PROMPT PARTICLE SWARM OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM

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1 PROMPT PARTICLE SWARM OPTIMIZATION ALGORITHM FOR SOLVING OPTIMAL REACTIVE POWER DISPATCH PROBLEM K. Lenin 1 Research Scholar Jawaharlal Nehru Technological University Kukatpally,Hyderabad , India gklenin@gmail.com Dr.B.Ravindranath Reddy 2 Deputy Executive Engineer Jawaharlal Nehru Technological University Kukatpally,Hyderabad , India Dr.M.Surya Kalavathi 3 Professor of Electrical and Electronics Engineering, Jawaharlal Nehru Technological University Kukatpally,Hyderabad , India Abstract In this paper prompt particle swarm optimization (PPSO) algorithm is proposed to solve the optimal reactive power dispatch (ORPD) Problem. The prompt particle swarm optimization (PPSO) algorithm is obtained by combining PSO and the Cauchy mutation and an evolutionary selection strategy. The idea is to introduce the Cauchy mutation into PSO in the hope of preventing PSO from trapping into a local optimum through long jumps made by the Cauchy mutation. The ORPD problem is formulated as a nonlinear constrained single-objective optimization problem where the real power I. INTRODUCTION In recent years the optimal reactive power dispatch (ORPD) problem has received great attention as a result of the improvement on economy and security of power system operation. Solutions of ORPD problem aim to minimize object functions such as fuel cost, power system loses, etc. while satisfying a number of constraints like limits of bus voltages, tap settings of transformers, reactive and active power of power resources and transmission lines and a number of controllable Variables [1, 2]. In the literature, many methods for solving the ORPD problem have been done up to now. At the beginning, several classical methods such as gradient based [3], interior point [4], linear programming [5] and quadratic programming [6] have been successfully used in order to solve the ORPD problem. However, these methods have some disadvantages in the Process of solving the complex ORPD problem. Drawbacks of these algorithms can be loss and the bus voltage deviations are to be minimized separately. In order to evaluate the proposed algorithm, it has been tested on IEEE 30 bus system consisting 6 generator and compared other algorithms reported those before in literature. Results show s that PPSO is more efficient than others for solution of single-objective ORPD problem. Keywords: Particle swarm optimization, Cauchy mutation, Swarm Intelligence, optimal reactive power, Transmission loss. declared insecure convergence properties, long execution time, and algorithmic complexity. Besides, the solution can be trapped in local minima [1, 7]. In order to overcome these disadvantages, researches have successfully applied evolutionary and heuristic algorithms such as Genetic Algorithm (GA) [2], Differential Evolution (DE) [8] and Particle Swarm Optimization (PSO) [9]. It is reported in those that evolutionary or heuristic algorithms are more efficient than classical algorithms for solving the RPD problem. During the last decades a lot of population-based Meta heuristic algorithms were proposed. Voltage stability evaluation using modal analysis [10] is used as the indicator of voltage stability. In the recent decades a number of optimization algorithms based on natural phenomena have been developed. Particle Swarm Optimization (PSO) was first introduced by Kennedy and Eberhart in 1995 [11,12]. PSO is motivated from the social behaviour of organisms, such as bird flocking and fish schooling. Particles fly through the search space by following the previous best positions of their

2 neighbours and their own previous best positions. Each particle is represented by a position and a velocity which are updated as follows: ( ) (1) (2) Where and represent the current and the previous positions of idth particle, V id and are the previous and the current velocity of idth particle, and are the individual's best position and the best position found in the whole swarm so far respectively. is an inertia weight which determines how much the previous velocity is preserved, η 1 and η 2 are acceleration constants, rand() generates random number from interval [0,1]. In PSO, each particle shares the information with its neighbours. The updating equations (1) and (2) show that PSO combines the cognition component of each particle with the social component of all the particles in a group. The social component suggests that individuals ignore their own experience and adjust their behaviour according to the previous best particle in the neighbourhood of the group. On the other hand, the cognition component treats individuals as isolated beings and adjusts their behaviour only according to their own experience. Although the speed of convergence is very fast, many experiments have shown that once PSO traps into local optimum, it is difficult for PSO to jump out of the local optimum. Ratnaweera et.al.[13] state that lack of population diversity in PSO algorithms is understood to be a factor in their convergence on local optima. Therefore, the addition of a mutation operator to PSO should enhance its global search capacity and thus improve its performance. A first attempt to model particle swarms using the quantum model (QPSO) was carried out by Sun et.al. [14]. In a quantum model, particles are described by a wave function instead of the standard position and velocity. The quantum Delta potential well model and quantum harmonic oscillators are commonly used in particle physics to describe the stochastic nature of particles. In their studies [15], the variable of gbest (the global best particle ) and mbest (the mean value of all particles previous best position) is mutated with Cauchy distribution respectively, and the results show that QPSO with gbest and mbest mutation both performs better than PSO. The work of R. A. Krohling et.al.[16][17] showed that how Gaussian and Cauchy probability distribution can improve the performance of the standard PSO. Recently, evolutionary programming with exponential mutation has also been proposed [18]. In order to prevent PSO from falling in a local optimum, a prompt PSO (PPSO) is proposed by introducing a Cauchy mutation operator in this paper. Because the expectation of Cauchy distribution does not exist, the variance of Cauchy distribution is infinite. Some researches [19][20] have indicated that the Cauchy mutation operator is good at the global search for its long jump ability. This paper shows that the Cauchy mutation is helpful in PSO as well. Besides the Cauchy mutation, PPSO chooses the natural selection strategy of evolutionary algorithms as the basic elimination strategy of particles. PPSO combines PSO with Cauchy mutation and evolutionary selection strategy. It has the fast convergence speed characteristic of PSO, and greatly overcomes the tendency of trapping into local optima of PSO.The performance of PPSO has been evaluated in standard IEEE 30 bus test system and the results analysis shows that our proposed approach outperforms all approaches investigated in this paper. The performance of PPSO has been evaluated in standard IEEE 30 bus test system and the results analysis shows that our proposed approach outperforms all approaches investigated in this paper. II. VOLTAGE STABILITY EVALUATION A. Modal analysis for voltage stability evaluation Modal analysis is one of the methods for voltage stability enhancement in power systems. In this method, voltage stability analysis is done by computing eigen values and right and left eigen vectors of a jacobian matrix. It identifies the critical areas of voltage stability and provides information about the best actions to be taken for the improvement of system stability enhancements. The linearized steady state system power flow equations are given by. [ ] [ ] (3) Where ΔP = Incremental change in bus real power. ΔQ = Incremental change in bus reactive Power injection

3 Δ = incremental change in bus voltage angle. ΔV = Incremental change in bus voltage Magnitude J p, J PV, J Q, J QV jacobian matrix are the submatrixes of the System voltage stability is affected by both P and Q. However at each operating point we keep P constant and evaluate voltage stability by considering incremental relationship between Q and V. To reduce (1), let ΔP = 0, then. Where [ ] (4) (5) ( ) (6) is called the reduced Jacobian matrix of the system. B. Modes of Voltage instability: Voltage Stability characteristics of the system can be identified by computing the eigen values and eigen vectors Let (7) The corresponding ith modal voltage variation is [ ] (13) It is seen that, when the reactive power variation is along the direction of i the corresponding voltage variation is also along the same direction and magnitude is amplified by a factor which is equal to the magnitude of the inverse of the ith eigenvalue. In this sense, the magnitude of each eigenvalue i determines the weakness of the corresponding modal voltage. The smaller the magnitude of i, the weaker will be the corresponding modal voltage. If i =0 the ith modal voltage will collapse because any change in that modal reactive power will cause infinite modal voltage variation. In (10), let ΔQ = e k where e k has all its elements zero except the kth one being 1. Then, (14) k th element of V Q sensitivity at bus k (15) Where, = right eigenvector matrix of J R = left eigenvector matrix of J R = diagonal eigenvalue matrix of J R and (8) From (5) and (8), we have (9) or (10) Where i is the ith column right eigenvector and ith row left eigenvector of J R. i is the ith eigen value of J R. The ith modal reactive power variation is, (11) where, (12) Where ji is the jth element of i the III. PROBLEM FORMULATION The objectives of the reactive power dispatch problem considered here is to minimize the system real power loss and maximize the static voltage stability margins (SVSM). This objective is achieved by proper adjustment of reactive power variables like generator voltage magnitude ( gi ) V, reactive power generation of capacitor bank (Qci), and transformer tap setting (tk).power flow equations are the equality constraints of the problems, while the inequality constraints include the limits on real and reactive power generation, bus voltage magnitudes, transformer tap positions and line flows A. Minimization of Real Power Loss It is aimed in this objective that minimizing of the real power loss (Ploss) in transmission lines of a power system. This is mathematically stated as follows. (16) Where n is the number of transmission lines, g k is the conductance of branch k, V i and V j are voltage magnitude at bus i and bus j, and ij is the voltage angle difference between bus i and bus j.

4 B. Minimization of Voltage Deviation It is aimed in this objective that minimizing of the Deviations in voltage magnitudes (VD) at load buses. This is mathematically stated as follows. Minimize VD = (17) Where nl is the number of load busses and V k is the voltage magnitude at bus k. C. System Constraints In the minimization process of objective functions, some problem constraints which one is equality and others are inequality had to be met. Objective functions are subjected to these constraints shown below. Load flow equality constraints: [ ] (18) [ ] (19) where, nb is the number of buses, P G and Q G are the real and reactive power of the generator, P D and Q D are the real and reactive load of the generator, and G ij and B ij are the mutual conductance and susceptance between bus i and bus j.generator bus voltage (V Gi ) inequality constraint: (20) Transformers tap setting (T i ) inequality constraint: (24) Transmission line flow (S Li ) inequality constraint: (25) Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers. IV. PROMPT PARTICLE SWARM OPTIMIZATION ALGORITHM WITH CAUCHY MUTATION AND NATURAL SELECTION STRATEGY A. Cauchy mutation From the mathematic theoretical analysis of the trajectory of a PSO particle [21-23], the trajectory of a particle X id converges to a weighted mean of P id and P gd. Whenever the particle converges, it will fly to the personal best position and the global best particle s position. According to the update equation, the personal best position of the particle will gradually move closer to the global best position. Therefore, all the particles will converge onto the global best particle s position. This information sharing mechanism makes PSO have a very fast speed of convergence. Meanwhile, because of this mechanism, PSO can t guarantee to find the global minimal value of a function. In fact, the particles usually converge to local optima. Without loss of generality, only function minimization is discussed here. Once the particles trap into a local optimum, in which P id can be assumed to be the same as P gd, all the particles converge on P gd. At this condition, the velocity update equation becomes: Load bus voltage (VLi) inequality constraint: (26) (21) Switchable reactive power compensations (QCi) inequality constraint: (22) Reactive power generation (QGi) inequality constraint: (23) When the iteration in the equation (26) goes to infinite, the velocity of the particle V id will be close to 0 because of 0 ω <1. After that, the position of the particle X id will not change, so that PSO has no capability of jumping out of the local optimum. It is the reason that PSO often fails on finding the global minimal value. To overcome the weakness of PSO discussed at the beginning of this section, the Cauchy mutation is incorporated into PSO algorithm. The basic idea is that, the velocity and positions of a particle are updated not only according to (1) and (2), but also according to Cauchy mutation as follows:

5 (27) opponent j is larger than particle i, then win[i]++. Step4: Select m particles that have the more winnings to be the next generation. (28) Where δ and δ id denote Cauchy random numbers since the expectation of Cauchy distribution doesn t exist, the variance of Cauchy distribution is infinite so that Cauchy mutation could make a particle have a long jump. By adding the update equations of (27) and (28), PPSO greatly increases the probability of escaping from the local optimum. In standard PSO, the position of a particle is updated according to equations (1) and (2). That is, for each particle,there is nowhere to move but following the direction of the best particle, and the flying direction is nearly determinate through the generation. From the above analysis of PSO, the particles incline to converge on a local optimum. B. Natural selection strategy In the standard PSO, all particles are directly updated by their offspring no matter whether they are improved. If a particle moves to a better position, it can be replaced by the updated. However if it moves to a worse position, it is still replaced by its offspring. In fact, the most particles fly to worse positions for most cases, therefore the whole swarm will converge on local optima. Like evolutionary algorithms, FPSO introduces an evolutionary selection strategy in which each particle survives according to a natural selection rule. Therefore, the particle s position at the next step is not only due to the position update but also the evolutionary selection. Such strategy could greatly reduce the probability of trapping into local optimum. The evolutionary selection strategy is carried out as follows. Assume the size of the swarm is m, pair-wise comparison over the union of parents and offspring (1,2, 2m) is made. For each particle, q opponents are randomly chosen from all parents and offspring with equal probability. If the fitness of particle i is less than its opponent, it will receive a win. Then select m particles that have the more winnings to be the next generation. The detail of the selection framework is as follows: Step1: For each particle of parent and offspring, assign win[i ]=0. Step2: Randomly select q particles (opponents) for each particle in parent and offspring. Step3: For each particle, compare it with its q opponents. For particle i, if the fitness of its C. PPSO Algorithm for solving reactive power dispatch problem Step1: Generate the initial particles by randomly generating the position and velocity for each particle. Step2: Evaluate each particle s fitness. Step3: For each particle, if its fitness is smaller than its previous best(p id ) fitness, update P id. Step4: For each particle, if its fitness is smaller than the best one (P gd ) of all the particles, update Pgd. Step5: For each particle, do 1).Generate a new particle t according to the formula (1) and (2). 2).Generate a new particle t according to the formula (27) and (28). 3) Compare t with t chose the one with smaller fitness to be the offspring. Step6: Generate the next generation according to the above evolutionary selection strategy. Step7: if the stop criterion is satisfied, then stop, else goto Step 3. V. SIMULATION RESULTS Proposed approach PPSO has been applied to solve ORPD problem. In order to demonstrate the efficiency and robustness of proposed PPSO approach based on Newtonian physical law of gravity and law of motion which is tested on standard IEEE30-bus test system.the test system has six generators at the buses 1, 2, 5, 8, 11and 13 and four transformers with off-nominal tap ratio at lines6-9, 6-10, 4-12, and and, hence, the number of the optimized control variables is 10 in this problem. The minimum voltage magnitude limits at all buses are0.95 pu and the maximum limits are 1.1 pu for generator buses 2, 5, 8, 11, and 13, and 1.05 pu for the remaining buses including the reference bus 1. The minimum and maximum limits of the transformers tapping are 0.9 and 1.1pu respectively. The optimum control parameter settings of proposed approach are given in Table 1. The best power loss and best voltage deviations obtained from proposed approach are MW and respectively. The results obtained from Proposed algorithm have been compared other methods in the literature. The results of this comparison are given in Table 2. The results in Tables 1 and 2 show s that the reactive dispatch and voltage deviations solutions specified by the proposed PPSO

6 approach lead to lower active power loss and voltage deviations than that by the ref. [1] simulation results, which confirms that the proposed approach is well capable of specification the optimum solution. Table 1 Best Control Variables Settings for Different Test Cases of Proposed Approach Table 2: Comparison of the Simulation Results for Power Loss Contro l Variabl es Setting PPSO GSA [24] VG VG VG VG VG VG T T T T Power Loss (Mw) Voltag e Deviati ons Individ ual Optimiz ations [1] Multi Object ive Ea [1] As Single Object ive [1] VI. CONCLUSION In this paper, one of the recently developed stochastic algorithms PPSO has been demonstrated and applied to solve optimal reactive power dispatch problem. The problem has been formulated as a constrained optimization problem. Different objective functions have been utilized to minimize real power loss Control Case 1: Case 2: Variables Power Loss Voltage setting Deviations VG VG VG VG VG VG VG VG VG VG Power Loss (Mw) Voltage deviations and the voltage profile has been enhanced within the limits. The proposed approach has been tested on the IEEE 30-bus power system and the simulation results indicate the effectiveness and robustness of the proposed algorithm to solve optimal reactive power dispatch problem. REFERENCES [1] M. A. Abido, J. M. Bakhashwain, A novel multiobjective evolutionaryalgorithm for optimal reactive power dispatch problem, in proc. Electronics, Circuits and Systems conf., vol. 3, pp , [2] W. N. W. Abdullah, H. Saibon, A. A. M. Zain, K. L. Lo, GeneticAlgorithm for Optimal Reactive Power Dispatch, in proc. EnergyManagement and Power Delivery conf., vol. 1, pp , [3] K. Y. Lee, Y. M. Park, J. L. Ortiz, Fuel-cost minimisation for both realandreactive-power dispatches, in proc. Generation, Transmission and Distribution conf., vol. 131, pp , [4] S. Granville, Optimal Reactive Dispatch Trough Interior PointMethods, IEEE Trans. on Power Systems, vol. 9, pp , 1994.

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