A NON-LINEAR STABILITY STUDY FOR THE DETERMINATION OF THE SPEED GOVERNOR PARAMETERS FOR THE ITAIPU 9A GENERATING UNITY

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1 A NON-LINEAR STABILITY STUDY FOR THE DETERMINATION OF THE SPEED GOVERNOR PARAMETERS FOR THE ITAIPU 9A GENERATING UNITY Paulo Renato Galveias Lopes ALTOM POWER BRASIL Av. Charles Scheneider, s/n Taubaté SP- Brasil Tel: , Fax: Abstract. This paper presents the results obtained in the methodology applied in the determination of the parameters to be used in the speed governor of the 9A generating unit of ITAIPU power plant. Two algorithms developed by ALSTOM POWER HYDRO based on MATLAB and SIMULINK of MATHWORKS were used. The first algorithm permits, through a linear analysis of the transfer's functions involved, the determination of the optimum parameters to be used in the speed governor of the generating unit. The second algorithm permits a non-linear simulation of load rejection and load acceptance making possible the validation of the results obtained in the first analysis. The main advantage of this methodology is to permit the verification of the turbine non-linearity's influences in the group stability, that are usually neglected in this kind of studies. Keywords. hydraulic turbines speed governors, stability studies, hydraulic power generation, computer simulation, control system optimization. Introduction Since 970 s, power increase of microprocessors and decrease of the cost of computers, along with the development of mathematical tools, have encouraged main actors in hydro-activity to create and enhance dynamic simulation tools for hydro generating units with their governing systems. Initial target was to determine by simulation the optimal parameters to be introduced into the governor so as to guarantee unit stability in an isolated grid. Nowadays the main aim of this kind of simulations is to determine the influences of the non-linearity involved in the process on the stability of the hydraulic turbines operation.. Era of linear Simulation Up to recent times, everyone only could treat linear simulation in the vicinity of an operating point. Among main reasons: - classical tools for solving continuos linear systems (Laplace transform, Bode and Nyquist diagrams) are well known and mastered; - when governors changed to digital devices, the tools for solving sampled linear systems were immediately available (Z transform) with similar use compared to the previous ones; - calculation facilities were still expensive and the simulation times were long, encouraging simplification hypotheses; - last but not least, the results were globally satisfying. A full model for studying the behavior of an unit on an isolated grid was easily described: - According to this model, a hydraulic turbine with the penstock may be modeled by: w x m Tw Tw 2 () wm - variation of turbine output; x - variation actual posicion of flow setting device; Tw - penstock hydraulic launching time; s - complex variable.

2 - equation for rotation parts and grid: ω w w A m r Ta + A (2) ω - variation of angular velocity; wr - power delivery to the grid; A - Grid auto-setting coefficient; T - Mechanical launching time. a - servo-positioner reduces to a first order low-pass filter combined with a pure time delay: x y e τ T i (3) y - positioning order from governor; τ - pure time delay; T - time characteristic of servo-mechanism for small displacements; i - Finally the speed governor is a simple PID governor with a pure time delay: y e ω B τ 2 s p Tn Tn T d N (4) B - permanent droop; Tn p T d N - derivative action time constant; - integral action time constant; - derivative action gain; governor y servopositioner x penstock and turbine hydraulic duct w m + - turbine rotating parts and network w r perturbation ω Figure. Model of hydraulic turbo-generator for studying the stability in an isolated grid. Using such a single variable model, fully described with Laplace transforms, classical criteria may be applied, e.g. gain and phase margins of the transfer function in open loop, to obtain optimal parameters of the governor. For simulating the hydraulic system, all we have to do is to convert the Laplace transform into a sampled relation (Ztransform) and to solving it using a sampling time step which is reasonably low compared to the smallest time constant of the system.

3 This method leads to software modules implementation having low memory requirements and however acceptable computational speed, even on the first generation of PCs. Nevertheless it implies some weaknesses : - the transfer function models the turbine as a simple orifice plate. It is realistic only for operating points where both the amplitude of perturbations and the gradient of efficiency are small enough for considering a constant efficiency value. In particular, this is widely false near the no-load operating points (0 to 30 % opening); - it cannot simulate turbines with more than one flow setting device; - results are false when the amplitude of perturbations imposes moving the flow setting device nearly as fast as allowed by its safety limitation; - penstock model does not account for water hammer waves; - the model is only accurate in the vicinity of the nominal rated speed. 2. Non-linear model for Francis turbines A decade ago software tools on micro-computers for modeling and simulating dynamic processes was developed, including the treatment of highly non-linear ones. From the user point of view, these tools are no longer computer coding languages as Fortran or C used to be, but truly automation languages : programming input can be composed of functional blocks that represent directly Laplace transforms, Z transforms, or identify the state of a given dynamic process to simulate. Furthermore, the user no longer has to worry about temporal solution which is handled by the software. All he has to do is to choose and parameter one of the proposed methods, depending on the level of nonlinearity and on time constants of the model he created. Allowing automatism to control the full process modeling by themselves, these tools promoted significant progress in that field. So ALSTOM Power Hydro set about non linear modeling of Francis turbines, targeting to account for the behavior (speed, torque, pressure) of the unit in any foreseen domain of operation (Boireau et Vuillerod, 2002). Turbine manufacturers characterize their prototypes at design stage for a wide range of operation from model tests performed in a hydraulic laboratory. Homology rules to derive the true prototype characteristics from model tests values are well known and codified. These results are "hill charts, that allow to calculate the discharge, the torque and the efficiency, given an operating point (speed, net head and opening of the flow setting device). Normalized variables for speed, discharge and torque used in the model are n, Qand T. hill :efficiency vs n and opening hill : torque vs n and opening efficienency reduced torque (c) opening reduced speed (n) reduced speed (n) opening Figure 2. Examples of hill charts. Aiming this kind of simulations ALSTOM power has developed two non-linear simulations tools based on hill charts. The former takes into account a hill chart in electronic table format covering all operation regions of a hydraulic turbines, including speed no load region and runway region. These full hill charts in electronic format are not always available either because that only more recently the speed no load regions are characterized during the hydraulic model tests or because only in the last years modern data acquisition system has permitted to acquire easily the hill charts to electronic table format. The later simulation tool comes justly to fill the cases where the full hill charts are not available, being based on general curve of efficiency vs overture, that can be inferred from the hill charts available, and on a general law of flow in function of the wicket gate position, flow (wicket gate position) nq. Since this law is not always known, nq 0.7 represent a realist value for the major part of the cases. The following figures show the models utilized by ALSTOM Power for simulations when the full hill charts are not available.

4 servo-positioner wg pos. flow net head frequency reference effic * wg - wiket-gate efficiency curve Figure 3. General view of the model utilized when full hill charts are not available. Figure 4. Detailed view of the servo-positioner representation. Figure (3) shows an overview of the model utilized in the simulation, the turbine part is based on water hammer effect theory presented by Bergeron (950). In this model the hydraulic duct is simplified in an equivalent penstock of constant section with the following formula: L Seq n i Li Si Li and Si are the length and section of i-th segment, each segment corresponding to a portion of the penstock of constant section. Of course, L n i Li. Figure (4) shows a detailed view of the ALSTOM positioner modeling. In this model, the servo-positioner has also a non-linear representation, accounting for limitations of motion speed of the servomotor. The validation of these models were ideally achieved on prototypes for which at the same time either accurate hill charts and field test results were available or calculations had been achieved with already validated tools, as shown for the Three Gorges case by Boireau et Vuillerod, Itaipu 9A Group stability study For the Itaipu 9A group stability case study, the second model based on approached laws to replace the hill charts was utilized, since Itaupu hydraulic model tests were made during 970 s when rarely the full operation range of the turbine was characterized. 3.. Itaipu Groups features The Itaipu Hydroelectric Power Plant, the largest one in operation in the world, is a binational undertaking developed by Brazil and Paraguay in the Paraná River. The installed power of the plant is 2,600 MW, with 8 generator units of 700 MW each.

5 Recently the Itaipu power plant capacity is being enlarged with the installation of two new generating groups (9A, 50Hz and 8A, 60 HZ). Among other things ALSTOM Power is supplying the digital speed governors for these groups, including the stability study for the determination of the parameters to be set on such equipment. The results of this study is presented hereafter. The main features of the new 9A group which are necessary to the stability study are listed in the table below. Table. Itaipu 9A group features. Turbine type Francis Nominal head (m) 8.4 Turbine nominal output (MW) 740 Rotational speed (rpm) 90.9 Nominal flow (m3/s) Turbine and generator inertia MR2 (T.m2) Wicket gate opening time (s) 9 Wicket gate closing time (s) 9 Runner outlet diameter (m) 8. Auto-setting coefficient 0 The typical variables are derived from these data: Table 2. Typical variables of Itaipu 9A group. Mechanical lauching time (s) Ta 0 Penstock hydraulic launching time (s) Tw.38 Alievi parameter (dimensionless) R Speed governor PID optimization The optimization of the PID speed governor parameters for the Itaipu 9A group case was realized through the utilization of Optim software developed by ALSTOM Power and which is based on MATLAB and SIMULINK of MATHWORKS. The methodology implemented in this software makes use of a linearized turbine transfer function derived from the complete one model presented in Fig. (3). The adjustment is determined within the frequency range (Nyquist plane) to ensure stability of closed loop with specified gain and phase margins. As mentioned before, the adjustment obtained by this method was then introduced into the complete model to analyze stability in the case of high disturbances. The complete model always processes the general case of wave water hammer effect. The optimization through Optim software resulted in: Table 3. Optimized PID speed governor parameters (N0 and A0) : Tn 0.64 s Td 7.98 s bt 5 %

6 3.3. Non-linear simulations results The simulations was performed at about 30%, 60% and 90 % of the maximum power output for three conditions of head 84 m, 8 m and 27 m. Firstly a 0 % of rated power load step down was applied at time 2 s, then a 0 % step up at 50 s. The results achieved for a head of 8m and 90 % of the maximum power are shown below:.04 speed net head.02 normalized data time (s) Figure 5. Speed and pressure variations for load steps of 0% of rated power output gate 0.9 normalized data time (s) Figure 6. Power and gate position for load steps of 0% of rated power. The results show that the maximum speed deviation was about 5 % and that the maximum recovery time was about 30s.

7 4. Conclusion Taking full benefit from the increased possibility of modern automation software tools combined with its strong experience in hydro-turbine dynamic behavior, ALSTOM Power Hydro is now able to propose digital governors containing high performance added-value and hence to master the most challenging non-linear transient conditions for stabilizing operation of hydro turbines. 5. References Boireau, C., Vuillerod, G., 2002, Non-linear modeling of a hydraulic generating unit. Application to stabilizing pumpturbines in speed-no-load turbine mode. Lausanne, proceedings of the Hydraulic Machinery and Systems 2 st IAHR Symposium, Sepetember 9-2, Bergeron, L., 950, Du coups de bélier en hydraulique au coup de foudre en éléctricité (From water hammer in hydraulics to lighting in electricity). Dunod: Paris (in French). 6. Copyright Notice The author is the only responsible for the printed material included in his paper.