Simulation of hydroelectric system control. Dewi Jones

Transcription

1 Simulation of hydroelectric system control Dewi Jones

2 Simulation of hydroelectric system control Dewi Jones Report GW2 October 28 GWEFR Cyf Technium CAST Ffordd Penlan Parc Menai Bangor Gwynedd LL57 4HJ Wales Tel: +44 () Care has been taken in the preparation of this report but all advice, analysis, calculations, information, forecasts and recommendations are supplied for the assistance of the relevant client and are not to be relied upon as authoritative or as in substitution for the exercise of judgement by that client or any other reader. Neither GWEFR Cyf, nor any of its personnel engaged in the preparation of this Report shall have any liability whatsoever for any direct or consequential loss arising from the use of this Report or its contents and give no warranty or representation (express or implied) as to the quality or fitness for the purpose of any process, product or system referred to in the Report. Copyright in this Report remains the sole property of GWEFR Cyf.

3 Contents Introduction... 2 Outline of the simulation Physical layout Model summary Simulation of hydraulic system with fixed rate GV opening Lumped parameter models Lumped parameter, inelastic water column model Linearised model Control system design Closed loop simulation with Grid connection Frequency control mode Power dial up mode Full load rejection Conclusion... 9

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5 Introduction This sample report outlines how the generic simulation package offered by GWEFR Cyf can provide valuable information about the dynamic response of a proposed hydroelectric installation, given just the basic information about the Plant s physical layout, its primary parameters and modes of operation. The objective is to increase understanding of how the proposed hydraulic and electrical systems affect the dynamic response of the Plant and hence the quality of power generation an approach advocated in IEEE Standard IEEE Guide for the Application of Turbine Governing Systems for Hydroelectric Generating Units []. The following section presents a simplified version of the hydraulic system, composed of a reservoir supplying two Grid connected Francis turbine/generators via a common tunnel, a manifold and two identical penstocks. This is used as the basis of a travelling wave model that estimates the pressure and flow variations in the water passages under defined rates of opening or closing of the turbine guide vanes. Comparisons are made between the responses produced by three different types of model, as suggested by an expert IEEE Working Group [2]: Nonlinear distributed parameter, elastic water column the most accurate model but computationally intensive; Nonlinear lumped parameter, inelastic water column requires less computation but does not represent travelling wave effects; Linearised model suitable for control design. Standard tuning rules are applied to design a PID controller based on the linearised model. The resulting governor is integrated with models for the hydraulics, turbine/generator and Grid to study system performance for 3 scenarios: Closed loop frequency control mode Dial up power mode with feed forward Full load rejection The report provides constructive input to the requirements capture stage of the project and forthcoming design efforts. 2 Outline of the simulation 2. Physical layout Figure shows the general layout of the hydraulic system assumed in the model. The reservoir is connected by a long supply tunnel to a manifold that separates into two identical penstocks, which feed identical turbine/generators. The tailrace is not included in the model at this stage. Note that the layout is not quite symmetrical because the water path to Unit #2 has additional length due to the manifold. The dimensions are shown in Figure 2.

6 Figure General layout of the hydraulic system. 53m 88m Øm m Ø9.5 m 3m Ø3.3m 3m Ø3.3m 8m Ø9.5m Figure 2 Cross section and plan views showing dimensions (datum taken at turbine inlet). 2

7 2.2 Model summary Hydraulic system: The 4 pipe sections are modelled using the Method of Characteristics [3] which is used to compute the time variation of piezometric pressure and uniform flow at discrete points in the network. All head losses except for friction are considered negligible. There are 5 boundary conditions: The head at the reservoir outlet is considered constant. At the branch between tunnel, manifold and penstock # the head is constrained to be identical for all 3 branches and continuity of flow is applied. Similar conditions are applied at the junction between the manifold and penstock #2. Both turbines are modelled as idealised lossless control components where the relationship between head and flow is given by: H u 2 V = Atb 2 G (2.) where H u = head across the turbine V = flow through the turbine A tb = turbine constant G = guide vane opening ( : ) For a flow, Q, the mechanical output power of a turbine is given by: P = QH ρg (2.2) m u Electrical system It is assumed that the generators are Grid connected and represented by a linearised version of the classic swing equations as given, for example, by Kundur [4]: ω = 2Hs ( P P K ω ) r m e D r K ω P = s ( ω P ) s e r L (2.3) where: ω G = Grid frequency ω r = Unit (electrical) speed P L = load change on Grid P m = turbine mechanical power P e = generator electrical power ω, ω r = rated and actual Unit speed K s = synchronising coefficient K D = speed damping coefficient H = inertia constant s = Laplace variable 3

8 Power system The U.K. Grid is modelled as a second order system based on the study by Jones [5]: 2 G( s) ( / βω ) n = 2 2 e( ) 2ζωn ωn ω P s s + s+ (2.4) where β is the Grid stiffness. Parameter values Along with the dimensions in Figure, the parameter values used in the simulation are summarised in Table : Table Simulation parameter values Parameter Symbol Value Per unit base of volumetric flow Q b 65 m 3 /s Per unit base of speed ω b π r/s Per unit base of head H b 53 m Acceleration due to gravity g 9.8 m/s 2 Density of water ρ kg/m 3 Per unit base of power P b = 6 Q b H b gρ 327 MW Wave velocity a 43 m/s Darcy Weisbach friction coefficient for tunnel & manifold f, f 2.5 m/ (m 3 /s) 2 Darcy Weisbach friction coefficient for penstocks f 3, f 4. m/ (m 3 /s) 2 Turbine constant A tb m/(m/s) 2 Turbine / Generator inertia parameter H 3.95 s Generator synchronizing torque coefficient K s.7 pu/pu Generator speed damping coefficient K D.3 pu/pu Grid stiffness β 723 MW/.Hz Grid natural frequency ω n.37 r/s Grid damping factor ζ.38 3 Simulation of hydraulic system with fixed rate GV opening In order to illustrate the kind of information that the simulation can produce, the hydraulic system was set up for the following scenario: Turbine #2, whose penstock is at the far end of the manifold, is set at a fixed guide vane (GV) opening to generate a fixed mechanical power of 288MW. Turbine #, whose penstock is at the tunnel end of the manifold, is initially set to generate MW and GV then opened to increase generation to 25MW. The rate of opening is set to a relatively conservative equivalent of full scale traverse in 2s. 4

9 Figure 3 shows that the flow velocity increases in penstock # as GV # opens with a small dip in the velocity in penstock #2 over the same period. Opening GV # causes the pressure at the turbine inlet to fall and, after a brief delay while the pressure wave travels through the 2 penstocks and the manifold, hydraulic coupling causes the pressure at turbine #2 to decrease. The response also exhibits a poorly damped oscillation with a period of.7s as the travelling wave traverses the tunnel. These effects are reflected in the generated power. Turbine # is subject to a brief time delay before the required power increase begins while turbine #2 is perturbed from its set level. 52 pressure head (m) turbine # turbine #2 branch 48 7 penstock velocity (m/s) penstock # penstock #2 mechanical power (MW) turbine # turbine #2 5 5 time (s) Figure 3 Variation of pressure head, penstock flow and mechanical power with GV # opening in 2.6s. Figure 4 shows the head and flow profiles in the pipe network shortly after the simulation is initiated. The pressure head increases, because of elevation, along the length of the tunnel and then flattens out over the length of the penstock (the red line shows the pressure in penstock #). The small notch of lower velocity between 88 and 96m is located at the manifold, which only carries the flow to penstock #2. An animated version of Figure 4 shows the pressure wave travelling up and down the tunnel as the simulation proceeds. This is a very simple model of the hydraulic system (it should be emphasised that it is not a substitute for a full study for structural engineering purposes) but quite satisfactory for representing the dynamics within the tolerance required for a control systems study. Clearly, in this particular example, the pressure variation is within bounds and quite satisfactory in this respect. 5

10 5 4 pressure head (m) flow velocity (m/s) Pressure head variation (normalised to max and min of range) distance along pipe (m) Figure 4 Pressure and flow profiles in the tunnel and penstocks. 54 pressure head (m) turbine # turbine #2 branch 7 penstock velocity (m/s) penstock # penstock #2 mechanical power (MW) turbine # turbine #2 5 5 time (s) Figure 5 Variation of pressure head, penstock flow and mechanical power with GV # opening in.s. 6

11 The result in Figure 5 was obtained by decreasing the opening time of GV # to.s. This leads to a much larger variation in penstock pressure head, which falls to.5% below and rises to +4.5% above the static head. There is a corresponding variation of the mechanical power produced by the turbines, with perturbation peaks for turbine #2 of 9.7% and +6.7% of the steady state value. 4 Lumped parameter models 4. Lumped parameter, inelastic water column model The model presented in this section is derived from the block diagram in Sec. 2.6 of [2]. The water column is considered rigid so travelling wave effects are omitted. The inertia of the water column is represented by parameters known as the water starting times. The model is multivariable, so hydraulic coupling effects are retained. It is also nonlinear and suitable for use when simulating large excursions of the variables. The water starting time for a section of pipe of length l is defined as: T W V b = ghb (4.) For the layout of Figure, the relationship between the head and flow rate variation in the two penstocks is: h TW + TW3 TW q = h2 TW TW TW2 T + + W4 q 2 (4.2) where: h, h 2 are the turbine heads q, q 2 are the turbine flows T W is the water starting time for the tunnel (.45s) T W2 is the water starting time for the manifold (.4s) T W3, T W4 are the water starting time for penstock # and #2 respectively (both.98s). Inverting this relationship allows it to be embedded in a Simulink block diagram, as shown in Figure 6, which incorporates the nonlinear turbine characteristics and head losses due to friction. The simulation was run with GV #2 set to operate at a fixed power of 288MW. GV# was set for an initial output power of 28MW and then opened linearly over 2.6s to produce a steady state power of 3MW. The distributed parameter model was run for the same conditions and Figure 7 shows that there is a good match between the power outputs. Because the simulation is run here with Unit # operating at a higher power level than in Figure 5, both models predict an initial decrease of output power at turbine # (the characteristic non minimum phase response). The obvious discrepancy of course is the absence of the travelling wave oscillation from the response predicted by the inelastic water column model. The difference becomes more evident as the GV opening or closing times become shorter. 7

12 Dw Product 4 damping no -load flow hf q G 2 G2 P.U. base head Divide Divide h Product h2 Product itmat * uvec friction head loss inverse water starting time matrix s Integrator s Integrator q q2 Pm Product 2 Pm2 Product 3 Pb P.U. base power Pinel hf q friction head loss 2 Dw2 Product 5 damping Inelastic water column, multiple penstock model in p.u. representation. Based on Figure 8 of the IEEE Wkg Group paper. Figure 6 Simulink block diagram for the inelastic water column model Mechanical power (MW) Unit # inelas Unit #2 inelas Unit # dist Unit #2 dist time (s) Figure 7 Comparison of responses for distributed parameter and lumped parameter models. 8

13 4.2 Linearised model It is useful to have a linear model so that classical control system methods can be applied to designing a basic controller. Using standard techniques, equations (4.2) and (2.) can be linearised to give a small signal state space model of the usual form: x = Ax+ Bu y = Cx+ Du (4.3) In per unit form, changes in output power ( P m ) are related to changes in flow ( q) and guide vane position ( G) around a fixed operating point (G ) by: q a a2 / G q a a2 / G G = 2 2 q2 a2 a22 / G + 2 q2 a2 a22 / G 2 G 2 Pm 3 q 2 G = P 3 q 2 G m2 2 2 (4.4) where the terms a... a 22 are the elements of the inverse of the water starting time matrix in (4.2). Equation (4.4) is a linear relationship whose characteristics change with the selected GV openings of the two turbines which define the operating points G and G 2. Finally, following Sec. 3 in reference [2], the generator model of equation (2.3) can be appended to the hydraulic model of equation (4.4) to relate changes in Unit speed ( ω r ) to GV position ( G). The composite model is then in a suitable form for designing a closed loop speed control system. 5 Control system design The cross coupling terms in equation (4.4) reflect the intrinsic hydraulic coupling of the penstocks in the physical system. It is known that significant coupling leads to loss of stability margin, so Jones [6] has proposed a multivariable controller which takes this into account. However, the traditional (and simpler) strategy of treating the penstocks as being separate is adopted here. This allows tuning rules for a single input single output (SISO) system to be applied to the design of a basic PID governor for speed control when supplying an isolated load a procedure recommended in Section F.3 of []. A Simulink block diagram for the SISO linearised model and governor is shown in Figure 8 (the power feedback loop is eliminated by setting the droop gain to zero). Tuning the gains K i, K p and K d for satisfactory closed loop response is done by applying the rules proposed by Hagihara et al [7], [2] as follows: "transient droop" R t = /K p = 5T W / 8H "transient droop washout time constant" T R = K p / K i = 3.333*T W K d = T W K p / 3 9

14 Power ref speed ref Sum Sum droop hydlin.mdl power speed_err Governor GV GV sys GV --> power P sys2 power --> speed S Speed, power and GV Simple linearised model 2 speed_err wl*kd.s s+wl derivative power Sum Ki s integral Sum GV Kp proportional Figure 8 Block diagram of the linear system arranged for closed loop speed control. Setting the operating point at % and knowing the Unit s inertia constant (H) and the water starting time, taken here as T W = (T W + T w3 ) =.343s for the tunnel and penstock #, allows the gains to be evaluated as K i = 6., K p = 8.4 and K d = 2.. It is common to include an additional pole in a PID compensator to limit high frequency noise due to the derivative term. Placing it at ω = r/s means that it has only a minor effect on the closed loop dynamics. The open loop Bode plot for the Plant, compensator and forward loop is shown in Figure 9, which conforms to the typical pattern for a hydroelectric Plant see [2] and sections F.4. and F.3 in []. The crossover frequency is about 3r/s. The effect of the right half plane zero on the Plant frequency response is evident in the range /T W to 2/T W (3 to 6 r/s), where the increasing phase lag is not accompanied by a corresponding fall in gain. This limits how much phase lead can be contributed by the compensator over this frequency range, because the associated increase in gain could make the forward loop gain start to increase. As pointed out in [2], this could easily cause a second crossover and closed loop instability. The root locus (see sections F..5 & F..6 in []) for this case is shown in Figure. It is possible, using proportional gain alone, to place the dominant closed loop poles at approximately the same locations as with the PID controller. However, the step response has a % steady state error and it is necessary to introduce the integral term to counter it. Figure shows the closed loop step response to a.2 p.u. step demand in speed. The very large initial dip in the generator s speed is due to the very large and rapid GV motion. In practice, a rate limiter would be included to moderate this action.

15 5 Magnitude (db) -5 Phase (deg) Plant Compensator Forward loop Frequency (rad/sec) Figure 9 Open loop Bode plot for the Plant, PID controller and forward loop. Root locus for Plant 5 Imaginary Axis Real Axis Root locus for Plant + Governor 5 Imaginary Axis Real Axis Figure Root locus for (i) the Plant with proportional gain only, (ii) with the PID controller (where the diamonds show the locations of the dominant closed loop poles).

16 .4 PID gains Ki = 6., Kp = 8.4, Kd = 2..2 Speed (pu) Power (pu) Guide vane (pu) time (s) Figure Step response of the linearised model with the governor tuned using the Hagihara rules. It is concluded that traditional analysis, using the linearised, SISO inelastic model gives a good first cut at a control system, sufficient at least to use in further simulation using the more accurate models. This approach is likely to be overly optimistic in estimating how much gain can be included in the controller. In a multi penstock installation, hydraulic coupling can have a significant adverse effect on stability. A long penstock has a large water starting time, which encourages the designer to include as much derivative action as possible, in order to improve closed loop bandwidth. However, this is precisely the case where travelling wave effects are significant and controller design becomes more critical. It is at least necessary to test the controller on a model with elastic water column, which may indicate that the PID gains must be reduced to prevent an underdamped response or even instability. In such cases, it would desirable to take the travelling wave effect into account during control system design. 6 Closed loop simulation with Grid connection In this section, three operational scenarios are considered. The simulations are performed using the distributed parameter model for the hydraulics and the PID governor designed in the previous section. In all cases, additional dynamics to represent the GV servo lags are included as the transfer function: Gs ( ) = u ( s) ( s+ 2.5) ( s+ 5.26) G (6.) 2

17 6. Frequency control mode. Control of Grid frequency typically involves several regulators, of varying capacity and speed of response, simultaneously connected to the power system. The role of a hydroelectric station in frequency control mode is to provide accurate and timely supply of its target power contribution to the Grid. Stable sharing of the load between all the regulating sources is achieved by including a speed regulation or droop characteristic in their governors. The objective is to test the response of the Plant to a simulated speed (frequency) step see section in []. The block diagram for the governor, based on the design in the previous section, is shown in Figure 2. The block diagram for the complete simulation is shown in Figure 3. PID Governor wl *Kd.s s+wl D term freq _err Kp prop gain control 2 power _ref droop droop Ki int gain s Integrator Kff 3 Pe s power transducer FF gain Figure 2 Simulink block diagram for the governor. There are two feedback loops: The electrical power generated by the individual Units is measured by a sensor and filter, represented by the transfer function. The power error ( P) is formed as the ( s + ) difference between a (fixed) reference power and the measured electrical power. This is fed through a droop gain (α) to the PI part of the controller. The frequency error ( f) is formed as the difference between the demanded and measured Grid frequency and also added as an input to the PI part of the controller. The frequency error is also fed forward into the GV control signal (u G ), via the derivative part of the controller, in order to reduce system response time. The GV control is therefore formed as: G ( α P ) ( ) u = PI + PID f (6.2) The sum of the electrical powers from the two generators is used as an input to the Grid model. 3

18 Freq dem # zero Power ref # ramp Freq dem #2 Power set # select indval index freq_err power_ref Pe control GVC Governor freq _err power_ref2 Pe2 control2 GVC 2 GVC G GVC2 G2 GV dynamics Hydroelectric plant with PID control system. Hydraulics represented using the Method of Characteristics. This simulation is called from hydraul _p3.m. G G2 S_moc _v Plant hydraulics Pm H Q /Pb p.u. base Terminator Terminator mech_power Grid_freq Generator # mech_power Grid_freq elec_power elec_power Generator #2 Pe Pe2 grid load Pelec / p.u. base change rs Grid model grid_freq Pe_2 Governor 2 Power ref #2 Figure 3 Simulink block diagram for the complete simulation 4

19 The simulation is initialised with Units # and #2 supplying MW and 288MW respectively. The test input is a step frequency change of.7hz (.4 p.u.). The result is shown in Figure 4. Guide vane (pu) Unit #2 power (MW) Unit # power (MW) Grid freq (pu) Freq. loop PID gains Ki = 6., Kp = 8.4, Kd = mechanical electrical mechanical electrical Unit # Unit # x -3 Grid stiffness =.83 pu time (s) Figure 4 Response to a step demand of.7hz in frequency control mode; droop of %. Raising the Grid frequency by.7hz requires an additional 56MW of power of which 42MW is picked up by Unit #. Figure 4 is known as the system s primary response characteristic. For generators on the England & Wales Grid, a typical specification requires 7 9% of the target power output to be achieved within s of the occurrence of the event leading to the frequency deviation [8]. This system achieves 36.5MW (87%) of its target contribution in s, which is within the specified range. Developing a more detailed specification (see [9]) and a more accurate simulation will yield an improved assessment of the system s capability. 5

20 6.2 Power dial up mode. In this mode, the goal is to change the generated power rapidly in response to a dialled in operator request see section in []. Instead of applying a step reference change directly to the PID, it is preferable to generate a ramp power reference (whose slope is essentially determined by the maximum allowable GV opening rate) and to feed this forward directly to the GV control via a gain K ff as well as to the PID. In general, the feed forward and PID gains would be optimised for this specific mode of operation but in this simulation the PID settings are kept at their previous values, with K ff =. Unit # is commanded to increase its output from MW to 25MW by ramping the GV control to its new setting in 2.6s. This case is similar to Figure 3 but not identical because of the intervening GV dynamics (6.). The results are presented in Figure Power ramp PI gains Ki = 6., Kp = 8.4. Feedforward Kff = Grid freq (pu) Guide vane (pu) Unit #2 power (MW) Unit # power (MW) 2 5 mechanical electrical mechanical electrical Unit # Unit # x -3 Grid stiffness =.83 pu time (s) Figure 5 Response to a ramp GV input in dial up mode. The graph shows that the required power change is achieved in 3s although there is an overshoot of 28 MW (27%) followed by a long decay to the final value. Experiment shows that both overshoot and settling time could be improved by reducing the feed forward gain. 6

21 6.3 Full load rejection. The objective here is to assess performance when the system is disconnected suddenly from the Grid, perhaps because of some fault condition (see section in []). When the Grid breaker trips, the reaction torque on the turbine vanishes and the excess torque causes it to accelerate, with a risk of over speeding, unless prompt action is taken to close the GV. However, the rate at which the GV can be closed is limited by over pressure transients in the penstock. The block diagram of Figure 6 includes a logic signal that simulates the Grid breaker by disconnecting the feedback loops for electrical power and Grid frequency from the turbine/generator block. It also initiates an open loop ramp down of both GV controls at a rate of /2 p.u./s. breaker GV ref # ramp GV ref #2 ramp GV set # GV set #2 GVC GVC 2 GVC G GVC2 G2 GV dynamics G G2 S_moc _v Plant hydraulics Pm H Q /Pb p.u. base Terminator Terminator mech_power Grid_freq elec_power breaker Generator # mech_power Grid_freq elec_power breaker Generator #2 Pe Pe2 Pelec / p.u. base change rs Grid model grid_freq Hydroelectric plant simulation for total load rejection. Hydraulics represented using the Method of Characteristics. This simulation is called from hydraul _p3.m. select earth 3 breaker mech _power /(2*H) inertia s Integrator sp wb *Ks sync torque s Integrator select elec _power KD earth damping coefficient 2 Grid _freq Figure 6 Simulink block diagram for simulating load rejection. In Figure 7, the electrical power reduces instantaneously to zero when the breaker is tripped and both GVs begin to close at a fixed rate, causing the mechanical power produced by the turbines to decrease. During the closure interval, Unit # peaks at 5% over speed and Unit #2 at 2% overspeed; the Grid frequency decreases to its nominal value. Figure 9 shows the predicted volumetric flow and head at the two turbine inlets. The flows decrease more or less in proportion to the GV opening, as expected. As the closure begins, there is a sharp increase in turbine head which starts a travelling wave oscillation of identical amplitude in both penstocks, because they are hydraulically coupled. Once GV # is fully closed, the mean level of the oscillation decreases and a further reduction occurs once GV #2 closes, leaving a long lived oscillation around the static head. Whether or not the pressure variation is acceptable depends on the structural strength of the installation. 7

22 5 Unit # power (MW) mechanical electrical Unit #2 power (MW) mechanical electrical Guide vane (pu) time (s) Unit # Unit #2 Figure 7 Full load rejection: GV closure and power at both Units..2.5 Generator speed (pu)..5 Unit # Unit # x -4 Grid stiffness =.83 pu 8 6 Grid freq (pu) time (s) Figure 8 Full load rejection, showing speed increase for both Units and drop in Grid frequency. 8

23 Unit # Unit #2 535 Head at turbine (m) Unit # Unit #2 Flow at turbine (m 3 /s) time (s) Figure 9 Full load rejection: pressure and flow in the two penstocks. 7 Conclusion The generic models used here provide a wealth of predictive information early in the project, based on a small set of critical parameters. Although model validation, by comparison with the real installation, would usually not be possible at this stage, the methods adhere to industry standards and are well tried and tested on other Plant. The main points of the study are: basic prediction of pressure and flow variation in the conduit, which can be used to enhance rule of thumb calculations and precede detailed hydraulic analysis; development of lumped and distributed parameter models to predict system performance; development of a basic PID governor; simulation under different operational scenarios: o in frequency control mode, the primary response characteristic is shown to comply with specification; o in power dial up mode, feed forward allows a very rapid change of power level to be made; o during full load rejection, the Unit over speed is limited to 2% during GV closure. 9

24 Clearly, predictions made on the basis of simplified models have a margin of error and need to be treated with caution. Nevertheless, they provide valuable results that can be used to make better informed decisions on later stages of the project. The generic package reported here is only the first stage of simulation based design. There is ample scope for extension and refinement of the models as more details of the installation become available. References Std 27 24, 'IEEE Guide for the Application of Turbine Governing Systems for Hydroelectric Generating Units', New York, IEEE IEEE, W. G.: 'Hydraulic turbine and turbine control models for system dynamic studies', IEEE Trans Power Systems, 992, 7, (), pp Larock, B. E., Jeppson, R. W. and Watters, G. Z.: 'Hydraulics of Pipeline Systems', 999 (CRC Press). 4 Kundur, P.: 'Power System Stability and Control', 994 (McGraw Hill). 5 Jones, D. I.: 'Dynamic parameters for the National Grid', Proc IEE Gener. Transm. Distrib., 25, 52, (), pp Jones, D. I.: 'Multivariable control analysis of a hydraulic turbine', Trans Inst MC, 999, 2, (2/3), pp Hagihara, S., Yokota, H., Goda, K., et al.: 'Stability of a hydraulic generating unit controlled by PID governor', IEEE Trans on Power Apparatus & Systems, 979, PAS 98, (6), pp Erinmez, I. A., Bickers, D. O., Wood, G. F., et al.: 'NGC experience with frequency control in England and Wales provision of frequency response by generators'. Proc IEEE Power Engineering Society Winter Meeting, New York. 999, pp Jones, D. I., Mansoor, S. P., Aris, F. C., et al.: 'A standard method for specifying the response of hydroelectric plant in frequency control mode', Electric Power Systems Research, 24, 68, (), pp