20 CHAPTER 2 DYNAMIC STABILITY MODEL OF THE POWER SYSTEM 2. GENERAL Dynamic stability of a power system is concerned with the dynamic behavior of the system under small perturbations around an operating condition and more specifically it is a phenomena of slow and poorly damped or sustained or even diverging power oscillations which are essentially due to varying system loads and ill controlled controllers of the system. Computer analysis of this problem requires mathematical models which simulate as accurately as the behavior of physical system but at the same time not very complex to handle. This chapter presents the development of the mathematical model for the dynamic stability analysis of Single Machine Infinite Bus System (SMIB) and Multi- machine power system. 2.2 SIMPLIFIED LINEAR MODEL FOR SMIB SYSTEM In stability analysis, the mathematical model is used for dynamic analysis of power systems. 2.2. Assumptions The following assumptions are made for the development of simplified linear model of SMIB system:
2. Damper windings both in the d and q axes are neglected. 2. Armature resistance of the machine is neglected. 3. Excitation system is represented by a single time constant. 4. Balanced conditions are assumed and saturation effects are neglected. 2.2 2 Classical Machine Model In the classical methods of analysis, the simplified model or classical model of the generator is used (Kundur 994). Here, the machine is modeled by an equivalent voltage source behind impedance connected to an infinite-bus as shown in Figure 2.. Infinite Bus An infinite bus is a source of invariable frequency and voltage (both in magnitude and angle). A major bus of a power system of very large capacity compared to the rating of the machine under consideration is approximately an infinite bus. x e Figure 2. One-line diagram of SMIB The state space classical machine model is shown in Figure 2.2.
22 K s Figure 2.2 Classical Machine Model The state equations of the classical model are given in equation (2.): p( ω ) = Tm K S D 2H δ ω p( δ ) = ω ω (2.) 0 State vector T x = ( ω δ ) (2.2) And when the effect of flux linkage is included, three states are used to model the generator: ω, δ and E q '. The state equations are given in equation (2.3) as follows: T K K D p ω= - δ- E - ω τ τ τ τ m 2 ' q j j j j p( δ ) = ω ω 0 - K p E = E + E - δ (2.3) ' ' 4 q ' q ' FD ' Kτ 3 do τdo τdo where τ j = 2H, H is the inertia constant.
23 State Vector of the SMIB system including the effect of flux linkage is given by equation (2.4) t x [ E' q] = ω δ (2.4) where the variables are E q ' ω δ - quadrature axis component of voltage behind transient reactance - angular velocity of rotor - rotor angle in radians K to K 6 is the Heffron Philips constants (Padiyar 2002). 2.3 EXCITATION SYSTEM REPRESENTATION The excitation system model considered is the simplified form of STA model shown in Figure 2.3. A high exciter gain, without derivative feedback, is used. By inspection of Figure 2.3, the state space equations can be written as, V V t KA + st A E FD Figure 2.3 Excitation System Representation p E FD = -/T A (K A V t + E FD ) (2.5) with T R is neglected, V ref =constant. And V t = K 5 δ + K 6 E q '
24 where E FD - Equivalent stator emf proportional to field voltage K A - Gain of the Exciter T A - Time constant of the exciter T R V t - Terminal Voltage Transuder Time Constant - Terminal voltage of the Synchronous machine V ref - Reference voltage of the Synchronous machine Combining the Equations (2.3) with the exciter equation (2.5), the complete state space description of SMIB system including exciter is given in equation (2.6). T K K D p ω= - δ- E - ω τ τ τ τ m 2 ' q j j j j p δ = ω0 ω - K p E = E + E - δ ' ' 4 q ' q ' FD ' Kτ 3 do τdo τdo pe FD = -/T A (K A K 5 δ + K A K 6 E q ' + E FD ) (2.6) The state vector is thus defined by Equation (2.7): t x [ E' q E FD] = ω δ (2.7) 2.4 SMIB SYSTEM REPRESENTATION WITH CPSS The block diagram of the CPSS is shown in Figure 2.4. The state equations for the same can be written as follows. p V 2 = K PSS p ω (/T w ) V 2 (2.8) p V s = (T /T 2 ) p V 2 + (/T 2 ) V 2 (/T 2 ) V s (2.9)
25 where K PSS - CPSS gain T, T 2 - Phase compensator time constants T w - Wash out time constant CPSS V t V s Figure 2.4 CPSS Representation State vector of the synchronous machine model including PSS is given by equation (2.0): T x = [ E q' EFD V2 V s] ω δ (2.0) The block diagram of simplified linear model of a synchronous machine connected to an infinite bus with exciter and PSS is shown in Figure 2.5. V t Conventional Power System Stabilizer Figure 2.5 State Space Model of SMIB system representation with CPSS
26 2.5 DYNAMIC STABILITY MODEL OF MULTI MACHINE POWER SYSTEM In stability analysis of a multi-machine system, modelling of all the machines in a more detailed manner is exceedingly complex in view of the large number of synchronous machines to be simulated. Therefore simplifying assumptions and approximations are usually made in modelling the system. In this thesis two axis model is used for all machines in the sample system taken for investigation. 2.5. Assumptions Made In this work the synchronous machine is modeled using the twoaxis model (Anderson and Fouad 2003). In the two-axis model the transient effects are accounted for, while the sub transient effects are neglected. The transient effects are dominated by the rotor circuits, which are the field circuit in d-axis and an equivalent circuit in the q-axis formed by the solid rotor. The amortisseur winding effects are neglected. An additional assumption made in this model is that in stator voltage equations the terms pλ d and pλ q are negligible compared to the speed voltage terms and that ω ω =p.u. The R block diagram representation of the synchronous machine in two-axis model is shown in Figure 2.6.
Figure 2.6 Block diagram representation of two axis model for synchronous machine 27
28 2.5.2 Synchronous Machine Representation Using the block diagram reduction technique and with the simplifying assumptions the state equations for the two-axis model in p.u. form pe d ' = {-E d ' - (x q -x q ') I q } / τ qo ' pe q ' = {E FD - E q ' - ( x d - x d ' ) I d } / τ do ' pω = {T m - Dω - T e } / τ j pδ = ω - (2.) where the state variables are E d ' - direct axis component of voltage behind transient reactance E q ' - quadrature axis component of voltage behind transient reactance ω - angular velocity of rotor δ - rotor angle in radians and T e = E d 'I d + E q 'I q (x q ' x d ' ) I d I q τ j = 4πfH x d - direct axis synchronous reactance x q - quadrature axis synchronous reactance x d ' - direct axis transient reactance x q ' - quadrature axis transient reactance τ do ' - direct axis open circuit time constant
29 τ qo ' - quadrature axis open circuit time constant T e - electrical torque of synchronous machine T m - mechanical torque of synchronous machine D - damping coefficient of synchronous machine E FD - Equivalent stator emf corresponding to field voltage I q - quadrature axis armature current I d - direct axis armature current H - inertia constant of synchronous machine in sec f - frequency in Hz A multi-machine power system is shown in Figure 2.7 and the network has n machines and r loads. The active source nodal voltages in Figure 2.7 are taken as the terminal voltages V i, i =.2.n instead of the internal EMF`s. The loads are represented by constant impedances and the network has n active sources representing the synchronous machines. Figure 2.7 Multi-machine with constant impedance loads
30 This network is reduced to a n-node network shown in Figure 2.8 in which the current and voltage phases of each node are expressed in terms of the respective machine reference frame. Figure 2.8 Reduced n-port network The objective here is to derive relations between v di and v qi, i=,2,.n, and the state variables. This will be obtained in the form of a relation between these voltages, the machine currents i qi and i di, and the angles δ i, i=,2,.n. For convenience we will use a complex notation as follows. For a machine i we define the phasors V and I i i as V = V + jv ; I = I + ji (2.2) i qi di i qi di
3 where V = v / 3 : V = v / 3 qi qi di di I = i / 3 : I = i / 3 qi qi di di and where the axis q i is taken as the phasor reference in each case. Then we define the complex vectors V and I by V V + jv q V V jv 2 + V = =...... V V + jv qn n d q 2 d 2 dn (2.3) I I I + ji q d I I + ji 2 q 2 d 2 = =...... I I + ji qn dn n (2.4) The voltage Vi and thecurrent I i are referred to the q and d axes of machine i. In the other words the different voltages and currents are expressed in terms of different reference. To obtain general network relationships, it is desirable to express the various branch quantities to the same reference which is given by equation (2.5): The node voltages and currents are expressed as Vˆ and ˆI i i, i =,2,..n, and ˆI = YV ˆ (2.5) where Y is the short circuit admittance of the network.
32 2.5.3 Converting to Common Reference Frame Let us assume that we want to convert the phasor Vi V jv qi di = + to the common reference frame (moving at synchronous speed). Let the same voltage, expressed in new notation, be ˆV = V + jv i Qi Di as shown in Figure 2.9. where, = + and ˆV = V + jv i Qi Di (2.6) Vi V jv qi di D ref V Di V i = V ˆi d i q i V qi V di δ i δ i Q ref V Qi Figure 2.9 Two frames of reference for phasor quantities From the Figure 2.9 V Qi = [Vqi cos δ i Vdi cos δ i ] (2.7) V = [V cos δ V cos δ ] Di di i qi i (2.8) V + V = (V cos δ V cos δ ) + (V sin δ + V cos δ ) Qi Di qi i di i qi i di i (2.9) ˆV i j i = V e δ i (2.20)
33 The equation (2.20) can be written in generalized matrix form as below jδ V jv e 0 0 + V jv Q D + q d jδ2 V + jv e 0 V jv Q2 D2 + = q2 d2 0...... 0 V jv + V qn jv Qn Dn j n + δ 0 0 0 0 e dn (2.2) jδ e 0 0 jδ2 e 0 T = 0 0 jδn 0 0 0 0 e (2.22) The equation (2.20) can be written as ˆV = TV (2.23) Thus T is a transformation that transforms the d and q quantities of all machines to the system frame, which a common frame is moving at synchronous speed. The transformation matrix T contains elements only at the leading diagonal and hence we can show that T is orthogonal, i.e. T - = T *. Now the equation (2.23) can be rewritten as * ˆ V=T V (2.24) Similarly for node current Î = TI (2.25)
34 * ˆ I = T I (2.26) we get Substituting equation (2.25) and equation (2.23) in equation (2.5), I = MV (2.27) where M = T - YT (2.28) Linearizing equation (2.27) and making necessary substitutions (Anderson and Fouad 2003), the following equations are obtained. I qi = G ii V qi B ii V di + n [ Y ii cos(θ ij - δ ij0 ) V qj ] j= - n [ j= Y ij sin (θ ij - δ ij0 ) V dj ] + n [ Y ij {sin(θ ij - δ ij0 ) V qj0 + cos(θ ij - δ ij0 )V dj0 ] δ ij j= ; i =,...n (2.29) I di = B ii V qi + G ii V di + n [ Y ij cos (θ ij - δ ij0 ) V dj ] j= + n [ j= Y ij sin (θ ij - δ ij0 ) V qj ] + n [ Y ij {sin (θ ij - δ ij0 )V dj0 - cos(θ ij - δ ij0 )V qj0 ] δ ij j= ; i =,...n (2.30) The state space model for linearized system is obtained by linearizing the differential and algebraic equations at an operating point. While doing this linearization process, additional terms involving terminal voltage components (which are not state variables) remain in the differential
35 equations. To express the voltage components in terms of state variables, the machine currents are also linearized and expressed in terms of state variables and voltage components. Finally the current components are eliminated using the interconnecting network algebraic equations. From the initial conditions, E d ' i0, E q ' i0, I qi0, I di0, E FDi0 and δ i0 are determined. Linearizing equation (2.) we get p E d ' i = {- E d ' i - (x qi - x q ' i ) I qi } / τ qo ' i ; i =,...n p E q ' i = { E FDi - E q ' i + ( x di - x d ' i ) I di } / τ do ' i ; i =,...n p ω i = { T mi (I di0 E d ' i + I qi0 E q ' i + E d ' i0 I di +E q ' i0 I qi )- D i ω i } / τ j ; i =,...n (replacing V by E'): p δ i = ω i ; i =, n (2.3) Substituting equations (2.29) and (2.30) in equation (2.3). p E d i = {[(x qi - x q ' i ) B ii ] E d ' i τ qo ' i + (x qi - x q ' i ) n [ Y ik {sin (θ ik - δ ik0 ) E d ' k -(x qi - x q ' i ) G ii E q ' i k= - (x qi - x q ' i ) n [ Y ik cos (θ ik - δ ik0 ) ] E q ' k k= - (x qi - x q ' i ) n [ Y ik cos(θ ik -δ ik0 )] E d ' k0 +Y ik sin ((θ ik - δ ik0 ) E d ' k0 ] δ ik } k= i =,2.n (2.32)
36 p E q ' i = τ ' do i {[(x di x d ' i ) B ii ] E q ' i + (x di x d i ) n [ Y ik {sin (θ ik - δ ik0 )] E d ' k + (x di x d ' i ) G ii E d ' i k= + (x di x d ' i ) n [ Y ik sin (θ ik - δ ik0 ) ] E q ' k k= - (x di x d ' i ) n [ Y ik cos(θ ik -δ ik0 ) E q ' k0 -Y ik sin((θ ik -δ ik0 )E d ' k0 ] δ ik + E FDi )} k= i =,2.n (2.33) p ω i = τ ji {[ T mi - D i ω i -[I di0 + G ii E d ' i0 - B ii E q ' i0 ] E d ' i - [ I qi0 + B ii E d ' i0 + G ii E q ' i0 ] E q 'i - n [ k= Y ik cos (θ ik - δ ik0 ) E d ' i0 - Y ik sin (θ ik - δ ik0 ) E q ' i0 ] E d ' k - n [ k= Y ik sin (θ ik - δ ik0 ) E d ' i0 + Y ik cos (θ ik - δ ik0 ) E q ' i0 ] E q ' k - n [ k= Y ik cos (θ ik - δ ik0 ) (-E q ' k0 E d ' i0 +E d ' k0 E q ' i0 ) + Y ik sin ((θ ik - δ ik0 ) (-E d ' k0 E d ' i0 +E q ' k0 E q ' i0 ) δ ik } i =,2.n (2.34) p δ i = ω - ω i (2.35) i = 2,3.n taking machine as reference.
37 The above set of equations (2.32 to 2.35) gives the state space model of n-machine system. 2.6 EXCITER REPRESENTATION The state space equation of the exciter can be derived from the block diagram of the exciter shown in the Figure 2.3. From the Figure 2.3, we get K A EFD = + STA V t (2.36) For n, number of exciters, the state equations is as follows: -K p E = - V + V - E ; ( ) Ai fdi Refi i fdi TAi TAi i=, n (2.37) Now the state vector of the n machine state model including exciter equation is as follows. X T i = [ E d ' i E q ' i ω i δ i E FD i ] ; i=, n (2.38) 2.7 CONVENTIONAL POWER SYSTEM STABILIZER REPRESENTATION The Conventional Power System Stabilizer (PSS) adds damping to the generator rotor oscillations by controlling its excitation using auxiliary stabilizing signals. To provide damping, the stabilizer must produce a component of electrical torque in phase with the rotor speed deviations.
38 The important blocks in a power system stabilizer are: Washout circuit. Phase compensator. Stabilizer gain. The state space equation for the power system stabilizer (PSS) can be obtained from the block diagram shown in Figure 2.0. Figure 2.0 Conventional Power System Stabilizer Structure (CPSS) From the wash out block, we get st V = (K ω) w 2 PSS + stw (2.39) p V 2i = K PSSi p ωi (/T wi ) V 2i ; i =,...n (2.40) From the phase compensator block we get + st V V + st 2 = s 2 (2.4) From equation (2.4) we get p V si = (T i /T 2i ) p V 2i +(/T 2i ) V 2i (/T 2i ) V si ; i =,...n (2.42)
39 The state vector of the complete system after the inclusion of power system stabilizer is as follows: x T i = [ E' di E' qi ω i δ i E FDi V 2i Vs i ] ; i=, n (2.43) 2.8 FUZZY LOGIC BASED POWER SYSTEM STABILIZER (FPSS) FPSS. Figure 2. shows the schematic block diagram of the system with d dt ω ω FPSS V s + V t - + V ref Power System Generator and Exciter ω Figure 2. Structure of the Power system with FPSS Speed Deviation of the synchronous machine ( ω) and its deviation ( ω ) are chosen as inputs to the FPSS. Simulation of the sample SMIB system without PSS is carried out for several operating conditions and different disturbances and the inputs are normalized using their estimated peak values. Seven labels are taken for both the inputs and output. The labels are LP (large positive), MP (medium positive), SP (small positive), VS (very small), SN (small negative), MN (medium negative) and LN (Large negative). Linear triangular membership function is used in the design of FPSS. In our design of FPSS, the fuzzy sets with triangular membership function for ω are shown in Figure 2.2. The membership function for similar to the above Figure 2.2. ω and Vs are
40 LN MN SN VS SP MP LP - -0.66-0.33 0 0.33 0.66 Figure 2.2 Triangular membership function of ω Khan 2000). Table 2. shows the rules of fuzzy logic based PSS (Lakshmi and Table 2. Rule Table of fuzzy logic PSS ω LP MP SP VS SN MN LN LP LP LP LP LP MP SP VS ω MP LP LP MP MP SP VS SN SP LP MP SP SP VS SN MN VS MP MP SP VS SN MN N SN MP SP VS SN SN MN LN MN SP VS SN MN MN LN LN LN VS SN MN LN LN LN LN
4 2.9 CONCLUSION Mathematical model of SMIB system for dynamic stability analysis is presented in this chapter. Various state variables with PSS, system matrix including static exciter and CPSS are included in this chapter. Block diagram of simplified linear model of SMIB including exciter and CPSS is also neatly presented in this chapter. Non linear mathematical model representing the dynamics of the multi machine power system combining the synchronous machine model, excitation system (IEEE Type ST A), with conventional power system stabilizers are described in this chapter. The fuzzy logic based PSS model is also described.