51 CHAPTER 3 FUZZIFIED PARTICLE SWARM OPTIMIZATION BASED DC- OPF OF INTERCONNECTED POWER SYSTEMS 3.1 INTRODUCTION Optimal Power Flow (OPF) is one of the most important operational functions of the modern energy management system. The purpose of the OPF is to find the optimum generation schedule among the committed units, such that the total generation cost is minimized while simultaneously satisfying the power flow equations and various other constraints in the system. The OPF solution gives the optimal settings of all controllable variables for a static loading condition. A number of mathematical programming based techniques have been proposed to solve the OPF problem. Methods based on successive linearization and interior point methods are popular. For medium size power systems, the conventional methods for OPF calculations may be fast and efficient enough. However, for large scale interconnected power systems the higher dimension of possible solution space and increase of constraints result in excessive computational burden. In order to reduce the computational burden Fuzzified PSO algorithm is developed to solve the OPF problem of multi-area network.
52 Electric power systems are interconnected due to the fact that it gives a better system to operate with more reliability, improved stability and less production cost than the isolated systems. The multi-area OPF problem is a large scale non-linear optimization problem with both linear and non-linear constraints. Multi-area OPF calculations determine the optimum generation schedule, optimal control variables and system quantities of each area with due consideration of generation and transmission system limitations. In order to verify the effectiveness of the proposed FPSO algorithm, first DC-model of power system network (Wood and Wollenberg, 1984) is used to solve multi-area power system network. In DCOPF, the objective function is non-linear and all the constraints are linear. DCOPF is used for the real time application in a de-regulated electricity market (Texas Electricity Market ERCOT). This chapter presents the formulation of Fuzzified Particle Swarm Optimization (FPSO) based algorithm for solving Multi-area DCOPF problem. Subsequently the FPSO based algorithm is applied on standard test system and the optimal results obtained are compared and analyzed with other stochastic optimization techniques. 3.2 PROBLEM FORMULATION OF MULTI-AREA DCOPF The Multi-area OPF problem is decoupled into equivalent single area sub problems (Bakirtzis et al 2003) by considering transmission line flows as new variables. The decoupling of Multi-area system into equivalent single area sub systems is shown in Figure 3.1.
53 Figure 3.1 Decoupling of Multi-Area Power Systems From Figure 3.1, the multi-area OPF problem is decoupled into single area OPF sub problems with two more variables namely, A T ij and AA T ji (Tie line flows). The single area OPF sub problems (considering area A) is formulated as follows, Objective: cng 2 2 A A A A A T Gi Gi Gi Gi Gi GS GS GS GS GS i1 Min F a P b P c a P b P c (3.1) Subject to: Power flow equations, B R T P D ; 0 (3.2) A A A A A A A ref
54 Generation active power limits, P P P for all generating units in area A (3.3) min A A max A Gi Gi Gi Slack bus generation limit, P P P (3.4) mina A maxa GS GS GS Transmission line flow constraints, 1 x ij LF for all lines in area A (3.5) A A max i j ij Tie line flow constraint, 1 T 0 A AA A i j ij x for all tie lines connecting area A (3.6) ij 3.3 PSO BASED ALGORITHM FOR MULTI-AREA DCOPF The detailed algorithmic steps for solving Multi-area DCOPF using PSO based algorithm are as follows: a) Initialization An initial swarm of particles I i of size n is generated randomly within the feasible range and the distributions of initial trial parents are uniform. The elements of each initial particle are the controllable real power outputs of committed cng generating units (excluding slack bus generator). A i A i A i A i Ii P G1,P G2,...,P Gj,...,P Gcng ;i 1,2,...,n (3.7)
55 The elements of I i are generated as, A i min A max A P U P, P ; j 1, 2,3,...,cng Gj Gj Gj (3.8) where Ux,y denotes a uniform random number between x and y. For each particle of the initial swarm, DC load flow is conducted using equation (3.2) and slack bus generation A P GS and line flows LF ij are computed. Evaluate fitness f i of all particles (i=1,2, n) in the swarm using equation (3.9) and obtain the Pbest of each particle and thereby compute the Gbest. Initialize the initial velocities of all particles to zero. nl NTie A i A ilim A ilim A i lim i FT 1 GS 2 LFk 3 Tk k1 K1 f k P k k (3.9) where, k 1, k 2 and k 3 are the penalty factors for constraints (3.4) to (3.6) violations respectively. P min A A i A i min A A i lim PGS P GS,if PGS PGS GS A i max A A i max A PGS P GS,if PGS PGS (3.10) LF A i lim k LF LF LF LF 0, otherwise A i max A A i max A k k,if k k (3.11) T 1 ( ) T (3.12) A i lim A AA A i k m n k x mn
56 b) Velocity updating Using the Pbest and Gbest compute the i th particle velocity as, v i iter+1 P j G i iter i A i w vp G j c1 rand Pbest j PGj A i c2 rand Gbest j PGj (3.13) c) Position updating as, Based on the updated velocities, each particle changes its position P P v (3.14) A i iter+1 A i iter i iter+1 Gj Gj P j G If a particle violates its position limits in any dimension, set its position at the violating limit. d) Updating Pbest and Gbest With the updated position evaluate fitness of all particles in the swarm and obtain the Pbest of each particle and thereby compute the Gbest. e) Stopping Criteria If the preselected maximum iteration is reached the Gbest is the global optimal solution, otherwise increment the iteration count and repeat steps b to d.
57 3.4 FPSO BASED ALGORITHM FOR MULTI-AREA DCOPF The various sequential steps involved in the FPSO based algorithm are same as that of Section 3.3 except the calculation of inertia weight factor in the velocity updating process as in Section 2.6. 3.5 OPTIMAL SOLUTION OF MULTI-AREA DCOPF The proposed FPSO algorithm is tested on a standard IEEE 30-bus system, an interconnected two area system (two identical IEEE 30-bus systems interconnected by a tie-line of scheduled interchange from Area 1 to Area 2) and an interconnected four area system as shown in Figure 3.2. The standard IEEE 30-bus system consists of 6 generating units, 41 lines and a total demand of 283.4 MW. The line data, bus data and generator data are presented in Tables A3.1, A3.2 and A3.3 respectively. The interconnected two area system has a scheduled tie-line power flow of 20 MW between buses 3 and 26 corresponding to Area 1 and Area 2 respectively. The interconnected four area system consists of 4 identical standard IEEE 30-bus systems with five tie-lines as shown in Figure 3.2. The scheduled tie-line interchange ST A-AA are as follows: ST A1-A2 = 70 MW, ST A2-A4 = 70 MW, ST A4-A1 = 60 MW, ST A1-A3 = 60 MW and ST A3-A4 = 60 MW. follows: The parameters used in the PSO and FPSO based algorithms are as c 1 = c 2 = 2.05, w max =2.0 and w min =0.2. The swarm size n for the PSO and FPSO is 100. The simulations were carried out on Pentium IV, 2.5 GHz processor.
58 Figure 3.2 Four Area Interconnected system The fuzzy logic data for inertia weight computation is same as in Table 2.2. The convergence characteristics of IEEE 30-bus system corresponding to EP, TS, PSO and FPSO algorithms based DCOPF (with same initial population) is shown in Figure 3.3. The convergence characteristics are drawn by plotting the minimum fitness value from the combined population across iteration index. From Figure 3.3 it is observed that the fitness function value converges smoothly to the optimum value without any abrupt oscillations, thus ensuring convergence reliability of the proposed FPSO algorithm and this algorithm has much better convergence than other techniques. The optimal solutions of IEEE 30-bus system using the proposed algorithm are compared with Successive Linear Programming (SLP), EP, TS and PSO techniques and the results are presented in Table 3.1.
59 Fitness Value 830 825 820 815 810 805 800 795 FPSO TS PSO EP 790 785 0 25 50 75 100 125 150 175 Iterations Figure 3.3 Convergence Characteristic of IEEE-30 Bus System using EP, TS, PSO and FPSO Algorithms Table 3.1 Optimal Solution of IEEE 30-Bus System Algorithm SLP EP TS PSO FPSO P G S (MW) 136.953 137.571 137.522 137.563 137.596 P G 2 (MW) 59.13 60.7081 60.6261 60.8202 60.6806 P G5 (MW) 23.9 23.0025 22.4942 22.2354 22.7319 P G 8 (MW) 25.89 31.4419 31.7184 31.7032 31.6482 P G 11 ( MW) 14.61 15.5786 14.8188 16.1062 15.039 P G 13 (MW) 22.88 15.0976 16.2201 14.9719 15.7047 Total Gen (MW) 283.4 283.4 283.4 283.4 283.4 Total Fuel cost ($/hr) 789.275 787.147 787.161 787.137 787.106 Max no of iterations 6 200 160 100 60 Computation time (ms) 15 100 80 60 45
60 From Table 3.1 it is inferred that for the same optimum solution the number of iterations for FPSO algorithm are lesser than EP, TS and PSO techniques respectively. Even though the number of iterations and CPU time are less for SLP, the optimal solution mainly depends on the initial conditions. It is observed that the deterministic SLP method initially suffers from oscillations and also the model becomes inaccurate when wider variations are allowed in the control variables. The optimal solutions of an interconnected two area system using the proposed FPSO algorithm are compared with PSO technique and the results are presented in Table 3.2. From Table 3.2 it is observed that fuzzy implemented FPSO algorithm has faster convergence. Table 3.2 Optimal Solution of Interconnected Two Area System using PSO and FPSO Algorithms Area Area 1 sub-problem Area 2 sub-problem Algorithm PSO FPSO PSO FPSO P G S (MW) 139.139 139.328 136.528 136.581 P G 2 (MW) 61.6068 60.9931 57.7171 58.0074 P G5 (MW) 25.2354 26.7094 21.6394 21.0951 P G 8 (MW) 32.3468 32.3227 21.8116 23.1588 P G 11 (MW) 19.5618 19.6465 12.9637 12.5579 P G 13 (MW) 25.5099 24.3998 12.7404 12 Total Gen (MW) 303.4 303.4 263.4 263.4 Total Fuel cost ($/hr) 865.05 865.025 713.39 713.376 Max no of iterations 100 60 100 60 CPU time in ms 60 45 60 45
61 The optimal solutions of an interconnected four area system using the proposed FPSO algorithm are compared with EP, TS and PSO techniques and the results are presented in Table 3.3 and Table 3.4. From Table 3.3 and Table 3.4 it is observed that fuzzy implemented FPSO algorithm has faster convergence. The computation time taken by the proposed FPSO algorithm is only 45%, 56% and 75% of the time taken by EP, TS and PSO methods respectively. Also it is inferred that the proposed algorithm can be used to solve any number of areas with larger number of buses. Table 3.3 Optimal Solution of Interconnected Four Area System for Area 1 and Area 2 Algorithm EP TS PSO FPSO Area 1 sub-problem P G S (MW) 142.83 141.307 141.55 142.131 P G 2 (MW) 65.34 57.35 58.3 60.5 P G5 (MW) 50 50 50 50 P G 8 (MW) 35 35 35 35 P G 11 (MW) 10 19.76 18.51 15.15 P G 13 (MW) 30.21 29.97 30.01 30.01 Total Fuel cost ($/hr) 1027.47 1027.09 1026.71 1026.2 Max no of iterations 200 160 100 60 CPU time in ms 100 80 60 45 Area 2 sub-problem P G S (MW) 136.46 136.668 136.5 136.79 P G 2 (MW) 57.01 57.187 57.21 59.38 P G5 (MW) 28.19 27.544 27.07 24.76 P G 8 (MW) 23.609 22.144 23.194 31.86 P G 11 (MW) 18.47 19.753 19.75 15.59 P G 13 (MW) 19.64 20.102 19.35 15.25 Total Fuel cost ($/hr) 791.208 791.136 790.645 790.268 Max no of iterations 200 160 100 60 CPU time in ms 100 80 60 45
62 Table 3.4 Optimal Solution of Interconnected Four Area System for Area 3 and Area 4 Algorithm EP TS PSO FPSO Area 3 sub-problem P G S (MW) 137.218 137.465 137.465 137.69 P G 2 (MW) 60.18 60.71 59.97 60.72 P G5 (MW) 24.23 22.77 23.08 22.95 P G 8 (MW) 27.55 29.28 33.58 32.16 P G 11 (MW) 19.21 17.08 14.13 16.29 P G 13 (MW) 15.01 16.07 15.16 13.57 Total Fuel cost ($/hr) 788.054 787.314 787.261 787.162 Max no of iterations 200 160 100 60 CPU time in ms 100 80 60 45 Area 4 sub-problem P G S (MW) 125.514 124.914 125.838 125.511 P G 2 (MW) 35.908 35.3 34.73 35.88 P G5 (MW) 15 15.5 15.84 16.54 P G 8 (MW) 10.572 12.64 11.98 10.26 P G 11 (MW) 11.74 10 10 10.19 P G 13 (MW) 12.65 12 12 12 Total Fuel cost ($/hr) 546.895 546.00 545.623 545.605 Max no of iterations 200 160 100 60 CPU time in ms 100 80 60 45 3.6 SUMMARY In this chapter EP, TS, PSO and proposed FPSO algorithms are implemented to solve multi-area DCOPF problem. The proposed FPSO method is demonstrated on standard test systems and its corresponding results are quite encouraging. The proposed FPSO has produced high quality solution, stable convergence characteristics and takes less computational time.