IAHR2001

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Revision received Februari 19, 2000. Open for discussion till February 28, 2002.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 4 429

DynamicoricemodelonwaterhammeranalysisofhighormediumheadsofsmhydropowerschemesModledoricedynamiquedanslanalyseducoupdeblierdespetitescentralehaute et moyenne chutes

H. RAMOS,Assistant Professor Dr. of Civil Engineering Department, Technical University of Lisbon, IST, 1049-001 Lisbon, Portugal

A.B. ALMEIDA,Full Professor Dr. of Civil Engineering Department, Technical University of Lisbon, IST, 1049-001 Lisbon, Portugal

ABSTRACTThe most severe hydropower transients are induced in long hydraulic circuits due to extreme operating conditions. A computational modeveloped in order to on one hand ensure waterhammer system control and, on the other hand, provide a more reliable and easier analysis forspecic speed turbines and alternative solutions of the system as a whole, through interaction between different hydraulic components. In turbines, runaway conditions and guide vane closure cause signicant discharge variations and pressure uctuations that can affect the dconveyance systems. A new approach for groups modelling as dynamic orices concept was developed enabling the characterisation of the insystem. The simulation results were compared with laboratory tests. This model can be used in the initial stages of civil works design as an way to better characterise the hydrodynamic behaviour of the system when equipped with reaction turbines.

RSUMLes transitoires hydrauliques les plus svres sont provoqus par des conditions extrmes de fonctionnement des systmes hydrolectriqparticulier dans le cas de longues conduites forces. Un logiciel a t dvelopp de faon simuler et assurer le contrle du coup de blier permettre une analyse plus simple et sre pour diffrents types de turbines et des solutions alternatives pour le systme hydraulique. Le prodvelopppermet linteractionde plusieurslments.Pour desturbines raction lesconditions,demballementet de fermeturedu distributeurpprovoquer des variations signicatives du dbit et des uctuations de pression dont il faut tenir compte lors de la conception des systmes dadUne nouvelle approche, base sur le concept dun orice dynamique quivalent, a t dvelopp pour modliser les groupes, ce qui pecaractrisation du systme intgr. Les rsultats obtenus ont t compars avec des essais en laboratoire. Cette technique de calcul peut trelors de lanalyse prliminaire des travaux de gnie civil, comme un moyen efcace de mieux caractriser le comportementhydrodynamique du sylorsquil est quip de turbines raction.

Introduction

Smallhydropowerschemes, with long conduits,arethesubject of safety and economic concerns due to the occurrence of severehydraulic transients. In waterhammer analysis, abnormal opera-tion under extreme conditions is the base of any design study.When equipped with reaction turbines of low inertiaandlow spe-cic speed, during abnormal conditions, runaway can easily oc-cur, whichcaninducedangerous overpressures,dependingon thetype of turbine runner. A serious safety problem can occur aftera full-loadrejectionwhen theoverspeed effect, causes maximumoverpressure, which can be attained very rapidly (in as little asthreeseconds).Thetotaldurationof guide vaneclosure will beanimportant parameter, but not the most important in this case.

An efcient model for transient analysis will enable the opera-tionalbehaviour of thehydro-system to be evaluated. Thisanaly-sismayleadtounconventionalsolutionsbyeliminatingconserva-tiveprotection devices such as a surge tank. Actually, at theinitialdesign, the stage of civil works conception, the characteristics of the groups are not yet well dened and not completely available.In order to overcome such lack of knowledge, a computationalmodel based on a dynamic orice concept was developed. Thespecic speed allows characterisation of the turbine behaviourthat will inuence hydrotransients along the conveyance system.

State-of-the-art

Thepredictionofoverpressuresinducedbyhydraulicturbineshasalways been an important objective in the development of pres-suretransientanalysis.Earlystudies focused on thecalculationof the maximum penstock overpressure due to a more or less rapidow stoppage induced by turbinenozzles orguide vanes. Simpleformulas were proposed based on the fundamental penstock andow parameters or time constants and on the time of gatemanoeuvre. An example is Michauds formula (1878), orVensanos formula in the U.S.A., for slow manoeuvres.Allievi (1903) was closely involved in unsteady ow behaviourand safety in hydroelectric schemes when developing his mathe-matical formulationof the waterhammer phenomenon. However,

as a boundary condition of a hydraulic system, a reaction turbinewill induce a compound effect on the ow during a full load rejection: the guide vaneeffect and the speed or runner effect. Sincethe sixties, improved computer methods have enabled more de-tailed analysis of pressure transients in pipe systems with differ-ent components and operating conditions. The complete charac-teristic curves of hydraulic machines, including pumps and tur-bines,togetherwithrotatingmass equationandby thecompatibil-ity equations of the method of characteristics, applied along thepipelines, cannow be solved with high accuracy using numericaltechniques.

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430 JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 4

(1)C

WC

o

M

TT

KHH =

One of the ways to represent reaction type turbine characteristicsis based on unit parameters combining discharge, speed, torque,power, head, and machine diameter. For transient analysis it isconvenient to make use of dimensionless homologous relation-ships such as the Suter parameters (Chaudhry, 1987; Wylie andStreeter, 1993). Two families of curves can then be developed foreach guide vane opening or position. Numerical interpolation ortransformation techniques can be applied to these non-linearcurves to solvethecoupled machine rotating massesequationandpenstock or pipeline equations (Boldy, 1976; Chaudhry, 1980).For a full-load rejection condition, design constraints are placedon both turbine overspeed and penstock overpressure. The gateclosure law needs to be specied to control both these effects.Optimisation techniques were developed by different authors todeal with this situation (Ruus, 1966; Wozniak and Fett, 1973;Nai-Xiang and Zu-Yan, 1989).Fordesignof hydraulic conveyance structures, the general speci-cations of the equipment and the overpressures along the pen-stocksorgalleriesmust bedetermined. Most approximate formu-las are based on the start-up (or inertia) time of the water mass atfull load, TW (e.g. Lein, 1965):

in whichHM = maximum head variation, Ho = turbine grosshead, TC = closing time and KC = a factor that depends upon theturbine specic speed Ns.Typically, KC values vary between 1.3and1.5. Equation (1)coin-cides with Michauds formula forKC= 2. In fact, according to the

theoretical studies of De Sparre (Remenieras, 1961) the maxi-mum overpressure due to a full gate closure will be less than thevalue obtained by Michauds formula. However, Gariel (1918)showed that the maximum maximorum overpressure inducedby a critical partial gate closure, with a lineardischargevariation,will be Michauds value (Remenieras, 1961). Due to the runneroverspeed effect in reaction turbines,HM will be a function of the inertia of the rotating masses or a function of the unit startingtime,Tm.Morerecently,Bahamonde(1991)presentedanapprox-imate analysis of the combined variation of pressure and runnerspeed rise during closure of the turbine gate. However, thismethod, based on Allievis methodology for a uniform rate of gate opening variation, does not take into consideration the rapiddischarge variation due to overspeed.The turbine behaviour as well as the dynamic response of the hy-draulic system will depend on the specic speed. Among the ef-fects induced by a particular type of turbine are pressure surgesdue to runner overspeed, and the peculiar S shape effect of thecharacteristic curves of low specic turbines, notably in pump-turbines and in double runner Francis turbines (Taulan, 1983).Brekke (1976) also showed the inuence of turbine characteris-tics on turbine governing.Two levels of tools are available for hydraulic transient analysis

of hydroelectric schemes: complete models based on the full setof machine equations obtained for each unit type, and classicalmethods based on the gate effect or on empirical data. The dy-

namic oricemodel developed in this paper sets out to ll the gapbetween these two approaches.

The dynamic orice model - new hydraulic approach for tur-bine analysis

The dynamic orice model offers a exible general model basedon a small number of parameters, the objective of which consistsin analysis of theextreme conditiongeneratedby runaway and/orguide vane closure as the most unfavourable condition, with re-gard to hydraulic circuit design (overpressures). It includes pre-diction of overspeed effect under runaway conditions on hydro-transient response all the way along conveyance system. Themodel can be seen as a useful tool for an integrated computeranalysis of any multi-component systems (reservoir, total pres-surised penstock, or a mixed circuit composed by a canal and aforebay with free-surfaceow anda penstock with pressure ow,and typical response of any type of turbo-generator groups).The dynamic orice model is based on the concept of the turbineacting as a hydraulic resistive component where the head lost bythe ow is characterised by a dimensionless orice formula thathasa dynamic dischargecoefcient.This coefcient is composedof two terms: a gate factor and a runner overspeed factor. Therst is the gate-opening coefcient, which denes the maximumturbine discharge for a given head and speed as a function of thegate opening. The second factor modies the discharge coef-cient as a function of the runner speed because, for reaction typeturbines at constant head and gate position, the discharge is afunction of the runner speed. The coupled response will dependon the specic turbine speed Ns (Bchi, 1957; Mataix, 1975;

Raabe, 1985) as well as the rated turbine speed NR, discharge QRand head HR.With low specic speed turbines the discharge decreases withrunner speed. Conversely, for high specic speed turbines thedischarge may increase with speed. This turbine behaviour has asignicant effecton thetransient response of theconveyancesys-tem after a full load rejection and must be taken into account inthe simulation of extreme operational conditions. Both the over-speed factor of the turbine discharge coefcient and the turbinehydraulic torquearebased on dimensionless relationships andona few simpleparameters, andarecharacterised by heuristic equa-tions that are approximations of the real characteristic curves.In impulse turbines the transient discharge is decoupled from therunner speed and the turbine discharge does not change withwheel speed so long as the gate or nozzle opening and the headremain constant. For this type of turbine, the overspeed factor isequal tounity and the model containsonly the pureoricemodel.When the dynamic orice is coupled to ow modelling along theconveyance system, the model has the capacity to simulate: pressureanddischarge variationsandthe head envelopesalong

the pipelines as well as, for mixed circuits,also the head varia-tion along the canal and forebay;

transient pressure and discharge variations in the powerhouse

forfull-loadrejectionand/orguide vane closure(or nozzleclo-sure); the rotational speed variation of the groups.

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N/N

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

1 2 3

Q / Q

0.78

0.95

1.12

1.29

1.38

y=1

y=0.8

y=0.6

y=0.4

1.0 1.5 2.0

y=0.2

H/H

=0.65R

R

R

R

Turbine simulation - Dynamic Orifice technique

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0-1.5-1.4-1.3-1.2-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0

runaway conditionsh=0.7h=0.8h=0.9h=1h=1.1h=1.2h=1.3h=1.4

Pump-Turbine Ns = 65 (m, kW) - Suter parameters

N/NR

TG =0

Turbine zone

Q/QR

Overspeed

zone

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

0 50 100 150 200 250 300 350

WH

WT

/

Suter parameters - Ns = 65 (m, kW)

Turbine zone

0.0

-0.4

-0.2

2.0

WH

WT

/R

Fig. 1. Analogy between pump characteristic curves based on Suter parameters for turbine zone under runaway conditions, with turbine simthrough Dynamic Orice modelling (for a low specic speed turbine ( R=0.65))

Dynamic orice equations

The dynamic orice model is based on two fundamental dimen-

sionless parameters: and , with QRW=RRW

R

Q

Q= R

RW

R

N

N=

turbine dischargeat runawayspeed,QR = rated turbine discharge,NRW = turbine runaway speed and NR = rated turbine speed, allfor the rated turbine head. In this paper the subscripts R and RW

indicate the rated and the runaway conditions (at rated head), re-spectively.These parameters depend on the turbine type: a low specicspeed Francis turbine (Figure 1) will have R 1 and a high spe-cic speed, Propeller and Kaplan turbines will have R > 1. Em-pirical relationships can be found in the literature allowing anapproximatevaluationof RandR asafunctionofNs (Fazalare,1991; Ramos, 1995 and Ramos and Almeida, 1996). In practicethe turbine manufacturers know the values of these parameters

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(4)RQ

Q= C g C s

RHH

(5)

+= 1

HH

NN

11

1C RRR

Rs

(6)( )

= 1 / 1

HHN

N1

C

HH

TT

RRR

R

g2 / 3

RH

H

R

NNR

(7)

(8)

=

RRW

RWo NN

NN

(9)

=

RRW

RWo NN

NNy2

(10)T T IddtH G

=

throughturbinetests.Basedonpublishedinformationandondatafurnished by turbine manufacturers, the following equation wasobtained for R:

R = 0.3 + 0.0024 Ns (2)

and forR , another equation was proposed by Fazalare, 1991:

R = 1.6 + 0.002 Ns (3)

both equations with Ns in (m, kW).

According todata obtained from real casestudies, for low inertiaunits in small hydroelectric powerplants equipped with reactionturbines,R is about 2 20%.Figure 1 shows the analogy between the turbine operation withthe pumpwhenoperating in turbine zone, aswell as the dischargereduction, in low specic speed machines,dueto runaway condi-tions.The dynamic orice equation, being a modied orice ow, isbased on the turbine head-discharge equation:

where Q stands for the turbine discharge; H is the turbine head;Cg is thegate coefcient varying between oneandzero accordingto gate opening, y; and Cs is the runner speed coefcient, whichequation can be written as

where N is the runner speed. In the model it is assumed that Cg =f (y).As presented in Figure 1, it was assumed a linear variation of thedischarge with rotating speed under runaway conditions. Thisapproach was veried through laboratory tests applied to a lowand high specic speed Francis turbine runner.In this way, the hydraulic torque (TH) is obtained through the fol-lowing equation:

where is the unit efciency and

The unit efciency for a load rejection condition can be consid-ered to vary according to the following approximate equations:

For y > 0.5 and N NR

For y 0.5 and N NR

withO the unit efciency for initial condition.Additional model conditions are introduced in order to considerthe turbine speed-no-load condition and the runner speed decayafter the guide vane closure.

Rotating mass equation

The unbalanced torque between turbine and generator changesaccording to the angular momentum equation for the rotatingmass according to the following equation

inwhich TH = net hydraulic turbine torque, TG = electromagneticresistant torque, I = total polar moment of rotational mass inertia(I = WR2 / g) and = angular velocity of the rotating mass.After a full load rejection the electromagnetic resistance torque,

TG, can beset equal tozero. According toequation (10), the polarmoment has a signicant inuence on the speed variation of therotating mass of the turbo-generators. For low inertia units therunner speed increases rapidly after a full-load rejection, andcanattain runaway conditions.

Canal and pipeline equations

Transient regimes in hydraulic conveyance circuits of hydro-power plants can be modelled by 1-D ow models when thereaches are straight and uniform and the cross-section is muchless than its length.In canals it is important to know the unsteady ow behaviourcausedby rapid strong discharge variations,with possible forma-tion of shockwaves, in order to dene the canal geometry. In thiscase there is a discontinuity in the water surface or bore, withrapid varied ow, and a gradual varied ow upstream and down-stream of the bore.Explicit methods with second order accuracy have proved to besuitable for ow modellingwith shocksandbores.Thefollowinggeneral assumptions are accepted for free-surface equations(Chaudhry, 1987, Almeida and Koelle, 1992 and Ramos, 1995): the transient ow is 1-D with the horizontal water surface and

uniform velocity in each cross-section; the streamline curvature is small and vertical accelerations areneglected, hence the pressure is hydrostatic;

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(11)( )[ ] += EK1K

c

(12)( )( )1211G12G22

AAA

hAhAgAc

=

(13)( )UDxUF

tU =

+

(14)( ) ( ) ( )=+== JigA0

UDgAh

AQ

QUF

Q

AU 2

(15)( ) ( ) === QQQ

JgA

0UD

gAH

QgAc

UFQ

HU

2

2 transientfrictionlossesandturbulencearemodelledby empiri-cal resistance laws (quasi-stationary assumption).

Theelasticmodelofpressurised ow forsmallhydroelectricpen-stock systems is compatible with the following assumptions(Chaudhry, 1987, Pejovic, Boldy and Obradovic, 1987, Almeidaand Koelle, 1992 and Wylie and Streeter, 1993): the ow is slightly compressible; the velocity and pressure follow a uniform distribution in each

cross-section ( = = 1); the rheological behaviour is elastic and linear; the convective terms in the basic equations are neglected com-

paring with the other terms.

Any disturbance induced in the ow is propagated with a celeritythat will strongly inuence thedynamic response in thehydrauliccircuit. For pressure ows the celerity of the elastic waves corre-sponds to the storage capacity of the uid compressibility andpipe deformation:

and in free-surface ows corresponds to the kinetic energy of theow:

where A stands for the ow cross section; E is Youngs modulusofelasticity of theconduitwalls;K thebulkmodulus ofelasticity;hG thedepthofthecentredareaA; thedimensionless parameterthat depends on the elastic properties of the conduit; and 1 and 2are upstream and downstream of wave front.For free-surface ow the complete dynamic model based on theSaint-Venant equations must be written in a conservative form inorder to simulate the propagation of bores (Franco, 1996):

where U, F(U) and D(U) are the following vectors:

The pressure transients in pipes are modelled by the well-knownwaterhammer equations (Chaudhry, 1987andWylieandStreeter,1993). The basic differential equations of unsteady pressurisedows can also be written in matrix form, yielding the following

vectors:

in which x = distance along the canal bottom or the pipe axis; t =time; A = cross-section ow area; Q = turbine discharge; h = wa-ter depth (canal); H = piezometric head (for penstock); i = chan-nel bottom slope; J = slope of the energy grade line; g = gravita-tional acceleration; c = wave celerity in open channel/pressurepipe.TheinteriorpointswillbesolvedusingtheMacCormackmethod.Following MacCormacks recommendation, the predictor andcorrector steps are used alternately with the nite forward andbackward differences. The method of characteristics transformsthese equations into a pair of ordinary differentialequationsvalidalong the characteristic lines in the (x, t) plane. These equationscan be replaced by algebraic equations that are solved togetherwith the other basic equations of the computational model. Noapproximate model based on the rigid column theory can be ap-plied due to the potential rapid ow velocity variation caused bythe overspeed effect. All the basic equations are solved togetherin order to obtain the variations of head (or pressure), discharge(or ow velocity) along the pipe sections, and turbine runnerspeed for a given guide vane motion.The dynamic orice equations, rotating mass equation and pipe-line (and canal) equations compose the complete computationalmodel.

Operational advantages and validation

The dynamic orice model has the advantage that the dynamicrunaway conditions induced by any type of reaction turbine unitscan be easily evaluated. This technique, together with computermodellingof theothercomponents of thesystem, enables an inte-grated hydrotransient analysis to be developed that has provedsufciently accurate for conveyance system design purposes, es-pecially for the feasibility and general design stages.Figure 1 shows the turbine discharge (Q) variations for differentvalues of head (H), guide vane opening (y) and rotational speed(N), based on the dynamic orice model (equations (4) and (5))applied to the turbine overspeed zone for a full-load rejection.The dynamic behaviour ofa low and a high specic speed turbinewas compared with predictions using the dynamic orice modeland lab tests. Examples of the transient variations of severalquantities are presented in Figures 2 and 3, respectively for lowand high NS runners placed downstream of the pipeline.

Waterhammer effects

Prediction of overpressures along the penstock and interactionwith forebay and canal response (e.g. for mixed hydraulic cir-cuits), after a full load rejection, including guide vane and

overspeed effects for any specic speed of reaction turbine, is

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0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25Time (s)

N(rev/min)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

H/HRQ/QR

mod - N exp - Nmod - H/Ho exp - H/Homod - Q/Qo exp - Q/Qo

overpressurefrom overspeed

overpressure fromguide vane closure

H/HR

N

Q/QR

Fig. 2. Comparisonbetweenexperimental anddynamic orice modelresultsofoverpressure,dischargeand runnerspeed, for a low specic speed Francis turbine ( R = 0.5; TE /Tm = 1.40; TW /TC = 0.12)

0

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12 14 16 18 20Time (s)

N(rev/min)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

H/HRQ/QR

mod - N exp - N

mod - H/Ho exp - H/Homod - Q/Qo exp - Q/Qo

guide vaneclosure effect

overspeed effect

N

H/HR

Q/QR

Fig. 3. Comparisonbetweenexperimental anddynamic orice modelresultsofoverpressure,dischargeand runnerspeed, for a high specic speed Francis turbine ( R = 0.88; TE /Tm = 1.50; TW /TC = 0.12)

possible with the complete set of equations (4) to (15), presentedhere in a simplied way.As an example of the results obtained through systematic com-puter simulations, Figure 4 shows the dimensionless maximumupsurge or overpressure values induced by full load rejection atthedownstreamendofasingleuniformpenstock,HM /H0,forR= 2.0, as a function of R.

The symbols are dened as H0 = gross head(m), TC = guide vanefull closure time (s), TE = pipeline elastic time constant (s), Tw =

pipeline hydraulic inertia time constant (s) and Tm = unit startingtime (s).This gure shows that for low specic speed Francis turbines orfor R

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0.05

0.15

0.25

0.35

0.45

0.6 0.7 0.8 0.9 1

H ____

Ho

M

T E /T m

Tw/Tc

R

0.140.100.05

4.60

3.402.00

= 2.0R

Fig. 4. Maximum upsurge induced by a full-load rejection based on the dynamic orice model technique

(16)( )h2HH R

wo

M 1 = + 20

TET

R

m

sults show thatHM /Ho become proportional to TW /TC, as inequation (1) with KC = 1.4.When runaway speed is attained in a very short time interval of order TE, the overpressure due to overspeed can be evaluated bythe following modied Joukowsky formula:

withhw theAllievi parameter (typically hw < 1 for high-head sys-tems).Equation (16) will give the maximum overpressure induced bythe overspeed effect, especially for low-inertia units. The dy-namic orice model is able to show the inuence of the groupsinertia on overspeed and pipe-uid elasticity through TE /Tm.It can be observed that, for each TW /TC value, the maximumoverpressure due to overspeed can even (for low R) exceed themaximumMichaudvaluecorrespondingtothecriticalpartialgateclosure, as deduced by Gariel (equation (1) with KC=2.0).

Conclusion

Empirical formulae or simpliedmodels forhydrotransient anal-ysis of a long hydraulic circuit, based only on the closure time of the guide vane, can not be accurate enough for design purposes.However, it can be very difcult to obtain the complete set of theturbine characteristic curves in the early stages of the design, de-pending on the available manufacturers data.From the practical operational point of view, the application of the dynamic orice technique, in real case studies, has enabledmore reliable and economical layout solutions for long hydraulic

conveyance systems, in most cases of small hydropower plantsavoiding costly protection systems against waterhammer. Com-

puter simulations based on DO model take few seconds or min-utes, depending on the type of the hydraulic system (e.g. a totalpressurised pipe or a mixed system, with part in open canal fol-lowedby thepenstock), using a PC computer, includingtheinter-action between different elements of the hydropower scheme.Thus, the dynamic orice conception is a powerful aid for hydro-transient analysis and for understanding the conveyance systemresponse(e.g. canalandpenstock) andpowerhousedesigndepen-

dence of hydraulic phenomena parameterisation. The allowablepenstock pressure, the guide vane manoeuvre time and the polarinertia moment can be specied before the full machine charac-teristics are known, in order to control excessive overpressures,especially for the critical full load rejection of low specic tur-bine speed.Thefeasibility of a small hydropower project depends heavily oncivil construction costs and on environmental impacts, requiringaccurate computational simulations and cost-benet studies inorder to mitigate these factors. With this integrated model moreefcient solutions can be selected from among different alterna-tives at an early but very important design stage.

References

Almeida , A.B., Koel l e , E. (1992) -Fluid Transients in Pipe Networks , C. M. P., Elsevier.

Bahamonde , R. (1991) -Predicting the Least Closing Time of Hydraulic Turbines . Water Power & Dam Construction, pp.43-47.

Brekke ,H.(1976) A Study of the Inuence of TurbineCharac-teristics on Turbine Governing. 2nd International Conferenceon Pressure Surges, BHRA, London, paper J2.

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Boldy , A. P. (1976) -Waterhammer Analysis in HydroelectricPumped StorageInstallations , 2ndInternational ConferenceonPressure Surges, BHRA, London, paper B1.

Buchi , G. (1957) -Le Moderne Turbine Idrauliche, ed . IRegolatori di Velocit, Hoepli.

Chaudhry , M.H. (1987) -Applied Hydraulic Transients . Sec-ond Edition. Van Nostrand Reinhold Company.

Chaudhry , M.H. (1980) -A Non-linear Mathematical Model for Analysis of Transients Caused by a Governed Francis Tur-bine. 3rd International Conference on Pressure Surges, BHRA,Canterbury, paper G1.

Franco , A. B. (1996) Computational and Experimental Mod-elling of ows provoked by Dam Break . Ph. D. Thesis (in Por-tuguese). IST, Lisboa.

Faza lare , R.W. (1991) -Five Technical Recommendations for Hydro Machinery . Water Power and Dam Construction, pp.18-20.

Lei n , G. (1965) -The Inuence of Waterhammer on the Designand Operation of Pumped Storage Plants . Proceedings of theInternationalSymposiumonWaterhammerin Pumped StorageProjects, ASME, pp.96-122.

Mataix ,Claudio(1975)-Turbomaquinas Hidrulicas .EditorialICAI, Madrid.

Nai -Xi ang , C., Zu -Yan , M. (1989) -A Numerical Method for Determining the Perfect Closure Rule of Turbine Guide Vanes ,in The Current State of Technology in Hydraulic Machinery,edited by A.P.Boldy and D. Chang Guo, Gower Technical andScience Press, Chapter 4.

Pe jovic , S., Boldy , A.P., Obradovic , D. (1987)- Guidelinesto Hydraulic Transients Analysis , Technical Press.

Raabe ,J.(1985)- Hydro Power: The Design, Use, and Functionof Hydromechanical, Hydraulic, and Electrical Equipment .Dusserldorf. VDI - Verlag.

Ramos , H. (1995)- Simulation and Control of Hydrotransientsat Small Hydroelectric Power Plants . Ph.D. Thesis. TechnicalUniversity of Lisbon, Portugal (in Portuguese).

Remenieras , G. (1961) Maurice Gariel and Water Hammer Research. La Houille Blanche,N 2,March-April,pp.156-167(in French)

Ruu s ,E.(1966)-Optimum Rate of Closure of Hydraulic TurbineGates , ASME - EIC Fluids Engineering Conference, Denver.

Taulan , J.P. (1983) -Pressure Surges in Hydroelectric Installa-tions: Peculiar Effects of Low Specic Speed Turbine Charac-teristics , 4th International Conference on Pressure Surges,BHRA, Bath, paper H2.

Wozniak ,L.,Fet t ,G.H.(1973)-Optimal GateClosureSched-ule for Hydroelectric Turbine System , Journal of Fluids Engi-neering, ASME, Vol. 45, n1.

Wyl i e , E., Streeter, V. (1993) -Fluid Transients in Systems .Prentice Hall.

Nomenclature

The following symbols are used in this paper:A area of pipe cross-section (m2);Cg gate coefcient (-);Cs runner speed coefcient (-);c wave celerity (m/s);g gravitational acceleration (m/s2);H net head (m);Ho gross head (m);HR turbine rated head (m);

hw Allievi parameter (-);

=

ow Hg2

cVh

I polar moment of inertia of rotating masses (kg m2);L pipeline length (m);N runner speed (rev/min);NR turbine rated speed (rev/min);N

RWturbine runaway speed at rated head (rev/min);

Ns specic speed (rev/min);4 / 5

R

2 / 1

mxRsH / PNN =

PR rated turbine power (kW);Pmax maximum power for rated head (kW);Q turbine discharge (m3 /s);QR turbine rated discharge (m3 /s);QRW turbine discharge at runaway speed (m3 /s);

R friction loss coefcient (m-5s2);2QxJ

R=

TC guide vane complete closure time (s);

TE elastic time constant (s);c

L2TE =

TG electromagnetic torque (N m);TH hydraulic motor torque (N m);... rated hydraulic motor torque (N m);

Tm machine starting time (s), with WD2 =3R

2R

2

m 10P3575NWD

T =4gI;

TW inertia time constant (s);o

W gHLV

T =t time (s);V ow velocity (m/s);WH head turbomachinecharacteristics (Suterparameters) (-);WT torque turbomachine characteristics (Suter parameters)

(-);x distance along the pipe (m);y dimensionless gate opening (-);HM maximum head variation (m); unit efciency (-);R unit rated efciency (-);O unit efciency for initial ow condition (-); water mass density (kg/m3);

angular velocity (rad/s).60

N2=

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IAHR2001