POWER FLOW SOLUTION METHODS FOR ILL- CONDITIONED SYSTEMS

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1 104 POWER FLOW SOLUTION METHODS FOR ILL- CONDITIONED SYSTEMS 5.1 INTRODUCTION: In the previous chapter power flow solution for well conditioned power systems using Newton-Raphson method is presented. The conclusions drawn from chapter 4 indicate that the standard Newton- Raphson method fails to obtain the solution when the R/X ratio is high. Researchers in the past indicated that the standard N-R method failed to converge due to the following reasons. 1. Selection of reference slack bus 2. Existence of negative line reactance 3. High R/X ratio 4. Choosing initial values In this chapter, load flow solution methods such as Iwamoto s optimal multiplier method, Runge-Kutta method and Newton s accelerated methods are implemented. The proposed methods are tested with 11 and 13-bus Ill-Conditioned test systems which have the characteristics as mentioned above. Solutions are obtained along with the incorporation of the FACTS devices. Conclusions drawn from these solutions are furnished. 5.2 RUNGE KUTTA METHOD MODELING It is well known that the forward Euler method, even with the variable time step, can be numerically unstable. Reference [52]

2 105 suggests that given analogy between the power flow equations (5.1) and ODE (5.2), any well-assessed numerical method can be used to integrate (5.2). It is thus intriguing to use some efficient integration method for solving (5.1).It is observed that, the computation of f=-[g] -1 g implies the inversion of power flow Jacobain matrix, only explicit integration methods are suitable and computationally efficient, since one does not need to compute the Jacobian matrix of f. F(x) =0 (5.1) i 1 i f f i x x (5.2). x f ( x) (5.3) For the sake of example, we use classical fourth order Runge- Kutta formula (RK4). A generic step of the RK4 is as follows: K1=f(x (i) ) (5.4) K2=f(x (i) +0.5 tk1) K3=f(x (i) +0.5 tk2) K4=f(x (i) + tk3) X (i+1) =x (i) + t(k1+ 2k2+ 2k3+ k4)/6 The time step t can be adopted based on the estimated truncation error of integration method [54]. For example, RK4 error can be estimated based on half step method. =max(abs(k2-x (i+1) )) (5.5) Then the time step t can be computed based on the following simple heuristic rules. If >0.01 then t=max(0.985 t,0.75) (5.6)

3 106 If 0.01 then t=min(1.015 t,0.75) Based on these rules, the time step is increased if the truncation is greater than a given threshold and decreased if the truncation error is lower than a given threshold. The minimum value of time step is limited to If the lower value of time step t is not limited, in the case of unsolvable power flow problems, the proposed algorithm provides a solution close to the feasibility boundary of power flow equations [53]. All thresholds and tuning parameters in equation (5.6) have been determined based on heuristic criteria STEPS TO IMPLEMENT RK 1. Initialize x (0) and set iteration count i=1, Δt=1 2. Solve (5.4) 3. If ε is greater than max(abs( x (i) )), then stop the iterations. Otherwise go to step 4 4. Update Δt using (5.5) and increment iteration count by If iteration count is more than maximum no.of iterations then stop the iterations. Otherwise go to step (2)

4 FLOW CHART Initialization X (0),i=1, t=1 Solve(5.4) Yes Є>max(abs( x (i) )) Stop Update t using (5.6) i=i+1 Yes i> i max Fig.5.1. Flowchart of the proposed RK4-based continuous Newton s method for solving the power flow analysis NUMERICAL EXAMPLE Let us consider the two variable problem given below. The problem can be solved easily using the RK method x 1 2x1x2 2x x 1 x1x2 2x (5.7) Where the initial estimate is X E, 1.0 And time interval is Δt=0.1 Using eq(5.4), K1, K2, K3, K4 and L1, L 2, L 3, L 4 are obtained as below. K1=0.24, K2=0.2877, K3=0.3442, K4=0.4758

5 108 L1=1.64, L 2=1.6161, L 3=1.5879, L 4= The new estimate is therefore x x (1) 1 (1) 2 x x (0) 1 (0) 2 t(k 1 t(l 1 2k 2 2L 2 2k 3 2L 3 k 4 L )/ )/ The same procedure is applied for the rest. The converged solution is as follows: X1= and X2= DERIVATION OF THE PROPOSED OPTIMAL MULTIPLIER Let us derive the optimal multiplier. Moving all the right-hand side of Taylor series to the 1eft hand side, we have Ys-y(xe)-J x-y( x)=0 (5.7) In order to adjust the length of the correction vector x, we multiply the scalar quantity μ by x. Then it follows that. Ys-y(xe)-Jμ x-y(μ x)=0 (5.8) In the above equation, μ in the third term can appear in front of J being a scalar, and the forth term can become μ 2 y( x), that is Ys-y(xe)-μj x-μ 2 y( x)=0 (5.9) Here we define the vectors a,b,c for simplicity. a1 b1 c1... a. ys y( xe ), b. J x, c. y( x) (5.10)... a n b n c n a+μb+μ 2 c=0 (5.11) In order to determine the value of the μ in a least squared sense, the following cost function is considered.

6 109 F 1 a 2 n Minimize 2 2 i 1 i b c (5.12) i i The optimal solution μ* of the above equation can obtained by solving the equation below. F 0 (5.13) Namely, 2 3 g g g g 0 (5.14) Where g g n n 2 aibi, g1 b i aici 0 2 i 1 i 1 n n 2 2 bi ci, g3 2 ci i 1 i 1 3 (5.15) It can be easily observed that the equation (5.14) is a scalar cubic equation with respect to μ. This equation can be solved for optimal value of μ Application of the Optimal Multiplier to the N-R Method The most widely used AC load flow calculation method is the N- R method, and our examination also revealed that the application of the optimal multiplier to the N-R method was most effective. Thus, we describe here how to apply the optimal multiplier to the N-R method. If applied to the N-R method, the solution never diverges but converges in such a manner that the value of the cost function always decreases. In the N-R method, the correction vector x(r) is obtained basically by triangulating the matrix J(r) in the following equation.

7 110 Ys-y(xe(r))=J(r) x(r) (5.16) The quantities required for calculating the optimal multiplier μ(r) * are given by (5.10) as below. A(r)=ys-y(xe(r)) (5.17) B(r)=-J(r) x(r)=-a(r) (5.18) C(r)=-y( x(r)) (5.19) te the important fact that b(r) =-a(r) in (5.18). These calculations are carried out automatically in the process of the N-R method, and thus no additional calculations are required EXAMPLE In order to illustrate the proposed method, first a simple example using two variables is considered Two Variables Example: Let us consider the problem mentioned earlier in this chapter. Although the problem can be solved easily using the RK method, it is used here just to demonstrate the application procedure of the proposed optimal multiplier x 1 2x1x2 2x x 1 x1x2 2x Where the initial estimate is (5.20) 0 X E Using the NR method, Δx (0) is obtained as below. X (0) A (0),b (0),c (0) are calculated from (5.21),(5.22),(5.23).

8 111 (0) 0.24 (0) 0.24 (0) a, b, c (5.21) g, g are from (5.19), (0) (0) (0) 0 g1, g 2, (0) 3 g (0) , g (0) (5.22) g (0) , g (0) The scalar cubic equation to be solved is (omitting (0) for simplicity) (5.23) Solving the equation, the optimal multiplier is obtained as follows. (0)* (5.24) The new estimate is therefore x (1) e x (0) e (0)* x (0) * (5.25) Using x (1) e, the next correction vector Δx (l) is obtained by the N-R method, and μ(l)* is calculated by the proposed method, then x (2) e (1) (1)* (1) xe x, and the same procedure is applied for the rest NR METHOD WITH ACCELERATED CONVERGENCE TWO STEP ALGORITHM This method is based on the numerical technique [63] and can be summarized mathematically as follows: y n n 1 J ( x ) B( x ) x (5.26) n n n 1 J ( y ) B( y ) (5.27) n n

9 112 x n 1 yn (2 C n ) n (5.28) Where and C are positive constants, n is the norm of the vector n and y is the intermediate solution vector THREE STEP ALGORITHM This method is an extension of two step algorithm [63] and can be summarized mathematically as follows: y n n 1 J ( x ) B( x ) x (5.29) n n z n n 1 J ( y ) B( y ) y (5.30) n n n 1 J ( z ) B( z ) (5.31) n n x n 1 zn (2 C n ) n (5.32) Where and C are positive constants, n is the norm of the vector n and y and z are the intermediate solutions CASE STUDY WITH RK METHOD In this section case study is presented for 13- and 11- Ill- Conditioned systems with and without devices. The data and Line data for the test systems are shown from Appendix-III to Appendix-IV. The initial data of devices for 13- and 11- Ill-Conditioned system is as follows:

10 113 Initial Values for 13-bus Ill-Conditioned system with STATCOM STATCOM is connected to bus : 3 Converter s reactance (p.u.), Xvr = 10 Target nodal voltage magnitude (p.u.)=1 Initial source voltage magnitude (p.u.), Vvr = 1 Initial source voltage angle (deg)=0 Lower limit of source voltage magnitude (p.u.)=1.1 higher limit of source voltage magnitude (p.u.)=0.95 Initial Values for 13-bus Ill-Conditioned system with SVC Susceptance Model SVC is connected to bus : 9 Initial SVC s susceptance value (p.u.)=0.02 Lower limit of variable susceptance (p.u.)=-0.25 Higher limit of variable susceptance (p.u)=0.25 Target nodal voltage magnitude to be controlled by SVC (p.u.)=1 Initial Values for 13-bus Ill-Conditioned system with SVC Firing Angle Model SVC is connected to bus : 13 Capacitive reactance (p.u.)=1.07 Inductive reactance (p.u.)=0.288 Initial value of SVC s firing angle (Deg)=140 Lower limit of firing angle (Deg)=90 Higher limit of firing angle (Deg)=180 Target nodal voltage magnitude to be controlled by SVC (p.u.)=1

11 114 Initial Values for 13-bus Ill-Conditioned system with TCSC Variable Impedance Power Flow Model TCSC is connected between bus-10 and bus-11 Reactance of TCSC=-0.05 Lower reactance limit=-0.09 Higher reactance limit=0.09 Power flow direction is taken from sending end to receiving end Active power flow to be controlled=0.1 Initial Values for 13-bus Ill-Conditioned system with TCSC Variable Firing Angle Power Flow Model TCSC is connected between bus-7 and bus-8 Capacitive reactance of TCSC (p.u.)=9.375 Inductive reactance of TCSC (p.u)=1.625e-1 Power flow direction is taken from receiving end to sending end Target active power flow (p.u.)=0.13 Initial firing angle (deg)=140 Firing angle lower limit (deg)=60 Firing angle higher limit (deg)=180 Initial Values for 13-bus Ill-Conditioned system with UPFC Shunt converter is connected to bus=7 Series converter is connected between bus-7 and bus-8 Inductive reactance of Shunt impedance (p.u.)=0.1 Inductive reactance of Series impedance (p.u.)=5 Power flow direction is taken from receiving end to sending end Target active power flow (p.u.)=0.4

12 115 Target reactive power flow (p.u.)=0.02 Initial value of the series source voltage magnitude (p.u.)=0.04 Initial value of the series source voltage angle (rad.)=-pi/4 Lower limit of series source voltage magnitude (p.u.)=0.001 Higher limit of series source voltage magnitude (p.u.)=0.2 Initial value of the shunt source voltage magnitude (p.u.)=1 Initial value of the shunt source voltage angle (rad.)=0 Lower limit of shunt source voltage magnitude (p.u.)=0.95 Higher limit of shunt source voltage magnitude (p.u)=1.1 Target nodal voltage magnitude to be controlled by shunt branch (p.u.)=1 Initial Values for 11-bus Ill-Conditioned system with STATCOM STATCOM is connected to bus : 4 Converter s reactance (p.u.), Xvr = 10 Target nodal voltage magnitude (p.u.)=1 Initial source voltage magnitude (p.u.), Vvr = 1 Initial source voltage angle (deg)=0 Lower limit of source voltage magnitude (p.u.)=1.1 higher limit of source voltage magnitude (p.u.)=0.95 Initial Values for 11-bus Ill-Conditioned system with SVC Susceptance Model SVC is connected to bus : 4 Initial SVC s susceptance value (p.u.)=0.02 Lower limit of variable susceptance (p.u.)=-0.25 Higher limit of variable susceptance (p.u)=0.25 Target nodal voltage magnitude to be controlled by SVC (p.u.)=1

13 116 Initial Values for 11-bus Ill-Conditioned system with SVC Firing Angle Model SVC is connected to bus : 11 Capacitive reactance (p.u.)=1.07 Inductive reactance (p.u.)=0.288 Initial value of SVC s firing angle (Deg)=140 Lower limit of firing angle (Deg)=90 Higher limit of firing angle (Deg)=180 Target nodal voltage magnitude to be controlled by SVC (p.u.)=1 Initial Values for 11-bus Ill-Conditioned system with TCSC Variable Impedance Power Flow Model TCSC is connected between bus-4 and bus-6 Reactance of TCSC= Lower reactance limit=-0.05 Higher reactance limit=0.05 Power flow direction is taken from sending end to receiving end Active power flow to be controlled=0.21 Initial Values for 11-bus Ill-Conditioned system with TCSC Variable Firing Angle Power Flow Model TCSC is connected between bus-4 and bus-6 Capacitive reactance of TCSC (p.u.)=9.375e-3 Inductive reactance of TCSC (p.u)=1.625e-2 Power flow direction is taken from receiving end to sending end Target active power flow (p.u.)=-0.45 Initial firing angle (deg)=145

14 117 Firing angle lower limit (deg)=90 Firing angle higher limit (deg)=180 Initial Values for 11-bus Ill-Conditioned system with UPFC Shunt converter is connected to bus=4 Series converter is connected between bus-4 and bus-5 Inductive reactance of Shunt impedance (p.u.)=0.1 Inductive reactance of Series impedance (p.u.)=5 Power flow direction is taken from receiving end to sending end Target active power flow (p.u.)=0.4 Target reactive power flow (p.u.)=0.02 Initial value of the series source voltage magnitude (p.u.)=0.04 Initial value of the series source voltage angle (rad.)=-pi/4 Lower limit of series source voltage magnitude (p.u.)=0.001 Higher limit of series source voltage magnitude (p.u.)=0.2 Initial value of the shunt source voltage magnitude (p.u.)=1 Initial value of the shunt source voltage angle (rad.)=0 Lower limit of shunt source voltage magnitude (p.u.)=0.95 Higher limit of shunt source voltage magnitude (p.u)=1.1 Target nodal voltage magnitude to be controlled by shunt branch (p.u.)=1

15 BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE Table 5.1: s and Phase angles Table 5.2: Complex Power Flows through Lines Sending end Power Receiving end Power e

16 BUS ILL-CONDITIONED SYSTEM WITH STATCOM The 13-bus Ill-Conditioned system is designed by Japanese researchers which exhibits high degree of Ill-Conditionality, and it is also reported in the literature that the conventional Newton-Raphson method fails to converge for this system. The same system is considered to demonstrate the effect of FACTS devices on the power flow and convergence behavior. The steady state mathematical model developed for STATCOM in chapter-3, is incorporated into the RK method to test for the existence of the solution for 13-bus Ill-Conditioned system. The STATCOM is connected at bus no 3 to regulate the voltage to 1.0 per unit.. The voltage profile before incorporation of the device is per unit. The STATCOM reactance is taken as 10 per unit on 100 MVA base. The initial source voltage angle is taken as 0 0. From the results obtained, it is observed that the device is able to regulate the voltage at bus no 3 to 1.0 per unit. The source voltage angle is degrees. The reactive power flow in the lines connected to the bus no 3 is significantly affected. The real power flow in the lines with and without the device remains the same. The real power mismatch variation is large in the first three iterations. The problem is converged in 18 iterations without device and 21 iterations with device. The large number of iterations for the solution may be due to system Ill- Conditionality as well as non linearities introduced by the device.

17 120 Table 5.3: s and Phase angles Table 5.4: Complex Power Flows through Lines Sending end Power Receiving end Power

18 BUS ILL-CONDITIONED SYSTEM WITH SVC- SUCCEPTANCE MODEL The steady state mathematical models developed for SVC in susceptance and firing angle modes in chapter-3 are incorporated into the Runge -Kutta algorithm to test for the existence of solution for a 13-bus Ill-Conditioned system The SVC is connected to bus no 9. The initial susceptance value is chosen as 0.02 per unit on 100 MVA base. With a susceptance of 0.03 p.u in susceptance, it is revealed from the study that reactive power flow variations are significant. The solution is converged in 20 iterations whereas the problem is converged in 18 iterations without device. This may be due to nonlinearities of the device model. The mismatch power variations are large in first three iterations.

19 122 Table 5.5: s and Phase angles Table 5.6: Complex Power Flows through Lines Sending end Power Receiving end Power

20 BUS ILL-CONDITIONED SYSTEM WITH SVC-FIRING ANGLE MODEL The SVC is connected to bus no 9. The initial values are chosen as capacitive reactance is 1.07 p.u., inductive reactance is p.u. and firing angle is 140 degrees. From the results it is observed that the reactive power flow variations are significant. The solution is converged in 20 iterations whereas the problem is converged in 18 iterations without device. This may be due to non-linearities of the device model. The mismatch power variations are large in first three iterations.

21 124 Table 5.7: s and Phase angles Table 5.8: Complex Power Flows through Lines From To Sending end Power Receiving end Power Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)

22 BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE IMPEDANCE POWER FLOW MODEL The steady state mathematical models developed for TCSC in susceptance and firing angle modes in chapter-3 are incorporated into the runge -kutta algorithm to test for the existence of solution for a 13-bus.The TCSC is connected between bus-10 and bus-11 with the reactance of p.u.and pre specified power flow in the line is set at 0.1 p.u. It is observed that with the incorporation of TCSC there is much difference in real power flows through the lines where as reactive power flows are not much affected. From the power mismatch versus iterations it is observed that there is a large variation in mismatch powers in the first two iterations and the variable impedance power flow model exhibits oscillations between fourth and fifth iterations.

23 126 Table 5.9: s and Phase angles Table 5.10: Complex Power Flows through Lines From To Sending end Power Receiving end Power Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)

24 BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING ANGLE MODEL The TCSC is connected between bus-7 and bus-8 with the following initial conditions: inductive reactance is p.u., capacitive reactance is p.u and pre specified power flow in the line is set at 0.13 p.u. The firing angle is degrees. It is observed that with the incorporation of TCSC there is much difference in real power flows through the lines where as reactive power flows are not much affected. From the power mismatch versus iterations it is observed that there is a large variation in mismatch powers in the first two iterations.

25 128 Table 5.11: s and Phase angles Table 5.12: Complex Power Flows through Lines Sending end Power Receiving end Power

26 BUS ILL-CONDITIONED SYSTEM WITH UPFC The steady state mathematical models developed for UPFC in chapter3 are incorporated into the runge -kutta algorithm to test for existence of solution for a 13-bus Ill-Conditioned system.the UPFC is incorporated between bus-7 and bus-8 with shunt converter connected close to bus 7 and series converter connected between buses 7 and 8 with inductive reactance of shunt converter taken as 0.1 p.u and inductive reactance of series converter taken as 5 p.u with specified active and reactive power flows set at 0.4 and 0.02 respectively. The shunt converter is set to maintain a target voltage of 1p.u.From the results it is observed that UPFC holds its target values with the given initial conditions. From the power flow results it is observed that UPFC is the only FACTS device which affects both real and reactive power flows through the lines.the maximum power mismatch is less when compared to the other facts devices discussed.

27 130 Table 5.13: s and Phase angles Table 5.14: Complex Power Flows through Lines Sending end Power Receiving end Power e

28 COMPARISON OF MAXIMUM POWER MISMATCH FOR 13- BUS ILL-CONDITIONED SYSTEM Fig 5.2 Comparison of maximum power mismatch using Runge-Kutta method for 13-bus Ill-Conditioned System

29 BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE Table 5.15: s and Phase angles Phase Angle Table 5.16: Complex Power Flows through Lines Sending end Power Receiving end Power

30 BUS ILL-CONDITIONED SYSTEM WITH STATCOM The steady state mathematical model developed for STATCOM in chapter3 is incorporated into the runge -kutta algorithm to test for the existence of solution for a 11-bus Ill-Conditioned system The STATCOM is connected at bus no 4 to regulate the voltage to 1.0 per unit.. The voltage profile before incorporation of the device is.9987 per unit. The STATCOM reactance is taken as 10 per unit on 100 MVA base. The initial source voltage angle is taken as 0 0. From results obtianed, it is observed that the device is able to regulate the voltage at bus no 4 to 1.0 per unit. The source voltage angle is 10 degrees. The reactive power flow in the lines connected to the bus no 4 is significantly affected. The real power flow in the lines with and without the device remains the same. The real power mismatch variation is large in the first three iterations.

31 134 Table 5.17: s and Phase angles Table 5.18: Complex Power Flows through Lines Sending end Power Receiving end Power

32 BUS ILL-CONDITIONED SYSTEM WITH SVC SUSCEPTANCE MODEL The SVC is connected to bus no 4. The initial susceptance value is chosen as 0.02 per unit on 100 MVA base. With a susceptance of 0.06 p.u in susceptance, it is revealed from the study that reactive power flow variations are significant. The solution is converged in 16 iterations whereas the problem is converged in 14 iterations without device. This may be due to non-linearities of the device model. The mismatch power variations are large in first three iterations.

33 136 Table 5.19: s and Phase angles Table 5.20: Complex Power Flows through Lines Sending end Power Receiving end Power

34 BUS ILL-CONDITIONED SYSTEM WITH SVC FIRING ANGLE MODEL The SVC is connected to bus no 11. The initial values are chosen as capacitive reactance is 1.07 p.u., inductive reactance is p.u. and firing angle is 140 degrees. From the results it is observed that the reactive power flow variations are significant. The solution is converged in 20 iterations whereas the problem is converged in 18 iterations without device. This may be due to nonlinearities of the device model. The mismatch power variations are large in first three iterations.

35 138 Table 5.21: s and Phase angles Table 5.22: Complex Power Flows through Lines Sending end Power Receiving end Power

36 BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE IMPEDANCE POWER FLOW MODEL The steady state mathematical models developed for TCSC in susceptance and firing angle modes in chapter-3 are incorporated into the runge -kutta algorithm to test for the existence of solution for a 13-bus.The TCSC is connected between bus-4 and bus-6 with the reactance of p.u. and pre specified power flow in the line is set at 0.21 p.u. It is observed that with the incorporation of TCSC there is much difference in real power flows through the lines where as reactive power flows are not much affected. From the power mismatch versus iterations it is observed that there is a large variation in mismatch powers in the first two iterations and the variable impedance power flow model exhibits oscillations between fourth and fifth iterations.

37 140 Table 5.23: s and Phase angles Table 5.24: Complex Power Flows through Lines Sending end Power Receiving end Power

38 BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING ANGLE MODEL The TCSC is connected between bus-4 and bus-6 with the following initial conditions: inductive reactance is 9.375e-3 p.u., capacitive reactance is 1.625e-2 p.u and pre specified power flow in the line is set at 0.45 p.u. The firing angle is degrees. It is observed that with the incorporation of TCSC there is much difference in real power flows through the lines where as reactive power flows are not much affected. From the power mismatch versus iterations it is observed that there is a large variation in mismatch powers in the first two iterations.

39 142 Table 5.25: s and Phase angles Table 5.26: Complex Power Flows through Lines Sending end Power Receiving end Power

40 BUS ILL-CONDITIONED SYSTEM WITH UPFC The steady state mathematical models developed for UPFC in chapter3 are incorporated into the runge -kutta algorithm to test for existence of solution for a 11-bus Ill-Conditioned system.the UPFC is incorporated between bus-4 and bus-5 with shunt converter connected close to bus 4 and series converter connected between buses 4 and 5 with inductive reactance of shunt converter taken as 0.1 p.u and inductive reactance of series converter taken as 5 p.u with specified active and reactive power flows set at 0.4 and 0.02 respectively. The shunt converter is set to maintain a target voltage of 1 p.u. From the results it is observed that UPFC holds its target values with the given initial conditions. From the power flow results it is observed that UPFC is the only FACTS device which affects both real and reactive power flows through the lines.the maximum power mismatch is less when compared to the other facts devices discussed.

41 144 Table 5.27: s and Phase angles Table 5.28: Complex Power Flows through Lines Sending end Power Receiving end Power

42 > Maximum Power Mismatch COMPARISON OF MAXIMUM POWER MISMATCH FOR 11- BUS ILL-CONDITIONED SYSTEM x Comparison of Maximum Power Mismatch Without Device With STATCOM With SVC-B With SVC-FA With TCSC-POWER With TCSC-FA With UPFC >.of Iterations Fig 5.3 Comparison of maximum power mismatch using Runge-Kutta method for 11-bus Ill-Conditioned System

43 CASE STUDY WITH OPTIMAL MULTIPLIER METHOD In this section case study is presented for 13- and 11- Ill- Conditioned systems with and without devices. The initial data for the devices is presented in section BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE Table 5.29: s and Phase angles Table 5.30: Complex Power Flows through Lines Sending end Power Receiving end Power

44 BUS ILL-CONDITIONED SYSTEM WITH STATCOM Table 5.31: s and Phase angles Table 5.32: Complex Power Flows through Lines Sending end Power Receiving end Power

45 BUS ILL-CONDITIONED SYSTEM WITH SVC- SUCEPTANCE MODEL Table 5.33: s and Phase angles Table 5.34: Complex Power Flows through Lines Sending end Power Receiving end Power

46 BUS ILL-CONDITIONED SYSTEM WITH SVC-FIRING ANGLE MODEL Table 5.35: s and Phase angles Table 5.36: Complex Power Flows through Lines Sending end Power Receiving end Power

47 BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE IMPEDANCE POWER FLOW MODEL Table 5.37: s and Phase angles Table 5.38: Complex Power Flows through Lines Sending end Power Receiving end Power

48 BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING ANGLE MODEL Table 5.39: s and Phase angles Table 5.40: Complex Power Flows through Lines Sending end Power Receiving end Power

49 BUS ILL-CONDITIONED SYSTEM WITH UPFC Table 5.41: s and Phase angles Table 5.42: Complex Power Flows through Lines Sending end Power Receiving end Power

50 Comparison of Maximum Power Mismatch for 13- Ill-Conditioned System Fig 5.4 Comparison of maximum power mismatch using Iwamoto method for 13-bus Ill-Conditioned System

51 BUS ILL-CONDITIONED SYSTEM WITHOUT ANY DEVICE Table 5.43: s and Phase angles Table 5.44: Complex Power Flows through Lines Sending end Power Receiving end Power

52 BUS ILL-CONDITIONED SYSTEM WITH STATCOM Table 5.45: s and Phase angles Table 5.46: Complex Power Flows through Lines Sendng end Power Recevng end Power

53 BUS ILL-CONDITIONED SYSTEM WITH SVC SUSCEPTANCE MODEL Table 5.47: s and Phase angles Table 5.48: Complex Power Flows through Lines Sending end Power Receiving end Power

54 BUS ILL-CONDITIONED SYSTEM WITH SVC FIRING ANGLE MODEL Table 5.49: s and Phase angles Table 5.50: Complex Power Flows through Lines Sending end Power Receiving end Power

55 BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE IMPEDANCE POWER FLOW MODEL Table 5.51: s and Phase angles Table 5.52: Complex Power Flows through Lines Sending end Power Receiving end Power

56 BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING ANGLE MODEL Table 5.53: s and Phase angles Table 5.54: Complex Power Flows through Lines Sending end Power Receiving end Power

57 BUS ILL-CONDITIONED SYSTEM WITH UPFC MODEL Table 5.55: s and Phase angles Table 5.56: Complex Power Flows through Lines Sending end Power Receiving end Power

58 > Maximum Power Mismatch COMPARISON OF MAXIMUM POWER MISMATCH FOR 11- BUS ILL-CONDITIONED SYSTEM Comparison of Maximum Power Mismatch Without Device With STATCOM With SVC-B With SVC-FA With TCSC-POWER With TCSC-FA With UPFC >.of Iterations Fig 5.5 Comparison of maximum power mismatch using Iwamoto method for 11-bus Ill-Conditioned System

59 162 The 13 and 11-bus Ill-Conditioned systems discussed above are tested with the same initial conditions using iwamoto optimal multiplier method with the incorporation of FACTS devices.the placement of the devices remains same as in the case with Rungekutta method. All the facts devices (STATCOM,SVC,TCSC and UPFC) hold their functional capabilities with the incorporation of optimal multiplier method for the steady state mathematical models presented in chapter 3. The results for voltage magnitudes and phase angles along with the power flows are tabulated above.the value of optimal multiplier μ stays nearer to one with the incorporation of devices. From the results obtained, it is observed that the number of iterations taken to converge is less when compared with runge- kutta method. The converging behavior of TCSC variable impedance power model is oscillatory.

60 Case Study with 2-Step Method In this section case study is presented for 13- and 11- Ill- Conditioned systems with and without devices. The initial data for the devices is presented in section BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE Table 5.57: s and Phase angles Table 5.58: Complex Power Flows through Lines Sending end Power Receiving end Power E

61 BUS ILL-CONDITIONED SYSTEM WITH STATCOM Table 5.59: s and Phase angles Table 5.60: Complex Power Flows through Lines Sending end Power Receiving end Power E

62 BUS ILL-CONDITIONED SYSTEM WITH SVC SUSCEPTANCE MODEL Table 5.61: s and Phase angles Table 5.62: Complex Power Flows through Lines Sending end Power Receiving end Power e

1 Introduction General Background The New Computer Environment Transmission System Developments Theoretical Models and Computer Programs

1 Introduction General Background The New Computer Environment Transmission System Developments Theoretical Models and Computer Programs Modeling Techniques in Power Systems 1 General Background The New Computer Environment Transmission System Developments Theoretical Models and Computer Programs 2 Transmission Systems Linear Transformation