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1 Available online at ScienceDirect IFAC-PapersOnLine (2015) Stochastic Economic Load Dispatch with Multiple Fuels using Improved Particle Swarm Optimization Mirtunjay. K. Modi*, A. Swarnkar*, N. Gupta*, K. R. Niazi*, R. C. Bansal** * Malaviya National Institute of Technology, Jaipur, India ( mirtunjay.modi@gmail.com, mnit.anil@gmail.com, nikhil2007_mnit@yahoo.com, krn152001@yahoo.co.in) ** Department of Electrical, Electronics and Computer Engineering, University of Pretoria, Pretoria, South Africa ( rcbansal@hotmail.com) Abstract: In this paper, Stochastic Economic Load Dispatch (ELD) problem with multiple fuels is solved using Improved Particle Swarm Optimization (IPSO). Generally, ELD problem is solved using deterministic models, but data required for such studies are rarely available with complete certainty. So uncertainties in unit s generation, load demand and cost coefficients should be considered to get actual scenario. Thus, stochastic model for ELD problems is more suitable than deterministic model from the utilities point of view. ELD problem with deterministic model is first solved using IPSO to examine the effectiveness of the proposed method. Then IPSO is applied for ELD problem with stochastic model to investigate the real generation cost. 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Economic Load Dispatch, Multiple Fuel Generation, Particle swarm optimization. 1. INTRODUCTION Due to highly competitive electric market, depletion of fossil fuels and rapid escalation of fuel prices, the main goal of Generation Companies (GenCos) is to generate the given amount of power at the lowest possible cost, i.e., by least use of fossil fuels. Economic Load Dispatch (ELD) is an important optimization problem in power system operation to allocate the total power demand among the committed units economically, while satisfying various constraints (Wood and Wollenberg, 1996). In old days, the cost function of each generator was approximated monotonically increasing in nature either piece-wise linear or quadratic. Using such cost functions, traditional methods like lambda iteration, base point participation factor (Wood and Wollenberg, 1996), gradient method and Newton method (Zhu, 2009), could solve ELD problems very effectively. But, practically generators have non-differential cost curve due to prohibited operating zones, valve point effects, and multi-fuel options. With these constraints and effects, the modern ELD became a complex optimization problem and traditional methods cannot provide quality solutions (Amjady and Rad, 2009). However, the performance of Dynamic Programming (DP) (Liang and Glover, 1992) does not depend on nature of the cost curve, but it suffers from the curse of dimensionality. As practical ELD problems are very difficult to solve mathematically, a large number of meta-heuristic methods like Genetic Algorithm (GA) (Chen and Chang, 1995), Evolutionary Programming (EP) (Jayabarathi and Sadasivam, 2000), Tabu Search (Lin et al., 2002), Adaptive Hopfield Neural Network (AH) (Lee et al., 1998), Particle Swarm Optimization (PSO) (Gaing, 2003), Bacterial Foraging (BF) (Panigrahi and Pandi, 2008), Ant Colony Optimization (Song and Chou, 1999), etc. have been successfully applied to solve them. Though these techniques do not guarantee to provide the global optimal solution, they can normally produce suboptimal solutions in a reasonable computational time. These methods in their original form have the problem of trapping into the local optima. Therefore, lots of modifications are suggested to these methods to improve the quality of solution (Vlachogiannis and Lee, 2009; Park et al., 2010; Amjady and Rad, 2009). Meanwhile some researchers have reported the use of hybrid approaches to solve ELD more effectively (Niknam, 2010; Bhattacharya and Chattopadhyay, 2010). However, in most of these studies, ELD is solved with deterministic models, which is not capable to represent the practical situation, due to the inaccuracy and uncertainties in forecasting and measurement. Therefore, it would be more preferable to construct the stochastic model, upon which load dispatch should be solved. A very few studies have been conducted to deal with the stochastic load dispatch problem. Bunn and Paschentis, 1986 solved ELD problem, where mismatch between actual load demand and dispatched generation is considered. Dhillon et al., 1993 solved the multi-objective load dispatch problem by weighted minimax. Kasangaki et al., 1995 solved unit commitment and economic load dispatch considering uncertainties in load demand and unit availability by using stochastic Hopfield artificial neural networks. Selvi et al., 2004 solved the ELD incorporating uncertainties in cost data by GA. Wang and Singh, 2008 formulated and solved ELD problems under different uncertain conditions by using modified PSO. Trivedi et al., 2013 solved generation scheduling under uncertain environment. In practice, many generating units are supplied with multifuel source, lead to the problem of determining the most economic fuel to burn (Lin and Viviani, 1984). In this paper, ELD problem with multiple fuels is solved considering its stochastic behaviour where load demand and fuel cost coefficients are treated as stochastic variables. An improved particle swarm optimization (IPSO) algorithm is initially used to solve the ELD problem with deterministic model , IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control /j.ifacol

2 Mirtunjay. K. Modi et al. / IFAC-PapersOnLine (2015) Comparative study is conducted to check the effectiveness of the proposed approach. Then, proposed IPSO is applied to solve the stochastic ELD problem. The results of deterministic and stochastic ELD are compared. Finally, the conclusion is drawn from the results obtained. 2. PROBLEM FORMULATION The typical formulations of power generation scheduling problems are considered to be deterministic, as these are assumed disturbance free and accurate (Dhillon et al., 1993). But, this assumption is not appropriate for practical applications, as uncertainties is available everywhere due to inaccuracies in the process of measuring and forecasting of input data and changes of unit performance during the period between measuring and operation. So deterministic model don t reflect the real situations in power generation scheduling problem and these deviation must be considered by generation utilities (Kasangaki et al., 1995). In this section, the stochastic model of objective in the ELD with multiple fuel and concerned constraints are presented. The load demand and fuel cost coefficients are normally distributed and inter-dependent and considered as random variables. 2.1 Stochastic Model for ELD with Multiple Fuels In real power plants, there may be generating units supplied with multiple fuels. In these cases, each generating unit has a set of cost curves, corresponding to the type of fuel being burned. Therefore unit s cost function is composed of a set of smooth fuel cost functions F i (P j ) (Amjady and Rad, 2009). The deterministic model of fuel cost is approximated by a set of quadratic function for each generator output P i FF(PP iiii ) = ii=1 (aa iiii PP 2 ii + bb iiii PP ii + cc iiii ) (1) where, ii shows generating units and j shows fuel type used, aa iiii, bb iiii and cc iiii are the cost coefficients of i th generator corresponding to j th fuels. The stochastic model of F i (P ij ) can be derived using Taylor s series expansion around the mean. Then the expected fuel cost can be obtained through the expectation of the expanded form (Wang and Singh, 2008): FF (PP iiii ) = ii=1[aa iiii PP ii2 + bb iiii PP ii + cc iiii + aa iiii vvvvvv(pp ii) + 2PP iicccccc(aa iiii, PP ii ) + cccccc(bb iiii, PP ii )] (2) where pp ii is the expected generation of i th generator, and aa iiii, bb iiii,and cc iiii are the expected fuel cost coefficients of i th generator for j th fuel. Then equation can be rewritten as: FF (PP iiii ) = ii=1[(1 + CC 2 PPii + 2RR aaiiii PPPPCC aaiiii CC PPiiii ) aa iiii PP 2 ii + (1 + RR bbiiii PPPPCC bbiiii CC PPiiii ) bb iiii PP ii + cc iiii] (3) where, CC PPii, CC aaiiii, and CC bbiiii are the coefficients of variation of the random variables PP ii, aa iiii and bb iiii, respectively. Coefficient of variation is defined as the ratio of standard deviation to the mean of the respective random variable. It measures the relative dispersion or uncertainty of the concerned random variable. The randomness of random variable is proportional to the coefficient of variation. RR aaiiii PPPP is correlation coefficient of the variables aa iiii and PP ii. RR bbiiii PPPP is correlation coefficient of the variables bb iiii and PP i. As transmission power loss is assumed to be zero, the real power generated must be equal to the demand throughout the system operations: ii=1 PP ii = PP DD (4) where, PP DD is the expected power demand. The expected power generation of each generator is restricted by its generator capacities: PP iimmmmmm PP ii PP iimmmmmm (5) PP iimmmmmm and PP iimmmmmm are the expected lower and upper power limit of the i th generator, respectively. 3. PROPOSED METHODOLOGY It will be very important to review the basics of the standard PSO first in order to introduce the IPSO. Similar to the other evolutionary algorithms Particle swarm optimization is a population based optimization search technique. PSO was proposed by Kennedy and Eberhart, which is inspired by the social phenomenon of fish schooling and birds flocking (Kennedy and Eberhart, 1998). In standard PSO, a swarm of feasible particle is randomly generated in the search space and velocity vector is initialized for each particle. Then, fitness value for each particle is calculated according to the objective function. Then particles basically utilize two important kinds of information sharing in decision process. The first one is their own experience; the second one is other particle s experiences. Then particle flies in the N-dimensional search space with its velocity, which is influence by the three components namely, inertial component, cognitive component and the social component (Kennedy and Eberhart, 1998). During evolution process each particle updates its velocity and position vector according to the following model: ww = ww mmmmmm ww mmmmmm ww mmmmmm iiiiiiii (6) iiiiiiii mmmmmm VV tt iiii = wwvv tt 1 iiii + CC 1 rrrrrrrr 1 ( ) (PP tt 1 bbbbbbbb PP tt 1 iiii iiii ) + CC 1 rrrrrrrr 2 ( ) (GG tt 1 bbbbbbbb PP tt 1 jj iiii ) (7) PP tt iiii = PP tt 1 tt iiii + VV iijj (8) w=inertial factor ww mmmmmm =Minimum value of inertial factor = 0.1 ww mmmmmm = Maximum value of inertial factor = 1 CC 1 =cognitive acceleration co-efficient=2 CC 2 =social acceleration co-efficient=2 rrrrrrrr 1 =random value between 0 and 1 rrrrrrrr 2 = random value between 0 and 1 where, i, j & t represents particle, dimension of the problem and iteration respectively. The previous best position of each particle is recorded as P best and the previous best among the particles is represented by

3 492 Mirtunjay. K. Modi et al. / IFAC-PapersOnLine (2015) G best. The inertial component guides the particle to fly in the original direction, cognitive component toward its personal best experience, P best and social component toward swarm s best experience, G best. All the terms are multiplied by appropriate parameters. In this way, PSO pursues local and global search to balance exploration and exploitation. Finally, the proposed algorithm is terminated if all particles reach to the global best position or predefined maximum iteration number is reached. 3.1 Proposed IPSO In standard PSO communication is very weak; as it depends only on P best and G best during the velocity updates. During the evolutionary process the diversity of the swarm might be lost. During later iterations particles close to current G best become inactive and lose both global and local searching capabilities. If the current G best is located in a local optimum, the evolution process will be stagnated. Then the swarm may have premature convergence, as all the particles will approach the current G best. As in various version of PSO till date, W, C 1 and C 2 are either constant or smoothly ascending or descending in nature, which do not reflect the real swarm behaviour as these are non-smooth in nature. So we introduce realistic swarm behaviour in proposed PSO by proposing non-smooth modulations in the inertia weight by improving compunication among best, worst and aggregate experience of the swarm. However, acceleration coefficients are taken same as in standard PSO. In addition, mutation operator similar to GA is used whenever the algorithm stagnates. GG bbbbbbbb = max (ffffffffffffff) PP aaaaaa = mean (ffffffffffffff) PP wwwwwwwwww = min(ffffffffffffff) RR mmmmmm = Maximum value of R = 1 RR mmmmmm = MMMMMMMMMMMMMM vvvvvvvvvv oooo RR = 0.1 RR 1 (iiiiii) = RR mmmmmm RR mmmmmm RR mmmmmm iiiiiiii mmmmmm iiiiiiii (9) RR 2 (iiiiii) = 1 GG bbbbbbbb PP aaaaaa (10) GG bbbbbbbb RR 3 (iiiiii) = 1 GG bbbbbbbb PP wwwwwwwwww (11) GG bbbbbbbb Where, GG bbbbbbbb is best fitness, PP aaaaaa is average fitness and PP wwwwwwwwww is worst fitness. ww(iiiiii) = RR 1 (iiiiii)+(rrrrrrrr RR 2 (iiiiii))+(rrrrrrrr RR 3 (iiiiii)) (12) (1+rrrrrrrr+rrrrrrrr) RR 1 (iiiiii) where, RR 1, RR 2 and RR 3 are the function of best fitness, average fitness and worst fitness. The proposed non-smooth inertial factor for a sample trial is shown in Fig. 1. W Iteration Fig. 1. Inertial factor Fig. 2. Flowchart of the Proposed IPSO When the fitness of the optimization problem gets stuck, mutation operator is called for Sequential operation on G best particle, so that the diversity of the optimization tool can be enhanced. Two new particles are generated by adding and subtracting dp, a random number in between 0 to1, to G best particle and fitness of both new particles calculated. Then by comparing we switch that power (dp) from feasible costliest generator to feasible most economical generator. G best particle is the input of mutation operator. The Fig. 2 illustrates the flowchart of the proposed PSO.

4 Mirtunjay. K. Modi et al. / IFAC-PapersOnLine (2015) SIMULATION RESULTS AND ANALYSIS The proposed IPSO is tested on a standard test system, having 10 multi-fuel generators. The system data are taken from (Lin and Viviani, 1984) and load demand is 2700 MW. In simulation, Number of Particles and Number of Generation are 20 and 500 respectively. The result obtained by proposed IPSO for deterministic model is shown in table 1. Then, the result is compared with that of other evolutionary methods available in literature presented in table 2. By comparison, proposed IPSO is found more promising optimizer for ELD problems. The convergence of PSO and IPSO is shown in Fig. 3. Then it is applied to the stochastic models of ELD with multiple fuels. Cost($/hr) Fig. 3. Convergence of PSO and Proposed IPSO PSO IPSO Iteration In the simulation for stochastic model, the coefficient of variation of all the involved variables is assumed 0.1, and the correlation coefficient of each pair of variables is assumed 1.0 (Wang and Singh, 2008). The result for stochastic model is shown in table 3. Finally results of deterministic models and stochastic models are compared in table 4. The proposed IPSO method, used in this paper, is implemented by using MATLAB 7.0 on a PC (core 2 duo, ram 2 GB) and CPU time was 0.86s. Table 1. Fuel combination and result with deterministic model Gen Fuel Power (Mw) Fuel Cost ($/h) Total Table 2. Comparison of Fuel Cost Method Total Cost ($/h) HM (Lin and Viviani, 1984) H (Park et al., 1993) EP (Jayabarathi and Sadasivam, 2000) AH (Lee et al., 1998) CGA-MU (Chiang, 2005) IGA-MU (Chiang, 2005) RCGA (Amjady and Rad, 2009) Proposed IPSO Table 3. Fuel combination and result with stochastic model Gen Fuel Power (MW) Fuel Cost ($/h) Total Table 4. Comparison of results for the deterministic and stochastic model Model Deterministic Stochastic Total Power (MW) Total Cost ($/h) CONCLUSION In the traditional ELD problems, inaccuracies and uncertainties are ignored as variables involved were considered as deterministic in nature. In this study, it has been observed that the stochastic model of ELD provides the higher optimal fuel cost than the deterministic model. Therefore, GENCOs must consider stochastic model for the economic operation of power system. The proposed inertia weight in IPSO is effective to generate better solutions. It happens due to improved communication in the swarm.

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