28 CHAPTER 3 MATHEMATICAL MODELING OF HYDEL AND STEAM POWER SYSTEMS CONSIDERING GT DYNAMICS 3.1 INTRODUCTION This chapter focuses on the mathematical state space modeling of all configurations involved in both single machine and multimachine hydel and steam power systems considering the effect of respective governor - turbine dynamics. Based on the state variables selected, the state matrices are developed for all the models considered. The damping controller model is also explained in this chapter. 3.2 POWER SYSTEM MODEL INVESTIGATED A classical model for synchronous generator is developed with the following assumptions (Prabha Kundur 2008). 1. The mechanical power input is taken as constant. 2. Natural damping (D) in the system is included in modeling. 3. The generator is modeled as a constant voltage source behind a transient reactance. A worst case analysis with the following modified assumptions is incorporated in the modeling of the system and in the simulation experiments conducted in this research work.
29 1. The mechanical power input (GT dynamics effect) variations are included in the modeling. 2. Natural damping (D) in the system is assumed to be very negligible. The third assumption is incorporated in this work, as considered in the classical model. Figure 3.1 Single machine infinite bus power system model model. Figure 3.1 represents the single machine infinite bus power system This model is developed by assuming the generator with constant voltage source behind a transient reactance Xd. Xd represents the direct axis transient reactance of the synchronous generator. Here the generator (hydel or steam) supply power to the infinite bus through an external impedance Z.The generator terminal voltage is represented by V T. Figure 3.2 represents the three machine nine bus power system model taken for analysis in this work.
30 Figure 3.2 Three machine nine bus power system model For all the mathematical modeling and simulation, the Heffron- Phillips block diagram that describes the complete system dynamics of synchronous generator has been implemented in this work. Figure 3.3 represents the Heffron-Phillips block diagram of the synchronous generator. Figure 3.3 Heffron-Phillips generator model with PSDC
31 Here the model is equipped with PSDC in the excitation feedback loop to provide supplementary signal to the system. The block EXC(s) represent the excitation system model provided in the system. The IEEE Type 1 excitation and IEEE ST1A type excitation system models are used in this work (IEEE Standards 2005).The detailed explanations of these two types are explained in section (3.4) of this chapter. All the data used for simulation of test power systems are listed in Appendix 2 of this work (Anderson and Fouad 2008). 3.3 DAMPING CONTROLLER MODEL The basic function of a power system damping controller is to provide required damping to the electromechanical oscillations by controlling its excitation using auxiliary stabilizing signal. The damping controller model is represented in Figure 3.4. It consists of the washout block, gain block and the cascaded identical phase compensation block(bikash Pal and Balarko Chaudhuri 2005).The Gain (Ks) and the time constants of the phase compensation block are tuned effectively, so that the damping controller provide the required damping torque to damp the power oscillations. The input to the controller is the rotor speed deviation and output is the damping control signal ( U E ) given to generator- excitation system feedback loop. Figure 3.4 Power system damping controller model
32 The transfer function of the damping controller model is given by U E st 1 st 1 st K 1 st 1 st 1 st w 1 3 s w 2 4 (3.1) where Ks = Damping controller gain Tw T 1, T 2, T 3, T 4 = Washout time constant = Time constants of the damping controller The washout block functions as a high pass filter, which prevents steady changes in speed from modifying the field voltage. The washout time constant may be anywhere in the range of 1 to 20 seconds (Prabha Kundur 2008,Abido 2002, Shayeghi et al 2008).Tw is taken as 20 seconds in this work. 3.4 STATE SPACE MODELING OF SINGLE MACHINE HYDEL AND STEAM POWER SYSTEMS The state space representation of power system provide an effective and compact way to model and analyze the small signal stability of the system models considered. The dynamic linearized state space equation used for modeling the power systems is given by x Ax Bu (3.2) where x = State variable vector A, B = State matrix and Input matrix respectively.
33 3.4.1 Modeling of Single Machine Hydel Power System In all the classical model analysis of single and multimachine systems, the mechanical input power is assumed as constant. But in this work, hydel governor and turbine model (mechanical power input) is included along with the hydel generator model for mathematical modeling, simulation and analysis. In case of single machine hydel power system modeling, a 247.5 MVA, 16 KV, 180 rev/min hydel generator equipped with governor and turbine is taken as the test system. In this work, the following two hydel models are taken for analysis. 1. Hydel model (H1) 2. Hydel model (H2) In H1 model, hydel generator is equipped with hydel governor and turbine. The transient droop compensation block is not included in the hydel governor model. The IEEE Type 1 excitation (rotating exciter type) is taken in this model. In H2 model, hydel generator is equipped with hydel governor and turbine. The transient droop compensation block is included in the hydel governor model. The IEEE ST1A excitation (static exciter type) is taken in this model. 3.4.1.1 Hydel system H1 model Figure 3.5 represents the hydel governor and turbine used in hydel H1 model.t GH and T WH are the time constants of hydel governor and turbine respectively. Here, the output Tm of hydel GT model is given as input to the
34 Heffron-Phillips generator model, as in Figure 3.3. R P represent the steady state speed droop. The transfer function for hydel governor and turbine are shown in Figure 3.5. Figure 3.5 Hydel governor and turbine (for H1 model) Figure 3.6 IEEE Type 1 Excitation system model For this model, the generator is equipped with IEEE Type 1 excitation system shown in Figure 3.6. This excitation model represents rotating exciter configuration and it consists of exciter, amplifier and Excitation system stabilizer(ess).the ESS is used to increase the stable region of operation of excitation system.k A and T A represent the gain and time constant of the amplifier. K E and T E represent the gain and time constant of the exciter. K F and T F represent the gain and time constant of ESS. The state space equation of model (H1) is given by
(3.3) x A x B u H1 H1 m H1 H1 m H1 35 where m = Number of hydel generators, here m=1. A H1,B H1 = State matrix & input matrix of hydel H1 model. x H1 = State variable vector of hydel H1model. Equation (3.4) and (3.5) indicate the state variables selected for this model for open loop and closed loop analysis. [x H1 ] OPEN = [ Eq, E FD V R V E Xe, Tm] T (3.4) [x H1 ] CLOSED = [ Eq, E FD V R V E Xe, Tm, P 1 P 2 U E ] T (3.5) where = Incremental change in Rotor speed = Incremental change in power angle Eq = Incremental change in generator voltage E FD = Incremental change in field voltage V R V E = Incremental change in amplified voltage = Incremental change in output of excitation system stabilizer Xe = Incremental change in output of hydel governor Tm = Incremental change in mechanical power input. In Equation (3.5), P 1 P 2 U E represent the state variables involved in the closed loop PSDC model, also presented in Figure 3.4. 3.4.1.2 Hydel system H2 model Figure 3.7 represents the hydel governor and turbine model connected to the Heffron generator model. In this H2 model, the transient droop compensation block is added between hydel governor and turbine
36 block. This block with a long resetting time is added to have stable control performance of the system. The resetting time constant (T R ) is typically 5 seconds (Prabha Kundur 2008).R T represents the transient state speed droop. Xe represents the incremental change in output of Transient droop compensation block. The hydel generator in this model is equipped with IEEE ST1A type excitation system. Figure 3.8 represents the IEEE ST1A type excitation system model. Figure 3.7 Hydel governor and turbine (for H2 model) This model represents the static potential source controlled rectifier excitation system. By considering K E =1 and T E =0 in the exciter block of IEEE Type 1 excitation system model, the IEEE ST1A model is developed. The main advantages of this model compared to the Type 1 excitation system are: Instantaneous response, inexpensive, easily maintainable and reduction of state variables in the modeling.
37 Figure 3.8 IEEE ST1A type Excitation system model Figure 3.9 represents the complete state space model of the hydel power system with hydel generator, governor- turbine and PSDC. Pd in Figure 3.9 indicate the load change disturbance given to the system for simulation and analysis. The state space equation of model (H2) is given by x A x B u (3.6) H2 H2 m H2 H2 m H2 where m = Number of hydel generators, here m=1. A H2 = State matrix of hydel system H2 model. x H2 and B H2 = State variable vector and input matrix involved. model. Equation (3.7) and (3.8) indicate the state variables selected for this [x H2 ] OPEN = [ Eq, E FD V E Xe, Xe, Tm] T (3.7) [x H2 ] CLOSED = [ Eq, E FD V E Xe, Xe, Tm, P 1 P 2 U E ] T (3.8)
Figure 3.9 Complete state space model of hydel generator with GT dynamics and PSDC 38
39 3.4.2 Modeling of Single Machine Steam Power System In case of single machine steam power system modeling, a 192 MVA, 18 KV, 3600 rev/min steam generator equipped with steam governor and turbine is taken as the test system. In this work, the following configurations are modeled and analyzed. 1. Steam Model (S1) 2. Steam Model (S2) In S1model, steam generator is equipped with steam governor, reheat steam turbine and IEEE Type 1 excitation system. In S2 model, steam generator is equipped with steam governor, non- reheat steam turbine and IEEE ST1A excitation system. 3.4.2.1 Steam system S1 model In this model, the single reheat steam turbine is used in the modeling. The transfer function of the reheat steam turbine is given in the block represented in Figure 3.10. Here, F HP represents the high pressure flow fraction, T CH represent the inlet and steam chest delay. P G represents the incremental change in output of steam governor model.t GS represents the time constant of the steam governor. Figure 3.10 Reheat steam turbine and governor model.
40 The responses of the reheat turbines are significantly slower than those that of non-reheat turbines. The main reason is that, the reheater time constant will have more time delay compared to non-reheat type turbines. The reheater time constant (T RH ) used in this modeling is 6 seconds (IEEE Standards 2005).This steam system S1 model is equipped with IEEE Type 1 Excitation system. The state space equation of model (S1) is given by x A x B u (3.9) S1 S1 n S1 S1 n S1 where n = Number of steam generators, here n=1. A S1 = State matrix of steam system S1 model. x S1 and B S1 = State variable vector and input matrix involved. this model. Equation (3.10) and (3.11) indicate the state variables selected for [x S1 ] OPEN = [ Eq, E FD V R V E P G Tm] T (3.10) [x S1 ] CLOSED = [ Eq, E FD V R V E P G Tm, P 1 P 2 U E ] T (3.11) 3.4.2.2 Steam system S2 model In this model, the single non-reheat steam turbine has been used in the modeling. The transfer function of the non-reheat steam turbine is given in the block given in Figure 3.11. By substituting, the time constant T RH =0 in the reheat turbine transfer function (Figure 3.10), the transfer function for the non-reheat turbine has been obtained. Here, the non-reheat time constant is represented by T TS.
41 Figure 3.11 Non-Reheat Steam turbine and governor model The state space equation of model (S2) is given by x A x B u (3.12) S2 S2 n S2 S2 n S2 where n = Number of steam generators, here n=1. A S2 = State matrix of this steam system model (S2). X S2 and B S2 = State variable vector and input matrix involved. Equation (3.13) and (3.14) indicate the state variables selected for this model. [x S2 ] OPEN = [ Eq, E FD V E P G Tm] T (3.13) [x S2 ] CLOSED = [ Eq, E FD V E P G Tm, P 1 P 2 U E ] T (3.14)
42 Table 3.1 Consolidated state variables and state order of hydel and steam power systems S. No System State variables for various system models H1 Model H2 model S1 model S2 model 1 Generator,,,, 2 Excitation system Eq, E FD, Eq, E FD, Eq, E FD, Eq, E FD, V R V E V E V R V E V E 3 GT Xe, Tm Xe, Xe, Tm P G, Tm P G, Tm 4 PSDC P 1 P 2, U E P 1 P 2, U E P 1 P 2, U E P 1 P 2, U E 5 Open loop State order 6 Closed loop State order 8x8 8x8 8x8 7x7 11x11 11x11 11x11 10x10 Table 3.1 represents the consolidated view of the various state variables involved in generator, excitation system, Governor-Turbine and damping controller of all the models involved in hydel and steam power systems. Figure 3.12 represents the complete state space model of the steam generator equipped with governor-turbine model and PSDC. The state matrices are developed for all the models involved in both hydel and steam power systems. The state matrices are given in Appendix 1.
Figure 3.12 Complete state space model of steam generator with GT dynamics and PSDC 43
44 3.5 STATE SPACE MODELING OF MULTIMACHINE HYDEL AND STEAM POWER SYSTEMS In this work, three machine nine bus power system model (Figure 3.2) is taken as the test system for modeling, simulation and analysis. The single machine Heffron-Phillips generator model (Figure 3.3) is extended to perform the modeling of multimachine system. Because of the interaction among various generators in the multimachine system, the branches and loops of the single machine generator model become multiplied. For instance, the constant K 1 in the single machine model becomes K 1ij,i=1,2 n; j=1,2.n in the multimachine modeling. In this work, n will be equal to 3, representing the number of generators in the multimachine system considered. Similarly all the K constants (K1 to K6), damping factor D,inertia M and the state variables used in the single machine model are generalized for n-machine notation. Figure 3.13 represents the state space model of multimachine power system. Figure 3.13 State space model of multimachine power system
45 For the linearized state space modeling of three machine nine bus system, the same state space Equations (3.3, 3.6, 3.9 and 3.12) used for various single machine hydel and steam configurations are implemented for the multimachine system modeling. 3.6 SUMMARY In this chapter, a detailed mathematical state space modeling of all the configurations involved in both single machine and multimachine hydel and steam power systems considering the effect of governor and turbine dynamics has been presented. Also, the modeling of damping controller has been presented. These various hydel and steam power system models form the basis for all the simulation experiments carried out in this work.