values below ho , this is
because the sum of the 3 waves is smaller than ho .
This example demonstrates that even for a simple case, made of one pipe, it is not
easy to predict the time evolution of the piezometric head h because it results from the
summation of all the wave propagating in the pipe. Therefore, it becomes very difficult to
predict accurately the piezometric head time evolution with dAlembert solution in complex cases made of several pipes, connections and junctions. Only numerical solution of
momentum and mass conservation equation can describe the phenomenon with sufficient
accuracy.
3.2.4
(3.44)
Expressing the specific impedance Z a (l) for x = l from the specific impedance Z a (0)
at x = 0 provides:
Z a (l) =
Z a (0) Z c tanh( l)
1
Z a (0)
Zc
tanh( l)
(3.45)
Using known boundary conditions such as Z a (0) = 0 (open end) or Z a (0) = (dead
end) it is possible to determine the specific impedance for any location x. It is then
EPFL  Laboratoire de Machines Hydrauliques
40
Ho
Qo
D
L
h+
h=ho+Gri=ho+h
a
0<t<L/a
x+
h+
Gri
h=ho+Gri+Fpr
a
L/a<t<2L/a
x+
a
Fpr
Gri
h+
h=ho+Gri+Fpr+Jrr
a
x+
a
Jrr
2L/a<t<3L/a
a
Fpr
Figure 3.4: Time evolution of the piezometric head h resulting from the closure of the
downstream valve inducing waterhammer.
possible to determine the specific impedance of one end of a system, by combining the
impedance of all hydraulic components starting from the other end where the impedance
is known. It is also possible to determine the specific impedance at x = 0 from the specific
impedance at x = l giving:
Z a (0) =
Z a (l) + Z c tanh( l)
1+
Z a (l)
Zc
tanh( l)
(3.46)
41
to the search of the natural frequencies of the system; i.e. the free oscillations analysis,
see the next section. For simplicity, the specific impedance will be denoted impedance in
the following sections of the document.
3.2.5
The study of the free oscillations regime of a hydraulic facility can be treated for a continuous system through 2 methods: (i) the transfer matrix method and (ii) the impedance
method. These calculation methods are described below.
Free Oscillation Analysis: Transfer Matrix Method
Using the transfer matrix of each hydraulic component, it is possible to combine them to
set up the global matrix of the system including the boundary conditions leading to the
following equation:
x = 0
[G]
(3.47)
Where the complex matrix [G] is the global matrix of the system, and
x is the state
vector. Obtaining a non trivial solution of equation 3.47 requires the determinant of
matrix [G] to be equal to zero:
det([G]) = 0
(3.48)
Equation 3.48 is the characteristic equation of the system whose k complex roots
x (t) =
x 1 es1 t +
x 2 es2 t +
x 3 es3 t + . . .
(3.49)
Z a (xn1 )
Z cn
(3.50)
If the boundary condition at this end is known, the problem to be solved boils down
to find the k complex frequencies sk = k + j k satisfying the following equation :
Z atot (xend , sk ) = Z end
(3.51)
Once the complex frequencies, i.e. the eigen frequencies, are known, the corresponding
eigen modes can be determined using equations 3.22.
42
3.3
Resolution of the Set of Hyperbolic Partial Differential Equation for Discrete System
The hyperbolic partial differential equation set 3.3, can be solved numerically leading
to the modelling as a discrete system. The state variables, i.e. the piezometric head h
and the discharge Q, cannot be determined for any x or t, but are known only for given
locations and given times according to the numerical scheme used to solve equation set
3.3.
3.3.1
The numerical integration in space and in time of equation set 3.3 requires appropriate
discretization. Regarding the spatial discretization, the centered scheme illustrated in
figure 3.5 can be used.
Control volume "i"
Node
i
i+1
x+
dx
Using the scheme centered at location i + 1/2 of the figure 3.5, one gets the following
expression for the piezometric head and discharge space partial derivatives:
h
hi+1 hi
i+1/2 =
x
dx
(3.52)
Q
Qi+1 Qi
i+1/2 =
x
dx
(3.53)
Considering the above numerical scheme and using the total differential in equation
set 3.3 expressed for a location i + 1/2 gives:
dhi+1/2
1 Qi+1 Qi
+ 0
=0
dt
C
dx
(3.54)
hi+1 hi + L0 dQi+1/2 + R0 Q
i+1/2 = 0
dx
dt
In order to ensure stability of the computation, a numerical scheme of Lax, based on
the mean value of the discharge, is used:
Qi+1/2 =
Qi+1 + Qi
2
(3.55)
43
dhi+1/2
C 0 dx
= Qi Qi+1
dt
0
L0 dx dQi+1 R0 dx
L dx dQi R0 dx
hi+1 +
+
Qi+1 = hi
+
Qi (3.56)
2
dt
2
2
dt
2
{z
} 

{z
}
hi+1/2
h
i+1/2
Noticing that the second part of the equation corresponds to the piezometric head for
the location i + 1/2, the system can be expressed under implicit matrix form:
C 0
0
hi+1/2
0 1
1
hi+1/2
0
0 L/2 0 d Qi + 1 R/2 0 Qi = hi (3.57)
dt
0
0 L/2
Qi+1
1 0 R/2
Qi+1
hi+1
Set of equations 3.57 is written using the hydroacoustic parameters obtained for a
length dx given by:
R = R dx
L = L0 dx
(3.58)
C = C 0 dx
The compact expression of equation set 3.57 is given by:
[A]
d
x
+ [B]
x =C
dt
(3.59)
Set of equations 3.57 can be integrated numerically using classical methods such as
Euler, RungeKutta, etc. CourantFriederichsLewy have demonstrated that there exists
a numerical stability criteria known as the CFL criteria linking the space and time
discretization, respectively dx and dt through the wave speed a [38]. This criteria ensures
the causality of the system because the information cannot transit faster than the wave
speed. The CLF criteria is given by:
dt <
dx
a
(3.60)
44
Boundary condition:
h 1 = cste
1
Boundary condition:
h = cste
Node
n+1
i+1
n+1
x+
dx/2
dx
dx
dx
dx/2
Equation set 3.57 corresponds to the equation describing the dynamic behavior of a
pipe of length dx. A pipe is modelled by n pipes of length dx and dx = L/n. The
corresponding spatial discretization is presented in figure 3.6.
For a pipe of length l, the boundary conditions are the piezometric head at both
ends of the pipe, whereas momentum equations can be merged 2 by 2, introducing the
piezometric head at each node i + 1/2. The matrices [A] and [B] of equations set 3.57
becomes for a pipe of length l:
C
(n)
(2n+1)
..
.
0
C
(n)
[A] =
(3.61)
L/2
..
0
L
L/2
(2n+1)
And:
[B] = (n)
1
(2n+1)
(n)
1
1
(2n+1)
1
..
.
1
1
1
R/2
1
R
..
..
.
1
1
1
.
R
(3.62)
R/2
One can notice that the equation set has the dimension (2n + 1) (2n + 1) and the state
vector comprises n piezometric heads along the pipe for the nodes 1 + 1/2 to n + 1/2 and
n + 1 discharges at both end of n pipes of length dx.
EPFL  Laboratoire de Machines Hydrauliques
45
(3.63)
C = 0 . . . 0 h1 0 . . . 0 hn+1
(3.64)
It is also important to notice that the matrix [B(Qi )] is function of the discharge and
introduces a nonlinear behavior. For the numerical integration of this set of equation, all
the discharge Qi in matrix [B(Qi )] are retrieved from the previous time step.
3.3.2
L/2
R/2
hi
Qi
hi+1/2
L/2
R/2
Qi +1
hi+1
This equivalent scheme is the model of a pipe with length dx where the hydroacoustic
parameters are defined as follows:
hydroacoustic capacitance C = dxgA
[m2 ], related to storage effect due to pressure
a2
increase and is therefore function of the wave speed ; in the wave speed equation 2.20,
the term /Ewater account for water compressibility storage and the term D/(eEc )
account for pipe wall deflection storage;
hydroacoustic inductance L =
hydroacoustic resistance R =
pipe.
dx
gA
dxQ
2gDA2
46
losses correspond to the piezometric head difference between the inlet and the outlet. The
steady state conditions lead to the following equation:
hi hi+1/2 =
hi+1/2 hi+1
R
Qi
2
R
= Qi+1
2
(3.65)
(3.66)
During transients, assuming a sudden increase of the downstream head hi+1 gives the
following equation for the second loop of the equivalent scheme:
hi+1/2 hi+1
L dQi+1
R
Qi+1 =
2
2
dt
(3.67)
For an increase of the downstream piezometric head hi+1 , the inequality is obtained:
hi+1/2 hi+1
dQi+1
R
Qi+1 < 0
<0
2
dt
(3.68)
The increase of the downstream piezometric head induces the decrease of the discharge
in the second loop, and as a result, the piezometric head in the middle of the pipe increases
because the discharge in the first loop has not changed after an infinitesimal time interval:
dhi+1/2
dhi+1/2
= Qi Qi+1
>0
dt
dt
(3.69)
This qualitative explanation describes how the equivalent scheme of a pipe represents
the propagation of pressure waves in a pipe when the downstream pressure increases
suddenly.
Generalized Representation of a Pipe
The implicit system of equation 3.57 can be derived directly from the equivalent scheme
of the pipe of figure 3.7 using Kirchhoffs law. The same approach can be used to model
a pipe of length l made of n pipes of length dx as presented in figure 3.6. Kirchhoffs law
applied to this system leads to equation 3.57.
It can be noticed that the piezometric heads are determined for the node i + 1/2 and
the discharges are determined for the loops i as indicated in figure 3.8. The corresponding
equivalent scheme is presented in figure 3.9 where n equivalent schemes are concatenated
together.
The 3step modelling procedure is summarized in figure 3.10: (i) a mathematical model
of the physical system is established, providing a set of hyperbolic partial differential
equations; (ii) a numerical integration scheme in space provides the structure of the
equivalent scheme that can be generalized, meaning that the set of total partial derivative
equations can be obtained directly from the equivalent scheme using Kirchhoffs law; (iii)
the set of total derivative equations is integrated numerically using standard algorithm
such as RungeKutta.
EPFL  Laboratoire de Machines Hydrauliques
Q1
Qi
Q2
h 2+1/2
h 1+1/2
h i+1/2
h i1/2
Boundary condition:
h n+1= cste
Qn+1
Qn
Qi+1
h n1/2
h i+1+1/2
47
h n+1/2
x+
dx/2
dx
dx
dx
dx/2
R/2
h1
Q1
L/2
Q2
h1/2
C h
1+1/2
C h
n11/2
Qn
L/2
C h
n1/2
R/2
Qn +1
hn+1
3.3.3
The free oscillation analysis of a hydraulic facility can be performed from its equivalent
circuit with 2 different approaches: (i) solving the eigen value/vectors problem, directly
from the total differential equation set 3.59; (ii) performing a numerical calculation of the
system impedance, and searching for the complex frequencies satisfying all the boundary
conditions.
Eigen Values/Vectors Problem
The free oscillation analysis of a discrete system modelled by n elements corresponds
to the problem of the determination of the eigen values/vectors of the system of partial
differential equations 3.59 using Kirchhoffs law:
d
x
[A]
+ [B]
x = 0
dt
(3.70)
[A] s
x + [B]
x = 0
(3.71)
(3.72)
Where [I] is the identity matrix. To ensure a nontrivial solution, the determinant of
the global matrix of the system must be zero:
det [I] s + [A]1 [B] = 0
(3.73)
EPFL  Laboratoire de Machines Hydrauliques
48
Numerical integration
of the matricial system
(RungeKutta for ex.)
Physical system
PDE system
(Partial differential
equations)
Multidimensional
Kirchhoff circuit
Physical modeling
Interpretation into
equivalent electrical
circuit
This equation is the characteristic equation of the system whose (2n + 1) roots comprising a null root and n double complex roots sn = n j n are the eigen values of
x (t) =
x 1 es1 t +
x 2 es2 t + . . . +
x n esn t
(3.74)
It can be noticed that the real part of the eigen values n are the damping coefficients;
negative values correspond to damped modes. Stability of the system is ensured only if all
the damping coefficient of the system are negative. This stability criteria is represented
graphically in the complex plane where the shadow area in figure 3.11 is the stable domain.
xk
Im
xk (t ) = xk e s k = xk e k e jk t
xk (t ) = xk Re(e s k ) = xk e k cos(k t )
s k = k + jk
Re
s k = k jk
e k
49
is calculated starting from one end where the boundary condition are known, for example
an open end or a dead end, and then calculated until another end. The calculation is done
by successive series and parallel equivalent computation of the branches of the system.
The first loop of a pipe with a load impedance Z load at the end is presented in figure 3.12.
R/2
Qn
hn
L/2
L/2
R/2
Qn +1
Z load
hn
Qn
Z equ
Figure 3.12: Equivalent impedance of the last loop of a pipe with load impedance Zload .
1
(L/2 s + R/2 + Z load ) Cs
= (L/2 s + R/2) +
1
(L/2 s + R/2 + Z load ) + Cs
(3.75)
If the pipe is modelled by n elements, the equivalent impedance is calculated recursively considering as load impedance Z load for the loop n 1, the equivalent impedance
Z equ of the loop n, given by :
1
L/2 s + R/2 + Z equi+1 Cs
Z equi = (L/2 s + R/2) +
(3.76)
1
L/2 s + R/2 + Z equi+1 + Cs
The computation is done for a given complex frequency. The problem to be solved
is therefore to find the complex frequency satisfying all the boundary conditions. This
leads to a minimization calculation based, for example, on NewtonRaphsons algorithm.
It is convenient to use a first guess obtained from frictionless system whose complex roots
become:
s=j
This approach allows computing the impedance of the system at one end for a given
range of pulsation and identifying which are satisfying the boundary conditions. The
typical boundary conditions are:
open end: Z x (j) = 0
dead end: Z x (j) =
The pulsations obtained from frictionless conditions are an excellent guess for the
research of the eigen values sk = k + j k of the dissipative system as the damping
affects only slightly the eigen pulsation of a system. Once the eigen frequencies are
known, the impedance can be computed along the system for the eigen value of interest
indicating the location of minima and maxima of the discharge and the head.
EPFL  Laboratoire de Machines Hydrauliques
50
3.4
3.4.1
Truncation Error
The spatial discretization of the partial derivative introduces truncation errors as the
Taylor development of a value u is given for a progressive scheme by:
ui+1 ' ui +
u
x
x +
i
2u
x2
(x)2
+
2
i
3u
x3
(x)3
+ ...
6
i
(3.77)
u
x
'
i
ui+1 ui
+ O(x)
x
(3.78)
Centered Scheme
The finite difference scheme developed above leads to an equivalent scheme, see figure
3.13, whose set of ordinary differential equations can be derived directly from Kirchhoffs
law.
R/2
hi
Qi
L/2
hi+1/2
L/2
R/2
Qi +1
hi+1
L dQi R
hi =
+ Qi + hi+1/2
2 dt
2
dhi+1/2
C
= Qi Qi+1
dt
L dQi+1 R
+ Qi+1 + hi+1
i+1/2 =
2
dt
2
(3.79)
Combining equations 3.79 yields to the transfer matrix of the equivalent circuit of
figure 3.13 :
2
2 h
h 2
ii
x
x
x
+1
1+ 2 +1
h
hi
2
2Cs
i+1
2
=
(3.80)
Qi+1
Qi
x
+1
C s
2
Where: x = dx.
Backward Scheme
Conserving the RL terms of the branch of the equivalent scheme of a pipe while moving
the capacitance upstream leads to the equivalent scheme of figure 3.14. This equivalent
scheme is based on backward numerical scheme without Lax scheme.
hi
Qi
hc
Qi +1
hi+1
hi = hi+1/2
dhi+1/2
C
= Qi Qi+1
dt
(3.81)
Combining equations 3.81 yields to the transfer matrix of the equivalent circuit of
figure 3.14:
2x
1
hi+1
hi
Cs
=
(3.82)
Qi+1
Qi
C s 2 + 1
x
52
Progressive Scheme
As it is done for the backward scheme, RL terms are conserved, but the capacitance is
moved downstream. This leads to the equivalent scheme of figure 3.15, corresponding to
a progressive numerical scheme without Lax scheme.
hi
Qi
hc
Qi +1
hi+1
L dQi R
hi =
+ Qi + hi+1/2
2 dt
2
dhi+1/2
C
= Qi Qi+1
dt
hi+1/2 = hi+1
(3.83)
Combining the above equations 3.83, yields to the transfer matrix of the equivalent
circuit of figure 3.15:
2 #
" 2
x
h
hi
x + 1
i+1
Cs
=
(3.84)
Qi+1
Qi
C s
1
3.4.2
The transfer matrix of the following 4 models are compared: (i) the continuous hyperbolic
model, (ii) the discrete centered model, (iii) the discrete backward model, and (iv) the
discrete progressive model. The transfer matrix and the equivalent scheme of these models
are summarized in figure 3.1.
Continuous hyperbolic
Table 3.1: Comparison of the transfer matrix of 4 models of a pipe of length dx.
Model
Equivalent scheme
Transfer matrix
i+1
h(dx)
h(0)
= [M ]
Q(dx)
Q(0)
dx
Z c sinh( dx)
cosh( dx)
[M ] =
Z1 sinh( dx)
cosh( dx)
Discrete progressive
Discrete backward
Discrete centered
R/2
hi
L/2
Qi
L/2
hi+1/2
hi+1
hi
= [M ]
Qi+1
Qi
R/2
Qi +1
hi+1
2
2 h
h 2
ii
x
x
+
1
1
+
+
1
2
2Cs
2
2
[M ] =
x
C s
+
1
2
x
hi
Qi
hi+1
Qi +1
1
hi
=
Qi
C s
2x
Cs
2x +
hi+1
Qi+1
1
hi
hc
Qi
hc
Qi +1
hi+1
" 2
hi
x + 1
=
Qi
C s
2x
Cs
#
hi+1
Qi+1
1
54
Assuming frictionless regime, the transfer matrix of equation 3.27 becomes for a length
dx:
)
j Zc sinh( dx
)
cos( dx
h(dx)
h(0)
a
a
=
j Z1c sinh( dx
)
cosh( dx
)
Q(dx)
Q(0)
a
a
(3.85)
a
2a
=
,
f
(3.86)
(3.87)
The key parameter for a study in the frequency domain is the ratio:
dx
(3.88)
=
(3.89)
Q(dx)
Q(0)
M 21 M 22
The impedance of the pipe of length dx is evaluated starting from the open end x = 0
until the end of the pipe x = dx using one element for the modelling of the pipe. The
resulting impedance amplitude are represented in figure 3.17. The impedance of the pipe
with open end is given by:
Z(dx) =
3.4.3
M 12 (dx)
M 22 (dx)
(3.90)
The amplitude of the impedance can be used to compare the accuracy of the 3 discrete
models, the continuous hyperbolic model being taken as reference. The error of the
impedance amplitude is represented for the 3 discrete models as a function of the rated
wavelength /dx in figure 3.18. The error is provided in table 3.2 for /dx = 10 and
/dx = 20.
The numerical scheme of the centered model is of the second order while the numerical
scheme of both the backward and the progressive scheme are of the first order. As a result,
the centered scheme features an error less than 3% for = 10 dx and even less than
1% for = 20 dx. This means that 20 nodes are required to model properly a standing
wave of one wavelength with less than 1% of error. The 4 terms of the transfer matrix
EPFL  Laboratoire de Machines Hydrauliques
M11
M12
6
5
3
/dx = 10
Amplitude/Zcar []
Amplitude []
f max = a*dx/10
4
3
2
1
0
0
Hyperbolic model
Backward
Progressive
Centered
2.5
2
1.5
1
0.5
1
2
3
Dimensionless frequency f/fmax []
0
0
1
2
3
Dimensionless frequency f/fmax []
M22
2.5
Amplitude []
Amplitude*Zcar []
M21
1.5
1
0.5
0
0
3
2
1
1
2
3
Dimensionless frequency f/fmax []
0
0
1
2
3
Dimensionless frequency f/fmax []
Figure 3.16: Comparison of the amplitude of the 4 terms of the transfer matrix of the 4
models.
of the centered scheme presented in figure 3.16 show good agreement with the hyperbolic
solution up to f = fmax corresponding to = 10 dx. Whereas for both first order
scheme the accuracy is much worst due to the non symmetry of the models. Thus, the
models behave differently if they are considered from one side or the other. The increase
of spatial resolution presents the drawback of increasing the size of the equation system
to be solved and reducing the integration time step. It is therefore very important to have
criteria to define the appropriate discretization offering a good balance between accuracy
and computational time.
To summarize, the spatial discretization of a hydraulic system should be defined prior
Error
= 10 dx
= 20 dx
Table 3.2: Error on the impedance amplitude obtained for a pipe of length dx with the 3
discrete models as a function of the rated wavelength /dx.
56
Zc = M12/M22
10
Hyperbolic model
Backward
Progressive
Centered
10
Z/Zcar []
10
10
2
10
4
10
0.5
1.5
2
2.5
3
Dimensionless frequency f/fmax []
3.5
Figure 3.17: Comparison of the magnitude of the impedance Z x=dx of the 4 terms of the
transfer matrix obtained for the 4 models.
10
Backward
Progressive
Centered
10
10
1
10
2
10
3
10
10
20
30
40
50
60
/dx []
70
80
90
100
Figure 3.18: Comparison of the error on the impedance amplitude of the 3 discrete models
regarding the continuous model.
57
to the simulation and setup according to the frequency of interest finterest and the wave
speed a of the pipe with the following resolution:
a
error < 3%;
10 finterest
a
dx =
error < 1%.
20 finterest
dx =
3.5
(3.91)
58
Physical system
Physical model
Analytical solution for
sinuisoidal fluctuations
Continuous system:
Discrete system:
Exact solution:
 transfer matrix
x 2 = M syst x1
 impedances
h = Z syst Q
Frequency domain
x i = M syst x exct
hi = Z i Q exct
or
Q i = h exct Z i
Set of partial
differential equation
 A M syst X = 0
det [A] [B ]+ [I ] = 0
1
Transient response:
1
h2 (t ) = Re Z c (tanh( l ) )e j (t ) d Q2 ( )d
0
0
Time domain
Figure 3.19: Summary of the methods for analyzing dynamic behavior of a pipe.
Chapter 4
Hydroacoustic Characterization of a
Pipe
4.1
Ho
a,
Kv, Aref
D
L
Qo
fo = a/(4L)
3
[] [m /s]
[Hz]
0.02
0.5
0.5
analysis consists of: (i) a free oscillation analysis, (ii) a forced response, (iii) an impedance
calculation of the system. The time domain analysis consists of: (i) the determination of
the maximum amplitude of the waterhammer overpressure, (ii) the graphical resolution
of the waterhammer problem by the method of the characteristics, (iii) the numerical
resolution of the waterhammer problem.
4.2
The hydraulic system of interest includes a valve characterized by a energetic loss coefficient Kv and a reference cross section Aref . The head losses through the valve for a given
discharge are given by:
Hr =
Kv
Q2 = Rv Q
2
2 g Aref
(4.1)
Kv
Q [s/m2]
2 g A2ref
(4.2)
The valve impedance Z v can be obtained by linearizing the energetic loss for a mean
discharge Q:
dHr =
Kv
Kv
dQ2 =
2 Q dQ
2
2 g Aref
2 g A2ref
(4.3)
H(j)
dHr
Kv
= Rv
=
2Q
=
Q(j)
dQ
2 g A2ref
(4.4)
For a given valve stroke position, the energetic losses can be expressed by:
Kv
Hro
=
2
Qo
2 g A2ref
(4.5)
4.2.1
Kv
2 Hro
2Q=
2
2 g Aref
Qo
(4.6)
Continuous System
The analytical expression of the eigen frequencies of the system can be determined using
the characteristic impedance of the system, equation 3.45. The eigen values of the system
are the complex frequencies s satisfying the following boundary conditions:
for x=0 ; Z a (0) = 0;
EPFL  Laboratoire de Machines Hydrauliques
61
for x = L ; Z a (L) = Z v .
Introducing the above boundary conditions in equation 3.45 yields to:
Z v cosh( l) + Z C sinh( l) = 0
(4.7)
(4.8)
Thus:
Zv
2 l = ln (1)
ZC + Zv
k ZC
+ i (k )
(4.9)
ka
2l
; k = 1, 2, 3, ...
(4.10)
And:
a
k ZC Zv
=
ln (1)
2l
ZC + Zv
(4.11)
As the attribute of logarithm functions must be positive, 2 cases are identified for real
positive values of Zc and Zv:
Z Z
a
Zv > ZC : = 2l
ln Z v +ZC and = ka
; k = 1, 3, 5, ...
2l
C
(4.12)
Zv < Z C : =
a
2l
ln
Z C Z v
Z C +Z v
and =
ka
2l
; k = 2, 4, 6, ...
(4.13)
Table 4.2: Three first eigen mode shape determined with the discrete model with n = 100
elements for open/open and open/dead boundary conditions.
open
open
open
0.2
0.15
Head H [m]
Head H [m]
0.2
0.1
0.05
0
0
x 10
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0.15
0.1
0.05
0
0
4
3
Discharge Q [m3/s]
Discharge Q [m3/s]
0.1
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
x 10
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
1
0
0
0.1
4
0.15
0.15
Head H [m]
Head H [m]
0.1
0.05
0
0
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0.1
0.05
0
0
4
3
Discharge Q [m3/s]
Discharge Q [m3/s]
x 10
0.1
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
x 10
0.15
Head H [m]
0.15
0.1
0.05
Discharge Q [m3/s]
x 10
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0
0
3
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0.05
4
0.1
0.4
0.5
0.6
Abscisa x/L []
0.1
0
0
0.3
Discharge Q [m3/s]
Head H [m]
0.1
0.2
1
0
0
0.1
4
0.2
0
0
dead
x 10
0.1
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
Abscisa x/L []
0.7
0.8
0.9
4
2
1
0
0
63
0.98
f/fo []
0.96
0.94
0.92
0.9
10
15
n []
20
Figure 4.2: First natural frequency fo as function of the number of elements n modelling
the pipe.
The confidence threshold is connected to the number of elements as follow:
a
an
=
=
(4.14)
dx
f dx
f L
Therefore:
f L
n=
(4.15)
dx
a
For the confidence threshold previously determined the following values are obtained:
confidence threshold of 3 % : n =
1.85% < 3% ok!;
confidence threshold of 1 % : n =
0.42% < 1% ok!.
dx
dx
f L
a
f L
a
= 10
0.5600
1200
= 20
0.5600
1200
The above results show good agreement with the confidence threshold.
4.2.2
The eigen frequencies of the system, see figure 4.1, can be also determined from a time
domain simulation with a frequency domain analysis. Therefore, a simulation model
made of the upstream reservoir, a pressure source excitation H(t), the studied pipe and
the downstream valve is set up. The time domain simulation is performed considering
a Pseudo Random Binary Sequence, PRBS, pressure source excitation. The pressure
source is in series, and therefore corresponds to a difference of pressure specified between
2 elements (and not the pressure at a given node).
A PRBS signal is a random sequence of number of value 0 or 1. Such a signal is a
good approximation of white noise excitation as the energy of the signal is distributed
almost uniformly in the frequency domain. The amplitude spectra of a PRBS signal with
period dT = 0.01s is presented in figure 4.4 and the corresponding energy spectra of the
signal are presented in figure 4.5. This PRBS signal is obtained using a shift register
as illustrated by figure 4.6. The energy spectra shows that the energy of the signal is
distributed uniformly in the range 0 to 50 Hz.
H(t)=PRBS(t)
a,
Ho
Kv, Aref
D
L
Figure 4.3: Case study including pressure source excitation for forced response analysis.
The simulation of the dynamic behavior of the case study with the pipe modelled
by n = 100 elements under PRBS pressure source excitation provides the pressure and
discharge fluctuations in the time domain. The waterfall diagram representation of the
pressure fluctuations reveals the eigen mode and frequencies of the system. The downstream valve opening is selected ranging from open end until dead end conditions, and
is characterized using the linearized impedance Zv . The resulting waterfall diagrams are
presented as a function of the rated valve impedance Zv /Zc in table 4.3; where Zc is the
EPFL  Laboratoire de Machines Hydrauliques
65
0.030
0.025
0.015
0.020
0.010
0.005
characteristic impedance of the pipe. As expected, the resulting eigen modes fulfil the
specified boundary conditions.
When Zv /Zc < 1, according to equation 4.12 the eigen frequencies obtained are given
by:
fk = k
a
2L
; k = 1, 2, 3, ...
(4.16)
When Zv /ZC > 1, according to equation 4.13 the eigen frequencies are given by:
fk = (2 k 1)
a
4L
; k = 1, 2, 3, ...
(4.17)
When Zv /ZC = 1, the boundary condition is anechoic, and there is no wave reflection
downstream the pipe and no piping mode shape can be excited. To summarize:
a
fk = k 2L
a
fk = (2 k 1) 4L
10
10
Energy [m 2 ]
10
10
10
10
10
10
10
2
3
4
5
6
7
8
9
10
10
20
30
40
50
Frequency [Hz]
67
Table 4.3: Waterfall diagram obtained from PRBS excitation of the pipe with downstream
valve.
Zv /ZC = 0
Zv /ZC = 0.96
Zv /ZC = 0.04
Zv /ZC = 3.13
Zv /ZC = 0.25
Zv /ZC = 10.1
4.2.3
System Impedance
The impedance of the hydraulic system of figure 4.1 can also be calculated using equation
3.76. The impedance of this system Z = Z(x, f ), is calculated assuming an upstream
tank impedance Z = 0. The magnitude of the impedance divided by the characteristic
impedance of the pipe Zc is represented as a 3D plot function of the rated frequency f /fo
and rated location x/L, see figure 4.7. This impedance reveals 2 pieces of information:
the impedance downstream the pipe, Z(x = L, f ), providing the determination of
the eigen frequencies fk ;
the longitudinal impedance, Z(x, f = fk ), providing the location of pressure/discharge
minima and maxima.
The downstream impedance corresponds to the front face of the 3D impedance while
the longitudinal impedance is a cut at an eigen frequency f = fk . The downstream
impedance is extracted from figure 4.7 and presented in figure 4.8.
Figure 4.7: Three dimensional representation of the magnitude of the impedance of the
pipe.
69
Impedance for x = L
400
350
300
Z(j)/Zc []
250
200
150
100
50
0
0
5
f/f []
10
extracted for the first 4 eigen frequencies identified from figure 4.8 are presented in table
4.4. The minima and maxima of pressure and discharge locations are determined knowing
that:
for Z x (fk ) = H(j) Q(j) = 0 are the minima of pressure and maxima of discharge;
for Z x (fk ) = H(j) Q(j) = are the maxima of pressure and minima of discharge.
The locations of pressure and discharge minima and maxima present good agreement
with the results of table 4.2.
Table 4.4: Longitudinal impedance of the pipe for the different eigen frequencies f /fo =
1, 2, 3 and 4.
f /fo = 1
f /fo = 2
Impedance along x of the pipe for the second eigen frequency
350
350
300
300
250
250
Z(j)/Zc []
400
Z(j)/Z []
200
200
150
150
100
100
50
50
0
0
0.1
0.2
0.3
0.4
0.5
x/L []
0.6
0.7
0.8
0.9
0
0
0.1
0.2
0.3
f /fo = 3
350
300
300
250
250
Z(j)/Zc []
350
Z(j)/Z []
400
200
150
100
100
50
50
0.3
0.4
0.5
x/L []
0.7
0.8
0.9
200
150
0.2
0.6
400
0.1
0.5
x/L []
f /fo = 4
0
0
0.4
0.6
0.7
0.8
0.9
0
0
0.1
0.2
0.3
0.4
0.5
x/L []
0.6
0.7
0.8
0.9
4.3
71
Waterhammer are produced by sudden closure of valves in piping systems. In this section, the waterhammer problem is treated for the case study of figure 4.1 via 3 different
time domain approaches: (i) analytical determination of the overpressure, (ii) graphical
resolution, and (iii) numerical simulation.
4.3.1
Applying mass and momentum conservation law to the volume of control V 3 of figure 4.9
leads to the determination of the overpressure due to the valve closure.
(aCo)t
V3
y, v
Ap
aCo
Co
Co+C
V1
V2
x, u
C
n dV +
t
C C
n dA = Fx
(4.20)
V 3
C n dV =
C n dV +
C n dV
(4.21)
t
t
t
V3
V1
V2
C C
n dA = ( + ) (Co + C)2 A A Co2
V 3
(4.23)
Introducing equations 4.22 and 4.23 in equation 4.20 and expressing the force resulting
from the pressure balance yields to:
A(aCo )t
[( + ) (Co + C) Co ]
+( + ) (Co + C)2 AACo2 = (p2 p1 )A
t
(4.24)
The mass conservation equation for the same volume V 3 can be written as:
Z
Z
dM
=
dV +
C
n dA = 0
dt
t
V3
(4.25)
V 3
[( + ) ]
A [Co ( + ) (Co + C)] = 0
t
(4.26)
Combining equations 4.24 and 4.26 leads to an expression for the overpressure as a
function of the velocity change:
p = AC
(4.27)
a C
g
(4.28)
For a total closure of the valve: C = Co, leading to the maximum overpressure
given by:
Hmax =
4.3.2
a Co
g
(4.29)
The hyperbolic partial differential equations 2.26 can be solved using the method of
characteristics [14]. The set of differential equations to be solved is given by:
Q
h Q Q
+ gA
+
=0
t
x
2DA2
h
Q
L2 = gA
+ a2
=0
t
x
L1 =
(4.30)
(4.31)
(4.32)
73
The piezometric head and discharge are both functions of the location x and the time
t, thus:
h dx h
dh
=
+
dt
x dt
t
dQ
Q dx Q
=
+
dt
x dt
t
(4.33)
(4.34)
One obtains:
dQ
dh Q Q
+ gA +
=0
dt
dt
2DA2
(4.35)
With:
=
1
a
dQ gA dh Q Q
+
+
=0
dt
a dt
2DA2
dx = a
dt
dQ gA dh Q Q
+
=0
dt
a dt
2DA2
dx = a
dt
(4.36)
(4.37)
(4.38)
The above equation sets 4.37 and 4.38 consists of a compatibility equation, which is
only a function of the time derivative and a characteristic equation given by dx/dt = a.
To summarize, the linear combination of equations 4.30 leads to the more convenient
compatibility equations, but their validity is restricted along the characteristic lines.
For graphical resolution purposes, the equation sets 4.37 and 4.38 are expressed in
terms of finite differences while head losses are assumed to be concentrated at one pipe
end, yielding to:
h
a
Q
gA
(4.39)
=a
t
h
a
=+
Q
gA
= a
t
EPFL  Laboratoire de Machines Hydrauliques
(4.40)
The equation sets 4.39 and 4.40 define straight lines in the diagram [Q, h] with the
slopes a/(gA) with respective straight lines in the diagram [x, t] with slopes a. Figures
4.10 and 4.11 presents the graphical resolution in both diagram [x, t] and [Q, h] of a
waterhammer resulting from a valve closure with tclosure = 4L/a. The resolution is based
on the survey of 2 observers leaving at same time locations A and B, respectively the
reservoir and the valve. The 2 observers travel with the wave speed velocity a along the
characteristic lines dx/dt = +/ a in the diagram [Q, h], see figure 4.11 left, going forth
and back from the valve characteristic Hv (Kv (ti )) and reservoir characteristic Ho Hr .
The combination of the journey of the 2 observers in terms of location and time in both
diagrams [x, t] and [Q, h], provides the time evolution of the head and the discharge at
the valve and reservoir. In addition, the [Q, h] diagram provides the maximum amplitude
of the pressure Hmax = a Co /g from the characteristic line starting from the steady state
operation A(t1) to the vertical abscissa H. From this characteristic line, it can be stated
that:
tclosure < 2L/a H = Hmax
(4.41)
It leads to the expression of the critical closure time of the valve given by:
tcrit =
2L
a
(4.42)
75
t/(L/a)
5
4
3
2
x
Ho
Figure 4.10: Diagram [x, t] for the graphical resolution by the method of the characteristic
Hv(Kv(t4))
Hv(Kv(t3)) Hv(Kv(t2))
Hv(Kv(t1))
Hv Qv
HoHr(Qo)+a*Co/g
(B,t4)
(B,t3)
(B,t5)
(B,t2)
(A,t6)
Ho
(A,t4)
(B,t7)
(A,t3)
HoHr
(A,t1)
(B,t1)
Qo
10 t/(L/a)
Figure 4.11: Diagram [Q, h] for the graphical resolution by the method of the characteristic
and resulting time evolution of the discharge and the pressure.
4.3.3
The equivalent scheme of the system presented in figure 4.1 is presented in figure 4.12.
The equivalent scheme is made of a pressure source Ho, a series of n Tshaped equivalent
schemes of elements of length dx and a variable resistance modelling the downstream
valve.
R/2
L/2
Ho
L/2 R/2
R/2
L/2
R/2
Rv(t)
Cn
C1
Reservoir
L/2
Pipe
Valve
(4.43)
With:
Tstart : time at the start of the closure [s]
Tclose : valve closure time [s]
The valve loss coefficient is defined as follows:
Kv (t) =
Kvo
y1 (t)2
(4.44)
Applying Kirchhoffs law to the equivalent scheme of figure 4.12 leads to a set of
ordinary differential equations given by:
[A]
dx
+ [B] x = C
dt
(4.45)
This system is solved by using the 4th order explicit RungeKutta method. The simulation parameters are summarized in table 4.5.
Table 4.5: Simulation parameters
n[] dt[s] Tclose [s]
50 0.005
2.1
The critical time of the valve closure for the case study is tcrit = 2L/a = 1 s. As a
result, the amplitude of the waterhammer overpressure with a valve closing time of 2.1 s is
EPFL  Laboratoire de Machines Hydrauliques
77
below the maximum overpressure Hmax . The time evolution of the head at the valve, Hv ,
of the discharge at the valve, Qv and of the discharge at the reservoir, Qb , are represented
in figure 4.13. It can be noticed that the discharge at the valve decreases until zero within
the valve closure time inducing an overpressure at the valve of amplitude 0.8 Ho. The
maximum overpressure occurs at the instant Tstart + 2L/a.
Hv/Ho
Qb/Qo
Qv/Qo
Figure 4.13: Time evolution in rated values of the head at the valve Hv /Ho, the discharge
at the valve Qv /Qo and of the discharge at the reservoir Qb /Qo resulting from a valve
closure in 2.1s, i.e. 4.2L/a.
The simulation results obtained with the equivalent scheme are compared with the
simulation results obtained by MOC solved using a finite difference method. The comparison of the simulation results is presented for the head at the valve and the discharge
at the reservoir in table 4.14. The results obtained with the equivalent scheme show good
agreements with the ones obtained with MOC.
The results of the simulation of the valve closure in 0.2s (0.1 tcrit ) are presented
in figure 4.15. The maximum amplitude of Hv /Ho = 3 is reached after the full valve
closure for t = 0.2s. The time evolution of the pressure exhibits square shape with period
T = 4L/a.
The piezometric line along the pipe is represented in figure 4.16 for 8 different times
extending from to + T /8 to to + T with T = 4L/a. This representation evidences the
pressure wave propagation and also shows the steep slope of the pressure wave propagating
in the pipe for a valve closure time below the critical time tcrit . The pipe experiences
successively overpressure and underpressure phases lasting each 2L/a. The 4 phases are
summarized in figure 4.17.
Figure 4.15 depicts numerical instabilities visible on the head signal and on the discharge signal at the valve. The longer the simulation, the higher the numerical instabilities. These simulation errors are amplified at each wave reflection owing to the fact that
the model of the pipe is of the second order in the middle of the pipe but only of the first
order at the end. Furthermore, the quicker the valve closure, the higher the amplitude
EPFL  Laboratoire de Machines Hydrauliques
2
Hv/Ho MOC
Hv/Ho Equ. scheme
Qb/Qo MOC
Qb/Qo Equ. scheme
1.6
1.2
0.8
0.4
0.4
Time [s]
Figure 4.14: Comparison of the time evolution of the pressure at the valve (left) and
discharge at the reservoir (right) obtained with equivalent scheme and method of characteristic (MOC).
Hv/Ho
Qb/Qo
Qv/Qo
Figure 4.15: Time evolution of the rated head at the valve Hv /Ho, of the rated discharge
at the valve Qv /Qo and of the rated discharge at the reservoir Qb /Qo resulting from valve
closure in 0.2s, (0.4 L/a).
79
of the numerical instabilities because the frequency content of the pressure wave shape
becomes higher and thus requires a higher number of elements to properly model the phenomenon. Figure 4.18 shows how numerical instabilities can be reduced by increasing the
number of elements n from 10 to 100. This means that for simulating an instantaneous
valve closure, an infinite number of elements would be required and therefore, infinitely
small time steps. The influence of the integration method on the numerical instabilities
is discussed in appendix A.
For the real cases encountered in the operation of hydropower plants, instantaneous
valve closures are not supposed to be experienced. Indeed, designing the pipe walls with
a thickness capable of supporting the maximum amplitudes of waterhammer overpressure
or underpressure is not possible for economical reasons. Thus, the appropriate spatial discretization must be a compromise between computational time and simulation accuracy.
Figure 4.16: Piezometric line along the piping for different time (T = 4L/a).
t=0
Ho
Co
C=0
C=0
Co
a
C=0
n=10
n=20
n=50
n=100
Figure 4.18: Comparison of the time evolution of the head at the valve for different spatial
discretizations of the pipe.
Chapter 5
Modelling of Hydraulic Components
5.1
The hydraulic circuits of hydroelectric power plants are made of several components that
have to be included to model its dynamic behavior. The modelling of the following
components are presented: (i) elastic or viscoelastic pipe, (ii) valve, (iii) surge tank, (iv)
surge shaft, (v) air vessel, (vi) cavitating flow.
5.1.1
Pipe
The pipe model is the key component of the hydroacoustic modelling. In most of industrial applications pipes are made of metallic materials with elastic behavior. However,
recent developments in the domain of polymer materials has brought a wide range of new
materials to hydraulic installations. PVC for industrial applications and fiber glass for
prototyping are 2 examples of polymers that are commonly used in the construction of
hydraulic circuits.
Many polymers materials exhibit a viscoelastic behavior that induces additional damping in the systems that has to be properly taken into account especially for stability
analysis purposes. Therefore, the model of the pipe presented in chapter 3 is extended
for taking into account possible viscoelastic effects of the pipe wall material. The concept
of viscoelastic behavior is also extended to the fluid.
Elastic Pipe
The model of the pipe derived from the momentum and mass equations leads to the
representation of pipe of length dx by an equivalent electrical circuit made of 2 resistances,
2 inductances and one capacitance as presented in figure 5.1.
This modelling approach can be extended to a full length pipe by considering n equivalents schemes in series as shown in figure 5.2.
Viscoelastic Pipe
Viscoelastic materials present dynamic behavior as stress in the material is not only
proportional to the rated deformation = dl/l but is also function of the rate of deforEPFL  Laboratoire de Machines Hydrauliques
82
met
pi
g
ric l
ine
pi +1
g
Qi
Qi+
hi
dx
Zi
hi+1
R/2
L/2
R/2
L/2
Zi+1
Qi
hi
Datum
hi+1/2
Qi +1
hi+1
Figure 5.1: Modelling of a pipe of length dx (left), with the corresponding equivalent
scheme (right).
R/2
h1
Q1
L/2
Q2
h1/2
C h
1+1/2
C h
n11/2
Qn
L/2
C h
n1/2
R/2
Qn +1
hn+1
mation d/dt. Figure 5.3 shows typical time evolution of the stress as a function of the
rated deformation for viscoelastic materials.
The modelling of the viscoelastic behavior can be achieved using rheologic models
made of springs and dashpots. Maxwells, KelvinVoigts and Standard models are 3
common rheologic models presented in table 5.1 with their equivalent electrical schemes.
Theses models are elementary models from which more advanced models can be derived.
Without loss of generality, a viscoelastic pipe accounting for both pipe material and
water viscoelasticity can be modelled by 2 KelvinVoigt models as presented in figure 5.4.
Therefore, considering first the pipe wall material viscoelastic behavior and assuming a
pipe perimeter deflection = dD/D due to pressure increase leads to:
= Epipe + pipe
d
dD
1 d(dD)
= Epipe
+ pipe
dt
D
D
dt
(5.1)
The total derivative of the volume of the pipe V of length dx is given by:
dV = d(D2 /4) dx =
D
dD
2
(5.2)
83
Table 5.1: Rheologic models of viscoelastic materials and their equivalent scheme.
Model
Rheologic model
Equivalent scheme
Equations
KelvinVoigt
Maxwell
U
2
i2
= 1 E =
U=
1
C
U1
i1 dt = R i2
= 1 + 2 = E +
U2
U = U1 + U2 =
2
E1
E2
d2
dt
Standard
i1
i1
U
C1
R
1
C
i dt + R i
= 2 E2 = 1 E1 +
i2
C2
= 1 + 2
U=
U=
1
C1
1
C2
i2 dt
i1 dt + R i1
i = i1 + i2
d
dt
d1
dt
84
Efluid
Epipe
Cfluid Rfluid
fluid
pipe
Cpipe Rpipe
Figure 5.4: Rheologic (left) and equivalent (right) models of a viscoelastic pipe with
contribution of water and and pipe material viscoelastic behavior.
dt
2e dt
(5.3)
Combining equations 5.1, 5.2 and 5.3 and introducing the stored discharge Qp = dV /dt
leads to:
dh
Epipe e
pipe e
dQp
=
Qp +
dt
ADgdx
ADgdx dt
(5.4)
By integrating equation 5.4, one gets KelvinVoigts equation of the pipe wall material:
h=
1
Cpipe
Qp dt + Rpipe Qp
(5.5)
Where the viscoelastic resistance Rpipe and capacitance Cpipe of a pipe of length dx
are given by:
Rpipe =
pipe e
ADgdx
Cpipe =
ADgdx
Epipe e
(5.6)
Then, considering the fluid compressibility from equation 2.14 and the second viscosity
f luid leads to:
dp
Ef luid d f luid d2
=
+
2
dt
dt
dt
(5.7)
Reintroducing the piezometric head h = p/(g) + z and the stored discharge due to
fluid compressibility Qf = V d/dt gives:
dh
Ef luid
f luid dQf
=
Qf +
dt
Agdx
Agdx dt
(5.8)
By integration of equation 5.8, one gets KelvinVoigts equation of the pipe fluid:
Z
1
h=
Qf dt + Rf luid Qf
(5.9)
Cf luid
EPFL  Laboratoire de Machines Hydrauliques
85
Where the viscoelastic resistance Rf luid and capacitance Cf luid of a pipe of length dx
are given by:
f luid
Agdx
Rf luid =
;
Cf luid =
(5.10)
Agdx
Ef luid
It can be noticed that:
both viscoelastic resistances are proportional to the invert of the length of the pipe
dx;
the viscoelastic losses are proportional to the discharge and not to the square of
the discharge.
It can be also noticed that, if, in the model of the pipe of figure 5.4, the 2 viscoelastic
resistances of the fluid and of the wall material are neglected, the 2 capacitances in parallel
are equivalent to the capacitance of the elastic pipe:
1
gAdx
D
+
=
(5.11)
Cequ = Cpipe + Cf luid = Agdx
Epipe e Ef luid
a2
In the same way, if compressibility effects are neglected, the 2 viscoelastic resistances
in parallel can be expressed as:
1
equ
1
1
=
=
D
(5.12)
Requ = 1
1
Agdx pipe e + 1
Agdx
+ Rf luid
Rpipe
f luid
From the strict modelling point of view, a viscoelastic pipe is modelled by considering
the equivalent scheme of figure 5.4 made of 2 KelvinVoigt models for both the pipe
material and the fluid instead of the single capacitance of the elastic pipe model of table
5.1 (right). From the practical point of view, the determination of either the second
viscosity of the fluid or the viscosity of the pipe material is very difficult to perform with
good accuracy. However, experiments described by Haban et al. [66] have provided data
for a pipe filled with water. In this case, the equivalent viscosity is determined rather than
the fluid viscosity as in the experiment both viscosity contributions can not be dissociated.
As a result, it is very convenient to use a model made only of one KelvinVoigt model of
the whole pipe and accounting for both the fluid and the pipe material. In this model the
capacitance is calculated according to equation 5.11 and the viscoelastic resistance Rve is
calculated according to equation 5.12. The resulting model is presented in figure 5.5.
5.1.2
Valve
A valve induces head losses in hydraulic systems which are function of the valve obturator
position s. The head losses through a valve are given by:
Hv =
Kv (s)
Q2i
2gA2ref
(5.13)
With Kv the valve head loss coefficient. Therefore the valve corresponds to a variable
resistance function of the obturator position. The valve hydraulic resistance is given by:
Rv (s) =
Kv (s)
Qi 
2gA2ref
(5.14)
86
L/2
R/2
Qi
hi
L/2
R ve
hi+1/2
R/2
Qi +1
hi+1
Piezome
tric
line
Hv
hi
hi+1
s
Rv
Qi
hi
Qi
hi+1
Figure 5.6: Valve modelling with the example of a butterfly valve characteristic [105].
5.1.3
Surge Tank
The surge tank is a protection device against waterhammer effect behaving as a free
surface for wave reflection but where the water level is function of the discharge time
history. Surge tanks sometimes feature a cross section being function of the elevation z.
The volume of the surge tank is therefore expressed as the integral of the cross section
A(Z) along the elevation Z and is given by:
Z
VST = A(z)dz
(5.15)
The time derivative of the surge tank volume is:
dVST
dz
= A(z)
(5.16)
dt
dt
Noticing that the volume variation of the water in the surge tank is equivalent to the
stored discharge Qc = dVST /dt and introducing the piezometric head hc leads to:
A(z)
dhc
= Qc
dt
(5.17)
87
Equation 5.17 evidences the capacitive behavior of the surge tank whose capacitance
is directly the surge tank cross section:
CST = A(z)
(5.18)
The flow incoming and leaving the surge tank is subject to sudden cross section changes
and therefore energetic losses. It is common to have a diaphragm at the surge tank inlet
in order to increase the damping of water level oscillations. The head losses through the
diaphragm or the sudden change of cross section are given by:
Hd =
Kd
Q2c
2gA2ref
(5.19)
With Kd , the diaphragm head loss coefficient []. The corresponding hydraulic resistance is given by:
Rd (Qc ) =
Kd (Qc )
Qc 
2gA2ref
(5.20)
(5.21)
It can be noticed that the diaphragm loss coefficient Kd is usually function of the
discharge amplitude and direction: Kd = Kd (Qc ). The equivalent scheme of the surge
tank is made of a resistance and a capacitance in series as presented in figure 5.7. The
discharge incoming into the surge tank Qc is equal to the difference of the discharges at
the Tjunction and is given by:
Qc = Qi Qi+1
5.1.4
(5.22)
Surge Shaft
Surge shafts are surge tanks with small cross sections. If the surge tank inductance
effects can be neglected as the inductance is inversely proportional to the cross section
L = l/(gA), it is not anymore the case for the surge shaft. The inductance related to the
water inside the surge shaft is given by integrating the inductance along elevation axis z
and is given by:
Zhc
LSS =
hc zmin
dz
=
g A(z)
gA
(5.23)
zmin
The capacitance and resistance of the surge shaft have the same expression as that of
the surge tank and are given by:
Rd (Qc ) =
Kd (Qc )
Qc 
2gA2ref
and
CSS = A(z)
(5.24)
(5.25)
88
Piezometr
ic li
ne
Hd
Qi Qi+1
HST hc
Qc
Rd
Qc
Qi
Qi+1
Hd
HST
CST
hc
Datum
Piezometr
ic li
ne
Qi Qi+1
Qc
HSS hc
Qi
Qi+1 HSS
Zmin
Datum
Qc
Rd
Hd
LSS
HL
CSS
hc
5.1.5
89
Air Vessel
Air vessels are used for mitigating pressure fluctuations induced in hydraulic systems by
pumps, vortex shedding, valves openning/closure, etc. The water level hc changes in the
vessel lead to capacitive behavior. The stored discharge Qc and the cross section are
linked as follows:
A(z)
dhc
= Qc
dt
(5.26)
(5.27)
The gas volume is varying due to water level changes. Assuming a polytropic transformation of the gas leads to:
hg Vgn = cste
(5.28)
(5.29)
=
= Qc
hg n dt
dt
(5.30)
Equation 5.30 evidences the nonlinear capacitive behavior of the gas volume for which
the capacitance is given by:
CAV (Vg , hg ) =
Vg
hg n
(5.31)
(5.32)
The equivalent scheme of the air vessel is therefore made of 2 capacitances in series as
presented in figure 5.8.
5.1.6
Cavitating Flow
The development of cavitation in fluid flows is known to be a source of instabilities for the
whole hydraulic system. It was found that cavitation does not only represent a passive
additional compliance to the flow [27] but can play a crucial role of self excitation source
like of the famous POGO effect in propulsion systems of aerospace aircrafts [128]. The
stability of such system was studied using an onedimensional approach to model the
cavitation development with lumped elements. The volume of a cavitation development
is function of the head and discharge, therefore the total derivative of the volume is given
by:
V (Q, h)i+1 dV =
V
V
dhi+1 +
dQi+1
hi+1
Qi+1
(5.33)
90
Piezometr
ic li
ne
hg
HAV hc
Qc
Qi
Qi+1
Qi Qi+1
HAV
Qc
CAV
hc
Cg
hg
Datum
dt
hi+1
dt
Qi+1
dt
(5.34)
Defining:
the cavity compliance C = V
;
h
the mass flow gain factor = QVi+1 .
yields to:
Qi Qi+1 = Qc = C
dQi+1
dhi+1
+
dt
dt
(5.35)
(5.36)
Cavitation development can be modelled using equations 5.35 and 5.36. The resulting
equivalent scheme of the cavitation development is made of 1 capacitance as presented
in figure 5.10. The representation of the mass flow gain factor is difficult as it is kind of
mutual inductance effect and it is represented in figure 5.10 only by the value referring
to the discharge Qi+1
Piezometr
ic
91
line
Vg
Qc
hi
Qi
Qi Qi+1
Qi+1
Datum
hi
Qc
C
hi+1
92
5.2
5.2.1
General
There are 3 different types of hydraulic machines that are commonly used in the context of
hydroelectric power production: (i) the Francis turbine, or pumpturbine, (ii) the Pelton
turbine and (iii) the Kaplan turbine.
For safety purposes, the transient behavior of the entire hydraulic system comprising the turbine must be undertaken with appropriate turbine models. However, during
the exploitation, these hydraulic machines are subject to off design operations where 3D
complex flow structures such as flow separations, secondary flows, reverse flows, vortices,
cavitation development arise. Mathematical modelling of such flows leads to Computational Fluid Dynamic codes, which are nowadays far from being able to provide the
transient turbine parameters at a reasonable computational time with sufficient accuracy
for all the flow regimes experienced during transients. Therefore, experimental data are
required for the modelling of the dynamic behavior of turbines and pumpturbines. It is
assumed that the transient behavior of the hydraulic machines can be accurately simulated by considering that the machines are experiencing a succession of different steady
state operating points and therefore, can be modelled using the static characteristic of
the machine [90]. These models are called quasistatic models.
An operating point of a hydraulic machine is characterized by 5 quantities: the specific
energy E, the discharge Q, the rotational speed N , the torque T , and the guide vane
opening y. Therefore, the graphical representation of a turbine characteristic requires the
elimination of one of these quantities by the use of the hydraulic machines similitude laws.
For efficiency purposes, where the rotational speed can be considered to be constant, it
is convenient to use the dimensionless representation with , and . For transients
analysis, it is more convenient to use dimensional factors where the specific energy E is
eliminated. These factors are given by:
N Dref
N11 = p
(E/g)
Q11 =
2
Dref
Q
p
(E/g)
T11 =
3
Dref
T
E/g
(5.37)
5.2.2
Francis PumpTurbine
Francis pumpturbines are reaction turbines, i.e. they convert both kinetic and potential
energy of the fluid into mechanical work. The Francis turbine features fixed blades and
therefore the discharge through the turbine is controlled by the distributor. Figure 5.11
presents a vertical cutting plan of a Francis turbine.
Figure 5.12 presents a 4 quadrants characteristic of a Francis pumpturbine having a
specific speed of = 0.217. The discharge and torque factors are represented as a function
of the speed factor with the guide vane opening y as parameter; all values are rated by
the best efficiency point (BEP) value.
Some of the curves Q11 = Q11 (N11 ) of the pumpturbine characteristics of figure 5.12
exhibit a typical pumpturbine S shape between the 1st and the 4th quadrants leading
to numerical troubles for the interpolation of the Q11 values in the surface characteristics
EPFL  Laboratoire de Machines Hydrauliques
93
as illustrated in figure 5.13. This problem has been successfully solved by Marchal et al.
[101] who used a polar representation of the turbine characteristics. The polar coordinates
are defined in the plane N11 Q11 . Accordingly the polar angle is given by:
Q11 /Q11BEP
= atan
(5.38)
N11 /N11BEP
After simplification:
Q/QBEP
= atan
N/NBEP
(5.39)
(5.40)
1
r()2
(5.41)
H/HBEP
(Q/QBEP )2 + (N/NBEP )2
(5.42)
T11
T11BEP
T /TBEP
(Q/QBEP )2 + (N/NBEP )2
(5.43)
94
0
y=0
y = 0.03
y = 0.07
y = 0.14
y = 0.21
y = 0.29
y = 0.43
y = 0.57
y = 0.71
y = 0.86
y = 1.00
1
2
3
2
T11/T11BEP
Q11/Q11BEP
1
N11/N11BEP
y=0
y = 0.03
y = 0.07
y = 0.14
y = 0.21
y = 0.29
y = 0.43
y = 0.57
y = 0.71
y = 0.86
y = 1.00
2
4
2
1
N11/N11BEP
d X
=
Text = Tturb Telect
dt
(5.45)
Where:
J: total inertia of the rotating parts [kgm2 ]
: rotational pulsation [rd/s]
Tturb : mechanical torque of the turbine [N m]
EPFL  Laboratoire de Machines Hydrauliques
95
r()
Q11/Q11BEP
1
2
3
2
1
N11/N11BEP
Figure 5.13: Multiple value problem due to S shape of the characteristics and definition
of the polar representation.
96
1000
100
WH()
100
10
10
WB()
y=0
y = 0.03
y = 0.07
y = 0.14
y = 0.21
y = 0.29
y = 0.43
y = 0.57
y = 0.71
y = 0.86
y = 1.00
y=0
y = 0.03
y = 0.07
y = 0.14
y = 0.21
y = 0.29
y = 0.43
y = 0.57
y = 0.71
y = 0.86
y = 1.00
0.1
1
0.01
0.1
/2
/2
[rd]
3/2
0.001
/2
/2
[rd]
3/2
Piezometr
ic lin
E/g
N
HI
Qi
Lt
HI
Zref
Datum
HI
Rt
Qi
5.2.3
97
Pelton Turbine
B
D ref
Do
D2
Zo
The Pelton turbine is modelled by Ninj times a single injector turbine characteristic.
As a result, the Pelton turbine is viewed from the hydraulic circuit only as Ninj valves in
parallel. The equivalent circuit modelling the Pelton turbine corresponds to an equivalent
resistance of all injectors single resistance, as illustrated in figure 5.17 and given by:
Rt =
1
N
inj
P
i=1
(5.46)
1
Rinji
The single injector resistance is calculated from the characteristic of the turbine Q11 =
Q11 (yinj ), see figure 5.18, and is given by:
Rinj =
Qinj 
2
4
Q11 (yinj ) Dref
(5.47)
The mechanical torque of the machine is calculated as the sum of the contribution of
the torque of each single injector as follows:
Ninj
T = Kt
3
T11 (N11 , yinji ) Dref
H
(5.48)
i=1
Where Kt is a torque coefficient that accounts for the unsteadiness of the torque during
operation.
This modelling neglects both the influence on the efficiency of the multiinjectors operation and the dynamic behavior of the piping of the flow repartitor. However, with this
EPFL  Laboratoire de Machines Hydrauliques
98
HI
H
HI
Ninj
Qtot = Qinj
i =1
D ref
Rt(Q11(yinj))
yinj
Qinj
HI
Do
Qtot
HI
D2
Zo
Zref
Ninj = 1,...,7
Datum
model, injectors can be put in operation or shut off using the same turbine characteristic.
As the main purpose of such model is to perform transient simulations and not performing
energy production optimization, the single injector model is the most suitable.
Deflector Modelling
Pelton installations usually feature very long penstocks. As a consequence, the piping
critical valve closure time are very long and usually in contradiction with flywheel time
constants of the rotating parts. Specific protection devices have been developed, consisting
of deflectors cutting or deviating the jet between the injector and the turbine runner, see
figure 5.19. Such system can be activated within a very short time, 1 3 seconds, and
inducing a quick drop to zero of the mechanical torque, providing time for a slow closure
of the injectors in order to minimize the waterhammer effects in the piping.
As it exists several different types of deflectors, it is suitable to use a general method
for its modelling allowing to take into account any kind of deflectors. Therefore, a deflector
coefficient is introduced and is given by the ratio between the discharge that effectively
reaches the turbine and the discharge of the injector and is given for the ith injector by:
Kdef i =
Qrunner i
Qinjector i
(5.49)
An equivalent nozzle stroke ydef (t) corresponding to the discharge of the deflector
times this deflector coefficient is determined from the nozzle characteristic y(Q11 ). Finally the torque of the turbine is determined not from the nozzle position but from the
equivalent nozzle position ydef . In this modelling the deflector function Kdef has to be
known. The algorithm for the consideration of the deflector is illustrated in figure 5.20.
1.6
3
y=0
y = 0.03
y = 0.10
y = 0.17
y = 0.29
y = 1.00
T11/T11BEP
Q11/Q11BEP
1.2
0.8
0.4
99
0.2
0.4
y []
0.6
0.8
1
0.5
0.5
N11/N11BEP
1.5
2.5
Figure 5.19: Different types of deflectors, deviating jet (left) and cutting jet (right) [127].
Q11
y(t)
Kdef(t)
T11
Q11(t)
interpolation
Q11(y(t))
y=
cst
Kdef*Q11(t)
y(t)
T11(t)
T11def(t)
Kdef*Q11(y(t))
interpolation
ydef(Kdef*Q11(y(t)))
ydef(t)
interoplation
T11(ydef(t), N11(t))
ydef(t)
y(t)
y []
N11(t)
N11
T(t)
100
5.2.4
Kaplan Turbine
Kaplan turbines are reaction turbines, converting both kinetic and potential energy into
mechanical work. Because Kaplan turbines are subject to high relative variations of the
available energy E/E, they feature a double control system comprising the distributor
and mobile blades in order to ensure high efficiency on the whole operating range, see
figure 5.21. As a consequence, the characteristics of the Kaplan turbine has an extra
parameter, compared to Francis turbines, i.e. the blade pitch angle . Thus, the turbine
characteristics is made of a family of characteristics defined for different blade angles.
Figure 5.23 presents 2 characteristics for 2 different blade angles.
Do
zo
Bo
zb
Di
De
The model of the Kaplan turbine is also based on the polar representation of Suter,
as for the Francis turbine model, but the WH (, ) and WB (, ) values are interpolated
linearly between 2 blades angles as illustrated in figure 5.23 [69]. The linear interpolation
is given by:
WH,B (, ) =
WH,B (, 2 ) WH,B (, 1 )
( 1 )
2 1
(5.50)
2.5
2
y=0
y = 0.375
y = 0.50
y = 0.63
y = 0.75
y = 0.88
y = 1.00
1.5
T11/T11BEP
Q11/Q11BEP
0.5
1.5
N11/N11BEP
0.5
1.5
2.5
1.5
2.5
N11/N11BEP
2
y=0
y = 0.375
y = 0.50
y = 0.63
y = 0.75
y = 0.88
y = 1.00
1.5
T11/T11BEP
Q11/Q11BEP
y=0
y = 0.375
y = 0.50
y = 0.63
y = 0.75
y = 0.88
y = 1.00
1
3
2.5
2.5
0
y=0
y = 0.375
y = 0.50
y = 0.63
y = 0.75
y = 0.88
y = 1.00
1
2
0.5
2
0.5
101
0.5
1.5
N11/N11BEP
2.5
3
0.5
N11/N11BEP
Figure 5.22: Kaplan turbine characteristic for 2 blades angles; = 0.60 (top) and = 0.87
(bottom).
102
2.5
= 0.87
= 0.6
WH()
WH(=0.74)
0.5
/4
[rd]
/2
3/4
Figure 5.23: Linear interpolation of WH between 2 different blade angles for given guide
vane opening y.
Piezometr
ic lin
E/g
HI
HI
Qi
Lt
HI
Rt
Qi
HI
Datum
5.3
103
5.3.1
General
Once the models of the main hydraulic components constituting the hydroelectric power
plants are established, it is important to implement them in a software enabling a fast
modelling of the system, robust and efficient time domain integration and results analysis
in order to perform systematic analysis and optimization of the system. The Laboratory
of Electrical Machines (LME) of the EPFL has developed a simulation software for the
analysis of electrical power networks and adjustable speed drives called SIMSEN [130].
This software enables the time domain simulation of the dynamic behavior of an electrical installation featuring an arbitrary topology including the electrical machines, the
mechanical inertias and the control devices.
The modelling of the hydraulic components with electrical equivalents offers the possibility to implement them in SIMSEN in an easy way as it is based on the same syntax
and conventions. Thus, all the hydraulic components models described above have been
implemented in SIMSEN with the following advantages:
treating systems with arbitrary topology;
modelling hydroelectric systems comprising hydraulic circuit, electrical installations,
mechanical inertias and control devices;
ensuring to properly take into account the interaction between all components of
the installation as there is only one set of differential equations to be solved within
the same integration time step.
However, there is a drawback resulting from the last point. Electrical systems feature
time constants of about elec = 0.001s while hydraulic system features time constants of
about = 0.1s, i.e. 100 times larger.
5.3.2
Structure of SIMSEN
The SIMSEN software enables to set up the simulation model of a system according to its
own topology using electrical, mechanical and control modules through a Graphical User
Interface (GUI). Once all the parameters of each components are defined, the software
builds up a global system matrix using Kirchhoffs laws of the following form:
d
x
+ [B]
x =C
[A]
dt
With:
[A], [B]: system global matrix;
x : state vector;
(5.51)
104
Equation system 5.51 is solved in SIMSEN with the procedure of RungeKutta 4th
order taking into account all the nonlinearities of the system as every parameter of each
component can be parameterized using external functions. Finally, all the simulation
results are stored in text files.
Electrotechnic Modules
SIMSEN offers a wide range of electrotechnic modules comprising:
electrical machines: synchronous machine, induction, permanent magnet and DC
motors;
mechanical inertias: rotor, stator, shaft stiffness and damping, clutch, gearbox;
threephase elements: voltage supply, transmission lines, circuit breaker, transformers, loads;
semiconductors: diode, thyristor, thyristor GTO, IGTB;
singlephase elements: voltage supply, resistance, inductance, capacitance, circuit breaker, linked inductor, transformer;
analog functions: program, stransfer function, regulators, points/grid functions,
external DLL;
digital functions: limiter, pulse, generator, ztransfer function, hysteresis, sample.
The above list is nonexhaustive. These modules have been successfully validated by
comparison with experimental data [131] as illustrated by the example presented in figure
5.25. This example is the simulation of a load acceptance of an asynchronous machine
with a frequency converter of the Threelevel Voltage Source Inverter (VSI) type. Figure
5.26 presents the comparison of the simulated and experimental results witnessing the
good agreement.
105
Figure 5.26: Comparison between simulation (bottom) and measurements (top) of voltages (left) and torques (right) resulting from load acceptance of the system of figure 5.25
[131].
106
a)
id rs
ud
xss
xsDf
xad
b)
xsf
rf
iq rs
if
iD
xsD
uf
rD
uq
xss
xaq
iQ
xsQ
rQ
c)
Figure 5.27: Modelling of a synchronous machine with decomposition in the direct and
quadrature axis; a) geometry of salient pole [123], b) direct and quadrature axis [35], c)
equivalent schemes [32].
107
Table 5.2: Synchronous machine equivalent scheme parameters, state variables and boundary conditions.
Parameters
Units Description
rs , Xxs
[p.u.] statoric resistance and leakage inductance
Xad
[p.u.] inductance of principal field in axis d
XsDf
[p.u.] exclusive mutual inductance between
dampers of d axis and excitation
rf , Xsf
[p.u.] excitation resistance and leakage inductance
rD , XsD
[p.u.] damper circuit resistance
and leakage inductance in axis d
Xaq
[p.u.] principal field inductance in axis q
rQ1 , XsQ1
[p.u.] resistance and leakage inductance of the
damper circuit in axis q
p
[]
number of pair poles
State variables
Units Description
id ,iq
[p.u.] statoric phase in axis d and q
iD
[p.u.] damper circuit current in axis d
if
[p.u.] excitation current
iQ1
[p.u.] damper circuit current in axis q
m
[rd]
angular position of the rotor
m = dm /dt
[rd]
angular pulsation of the rotor
Boundary conditions Units Description
ud ,uq
[p.u.] statoric voltages in axis d and q
uD 0
[p.u.] damper circuit voltage in axis d
uf
[p.u.] excitation voltage
uQ 0
[p.u.] damper circuit voltage in axis q
108
Uabc: given
Uf: adapted
RLC: given
P, Q: obtained
RLC: adapted
Uabc: given
P, Q: given
Uf: adapted
Figure 5.28: Load flow scenarios according to available data; excitation voltage adapted
to the demand (top) or load parameters adapted to the production (bottom).
5.3.3
Hydraulic Modules
A hydraulic extension comprising all the hydraulic models presented above and called
SIMSENHydro, has been implemented in SIMSEN. These hydraulic models are summarized in figure 5.29 for the hydraulic circuit components and in figure 5.31 for the hydraulic
turbines.
Turbine Characteristics Interpolation
The modelling of the turbines is based on their characteristics curves. Therefore it is
necessary to perform interpolation of the WH,B values for the given abscissa y and . The
interpolation method implemented in SIMSENHydro is based on a Delaunay triangulation in the plane y . To each vertex of the triangle is associated the corresponding
WH,B values. Then, a planar interpolation is performed from the equation of the plan
in three dimensions. This method ensures the continuity of order 0 on the whole turbine
characteristic. The representation of the 2D triangulation and the resulting 3D surface
of the pumpturbine characteristic of figure 5.14 are presented in figure 5.31.
Initial Conditions Determination
Similarly to electrical systems, initial conditions of a hydraulic system simulation should
be determined according to the system boundary conditions prior to perform time domain
simulation. In SIMSENHydro, the initial condition procedure is not achieved by NewtonRaphson algorithm but by performing a fast simulation of the transient behavior of the
system leading to steady state conditions. However, to speed up the stabilization of the
EPFL  Laboratoire de Machines Hydrauliques
Description
Scheme
109
Equivalent scheme
R/2
Elastic pipe
hi
Qi
R/2
Viscoelastic
pipe
hi
Qi
L/2
L/2
Qi +1
hi+1/2
L/2
L/2
C
R ve
hi+1/2
R/2
hi+1
R/2
Qi +1
hi+1
Rv
Valve
hi
hi+1
Qi
Qi Qi+1
Qc
Rd
Surge tank
Hd
HST
CST
hc
Qi Qi+1
Qc
Rd
Hd
LSS
HL
CSS
hc
HSS
Surge shaft
Qi Qi+1
HAV
Air vessel
Vg
Cavitating
flow
Qc
CAV
hc
Cg
hg
Qi Qi+1
hi
Qc
C
hi+1
110
Description
Equivalent scheme
Scheme
N
Lt
Francis
pumpturbine
Rt
Qi
HI
HI
Rt(Q11(yinj))
Pelton
turbine
Qtot
HI
yinj
HI
Zo
Lt
Kaplan
turbine
HI
HI
system, the system parameters are optimized. Basically, 3 optimizations are undertaken:
an additional damping is introduced;
large capacitances leading to high period oscillations are reduced;
turbine characteristics are bounded in order to avoid errors due the search of a point
outside the turbine characteristic.
The additional damping is introduced by setting the viscoelastic resistance of the pipes
according to the system limit time constants. The time constant of the RC elements in
series in the Tbranch of the pipe model is given by:
RC = Rve C
(5.52)
The value of the viscoelastic resistance can therefore be determined with respect to the
integration time step to avoid numerical integration troubles by setting the integration
EPFL  Laboratoire de Machines Hydrauliques
4.5
111
4
6
3.5
log(W H ())
[rd]
2.5
2
4
2
0
1.5
2
0
1
0.5
0.5
0
0.5
0.2
0.4
0.6
y []
0.8
y []
1
[rd]
Figure 5.31: Delaunay triangulation of the pumpturbine characteristic of the figure 5.14;
2D (left) and 3D (right).
time step dt equal to the RC time constant divided by two leading to:
RC = Rve C > 2 dt
(5.53)
Hence:
Rve =
2 dt
C
(5.54)
112
Z: given
Z: given
T, : given
Q: result
y, : given
Q: result
y: adapted
T: result
Figure 5.32: Hydraulic load flow scenarios for generating mode (left) and pumping mode
(right).
Chapter 6
Analytical Analysis of Simplified
Hydraulic Systems
6.1
General
The modelling by electrical equivalent enables to set up simplified models of hydraulic installation to study their global dynamic behavior. The simplified models are preferably of
a low order to obtain the analytical solutions of the related differential equation set. Such
solutions provide the main dynamic quantities of the system in terms of eigen frequencies
and damping [1], [3], [34].
6.2
In the following sections, the eigen frequency related to mass oscillation problems is
analyzed for various types of tanks, i.e. (i) a surge tank, (ii) a surge shaft and (iii) an air
vessel.
6.2.1
The dynamic behavior of the hydraulic circuit shown by figure 6.1 (left) comprising an
upstream reservoir, a gallery, a surge tank and a penstock with a downstream valve is
investigated. Focusing only on low frequencies permits to neglect the compressibility of
both pipes. Thus the equivalent circuit of this system is made of a pressure source Ho ,
the gallery inductance LG and resistance RG , the surge tank capacitance CST and the
diaphragm resistance Rd , the penstock inductance LP and resistance RP and the valve
resistance Rv as presented in figure 6.1 (right).
The consequence of a sudden closure of the valve is analyzed assuming an initial steady
state condition with constant valve opening. Closing the valve yields to open the right
hand loop of the equivalent scheme related to Q2 . The differential equations written using
Kirchhoffs law applied to the left hand loop leads to:
dQ1
H o = LG
+ (Rd + RG ) Q1 + hST
dt
(6.1)
CST dhST = Q1
dt
EPFL  Laboratoire de Machines Hydrauliques
Surge tank
LG
Q1
Ho
Q2
Gallery
Ho
Penstock
Q1
RG
LP
Rd
hST
CST
RP
Q2
Rv
Valve
+
hST = 0
+
2
dt
L
dt
C L
 {zG }
 ST{z G}
2
(6.2)
o2
(6.3)
With:
1 = o
1 2
(6.4)
(6.5)
(6.6)
Where:
lG : length of the gallery [m]
AST : surge tank cross section [m2 ]
AG : gallery cross section [m2 ]
The period To is usually very low as the surge tank cross section and the gallery length
are large and the gallery cross section is small. This period is called the mass oscillation
period related the oscillation of the discharge in the gallery between the reservoir and the
surge tank.
EPFL  Laboratoire de Machines Hydrauliques
115
The amplitude of the water level oscillations in the surge tank, assuming a constant
surge tank cross section and a frictionless system, is obtained considering a solution of
type hST (t) = hST o sin(o t + ) whose first derivative introduced in equation 6.1 gives
the oscillation amplitude:
hST o =
Q1o 1
= C1o
CST o
Where C1o =
6.2.2
Q1o
AG
lG AG
g AST
(6.7)
The dynamic behavior of the hydraulic circuit presented by figure 6.2 (top) comprising
an upstream reservoir, a first part of gallery, an upstream valve, a surge shaft, a second
part of gallery, a surge tank and a penstock with a downstream valve is investigated.
Focusing again only on low frequencies, the compressibility of the pipes are neglected.
The system is considered frictionless. Thus the equivalent circuit of this system is made
of a pressure source Ho , the first gallery inductance LGo , the valve resistance Rv1 , the surge
shaft capacitance CSS and inductance LSS , the second gallery inductance LG1 , the surge
tank capacitance CST , the penstock inductance LP and the downstream valve resistance
Rv2 as presented in figure 6.2 (bottom).
Surge tank
Surge shaft
Qo
Ho
Valve1
Q1
Q2
Gallery
Penstock
Valve2
L Go R v1
Ho
Qo
L G1
L SS
CSS
Q1
LP
CST
Q2
R v2
Figure 6.2: Hydraulic circuit with surge tank and surge shaft.
dQ1
+ hST hSS = 0
(LG1 + LSS )
dt
dhST
(6.8)
CST
= Q1
dt
CSS dhSS = Q1
dt
Combining the 3 above equations leads to the following characteristic equation:
d2 Q1
1
1
1
Q1 = 0
(6.9)
+
+
2
dt
CST
CSS
LSS + LG1
{z
}

o2
1
CSS
1
CST
(6.10)
LSS + LST
(6.11)
Once the 2 valves are closed the system constituted of the surge shaft, the gallery and
the surge tank undergo mass oscillations. The mass of water oscillates between the surge
shaft and the surge tank with amplitudes driven by the tanks cross sections. It can be
seen that the inertia of the water in the surge shaft may strongly influence the oscillation
period depending on the inductance ratio between the surge shaft and the gallery.
6.2.3
The dynamic behavior of the hydraulic circuit presented by figure 6.3 (left) comprising
an upstream reservoir, a gallery, an air vessel and a penstock with a downstream valve
is investigated. Focusing only on low frequencies permits to neglect the compressibility
of both pipes. The system is considered frictionless. Thus, the equivalent circuit of
this system is made of a pressure source Ho , the gallery inductance LG , the 2 air vessel
capacitances CAV and Cg , the penstock inductance LP and the valve resistance Rv as
presented in figure 6.3 (right).
The consequence of a sudden closure of the valve is analyzed assuming an initial steady
state condition with constant valve opening. Closing the downstream valve yields to open
loop related to Q2 . The differential equations written using Kirchhoffs law applied to the
first loop leads to:
dQ1
LG
+ hAV + hg = Ho
dt
dhAV
(6.12)
CAV
= Q1
dt
Cg dhg = Q1
dt
EPFL  Laboratoire de Machines Hydrauliques
117
Air vessel
LG
Q1
Ho
Q2
Gallery
Ho
Penstock
Q1
LP
CAV
Cg
Q2
Rv
Valve
+
+
Q1 = 0
(6.13)
2
dt
CAV
Cg
LG

{z
}
o2
1
CAV
1
Cg
(6.14)
LG
(6.15)
Hg n
Comparing the mass oscillation period obtained for a system with air vessel with the
period obtained for a system with a surge tank, given by equation 6.5, it can be noticed
that:
if the gas volume tends to infinity, the period of the system with air vessel corresponds to the period of the system with a surge tank;
if the volume of gas becomes small, the mass oscillation period decreases;
for small gas volume and high oscillation amplitudes, the system becomes strongly
nonlinear as the air vessel capacitance is function of the gas volume and pressure.
To illustrate the influence of the gas volume on the mass oscillation period and time
evolution, the simulation of the sudden closure of the downstream valve is performed for
3 different gas volumes. The dimensions of the hydraulic circuit are summarized in table
6.1. The time domain evolution of the gas piezometric head hg , the water level in the air
vessel hAV and the total piezometric head h = hg + hAV are represented in dimensionless
values in figure 6.4. The above 3 statements are clearly confirmed by the simulations.
Especially the nonlinear behavior induced by the gas volume is clearly pointed out for
the simulation results with Vg = 50 m3 .
EPFL  Laboratoire de Machines Hydrauliques
Table 6.1: Parameters of the hydraulic installation with air vessel of figure 6.3.
Gallery
Air vessel
Penstock
Nominal values
2
l = 1100 m
A = 38.48 m
l = 1100 m
Ho = 700 m
a = 1100 m/s hgo = 100 m a = 1100 m/s
Q1o = 30.9 m
D = 3.57 m hAV o = 598 m D = 2.52 m
6.3
In the following sections, stability criteria are determined for the following installations:
(i) power plant with regulated turbine and surge tank; (ii) piping with cavitation development; (iii) piping with valve leakage; (iv) pumping system; and (v) pumpturbine power
plant.
6.3.1
dQ1
LG
+ RG Q1 + hST = Ho
dt
(6.16)
(6.17)
From the second loop, it can be stated that the piezometric head of the surge tank corresponds to the turbine head Ht which is equal to the initial head Hto plus a perturbation
z and is therefore given by:
hST = Ht = Hto + z
(6.18)
(6.19)
Then the discharge of the turbine can be expressed with equation 6.18 and then
expressed with the limited development (1/(1 + x) = 1 x + x2 x3 ...) leading to:
Hto
z
Q2 = Q2o
' Q2o 1
(6.20)
Hto + z
Hto
EPFL  Laboratoire de Machines Hydrauliques
119
(hg + hAV)/Ho
hAV/Ho
hg/Ho
Figure 6.4: Mass oscillations with air vessel of volume: Vg = 5000 m3 (top), Vg = 500 m3
(middle), Vg = 50 m3 (bottom).
dt
Hto
dt
(6.21)
+ CST
Hto
dt
dt2
(6.22)
The head loss in the gallery is a nonlinear term that requires to be developed, and
therefore the head losses are expressed as follows:
0
RG Q1 = RG
Q21
(6.23)
With equation 6.17 and 6.20, the discharge in the gallery gives:
Q21
= (QST + Q2 ) =
QST
hST
+ Q2o 1
Hto
2
(6.24)
Surge tank
L G RG
Q1
Ho
Q2
Gallery
Penstock
Ho
Q1
CST
Q2
Ht
Turbine
Figure 6.5: Hydraulic system with surge tank and regulated turbine.
Q2o
Hto
Q2o
Hto
(6.25)
Introducing equation 6.16, 6.21 and 6.25 in equation 6.22 leads to the characteristic
equation:
0
0
RG
Q2o
Q2o
RG
Q22o
dhST
1
d2 hST
+ 2
+ 12
hST =
2
dt
LG
Hto CST
dt
Hto
LG CST


{z
}
{z
}
2
o2
0
Q22o
Hto RG
LG CST
(6.26)
The stability of the system is ensured when 2 > 0, leading to the following stability
criteria:
CST >
LG
Q2o
Hto 2 RG
(6.27)
After expressing the inductance, resistance and capacitances, the stability criteria
gives the Thoma cross section:
AST
Q22o
lG
>
2 g Hto HrGo AG
(6.28)
Where HrGo are the head losses in the gallery calculated with the initial discharge Q2o ,
lG and AG are respectively the length and the cross section of the gallery, and Hto is the
initial head of the turbine.
The Thoma cross section is the surge tank limit cross section below which the system
becomes unstable after a perturbation induced by the turbine [145].
6.3.2
121
H1
Q1
Q2
H2
H1
Q1
Q2
H2
The system of differential equations of the equivalent scheme of the hydraulic system
is given by:
dQ1
+ R Q1 + Hc
H1 = L
dt
dQ2
dHc
(6.29)
+C
= Q1 Q2
dt
dt
Hc = L dQ2 + R Q2 + H2
dt
The determinant of this set of equations written in matrix form leads to the characteristic equation:
R
2
2
+
+
+
(6.30)
=0
L +
L LC
L{z
C}
{z}


{z
}
1/
o2
Where is the eigen value of the set of equation 6.29. The free motion time constant
of the fluid in the pipe can be identified in the left hand term of equation 6.30. The
frictionless eigen pulsation of the system o and the stability criteria of the system given
by 2 > 0 are determined from the right hand term. The stability criteria leads to the
following criteria:
R <
(6.31)
Unstable domain
6.3.3
Hydraulic circuit comprising an upstream reservoir, a pipe and a downstream valve may
present an unstable behavior after perturbations [86], [112]. Such a system can be modelled with a first order equivalent scheme of the pipe made of an inductance Lp and a
capacitance Cp while the valve is modelled by a variable resistance Rv as presented in
figure 6.8. The corresponding set of differential equations is given by:
Lp
Ho
Q2
Ho
Pipe
Valve
Q1
Cp
Q2
Rv
dQ2
Lp
+ Rv Q2 + hc = 0
dt
Cp dhc = Q1 Q2
dt
(6.32)
+
Q2 = 0
dt2
Lp dt
Cp Lp
{z}
 {z }
2
(6.33)
o2
123
The system remains stable for 2 > 0, leading to the following stability criteria:
Rv > 0
(6.34)
6.3.4
Pumping systems comprising an air vessel may feature instabilities as reported by Greitzer
[65]. Such a system can be simplified for investigation purposes to a system comprising a
downstream reservoir, a pump with fixed rotational speed, a pipe, a valve, an air vessel
and an upstream reservoir. The equivalent scheme of this system is made of the pump
pressure source Hp (Q), the pipe model with the pipe inductance Lp and resistance Rp ,
while the air vessel is modelled by its capacitance CAV and the valve by its resistance
Rv , see figure 6.10. The compliance of the pipe is neglected with respect to the air vessel
compliance.
Rv
Air vessel
Q1
Ho
Valve Q
Ho
Q1
CAV
Rp
Q2
Lp
Hp (Q2 )
Pipe
dQ2
Lp
+ Rp Q2 + Rv Q2 + Hp (Q2 ) = hAV
dt
CAV dhAV = Q1 Q2
dt
(6.35)
The head of the pump can be linearized around the operating point Q2 of interest as
follows:
Hp (Q2 ) = Hp +
dH
Q (Q2 Q2 )
dQ2 2
 {z }
(6.36)
RQ2
Combining the 2 above equations and assuming that the upstream discharge fluctuations are negligible leads to the following characteristic equation:
d2 Q2 Rv + Rp + RQ2 dQ2
1
+
Q2 = 0
2
dt
Lp
dt
CAV Lp

{z
}
 {z }
2
(6.37)
o2
The system remains stable for 2 > 0, leading to the following stability criteria:
Rv + Rp >
dH
dQ2
(6.38)
The above stability criteria shows that a negative slope of the characteristic curve of
the pump Hp = Hp (Q) considering a negative discharge in pump mode may lead to system
EPFL  Laboratoire de Machines Hydrauliques
125
instabilities. Then, for system whose slope of energetic losses in the pipe and the valve
are higher than the slope of the pump characteristic, the system is stable, as illustrated
in figure 6.11 left. But, if the slope of the energetic losses are below the energetic losses,
as illustrated in figure 6.11 right, the system is unstable.
Stable
Unstable
E
(Rv'+Rp')Q2
(Rv'+Rp')Q2
=
cst
e
cst
e
Q
Q
6.3.5
A hydraulic circuit comprising an upstream reservoir, a pipe and a noncavitating pumpturbine may exhibit an unstable behavior after perturbations (Greitzer [65], Martin [102],
Jacob [79], Huvet [75]). For the stability analysis of such a system, the rotational speed
changes of the pumpturbine has to be taken into account (Martin [102], Huvet [75]) and
therefore the model should consider both the hydraulic and mechanical properties. The
hydraulic part of the system can be modelled with a first order equivalent scheme for the
pipe made of a resistance Rp and an inductance Lp , compressibility effects being neglected,
while the pumpturbine is modelled by a variable pressure source Hpt as presented in figure
6.12. For the mechanical model, the angular momentum law is applied to the inertia J
accounting for both the rotor of the generator and the turbine inertia.
Lp
Ho
Q
Pipe
Rp
Ho
Hpt (Q,)
Pumpturbine
dQ
H o = Lp
+ Rp Q + Hpt
dt
J d = Tpt + Tel
dt
EPFL  Laboratoire de Machines Hydrauliques
(6.39)
dH
dH
Q (Q Q) +
 ( )
dQ
d{z }

 {z }
R
RQ
dT
dT
Tpt (Q, ) = Tpt +
Q (Q Q) +
 ( )
dQ
d{z }
 {z }
(6.40)
KQ
Combining equations 6.39 and 6.40 leads to the following matrix equation:
#
" RQ +R
R
d
Q
Q
H Hpt + R + RQ Q
Lp
Lp
+
(6.41)
kQ
k
Tpt + Tel k kQ Q
dt
J

{z J
}
[A]
The characteristic equation of the system 6.41 is obtained with the determinant
det([A] [I] ), leading to:
RQ + Rp k
kQ R
2
=0
(6.42)
+
+
Lp
J
J Lp

{z
}
 {z }
2
o2
The system remains stable for 2 > 0, leading to the following stability criteria:
RQ + Rp
k
>
Lp
J
(6.43)
+ Rp
Lp
{z
1/f Q
>
dT

d
(6.44)
J }
 {z
1/m
In equation 6.44 above, the local fluid time constant f Q evaluated for a given Q
and the local mechanical time constant m evaluated for a given are introduced. The
above stability criteria indicates that it is necessary that m > f Q . The US Bureau of
Reclamation even specifies that the stability of the power plant is ensured if m > 2 f2
[98]. Where:
m =
f =
JBEP
TBEP
lp QBEP
:
gHBEP Ap
127
Figure 6.13: Stability analysis in the plane Q11 (n11 ) and T11 (n11 ) for a low specific speed
Francis turbine [103].
Chapter 7
Transient Phenomena in
Hydroelectric Power Plants
7.1
General
Once the models and the related equivalent schemes of the hydraulic components are
established, transient simulation can be performed to highlight the possible hydroelectric
interactions. The following aspects are treated in the following sections:
validation of the hydraulic modules of SIMSENHydro;
investigation of the impact of classical hydraulic or electric disturbances on the
dynamic behavior of a hydroelectric power plant;
comparison of the stability of the turbine speed governor using a hydraulic model
versus a hydroelectric model in isolated production mode;
investigation on the turbine speed governor stability in islanded production mode.
7.2
7.2.1
Validation of SIMSENHydro
Case Study Definition
130
28 24
PT4
20 16
12
19 15
11
32
31
27 23
PT3
33
Lower reservoir
(192.6193.2 m)
30
26 22
29
25 21
PT2
18 14
10
17 13
34
35
PT1
Figure 7.1: Layout and pipe numbering of the pumped storage plant test case.
Table 7.1: Rated values of the pumpturbines of the validation test case.
Hn
Qn
Pn
Nn
Dref
Jtot
3
2
[m] [m /s] [M W ] [rpm] [m] [kg m ]
[]
6
305
109
315
300
5.08 2.77 10 0.272
The pumpturbine characteristic of the pumped storage plant are represented in figure
7.3. The modelling of the pumpturbines takes into account kinematic relation between
the servomotor stroke and the guide vane opening. The spiral case and draft tube are
modelled as pipes of equivalent length and diameter and correspond respectively to pipes
N 17 20 and N 21 24. During the load rejection, the circuit breaker between the
generator and the transformer is opened, which leads to an electromagnetic torque equal to
zero. Consequently, the electrical installations are assumed to have no significant influence
Pipe
L[m]
D[m]
a[m/s]
[]
Pipe
L[m]
D[m]
a[m/s]
[]
131
El. 206.35m
Acommon: 1154 m2
El. 196. 6m
Acircular: 122.7 m2
El. 169.16m
Ariser: 14.53 m2
on the transient and are modelled as an external torque acting on the mechanical inertias.
T11/T11BEP
Q11/Q11BEP
y=0
y = 0.12
y = 0.24
y = 0.35
y = 0.42
y = 0.53
y = 0.65
y = 0.76
y = 0.88
y = 1.00
1
2
3
2
1
0
y=0
y = 0.12
y = 0.24
y = 0.35
y = 0.42
y = 0.53
y = 0.65
y = 0.76
y = 0.88
y = 1.00
2
N11/N11BEP
4
2
1
N11/N11BEP
132
7.2.2
Simulation Results
N/NR
133
H/HR
[]
y
Q/QR
T/TR
a)
H, hST [m]
QST [m3/s]
QST
b)
hST [m]
hST2
hST1
hST3,4
c)
Figure 7.4: Transient behavior of the pumped storage plant resulting from an emergency
shutdown in generating mode; transient of the pumpturbine Unit 1 in rated values a),
transient of surge chamber of Unit 1 b), transient of the 4 surge tanks c).
EPFL  Laboratoire de Machines Hydrauliques
134
T11/T11BEP
Q11/Q11BEP
1.5
0.5
1
0.5
0.4
0.8
1.2
N11/N11BEP
1.6
2
0.4
0.8
1.2
N11/N11BEP
1.6
Figure 7.5: Operating point trajectory in the plan Q11 (N11 ) and T11 (N11 ) during the
emergency shutdown in generating mode for Unit 1.
[]
T/TR
135
H/HR
y
Q/QR
N/NR
H, hST [m]
QST [m3/s]
QST
hST2
hST [m]
hST3,4
hST1
Figure 7.6: Transient behavior of the pumped storage plant resulting from an emergency
shutdown in pumping mode; transient of the pumpturbine Unit 1 in rated values a),
transient of surge chamber of Unit 1 b), transient of the 4 surge tanks c).
EPFL  Laboratoire de Machines Hydrauliques
T11/T11BEP
Q11/Q11BEP
136
1
2
3
2
1.5
1
0.5
N11/N11BEP
0.5
1
2
1.5
1
0.5
N11/N11BEP
0.5
Figure 7.7: Operating point trajectory in the plan Q11 (N11 ) and T11 (N11 ) during the
emergency shutdown in pumping mode for Unit 1.
7.2.3
137
Validation
During the transients tests carried out at the commissioning of the pumped storage plant,
the piezometric head time history was recorded on Unit 1 at the spiral case inlet, at the
draft tube mandoor, and the surge chamber. The simulation results in generating and
pumping mode time evolution are compared for these 3 values and represented in table
7.3.
Table 7.3: Comparison of simulation results with transient tests carried out on Unit 1
during an emergency shutdown in both generating and pumping mode; simulation in light
red line and measurements in bold black line.
Generating Mode
Pumping Mode
By comparing the simulation results with the measurements on site during an emergency shutdown in generating mode it can be noticed that:
the steady state conditions before the transients presents a very good agreement;
EPFL  Laboratoire de Machines Hydrauliques
138
the amplitudes and time evolution of water level in surge tank 1 and piezometric
head in draft tube of unit 1 present also a very good agreement;
the overpressure amplitude of the piezometric head at the unit 1 inlet presents a
discrepancy of 30%, however the time evolution of the piezometric head shows a
good agreement.
By comparing the simulation results with the measurements on site during emergency
shutdown in pumping mode it can be noticed that:
the steady state conditions of the simulation presents small discrepancies that can
be due to errors on the downstream water level, difference in the friction losses
in pumping and generating mode or difference between scale model and prototype
characteristics;
the amplitudes of the piezometric head at the inlet of pumpturbine 1 presents a
good agreement;
the waterhammer at the pumpturbine 1 inlet does not damp out as quickly as the
in measurements;
the time evolution of the water level in the surge tank 1 and the piezometric head in
the draft tube of Unit 1 presents a discrepancy on the period of the mass oscillation
but the amplitudes fit roughly.
In order to identify the origin of the discrepancies of the surge tank water level time
evolution during the emergency shutdown in pumping mode, the steady state discharge
obtained by simulation is compared to the discharge measured on site. The comparison
of the steady state discharge for generating and pumping mode is presented in table 7.4.
The measured values of the discharges are given with a confidence range of 8m3 /s
because of the poor resolution of the graphical representation of the time evolution of the
discharge from which the value is deduced.
Table 7.4: Initial steady state conditions of discharge
Test Case
Simulation Measurements
Q [m3 /s]
Q [m3 /s]
Generating mode
166.8
155.7 8
Pumping mode
98.1
120 8
for Unit 1.
Error
[]
+0.07
0.18
The comparison of the simulated and measured discharges points out an error on the
discharge in generating mode that is in the range of uncertainties of the measured value.
However, a difference of about 18% is found on the initial conditions of the discharge
in pumping mode. Given small error in generating mode, the error in pumping mode is
probably not due to errors on the friction parameters of the hydraulic circuit. The error
is probably due to differences of the pumpturbine characteristic between the scale model
and the prototype.
Moreover, the mechanical power obtained in generating mode differs between the
simulations and the measurements. The measurements report a power of 390 MW for
EPFL  Laboratoire de Machines Hydrauliques
139
the chosen operating point while the simulation predicts only 362 MW. This corresponds
to an efficiency difference between the scale model and the prototype: the prototype
efficiency is higher than the scale model efficiency. This fact is commonly admitted.
As the initial discharge in pumping mode is smaller in the simulation compared to the
experiments, the amplitudes are also smaller, thus reducing the oscillation period due to
the nonlinear cross section of the surge chambers. It also explains the difference of the
initial conditions of the piezometric head which value would be reduced in the case of a
higher discharge and thus would present a better fit with the experiments. Therefore, if
the discharge were higher in the simulation, the waterhammer in the spiral case would
show higher amplitudes in similar fashion as in generating mode.
The error on the amplitude of the waterhammer is probably also due to differences
of the pumpturbine characteristics that strongly influences the overpressure of the emergency shutdown of the pumpturbine. Indeed, as the turbine goes through the S shape
of the pumpturbine during such a transient, see figures 7.5 and 7.7, the discharge becomes
negative during the closure of the guide vanes and therefore induces a higher overpressure
than in the case of turbines without S shape characteristic.
Despite the differences between the simulation and the experiments found in pumping
mode, the models of hydraulic components developed and implemented in SIMSENHydro
are satisfactory.
7.2.4
Numerical Instabilities
The simulation of the emergency shutdown resulting from a full load rejection of a pumpturbine shows that the pumpturbine is going through the unstable part of the characteristic, i.e. the Sshape. The numerical integration of the differential equation set of the
whole system using RungeKutta 4th order has proven its robustness, see appendix A.
However, when simulating the load rejection, numerical instabilities have been pointed
out during the transit through the S. To overcome this numerical problem, it has been
identified that the CFL criteria, equation 3.60, must be fulfilled as follow:
dx
=k
a dt
; k = 1, 2, 3...
(7.1)
Then, for each pipe, the wave speed has to be adapted in order to satisfy the above
criteria. It is acceptable to adapt the wave speed as there are uncertainties on this value
because its analytical value presents some dispersion compared to the experimental data
due to errors on the wall material properties, dimensions, contact between the pipe and
the support or concrete, or air content. A comparison of the simulation results of a load
rejection of a pumpturbine with constant guide vane opening y is presented in figure 7.8
for both cases: wave speed adapted or not. It appears, that when the turbine is going
through the unstable part of the characteristic, i.e. negative torque, the torque and the
head start oscillating for the simulation without adaptation of the wave speed, while,
the torque and head evolution stay stable over time evolution for the simulation with
adaptation of the wave speed.
[]
140
[]
T/TR adapted
wave speed
T/TR
Figure 7.8: Simulation of a full load rejection with constant guide vane opening with and
without adaptation of the wave speed.
7.3
141
Hydroelectric Transients
7.3.1
UNIT 1
+ Uc
KU(s)
2x86 MW Hydroelectric
power plant
Pc +
(1), (3)
Infinite network
U=205kV, f=50Hz
(2)
KP(s)
UNIT 2
KU(s)
Pc +
+ Uc
(1)
KP(s)
To investigate the dynamic behavior of the power plant, 3 simulation models are
used: (i) a hydraulic model, (ii) an electric model and (iii) a hydroelectric model. Then,
the hydroelectric model simulation results are compared either with the results obtained
with the hydraulic model for hydraulic disturbances or with the results obtained with the
electric model for electric disturbances. Therefore, 3 standard disturbances are simulated:
a total load rejection: the 2 circuit breakers between the transformers and the
synchronous machines of both Units, see figure 7.9 (1), are opened at t = 1 s, while
the guide vanes of the 2 Francis turbines are closed linearly in 7 s; hydraulic and
hydroelectric models are compared;
EPFL  Laboratoire de Machines Hydrauliques
142
Turbine
Hn = 82 m
Qn = 114 m3 /s
Nn = 200 rpm
Tn = 4.11 106 N m
Jt = 8.415 104 kg m2
Generator
Sn = 98 M V A
Un = 17.5 kV
f = 50 Hz
pair pole = 15
Jg = 1.683 106 kg m2
Kshaf t = 1.27 1010 N m
an earth fault: the 3 phases between the synchronous machine and the transformer
of Unit 1 are connected to the ground by closing the circuit breaker (2) of figure
7.9, electric and hydroelectric models are compared;
an out of phase synchronization: the circuit breaker between the transformers and
the synchronous machines of Unit 1, see figure 7.9 (3), is closed with an error of
synchronization with the infinite network, only the hydroelectric model is used.
7.3.2
Load Rejection
The first investigation deals with the case of a total load rejection where the circuitbreaker between the transformer and the generator is tripped. Simultaneously, the guide
vanes of the two Francis turbines are closed linearly in 7 seconds. The evolution of the
main variables during the total load rejection is presented in figure 7.10.
At the outset, the electromagnetic torque of the generators drops to zero instantaneously, as a result the rotational speed of the 2 Units increases. The closure of the guide
vanes reduces the hydraulic torque, quickly limiting the maximum rotational speed. The
guide vanes closure induces a waterhammer effect in the adduction part of the power plant
and a mass oscillation between the reservoir and the surge tank.
The comparison between the hydraulic and hydroelectric simulation results are shown
in figure 7.11 for the rotational speed and the pressure at the turbine inlet. In the
hydraulic model, the electrical installation is modelled by a constant torque dropping
to zero instantaneously at t = 1s. It can be seen that the 2 simulation results are
identical and therefore a hydroelectric model is not required for simulating such transient
phenomena.
143
T/Tn Turbine
1
N/Nn
hinlet/Ho
Q/Qn
QST/Qn
ZST/Ho
T/Tn
Figure 7.10: Evolution of the turbine 1 and surge tank variables during total load rejection.
Figure 7.11: Comparison of the evolution of the turbine 1 rotational speed n (left) and
inlet piezometric head H (right) obtained with two simulations: simulation with hydraulic
model and simulation with the hydroelectric model.
144
7.3.3
Earth fault
The effect of an earth fault occurring between the generator and the transformer of Unit
1 is evaluated using both the electric and hydroelectric simulation models. Depending
on the duration of the fault, the synchronization is maintained or lost after the fault
is removed, leading to the critical time tc . Figure 7.12 presents the comparison of the
simulation results obtained using the two models, for a duration inferior and superior to
the critical time tc . It is pointed out that the critical time tc is underestimated by 2%
using the electric model in which the turbine torque is assumed constant. The difference
between the 2 simulation results is due to the action of the turbine power regulator that
is taken into account by the hydroelectric model. For the simulation, it can be seen that
the maximum amplitudes are well predicted by the electric model but the time history
is more realistic using the hydroelectric simulation as the influence of the turbine speed
governor is properly taken into account. However, its influence on the critical time of the
duration of the fault is negligible.
Figure 7.12: Comparison of the effect of an earth fault on Unit 1 with a duration under
and over critical time tc obtained with two simulation models: electric model and the
hydroelectric model. On the left the synchronism is kept and on the right it is lost.
7.3.4
Three conditions are required for the success of the synchronization of the generator to the
power network during the group startup: the frequency, the phase and the magnitude
have to match the corresponding network conditions before the closure of the circuitbreaker (3). The worst synchronization cases occur when the generator and the network
are 120 and 180 out of phase. The simulation results obtained with the hydroelectric
model are presented in figure 7.13 for Unit 1, and the impact on Unit 2 is presented in
figure 7.14 for the out of phase synchronization of 120. For these simulations a turbine
speed governor is used in state of the turbine power governor.
In the case of a 120 out of phase synchronization, the closure of the circuitbreaker
induces a strong transient electromagnetic torque up to 6 pu that produces rotational
speed variations. This induces a reaction of the turbine speed governor acting on the
EPFL  Laboratoire de Machines Hydrauliques
145
Ttem1/TR
Ht1/H
nt1/nR
Figure 7.13: Evolution of the electromagnetic torque, the head of the turbine and rotational speed of the group number 1 during synchronization fault of 120electrical degree.
guide vane opening in order to keep the rotational speed constant. Both effects contribute
to the variation of the inlet pressure variations of the Unit 1 turbine. In addition, the
first peak of the electromagnetic torque produces free torsional vibrations at 63 Hz of
the shaft line constituted of the turbine and the generator inertias linked through the
connecting shaft. This dynamic response of the structure is observable on the pressure
at the inlet of the turbine of Unit 1 which evidences the coupling between the hydraulic
and mechanical parts. It is interesting to notice that Unit 2 is also affected by the fault
on Unit 1 through both the piping system and the electrical lines. Thus, the head of the
turbine and the current of the stator of the generator of Unit 2 are disturbed by the out
of phase synchronization of Unit 1.
The simulation results of the 180out of phase synchronization are presented for Unit
1 in figure 7.15. It can be noticed that, as expected, this fault produces stronger current
variations in the stator than for 120. The statoric current reaches 8 pu while transient
electromagnetic peak is reduced to 4 pu.
The simulation of the out of phase synchronization clearly points out the interaction
between the hydraulic and electric parts of the installation due to the link between the
2 Units through the piping and the electrical lines. Such influence can only be analyzed
with a hydroelectric simulation model.
146
Ht2/H
R
nt2/nR
Tem2/T
Figure 7.14: Effects of 120 out of phase fault of the Unit 1 on the Unit 2.
Ttem1/TR
Ht1/HR
ib1/in
nt1/n
R
Figure 7.15: Time evolution of the electromagnetic torque, the head of the turbine and
rotational speed of Unit 1 due to 180 out of phase synchronization.
7.4
147
Hydroelectric interactions have been pointed out by simulating classical faults that occur
on hydroelectric power plant such as total load rejection, earth fault and out of phase
synchronization. The maximum amplitudes of dimensioning values such as current, pressure, and rotational speed, are properly predicted by either a single electric or a single
hydraulic model as they mainly depend on short term transients. On the other hand, differences on long terms transients are highlighted. As a consequence, the difference in the
dynamic behavior of the single electric or hydraulic model compared to the hydroelectric
model becomes more significant when the focus is put on the stability of the turbine speed
governor.
This is the reason why the stability of the turbine speed governor during load rejection is investigated using 2 different simulation models: (i) a hydraulic model and (ii) a
hydroelectric model. In order to emphasize the possible interactions, the case of an islanded production mode resulting from the disconnection from the infinite electrical grid
is simulated using the hydroelectric model.
7.4.1
The investigated hydroelectric power plant comprises an upstream reservoir, a 1100 meters
long penstock, a 230 MW Francis turbine connected by mechanical inertias to a 250
MVA synchronous generator linked to a 205 kV infinite network through a 17.5/205 kV
transformer. A passive RL load is also connected on the high voltage side. The layout
of the hydroelectric power plant is presented by figure 7.16. The main dimensions of the
power plant are summarized in table 7.6. The installation is driven by a turbine speed
governor and a generator voltage regulator, both of the PID type.
UNIT 1
+ Uc
KU(s)
1x230 MW
Hydroelectric
power plant
Nc +
Infinite network
(1) U=205kV, f=50Hz
Passive
RL load
KN(s)
148
(ii) hydraulic model: the dynamic behavior of the synchronous generator is modelled
by a constant torque dropping by 25% instantaneously at t = 10s.
The initial conditions in terms of power distribution are given in table 7.7.
Table 7.7: Initial conditions of power distribution.
Element
Active Power Reactive Power
Generator
P = 200 MW Q = 100 MVAR
Passive Load P = 150 MW
Q = 50 MVAR
Network
P = 50 MW
Q = 50 MVAR
7.4.2
The simulation results of the 25% load rejection obtained with the hydraulic and the
hydroelectric models are presented respectively in figure 7.17 top and bottom. The time
evolution of the rated head H/Hn , discharge Q/Qn , rotational speed N/Nn , torque T /Tn ,
guide vane opening y and electromagnetic torque Tel /Tn are represented. It can be seen
that the simulation results obtained using the hydraulic model are fully stable and stable
operating conditions are recovered 40s after the disturbance. However, using the same
turbine speed governor settings with the hydroelectric model leads to a dynamic response
at the limit of stability of the system, and after 90s, the system still not recovers stable operating conditions. This difference is due to the strong influence of the dynamic behavior
of the electrical installation in the isolated production mode. This difference is evidenced
by the time evolution representation of the rated electromagnetic torque Tel /Tn obtained
with the hydroelectric model compared with the one obtained with the hydraulic model.
This means that for isolated production modes, the set of parameters of a turbine speed governor cannot be determined with a single hydraulic simulation
model.
In order to analyze the dynamic behavior of both models in more detail, the transfer
function of the turbine G(s) is identified by simulation considering the guide vane opening
y as input variable and the turbine rotational speed N as output variable. The transfer
function of the turbine is expressed in the Laplace domain as follows:
G(s) =
N (s)
Y (s)
(7.2)
149
H/Hn
N/Nn
Tel/Tn
Q/Qn
T/Tn
H/Hn
N/Nn
Q/Qn
Tel/Tn
y
T/Tn
Figure 7.17: Simulation results of the transient of the Francis turbine resulting from a 25%
load rejection based on the hydraulic model (top) and the hydroelectric model (bottom).
150
As a consequence, to guarantee the stability in the islanded production mode with the
hydroelectric model, the integrator time constant Ti of the PID turbine speed governor
is increased to reduce the amplification at low frequencies. The initial integration time
constant was Ti = 3.7 s and is increased to Ti = 14 s to have an efficient governor; the
gain and the derivative time constant remain unchanged (Kp = 1, Td = 1.21 s). The
simulation results of the dynamic behavior of the installation resulting from a 25% load
rejection using the hydroelectric model is presented in figure 7.19. It can be seen that, as
expected, the system is again fully stable in the isolated production mode.
Eigen frequencies of
mechanical inertias
Figure 7.18: Transfer function of the turbine G(s) = N (s)/Y (s) in the range 0 25Hz
on the left, and 0 1Hz on the right.
N/Nn
H/Hn
Tel/Tn
Q/Qn
T/Tn
Figure 7.19: Simulation results of the transient of the Francis turbine resulting from a 25%
load rejection with the hydroelectric model and the modified integration time constant of
the turbine speed governor.
7.5
Islanded power networks feature a small number of power plants and loads leading to
high interactions between all the components. Therefore high performance governors
must be used for each power unit. It is even more important for hydraulic power plants
which dynamic behavior is strongly related to the hydraulic circuit layout. Since every
hydraulic power plant is unique, no standard governor setting can be used. Hydraulic
power plants featuring a long penstock and a surge tank need to be properly modelled
in order to take into account waterhammer, surge tank water level oscillations and the
effects of the turbine characteristics. All these phenomena present a nonlinear behavior,
thus restricting the performance of the turbine speed/power governor. To ensure the
stability of the turbine governor, the governor parameters have to be validated by a time
domain simulation. The order of the model of the hydraulic installation has to be adapted
to the hydraulic layout and the investigated case, as advised by the working group on
prime mover and energy supply models for system dynamic performance studies [155].
It appears that hydroelectric power plants featuring a surge tank and a long penstock
connected to an islanded power network require a high order model. Such an installation
is investigated in this section.
7.5.1
7.5.2
152
+ Uc
KU(s)
Nc +
1 GW Hydroelectric
power plant
KN(s)
Adapted LR load
4 x 1.3 GW Thermal
power plants
200 MW LR load
reheater
Nc +
Uc
KN(s)
N 
KU(s)
po
UlRMS
HP
LP1
LP2
LP1
LP2
LP1
LP2
LP1
LP2
+

reheater
po
HP
reheater
po
HP
reheater
po
HP
Surge tank
Penstock
Francis turbine
Generator
Coupling shaft
The gallery and the penstock are respectively discretized into 22 and 31 elements. The
turbine draft tube is modelled by 2 pipe elements. The Francis turbine characteristics,
discharge and torque factors versus the speed factor are presented for different guide vane
opening values y, see figure 7.21.
The block diagram of the PID governor of the Francis turbine is presented in figure
7.22. The governor structure includes both speed and power feedbacks. The network
frequency feedback is neglected in this study because only islanded and isolated production
modes are considered. The servomotor of the guide vanes is modelled using a first order
transfer function with a time constant of sv = 0.1 s.
Turbine Governor Parameters Determination
The transfer function of the system to be regulated, G(s), should be determined for setting
the parameters of the turbine governor. The system consists of a turbine with the guide
vane opening y as input parameter and the rotational speed N as output parameter. The
transfer function of the turbine is identified for 4 different power levels: P/Pn = 0.4, 0.7, 1
EPFL  Laboratoire de Machines Hydrauliques
154
2
GVO = 0.00
GVO = 0.08
GVO = 0.15
GVO = 0.23
GVO = 0.31
GVO = 0.39
GVO = 0.46
GVO = 0.54
GVO = 0.62
GVO = 0.65
GVO = 0.69
GVO = 0.73
GVO = 0.77
GVO = 1.00
1.2
0.8
T11/T11BEP
Q11/Q11BEP
1.6
GVO = 0.00
GVO = 0.08
GVO = 0.15
GVO = 0.23
GVO = 0.31
GVO = 0.39
GVO = 0.46
GVO = 0.54
GVO = 0.62
GVO = 0.65
GVO = 0.69
GVO = 0.73
GVO = 0.77
GVO = 1.00
0
0.4
0.4
0.8
1.2
1.6
N11/N11BEP
1
0.4
0.8
1.2
N11/N11BEP
1.6
Figure 7.21: Turbine characteristics Q11 = Q11 (N11 ), left, and T11 = T11 (N11 ), right, for
different GVO opening.
and 1.15 p.u., in order to take into account the influence of the local gradient of the
turbine characteristics.
The model of the hydraulic installation set up using SIMSENHydro is of a high order
(up to 150 ODE), however the transfer function of the system cannot be directly inferred
from them. The transfer function of the turbine including the hydraulic circuit is identified
through a time domain simulation considering a white noise excitation. A PRBS signal
of 5% amplitude around a mean value of guide vane opening is used as excitation signal,
see figure 7.23. The mean value is set in accordance with the turbine power for which
the transfer function is identified. The PRBS excitation function [153] is preferred to
an indicial response because of its higher frequency content that evidences all natural
frequencies of the system. The PRBS signal is obtained using a shift register [60].
The amplitude and phase spectra of the turbine transfer function obtained from the
PRBS identification, for P/Pn = 0.4 is presented in figure 7.24. The amplitude of the
transfer function reveals that the hydraulic system natural frequencies are mainly related
to the piping system, the mechanical masses and the surge tank. The first natural frequency of the piping system is fo = 0.2 Hz and corresponds to the fourth wavelength
free oscillation mode of the penstock given by f = a/(4l). The natural frequencies above
correspond to higher mode eigen frequencies of the piping system. The natural frequency
of the mechanical masses presents a high amplitude at fm = 7.6 Hz and is given by [117]:
2
m1,2
1
= 21 + 22
2
s
22 21
2
2
+ 412
(7.3)
Where :
21
Kshaf t
=
JG
22
Kshaf t
=
JT
412
2
Kshaf
t
=
JG JT
(7.4)
The antiresonance of the generator inertia corresponding to 1 is also visible for fm = 2.44
Hz. The eigen frequency of the mass oscillation between the upstream reservoir and
EPFL  Laboratoire de Machines Hydrauliques
P
integral
Pc +
KW
Nc +
1
T3 s
K2
+
K1
ARW
proportional
servomotor
position
turbine
limits
1
1 + sv s
filtered derivative
1 + T1s
1 + T2 s
G(s)
P
N
tachometer
1
1+ s
G(s)
N ( s)
G (s) =
Y (s)
N(s)
GVO
N/Nn
Y(s)=PRBS
the surge tank is calculated with equation 6.6 and is equal to fST = 0.00866 Hz, i.e.
TST = 115.5 s. This frequency is visible also as an antiresonance on the spectrum. The
antiresonance behavior results from the fact that when more power is required, the guide
vanes open, resulting in an increase of the discharge coming from the surge tank but
reducing its water level, thus the available head at the turbine inlet. As a consequence,
the hydraulic power may decrease depending on the governor action. Similarly, the gate
opening induces a pressure drop at the turbine inlet resulting from a waterhammer effect
leading to a nonminimum phase visible on the linlin phase plot. The nonminimum phase
corresponds to half a period of the penstock, i.e. 2.5 seconds. This nonminimum phase is
clearly pointed out in figure 7.25 where the guide vane step responses are simulated for the
Figure 7.23: Open loop transfer function identification with a PRBS excitation for P/Pn =
0.4.
156
4 different power levels. It can be noticed that even if the guide vanes open, the rotational
speed does not increase immediately due to the waterhammer under pressure. It can also
be noticed that the higher the power level the higher the nonminimum phase influence.
Therefore, this nonminimum phase is strongly restrictive for the governor performance.
A large integrator time constant Ti is consequently required to ensure the system stability.
Amplitude []
Rotating inertias
Surge tank
Pipe
Generator inertia
Frequency
Frequency [Hz]
Frequency [Hz]
Figure 7.24: Amplitude (left) and phase (right) of the transfer function of the Francis
turbine for P/Pn = 0.4 including the hydraulic circuit and mechanical inertia.
The transfer function of the turbine is identified for the 4 power levels considered
P/Pn = 0.4, 0.7, 1 and 1.15 p.u. and are represented in figure 7.26. As the 4 transfer
functions are different because of the turbine characteristic, the determination of the PID
parameters is based on the most restrictive one, i.e. on the most critical behavior of the
system. The parameters of the governor are set in order to ensure a phase margin of
60 90, a gain margin of 6 9 dB, a cutoff frequency of 0.02 Hz to avoid resonance
amplification and a slope of 20 dB/decade at the cutoff frequency.
Then, the assessment of the regulator performance is performed by simulating the
dynamic behavior of the hydraulic power plant resulting from a successive 6% load rejection and acceptance for P/Pn = 0.4, 0.7, 1 and 1.15 p.u.. The electromagnetic torque is
modelled by an external torque function which does not take into account the dynamic
behavior of any electrical installation. The resulting time evolution of the rotational speed
N/Nn and guide vane position y are presented in figure 7.27 for P/Pn = 0.7 and 1.15 p.u..
The speed deviations obtained from the simulation correspond to 1.5% and 4.5% of the
nominal speed. For P/Pn = 0.7 p.u. the speed is stabilized after 10 seconds while it takes
150 seconds to stabilize the speed for P/Pn = 1.15 p.u. The full load appears to be more
critical because of the influence of the surge tank. A surge phenomenon occurs between
the PID governor and the surge tank. It is clearly visible from the time evolution of the
guide vane opening which fluctuates at the mass oscillation period (TST = 115 s). The
cross section of the surge tank is far from the Thoma section calculated to be 80m2 according to equation 6.28. It appears clearly that this criterion is not sufficient, especially
at full load where the turbine efficiency gradient becomes negative (d/dQ < 0). The
governor parameter values have to be validated by a time dependent simulation to ensure
EPFL  Laboratoire de Machines Hydrauliques
y1.15
N/Nn
y1
T=2.45s
y0.7
y0.4
Figure 7.25: Nonminimum phase of the turbines evidenced by rated transient rotational
speed N/Nn resulting from a guide vane opening step response for different power levels
P/Pn = 0.4, 0.7, 1, 1.15.
the stability on the whole operating range, even for large stroke variations of the guide
vanes.
7.5.3
As the turbine governor parameter values are validated only with the hydraulic simulation
model, a model of a thermal power plant is set up to investigate the influence of the
connection to an islanded power network. The thermal power plant is modelled with a
constant pressure tank, the steam generator dynamics being neglected, feeding the High
Pressure steam turbine (HP) through a regulating valve. Then the vapor flux transits
through a reheater before feeding 2 Low Pressure (LP) steam turbines as presented in
figure 7.28. The model of this thermal power plant is made of a proportional governor
with a frequency drop feedback and a speed feedback, a valve model, the model of the HP,
LP1 and LP2 steam turbines and the mechanical masses as presented in figure 7.29. The
parameter values of the model are given in table 7.9. The model of the steam turbines
is made of 3 parallels branches, one for each steam turbine, driven by the valve opening.
A time delay of b = 4 seconds is considered between the high pressure steam turbines
and the low pressure steam turbines to take into account the transit time of the steam
through the reheaters. The 3 turbines are modelled by first order transfer functions with
a short time constant for the high pressure and a long time constant for the low pressure
steam turbines. The model includes also simplified steam turbine characteristics that
are deduced from [17]. The inertia, stiffness and damping of the mechanical shaft are
obtained from [62]. The synchronous machine excitation is controlled by an ABB Unitrol
EPFL  Laboratoire de Machines Hydrauliques
158
Phase of G(s)
P/Pn = 0.4
Amplitude of G(s)
Frequency [Hz]
Frequency [Hz]
Frequency [Hz]
Frequency [Hz]
Frequency [Hz]
Frequency [Hz]
Frequency [Hz]
P/Pn = 1.15
P/Pn = 1.0
P/Pn = 0.7
Frequency [Hz]
Figure 7.26: Amplitude (left) and phase (right) of the transfer function of the Francis
turbine for P/Pn = 0.4, 0.7, 1 and 1.15 p.u..
EPFL  Laboratoire de Machines Hydrauliques
GVO
N/NR
GVO
N/NR
Figure 7.27: Time evolution of rotational speed N/Nn and guide vane opening GV O of
the Francis turbine during a 6% load rejection at 0.7 p.u. (top) and 1.15 p.u. (bottom)
power level.
reheater
Nc +
Uc
KN(s)
N 
KU(s)
po
+

UlRMS
HP
LP1
LP2
Network
The dynamic response of the thermal power plant is investigated for a 6% load rejection. The time evolution of the rotational speed N/NR , the electromagnetic torque
Tem /TR and the valve opening yvalve are represented in p.u. in figure 7.30. The rotational
speed recovers stability after 10 seconds, demonstrating the fast dynamic response of the
EPFL  Laboratoire de Machines Hydrauliques
160
thermal power plant. However, due to the proportional nature of the governor, a static
error can be noticed after full stabilization of the system. The pressurized steam reservoir
enables the thermal power plant to have short time constant. The static error is not compensated because the steam generator dynamics are neglected. The transient response
of this thermal power plant was validated by comparison with measurements on a power
plant performed by EDF [39].
Table 7.9: Main dimensions of the thermal power plant.
Element Dimensions
Steam turbines model HP = 0.5 s
LP = 12 s
b=4s
Kp = 25
Mechanical inertias JHP = 1.867 104 kg m2
JLP 1 = 1.907 105 kg m2
JLP 2 = 2.136 105 kg m2
Jgen = 5.223 104 kg m2
Mechanical shaft stiffness K1 = 3.614 108 N m/rd
and damping K2 = 8.206 108 N m/rd
K3 = 4.116 108 N m/rd
1 = 6.719 103 N ms/rd
2 = 7.06 103 N ms/rd
3 = 7.06 103 N ms/rd
Generator Sn = 1400 M V A
Un = 28.5 kV
f = 50 Hz
Polepaire: p = 2
Stator windings: Y
7.5.4
To investigate the influence of the power network, a simulation model of an islanded power
network is build up. The islanded power network of concern, see figure 7.20, is composed
of the hydraulic power plant, 4 thermal power plants and 2 passive consumer loads. The
5 power plants represent all together a 6.2 GW power network. Figure 7.31 presents
the topology of this power network where the power plants are connected to 2 passive
consumer loads through 400 kV transport lines. The LR parameters of the 2 consumer
loads are set in order to absorb the production of all the power plant of the network. One
is a 200 MW load and the other one represents the complement to the total power of the
islanded power network. In order to investigate the transients due to a load rejection, the
200 MW consumer load is disconnected by tripping the circuit breaker considering the 5
different network power levels given in table 7.10.
For completeness, the electrical installation of the hydroelectric installation is included
in the simulation model. The 4 hydrogenerators are 270 MVA synchronous machines and
EPFL  Laboratoire de Machines Hydrauliques
Governor proportional
Frequency drop
compensation
Pc
fnetwork
P
Pc
Valve opening
prediction
50
Valve
P=Pc +Kp(ffo)
Nc
Tachometer
1
Nc
Kp
rate
limiter
position
limits
d/dt
1
1+ s
Steam turbines
+ reheater
HP
reheater
G( z) = e
bz
G ( z ) = e bz
THP
LP1
reheater
1
1 + HP s
1
1 + LP s
LP2
1
1 + LP s
TLP1
TLP2
JHP
K1
1
JLP1
K2
2
JLP2
K3
3
JGEN
Mechanical shaft
Figure 7.29: Block diagram of the model of the thermal power plant.
N/NR
yvalve
Tem/TR
162
Figure 7.30: Transient response of the 1300 MW thermal power plant resulting from a
6% load rejection.
are connected to the islanded network through 18.5/400 kV Yd5 transformers. The synchronous machine excitations are controlled by ABB Unitrol voltage regulators. The
synchronous machines are of the laminated rotor type. The model is considering saturation, leakage and damping effects of windings, allowing taking into account a subtransient
behavior.
The Power System Stabilizers PSS of the hydroelectric power plant are not considered
in this investigation in order to focus on the performances and stability of the turbine speed
governors. The use of PSS would help reducing the speed deviations as demonstrated by
Kamwa et al. [88] but the influence of such device is out of the scope of the present work.
Frequency Analysis
The transfer function of the turbine connected to the power network is reidentified for
3 different network power levels: Ptot = , 6.2 and 2.3 GW, the power of the hydraulic
power plant being 1 GW. The amplitude spectra of the resulting transfer function are
presented in figure 7.32. From the new transfer function an additional natural frequency
with high amplitude is pointed out at 1.36Hz. This corresponds to the synchronous
machines natural frequency depending on the operating point of the generator, voltage
regulator and system configuration [91]. The source of this frequency was checked by
introducing a sinusoidal excitation added to the wicket gate mean value in open loop
conditions.
A high stabilization effect of the power network can be noticed on the amplitude
spectrum of the turbine transfer function for low frequencies. The attenuation and the
frequency range affected by the network become smaller as the level of the power network
decreases. The power network behaves like a highpass filter. It means that the perturbations of the rotational speed at low frequencies are reduced by the power network
stiffness and that the higher the power level of a network, the lower the performances
of the governor required.
EPFL  Laboratoire de Machines Hydrauliques
Hydro Power
Plant 1 GW
H
200 MW Consumer
Load
T
T
T
Transient Analysis
A time domain simulation of the tripping of the 200 MW consumer load is performed
without considering the synchronous machines natural frequency in the turbine speed
governor design. The simulation is done for Phydro /Ptot = 0.43, 0.28, 0.20, 0.16, 0, and the
resulting turbine rotational speed evolutions are presented figure in 7.33. As expected,
whatever the power level of the network, the turbines operation is unstable and the rotational speed exhibits a fluctuation at 1.36 Hz. To avoid a resonance between the turbine
speed governor and the synchronous machine, the filter time constant of the turbine rotational speed is reduced to = 0.3 s. Consequently, the turbine rotational speed remains
stable as presented in figure 7.34. As expected from the turbine transfer function analysis, the islanded network power level has a strong influence on the speed deviation. For
infinite power level the tripping of the 200 MW load is almost not perceptible while for a
ratio PHydro /Ptot = 0.43 the speed deviation reaches 3.5%. The amplitudes of the speed
deviation are reported in figure 7.35 as a function of the ratio between the power of the
hydraulic part (1 GW here) and the power of the whole network.
7.5.5
Concluding Remarks
This investigation demonstrates the importance of a high order modelling in the case of
a hydroelectric power plant featuring a surge tank with a small diameter and a long penstock in islanded power networks. A detailed setup and an analysis procedure necessary
to ensure the hydraulic turbine governor stability are presented. The case of a 1 GW
hydroelectric power plant connected to a 5.2 GW islanded power network is investigated.
The transfer function of the hydraulic turbine is identified using a time domain simulation with a PRBS excitation. This method has the advantage to evidence all the natural
frequencies of the hydraulic system. The governor performances are tied to the natural
frequencies of the power plant. The analysis of the transfer function of the turbine in
different islanded production levels, evidences a stabilization effect below 1 Hz in larger
EPFL  Laboratoire de Machines Hydrauliques
164
fo Generator = 1.36Hz
Pnetwork = infinity
tio
iza
bil
ct
ffe
of
ork
etw
n
e
th
ne
Sta
Frequency
Pth = 5.2 GW
Frequency
Pth = 1.3 GW
Frequency
Figure 7.32: Francis turbine transfer function considering connection to an infinite power
network (top), 5.2 GW power network (middle) and 1.3 GW power network (bottom).
Pth = 1.3 GW
Pth = 2.6 GW
Pth = 3.9 GW
Pth = 5.2 GW
Figure 7.33: Francis turbine rotational speed evolution after 200 MW load tripping for
PHydro /Ptot = 0.43, 0.28, 0.2 and 0.16 without filter modification.
power networks. In the case of weak power networks, the stability of hydraulic installations with a long penstock featuring natural frequencies in this frequency range can be
strongly affected. A study taking all these considerations into account eases the final
tuning of governors and hence reduces the commissioning delay.
166
Pth = 1.3 GW
Pth = 2.6 GW
Pth = 3.9 GW
Pth = 5.2 GW
Pth = oo
Figure 7.34: Time evolution of the turbine rotational speed resulting from the tripping of
a 200 MW load for different thermal power plant levels, with filter modification.
N/Nn [%]
0
0.1
0.2
0.3
Phydro/Ptot
0.4
0.5
Figure 7.35: Speed deviation due to a 200 MW load tripping as a function of power ratio
between the hydraulic power plant and the total power of the islanded power network.
Part II
Modelling of Pressure Fluctuations
in Francis Turbines
Chapter 8
Pressure Excitation Sources in
Francis Turbines
8.1
General Remarks
8.2
From mathematical point of view, there is distinction between 3 types of excitations can
be distinguished: (i) forced excitation, (ii) parametric excitation, (iii) self excitation.
These 3 types of excitation mechanisms are illustrated with basic equivalent electrical
circuits. The characteristic equations derived from the circuits point out the nature of
the excitation in the following subsections.
Forced Excitation
The figure 8.1 presents the equivalent scheme of a hydraulic system subject to a forced
excitation. The system is made up of a pressure excitation source H(t), a resistance R,
an inductance L and a capacitance C. The related equation set is given by:
dQ
L
+ R Q + hc = H(t)
dt
C dhc = Q
dt
EPFL  Laboratoire de Machines Hydrauliques
(8.1)
170
L
Q
H(t)
+
hc = 0
+
2
dt
L dt
C}
{z}
L{z
2
(8.2)
o2
The harmonic response of the system of figure 8.1 is presented in figure 8.2 for different
values of the relative damping = R/(2 L o ). The resonance occuring for = 0 is
clearly evident and the influence of the relative damping is pointed out.
10
= 0.00
= 0.10
= 0.15
R
2 Lo
= 0.25
hc/H []
= 0.50
1
= 0.71
= 1.00
0.1
/o []
171
Parametric Excitation
The figure 8.3 presents the equivalent scheme of a hydraulic system subject to parametric
excitation. The system is made of a constant pressure source Ho , a resistance R, an
inductance L and a time dependant capacitance C(t). The related equation set is given
by:
dQ
L
+ R Q + hc = Ho
dt
(8.3)
C(t) dhc = Q
dt
L
Q
Ho
C(t)
+
hc = 0
2
dt
L dt
L C(t)
{z}
 {z }
2
(8.4)
o2
dQ
L
+ R Q + hc = Ho
dt
(8.5)
C dhc + dQ = Q
dt
dt
The characteristic equation of the above system is expressed as follow:
(Q) R dhc
1
d2 hc
hc = 0
+ (1
)
+
2
dt
C{z
R L} dt
L}

C{z
2
o2
(8.6)
172
C
Co o
( o )
L
Q
Ho
Considering a system for which the mass flow gain factor is a quadratic function of the
discharge expressed as = o Q2 leads to the Van der Pol equation. The unsteady behavior related to this equation is presented in figure 8.6. Such a system features amplitude
amplification for o Q2 /(C R) < 1 and amplitude damping for o Q2 /(C R) > 1.
4
hc/H []
2
0
R/L =1
o/(RC) =1
2
4
10
20
30
40
50
t.o []
Figure 8.6: Time evolution of hc of system of figure 8.5 leading to the Van der Pol equation.
173
8.3
The complex flow structure developing in a Francis turbine leads to several types of
excitation related to cavitation, vortices (cavitating or not) and rotorstator interactions.
The typical excitations source induced by the flow in a Francis turbine are the following:
part load vortex rope excitation: in the frequency range of 0.2 0.4 n [126];
upper part load vortex rope excitation: in the frequency range of 2 6 n [44];
full load vortex rope excitation: in the frequency range of 0.1 1 n [79];
interblade cavitating vortices: in the frequency range of 8 12 n [100];
rotor stator excitation: in the frequency range of 1 4 Zb n [22];
von Karman vortex shedding: in the Strouhal range of St = 0.1 0.3 [18];
bubble and sheet cavitation: in the frequency range of 0.5 12 kHz [25].
If all these excitation sources may result in unacceptable solicitations of the turbine
structure, only a small part of them is expected to result in strong interaction with the
hydraulic system. Indeed, many of these excitations are in a high frequency range with
respect to the size of the hydraulic circuit of a power plant. However, some high
frequency phenomena can excite an eigen frequency of a scale model test rig, while it
does not on the prototype because of the difference of hydroacoustic parameters of both
installations. Especially, a given rated frequency on a scale model corresponds usually to
a higher rank of system eigen frequency harmonics of the actual power plant, and then
benefits from higher damping on the prototype.
Therefore, the excitations of particular interest regarding the risk of hydroacoustic
resonance during prototype operation, are the part load, upper part load and full load
vortex rope excitations, and the rotorstator excitations. The risk of resonance with
interblade vortices potentially exists, but was never reported on prototype or scale model.
However, the related cavitation compliance may play a strong role in the determination
of the eigen frequencies of a hydraulic installation.
Consequently, the focus of the hydroacoustic modelling of the present work is put
particularly on the vortex rope excitations and the rotorstator excitations. The basics of
the related excitation mechanisms are described in the following sections.
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8.4
The cavitating vortex rope developing in the draft tube features different patterns in
terms of shape and related pressure pulsation frequencies and amplitudes. The figure
8.7 presents a waterfall diagram of the wall pressure fluctuations as function of the rated
frequency and discharge coefficient measured in the draft tube cone of a Francis turbine
of specific speed = 0.22 for the best efficiency energy coefficient = BEP , [79]. From
the waterfall diagram it can be seen that the part load and full load pressure fluctuations
are paramount.
Pressure transducer
location
= BEP
Figure 8.7: Waterfall diagram of pressure excitation and related vortex rope shape [79].
The figure 8.8 [43] presents the typical patterns of the vortex rope with respect to
the rated flow. It can be seen that the occurrence of forced excitations are common
for part load operation and are occasional for full load operation. Self excitations are
occasional at lower part load and full load as it is also mentioned by Jacob [79]. Pressure
shocks resulting from the impact of the cavitating vortex rope on the draft tube wall are
occasionally expected for a large part of part load and beginning of full load operation.
But it is usually at the upper part load that they may feature high amplitudes and are
coming with strong noise and mechanical vibrations [44].
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Figure 8.8: Draft tube flow patterns and associated excitations types [43].
8.4.1
The part load pressure fluctuations induced by the vortex rope are of major concern
as they are regularly expected on both scale model and prototype and can feature high
amplitudes. Hydroacoustic resonance is expected because these pressure fluctuations are
of the forced type. In the following subsections the excitation mechanism associated
with the part load vortex rope is presented, then the possible modelling approaches are
discussed.
From the Vortex Breakdown to the Draft Tube Surge
The draft tube of a Francis turbine is characterized by the fact that it is divergent, with
elbow and is subjected to high swirl momentum induced by the runner at part load.
Consequently, the flow extending in the draft tube is fully 3 dimensional, unsteady, may
feature separation and cavitation. Therefore, it is suitable to analyze the onset of the
cavitating vortex rope step by step.
Initially, the focus was on the identification of the different flow patterns that may
result from a swirling flow developing in an axial tube. Swirling flow is defined to result
from both axial and vortex motion. Increasing the swirl momentum, the flow starts from
purely steady axial flow and suddenly becomes unsteady and features a precessing vortex.
Early analytical investigations were able to predict the occurrence of the famous phenomenon known as the vortex breakdown and are expected when reverse flow occurs along the
axis, see Benjamin 1962 [12]. Such phenomenon was soon confirmed experimentally by
Harvey, 1962 [71], with an apparatus generating an air swirling flow extending in a cone
and evidencing the vortex breakdown by smoke injection, see figure 8.9. The precessing
motion of the vortex induced by the swirling flow was also evidenced. Using a similar
apparatus, Cassidy and Flavey, 1970 [33], have highlighted the dependency of the vortex
precession frequency on the ratio between the axial and swirl momentum. Therefore the
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Dref
Q2
(8.7)
(8.8)
In addition, they have pointed out that wall pressure fluctuations are induced by the
helical vortex precessing about the axis of the tube. It was also shown analytically that
such precessing flow is capable of sustaining an axisymmetric standing wave, see Benjamin
1967 [13]. The method of swirl momentum was then applied by Uldis and Palde, 1974
[114], to Francis turbine scale models in order to try to predict the surging operating
range and the related frequency of pressure fluctuations on the prototype. The method
showed that improvement was required, probably because the influence of the cavitation
number was not considered.
177
Figure 8.10: Conical elbow draft tube with swirl generator at inlet without runner [108].
Figure 8.11: Helicoidal vortex rope precessing around a recirculating zone in the conical
elbow draft tube with swirl generator at inlet without runner [110].
In parallel many investigations were undertaken on scale model conical elbow draft
tubes of Francis turbines under cavitating and non cavitating conditions, see Deriaz 1960
[47], Ulith et al. 1974 [149], Henry et al. 1981 [74], Muciaccia 1984 [107], Jacob 1996 [81]).
Particularly, investigations in the framework of the FLINDT project, FLow INvestigation
in Draft Tube, pointed out analytically the unstable nature of the flow in a conical draft
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Figure 8.12: Draft tube pressure pulsation as function of the cavitation number [111].
Figure 8.13: Decomposition of draft tube pressure pulsation in rotating and synchronous
component [5].
tube, see Avellan 2000 [9] and Resiga et al. 2006 [139]. It was shown that the swirling
flow can be accurately decomposed in a sum of 3 Batchelor vortices, Batchelor 1967 [11].
The solution of the eigen value problem of the related swirling flow, keeping the discharge
flow coefficient as a parameter, shows the unstable nature of the flow for low discharge.
Such an approach provides the insights for the optimization of the runner geometry with
respect to the stability of the draft tube flow.
Modelling of Part Load Pressure Fluctuations
The understanding of the onset conditions of the vortex breakdown and the related pressure excitation was a helpful approach for developing models of the excitation related to
the part load vortex rope and assessing the risk of resonance on the prototype. It was
understood early on that a resonance risk evaluation should be conducted directly for
the power plant installation itself and cannot be directly transposed from scale model to
prototype. Thus the method of transfer matrices was applied by Zielke, 1972 [160], for
determining the eigen frequencies of the system. Then in 1980, [159], [158], assuming
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179
Vcavitation
H
(8.9)
Where Vcavitation is the volume of the cavitation and H is here the static pressure
expressed in [mW C].
This concept of cavitation compliance was successfully introduced by Ghahermani and
Rubin, in 1971 and in 1966 [59], [128], for the understanding of the so called POGO effect
in the liquid rocket propulsion system. Brennen and Acosta, in 1973 [27], then extended
this method by evaluating the cavitation compliance using quasi potential flow applied
to a cavitating cascade and investigating the influence of both the cascade geometry and
the cavitation number.
Similarly, Dorfler [40] determined the cavitation compliance as function of the cavitation number as illustrated in figure 8.14. For transposition purposes, he introduced the
following dimensionless compliance:
C0 =
C Hkin
3
Dref
(8.10)
Q2
2gA
(8.11)
Using this method, he obtained results with very good agreement with the measurements as illustrated in figure 8.15 where pressure and torque fluctuations amplitudes are
presented. The torque fluctuations are obtained by linearizing the turbine characteristic,
Dorfler 1980 [45]. Later, Tadel and Maria 1986 [142], used the impedance model of Dorfler
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to show the influence of upstream and downstream arrangement of the power plant on
the pressure pulsation amplitudes but also on mechanical torque.
However, the impedance model of Dorfler is based on the modelling of the vortex rope
compliance by a single lumped capacitance C. Therefore, this modelling is restricted
to low frequency phenomena, i.e. below 10 Hz, as it was shown by Couston and
Philibert [120]. These authors proposed a new matrix model of the entire draft tube
including the cavitating vortex rope based on the analytical expression of the vortex rope
diameter as function of the curvilinear abscissa. This model presented good agreement
with measurements.
The parametric excitation model was used by Koutnik and Pochyly, 2002 [96], for
the understanding of atypical pressure fluctuation frequencies measured on pumpturbine
power plants. In this modelling, a time dependant solution was necessary to investigate
the influence of sinusoidal variation of the cavitation compliance.
Some recent work has been carried out with the aim to couple CFD computations of
the vortex rope extending in the draft tube with a hydroacoustic model of the power plant
piping for the prediction of resonance by Ruprecht and Helmrich [129]. However, the CFD
computation does not take into account the compressibility effect in the draft tube, which
is significant as very low wave speed is expected in this component. Consequently it does
not represent the hydroacoustic behavior of the draft tube.
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Figure 8.15: Comparison of measurements (dots; black/white decreasing/increasing cavitation number) and model (solid lines) at the vortex rope frequency frope /n = 0.224
[40].
8.4.2
While the vortex rope precession frequency commonly corresponds to 0.2 to 0.4 times
the turbine rotational frequency n, pressure fluctuations may occur in a higher frequency
range, 2 n to 4 n, for high specific speed turbines at upper part load range as described
by Fischer in 1980 [52], Dorfler in 1994 [44] and Jacob in 1996 [81]. Figure 8.16 shows
a waterfall diagram where upper part load pressure fluctuations are pointed out at the
point number (3). Moreover, a shock phenomenon may occur in the same operating
range and induces structural vibrations due to vortex rope impacts on the draft tube wall,
see Dorfler 1994 [44].
However, from the modelling point of view, no investigations on the upper part load
pressure fluctuations are reported in the literature. In addition, it is not very well understood why on the prototype this phenomenon is not present or results in small amplitudes
only [52], [44].
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Figure 8.16: Waterfall diagram of pressure pulsations at turbine draft tube cone at constant of a medium head Francis turbine [81].
8.4.3
183
Figure 8.17: Stability diagram of a power plant with respect to the cavitation compliance
C and mass flow gain factor [97].
A similar approach of cavitation parameter mapping was also successfully applied for
the explanation of inducer instabilities by Tsujimoto et al. in 1997 [148] and of propeller
instabilities in cavitation tunnels by Duttweiler and Brennen in 2002 [49] and by Watanabe
and Brennen in 2003 [154].
8.5
184
head cover see Fischer 2004 [53] and Franke 2005 [57], and may induce strong vibrations,
noise, fissures or guide vane bearing ruins. The second phenomenon may cause resonance
with the power house structure that generates unacceptable vibrations and noise levels,
see Ohura 1990 [113]. The standing wave phenomenon may affect also the penstock,
Dorfler 1984 [42], Den Hartog 1956 [70], which evidences the potential interaction of the
hydraulic machine with the hydraulic circuit.
Figure 8.18: Power house structure vibrations measured by Ohura et al. [113].
The prediction of such phenomena is a challenging task during the early stage of the
design of a reversible pumpturbine unit for a hydroelectric power plant. Some analytical
models have been developed by Bolleter in 1988 [22] and Ohura et al. in 1990 [113] for
the diametrical mode shape and by Chen in 1961 [37] and Dorfler in 1984 [42] for the
standing wave to predicts the risk of occurrence of these phenomena. The prediction of the
occurrence of the standing wave is based on the analysis of the travelling time of pressure
waves propagating in a onedimensional system modelling the pumpturbine according to
its topology. Recently, Haban et al. in 2002 [66] have developed more sophisticated onedimensional models based on the transfer matrix method that have shown their capability
of predicting spiral casing standing wave patterns. By the use of such models Francke
et al. in 2005 [57] have pointed out the link between the standing waves in the spiral
casing and the penstock and the diametrical pressure mode rotating in the vaneless gap
by performing a forced response analysis in the frequency domain. However this approach
requires the identification of the excitation pattern by the method described by Bolleter
[22] and Ohura et al. [113].
The simulation of the incompressible 3D unsteady flow of a vaneless centrifugal pump
performed by Gonzalez et al. in 2002 [61] using a commercial CFD tool has shown the
capability of CFD to predict accurately the unsteady convective field related to the RSI
phenomenon at the blade passing frequency which is dominant close to the nominal operating point. However some discrepancies appeared for offdesign operating conditions
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186
Chapter 9
Upper Part Load Pressure
Fluctuations Analysis on a Scale
Model
9.1
General Remarks
Upper part load pressure pulsations are commonly reported from scale model tests and
are usually attributed to test rig resonance. However, the excitation source mechanism is
not well understood. Does the pressure pulsation result from vortex rope pressure source
excitation or is it of the self excited nature? What is the role of the shock phenomenon
mentioned in chapter 8.4.2 ?
The upper part load pressure fluctuations phenomenon was observed and measured
in the framework of the FLINDT project, providing the opportunity to set up a hydroacoustic model of the test rig and to assess resonance conditions of the full hydraulic
system. However, the analysis on the FLINDT project is done for only one operating
point. Therefore, to clarify some unanswered questions on the role of the cavitation and
Froude numbers on the upper part load pressure fluctuations, an experimental investigation is carried out on a similar scale model Francis turbine by performing vortex rope
visualizations.
9.2
9.2.1
In the framework of the FLINDT project [9], wall pressure fluctuation measurements
were carried out on the draft tube of a scale model of a medium head Francis turbine of
specific speed = 0.56 for several operating points by Arpe [6]. For the upper part load
under low cavitation number conditions, = 0.38, pressure fluctuations were recorded at
a frequency value of about 2.5 n. For these operating conditions, structural vibrations
of the whole test rig and strong noise were also observed.
This section presents the analysis of the pressure fluctuations measured in the draft
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188
tube at part load operation under low cavitation number. First a phase shift analysis is
carried out, pointing out a pressure excitation source in the elbow of the draft tube. The
wave speed along the draft tube is deduced from this analysis enabling to carry out a
hydroacoustic model of the entire test rig using SIMSEN. Then, a free oscillation analysis
as well as a forced oscillation analysis are performed in order to explain the pressure
fluctuation mechanism.
9.2.2
The experimental investigations are performed on a scale model of a high specific speed
Francis turbine, = 0.56 , see figure 9.1. The draft tube is equipped with 292 pressure
taps, distributed in 13 sections. The draft tube is composed of a cone made in plexiglas,
an elbow in glass fiberepoxy resin and a diffuser with a pier made of welded stainless
steel plates. A set of 104 miniatures Keller Piezoresistive pressure transducers series 2MI
are mounted on the draft tube wall. The transducers range covers 0 to 3 bars to measure
low pressures that extend in the whole draft tube. The scale model equipped with the
FLINDT draft tube is installed in the test rig PF3 of the EPFL Laboratory for Hydraulic
Machines and the tests are carried out according to IEC 60913 standards [77].
To capture phenomena of interest at low flow rate turbine conditions, all pressure
signals are acquired simultaneously with a HPVXI acquisition system using a sampling
frequency of 200 Hz and 214 samples. This setup enables recording 430 vortex passages
providing a number of segments acceptable for the averaging process.
9.2.3
Phenomenon of Interest
The turbine operating point selected for the investigation at low discharge is given in
table 9.1.
Table 9.1: Studied operating point.
Specific speed /BEP /BEP
N
Guide Vane Opening
[]
[]
[]
[rpm]
[]
0.56
0.703
1.06
750
16
The spectral analysis of the pressure fluctuations measured in the cone are presented
for different values in figure 9.2. Pressure fluctuations between 2 n and 4 n appear for
low values. The amplitudes decrease and the frequencies increase as is increasing as
shown in figure 9.3 (right). These fluctuations are apparently modulated by the vortex
precession and give pressure fluctuation amplitudes for f = 2.5 n, 2.5 n frope , 2.5
n 2 frope , etc. During measurements, strong noise and vibration of the draft tube
were reported. Concerning the pressure fluctuations at the vortex rope frequency, their
amplitudes increase with while the vortex rope frequency drops slightly, see figure 9.3
left. Moreover, the amplitude spectra following 4 paths along the draft tube are presented
in figure 9.4. It can be noticed that the pressure fluctuations associated with the vortex
rope and its harmonics are restricted to the area of the draft tube cone and elbow, sections
(1) to (3). While the pressure fluctuations at 2.5 n are present in the entire draft tube.
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189
A maximum of pressure amplitudes is visible at the section (3) and minimum at section
(11), the draft tube outlet. The spectra along each path is quite similar. This suggests
that it is a onedimensional pressure fluctuation present in the entire draft tube.
f/n []
Figure 9.2: Influence of on the pressure fluctuations measured in the draft tube cone
[6].
The pressure fluctuations spectrum provides the phase angle between two signals at
the frequency 2.5 n. The phase shift is calculated for every pressure signal with respect
to a reference signal in the cone, see Arpe [7]. Those phase shifts are presented in an
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190
f/f
[]
f/n
n []
[]
f/ff/n
[]
n
frequency
amplitude
frequency
amplitude
Higher frequency
Figure 9.3: Influence of on the frequency and amplitudes of the pressure fluctuations
for the vortex rope precession (left) and the higher frequency (right) [6].
f/n []
f/n []
f/n []
f/n []
Figure 9.4: Spectra of pressure fluctuations along 4 paths for = 0.38 [6].
191
unfolded draft tube representation, see figure 9.5 left. It can be observed that in the cone,
there is no phase shift between pressure signals recorded in the same section pointing out
a synchronous pressure fluctuation. On the other hand, there is a maximum of phase
located at the inner part of the elbow. It corresponds to the location where impacts
between vortex rope and the wall occur. The phase evolution indicates that pressure waves
start from this point towards both upstream and downstream. These results indicate a
source of pressure waves as it is illustrated on the 3D phase representation on the draft
tube wall in figure 9.5 right.
The calculated phase shifts can be expressed in time for the studied frequency. The
resulting wave speed are presented along the 4 paths defined in figure 9.4 in figure 9.6.
The wave speed starts from 20 m/s in the cone, increases to about 450 m/s in the elbow
and decreases to 200 m/s in the channels. Generally, the acoustic wave speed depends
mainly on the diameter of the crosssection, the pipe wall thickness and material and
strongly on the vapor content. The low value of the wave speed in the cone is due to
the low Young modulus of the plexiglass but mainly due to high vapor content due to
the vortex rope. In the elbow, the wave speed increases because of the section stiffness
and the decreasing vapor content. The wave speed decreases in the draft tube channels
mainly because of the lateral deformations of the rectangular sections of the channels. It
can be also noticed that the wave speed is higher in the outer elbow than in the inner
elbow because of the difference of travel length.
During experiments, impacts between the vortex rope and the inner part of the elbow
wall were noticed. Those impacts are producing strong acoustic noise similar to hammer
strike. This shock phenomenon, described by Dorfler [44] and Jacob [81], can be assumed
to be a broad band noise source supplying the system with energy distributed on a wide
frequency range. This assumption is confirmed by the comparison between pressure fluctuation energy at the pressure source location and at the upper cone part in figure 9.7.
One can notice that energy is uniformly distributed for the pressure source with higher
amplitudes at the vortex rope precession harmonics. Whereas, at the upper part of the
cone, there is energy only at 2.5 n, the modulation frequencies, 5 n and the vortex
rope precession. This evidences the white noise excitation characteristic of the pressure
source.
9.2.4
To identify the origin of the pressure fluctuations at the frequency 2.5n, the hydroacoustic
behavior of the entire test rig is investigated. The test rig is simulated using SIMSENHydro including the Francis turbine scale model, the free surface downstream tank, the
connecting pipes and the 2 circulating pumps operating in parallel mode, see figure 9.8.
The hydroacoustic characteristics of the simulation model are given in table 9.2. The
pipes spatial discretization is set according to equation 3.91 to ensure an error below 1%
at the frequency of 2.5 n.
The wave speed of the pipe L6 is determined experimentally and the same value
is also considered for the pipes L1 to L5. The cross section and equivalent length of
the circulating pumps are obtained from [22]. The wave speed through the turbine is
determined using the phase shift at the frequency 2.5n between the turbine intake and
the runner outlet pressure transducers. The geometrical dimensions are deduced from the
drawings.
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192
Figure 9.5: Field of the phase shift at the wall of the cone and the elbow at 2.5 n;
unfolded mapping (left) and 3D distribution (right) [6].
modelling the
Dequ Aequ
[m]
[m2 ]
0.610 0.292
0.610 0.292
0.610 0.292
0.484 0.184
0.508 0.203
0.529 0.220
0.270 0.06
0.418 0.137
193
Figure 9.6: Distribution of the wave velocity along the draft tube along 4 different paths
[6].
ltot
dx
=
gA(x)
gA
(9.1)
Where:
A: average area of the pumpturbine cross section [m2 ];
ltot : total length of the pumpturbine in terms of curvilinear abscissa [m].
The capacitance value is a function of the equivalent wave speed in the hydraulic
machine. The evaluation of this term requires the knowledge of the bulk modulus of
the water, Ew , and the variation of the cross section area under the pressure increase
A/(A P ). Bolleter [23] performed investigations for a centrifugal pump and determined the RLC terms of the corresponding hydroacoustic model. He demonstrated the
good agreement between measurements and the model for the inductance and the resistance terms. However, the model overpredicted the capacitance by twice the measured
value. It allows estimating the wave speed for such a pump:
1200
a
a = = = 850m/s
2
2
(9.2)
194
f/n []
Figure 9.7: Energy spectra of the pressure fluctuation for the upper cone and for the
pressure source location.
Turbine
Air vessel
Feeding pumps
Turbine
Spiral case
and runner
Draft tube
L2
Centrifugal
pumps
L3
L6
Connecting
pipe
L1
L4
L5
Air vessel
Lequ / 4
QI
hI
195
QI
Qt
hI
The 3 models are illustrated in figure 9.10. The concentrated compliance model is made
of one pipe with one node, whose RLC terms are calculated with the mean hydroacoustic
parameters obtained by integration between the inlet and the outlet of the draft tube.
The corresponding equivalent scheme is presented in figure 9.11 with the parameters of
equation 9.4.
The distributed mean compliance model is made up out of 2 pipes with constant
hydroacoustic parameters. The first pipe accounts for the inlet part of the draft tube
where the wave speed is increasing and comprises 9 nodes. The second pipe accounts for
the second part of the draft tube where the wave speed is decreasing and comprises only 5
nodes. This node repartition results from the fact that high number of nodes are required
to model pipes with low wave speed. For both pipes, the hydroacoustic parameters are
obtained by integration along the curvilinear abscissa.
The distributed compliance model is made up out of 26 pipes with constant parameters. The values of the parameters are obtained as a function of the curvilinear abscissa
as illustrated by the equivalent scheme of figure 9.12 and the related equations 9.5.
The mean wave speed of a part of any of the 2 first models is determined assuming a
linear change of the wave speed, and is thus given between the point i and i + 1 by:
a=
L
L
ai ai+1
= i+1
=
R dx
dt
ln(ai /ai+1 )
i
(9.3)
a(x)
The cross section and wave speed distribution along the curvilinear abscissa obtained
for the 3 different models are reported in figure 9.14 respectively left and right. The draft
tube wave speed distribution is deduced from figure 9.6.
Rd =
Qd  ld
2gDA
R(x) =
Ld =
(x) Q dx
2 g D(x) A(x)2
ld
gA
L(x) =
Cd =
g A ld
a2
dx
g A(x)
C(x) =
(9.4)
g A(x) dx
a(x)2
(9.5)
196
Nb = 1
Nb = 5
Nb = 9
26 x Nb = 1
Ld
hin
Cd
Qd
Rd
hout
R(x)/ 2 L(x)/ 2
L(x)/ 2 R(x)/ 2
Qi
hi
197
C(x)
Qi +1
hi+1
Figure 9.12: Model of an infinitesimal pipe cross section for high order modelling of the
pipe with parameters function of the location x.
0.8
450
concentrated compliance
distributed mean compliance
distributed compliance
0.7
concentrated compliance
distributed mean compliance
distributed compliance
400
350
0.6
0.5
0.4
300
250
200
150
0.3
100
0.2
50
0.1
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Figure 9.13: Draft tube model parameters: cross section A(x) (left) and wave speed a(x)
(right) for the 3 different models of the draft tube.
For comparing the 3 different proposed models, the impedance of the draft tube is
evaluated using the discrete impedance calculation of equation 3.76. Starting from impedance equal to 0 at the draft tube outlet, x = 0, up to the draft tube inlet (runner
outlet) for x = 2.59 m. The magnitude of the 3 impedances obtained for x = 2.59 m are
presented in figure 9.14. As expected, the concentrated compliance model features only
one eigen frequency at low frequency about 8 Hz and the frequencies above 16 Hz are all
damped. The 2 distributed compliance models exhibits eigen frequencies at least up to
60 Hz. The distributed mean frequency model impedance fits the distributed compliance
model rather well until 40 Hz and then high discrepancies appear. These results present
good agreements with the transfer matrix model proposed by Philibert and Couston [120].
Although the distributed compliance model is more accurate, it requires more effort to
be set up. Therefore, the distributed mean compliance model is a good compromise for
investigations until 40 Hz. Therefore the frequency response and the forced oscillation
analysis will be performed with distributed compliance and distributed mean compliance
models and compared.
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10
10
10
1
Z/Zc []
10
2
10
3
10
4
10
5
10
concentrated compliance
distributed mean compliance
distributed compliance
6
10
10
20
30
Frequency [Hz]
40
50
60
Figure 9.14: Comparison of the draft tube impedance evaluated form draft tube outlet
(x = 0 m) until the draft tube inlet (x = 2.59 m) for three models: concentrated compliance (doted line), distributed mean compliance (dashed line) and distributed compliance
(solid line).
9.2.5
To determine the eigen frequencies of the entire test rig, a PRBS excitation is imposed at
the pressure source location. The spectral analysis of the pressure fluctuations obtained
at every spatial node with both the distributed mean compliance and the distributed
compliance models, are presented as a waterfall diagram in figure 9.15 respectively left
and right. The waterfall diagrams point out the eigen frequencies of the test rig and the
related eigen mode shapes. The draft tube extends over on the first 17 nodes for the
distributed mean compliance model while it extends over on the first 26 nodes for the
distributed compliance model.
In the frequency range 0 to 3 n, the system exhibits 12 eigen frequencies with one at
2.46 n, which corresponds to the pressure fluctuation peaks measured at 2.5 n. The
corresponding mode shape exhibits 5 pressure maxima along the test rig. One is located in
the draft tube elbow. A simulation of the test rig dynamic behavior using the distibuted
mean compliance model and considering a sinusoidal excitation at the frequency of 2.46n
in the draft tube elbow provides the corresponding eigen mode shape, see figure 9.16 left
and right. The amplitude of the excitation is optimized to fit the experimental pressure
amplitudes in the draft tube elbow, leading to a pressure excitation source amplitude for
the frequency 2.46n, equal to 0.5% of the reference energy Eref . The pressure fluctuation
profile in the draft tube at the frequency 2.46 n obtained by simulation presents a good
agreement with measurements of figure 9.4. Moreover, pressure fluctuations at the turbine
intake, element node 17, are found to correspond to a pressure maximum, which is in
EPFL  Laboratoire de Machines Hydrauliques
f/n []
199
f/n []
Figure 9.15: Frequency response of the test rig resulting from draft tube PRBS excitation
with the distributed mean compliance (left) and distributed compliance model (right).
good agreement with the measurements too. Finally, one can notice the high amplitudes
of discharge fluctuations in the draft tube cone, see figure 9.16 left, due to the vortex rope
compliance.
Figure 9.16: Eigen mode shape of the hydraulic system for f /n = 2.46 for t = to (left)
and for t = to + T /4 (right).
9.2.6
The pressure excitation source identified experimentally, located in the draft tube elbow,
is related to both the shock phenomenon and the vortex rope precession, see figure 9.18.
Therefore, a forced excitation analysis simulation is performed, considering both synchronous pressure fluctuations and Dirac impulses at the frequency of the vortex rope. The
synchronous excitation is considered for frope and the corresponding harmonics which are
obtained experimentally by a decomposition procedure of the pressure fluctuations under
cavitation free operating conditions as prescribed by Dorfler [41]. The pressure excitation
EPFL  Laboratoire de Machines Hydrauliques
200
source used for the simulation is presented in figure 9.17, and is expressed as follow:
Hs (t) = H
X
k=1
(t k
1
frope
(9.6)
The amplitude of the Dirac term of the pressure source excitation is obtained by
minimization procedure in order to obtain 0.5% of Eref in the spectral decomposition of
the excitation signal at the frequency of 2.46 n. The amplitude of the Dirac impulses is
found to be 5% of Eref and the duration is found to be 1/18 of the vortex rope precession
period, i.e. TDirac = 0.0123 s. The pressure source imposed for the simulation is a
difference of pressure between two pipes (and not a pressure specified at a node). The
resulting pressure fluctuation spectra obtained for every element node are presented as a
waterfall diagram, again for both the distributed mean compliance and the distributed
compliance models in figure 9.19. One can notice, that even if the test rig presents
several eigen frequencies in the frequency range 0 to 10 n, the only significant pressure
amplitudes correspond to 0.72 n, 2.46 n, 5 n and 7.5 n, and are in good agreement
with the measurements.
A comparison between simulation and measurement of pressure fluctuations resulting
from forced excitation in the upper part of the cone using the distributed mean compliance
model is presented in figure 9.20. The numerical simulations at the upper part of the cone
show good agreement with the experimental values for the frequencies of interest, i. e.
vortex rope precession and all the harmonics as well as eigen frequencies of the hydraulic
system.
Figure 9.17: Time evolution of the pressure excitation imposed in the draft tube for the
forced response analysis according to equation 9.6.
9.2.7
The evolution of the wave speed obtained experimentally, see figure 9.6, is compared
to a theoretical value, that takes into account only pipe wall deformation and water
EPFL  Laboratoire de Machines Hydrauliques
201
Figure 9.18: Vortex rope shock on the draft tube wall at inner elbow part.
f/n []
f/n []
Figure 9.19: Waterfall diagram of pressure fluctuations obtained by forced response of the
test rig resulting from draft tube forced excitation with the distributed mean compliance
(left) and distributed compliance model (right).
f/n []
f/n []
Figure 9.20: Comparison of simulated and measured pressure fluctuations at the pressure
source (left) and upper cone (right).
202
compressibility, see figure 9.22. The wave speed accounting only for wall deformation
and water compressibility is much higher than the measured wave speed, the difference
between those 2 values being due to the vortex rope compliance. To properly model
a draft tube cross section with vortex rope, the capacitive term of the electrical model
must include the vortex rope compliance. Figure 9.21 presents the equivalent capacitance
Cequ corresponding to 2 capacitances: (i) one for the wall deformation and the water
compressibility, Co , and (ii) one for the rope compliance, Crope .
Cequ = Co + Crope
(9.7)
The capacitances are in a parallel scheme because the rope compressibility effects and
the pipe wall plus the water compressibility effects are related to the same pressure field.
The two discharges related to Co and Crope correspond to the discharge stored respectively
by both effects. Introducing the expression of the capacitance of a pipe in equation 9.7
and assuming a barotrop behavior of the vapor volume gives [24]:
dx g A
Vrope
dx g A
=
+
2
2
aequ
ao
Hrope
(9.8)
(9.9)
Finally, the ratio between the vortex rope area and the draft tube cross section area is
deduced:
1
1
Arope
= g Hrope
(9.10)
A
a2equ a2o
Where:
Arope : is the vortex rope cross section [m2 ];
Hrope : is the head of the vapor inside the rope [m]; (Hrope = 0.24 m for T=20C);
: is the polytropic coefficient ( = 1.4 for vapor).
One may notice, that, if the equivalent wave speed in the draft tube aequ corresponds
to the theoretical wave speed without vapor ao , the vortex rope section is zero. Using
the estimated and measured wave speeds in the draft tube and the vapor properties at
T = 20 C given above, the diameter of the vortex rope is calculated along the draft
tube curvilinear abscissa using equation 9.10 and presented in figure 9.22. The value of
the rated rope diameter Drope /D is found to be 9% in the cone, which presents good
agreement with the rated diameter determined using 3D PIV on the same turbine scale
model by Illiescu [76], and also found to be 9% for the same operating conditions. So it
means that knowing the rope diameter for a given operating point allows estimating the
wave speed in the draft tube cone for cavitating conditions.
EPFL  Laboratoire de Machines Hydrauliques
Cequ
Co
203
Crope
Figure 9.21: Equivalent capacitance accounting for both cavitating vortex rope compliance
and pipe compliance.
600
0.1
a equ
ao
Drope/D
400
0.06
0.04
200
0.08
0.02
0
2.6
2.4
2.2
1.8
0
1.6
Figure 9.22: Representation of the calculated wave speed without cavitation (ao ), the
equivalent wave speed from measurement (aequ ) and deduced rated vortex rope diameter
(Drope /D) as function of curvilinear abscissa of the draft tube x.
9.2.8
This case study presents the analysis of the origin of the pressure fluctuations measured
in the elbow draft tube of a Francis turbine scale model for low discharge operation. A
component of pressure fluctuation at 2.5 n frequency is identified all along the draft tube
walls, the source of those pressure fluctuations being located at the inner part of the draft
tube elbow. The energy of the pressure source extends over a wide frequency range and
results from the impact of the vortex rope on the draft tube wall at this location. The
analysis of the pressure fluctuations phases for the 2.5 n frequency provides a way to
determine experimentally the wave speed along the draft tube, which is the key parameter
for a numerical simulation of the hydroacoustic behavior of the test rig. The simulation
carried out for the full test rig, taking into account piping, circulating pumps and the
scale turbine model with the elbow draft tube shows that the 2.5 n frequency value
corresponds to an eigen frequency of the system. Therefore, a model of the excitation
EPFL  Laboratoire de Machines Hydrauliques
204
9.3
9.3.1
In the framework of the FLINDT project, the operating point investigated previously
is very particular as it is a case of resonance between the vortex rope and the test rig.
Moreover, it was noticed that the 7th harmonic of the vortex rope precession matches
an eigen frequency of the test rig under the chosen operating conditions. Unfortunately
no investigations on the influence of the rotational speed, i.e. the Froude number, or
of the cavitation number where performed. Thus, new investigations on upper part
load pressure fluctuations performed by Koutnik et al. in 2006 [94] have pointed out
the influence of these 2 parameters noticing that upper part load pressure fluctuations
can also occur even if the predominant amplitudes are not found at a frequency being a
multiple of the vortex rope precession. Therefore, the origin of these pressure fluctuations
cannot only be induced by the shock phenomenon. In addition, the physical modulation
process was identified by Koutnik et al. [94] to be related to the elliptical shape of the
vortex rope cross section observed on test rig at upper part load. The motion of the
elliptical vortex rope at upper part load that can be decomposed in precession movement
with pulsation rope and the self rotation of the rope with pulsation as it is illustrated
in figure 9.23.
rope
Figure 9.23: Elliptical vortex rope precessing in the draft tube cone at upper part load.
Thus, other investigations have been carried out on a Francis turbine similar to the
FLINDT Francis turbine, = 0.5 instead of = 0.56, to clarify by experiments the role
of rotational speed and of the cavitation number on the upper part load vortex rope. The
investigations have been undertaken by means of both: (i) vortex rope visualization using
high speed camera; (ii) simultaneous pressure fluctuations records.
9.3.2
The waterfall diagram of the wall pressure fluctuations measured at the downstream
cone of the draft tube of the Francis turbine of = 0.5 investigated here is presented
in figure 9.24 as function of the discharge coefficient /BEP . This diagram points out
EPFL  Laboratoire de Machines Hydrauliques
206
upper part load pressure pulsations in the range of 1 to 3 times the rotational speed for
a rated discharge coefficient around 0.8. The operating point of table 9.3 is chosen for
investigating: (i) the influence of the cavitation number ; (ii) the influence of the Froude
number F r.
Waterfall diagram
Channel
Sigma
1 Downstream cone
= 0.17974
/BEP
= 1.00
1.3
p/(E)
2
[%rms]
1.2
1.1
1.5
1
0.9
0.5
0
0
/BEP []
0.8
0.7
2
0.6
0.5
6
f/n
0.4
Figure 9.24: Waterfall diagram of the pressure fluctuations measured at the downstream
cone of the draft tube for /BEP = 1 and = 0.18.
Table 9.3: Parameter of the scale model Francis turbine and investigated operating point.
Specific speed /BEP /BEP
N
Guide Vane Opening
[]
[]
[]
[rpm]
[]
0.50
0.832
1.00
700
21.5
9.3.3
Experimental Apparatus
To perform the visualization of the vortex rope, a high speed camera Photron is used. The
frame rate used for the investigation is 4000 frames/seconds using a diaphragm aperture
time of 1/4000 s and a number of 3600 frames per movie. The images are obtained with a
EPFL  Laboratoire de Machines Hydrauliques
Spirale case
pressure transducer
Downstream cone
pressure transducer
Upstream cone
pressure transducer
Figure 9.25: Scale model installed on the test rig for high speed camera visualization.
9.3.4
Results
208
Fast camera
Oscilloscope
+ trigger
trig 5V
trig 1V
Pressure
transducer
Kistler 701
Draft tube
cone
Amplifier
HP acqusition system
Amplifier
Similarly to the FLINDT case, the frequency of the pressure fluctuations in the range
of 1 to 3 times the rotational speed increases with the cavitation number. The frequencies
and amplitudes of these pressure pulsations are reported in figure 9.28. It can be noticed
that frequency increases almost linearly with the cavitation while the amplitude of the
pressure pulsations rises quickly up to 2.5% of the specific energy E for = 0.3. The
amplitudes of pressure pulsations for this cavitation number are similar for the 3 pressure
transducers. In addition, strong mechanical vibrations and noise were noticed during
the measurements. The shock phenomenon was also strongly present for these operating
conditions. This operating point corresponds again to the resonance between the vortex
rope and the test rig.
The figure 9.29 depicts the elliptical shape of the vortex rope self rotating at the
pulsation and precessing with a pulsation rope for the resonance operating conditions,
i.e. = 0.3. For the 6 operating points investigated here, the pictures resulting from
the movie records of the vortex rope with synchronized pressure records are reported in
figures 9.30 to 9.35. For each operating point, 5 successive pictures are extracted from
the movie and correspond to one period of the pressure fluctuations of interest T ; i.e.
from t = to to t = to + T . The influence of the pressure on the dimensions of the
vortex rope appears clearly, as for t = to, when the pressure is high, the vortex rope
features small diameter while the diameter is the highest for t = to + T /2 when the
pressure is the lower. This pressure dependence is identical for the 6 operating points. It
can be noticed that, as expected, the mean diameter of the vortex rope increases as the
cavitation number decreases. But for small values of cavitation number, the pressure
amplitudes becomes smaller, even if locally they can be biggerlarger. This phenomenon
can be correlated with the apparition of cavitation at the trailing edge of the runner
blades. This cavitation is visible on the pictures but also on the pressure time history
where very high frequency is visible, see figures 9.34 and 9.35. So it means that the higher
EPFL  Laboratoire de Machines Hydrauliques
Downstream cone
Resonance
p/(E)
[%rms]
Fr
= 8.47[]
Resonance
Upstream cone
p/(E)
[%rms]
Resonance
p/(E)
[%rms]
Figure 9.27: Waterfall diagram of the pressure fluctuations measured at the spiral case,
upstream and downstream cone as function of the frequency and cavitation number for
/BEP = 1, /BEP = 0.833 and F r = 8.47.
210
Table 9.4: Operating parameters for the influence of the cavitation number.
N
Froude
N
Remarks
[]
[]
[rpm]
1 0.30
8.47
700 Hydroacoustic resonance,
strong vortex rope shock on draft tube wall
and mechanical vibrations
2 0.26
8.47
700 Random hydroacoustic resonance
vortex rope shock on draft tube wall
and mechanical vibrations
3 0.22
8.47
700
4 0.18
8.47
700
5 0.14
8.47
700 Low interblade
cavitation development
6 0.10
8.47
700 Strong interblade
cavitation development
content of cavitation bubbles and development in the runner outlet and draft tube helps
to mitigate the pressure pulsations appearing in the range of 1 to 3 times the rotational
speed. This result can be therefore also obtained by air injection in the flow. As illustrated
in figure 9.36 where the cavitation number is = 0.18 with air injection but the pressure
pulsations present in figure 9.33 almost vanished.
The pictures of the vortex rope for the 6 operating points considered, feature the
expected elliptical cross section of the vortex rope. When the cavitation number is the
lowest, i.e. for = 0.14 and 0.1, see figure 9.34 and 9.35, the elliptical cross section is
evidenced by 2 vortex ropes rotating in the same directions. The scheme of figure 9.37
depicts the complex structure of the vortex rope which under these conditions presents
small vortices attached to the runner nose and rotating on themselves at the pulsation
2 , while 2 vortex ropes rotating on themselves at a pulsation 1 constitute the main
body of the rope self rotating at the pulsation and the rope being animated of the
precession pulsation rope . By Analyzing the pictures of figures 9.30 to 9.35 it is found in
accordance with Koutnik et al. [94], that the pulsation of the self rotation of the elliptical
vortex rope is half the pulsation of the measured pressure fluctuations. Therefore, the
combination between the rope precession rope and the self rotation of the rope 1 leads to
the modulation process already described by Arpe in 2003 on FLINDT [6]. The elliptical
shape of the vortex rope make apparent the nonuniformity of the pressure distribution
in the cross section of the cone. Consequently, if the elliptical vortex rope rotates with
the pulsation , the associated pressure fluctuation features a pulsation of = 2 ,
as illustrated in figure 9.39.
4
f/n
p/( E)
0
0.08
0.12
0.16
0.2
0.24
0.28
0.32
Sigma []
Figure 9.28: Evolution of the frequency and amplitude of the pressure pulsations as
function of the cavitation number for /BEP = 1, /BEP = 0.833 and F r = 8.47 at
the downstream cone.
rope
Figure 9.29: Vortex rope precession at rope and self rotating at for operating point
N1: = 0.30, F r = 8.47.
212
T*
p/( E) []
0.04
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0
0.02
4
5
6
7
Runner revolutions []
t=to
t=to+T*/4
t=to+T*/2
t=to+3T*/4
t=to+T*
T* = 1/f*
f*/n = 3.04
= 0.30
Fr = 8.47
10
11
Figure 9.30: Vortex rope development for operating point N1: = 0.30, F r = 8.47.
T*
p/( E) []
0.04
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0
0.02
4
5
6
Runner revolutions []
t=to
t=to+T*/4
t=to+T*/2
t=to+3T*/4
t=to+T*
T* = 1/f*
f*/n = 2.66
= 0.26
Fr = 8.47
10
11
Figure 9.31: Vortex rope development for operating point N2: = 0.26, F r = 8.47.
214
T*
0.04
p/( E) []
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0
0.02
0
4
5
6
7
Runner revolutions []
t=to
t=to+T*/4
t=to+T*/2
t=to+3T*/4
t=to+T*
T* = 1/f*
f*/n = 2.42
= 0.22
Fr = 8.47
10
11
Figure 9.32: Vortex rope development for operating point N3: = 0.22, F r = 8.47.
T*
0.04
p/( E) []
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0
0.02
0
4
5
6
7
Runner revolutions []
t=to
t=to+T*/4
t=to+T*/2
t=to+3T*/4
t=to+T*
T* = 1/f*
f*/n = 1.96
= 0.18
Fr = 8.47
10
11
Figure 9.33: Vortex rope development for operating point N4: = 0.18, F r = 8.47.
216
T*
p/( E) []
0.04
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0
0.02
0
4
5
6
t=to
t=to+T*/4
t=to+T*/2
t=to+3T*/4
10
11
Cavitation
t=to+T*
T* = 1/f*
f*/n = 1.74
= 0.14
Fr = 8.47
Figure 9.34: Vortex rope development for operating point N5: = 0.14, F r = 8.47.
T*
p/( E) []
0.04
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0
0.02
0
4
5
6
t=to
t=to+T*/4
t=to+T*/2
t=to+3T*/4
10
11
Cavitation
t=to+T*
T* = 1/f*
f*/n = 1.11
= 0.10
Fr = 8.47
Figure 9.35: Vortex rope development for operating point N6: = 0.10, F r = 8.47.
218
0.04
0.02
Downstream cone
Upstream cone
Spiral case
Camera trig
0.02
Cavitation
Figure 9.36: Vortex rope development for operating point N4 with air injection: =
0.18, F r = 8.47.
2 2
1
*
rope
Figure 9.37: Vortex rope development for operating point N5: = 0.14, F r = 8.47.
Cref = E, as follows:
s
H
(9.11)
Fr =
Lref
The Froude number affects the distribution of cavitation in the flow as it determines
the pressure gradient relatively to the size of the machine. The relation between the
position of the vapor pressure pv can be expressed as a function of the Froude number,
see [56], neglecting Reynolds effects, assuming the same cavitation number as a function
of the reference position Zref as follows:
Zref Z1
F r12
=
Zref Z2
F r22
(9.12)
The Froude number being usually smaller on prototype than in the model, the elevation
of the position of the cavitation is higher on prototype than in the model, [56], [80]. Due
to the difference of Froude numbers, the vortex rope on scale model is more narrow and
longer than on prototype [41], as illustrated in figure 9.38.
The selected values of operating conditions as well as observations remarks are given
in the table 9.5. The waterfall diagram of the wall pressure pulsations measured at the
downstream cone, upstream cone and spiral case are presented as function of the frequency
and Froude number in figure 9.40.
Table 9.5: Operating conditions for the influence of the cavitation number.
N
Froude
N
Remarks
[]
[]
[rpm]
1 0.30
8.47
700 Hydroacoustic resonance
and mechanical vibrations
7 0.30
7.86
650 Hydroacoustic resonance
and mechanical vibrations
8 0.30
7.26
600 Low hydroacoustic resonance
and mechanical vibrations
9 0.30
6.64
550 Low hydroacoustic resonance
and mechanical vibrations
First, it can be noticed from the observations of table 9.5 that hydroacoustic resonance and mechanical vibrations are observed for all tested conditions, even if the level
EPFL  Laboratoire de Machines Hydrauliques
220
Prototype
Model
Zref
p=pv
ZP
ZM
FrM
ZM
>
<
FrP
ZP
Figure 9.38: Difference of the cavitation development between model (M) and prototype
(P) due to difference in Froude numbers.
of resonance and related vibration was lower for lower values of the Froude number. This
is due to the fact that increasing the Froude number leads to a reduction of the specific
energy and therefore of the absolute magnitude of the pressure fluctuations. However, as
it is reported by the waterfall diagram of figure 9.40, the relative amplitudes of pressure
fluctuations are not affected in the same way. The figure 9.41 represents the amplitude
and frequency of the pressure pulsations of interest as function of the Froude number.
It can be noticed that the frequency of the pressure pulsations are proportional to the
runner rotational speed, in accordance with to Koutnik [94], and that the amplitudes are
of the same order of magnitude for the values of Froude equal to F r = 8.47 and 7.86,
while they are divided at least by a factor 2 for F r = 7.26 and 6.64. The reason for
this difference is pointed out by the representation of one of the eight time histories used
for the calculation of the amplitude spectrum for each Froude number value as presented
in figure 9.42. It can be seen from the time history of the pressure, that the resonance
phenomenon features more random behavior for F r = 7.26 and 6.64 than for F r = 8.47
and 7.86. Consequently the values obtained in the waterfall diagram are smaller for the
higher values of Froude number.
No visualizations of the vortex rope are presented here as the shape of the vortex rope
EPFL  Laboratoire de Machines Hydrauliques
rope
+
+

Figure 9.39: Elliptical vortex rope with related pressure distribution rotating with the
pulsation .
222
Downstream cone
p/(E) 2.5
[%rms]
8.4
8.2
2
8
1.5
7.8
Froude []
7.6
0.5
7.4
0
0
7.2
7
50
6.8
f [Hz]
100
Upstream cone
p/(E)
[%rms]
8.4
2.5
8.2
2
8
1.5
7.8
Froude []
7.6
0.5
7.4
0
0
7.2
7
50
6.8
f [Hz]
100
p/(E) 2.5
[%rms]
8.4
8.2
2
8
1.5
7.8
7.6
0.5
Froude []
7.4
0
0
7.2
7
50
6.8
f [Hz]
100
Figure 9.40: Waterfall diagram of the pressure fluctuations at the spiral case, upstream
and downstream cone as function of the frequency and Froude number /BEP = 1,
/BEP = 0.833 and = 0.3.
4
f/n
p/( E)
0
6.4
6.8
7.2
7.6
8.4
8.8
Froude []
Figure 9.41: Evolution of the frequency and amplitude of the pressure pulsations as
function of the Froude number for /BEP = 1, /BEP = 0.833 and = 0.3 at the
downstream cone.
224
Froude = 8.47
20
Downstream cone
p/(E) [%]
20
20
Upstream cone
0
20
20
0
20
Time [s]
Time [s]
Time [s]
Time [s]
Froude = 7.86
20
Downstream cone
p/(E) [%]
20
20
Upstream cone
0
20
20
0
20
Froude = 7.26
20
Downstream cone
p/(E) [%]
20
20
Upstream cone
0
20
20
0
20
Froude = 6.64
20
Downstream cone
p/(E) [%]
20
20
Upstream cone
0
20
20
0
20
Figure 9.42: Time history of the pressure pulsations at the spiral case, upstream and
downstream cone for different Froude values and for /BEP = 1, /BEP = 0.833 and
= 0.3.
20
Downstream cone
p/(E) [%]
0
20
20
Upstream cone
0
20
20
0
20 0
0.5
1.5
2.5
3.5
4
Time [s]
FFT
4
frope
p/(E) [RMS%]
frope
0
4
2x
frope frope
f*
2x
frope
2 x f*
2
0
4
2
0
20
40
60
f*=7.5 x frope
80
100
Frequency [Hz]
Figure 9.43: Modulation of the pressure pulsation at the vortex rope precession frequency
frope and pressure pulsation f for /BEP = 1, /BEP = 0.833, = 0.3 and F roude =
7.86.
226
9.3.5
Chapter 10
Modelling of Over Load Pressure
Pulsations on a Prototype
10.1
For higherload operating conditions, Francis turbines can feature onset of a cavitating
vortex rope in the draft tube cone which is generated by the incoming swirling flow
[109]. At full or overload operating conditions, the cavitating rope may under certain
conditions act as an energy source, which leads to selfexcited pressure oscillations in the
whole hydraulic system [79]. These pressure oscillations can jeopardize the mechanical
and hydraulic system [85]. Depending on the turbine relative location in the circuit, the
turbine head may oscillate and generate power swings.
These power and pressure oscillations have been experienced at over load during commissioning tests carried out with Unit 4 of a PumpedStorage Plant, PS Plant, located
in the southeastern United States featuring four 400 MW Francis pumpturbines. The
pressure surge event was well monitored and the time histories of power and wall pressure
in the spiral case and draft tube cone mandoors were simultaneously recorded, see figure
10.1. The event was apparently initiated by the shut down of Unit 2, with Unit 1 and
Unit 3 at rest. The time sequence of the surge event onset is the following:
Unit 4 and Unit 2 in approximately steady generating mode with Unit 1 and Unit
3 at rest;
observation of low amplitude pressure pulsations at 2.9 Hz frequency for over load
operating conditions;
normal shut down of Unit 2;
onset of high amplitude pressure pulsations at about 2.2 Hz frequency;
generator power swing followed by pressure oscillations;
standing pressure oscillations at about 2.2 Hz everywhere in the power plant piping.
The aim of this investigation is to present the hydroacoustic analysis of the PS Plant
and to show the self induced oscillation nature of the observed pressure surge event. First
EPFL  Laboratoire de Machines Hydrauliques
80
60
40
Shut down U2
20
520
560
600
Time [s]
640
448
444
440
436
432
428
424
580
584
Time [s]
588
592
228
80
60
40
20
580
584
588
592
580
584
588
592
Time [s]
220
200
180
160
Time [s]
Figure 10.1: Recorded time history of pressure at the draft tube (top), pressure at spiral
case inlet (bottom right) and power (bottom left) for Unit 4.
the main physical parameters of the system stability are introduced by using a simplified
1D mathematical model of the higher load vortex rope. Then, a free oscillation analysis of
the full hydraulic circuit is performed to determine the stability domain of the PS Plant
as a function of the vortex rope parameters. Following this, a hydroacoustic model of
the PS Plant including the hydraulic circuit, the mechanical masses, the generators, the
transformers, and the voltage regulators is developed and validated for the simulation and
analysis of the above described time sequence of the pressure surge event onset, providing
the cause of this pressure surge event.
10.2
10.2.1
The investigated PS plant corresponds to the upgrade of the power plant of the SIMSEN
validation presented in the section 7.2. The layout of the PS Plant is repeated below in
EPFL  Laboratoire de Machines Hydrauliques
229
figure 10.2 including the layout of the electrical installation. A 590 meters long penstock
feeds four Francis pumpturbines with four downstream surge chambers of variable cross
section all connected to a 304 meters long pressurized tailrace water tunnel. The design
values of the upgraded pumpturbines are given in table 10.1.
The simulation model set up with SIMSENHydro takes into account the full hydraulic
system including pipes, spherical valves, downstream surge chambers, and pumpturbines.
Moreover, the inertia of the rotating shaft line, the generators with saturation effects, the
voltage regulators, the circuit breakers, and the transformers are included in the model.
In this study, the electrical grid is considered as an infinite grid at 60 Hz fixed frequency.
Infinite power
network
Uc +
KU(s)
Unit 4
Unit 3
Unit 2
Upper reservoir
(466.3509.6 m)
Unit 1
PT4
28 24
32
PT3
31
27 23
33
Lower reservoir
(192.6193.2 m)
35
26 22
29
25 21
PT1
34
12
19 15
11
4
PT2
30
20 16
18 14
10
17 13
Table 10.1: Rated values of the pumpturbines of the validation test case.
Hn
Qn
Pn
Nn
Dref
Jtot
230
10.2.2
As the Francis turbine features blades with constant pitch, the outlet flow in the draft
tube cone is optimal, almost purely axial, only for the best efficiency point. For off
design operation, i.e. part load or full load, the circumferential components of the flow
velocity are respectively positive or counter rotating, inducing a swirling flow [109]. For
full load operation and a given setting level of the machine, the swirling flow results in
the development of an axisymmetric cavitation vortex rope below the runner, see figure
10.3 left.
As the frequency of the phenomenon of interest is very low, 2.2 Hz, the vortex rope
is modelled by a lumped element using a cavitation flow model presented in section 5,
located at the outlet of the pumpturbine of Unit 4. The lumped component modelling of
the vortex rope gaseous volume is based on the assumption that the gaseous volume is a
function of the two state variables H and Q, the net head and the discharge respectively,
[27], [29] and [97]. As presented in section 5 the rate of change of the gaseous volume
is given by the variation of discharge between the 2 fluid sections limiting the rope, see
figure 10.3 right, and is therefore given by the equation repeated below:
dQi+1
dhi+1
+
(10.1)
Qi Qi+1 = Qc = C
dt
dt
Where the lumped parameters are defined as:
the cavity compliance C = V
h
the mass flow gain factor = QVi+1
10.2.3
This investigation is related to the validation test case of section 7.2 for which load rejection in both generating and pumping mode are simulated with satisfactory agreement
compared with available transient measurements carried out for Unit 1 during commissioning tests of the original runners in 1979. However, for the present investigation, the
best efficiency point of the turbine characteristic is modified according to the new best
efficiency point of the upgraded pumpturbines.
EPFL  Laboratoire de Machines Hydrauliques
231
10.3
10.3.1
The study of the stability of a dynamic system can be efficiently carried out performing
modal analysis. This analysis is performed to determine the stability domain of the
hydraulic installation with respect to the lumped parameters of the vortex rope, i.e. the
rope compliance and mass flow gain factor. The Transfer Matrix Method described in [138]
is used with advanced models of both friction losses and damping, [66]. The eigenvalues of
the corresponding system global matrix have been computed to derive both the natural
frequencies and the damping or amplification factors of the corresponding mode from,
respectively, their imaginary and real parts [95]. The stability domain is defined as a
function of both the vortex rope compliance and the mass flow gain factor in figure 10.4,
where the blue dots indicate unstable areas.
Given that Unit 4 experienced pressure oscillations at 2.9 Hz just prior to the event
and at 2.2 Hz after the event, and assuming that the event was caused by a hydroacoustic
instability the corresponding values of the vortex rope compliance and the absolute value
of the mass flow gain factor can be identified in figure 10.4. With these values for the
lumped parameters of the cavitation vortex rope, for each calculated natural frequency,
both the pressure and discharge mode shapes can be computed, as shown in figure 10.5
for 2.2 Hz. The modal analysis confirms that the maximum pressure oscillations occur
close to Unit 4, which is in agreement with the field tests, and that both the spiral casing
and the draft tube experience pressure oscillations in phase, see figure 10.1.
10.3.2
The time domain analysis is performed with the complete SIMSEN model featuring the
vortex rope model for Unit 4, the inertia of the rotating shaft line, the generators with
saturation effects, the voltage regulators, the circuit breakers, and the transformers. The
event is simulated according to the sequence observed on site, i.e. Unit 2 operating
normally, the modernized Unit 4 in operation at overload conditions, and Unit 1 and
Unit 3 at rest with guide vanes closed, and then Unit 2 is shutdown according to the
normal procedure. The normal shutdown consists of closing in 24 seconds the guide vanes
following a twoslope closing law and tripping the generator when 10% of the generator
power is reached.
For this analysis the vortex rope mass flow gain factor value is taken from the stability
diagram, see figure 10.4, for the point corresponding to the stability limit at 2.9 Hz. To
take into account the influence of the operating condition on the volume of the vortex
rope, the rope compliance was determined using the results of the CFD simulations for the
modernized units [95]. The draft tube flow is computed for different guide vane openings
and the cavitation vortex rope volume is assumed to correspond to the volume limited
by the vapor pressure isosurface, enabling estimation of the cavitation volume as a
function of , the Thoma number, and the guide vane opening, see figure 10.6 left. A
power law is fitted with the corresponding 80% guide vane opening overload conditions:
Vrope = A B
EPFL  Laboratoire de Machines Hydrauliques
(10.2)
232
Unstable
Estimation from
measurement
At the event
Shut down of U2
Figure 10.4: Stability diagram of the absolute value of the mass flow gain factor (top)
and the unstable frequency (bottom) as function of Vortex rope compliance.
Surge tanks
Turbine
Penstock
Upper reservoir
Unit 4
Unit 3
Unit 2
Lower reservoir
Unit 1
Figure 10.5: Pressure mode shape in the waterways, for natural frequency of 2.2 Hz.
233
N P SE
Ha (ZRef ZTail Water ) Hv
E
H
(10.3)
V
HDraft Tube
V
ZTail Water
B1
H
= A B
(10.4)
The resulting estimated compliance is presented in figure 10.6 right. However, this
approach is only applicable for full load operation where pressure isosurfaces of the flow
captured by CFD computation are close to the cavity limits observed in reducedscale
model tests. For part load operation, the isosurfaces computed by current CFD models
are not close to the cavity limits observed in model tests.
1
1000
0.8
0.6
0.4
0.2
0
0.04
Rope Volume V
Rope compliance dV/dh
100
GVO: 0.63
GVO: 0.66
GVO: 0.71
GVO: 0.73
GVO: 0.75
GVO: 0.77
GVO: 0.80
Fit to GVO: 0.80
GVO: 0.85
10
0.1
0.01
0.08
0.12
0.16
Cavitation number []
0.2
0.001
0.04
0.08
0.12
0.16
0.2
Figure 10.6: Vortex rope volume at full load computed with CFD (left), and deduced
vortex rope compliance (right).
The results of the timedomain analysis are presented for Unit 2 and Unit 4 in figure
10.7 for the vortex rope parameters indicated as case C in table 10.2, which define the
system stability limit. The transient resulting from the Unit 2 shutdown induces a slow
downstream pressure drop due to downstream mass oscillation. As a result, the Thoma
number value of Unit 4 decreases and the vortex rope compliance increases as the vortex
rope volume increases. Then, considering a constant mass flow gain factor, the vortex
rope compliance increases until the unstable domain of operation is reached; leading to
the pressure oscillation onset in the draft tube; the system being in selfexcited conditions.
Then the system remains unstable as long as the sigma value, i.e. the vortex rope size,
does not reach the same steady value as prior to the event. The calculated oscillation
frequency corresponds well with the measured frequency value, whereas the agreement
between the calculated and measured amplitudes is less good.
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234
A parametric study of the influence of the vortex rope is carried out using the values
given in table 10.2. For case A, the mass flow gain factor and compliance are both
kept constant. For cases B, C and D, the cavity compliance is taken as a function of
the Thoma number and the head, and the mass flow gain factor is still kept constant.
The resulting draft tube piezometric heads are presented in figure 10.8. In case A, the
amplitudes increase dramatically. Case B exhibits amplitudes that are strongly affected
by the compliance dependence on the Thoma number value. Case C corresponds to the
limit of stability with bounded amplitudes. In case D, the selfexcitation is initiated but
the oscillations are damped once the surge chamber level recovers its initial value. This
sensitivity analysis shows clearly that the shutdown of Unit 2 initiates the selfexcitation
mode by decreasing the Unit 4 draft tube pressure and, consequently, the Thoma number
value.
Table 10.2: Vortex rope parameters for 4 simulation cases.
Cases
A
B
C
D
2
2
Compliance C [m ]
10
C = C(, H) C = C(, H) C = C(, H)
Mass flow gain factor [s] 0.005
0.005
0.004
0.003
10.4
Concluding Remarks
An overload surge phenomenon was experienced at a pumped storage plant during over
operating range tests. In order to have a better understanding of the onset condition of this
phenomenon, a onedimensional model of the power plant, including a model of the vortex
rope, is carried out. The vortex rope at higher load conditions is modelled by defining a
compliance parameter and a mass flow gain factor for the cavitation volume. Using this
simplified model, the stability domain of the power plant with respect to the vortex rope
parameters is determined through modal analysis and the vortex rope parameters for the
power plant are estimated by locating the observed test conditions within the stability
domain. Then, a time domain simulation of the dynamic behavior of the power plant
according to the event time sequence observed on site is performed. The simulation model
includes the hydraulic system with a nonlinear vortex rope model, the rotating inertias,
the electrical components, and the governors. In an original approach, the vortex rope
cavitation compliance is modelled as a nonlinear function of the Thoma number and
the head, where the function is determined by comparison of results from scalemodel
experiments and CFD modelling. The simulation results show the role played in this
case by the shutdown of a neighboring unit, which decreases the pressure downstream
of the affected unit and, therefore, the cavitation number and eventually initiating the
selfexcited surge.
The comparison of calculated and measured results shows surprisingly good agreement
considering the relatively simple model used for the complex flow  vortex rope structure
within the pumpturbine. Moreover, this investigation provides a deeper insight into
the physical phenomena underlying the onset of overload pressure surge. However, to be
predictive, the hydroacoustic modelling requires a very accurate set of data provided from
either reduced scale model measurements or advanced unsteady twophase flow analysis.
EPFL  Laboratoire de Machines Hydrauliques
235
H/Ho
N/Nn
[]
T/Tn
Pumpturbine U2
y
Q/Qn
Pumpturbine U4
N/Nn
T/Tn
Pumpturbine U4
hSP4
hDT4
Generator U4
P/Pn
H/Ho
Q/Qn
Rated reactive power []
[]
Q/Qn
Figure 10.7: Simulation results of the transient of the pumpturbine of Unit 2 and Unit
4.
EPFL  Laboratoire de Machines Hydrauliques
hDT4
C=dV/dH
Cavity compliance [m2]
Cavitation number []
hDT4
C=dV/dH
hDT4
C=dV/dH
C=dV/dH
hDT4
236
CHAPTER 10. MODELLING OF OVER LOAD PRESSURE
PULSATIONS ON A PROTOTYPE
Figure 10.8: Simulation results of the draft tube pressure oscillations of Unit 4 for the
rope parameters of table 10.2.
Chapter 11
Modelling of RotorStator
Interactions on a Scale Model
11.1
General Remarks
This section aims to present the numerical simulation of the hydroacoustic part of the
RSI phenomenon based on a onedimensional hydroacoustic model. Therefore the case
of 20 guide vanes and 9 impeller blades high head Francis pumpturbine is investigated.
First the RSI patterns of the pumpturbine are described. Then, the determination of
the pumpturbine hydroacoustic parameters is described. The results obtained by the
simulations in time domain indicates the RSI patterns of the pumpturbine of interest.
A parametric study is presented and the influence of blade thickness, wave speed and
rotational speed is investigated.
11.2
The flow field leaving the guide vane of a Francis pumpturbine in generating mode is
characterized by the velocity defect caused by the guide vanes. The pressure field attached
to the rotating impeller blade induces also incoming flow field distortions. No matter how
complex these two periodic flow fields are they can be expressed as Fourier series. Then,
both the stationary and rotating pressure fields can be expressed as:
ps (r , t) =
Bn cos (n z0 r + n )
(11.1)
Bm cos (m zb r + m )
(11.2)
n=1
pr (r , t) =
X
m=1
The resulting velocity field is characterized by a strong modulation process as illustrated in figure 11.1. The pressure in the area of the vaneless gap between the guide
vanes and impeller blades can therefore be expressed as the product of both rotating and
stationary fields of pressure leading to the summation of every pmn component:
pmn (, t) = Amn cos (n zo s + n ) cos (m zb r + m )
EPFL  Laboratoire de Machines Hydrauliques
(11.3)
pmn (s , t) =
(11.4)
Moreover, the impeller angle coordinate is related to the stationary frame of reference
as r = s t, then the pressure field in the stationary coordinates becomes:
Amn
cos (m zb t (m zb n zo ) s + n m )
2
Amn
cos (m zb t (m zb + n zo ) s n m )
+
2
pmn (s , t) =
(11.5)
Figure 11.1: Modulation process between impeller blade flow field and guide vanes flow
field.
This equation describes the RSI pressure field in the vaneless gap which is function of
time and space [57]. This pressure field represents 2 diametrical pressure modes having
the following numbers of minima and maxima :
k1 = m zb n zo
k2 = m zb + n zo
(11.6)
rotating with the respective spinning speed in the stationary frame of reference.
1 = m zb b /k1
2 = m zb b /k2
(11.7)
Furthermore, the sign of the diametrical mode numbers k1 and k2 indicates that the
diametrical mode is rotating in the same direction as the impeller when positive and
counterrotating when negative. It is also important to note that lower amplitudes are
expected for higher k values, because of the high harmonic number. As a result, k2 is
usually not relevant. The figure 11.2 presents an illustration of the meaning of the k
values.
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239
11.3
The case investigated is a pumpturbine scale model installed on the PF3 test rig of
the EPFL Laboratory for Hydraulic Machines in the framework of the research project
Hydrodyna. The scale model of this pumpturbine is presented in figure 11.3 and the
rated efficiency hill chart is presented in figure 11.4. This pumpturbine features a 20
guide vane and 9 impeller blade rotorstator arrangement. The main characteristics of
this pumpturbine are summarized in table 11.1.
zo
zb
Qn
Hn
Nn
3
[] [] [] [m /s] [m] [rpm]
0.17 20
9
0.23
45
980
The pumpturbine has a specific speed of = 0.17, 20 guide vanes, 9 impeller blades
EPFL  Laboratoire de Machines Hydrauliques
/BEP
VO
/BEP
/BEP
Figure 11.4: Rated efficiency hill chart /BEP = /BEP (/BEP , /BEP ) of the pumpturbine.
and an outlet diameter, in turbine mode, of D1e = 400 mm. According to this rotorstator
blade arrangement the RSI patterns of this pumpturbine are determined analytically
using relations 11.5, 11.6 and 11.7, see table 11.2.
Table 11.2: RSI patterns of the pumpturbine (zo = 20; zb = 9).
Stationary Frame
Rotating frame
n m k1 k2 1 /b 2 /b f /fb 1 /b 2 /b f /fb
1 2 2 38
9.0
0.5
18
10.0
0.5
20
1 3 7 47
3.9
0.6
27
2.9
0.4
20
2 4 4 76
9.0
0.5
36
10.0
0.5
40
2 5 5 85
9.0
0.5
45
8.0
0.5
40
The table 11.2 points out the 4 diametrical rotating modes expected to present the
most significant amplitudes. These diametrical modes are k1 = 2, 7, 5, 4 with the
related frequencies in the stationary frame f /fb = m zb = 18, 27, 36 and 45. In the
rotating frame of reference these frequencies give f /fb = m zo = 20 for the first 2 modes
and f /fbo = 40 for the second 2 modes.
241
11.4
11.4.1
The plane view of the 20 guide vanes and 9 blades Francis pumpturbine scale model of
interest is presented in figure 11.5 left. The related hydroacoustic model of this pumpturbine is made up out of a pipe network, see figure 11.5 right.
SP11
SP10
SP12
SP9
SP13
D1
D9
D12
D11
D10
SP8
D8
R5
D7
R4
D1
R6
SP7
R7
SP6
D6
D15
SP15
R3
R8
SP5
SP14
D5
D16
SP16
R2
R1
R9
D17
D4
D1
8
SP17
D3
SP4
D1
D2
9
D1
D20
SP3
SP19
SP2
SP18
SP1
B0
A1
Figure 11.5: Plane view of the pumpturbine (left) and corresponding hydroacoustic model
(right).
The hydroacoustic model is made of 20 pipes for the guide vanes (pipes D1 to D20),
9 pipes for the impeller (pipes R1 to R9) as well as 19 pipes for the spiral casing (pipes
SP1 to SP19). The first part of the spiral casing between the turbine inlet and the guide
vane N1 is modelled by the pipe B0. The diffuser of the pumpturbine is modelled by the
pipe A1. The energy transfer through the impeller is modelled by the pressure source
VS1 for which the head is a function of the discharge H = H(Q) according to the slope
of the pumpturbine characteristics linearized around the operating point of interest, see
figure 9.9. The connection between the stationary part and the rotating part is achieved
through 180 valves connecting each guide vane to each impeller vane. The 180 valves
are controlled by the flow distribution between the stationary part and the rotating part
according to the impeller angular position. The valve head loss is calculated to ensure
the idealized discharge evolution presented in figure 11.6.
Assuming a constant impeller angular speed, the discharge evolution between one guide
vane and one impeller vane is function of the connection area between them. During the
rotation of the impeller 4 phases are identified:
phase 1: the impeller blade starts passing in front of the first blade of a guide vane
channel, the discharge between the guide vane channel and the impeller vane channel
EPFL  Laboratoire de Machines Hydrauliques
es
er
er
es
es
er
er
Q[m3/s]
t=q/Wn [s]
1
Figure 11.6: Idealized discharge evolution between an impeller vane and a guide vane as
function of the spatial coordinate of the impeller .
increases linearly according to the connection area increase until the impeller blade
reaches the second blade of the guide vane;
phase 2: the discharge between the guide vane and the impeller vane channels
remains constant until the second impeller blade reaches the first guide vane, the
connection area being constant;
phase 3: the discharge between the guide vane channel and the impeller vane channel
decreases linearly according to the connection area decrease;
phase 4: the discharge between the guide vane channel and the impeller vane channel
is kept to zero as the connection area is zero until the phase 1 is reached again.
The discharge evolution described above acts like a sliding slot between the 20 guide
vanes and one impeller vane. As a result there are 9 slots for the full pumpturbine,
one for each impeller vane. Each slot angle being shifted by 2 /zb . The discharge law
can be modified in order to take into account the thickness of both the guide vanes and
impeller blades, eo and eb respectively. The thickness can be expressed as equivalent angle
measured in degrees at the vaneless gap radius. For the first part of the investigations, the
thickness of the impeller blades is taken arbitrarily equal to 4, the real one being 3. The
consideration of the blades thickness induces discontinuity in the overall discharge law.
It means that the point 3 of the discharge law of an impeller blade does not correspond
to the point 2 of the following impeller blade but is shifted of the value of eb + eo . The
lack of discharge between two consecutive impeller blades is the source of the excitation
mechanism of this RSI model.
11.4.2
For the determination of the RLC terms of the hydroacoutic model of a pipe, the following
values must be determined for each pipe: (i) the length l; (ii) the cross section area A;
(iii) the friction coefficient ; (iv) the wave speed a.
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243
The determination of the length and of the cross section is done using the structural
characteristics of the pumpturbine scale model. The friction coefficient of the pipes are
evaluated to be = 0.02 for all the pipes. The wave speed of a pipe is given by:
1
a2 =
1
Efluid
1 A
A p
(11.8)
Where Ef luid is the bulk modulus of the water and A/(A p) is the rated area
increase due to pressure increase. This term can be determined analytically for circular
pipes but should be estimated for pipes having complex cross sections. In the case of the
pumpturbine, the wave speed of the spiral casing and of the distributor are estimated
using Finite Element Method, FEM, calculations. The determination of the wave speed
of a given cross section of the spiral casing is carried out in two steps as presented in
figure 11.7: (i) the distributor channel area increase due to a constant pressure in the
distributor channel only is determined; (ii) the spiral casing area increase due to constant
pressure in the spiral casing only is determined.
The FEM analysis is carried out with the commercial code ANSYST M . First, the stay
vanes and the assembling bolts are considered individually to determine their stiffness.
Then, by assuming that each spiral casing part of concern is axisymmetrical, the area
increase is determined by including the stiffness of the stay vanes and of the bolts previously determined. The number of cells in the radial direction of the spiral casing section
was set to 3 cells of mesh. This value was determined by comparison of circular pipe
wave speed determined by both FEM and analytical calculation. The difference for such
geometry was found to be 0.1%. The mechanical properties of the materials of the spiral
casing and distributor channels are summarized table 11.3.
Figure 11.7: Determination of the wave speed in the distributor channel (left) and in the
spiral casing (right).
The spiral casing material is a composite made of Epoxy resin matrix and fiberglass.
The equivalent Young modulus is calculated according to the volume fraction of both
components of the composite and by taking into account the main stress direction corresponding to the uniform loading of the hydrostatic pressure. Therefore, the fiberglass
EPFL  Laboratoire de Machines Hydrauliques
For the 4 cross sections presented in figure 11.8, the corresponding values of the wave
speed are calculated, and reported in figure 11.9. As expected, due to a constant spiral
casing wall thickness, the wave speed is increasing from the inlet up to the tongue as the
cross section decreases. For the determination of the wave speed of the 19 elements of the
spiral casing model a 2nd order polynomial interpolation is used, see figure 11.9.
Section1
Section2
Section 2
Section 1
Section 3
Section3
Section4
Section 4
Figure 11.8: FEM model of 4 different cross sections of the spiral casing.
The wave speed of the distributor channels is found to be 800 m/s with small deviation
between the 4 sections. The same value is chosen for the wave speed of the impeller blades.
11.5
The hydroacoustic model of the test rig includes the pumpturbine scale model, the 2
feeding pumps in series, the piping system and a downstream tank as presented in figure
11.10. The data related to the model of the test rig are given in table 9.2. The chosen operating condition for the simulation of the hydracoustic behavior of the test rig
corresponds to the nominal operating point in turbine mode with no cavitation. Each
pipe of the spiral casing is modelled with 1 element, and each guide vane and impeller
vane is modelled with 3 elements. This spatial discretization ensures that the confidence
EPFL  Laboratoire de Machines Hydrauliques
245
1200
1160
1120
1080
1040
1000
/2
[rd]
3/2
Figure 11.9: Wave speed evolution along the angular position of the spiral casing, calculated points (dot) and fitted line (solid line).
threshold of equation 3.91 is fulfilled with an error below 1%. In agreement with the CFL
criteria, an integration time step corresponding to 0.5 of impeller rotation is selected.
After convergence of the simulation to a periodic behavior, the simulation is continued
for 10 impeller revolutions.
The resulting pressure fluctuations in the spiral casing and in the guide vanes close to
the vaneless gap are properly non dimensionalized and presented in a waterfall diagram
as a function of the dimensionless frequency f /fb and the angular coordinate , see figure
11.11. In these diagrams the angular origin is taken at the spiral casing inlet and therefore
= 360 corresponds to the tongue.
The pressure fluctuation in the vaneless gap, figure 11.11 right, presents significant
amplitudes (less than 0.5% of the nominal head) for the expected frequencies f /fb =
9, 18, 27, 36, 45, etc. The analysis of the phase, not presented here, of these pressure
fluctuations shows clearly the diametrical pattern of the pressure mode shapes with for
example 2 minima and 2 maxima, i.e. 4 phase shift, for the frequency of f /fb = 18.
The pressure fluctuations in the spiral casing, of figure 11.11 left, indicates a standing
wave for the frequency f /fb = 18 with a wavelength of approximately 4/3 of the spiral
casing length. A pressure node is located at = 150while amplitude maxima take place
at = 20 and = 230. This standing wave influences the amplitude of the diametrical
mode in the vaneless gap at the frequency f /fb = 18, see figure 11.11 right.
The visualization of the different diametrical modes and related possible standing
waves in the spiral casing is carried out by filtering the time signal of the pressure fluctuations of each spatial node in the pumpturbine using a pass band filter around the
frequency of interest. For example, for k1 = 2, in the stationary frame it corresponds
to f /fb = 18 while for the rotating frame it corresponds to f /fb = 20. The figures 11.12
to 11.15 presents the resulting pressure fluctuation distributions in the entire machine
EPFL  Laboratoire de Machines Hydrauliques
Pumpturbine
Tank
Feeding pumps
Pump turbine
Connecting pipe
Air vessel
Centrifugal pumps
Figure 11.10: EPFL test rig PF3 including the pumpturbine scale model (top) and
corresponding hydroacoustic model (bottom).
Figure 11.11: Waterfall diagram of the pressure fluctuations in the spiral casing (left) and
in the guide vanes close to the vaneless gap (right).
247
respectively for k1 = 2, +7, 4 and +5. The representation is done for the initial time,
and for a 1/4 period later. The pressure amplitude of every node of the pumpturbine is
represented on the z axis for the spatial position [x, y] of the node. The bold solid line is
the connection between each last node of the guide vane and points out the diametrical
mode shape pattern expected with the related number of maxima/minima. The diametrical mode rotates in the opposite direction of the impeller rotation for k1 = 2 and 4 and
in the same direction as the impeller for k1 = +7 and +5 as predicted by the analytical
analysis above. For all the cases, the predicted number of minima and maxima also fits
with the predictions. However, it can be noticed that the k1 = +7 diametrical mode is
the last resolvable mode with a resolution of 20 pressure signals. As already presented in
figure 11.11 left, there is no standing wave in the spiral casing for the cases k1 = +7, 4
and +5.
For k1 = 2, the diametrical mode rotates with a spinning speed of 1 /b = 9 and
features 2 maxima. It means that using a pressure transducer in the vaneless gap, the frequency of f /fb = 18 is measured. The representation clearly points out the standing wave
in the spiral case with the related positions of pressure nodes and maxima. The interaction between the standing wave and the diametrical mode are clearly pointed out through
the pressure in the guide vanes where pressure fluctuations have similar amplitudes.
From these results it can be concluded that the hydroacoustic model of the pumpturbine enables to obtain and visualize the RSI pressure fluctuation patterns. Standing
wave and diametrical modes are pointed out and corresponding spinning speeds are properly obtained too. The high amplitude of the standing wave in the spiral casing for
f /fb = 18 is probably related to a natural frequency of the hydraulic system which is
close. However, it is important to notice that the rotating impeller corresponds to an
unsteady boundary condition at the end of the guide vanes. This makes it difficult to do
modal analysis using linearized approaches. One of the major advantages of simulating
the dynamic behavior of such a system in the time domain is that it offers the possibility to take into account such nonlinearities. The modelling based on a valve network
driven according to the flow distribution between stationary and rotating parts appears
to be efficient for the simulation of the hydroacoustic part of the RSI phenomenon. This
approach has the advantage to provide also the pressure fluctuations due to RSI in the
rotating impeller.
11.6
Parametric Study
k=2
t=0
y coordinates [mm]
x coordinates [mm]
t = T/4
y coordinates [mm]
x coordinates [mm]
Figure 11.12: Pressure fluctuation patterns in the pumpturbine for k1 = 2, f /fb = 18,
for t = 0 (top) and t = T /4 (bottom).
249
k=4
t=0
t = T/4
Figure 11.13: Pressure fluctuation patterns in the pumpturbine for k1 = 4, f /fb = 36,
for t = 0 (top) and t = T /4 (bottom).
k=+5
t=0
t = T/4
Figure 11.14: Pressure fluctuation patterns in the pumpturbine for k1 = +5, f /fb = 45,
for t = 0 (top) and t = T /4 (bottom).
251
k=+7
t=0
t = T/4
Figure 11.15: Pressure fluctuation patterns in the pumpturbine for k1 = +7, f /fb = 27,
for t = 0 (top) and t = T /4 (bottom).
11.6.1
The model of the excitation is based on the flow distribution between the stationary
part and the rotating parts. The resulting discharge law between the 20 guide vanes and
the 9 impeller blades can account for the guide vane and impeller blade thickness. The
equivalent thickness of these blades affects the evolution of the connection area between a
guide vane and an impeller vane. The figure 11.16 presents the 4 possible configurations
which are : (A) no blade thickness, (B) consideration of the guide vane blade thickness
only, (C) consideration of the impeller blade thickness only and (D) consideration of both
guide vane and impeller blade thickness. The 4 different evolutions of the discharge law of
the configurations (A), (B), (C) and (D) are represented on the right hand side of figure
11.16.
Q[m3/s]
Q[m3/s]
B
3
Q[m3/s]
C
1
Q[m3/s]
D
1
Figure 11.16: Discharge law resulting from the guide vane and impeller blade thicknesses
consideration.
Considering the configurations (B), (C) and (D) 3 series of simulations are performed
considering different blade thicknesses eb . The resulting amplitudes of the pressure fluctuations in the vaneless gap are presented in figure 11.17. Configurations (B) and (C)
provide similar pressure amplitudes in the vaneless gap for identical thicknesses. The
combination of both thicknesses, configuration (D), induces much higher pressure amplitudes. These results show that the higher the blockage effects due to blade thickness, the
higher the pressure amplitudes in the vaneless gap.
A dynamic pressure amplification factor is used to evaluate the influence between
the vaneless gap pressure amplitudes and the amplitudes in the spiral casing. It enables
separating the influence of any parameter on the RSI excitation and on the dynamic
response of the system. The amplification factor is defined as the ratio between the
pressure at any point in the spiral case and the average pressure in the vaneless gap and
is expressed as:
DA =
pspiral
pvaneless gap
(11.9)
0.04
[]
253
p/(E)
D
0.03
0.02
0.01
C
eb
0
6 []
Figure 11.17: Pressure fluctuation amplitudes in the vaneless gap for the blade thickness
arrangements of figure 11.16 for f /fb = 18.
The figure 11.18 presents the dynamic amplification factor for 3 different guide vane
thicknesses. It appears that even if the absolute amplitudes of the pressure fluctuations
increase as the excitation in the vaneless gap increases, the dynamic amplification factor
remains constant for the 3 tested thicknesses. One interesting consequence of this result
is that even if the excitation parameters are arbitrary, the dynamic amplification will
be meaningful. It also means that if the pressure fluctuations can be measured in the
vaneless gap during the scale model tests, it would be possible to predict the amplitude
of pressure fluctuations in the spiral case.
2.0
0.020
1.8
1.6
0.015
DA []
1.4
1.2
0.010
1.0
0.005
0.4
0.2
0
18
36
54
72
90 108 126 144 162 180 198 216 234 252 270 288 306 324 342
0.000
Figure 11.18: Influence of the distributor blade thickness on the dynamic amplification
between spiral case and vaneless gap for f /fb = 18.
11.6.2
The influence of the impeller vane wave speed is investigated by simulating the dynamic
behavior of the system considering three different values for this parameter. The chosen
values for the impeller wave speed are 700 m/s, 800 m/s and 900 m/s. The figure
11.19 presents the dynamic amplification factor DA for the 3 different impeller vane wave
speeds. It is clearly shown that the influence of the impeller vane wave speed is negligible.
It means that for pumpturbines the combined effect of small vaneless gap and the impeller
rotation acts like a dead end for such a frequency. However, this is right only as long as
the diffuser remains short compared to the system dimensions.
2.0
1.8
1.6
DA []
1.4
1.2
1.0
0.8
0.6
a=700 [m/s]
a*=800 [m/s]
a=900 [m/s]
0.4
0.2
18
36
54
72
90 108 126 144 162 180 198 216 234 252 270 288 306 324 342
Figure 11.19: Influence of the impeller vane wave speed on the dynamic amplification
between spiral case and vaneless gap for f /fb = 18.
11.6.3
Four different values of the guide vane wave speed are chosen to investigate the influence
of this parameter on the dynamic amplification between vaneless gap and spiral casing.
The chosen values for the guide vane wave speed are 700 m/s, 800 m/s, 900 m/s and
1000 m/s. The resulting dynamic amplification DA and the mean value of the vaneless
gap pressure fluctuation are presented in figure 11.20. The mean pressure fluctuation in
the vaneless gap remains constant for the 4 cases. However, the dynamic amplification
is strongly influenced by the guide vane wave speed. This is related to the fact that
travelling time between vaneless gap and spiral casing affects strongly natural frequencies
of the hydraulic system. It seems that, in this case, reducing the guide vane wave speed
makes a natural frequency of the hydraulic system closer to f /fb = 18.
EPFL  Laboratoire de Machines Hydrauliques
255
3.0
a=700 [m/s]
*a=800 [m/s]
a=900 [m/s]
a=1000 [m/s]
2.5
2.0
DA []
1.5
1.0
0.5
18
36
54
72
700[m/s]
800[m/s]
900[m/s]
1000[m/s]
90 108 126 144 162 180 198 216 234 252 270 288 306 324 342
Figure 11.20: Influence of the guide vane wave speed on the dynamic amplification between
spiral case and vaneless gap for f /fb = 18.
11.6.4
Four different impeller rotation speeds are chosen to investigate the influence of this
parameter. The chosen values corresponds to f = 18 fb = 292, 296, 300, 304 Hz. The
resulting dynamic amplification DA and the mean value of the vaneless gap pressure
fluctuations are presented in figure 11.21. The impeller rotating speed strongly influences
the dynamic amplification between the vaneless gap and the spiral case. This is due to
the change of excitation frequency, which seems to become closer to a natural frequency
of the hydraulic system when the rotating frequency decreases. This result is coherent
with the influence of the guide vane wave speed. In one case, the natural frequency of
the hydraulic system increases with the increase of the guide vane wave speed, and in
the second case the excitation frequency is reduced until it matches the hydraulic natural
frequency. However, in both cases the amplitudes of the pressure fluctuations in the
vaneless gap remain constant.
11.7
Concluding Remarks
2.0
1.8
18*n=292 [Hz]
18*n=296 [Hz]
18*n=300 [Hz]
18*n=304 [Hz]
1.6
DA []
1.4
1.2
0.008
1.0
0.8
0.6
0.4
0.2
0
18
36
54
72
90 108 126 144 162 180 198 216 234 252 270 288 306 324 342
Figure 11.21: Influence of the impeller rotating speed on the dynamic amplification between spiral case and vaneless gap for f /fb = 18.
The resulting RSI model comprises a network of 180 valves connecting each guide
vane with each impeller vane. The valve opening functions are driven according to the
flow distribution between the stationary and the rotating parts. The analysis of the
pressure fluctuations resulting from the rotorstator excitation shows that RSI patterns
like rotating diametrical pressure modes and standing waves are properly simulated. The
influence of the standing wave on the diametrical pressure mode is pointed out. The
chosen excitation model appears to be realistic for the investigation of the hydroacoustic
part of the RSI.
Moreover, a parametric study investigating the influence of blade thickness, impeller
blades and guide vane wave speeds and impeller angular speed is carried out. It results
that the wave speed in the impeller blades has not significant influence. The influence
of the guide vane wave speed and of the impeller rotating frequency tends to show that
the amplitude of the standing wave in the spiral casing for the frequency f /fb = 18 is
strongly affected by these parameters because of the closeness of a natural frequency of
the hydraulic system.
Overall, this model gives satisfactory qualitative results and needs to be validated experimentally. However, if the dynamic amplification gives reliable trends, the determination of the absolute pressure fluctuation amplitudes is more challenging. The excitation
model is based on the flow distribution between the stationary parts and the rotating
parts of the machine. Its parameters are only related to connection area between guide
vanes and impeller blades evolution due to impeller rotation. To obtain a more reliable
quantitative result, the excitation model should be improved and be able to take into
consideration all the parameters affecting RSI like the gap amplitude, the wake angle of
the impeller blade, the blades thickness, etc. This could be achieved through the coupling
of a onedimensional code for the modeling of the hydraulic system with a CFD tool for
the modeling of the RSI excitation. This approach seems realistic regarding the negligible
influence of the impeller hydroacoustic parameters.
Chapter 12
Prediction of Pressure Fluctuations
on Prototype
12.1
General
Direct computation of the prototype resonance due to vortex rope excitation using CFD
computation requires taking into account fluid compressibility, wall deflections, cavitation
modelling, and the geometry of the entire power plant. Such an approach is not yet feasible
with current computation performance and models. Therefore, there is still a large place
for prediction methods based either on scale models to prototype transposition or on
identification based modelling.
12.2
12.2.1
The study of the flow in a prototype using scale model requires to ensure the same velocity
distribution at both the inlet and the outlet of the turbine. Therefore, the geometries
should be homologous and the velocity triangles should be identical. This condition is
fulfilled if the discharge and energy coefficient are identical on both machines. Indeed,
these coefficients define the ratio between the peripheral velocity
U , the discharge velocity
Cm and the absolute velocity taken here arbitrarily as C = 2 E and can be expressed
as follows:
=
Cm
Q
=
3
U
Rref
2E
C2
= 2
2
U
Rref 2
(12.1)
Then the transposition of the internal and external forces acting on the fluid can be
investigated through the NavierStokes equations which can be written in the compact
derivative form as follows:
DC
Dt
{z }
inertia forces
1
= p +
f
+  {z2 C}
{z}
viscous forces
 {z }
body forces
pressure forces
(12.2)
258
Expressing the particle derivative and rearranging the above equation gives:
C
+ ( C ) C = ( + g Z) + 2 C
t
(12.3)
For expressing the NavierStokes equations in the dimensionless form, let introduce
the dimensionless values:
C
Cref
0
= Lref ; t = t
; C =
(12.4)
Lref
Cref
2
Introducing equations 12.4 in equation 12.3 and multiplying by Lref /(Cref
) yields:
0
0
0
C0
p pref g Lref Z Zref
(
)
+ ( C ) C =
+
+
02 C 0
2
2
0
t
Cref
Cref
Lref
Cref
Lref {z
 {z
}
}  {z
}
1/F r 2
Eu
1/Re
(12.5)
From the dimensionless NavierStokes equation 12.5 the following dimensionless numbers are derived:
Euler number: Eu =
ppref
,
2
Cref
C
Froude number: F r = ref , ratio between the inertia and gravity forces;
gLref
Reynolds number: Re =
Cref Lref
,
In addition, one can also express 2 others useful dimensionless numbers that are:
the Mach number M a = C
dp/d
C
:
a
pressure forces;
the Strouhal number St =
Cref t
:
Lref
Fulfilling all the similitude laws leads to preform full scale experiments. Consequently
only some of them can be satisfied during scale model tests and have to be chosen carefully
according to the focus of the experiments.
12.2.2
Similitude of Cavitation
Cavitation problems require fulfilling the Thoma and Froude similitudes in addition to
the kinematic similitude. The first similitude ensures having the same pressure at a
given point with respect to a reference pressure, while the second ensures having the
same pressure gradient p/l along the curvilinear abscissa of the machine. In addition,
the number of cavitation germs in the water, i.e. the number of nuclei by m3 should
be identical [73]. However, while the Thoma and the nuclei density similitudes can be
EPFL  Laboratoire de Machines Hydrauliques
259
achieved on the test rig, it is more problematic to achieve the Froude similitude. The
Froude similitude leads to the following statement between model and prototype:
E
E
=
(12.6)
Lref M
Lref P
The smaller dimension of the scale model results in smaller test specific energy which
combined with the Thoma similitude requires upstream pressure below atmospheric pressure. In addition, amplitudes of the pressure fluctuations become comparable to the
pressure noise of the test rig. Consequently, the Froude similitude is very difficult to
achieve for high specific speed Francis turbines.
12.2.3
The similitude of pressure pulsation requires satisfying both Strouhal and Euler similitudes [80]. These 2 statements provide the frequency and amplitudes of the pressure
pulsation from the following equations:
fpuls
fpuls
p
p
=
and
=
(12.7)
n M
n P
E M
E P
However, the 2 above transposition rules are true only if interactions with the hydraulic
system are small. If interactions occur, the impedance similitude should also be satisfied.
12.2.4
Similitude of Impedances
12.2.5
The aim of the transposition method is to have direct similitude between pressure pulsations measured on scale model and those measured on prototype. The difficulty arises
from the cavitation vortex rope similitude. The similitude of impedance shows that the
EPFL  Laboratoire de Machines Hydrauliques
260
wave speed must be in the ratio of the square root of the specific energies E. However,
if this condition can be fulfilled for adduction and tailrace tunnel, it is more complex for
the draft tube itself whose wave speed is strongly driven by the diameter distribution of
the vortex rope, i.e. the Froude similitude, the wall deflection and also the dissolved air.
Consequently it is not possible to adjust the wave speed on the model, and particularly
because the wave speed on prototype is unknown.
Fischer et al. [52] also concluded from model tests, that model tests are representative
as long as there is no interaction between the turbine and the hydraulic circuit on both
sides: on the test rig and on the prototype. Nevertheless, the prediction of resonance on
the prototype is precisely the goal.
Another approach for predicting the pressure pulsation on the prototype is to set
up a hydroacoustic model of the prototype installation, and to use different identification
methods, experimental or numerical, to determine the parameters of the model. Regarding
the experimental approaches, Dorfler [40], Jacob [82] and also Fritsch and Maria [58]
proposed solutions for the identification of vortex rope compliance and excitation source.
A similar approach is proposed in the following section.
12.3
12.3.1
General
12.3.2
261
KU(s)
+ Uc
Unit 4
Infinite grid
50 Hz
500kV
KN(s)
Surge tank
Gallery
Unit 3
Pe
n
sto
ck
Unit 2
Unit 1
262
a=a(Vrope) HRope(t)
a=a(Vrope)
Figure 12.2: Model of the draft tube for resonance risk assessment.
tube wave speed to an equivalent rope diameter, resembling the approach developed by
Philibert and Couston, [120], in order to link the wave speed to a physical dimension.
Assuming a cross section of the draft tube with diameter D and with a rope diameter
DR , one can determine the gas volume fraction of the cross section as follows:
ARope
=
=
Atot
DR
D
2
(12.10)
Where Arope is the cross section of the rope and A is the total draft tube cross section
for a given curvilinear abscissa. The wave speed in the liquid gas mixture is given by
Wallis (1969) [152]:
a2m =
1
m
g a2g
(1)
L a2L
(12.11)
The wave speed of the liquid gas mixture is represented as a function of the gas
volume fraction, see figure 12.3 left, and as function of the cavitation rope rated diameter
by combining equations 12.10 and 12.11, see figure 12.3 right. The wave speed of the
mixture drops to very low values with increasing cavitation rope diameter. Cavitating
rope diameters up to D/DR = 0.1 are common in part load operation of Francis turbines,
see Jacob [79]. For such rated diameters, the draft tube cross section wave speed would
be below 100 m/s. However, it should be noticed that this model assumes a cylindrical
vortex rope with constant diameter, from the draft tube inlet to outlet, whereas the real
vortex rope is helicoidal and conical. Therefore, the presented approach is only useful to
link the wave speed to a physical dimension. In addition, thermodynamic effects related
to the cavitation phenomenon are neglected.
12.3.3
The main natural frequencies of the piping system feeding the 4 turbines of the power plant
can be estimated by means of the analysis of the natural frequencies of an equivalent pipe
EPFL  Laboratoire de Machines Hydrauliques
10000
263
1200
1000
1000
100
800
600
400
200
10
0.2
0.4
0.6
0.8
0.02
0.04
0.06
0.08
0.1
Figure 12.3: Wave speed in vapor/water mixture (left) and wave speed as function of
rated diameter (right).
of the adduction system. Because of the longitudinal symmetry of the piping, a simplified
model of the piping can be used. The simplified model, presented in figure 12.4, comprises
2 pipes: the adduction, and the draft tube. Thus the influence of the draft tube wave
speed change on the natural frequencies can be qualitatively investigated.
L1,
L1+L2, a
a1
Turbine
f1
L2, a2
f2
L1 + L2
L1
+ La22
a1
(12.12)
Considering both upstream and downstream free surface boundary conditions of the
EPFL  Laboratoire de Machines Hydrauliques
264
piping, the equivalent wavelength of the eigen modes of the piping are given by:
i =
2
(L1 + L2 )
i
(12.13)
a
i
=
i
2 La11 +
L2
a2
(12.14)
The 10 first natural frequencies of the simplified piping of figure 12.4 are computed
for wave speeds in the draft tube ranging from a = 1200 m/s to a = 50 m/s. The
lengths and wave speeds of the simplified model are summarized in the table 12.1. The
natural frequencies obtained for different rated cavitating rope diameters are presented
in figure 12.5. As expected, the natural frequencies of the piping system decrease with
decreasing the wave speed. The natural frequency of the generator fgenerator = 1.21 Hz
is also represented in figure 12.5. The graph indicates an intersection between the 4th
piping natural frequency and the generator natural frequency for a draft tube wave speed
of a = 77 m/s. This intersection is in the range where pressure pulsations induced by the
cavitating vortex rope extending from 0.2 to 0.4 times the turbine rotating frequency, n,
are expected. This situation corresponds to one of the worst cases for the power plant,
because there is a coincidence of the piping natural frequencies and the generator natural
frequency. However, this requires that the pressure pulsation induced by the draft tube
flow matches this frequency. In addition, the influence on the electrical power fluctuation
is strongly dependant on the turbine position, relatively to the pressure mode shape
corresponding to this 4th natural frequency. Nevertheless, this simplified model shows the
importance of investigating carefully the resonance risk for the power plant of interest.
12.3.4
0.3
1.2
265
0.2
0.1
a = 77 m/s
Frequency f / n []
1.0
0.8
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
fo gen.
D/DR
0.6
0.4
0.2
400
800
1200
Figure 12.5: Power plant natural frequencies estimation with simplified model (n =
5.555Hz).
Generator f=1.21 Hz
Mechaincal
masses 7.5 Hz
f/n
Figure 12.6: PRBS mechanical torque excitation of the generator (left) and resulting
generator transfer function between mechanical and electromagnetic torque (right).
266
Piping Resonance
The full hydroelectric simulation model is taken into account to calculate the transfer
function between the draft tube pressure source excitation and the electromagnetic torque.
A PRBS draft tube pressure source excitation in the draft tube of the turbine of Unit 4 is
considered for the time domain simulation. For this simulation the turbine speed governor
is removed and the guide vanes are kept with constant opening of 40%, corresponding to
65% of the nominal discharge.
The resulting pressure oscillations in the piping system are represented as a waterfall
diagram for draft tube wave speed values of: 200, 100, 77, and 50 m/s in figure 12.7.
The pressure pulsations are represented as a function of the x coordinate starting from
the surge tank, node 1, and extending along the piping until the Unit4 downstream tank,
node 289, and the rated frequency f /n. It can be noticed that not all piping natural
frequencies are excited; this is due to the draft tube excitation source relative position in
the piping. It is pointed out that the value of natural frequencies drops with decreasing
the draft tube wave speed. In addition, because the wave speed of the draft tube affects
the pressure node and maxima location, these are not the same mode shapes that are
excited for the different wave speeds. Regarding the frequency range of interest, 0.2 to
0.4 n, it can be seen that the lower the draft tube wave speed, the higher the number of
natural frequencies in the frequency range of interest. Comparing these results with the
simplified model it can be seen that the natural frequency at 77 m/s that was pointed
out as critical is not excited by the draft tube pressure source and does not appear on the
waterfall diagram, demonstrating the limitation of such simplified models.
Hydroelectric Resonance
The influence of the draft tube excitation on the mechanical and electromagnetic torques
is evaluated by calculating the transfer function between the draft tube pressure excitation
and both mechanical and electromagnetic torques. The 2 transfer functions are presented
in figure 12.8 left and right respectively. Mechanical torque oscillations occur for all excited
piping mode shapes. Moreover, the electromagnetic torque pulsations are strongly affected
by the generator natural frequency. A clear amplification effect between the mechanical
and the electromagnetic torque appears for a draft tube wave speed of 50 m/s, where
there is coincidence of a piping natural frequency and generator natural frequency. Thus,
the worst conditions for the power plant is when the draft tube excitation is at 0.217
n with a draft tube wave speed of 50 m/s. However, for this draft tube wave speed,
electromagnetic torque pulsations are expected in almost the whole vortex rope frequency
range due to the presence of 2 piping natural frequencies in this range, i.e. 0.217 n and
0.37 n.
Comparison with Experimental Data
Once the piping resonance frequency and mode shape as well as the generators resonance
frequency are known for different wave speeds, these results have to be compared with
measurements of draft tube pressure pulsations performed during scale model tests in
order to evaluate the risk of resonance. The model tests, see figure 12.9, provide: (i)
the vortex rope pressure pulsations frequencies and amplitudes [79]; (ii) the vortex rope
EPFL  Laboratoire de Machines Hydrauliques
f/n
f/n
a=200 m/s
267
a=100 m/s
f/n
a=77 m/s
f/n
a=50 m/s
Figure 12.7: Piping pressure oscillations for different draft tube wave speed.
f/n
f/n
Figure 12.8: Transfer function between the draft tube pressure source and the turbine
mechanical torque (left) and transfer function between the draft tube pressure source and
the generator electromagnetic torque (right).
268
diameter by means of photography, or the vortex rope compliance by the method described
by Dorfler [40].
Finally, the possible resonances with the piping system and with the generator are
pointed out from cross checks between parametric simulations results and the model test
data providing the set of resonant conditions. The amplitudes of pressure oscillations in
the system and generator power swing can be estimated from the time domain simulation
with the set of resonant parameters. The overall methodology is summarized in the
synoptic scheme of figure 12.10.
Drope/D
(Jacob, 1993)
= 0.220
BEP
= 0.074
fexct
f/n
269
Simulation model
Prototype installation layout/datasheet
RESONANCE ?
no
ok
yes
Pressure/power oscillations amplitudes
prediction by time domain simulation
for resonant conditions
270
CFD
Compliance identification
1000
Rope Volume V
Rope compliance dV/dh
ppuls
psynchr
100
10
Vrope
0.1
dVrope/d
0.01
0.001
Hydroacoustic
0.04
0.08
0.12
0.16
0.2
HRope(t) a=a(dV/d)
12.3.5
Concluding Remarks
A methodology for the assessment of the risk of resonance of a hydroelectric power plant
operating at part load and subject to draft tube pressure pulsations is presented. This
method is based on a time domain simulation of the dynamic behavior of the whole
hydroelectric power plant considering a white noise draft tube pressure excitation. This
simulation is done for different draft tube wave speeds. The simulation results reveal
the piping natural frequencies that are excited by the draft tube pressure source. In
addition, the transfer function between the draft tube pressure source and the generator
electromagnetic torque points out the risk of electrical power swing. However, the risk can
be really evaluated only knowing the pressure excitations and the draft tube wave speed.
If the former can be obtained from scaled model testing, the latter has to be estimated,
either experimentally from vortex rope photography, or by CFD.
Nevertheless, the presented methodology is a helpful tool for predicting the risk of
resonance at the early stages of predesign or as a help for on site diagnostic purposes.
271
H/Hn
N/Nn
Q/Qn
T/Tn
a)
N/Nn
Rated reactive power
b)
H draft tube inlet
H spiral case outlet
c)
Figure 12.12: Time evolution of the turbine variables a), generator variables b) and head
at the turbine inlet/outlet c) of Unit 4 resulting from full hydroelectric resonance.
272
Conclusions
Chapter 13
Summary, Conclusions &
Perspectives
13.1
Summary
This work is a contribution to the modelling of the dynamic behavior of Francis turbine
hydroelectric power plants. An original modelling approach based on an electrical equivalent scheme of hydraulic conduits is generalized to hydraulic systems. The modelling is
onedimensional and takes into account compressibility effects in order to model propagation phenomena in hydraulic systems: the hydroacoustic modelling. The present work
is composed of two parts, one dealing with the modelling methodology and application to
transient phenomena, and the other one is devoted to the modelling of pressure fluctuation
problems in Francis turbines.
13.1.1
276
Hydroelectric transients
All these models are implemented in the EPFL software SIMSEN developed initially for
the analysis of the dynamic behavior of electrical installations. This hydraulic extension offers the possibility to model an entire hydroelectric power plant comprising the
hydraulic circuit, the electrical installations, the mechanical inertias and the control command systems enabling investigation of hydroelectric transients. Interactions between
hydraulic circuit and electrical installation are shown to be of major concern for turbine
speed governor parameters optimization.
The investigation of hydroelectric transients in islanded power networks also shows the
restriction of speed governor performances regarding the hydraulic layout. Particularly,
power plants featuring long penstocks and small diameter surge tanks require high order modelling to take properly into account waterhammer, mass oscillations and turbine
characteristic nonlinearity effects. A model of an islanded power network constituted of
a thermal power plant and passive consumer load points out the positive stabilization
effect up to 1 Hz of a large power network.
13.1.2
The review of pressure excitation sources associated with the operation of Francis turbines
shows that the vortex rope and rotorstator induced pressure fluctuations are the most
likely to lead to resonance phenomena with the hydraulic circuit. Good modelling was
already obtained in the past for part load vortex rope excitation on scale models [40],
however, improvements have been made in the field of the modelling of the upper part
load and the full load vortex rope as well as for the rotorstator phenomenon.
Upper Part Load Analysis
Upper part load pressure fluctuations were recorded on a Francis turbine scale model
in the framework of the FLINDT project. Onedimensional pressure fluctuations were
shown by Arpe [6] but the origin of these pressure fluctuations was not identified. The
modelling of the test rig based on the experimental determination of the wave speed in
the draft tube shows that the pressure fluctuations measured in the whole draft tube by
Arpe correspond to an eigen frequency of the test rig. As the pressure fluctuations are
of a relatively high frequency, 32.5 Hz, a concentrated compliance model of the vortex
rope is not sufficient. Therefore, the draft tube is modelled with distributed compliance in
terms of variable wave speed along the diffuser. Regarding the modelling of the excitation
source, during experiments, a shock phenomenon of the vortex rope on the wall of the draft
tube occurring at the vortex rope frequency was noticed. The analysis of the frequency
content at the location identified as the pressure source origin, in the inner elbow, revealed
a signal with energy distributed in a wide frequency range especially for the vortex rope
harmonics, up to the 10th order. Therefore the excitation source is simulated with pressure
pulse at the frequency of the vortex rope. The duration and amplitude of these pulses are
determined by error minimization. The result of the time domain simulation shows very
good agreement with the measurements.
However, the role of Froude similitude and cavitation number was not investigated
deeply in the FLINDT project. Therefore, visualizations of the vortex rope with simultaEPFL  Laboratoire de Machines Hydrauliques
13.1. SUMMARY
277
neous pressure records were carried out on a Francis turbine with similar specific speed.
The visualization indicates an elliptical shape of the vortex rope rotating with a frequency
being the half of the resonance frequency. The volume of the cavitation rope appeared to
change according to the pressure evolution. Low cavitation numbers show also the very
complex structure of the rope constituted of multiple vortices. The Froude investigation
demonstrated that the excitation frequency is proportional to the runner frequency and
thus the rope can be associated with a onedimensional excitation source. The additional
study of the effect of Froude and cavitation number influence also shows that in the case
of the FLINDT project, the operating point investigated was very particular point. For
this operating point the shock phenomenon provides energy at harmonics of the vortex
rope whose 7th harmonic coincides with the half of the frequency of the self rotation of the
rope and also with an eigen frequency of the test rig. For such a case, the ratio between
the two excitation phenomena cannot be deduced from available experiments. However,
additional work is necessary to separate the contribution of the influence of the shock
phenomenon of the 1D pressure source in the excitation source model.
Full Load Investigation
The full load surge is investigated in the case of a pumpedstorage plant where strong
pressure fluctuations were recorded for over load conditions. This phenomenon is of the
self excitation type. The stability of the whole system depends on the ratio of the 2
parameters of the vortex rope: the cavitation compliance and the mass flow gain factor.
The stability diagram of the power plant represented as function of these 2 parameters is
calibrated with the measurements and shows the self excited nature of the fluctuations.
A simulation model of the entire power plant comprising the hydraulic circuit, with a
model of the full load cavitation, the electrical installation, the mechanical inertias and the
control system is set up for time domain simulation purposes. The vortex rope compliance
is determined by CFD computation enabling the inclusion of the nonlinear behavior of
the vortex rope in the simulation. This approach, points out the role of the shutdown of
a neighboring unit, decreasing the cavitation number and conducting the system in the
unstable operation domain.
RotorStator Interactions
The rotorstator resonance is investigated for a scale model pumpturbine. The simulation
model is set up on the basis of the flow distribution between stationary and rotating parts.
The 20 guide vanes connected to the 9 runner vanes are all modelled by pipes and are
connected all together throughout 180 valves driven according to the flow distribution
during rotation. The induced pressure waves combine together and result in a standing
wave in the spiral case at the frequency of twice the runner blade passing frequency. The
simulation provides also all expected diametrical modes rotating in the vaneless gap with
the predicted patterns. A parametrical study shows the important role of the closeness
of eigen frequencies on dynamic amplification of the standing wave in the spiral case.
Resonance Risk Assessment
Finally, a methodology is proposed for the assessment of the risk of resonance due to
part load vortex rope excitation. The method is based on a parametric study where the
EPFL  Laboratoire de Machines Hydrauliques
278
hydraulic resonance and power fluctuations are determined as function of the draft tube
wave speed. A white noise excitation is used for the draft tube excitation in order to
reveal all possible resonances. The possible resonances should be compared either with
experimental data or with CFD computations of the excitation source and vortex rope
compliance.
13.2
Conclusions
The onedimensional modelling approach developed in the framework of the present work
appears to be a powerful tool for the simulation, the analysis and the optimization of
the dynamic behavior of hydroelectric power plants. This approach, based on equivalent
schemes representation, enables qualitative and quantitative analysis of hydraulic system
dynamics in an intuitive way. The proposed modelling can be used either for frequency or
time domain analysis using the same model formulation and breaks the rupture between
frequency and time domain investigations.
The implementation of these models in the simulation software SIMSEN opens the
door to multiphysics modelling of hydroelectric power plants with high level of complexity
for the entire system. Such a tool meets the requirement of improving the accuracy
of solicitation prediction for increasing specific power of Francis turbines and reducing
mistuning and troubleshooting on site during commissioning.
Using the same modelling, the focus can be put on either transient phenomena of
hydroelectric power plants or on specific resonance or stability problems related to the
operation of Francis turbines. The model of a Francis turbine can be set up for an
investigation according to the topology of the machine taking advantage of the modularity
of the onedimensional modelling. This concept has been successfully applied for the
modelling of upper part load pressure pulsation, requiring a high frequency model, for
full load instabilities, requiring appropriate damping and cavitation model, and also for
rotorstator interaction where the model should take into account the guide vane and
runner vane arrangement of the machine.
The onedimensional approach has the advantage of providing simulation results in
a short time for optimization and parametric studies, but requires a correct set of parameters. The amount of parameters increases with the complexity of the model and
of the studied physics. Therefore, appropriate identification methodologies should be
used. Propositions are made for the determination of wave speed in the draft tube with
cavitation development based on either photography or on CFD computation. If the determination of the cavitation compliance at full load seems to be valid, the method should
be validated for the mass flow gain factor and for part load problems.
13.3
Perspectives
13.3.1
Improving Models
There is always a need for developing and implementing new models in softwares like
SIMSEN. Regarding the hydraulic side, the implementation of the following models would
be interesting:
EPFL  Laboratoire de Machines Hydrauliques
13.3. PERSPECTIVES
279
13.3.2
For high frequency problems like rotorstator interactions, it is clear that the assumption of
propagation of planar waves reaches its limits. For such a kind of problem, a generalization
of the equivalent scheme representation from one to two or three dimensions as proposed
by Bilbao [16] would be an interesting approach. The two dimensional approach was
already successfully applied by Timouchev et al. [146], [147] to rotorstator problems.
The figure 13.1 shows the representation of the equivalent scheme for two dimensions.
13.3.3
As mentioned above, the difficulty with problems involving complex physics is not only to
have appropriate models but also to have reliable data for the parameter settings. Regarding the draft tube surge modelling, 2 key parameters are necessary: the wave speed along
the draft tube and the excitation source amplitude and frequency. Recent improvements
in the CFD computation of the part load vortex rope [104], [134] and [136], indicate that
EPFL  Laboratoire de Machines Hydrauliques
280
determination of missing parameters would be possible. The determination of these parameters can be done by a separate computation, as it is done for the full load vortex rope
in chapter 10, or done concurrently. The concurrent methodology approach is illustrated
in figure 13.2 where the CFD computation provides excitation sources and enables the
determination of the cavitation volume for the wave speed determination, while the hydroacoustic computation provides realistic boundary conditions for the CFD computation
[99], [129], [54] at each time step. However, direct coupling of CFD and hydroacoustic
codes makes sense only if the CFD domain represents a short hydroacoustic domain, which
is not the case for draft tube problems where wave speed can drop drastically.
pin
pout_average 
3D system Qout
Qin
Hydroacoustic
parameters
identification
p(t)
Zequ_down 1D hydroacoustic model
Zequ_up
hdown
Zequ(t)
hup
Qsource(t)
hsource(t)
1D equivalent model
of 3D the system
13.3.4
New Applications
The hydroacoustic modelling approach is applicable to other domains such as the biomedical or engine applications. Regarding biomedical applications, equivalent schemes
have already been used for a long time for the modelling of the human cardiovascular
systems [121], [54]. In the field of automotive engines, there is a large place for the
optimization of the admission and exhaust systems as well as injection devices [138].
APPENDICES
Appendix A
Numerical Integration Methods
The set of ordinary differential equations (ODE) to be solved in SIMSEN is the following:
[A]
dx
+ [B(t, x)] x = c
dt
(A.1)
The integration method used in SIMSEN to solve the system of equation A.1 is the explicit 4th order RungeKutta method (RK). This method might presents some restrictions
in terms of stability for solving stiff problems. Stiff problems are characterized by:
high stiffness in the problem resulting in very high eigen frequencies of the system
to be solved;
or, system presenting large differences of order of magnitudes leading to large differences between the smallest and the highest eigen values.
Stiff systems might be encountered in multiphysics systems such as hydroelectric
power plants where electrical devices present time constants of = 0.001 s and hydraulic
mass oscillation periods of T = 500 s. In addition, the explicit RK method presents
a stability domain covering the main part of the left hand complex plan that excludes
system physically unstable. Therefore, a more robust method for checking the validity of
simulation results is necessary. Implicit integration methods are usually used to overcome
integration stability problems. Three test cases are considered for comparing the explicit
and the implicit methods: (i) waterhammer effect in a pipe; (ii) surge effect due to physical
instabilities; and (iii) van der Pol equation.
A.1
Integration Methods
The differential equation set of interest can be expressed in the following compact form:
dy
= f (x, y) ;
dx
y(xo ) = yo
(A.2)
To solve this ordinary differential equation set, explicit an implicit RK methods are
investigated.
EPFL  Laboratoire de Machines Hydrauliques
284
A.1.1
The explicit RungeKutta sth order method applied to the system A.2 gives:
k1 = f (xo , yo )
k2 = f (xo + c2 h, yo + h a21 k1 )
k3 = f (xo + c3 h, yo + h (a31 k1 + a32 k2 ))
...
ks = f (xo + cs h, yo + h (as1 k1 + ... + as,s1 ks1 ))
y1 = yo + h (b1 k1 + ... + bs ks )
(A.3)
Where h = x1 xo and the parameters aij , bi , ci are given in the table A.1.
Using the simplified approach described by Butcher for a 4th order method, the parameters of table A.1 gives the parameters table A.2:
Table A.2: Parameters of the RungeKutta explicit method 4th order.
0
1/2 1/2
1/2 0 1/2
1
0
0
1
1/6 2/6 2/6 1/6
(A.4)
The triangular structure of the table indicates that a kn can be calculated directly from
the previous ki<n . No iterations are necessary for the computation of y1 . The method is
named ERK.
EPFL  Laboratoire de Machines Hydrauliques
A.1.2
285
The implicit sstage RungeKutta method [67] leads to the following expression:
ki = f (xo + ci h, yo + h
s
P
aij kj ) ;
i = 1, ..., s
j=1
y1 = yo + h
s
P
(A.5)
b i ki
i=1
The table used for the explicit method is extended to the aij above the diagonal. The
method is named IRK method. Then a given kn is computed by iteration considering ki>n .
The ki values converge if the Lipschitz condition (with the L is a constant) is fulfilled:
h<
1
P
L maxi
aij 
(A.6)
Several set of aij , bi , ci parameters can be used and present different stability patterns
and have to be chosen by experience depending on the nature of the system to be solved.
Here 3 methods are tested, the Euler and Radau IIA 3rd and 5th order method.
Implicit Euler Method
The Euler implicit method provides the parameters of table A.3:
Table A.3: Parameters of the Euler implicit method.
1 1
1
(A.7)
286
Table A.5: Parameters of the RungeKutta Radau IIA 5th order method.
4 6
10
4+ 6
10
1
1
A.2
887 6
360
296+169 6
1800
16 6
36
16 6
36
296169 6
1800
88+7 6
360
16+ 6
36
16+ 6
36
2+3 6
225
23 6
225
1
9
1
9
Three test cases have been selected to compare performances and stability of the 3 integrations methods described above. The test cases are the following: (i) the waterhammer
effect in a pipe due to a valve closure; (ii) the surge effects due to physical instabilities
related to the mass flow gain factor; and (iii) the van der Pol equation.
For both hydraulic cases, the parameters of the piping system are the same. The test
case piping layout is presented in figure A.1 and the related parameters are given in table
A.6. The number of nodes used to model the pipe is 10. To solve the differential equation
set with RK methods, the equation set A.1 is reordered as follows:
dx
= [A]1 c [A]1 [B(x, t)] x

{z
}
dt
(A.8)
f (t,x)
Ho
a,
Kv, Aref
D
L
Qo
fo = a/(4L)
3
[m] [m] [m/s] [] [m /s]
[Hz]
600 0.5 1200 0.02
0.5
0.5
A.2.1
Waterhammer Phenomenon
First, the waterhammer effect is simulated for 10 seconds for a partial closure law from
fully opened valve to 10% valve opening within 2.1 seconds. The comparison of head and
EPFL  Laboratoire de Machines Hydrauliques
287
H1 IRKRIIA / Ho
H1
/ Ho
ERK
H1
/ Ho
EulerE
H1 MOC / Ho
1.02
0.8
0.6
0.4
0.98
0.2
0.96
0.94
0
Q1IRKRDIIA / Qo
Q1
/ Qo
ERK
Q1
/ Qo
EulerE
Q1MOC / Qo
Q / Qo []
1.04
H / Ho []
Reservoir discharge
1.2
0
1
t [s]
0.2
0
Q2IRKRDIIA / Qo
Q2ERK / Qo
Q2EulerE / Qo
Q2MOC / Qo
1.2
1.4
1.2
1
0.8
0.6
0
Valve discharge
H / Ho []
1.6
t [s]
1.4
H2 IRKRIIA / Ho
H2 ERK / Ho
H2 EulerE /Ho
H2 MOC / Ho
1.8
Closure Law
0.8
0.6
0.4
0.2
t [s]
0
0
t [s]
discharge at both end of the pipe are presented in figure A.2. The Method of Characteristic
(MOC) is taken as reference.
Secondly, a fast valve closure of 0.2 seconds is simulated. The comparison of the
simulation results is presented in figure A.3 and in figure A.4.
The comparison of the different integration method shows that there is only a little
difference between all the integration methods. The number of iterations with the IRK5th method to reach an error of = 105 , is 4.71 if 10 nodes are used to model the pipe.
Increasing the number of nodes to 20 implies 5.94 iterations to reach the same error limit.
Thus, the computational time required for a simulation is 5 times greater. In addition,
when the number of nodes increases, the integration time step has to be reduced in the
same proportion to fulfil CFL criteria. The IRK methods can provide more accurate
solutions than ERK methods but are detrimental with respect to the computational time.
288
Reservoir discharge
1.6
H / Ho []
1.4
1.3
H1
/ Ho
IRKRIIA
H1
/ Ho
ERK
H1
/ Ho
EulerE
H1
/ Ho
MOC
0.5
Q / Qo []
1.5
1.5
1.2
1.1
1
0
0.5
0.9
0.8
0
0.2
0.4
t [s]
0.6
0.8
1
0
Q1IRKRDIIA / Qo
Q1ERK / Qo
Q1EulerE / Qo
Q1MOC / Qo
0.2
2
1.5
H2 IRKRIIA / Ho
H2 ERK / Ho
H2 EulerE /Ho
H2 MOC / Ho
0.2
0.4
t [s]
0.6
0.8
Q2IRKRDIIA / Qo
Q2ERK / Qo
Q2EulerE / Qo
Q2MOC / Qo
1.2
2.5
H / Ho []
t [s]
Valve discharge
0.5
0
0.4
Closure Law
0.8
0.6
0.4
0.2
0.6
0.8
0
0
0.2
0.4
t [s]
0.6
0.8
H2
/ Ho
IRKRIIA
H2 ERK / Ho
H2 EulerE /Ho
H2
/ Ho
MOC
H / Ho []
2.65
2.6
2.55
2.5
2.45
2.4
2.35
0.12
0.14
0.16
0.18
t [s]
0.2
0.22
0.24
A.2.2
289
Surge Phenomenon
V
V
dH2 +
dQ2
H2
Q2
(A.9)
dt
H2 dt
Q2 dt
(A.10)
Lets define:
the cavity compliance C =
V
H
V
Q
It yields to:
Q2 Q1 = C
dQ2
dH2
+
dt
dt
(A.11)
When the mass flow gain factor is negative, the hydraulic system might become unstable. In this test case, the limit of the stability is reached for = 0.035. The simulation
of the unstable behavior of the pipe is initiated by a 50% valve closure is presented in
figure A.5. This case features differences between the Euler method and the RK methods.
However, again, no big differences between ERK and IRK are visible.
Valve discharge
100
H / Ho []
50
H2 IRKRIIA o5 / Ho
H2
/ Ho
ERK
H2
/Ho
EulerE
50
100
0
t [s]
0
Q2IRKRDIIA o5 / Qo
Q2
/ Qo
ERK
Q2
/ Qo
EulerE
Closure law
5
0
t [s]
290
A.2.3
The van der Pol equation is chosen as a test function to evaluate performances of the
different integration methods. The van der Pols equation is given by:
d2 y
dy
+y =0
+ (y 2 1)
2
dt
dt
(A.12)
In order to have a first order ODE, equation A.12 can be rearranged as follows:
dy1
= y2
dt
dy2
+ (y12 1) y2 + y1 = 0
dt
+
2
y2
0
0 1
1 (y1 1)
dt y2
The system is solved considering the following initial conditions:
2
y1 (0) = 2 ; y2 (0) = 0 yo =
0
(A.13)
(A.14)
(A.15)
The van der Pol equation represents a dynamic system whose damping is negative for
small amplitudes (y < 1) and is positive for high amplitudes (y > 1). The parameter
enables emphasizing the instability of the system; the higher the , the more unstable
is the system. The simulation results obtained for = 10 using 5 different integration
methods are presented in table A.7 for different integration time steps and tolerances.
For high integration time steps, the Eulers methods, explicit or implicit, do not provide an accurate solution. The frequency of the oscillation is underestimated by the
implicit method while it is overestimated by the explicit method. However, all RK methods, implicit or explicit, provide reasonable solution. The solution of ERK is almost
identical to the IRK RADAU IIA order 3. The tolerance of the implicit method does not
significantly influence the results but increases the number of loops per time step. Decreasing the integration time step strongly improves the simulation results of the Eulers
methods. To reach the accuracy of RK method, the time step has to be divided by 100.
So Eulers methods are not suitable for the simulation of unstable systems. Compared to
the IRK methods, the ERK method provides good results in a reasonable computation
time. Figure A.6 shows that all the methods provide the same results only for very small
time step and tolerance. Between the IRK methods, the 3rd order method is the most
accurate method even for large tolerance and time steps as the solution is only slightly
influence by the reduction of the integration time step. The solution of the 5th order
converges to the solution of the 3rd order when the time step is reduced.
The ERK method is found to be the most suitable integration for the integration of
hydraulic systems. IRK RADAU IIA o3 might be useful for the verification of the solution
of the simulation results of hydraulic system physically unstable.
291
Table A.7: Comparison of simulation results for the van der Pol equation.
T olerance = 0.01
T olerance = 0.0001
Van der Pol Equation for = 10 ; Tol = 0.01 ; dt = 0.01
2.5
EULER E
EULER I
ERK
IRKRDIIA o3
IRKRDIIA o5
2
1.5
EULER E
EULER I
ERK
IRKRDIIA o3
IRKRDIIA o5
dt = 0.01s
y1 []
y1 []
0.5
0
0.5
1
1
1.5
2
2.5
0
2
nloop EU I = 2.3348 ; n
10
loop IRK1
= 2.2045 ; n
15
loop IRK2
20
t [s]
= 2.1622
25
nloop EU I = 3.2243 ; n
30
35
3
0
40
= 2.6058
25
30
35
40
EULER E
EULER I
ERK
IRKRDIIA o3
IRKRDIIA o5
1.5
1
0.5
0.5
0
0.5
0
0.5
1
1
1.5
1.5
2
loop IRK2
2.5
0
20
t [s]
y1 []
y1 []
dt = 0.001s
1.5
= 2.7603 ; n
15
2.5
EULER E
EULER I
ERK
IRKRDIIA o3
IRKRDIIA o5
loop IRK1
10
nloop EU I = 2.1043 ; n
10
loop IRK1
= 2.0768 ; n
15
loop IRK2
20
t [s]
2
= 2.0686
25
30
35
nloop EU I = 2.3655 ; n
2.5
0
40
2.5
loop IRK1
10
= 2.2811 ; n
15
20
t [s]
loop IRK2
= 2.2491
25
30
35
40
EULER E
EULER I
ERK
IRKRDIIA o3
IRKRDIIA o5
2
1.5
1
y1 []
0.5
0
0.5
1
1.5
2
2.5
0
10
15
20
25
30
35
40
Figure A.6: Comparison of simulation results for the van der pol equation with small
integration time step.
292
A.3
Let consider the smooth solution (x) of equation A.1, dy/dx = f (x, y) . Linearizing
f (x, y) in its neighborhood as follows [68]:
f
dy(x)
= f (x, (x)) +
(x, (x)) (y(x) (x)) + ...
dx
y
(A.16)
(A.17)
As a first approximation, considering the Jacobian J(x) constant and neglecting the
errors terms yields to:
dy(x)
=J y
dx
(A.18)
(A.19)
zs
z2
+ ... +
+ O(z p+1 )
2!
s!
(A.20)
The stability function can be interpreted as the numerical solution of the Dalhquist
test equation after one step for:
dy
=y
dx
yo = 1 ;
z =h
(A.21)
(A.22)
1
1z
(A.23)
1 + z 2/5 + z 2 /20
1 z 3/5 + z 2 3/20 z 3 /60
(A.24)
293
The stability domains related to ERK methods of order 1 to 4 are presented in figure
A.7 while stability domain of implicit Euler and IRK RADAU IIA order 5 are respectively
presented in figure A.8 left and right.
So these integration methods are stable if R(h k ) 1.
This inequality shows the influence of between the integration step h. The explicit
methods feature more stable behavior for higher order method that encloses region close to
the origin and higher frequencies. Highly damped systems might also present instabilities.
However, implicit method do cover the whole part of the left hand side an extended part
of the right hand part of the complex plan. The Euler implicit method appears to be a
very stable method. More than the IRK RADAU IIA 5th order method.
Imaginary
Imaginary
1
1
2
2
3
3
2
1
0
Real
3
3
1
2
2
2
1
0
Real
0
Real
1
3
3
1
Imaginary
Imaginary
2
3
3
2
1
0
Real
294
10
2
5
Imaginary
Imaginary
1
0
1
5
2
3
3
2
1
0
Real
10
10
5
0
Real
10
Figure A.8: Stability domain of the Euler implicit method (left), and stability domain of
IRK RADAU method order 5 (right).
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Index
Active control, 16
Air injection, 16, 182, 210
Automotive engines, 280
Biomedical, 280
Cavitating pipe, 279
Cavitation
bubble, 14, 15, 173
compliance, 89, 121, 179, 182, 193,
202, 230, 260, 275
sheet, 173
CFDhydroacoustic coupling, 280
CFL criteria, 43, 139, 287
Continuity
equation, 27, 32, 71, 275
simplified equation, 29
Dalhquist test, 292
Delaunay triangulation, 108
Dirac impulses, 199, 200
Draft tube fins, 16
Efficicency hill chart, 12, 240
Euler
equation, 10
integration method, 43
number, 258
similitude, 259
Fourrier inverse transform, 57
Froude
number, 205, 219, 258
similitude, 258, 260, 276
General User Interface, (GUI), 103
Greenhouse gases, 3
Jacobian, 292
KelvinVoight model, 82
Kirchhoff law, 46, 47, 50
310
INDEX
BIOGRAPHY
311
List of Publications
Papers
1. NICOLET, C., GREIVELDINGER, B., HEROU, J.J., ALLENBACH, P., SIMOND,
J.J. and AVELLAN, F., High Order Modeling of Hydraulic Power Plant in Islanded
Power Network, IEEE Transaction on Power Systems, Submitted for publication
January 2007.
Conferences
1. KOUTNIK, J., NICOLET, C., SCHOHL, G. A. and AVELLAN, F., Overload Surge
Event in a PumpedStorage Power Plant, Proceedings of the 23rd IAHR Symposium
on Hydraulic Machinery and Systems, Yokohama, Japan, October 18  21, 2006.
2. RUCHONNET, N., NICOLET, C. and AVELLAN, F., OneDimensional Modeling
of Rotor Stator Interaction in Francis PumpTurbine, Proceedings of the 23rd IAHR
Symposium on Hydraulic Machinery and Systems, Yokohama, Japan, October 18 21, 2006.
3. SIMOND, J.J.,ALLENBACH, P., NICOLET, C. and AVELLAN, F., Simulation
tool linking hydroelectric production sites and electrical networks, Proceedings of
the 27th Int. Conf. on Electrical Machines, ICEM, Chania, Greece, September 25,
2006.
4. NICOLET, C., HEROU, J.J., GREIVELDINGER, B., ALLENBACH, P., SIMOND,
J.J and AVELLAN, F., Methodology for Risk Assessment of Part Load Resonance
in Francis Turbine Power Plant, Proceedings IAHR Int. Meeting of WG on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Barcelona, 2830
June 2006.
5. RUCHONNET, N., NICOLET, C. and AVELLAN, F., Hydroacoustic Modeling of
Rotor Stator Interaction in Francis PumpTurbine, Proceedings IAHR Int. Meeting
of WG on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems,
Barcelona, 2830 June 2006.
6. NICOLET, C., GREIVELDINGER, B., HEROU, J.J., ALLENBACH, P., SIMOND,
J.J. and AVELLAN, F., On the Hydroelectric Stability of an Islanded Power Network, Proceedings IEEE Power Engineering Society General Meeting, 1822 June
2006, Montreal, Canada, paper 623.
312
BIOGRAPHY
BIOGRAPHY
313
Curriculum Vitae
Christophe NICOLET
Av. de Mategnin 104
CH1217 Meyrin
Ne le 3 janvier 1977, celibataire
Email: christophe.nicolet@a3.epfl.ch
FORMATION
Doctorat `
es sciences techniques
20012007:
Ecole
Polytechnique Federale de Lausanne (EPFL)
19972001:
Dipl
ome Ing
enieur en G
enie M
ecanique, EPF
Ecole polytechnique federale de Lausanne (EPFL)
Prix Rhyming et diplome dhonneur Genie Mecanique
Dipl
ome au Laboratoire de Machines Hydrauliques
19921997:
Dipl
ome dIng
enieur en G
enie M
ecanique, HES
Ecole dIngenieurs de Gen`eve (EIG)
Mention: bien; Prix Institut Batelle
Dipl
ome au Laboratoire dEnergetique
EXPERIENCES
PROFESSIONNELLES
Ecole
Polytechnique F
ed
erale de Lausanne (EPFL), Suisse.
20012007:
Laboratoire de Machines Hydrauliques (LMH).
Activit
es de recherche et de d
eveloppement:
Th`ese de Doctorat: Modelisation des phenom`enes hydroacoustiques,
application aux turbines Francis:
 simulation comportement dynamique dinstallations hydrauliques;
 develloppement de mod`eles hydrauliques pour implementation dans SIMSEN;
 etudes transitoires/hydroacoustiques pour Sulzer Pumps,
Meditecnic, les Forces Motrices de la Gougra et Colenco.
Activit
es denseignement:
 participation au cours dEcoulements Transitoires (Master; 5x2h/an);
 participation au cours de Specialisation en Machines Hydrauliques
(Postgrade; 2h/an);
 encadrement de 6 travaux pratiques de Master.
1999:
19972000:
314
BIOGRAPHY
1998:
1997:
19951996:
COMPETENCES
INFORMATIQUES
Program.:
MATLAB, Delphi, FORTRAN, Maple.
SIMSEN.
Transitoires:
Fluent, Gambit.
CFD:
ANSYS.
FEM:
Euclide (Dassault Syst`emes), PROEngineer, Me10 (HP).
CAO:
Acquisition:
HPVEE.
Bureautique: MSOffice, Latex, Adobe Illustrator et Photoshop, Grapher.
LANGUES
Francais
Anglais
Allemand
Langue maternelle.
Parle, lu et ecrit: couramment, Cambridge First Certificat (2001, Perth).
Parle, lu et ecrit: connaisances scolaires.
LOISIRS
Loisirs