OPTIMAL DISPATCH OF REAL POWER GENERATION USING PARTICLE SWARM OPTIMIZATION: A CASE STUDY OF EGBIN THERMAL STATION

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1 OPTIMAL DISPATCH OF REAL POWER GENERATION USING PARTICLE SWARM OPTIMIZATION: A CASE STUDY OF EGBIN THERMAL STATION Onah C. O. 1, Agber J. U. 2 and Ikule F. T. 3 1, 2, 3 Department of Electrical and Electronics Engineering, University of Agriculture Makurdi, Nigeria Abstract- Optimal dispatch is one main option for scheduling generation to find an effective real and reactive power scheduling to power plants to meet load demand as well as to minimize the operating cost. Therefore, this paper presents Particle Swarm Optimization (PSO), an efficiently reliable nonlinear optimization and population based stochastic technique, for solving the real power optimum dispatch problem including transmission loss, for six steam generating units in Egbin thermal plant, with constraints satisfaction and operating generation cost minimization. The loss coefficient or B-matrix, the generators operating limits, the quadratic cost function of the generating units together with other PSO parameters like the inertia weight, acceleration constants etc are used to set up the PSO program in MATLAB environment. The results obtained by the stochastic approach show high proficiency, ability for fast convergence, easy computation and implementation of the code and robustness to cope with the nonlinearity of optimal load dispatch problem, in obtaining the global optimum dispatch solution. Keywords- Optimal Load Dispatch, Thermal Power generation, PSO, Loss coefficient, MATLAB, Stochastic I. INTRODUCTION One of the most significant operational functions of modern day energy management system is Optimal Load Dispatch (OLD).The size of electric power system is increasing at a great speed to meet the energy requirements. OLD pertains to optimum generation in an interconnected power system to minimize the cost of generation subject to relevant system constraints [1]. With the development of grid system, it becomes necessary to operate the plant unit most economically. This paper presents an optimization method (PSO), which would be used to solve complex optimization problems of Egbin thermal station, that are nonlinear, non-differentiable and multimodal and also to find optimal solution to the OLD problem including losses and generating operational limits. PSO parameters are selected to significantly determine the efficacy and computational behavior in optimizing the problem. Finally, Matlab program is developed to solve the OLD problem of a six unit plant using PSO technique. II. PROBLEM FORMULATION The fundamental objective of optimal load dispatch problem is to minimize the total fuel cost while satisfying the operational constraints of the power system. In OLD problem, the allocation of optimal power generation among the different generating units at minimum possible cost is done is such a way as to meet demand constraints and generating constraints. The OLD problem is formulated as the minimization of total fuel cost of generating units for the entire scheduling period subject to variety of constraints. The formulation of OLD problem is as follows. A. Objective Function Aggregating the objective and constraints, the problem can be mathematically formulated as a nonlinear constrained single objective optimization problem as shown in equation (1). Minimize [ (P), (P)] Subject to g (P) = 0 (1) h (P) 0 DOI: /IJMTER BZE9N 1

2 where g is the equality constraint representing the power balance, h is the inequality constraint representing the unit generation capacity, is the total generation cost or fuel cost and is the total power loss or transmission loss in the system. The overall operating cost of the network is equal to the summation of all generation units fuel cost function, in a power system as given in equation (2). Minimize The cost function in equation (2) can be approximated to a quadratic function of the power generation as shown in equation (3) and (4) respectively. where is the fuel cost function of the generating units in (N/h). are the fuel cost coefficient of the i-th generator and is the generated real power output by the i-th generator (MW). is the total fuel cost and n is the number of generators including the slack bus. B. Equality Constraint Power balance constraint, otherwise known as the Equality constraint is well thought out in two ways. The first excludes transmission loss while the second includes transmission losses in the system. In the first case, balance is met when the sum of generation equals the sum of load, considering the equation as loss-less as represented in equation (5): In case two, balance is met when the sum of generation equals the sum of system load and total transmission power losses [2]: where is the system load demand and is the transmission line loss. The loss coefficient method which was developed by Kron [3] and popularized by [4], is used to include the effect of transmission losses. B-matrix, which is also known as the transmission loss coefficients matrix is a square matrix with dimension of, where is the number of generation units in the system. Applying B-matrix gives a solution of generated powers for different units as the variables. Equation (7) shows the function for calculating using B-matrix method [5]. where is the total transmission loss in the system, is the generated power by the i-th and j-th generating units respectively and is the element of the B-matrix between i-th and j-th generating units. C. Inequality Constraints Inequality constraint is also known as power generator capacity constraint. The power output of each generating unit has minimum and maximum generation capacity according to its machine ratings and unit power lies in between these capacities. If the power output of a generator for optimum operation of the system is less than a pre-specified value, the unit is not put on the bus All rights Reserved 2

3 because it is not possible to generate that low value of power from the unit. This is shown as an inequality constraint in equation (8): where is the minimum and maximum power output limit of the i-th generator. III. PARTICLE SWARM OPTIMIZATION PSO is a population-based stochastic search optimization technique with most recent developments in the category of combinatorial meta-heuristic optimization first developed by Kennedy and Eberhart in 1995 [6]. PSO is inspired by social behavior of bird flocking or fish schooling. Amongst various versions of PSO, the most familiar version was proposed by Shi and Eberhart in 1998 [7]. A PSO algorithm searches in parallel using a swarm consisting of a number of particles to search out optimal solutions. Each particle s position represents a candidate solution to the optimization problem. Each particle is initialized with a random position and random velocity, and searches for optimal solution within the feasible range by updating generations. A fitness evaluation function is used to assign the fitness value of each particle. The best position among all particles is assigned, and the best position of each particle up to the current iteration is also assigned. At every iteration, each particle update its position based on its own best position called and the swarm overall best position called assigned at the previous iteration, and its previous velocity. In a PSO system, particles fly around in a multi-dimensional search space. During flight, each particle adjusts its position according to its own experience and the experience of the neighboring particles, making use of the best position encountered by itself and its neighbors [8]. In the multi-dimensional space, where the optimal solution is sought, each particle in the swarm is moved toward the optimal point by adding a velocity with its position. The velocity of a particle is influenced by three components, namely, inertial, cognitive, and social. The inertial component simulates the inertial behavior of the bird to fly in the previous direction. The cognitive component models the memory of the bird about its previous best position, and the social components model the memory of the bird about the best position among the particles. The particle moves around the multidimensional search space until they find the optimal solution. The modified velocity of each agent can be calculated using the current velocity and the distance from and PSO has been successfully applied to global optimization problems with nonconvex or nonsmooth objective functions. In addition, PSO has demonstrated good properties and is easy in its concept and implementation and has few parameters to adjust. PSO, unlike most other stochastic optimization techniques requires relatively less computational burden or time. IV. PARAMETER SELECTION IN PARTICLE SWARM OPTIMIZATION PSO has a number of parameters that determine its behavior and efficacy in optimizing a given problem. A. Velocity Velocity of each particle can be modified by the following equation: where - Modified velocity of particle i at iteration t+1 is the weighting function, is the velocity of particle i at iteration t, - Cognitive acceleration constant, - Social acceleration constant, is the random number between 0 and 1, is the current position of particle i at iteration All rights Reserved 3

4 is the of particle i and is the of the group. The term is called the particle memory influence and is the swarm influence. where i = 1 n, n - Population size B. Position Modifications - modified position of particle i at iteration (t+1) change in time, measured in iteration step and time increment of iteration is 1. C. Acceleration Constant The learning factors and determines the impact of the, and the respectively. When the value of cognitive acceleration coefficient (C 1 ) increases, it enhances particles' attraction towards and decreases their attraction towards Also, increasing social acceleration coefficient in relation to cognitive acceleration coefficient increases attraction of particles towards Ozcan and Mohan (1999) [9] proposed setting C 1 = C 2 = 2 as a generally acceptable setting for most of the problems and is widely used in practical applications of PSO. D. Inertia weight Inertia weight in PSO plays an important role, because of its control on particle speed. The values = 0.9 and = 0.4 are widely accepted in literature. In current study, the value of inertia weight decreases linearly from 0.9 to 0.4 during a run time. The general selection of inertia weight is set according to the following equation: Where : Final inertia weight; : Initial inertia weight; The maximum number of iterations which is arbitrarily set as 500; : The iteration which is considered as the current iteration. E. Swarm Size Swarm size affects performance of PSO. Too few particles prompt the algorithm to get trapped in local optima, while too many particles slow down the algorithm. It is a problem dependent phenomenon and varies from problem to problem. F. Initialization Technique Random initialization of particles may facilitate the PSO algorithm to effectively explore the search space of various regions, detect solutions of better quality and enhance computational behavior of PSO. G. Number of Particles It is problem dependent. It is initialized with a few numbers of particles which is gradually increased. This will give the ideal number of particles. For the problem at hand, the number of particles chosen is 200. H. Dimension of Particles Dimension of particles would be specified by the problem to be optimized: D = (12) where number of particles and number of All rights Reserved 4

5 I. Stopping Criteria The maximum numbers of iterations that PSO accomplishes or the minimum error requirement are the stopping conditions. If the number of iteration reaches the maximum number of iteration set in PSO, then the latest is the optimal generation power unit, with minimum total generation cost at the maximum evaluation function iteration. Start Define Parameters: P min P max a, b, c, B, E, λ, P d, np, ng, It, ω, C 1 C 2 Initialize particle swarm with random position (P) and velocity vectors For each particle (i = 1,2 np), evaluate fitness Select the first particle as the global Set P i resulted so far as the Pbest for each I t 0 Set It = It + 1 Compute ω using equation (11) Update the velocity and position of the particles according to equations (9) and (10), ensuring all constraints are met Calculate the fitness of the new particles Check if Pnew < Pbest, if yes then Pbest = Pnew else maintain Pbest If Pbest < Gbest, then Gbest = Pbest otherwise Gbest = Gbest NO Is It = It YES Stop Figure 1. Flow Chart of Basic All rights Reserved 5

6 J. Algorithm of PSO The step-by-step algorithm for the proposed method is explained below: Step 1: Define parameters of PSO constants, C 1, C 2, n g, inertia weight and specify the maximum and minimum limits of generation power of each generating unit, maximum number of iterations to be performed, error, lambda, power demand, loss coefficient matrix and fuel cost co-efficient of each unit. Step 2: Initialize randomly the individuals of the population of all units according to the limit of each unit including individual dimensions, searching points and velocities. Step 3: Evaluate the fitness function of each particle using equation (13): (13) where F is the particle s fitness function, is lambda assumed to be 100 and E is the particle s error: Step 4: Assume minimum cost as the global best, that is, Step 5: Set P i obtained so far as the for each particle and the cost arising from them as cost. Step 6: Save the global best and its real power generation. Step 7: Set iteration count. Step 8: Compute the inertia weight according to equation (11). Step 9: Update particle s velocity using equation (9). If the velocity is out of range, then clamp the velocity of each particle: If (15) (16) Step 10: Modify the particle s new position using Step 11: Evaluate the fitness of the particle s new position. Step 12: For each individual particle, compare the particles fitness value with If the current fitness value is better than, then set the value equal to the current value and the position equal to the current particle s position. Step 13: Compare the best current fitness evaluation with the population. If the current value is better than the population, then reset the to the current best position and the fitness value to current fitness value. Step 14: Repeat steps 3-9 until a stopping criterion with maximum iteration is met. In Table 1, Oke-Aro and Ajah buses both have double circuit 330kV transmission lines, L1, L2 and L3, L4 respectively. They are both connected in parallel and hence, share the load and other parameters equally. Table 1. Bus names and their types Bus No. Bus Name Remark 1 Egbin Slack bus 2 Oke-Aro L1 and L2 PV bus 3 Ikeja West L3 PV bus 4 Benin L8 PV bus 5 Ajah L3 and L4 PV bus Table 2 presents the installed and generated capacities of the generating power units collated at Egbin Power Station on 24 th January, Unit ST1 was on outage, due to the fact that the All rights Reserved 6

7 step-up transformer was damaged, the generating capability of Unit ST2 decreased due to Vacuum problem and Unit ST6 was on 6 to 7 days maintenance. Unit Table 2 Egbin thermal statios installed and generated capacities Installed Capacity (MW) Installed Capacity (MWh) Generated Capacity (MW) Generated Capacity (MWh) ST OUT - ST ST ST ST ST Total Table 3. Cost coefficient and power limits of Egbin power plant Unit No. ai (N/hr) bi(n/mwhr) ci(n/mw 2 hr) Minimum Power (P min ) Maximum Power (P max ) ST ST ST ST ST ST The power limits in Table 3 is operated at a range of 25 to 100 percent maximum continuous rating (MCR). Table 4. Bus data Bus Name Bus No. Voltage, V Angle, P (pu) Q (pu) (pu) (degree) Egbin Oke-Aro Ikeja West Benin Ajah Table 5. Loss coefficient matrix VI. RESULTS AND DISCUSSION The simulation result for the cost of real power generation scheduled for different load demand is displayed in TABLE 6. Table 6. Best power output for six generating units at different load demands Power Demand All rights Reserved 7

8 Fitness in Naira/hr Fitness in Naira/hr Fitness in Naira/hr Fitness in Naira/hr International Journal of Modern Trends in Engineering and Research (IJMTER) (MW) P1 (MW) P2 (MW) P3 (MW) P4 (MW) P5 (MW) P6 (MW) Total Power Output (MW) Total Generation 57, , , , , Cost (N/hr) Transmission Loss (MW) Count Elapsed Time (secs) Plots for the optimized power output and the number of iteration for different load demands are depicted in Fig 3 to Fig 6. Simulated results shows optimal reduction in the fitness level of the particle or generation cost for 500 iterations. As the load demand increases, the transmission loss and generation cost also increases but at an optimal rate. The Figures thereby shows that, the proposed algorithm improves the quality of the solution as well as found a better optimal solution to the OLD problem for different load demands. 5.9 x 104 Plot of Fitness of Best Particle per 6 x 104 Plot of Fitness of Best Particle per Figure 2. Plot of fitness against number of iterations at 991MW load 6.08 x Plot of Fitness of Best Particle per 5.9 Figure 4. Plot of fitness against number of iterations at 1010MW load Figure 3. Plot of fitness against number of iterations at 1000MW load 6.09 x Plot of Fitness of Best Particle per 6 Figure 5. Plot of fitness against number of iterations at 1021MW All rights Reserved 8

9 Fitness in Naira/hr International Journal of Modern Trends in Engineering and Research (IJMTER) 6.18 x 104 Plot of Fitness of Best Particle per Figure 6. Plot of fitness against number of iterations at 1029MW load VII. CONCLUSION The developed PSO optimization technique has been successfully applied for the solution of the optimal dispatch in power system in this paper. The successful implementation of the proposed PSO algorithm on Egbin thermal station considering transmission losses proved to be the required method for solving optimal dispatch of real power generation problem. It has been observed that the PSO technique is capable of optimizing any given OLD problem irrespective of load demand. From the analysis of the proposed PSO technique which was implemented in MATLAB environment using Egbin six generator systems as case study considering transmission losses, proves that PSO is highly efficient, accurate and has capacity to minimize the fuel cost of generators and satisfies each and every constraint. Thus, PSO technique can be successfully applied to solve OLD problems in the real world power systems. REFERENCES [1] S. Prabakaran and S. V. Kumar, Security Constrained Optimal Load Dispatch using HPSO Technique for Thermal Scheduling Problems, International Journal of Research in Engineering and Technology, vol. 02 (05): , 2013 [2] K. Balamurugan, R. Muralisachithnndam and S. R. Krishnan, Differential Evolution Based Solution for Combined Economic and Emission Power Dispatch with Valve Loading Effect, International Journal on Electrical Engineering and Informatics, vol. 6 (1): 74-92, 2014 [3] G. Kron, A Set of Principles to Interconnect the Solution of Physical Systems, Journal of Applied Physics, vol. 24 (8): , 1953 [4] L. K. Kirchmayer, H. H. Happ, G. W. Stagg and J. F. Hohenstein, Direct Calculation of Transmission Loss Formula, AIEE Transaction vol. 79 (3): , 1960 [5] R. Rasoul, F. O. Moh d, Y. Rubiyah and K. Marzuki, Solving Economic Dispatch Problem using Particles Swarm Optimization by an Evolutionary Technique for Initializing Particles, Journal of Theoretical and Applied Information Technology, vol. 46 (2): , 2012 [6] J. Kennedy and R. Eberhart, Particle Swarm Optimization, Proceedings of IEEE International Conference on Neural Networks IV, , 1995 [7] Y. Shi and R. C. Eberhart, A Modified Particle Swarm Optimizer, Proceedings of IEEE International Conference on Computational Intelligence, 69-73, 1998 [8] M. J. Khan and H. Mahala, Particle Swarm Optimization by Natural Exponent Inertia Weight for Economic Load Dispatch, International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, vol. 3 (12): , 2014 [9] E. Ozcan and C. Mohan, Particle Swarm Optimization: Surfing the Waves, Proceedings of IEEE International Congress on Evolutionary Computation, , All rights Reserved 9