MODELING OF THE DYNAMIC RESPONSE OF A FRANCIS TURBINE

Transcription

1 MODELING OF THE DYNAMIC RESPONSE OF A FRANCIS TURBINE Paolo PENNACCHI, Steven CHATTERTON *, Andrea VANIA Politecnico di Milano, Deartment of Mechanical Engineering - Via G. La Masa, Milano, Italy. KEYWORDS: hydroelectric ower lant model, ressure waves, Francis turbine. ABSTRACT The aer resents a detailed numerical model of the dynamic behaviour of a Francis turbine installed in a hydroelectric lant. The model considers in detail the Francis turbine with all the electromechanical subsystems, such as the main seed governor, the controller and the servo actuator of the turbine distributor, and the electrical generator. In articular, it reroduces the effects of ieline elasticity in the enstock, the water inertia and the water comressibility on the turbine behaviour. The dynamics of the surge tank on low frequency ressure waves is also modeled together with the main governor seed loo and the osition controllers of the distributor actuator and of the hydraulic electrovalve. Model validation has been made by means of exerimental data of a 75 MW - 47 m hydraulic head - Francis turbine acquired during some starting tests after a artial revaming, which also involved the control system of the distributor. NOMENCLATURE ω Deviation of rotational seed in.u. G Deviation of gate oening in.u. H t Deviation of turbine hydraulic head in.u. P m Deviation of mechanical ower in.u. U t Deviation of water velocity in.u. φ Friction energy term of the enstock φ Friction coefficient of the tunnel c * Corresonding author: steven.chatterton@olimi.it, h: , fax:

2 ρ Volumetric mass of water τ Time constant of the servovalve ω Rotational seed in oerating condition A Penstock area,,, b, bt Tx Td T 3 Gains and time constants of the seed controller in oerating mode C I Proortional gain of the gate oening controller C x Flow gain of the servovalve c Wave velocity in the enstock D Daming factor D ck Diameter of the k-th tunnel (k =,2) D Penstock diameter e ω Rotational seed error in.u. E Young s modulus of elasticity of ie material f Thickness of ie wall g Gravitational acceleration G, G Gate oening reference signal and.u. form G Actual gate oening H,( H ) Hydraulic head at gate (in oerating condition) I Current command inut of the servovalve J Generator inertia k Gain constant of the servovalve k b Hydraulic cylinder net area constant k f Friction coefficient of the enstock k, k i Gains of the PI seed controller in starting mode K Bulk modulus of water comression K i Inertia constant Ku, K Velocity and ower constants L ck Length of the k-th tunnel (k =,2) L Penstock length Pm,( P m) Turbine mechanical ower (in oerating condition) t Water ressure at gate q oil Oil flow in the hydraulic cylinder Q Turbine water flow in oerating condition T a Mechanical starting time T Electrical load torque in.u. e T Elastic time of the enstock e T Mechanical torque of the turbine in.u. m T Time constant of the surge tank s T Starting time of the tunnel Wc 2

3 T W Water starting time of the enstock U,( U ) Water velocity (in oerating condition) x Pilot valve sool osition Z Penstock normalized hydraulic surge imedance Z Hydraulic surge imedance of the enstock INTRODUCTION The accurate dynamic model of turbine units in hydroelectric lants assumes great imortance in case of renewal of turbine comonents, such as the control and the actuator systems, or in case of eriodic and required insection of the emergency systems. In this sense, a model of the entire system revents dangerous conditions during the tuning of the control system and rovides a reference resonse for insections. This kind of model could be also used during design hases, for enstock dimensioning, for the overseed estimation and for the analysis of new control strategies of the seed governor. The first studies about ower system of hydro turbines date back to the early 7s, when a task force on overall lant resonse was established in order to consider the effects of ower lants on ower system stability and to rovide recommendations regarding roblems not already investigated. The outcome of this effort has been a first simle model consisting in transfer functions of the seed governing and the hydro turbines systems []. However, these early models were inadequate to study large variations of ower outut and frequency. For instance, they were not reliable in the very low frequency range, as they did not account for the water mass oscillations between the surge tank and the reservoir, and at high frequency [2], as they did not reroduce water hammer effects. To solve these limitations, some imrovements of both turbine and seed control models were made by the Working Grou on Prime Mover and Energy Suly Models for System Dynamic Performances Studies [3]. In this aer, the authors roose a detailed and comlete numerical model of the dynamic behaviour of a Francis turbine of a hydroelectric lant. The model considers in detail the Francis turbine with all the electromechanical subsystems, such as the main 3

4 seed governor, the controller and the servo actuator of the turbine distributor, and the electrical generator. In articular it reroduces the effects of ieline elasticity in the enstock, the water inertia and the water comressibility on the turbine behaviour. Among the different ossible aroaches for the ressure wave modelling, the transfer function method is emloyed in this aer, allowing testing different closing manoeuvres of turbine distributor erformed by the control system. The dynamics of the surge tank and of the reservoir on low frequency ressure waves are also modelled together with the main governor seed loo and the osition controllers of the distributor actuator and of the hydraulic electrovalve. The roosed model has been successfully validated by means of exerimental data of a 75 MW - 47 m hydraulic head - Francis turbine of a hydroelectric lant, acquired during some starting tests after a artial revaming, which also involved the control system of the distributor. 2 PLANT DESCRIPTION The Francis turbine unit here considered belongs to a comlex hydroelectric basin, in southern Italy, consisting of three hydroelectric lants, laced in series. Surge tank 2 Surge tank Tunnel Reservoir Tunnel 2 H Penstock Turbine Turbine 2 Figure Hydraulic scheme of the first lant. The first hydroelectric lant, shown in Figure is constituted of two 75 MW Francis turbines working at 6 rm, with a hydraulic head of about 47 m. The two verticalaxis turbines are sulied by means of a single steel enstock from two interconnected surge tanks, each of them connected to a lake reservoir by two rocky tunnels. 4

5 The discharged water of the first lant sulies a second reservoir used by the second lant and so on also for the last lant. The main data of the lant are reorted in Table. Table. Data of the hydroelectric central L c Tunnel length 49 m D c Tunnel diameter 2.5 m L c2 Tunnel 2 length 4 m D c2 Tunnel 2 diameter 3.2 m L Penstock length 22 m D Penstock diameter 3 m f Thickness of ie wall.2 m Q t Penstock overall rated flow 35 m 3 /s H Hydraulic head m J 2 PD Generator inertia PD kgm 2 Q Single turbine rated flow 7-8 m 3 /s P r Single turbine electric ower MW ω Rated seed 6 rm A sherical valve, moved by a hydraulic actuator, is laced at the end of the enstock as shown in Figure 2. This valve oerates only during starting, emergency hases and stoing. 5

6 Figure 2 Sherical valve at the end of the enstock. The generator is an 8 MVA- kv three-hase synchronous machine running at 6 rm. The uer art of the exciter system of the generator is shown in Figure 3. Figure 3 Uer art of the generator exciter system. 6

7 The considered lant is an early restoration lant, namely one of the hydroelectric lants able to restore a ower system after a blackout through reestablished ath called restoration lines. The turbines of these stations are equied with secial seed governors for that urose. The rotational seed of the Francis turbine is controlled by modifying the water flow in the turbine, by means of the distributor system. The distributor consists of steerable blades laced before the inlet of the runner and interconnected together by means of a ring. The rotation of the blades, erformed by the ring, modifies the section areas of the guide vanes that is the flow in the turbine. The rotation of the ring is realized by means of a hydraulic cylinder (servomotor) as shown in Figure 4. Figure 4 Distributor system of the Francis turbine. As said before, the exerimental tests, described in the next section, were erformed after the renewal of the control system unit (Figure 5) of the servomotor. 7

8 Figure 5 Hydraulic control unit of the servomotor. 3 MODEL OF THE SYSTEM The functional block diagram of the hydroelectric lant is shown in Figure 6, where only a single turbine unit is considered [4]. It mainly consists of the seed changer, the seed regulator, the distributor, the turbine and the generator. For the safety of the electric network, the frequency should remain almost constant. This is ensured by keeing constant the seed of the synchronous generator. The seed changer (external central control) gives the reference seed signal, deending on the ower network requirements, since the frequency deends on the active ower balance of the network. The rotational seed of the turbine is fed back and modified by the seed governor acting on the turbine distributor (gate) by means of the hydraulic servomotor. The turbine mechanical ower is essentially a function of gate osition. 8

9 External seed changer Frequency - Power Water suly Electric network Seed controller Servomotor Distributor Turbine Generator Rotational seed Figure 6 Functional block diagram of the overall system. The details of each submodel shown in Figure 6 are analysed in the following sections. 3. Seed controller model For a stable division of the load between two or more generating unit oerating in arallel, the seed governors are generally rovided with a ermanent droo characteristics so that the seed dros when the load decrease [5],[6]. The simle resence of the droo characteristics on the regulator causes a steady-state frequency error that could be reduced oerating comensation in the regulator scheme. A large transient (temorary) droo [7] with a long resetting time is also included in the regulator transfer function by means of a suitable transfer function in order to imrove the stability of the control. The actual seed regulator imlemented consist of two oerating modes: the starting mode for run-us from standstill to steady-state without load and the oerating mode for nominal oerating condition at rated seed and with an active load. The switching from the two oerating mode is done in a bumless way. The starting mode consists of a simle PI regulator including an integral anti windu in the imlemented algorithm: G ki Fstarting ( s) = = k + () e s ω where G is the normalized gate oening reference signal and e ω is the error of the normalized rotational seed. The oerating mode is reresented by a two-ole and two-zero transfer function: 9

10 F oerating ( s) T x s + G bt eω b T Ts x 3 s + b = = ( Ts d + ) ( + ) The final control scheme including the seed comensations is shown in Figure 7, where two comensation terms are added to the reference signal of the gate oening. The steady-state comensation allows reducing the steady-state rotational seed error due to the droo characteristics, whereas the variable comensation is a feed-forward term deending on the electrical load. (2) starting mode k i s l u l low Variable (load) comensation k + + e ω b T x s + bt T x s + b ( Ts d + ) ( Ts+ ) Steady-state comensation G oerating mode Figure 7 Seed controller block diagram. 3.2 Servomotor system model The outut of the seed regulator is roortional to the reference gate oening signal. The gate oening is actuated by the hydraulic cylinder (servomotor) that rotates the ring of the distributor allowing flow changes. The servomotor is osition controlled, by acting on the cylinder oil flow by an electrovalve that receives as inut the reference gate oening signal. The seed of the cylinder stem is roortional to the sulied oil flow in the cylinder chamber. Thus, the simlest first order transfer function of the cylinder that relates the actual gate oening G to the oil flow q oil could be written as:

11 ( ) ( ) G s Fc( s) = = k (3) b q s s oil where k b is a suitable constant reresenting the cylinder net area and the linear assumed relationshi between the cylinder osition and the gate oening. sool: The oil flow could be also assumed roortional to the osition x of the ilot valve q oil = Cx (4) x where C x is the flow gain of the electrovalve. A osition control is also actuated to the sool of the servovalve. The behaviour of the servovalve could be described by a first order transfer function relating the sool osition x to the current inut I : ( ) ( ) I s k FI x( s) = = x s τ s+ (5) where τ is the time constant of the servovalve and k a suitable constant. A simle roortional osition control of the gate oening is imlemented in the lant controller: ( ) ( ) I s = C G G (6) I The block diagram of the gate oening control including the servovalve and the servomotor transfer functions is shown in Figure 8. gate controller servovalve x max G k G max max G + I C x qoil G I C τ s + x C b G x s min G min G min servomotor Figure 8 Block diagram of the gate osition control loo. The nonlinearities due to the limits on the osition of the servovalve and to the limits on the seed and osition of the gate are also reorted in Figure 8.

12 3.3 Turbine model The dynamic behaviour of a hydraulic turbine oerating at full load is widely described by a transfer function that relates the deviation of the outut mechanical ower to the deviation of gate oening [5],[8]. The turbine-distributor system is modelled as a valve, where the velocity U of the water at gate is given by: where U= KG u H (7) The turbine mechanical ower is roortional to the roduct of ressure and flow: K u and P m = K HU (8) K are roortional constants, G is the gate (distributor) oening, and H is the hydraulic head at the gate. By linearizing and considering both small dislacements about the oerating oint and.u. exressions, it follows: + Pm ( s) F( s) = G( s) 2 F s ( ) (9) where Pm = Pm Pm, is the.u. small deviation of the hydraulic (or mechanical) ower (corresonding to the.u. mechanical torque), the gate oening and F( s) U ( s) H ( s) t t G is the.u. small deviation of = is the transfer function that relates the normalized flow (corresonding to the.u. water seed) to the normalized hydraulic head deviation at gate Classical model The classical model of the ideal turbine is obtained considering [],[9]: ( ) TWs F s = () where T W is the water starting time (or water time constant) of the enstock at rated load, that deends on the load conditions. The starting time reresents the time required 2

13 for a total head H to accelerate the water in the enstock of length to the velocity U : L from standstill T LU W = () gh Different values of T W are used in the model deending on the load condition: 3 T =.65s for full load condition ( Q, full = 7.5 m s ), whereas T =.65s for the W 3 idling condition ( Q, idling =.75 m s ). Equations (9) and () describe the behaviour of a simle ideal linear model of hydraulic turbine for small deviations from steady-state oerating oint, considering negligible hydraulic resistance, incomressible water and inelastic enstock ie. This model is only a medium-low frequency aroximation, with relevant errors in the high frequency range, because it does not include the water hammer henomenon. The model is not also entirely adequate in the very-low-frequency range, as it does not account for the water-mass oscillations in the tunnel (considered as inelastic) that connects the surge tank and the reservoir [9]. W Detailed model The effects of the travelling waves owing to the elasticity of enstock steel and of the water comressibility could be considered by means of the method of characteristics [],[] or by means of a transfer function aroach. The first method requires the integration of artial differential equations by means of finite difference method and allows the time evaluation of the hydraulic head and seed in several oint of the enstock. Instead, the transfer function aroach allows only the evaluation of the same quantities at the turbine level [2]. Considering the resence of the surge tank, the overall and detailed transfer function to be used in eq. (9) becomes [5]: ( ) F s t = = ( ) F s + tanh( Tes) U Z H φ + F s + Z T s ( ) tanh( ) t e F ( s ) is the transfer function that describes the tunnel and surge tank interaction: (2) 3

14 ( ) F s H φ + st = = r c Wc 2 U + stsφc + s TWcTs (3) T Wc is the starting time of the tunnel, T s the time constant of the surge tank and φ c the friction coefficient of the tunnel; T e is the elastic time of the enstock of length A = π D 4 : 2 L, diameter D and area c is the wave velocity in the enstock given by: T L e = (4) c c g = (5) α α considers the water comressibility and the ie elasticity: D g K Ef α ρ = + K is the bulk modulus of water comression, E the Young s modulus of elasticity of ie material and f the thickness of ie wall; Z is the normalized value of the hydraulic surge imedance of the enstock Z given by: (6) Z Q = Z H Z is the hydraulic surge imedance of the enstock: (7) φ reresents the friction energy term of the enstock: Z c = (8) Ag The term tanh( e ) φ = 2k U (9) T s in eq. (2) could be written as: f 4

15 tanh ( Tes) 2 ste st 2 e + Tes e n nπ = = = 2Tes 2 + e 2sT e + n= ( 2n ) π The elastic time is related to the time constant by: (2) where T = ZT (2) W e The ressure at gate could be easily evaluated by: t ( t ) = H + H (22) Ht, deending on the gate oening, could be evaluated by the following transfer function: F ( s) H G t = = ( ) F s 2 (23) Simlified model Excluding the surge tank and considering in this way only the high-frequency effects of the water hammer, the following equation should be used [5]: ( ) F s Ut = = H φ + Z tanh T s ( ) t e (24) 3.4 Exerimental enstock ressure All the quantities resented in the revious equation have been evaluated by means of available data and by exerimental data. For instance, the exerimental gate oening described by the servomotor osition and the exerimental and the simulated enstock ressure at gate are reorted in Figure 9. In articular, the model of the lant including the turbine, the enstock, the surge tank and the reservoir has been tuned using the exerimental gate oening reorted in the uer art of Figure 9 as inut of eq. (22) and evaluating the simulated enstock ressure at gate by means of eq. (23). In the lower art of Figure 9, both the exerimental and the simulated enstock ressures at gate are 5

16 reorted. It is ossible to observe the long eriod oscillation of the water ressure at gate of about 225 s, due to the reservoir and the surge tank interaction, and the water hammer henomena at time t=375 s, t=45 s and t=75 s. The water hammer henomena at t=5 s, t=575 s and t=8 s have not been identified by the model and could be ascribed to the interaction of the water with the other elements of the systems not included in the model such as the by-ass conduit, the main sherical valve and the conduit of the second ower unit..2.5 Servomotor osition Ref. [%] [bar] Penstock ressure Ex. Mdl time [s] Figure 9 Exerimental gate oening (uer) and simulated and exerimental enstock ressure at gate (lower). 3.5 Load model The simle torque equilibrium equation at the turbine shaft allows the insertion in the model of both the electrical load torque T e (.u.) and the inertia of the rotor: ω = Ts+ D a ( Tm Te) (25) where T m is the.u. mechanical torque given by the turbine, D the daming factor, 6

17 ω the.u. rotational seed deviation and T a the mechanical starting time. The mechanical starting time is obtained by the inertia constant K = T 2 that it is defined by the ratio of the kinetic energy of the rotor at rated seed ω and the rated electrical ower of the generator ( VA ) : 2 Jω Ki = (26) 2 ( VA) Inertia J and daming D change deending on the oerating conditions in the model. For instance, during a closure manoeuvre of the gate, both J and D decrease due to a reduction of the water resent in the runner. The mechanical starting time T a linearly changes in the range 8.5 9s, whereas D in.6. as function of the normalized gate oening limits. In this way, the model is linearized about different steady-state oerating oints. A change of the daming factor also includes nonlinear variation of the turbine efficiency for different oerating oints [2]. Regarding the electrical load, all electrical devices such as the generator, the transformers and the ower network have not been modelled in detail [3]. The exerimental electrical load T e has been used as inut in the model simulations. i a 4 EXPERIMENTAL TESTS As above mentioned, the considered lant is an early restoration lant and a artial revaming involved the control system of the distributor. For these reasons, the exerimental data are acquired during some insection tests of restoring rocedures of the lant control system. In articular, they were aimed at verifying the roer working of the overall system and have been erformed by sending suitable signals to the control system. For instance, comlete closures of the main sherical valve or some disturbances on the reference seed signal have been simulated. In articular, two different sets of tests have been carried out only on one of the two units, with the sherical valve of the second unit comletely closed. The first set has regarded the correct working of the revamed control system of the servomotor, whereas the latter has been carried out on the overall system in order to test several oerating conditions. 7

18 4. Tests on the control system of the servomotor The tests on the revamed servomotor control system have been carried out in standstill conditions with the main valve closed and without electrical load. The exerimental data acquired during these tests have been used to tune the arameters and the time constants of the described models for the gate osition controller, the servovalve and for the servomotor. Considering the control block diagram of the gate osition reorted in Figure 8, the tests have been erformed alying square and triangular wave reference signals as inut of the outer gate osition control loo G or directly as current inut I of the servovalve in the inner loo. The exerimental simulated data for reference current signals alied as inut of the servovalve controller are reorted in Figure. The osition of the sool of the servovalve and the osition of the stem of the hydraulic cylinder that moves the distributor are reorted in the figure. In the left art of Figure, the reference signal of the servovalve is a full square current wave in the range of 4-2 ma with a eriod of 6 s. The servovalve sool reaches its hysical limit of - to mm, where the stem cylinder shows its maximum ositive and negative seeds. The maximum stroke of about 35 mm of the cylinder corresonding to gate oening of G =, is considered in the model. The reference signal of the servovalve is a triangular current wave in the range of ma with a eriod of 2 s in the right art of Figure. 8

19 I - Servovalve current inut I - Servovalve current inut [ma] Ref Ref. 5 5 x - Servovalve sool osition x - Servovalve sool osition.5.5 [mm] G - Servomotor osition G - Servomotor osition [mm] 5 Ex. Mdl time [s] 5 Ex. Mdl. 5 5 time [s] Figure Tests on the inner loo of gate osition control scheme: % square current wave (left) and triangular wave (right) reference signal as inut of the servovalve. The exerimental characteristic of the servovalve, which relates the servovalve sool osition x to the oil flow q oil in the cylinder, as exressed by the ideal and linear relation of eq. (4), has been obtained by means of tests erformed on the inner osition loo of the servovalve. These results are shown in Figure, where it is ossible to observe the nonlinearity across the middle oint of the servovalve sool. 9

20 .8 Servovalve characteristics q oil x Figure Characteristics curve of the servovalve obtained from exerimental data. 4.2 Tests on the overall system The second set of tests has been carried out on the overall system in order to test several oerating conditions: A) starting rocedure from standstill to rated seed, idling electrical load; B) ste reference seed signals at rated seed then sto in idling electrical load; C) full sto from rated condition (seed and load) closing both to the distributor and the sherical valve; D) full load cut out at rated seed. For the sake of brevity, only the first and the last interesting oeration conditions are reorted in the following subsections Starting from standstill to rated seed condition The rotational seed, the gate oening and the ressure in the enstock at the sherical valve are reorted in Figure 2, for the starting rocedure from standstill to the rated seed in absence of the electrical load where the seed controller is the PI of eq. (). The rotational seed reaches the rated value in about s, where the model shows a negligible seed overshoot with resect to the exerimental data. The rated 2

21 condition in absence of the load is reached with a artial oening of the gate (about 2%). [rm] Rotational reference seed Rotational seed 6 Ref. [rm] [%] Gate oening 4 Ex. 2 Mdl. [bar] Penstock ressure time [s] Figure 2 Rotational seed, gate oening and enstock ressure at gate for the starting test from standstill to rated seed. An emergency condition is simulated at time 7 s, by utting to zero the reference seed signal of the seed regulator and consisting in a comlete closure of the distributor in about.5 s; accordingly, the rotational seed decreases. As already mentioned in the revious section, some water hammer henomena, not described by the model, occur at about time t=5 s and t=575 s. 2

22 4.2.2 Full load cut out at rated seed condition The electrical load cut out at rated seed is analysed in this case. This condition is extremely dangerous because a turbine overseed occurs, as shown in the uer diagram of Figure 3, where the rotational seed, the gate oening, the enstock ressure and the active electrical ower are also reorted. The electrical load is linearly alied starting from time t=72 s till t=95 s and is comletely removed at time t=37 s as aears from the last diagram of the figure, by disconnecting the generator from the ower network. The exerimental overseed is about 4% whereas the simulated one is about 5% of the rated seed. The difference between the exerimental and the simulated overseed could be ascribed to the nonlinearity of the daming of the system described by eq. (25). 22

23 [rm] Ex. Mdl. Rotational seed Gate oening [%] 5 [bar] Penstock ressure Active ower [MW] time [s] Figure 3 Rotational seed, gate oening, enstock ressure at gate and active electrical ower for the load cut out at rated seed condition. The actual and the ideal straight line seed-droo characteristics of the considered Francis turbine are shown in Figure 4. 23

24 Frequency [Hz] Power outut [MW] Figure 4 Actual and ideal seed-droo characteristics of the Francis turbine. 5 CONCLUSIONS A detailed model of the overall lant of a Francis turbine hydroelectric unit has been designed and successfully tested as described in the aer. The model is develoed with the transfer function aroach and includes the servovalve-servomotor control loo, the effects of the travelling waves between the reservoir and the surge tank as well as the water hammer henomenon. It was tuned and verified by means of exerimental data acquired during some insection tests erformed after a revaming of the distributor control unit and used as reference resonse to validate the required test of restoring rocedure of the same lant. The model is very useful for simulating critical conditions as overseed and water hammer henomenon. 6 REFERENCES [] Byerly R.T., Aanstad O., Berry D.H., Dunlo R.D., Ewart D.N., Fox B.M., Johnson L.H., Tschaat D.W., Dynamic models for steam and hydro turbines in ower system studies, IEEE Transactions on Power Aaratus and Systems, Vol.PAS-92, No.6, (973), [2] Oldenburger R., Donelson Jr. J., Dynamic Resonse of a Hydroelectric Plant, 24

25 Transaction of the American Institute of Electrical Engineers, Vol.8, No.3 (962), [3] Working Grou on Prime Mover, Hydraulic turbine and turbine control models for system dynamic studies, IEEE Transactions on Power System, Vol.7, No. (992), [4] De Jaeger E., Janssens N., Malfliet B., Van De Meulebroeke F., Hydro Turbine model for system dynamic studies, IEEE Transactions on Power Systems, Vol.9, No.4, (994), [5] Kundur P., Power system stability and control, (994), EPRI Editors. [6] Schleif F.R., Wilbor A.B., The coordination of hydraulic turbine governors for ower system oerations, IEEE Transactions on Power Aaratus and Systems, Vol.PAS-85. No.7, (965), [7] Undrill J.M., Woodward J.L., Nonlinear hydro governing model and imroved calculation for determining temorary droo, IEEE Transactions on Power Aaratus and Systems, Vol.PAS-86, No.4, (967), [8] Hannett L.N., Feltes J.W., Fardanesh B., Field tests to validate hydro turbinegovernor model structure and arameters, IEEE Transactions on Power System, Vol.9, No.4, (994), [9] Ramey D.G., Skooglund J.W., Detailed hydrogovernor reresentation for system stability studies, IEEE Transactions on Power Aaratus and Systems, Vol.PAS- 89. No., (97),.6-2. [] Larock B.E., Jeson R.W., Watters G.Z., Hydraulics of ieline system, CRC Press, (2). [] Afshar M.H., Rohani M., Water hammer simulation by imlicit method of characteristic, International Journal of Pressure Vessels and Piing, Vol.85, No.2, (28), [2] Tzuu Bin Ng, Walker G.J., Sargison J.E., Modelling of transient behaviour in a Francis turbine ower lant, 5th Australian Fluid Mechanics Conference, (24),.-4. [3] Okou F.A., Akhrif O., Dessaint L., A robust nonlinear multivariable controller for multimachine ower systems, Proceedings of the 23 American Control Conference, Vol.3, (23),

26 Figure Cations: Figure Hydraulic scheme of the first lant. Figure 2 Sherical valve at the end of the enstock. Figure 3 Uer art of the generator exciter system. Figure 4 Distributor system of the Francis turbine. Figure 5 Hydraulic control unit of the servomotor. Figure 6 Functional block diagram of the overall system. Figure 7 Seed controller block diagram. Figure 8 Block diagram of the gate osition control loo. Figure 9 Exerimental gate oening (uer) and simulated and exerimental enstock ressure at gate (lower). Figure Tests on the inner loo of gate osition control scheme: % square current wave (left) and triangular wave (right) reference signal as inut of the servovalve. Figure Characteristics curve of the servovalve obtained from exerimental data. Figure 2 Rotational seed, gate oening and enstock ressure at gate for the starting test from standstill to rated seed. Figure 3 Rotational seed, gate oening, enstock ressure at gate and active electrical ower for the load cut out at rated seed condition. Figure 4 Actual and ideal seed-droo characteristics of the Francis turbine. 26

27 Table Cation: Table. Data of the hydroelectric central 27