2013 International Conference on Power, Energy and Control (ICPEC) 978-1-4673-6030-2/13/$31.00 ©2013 IEEE 405 Abstract- This research work presents a Chaotic Self Adaptive Particle Swarm Optimization (CSAPSO) algorithm in order to solve the Economic Dispatch (ED) problem. The main purpose of the work is to derive a simple and effective method for optimum generation dispatch to minimize the generation cost power networks by considering several non-linear characteristics of the generator such as valve point effect, prohibited operating zones and ramp rate limits. A chaotic local search operator is introduced in the proposed algorithm to avoid premature convergence. Simulation studies are carried out, using MATLAB software, to show the effectiveness of the proposed optimization method. The applicability and high feasibility of the proposed method is validated on three different test systems. Results show that the CSAPSO is more powerful than other algorithms. Key words: Economic Dispatch (ED), Chaotic Self Adaptive Particle Swarm Optimization (CSAPSO), ramp rate limits, valve point effect. I. INTRODUCTION Today’s power systems are highly complex and their operations are unpredictable. The primary objective in the planning and operation of power systems is to provide quality supply to consumers at economical cost. The purpose of Economic Dispatch(ED) is to determine the optimal combination of power generations that minimizes the total operating cost while satisfying all equality and inequality constraints. It is difficult to solve the ED problemsince the ED problem is a highly non-linear, non- convex and discrete optimization problem (see list of references herein).The increasing energy demand and the decreasing energy resources have made optimization a great necessity in power system operation and planning. Traditionally, electric utilities dispatch generation using minimum fuel cost as the main criterion. It therefore makes it difficult to handle such problem by conventional approach. The ED is the optimization scheme of generation system to determine the best generation schedule to supply a given load with minimum fuel cost. Because of the increasing size and complexity of power system networks, more attention is being given to develop various optimization methods. Previous efforts on solving generation dispatch problem that have employed the conventional methods includes the linear programming method[1] lambda iteration and gradient method, dynamic programming algorithm,quadratic programming algorithm, lagrangian relaxation method etc.But unfortunately these methodshave the following drawbacks: It would generate large errors, for quadratic programming method,the objective function should be differentiable;thelagrangian relaxation algorithm may lead to the phenomenon of solution oscillation and the dynamic programming algorithm may result in ‘curse of dimensionality’. Recently, with the development of computer science and technology, many modern stochastic search algorithms like,simulated annealing (SA) [2],Tabu search , particle swarm optimization (PSO)[4], evolutionary programming method (EP) [5] genetic algorithms (GA) [12], and neural networkmay prove to be very effective algorithm in solving non-linear ED problem without modifying the shape of its fuelcost curves.Allthesemethods use probabilistic rules to update their particle positions and velocity in the solution space. Thesealgorithms are capable of providing a fast and reasonable solution, when handling ED problems, [12],[16],[17].Among the above population based algorithms, the PSO is one of the modern heuristic algorithms and has great potential to solve highly complex optimization problems. Compared to other optimization methods, PSO has fast converging characteristics [4], [5] [7], [11].However the performance of the traditional PSO greatly depends on its parameters and it often suffers the phenomenon of the premature convergence,lacks of mechanism to deal with various constraints in ED problem[8]. In this work,aCSAPSO algorithm based on the logistic equation[6], [8]has been proposed for solving ED problem with the aim of overcoming some of the above- mentioneddisadvantages. Additionally, the algorithm takes Chaotic Self Adaptive Particle Swarm Approach for Solving Economic Dispatch Problem with Valve- Point Effect C. Rani., MIEEE Vellore Institute of Technology VIT University,Vellore,India
[email protected] D.P.Kothari, Fellow, IEEE J B Group of Institutions Hyderabad, India
[email protected] K. Busawon Northumbria University, Newcastle Upon Tyne, UK
[email protected] 2013 International Conference on Power, Energy and Control (ICPEC) 406 into consideration the ramp rate limits of the generators and its valve point effect.The results confirm the effectiveness and applicability of the CSAPSO algorithm for real-world applications. An outline of the paper is as follows: In the next section, the ED problem is formulated. In Section 3, an overview of the proposed algorithm is presented. In Section 4the implementation of the CSAPSO for the ED problem is considered.Section 5 is devoted to three case studies. Finally, conclusions are drawn in Section6. II. ED PROBLEM FORMULATION The ED problem can be formulated as a mathematical optimization problem which is highly non-linear. The objective of ED is to minimize the fuel cost of power productionin order to meet the load demand of a power system over some appropriate periods while satisfying various equality and inequality constraints.The following constraints are considered in the formulation of the ED problem [17]. The objective of the dispatch problem is to minimize the total fuel cost for running ‘N’ generating units. For more practical and accurate model of the fuel cost function, the valve point effect of turbine generators is also to be considered which is expressed as the sum of quadratic and sinusoidal function. The total fuel cost in terms of its real power output can be expressed as | sin _ | 1 Where , , , , are the fuel cost curve coefficients of ith generating unit. is the output power of ith generating unit , and are the lower and upper generationlimits and is the total number of generating units . a) Real Power balance constraint The total generating power must be equal to the sum of load demand and the transmission loss: 0 2 Where is the load demand, is the transmission line losses, which is a function of all the generators. The transmission losses are calculated by the following formula: 3 Where , and are the transmission network power loss coefficients. As the second and third terms are relatively small, they are neglected in this paper for simplicity. b) Capacity limit Constraint The power output level of generator ‘i’ should be between its and power output limits c) Generating unit Ramp Rate limit The ramp rate limits are restricting the operating range of all the committedunitsfor adjusting the generator operation between twooperating periods. The generation may increase or decrease with corresponding upper anddown ramp rate limits. Therefore, generating units are constraineddue to these ramp rate limits as follows: (a) As generation increases (b) As generation decreases and are ramp rates of ith generator (MW/hr) and is power generation at previous hour. The inclusion of ramprate limits modifies the generator operation constraintsas follows: max , min , 4 d) Prohibited operating zone constraints Each unit must avoid operation in prohibited zones. , , , 2,3 … . , Where , and , are the lower and upper bounds of the jthprohibited zones of unit i and Ziis number of prohibited zone of unit i. III. OVERVIEW OF PROPOSED METHOD A. Particle swarm optimization Kennedy and Eberhart first introduced PSO in year of 1995. PSO is motivated from the simulation of the behavior of social systems such as fish schooling and birds flocking [18]. The PSO algorithm requires less computational time and less memory. The basic assumption behind the PSO algorithm is, birds find food by flocking and not individually. This leads to assumption that information is owned jointly in flocking. The proposed method can be split into three major processes a) Initialization Process Let P be the ‘particle’ coordinate (position) and V its speed (velocity) in search space. Consider ‘i' as a particle in the total population (swarm).Now, the ith particle position can be represented as Pi = [Pi1, Pi2,…..PiN] in the N dimensional space. 2013 International Conference on Power, Energy and Control (ICPEC) 407 b) Fitness Evaluation Process: In this fitness evaluation process, each particle in the population is evaluated using the fitness function in the first iteration. The best previous position of ith particle is stored and represented as . All the are evaluated by using a fitness function. The best particle among all is represented as . c) Updating Process In this updating process, modify the each individual velocity V of the each particle Pi according to the equation shown below: 5 The use of linearly decreasing inertia weight factor has provided improved performance in all the application. Its value is decreased linearly from about 0.9 to 0.1 during a run. After this velocity updating process of the entire individual in each particle, modify the position (generator output level) of each individual in the particle Pi according to the following equation 6 At the end of updating process, if the evaluation value of each individual is better than the previous , the current evaluation value is set to be as , if the best Pbest is better than , that value is set to be . . B. Chaotic Self Adaptive PSO method This CSAPSO combines traditional PSO with self adaptive inertia weight factor(SAIWF) and chaotic local search(CLS) based on Logistics equation [6]. Suitable selection of SAIWF provides a balance between global and local exploration and results in less iteration on average to find a sufficiently optimal dispatch solution.Its value is set according to the following equation [8]. 7 Where and represent the maximum and minimum value of ω.f is the current objective value of the particle, fmax,fmin, favg are the maximum, minimum and average objective values of all particles. The logistic equation,which exhibits sensitive dependence on initial conditions,is introduced in the process of chaotic local search defined by the following equation [8]. 1 8 Wherecxi denoted the ith chaotic variable,‘iter’ represent the iteration number and is control parameter.The procedure of CLS based on the logistic equation can be illustrated as follows. Step1: Setiter=0 and mapping the decision variables ,chaotic variables located in the interval(0,1) using the following equation. 9 Step 2:Determine the chaotic variable for the next iteration using the logistics equation. Step3: Convert chaotic variables to the decision variable 10 Step 4: Evaluate the new solution with decision variables. Step5: If the new solution is better than xo or the predicted maximum iteration is reached,output the new solution as the result of the CLS;otherwise let iter=iter+1 and go to Step 2. IV. IMPLEMENTATION OF CSAPSO FOR ED Step1: Set all parameter of generating units and power demand. Step 2: Generate the particles randomly between the maximum and minimum operating limits of the units. Pi = [Pi1, Pi2,…..PiN] in the N dimensional space, i=1,2,3…..N. Step 3: For each individual Pg in the population,calculate the transmission loss PL using the B-coefficients loss formula. Step 4: Evaluate the fitness of each individual Pgi according to the equation (1). Step 5: Compare each individual’s evaluation value with its and then identify best value among the all be . Step 6: Update the velocity of each particle based on the equation (5). Step7: However, after update the velocity, the individual velocity may violate its Velocity maximum, minimum constraints. This violation is corrected as follows , , , , , , Step 8: Update the position according to the equation (6). Step9: Check the position of each individual and adjust to its maximum and minimum limits. Step 10: If the evaluation value of each individual is better than the previous , the current value is set to be . If the best is better than , the value is set to be . Step11: Implement the CLS for the best particle and update. 2013 International Conference on Power, Energy and Control (ICPEC) 408 Step12: If the stopping criteria are met, then go to step 13, otherwise, let iter=iter+1 and go to step 3. Step 13: The individual that generates the latest is the optimal generation power of each unit when the system reaches the minimum total generation cost. V. CASE STUDY AND RESULTS In order to validate the feasibility and effectiveness of the proposed method, the algorithm is implemented in three test systems and its performance is compared to other optimization techniques such as theCEP [5], GA [19] and NPSO [9].In all three cases,the ramp rate limits, prohibited operating zones andvalve point effects are considered. All simulations are carried out by MATLAB 7.01 on Pentium IV, 3GHz personal computer.The results are given below; demonstrate the effectiveness and applicability of the proposed method. The results represent the average of 30 trials.The parameters of CSAPSO approach for all test systems: Inertia weight factor ωmax=0.9;ωmin=0.2 Velocity limits: -0.5 Vmin and 0.5 Vmax. Acceleration coefficients: C1=C2=2.05 Control parameter σ = 4 A. Description of test systems Case 1: 6-generator system. The 6-generatortest system with non-linear operating cost ,the usable B loss coefficient matrix and the total power capacities were committed to meet the load demand was taken from [ 14]. Through the evolutions process of the proposed methods, their best solutions are shown in Table I, which satisfy the various system constraints. Case 2: 15-generator system The system contains 15 thermal units whose characteristics and the total power capacities were committed to meet the load demand was taken from [8]. To simulate this, each individual Pg contains 15 generator outputs. The experimental results are shown in Table VI. Transmission losses are neglected for the purpose of comparison with already published results. Case 3: 40-generator system This system contains 40 thermal units whose fuel cost and emission coefficients, active power limits,ramp rate limits and prohibited operating zones are taken from [22]. The simulation results are performed at 10500 MW load demand.In this case, transmission losses are neglected for simplicity. B. Simulation results obtained for three test systems: a) Test system I The sample test results for CSAPSO, GA[18] and NPSO-LRS [9] for 1263 MW load with 300 iterations areillustrated in Table I. The simulation results show the total cost obtained from the CSAPSO is lower and thus shows that the CSAPSO yields better solution than other methods. TABLE I SIMULATION RESULTS OF 6 GENERATOR SYSTEMS Unit (MW) GA[19] NPSO-LRS[9] CSAPSO Pg1 474.8066 446.96 447.7685 Pg2 178.6363 173.3944 172.0578 Pg3 262.2089 262.3436 261.7253 Pg4 134.2826 139.5120 144.5623 Pg5 151.9039 164.7089 162.8342 Pg6 074.1812 089.0162 086.44287 Ploss 013.0217 012.9361 012.391 Total Generation (MW) 1276.03 1275.94 1275.391 Generation cost ($/hr) 15459 15450 15315.903 In order to discuss the influence of the proposed techniques to enhance the convergence characteristics of the CSAPSO, fig 1 is presented. From graph, it is clear that the value of the cost function converges smoothly to the optimum solution without any sudden oscillations for CSAPSO. Fig 1.Convergence characteristics of 6 generator system. TABLE II COMPARISON OF COMPUTATION EFFICIENCY Population size Generation cost ($/hr) Time need for convergence(s) Number of Iterations 20 15316.83 0.94 150 50 15315.47 1.56 202 100 15315.79 3.92 258 200 15315.25 7.84 279 300 15315.85 12.71 298 It is worth noting that, the best obtained solution of total cost using GA [18] is 15459 $/hr while the proposed method provided 15315 $/hr which confirms the efficiency and applicability of the proposed method. As mentioned in Table I,total generation cost using the proposed algorithm can be reduced about 144 $/hr. If every hours can save thesame cost, and operating hours per year is 8760hr, then operating by CSAPSO methods can save 12,61,440 $/year. 0 50 100 150 200 250 300 1.53 1.532 1.534 1.536 1.538 1.54 1.542 1.544 1.546 x 10 4 FINDING OPTIMAL COST FOR 6 GENERATOR ED PROBLEM No of Iteration G en er at io n C o st ( $/ h r) 2013 International Conference on Power, Energy and Control (ICPEC) 409 TABLE III CSAPSO PARAMETERS TUNING RESULTS (Average of 30 trials) C1 C2 Generation cost($/hr) 0.6 3.4 15315.867 1.1 2.9 15314.984 1.4 2.6 15314.532 1.8 2.2 15315.645 1.9 2.1 15315.085 1.7 2.3 15314.809 2.5 1.5 15315.983 3.2 0.8 15315.361 The performance of CSAPSO with population size is varied from 20 to 300 is given in Table II. It can be seen that the solutions obtained from CSAPSO depends on the population size. Increasing population size will provide a decrease in total production cost but takes longer computing time andit is evident that the population size significantly affects the well-distributed non-dominated solutions. The effect of CSAPSO tuning parameters C1 and C2 for generation cost is shown in Table III. b) Test system II In this test system, the 15 generating units are considered under the load demand of 2650 MW. The comparative test results are given in Table IV. The results are compared with GA [19] and CEP [5] algorithms. It is worth noting that, the best obtained solution of total cost using GA [19] is 32517 $/hr while the proposed method provided 32506 $/hr which confirms the efficiency and applicability of the proposed method. TABLE IV SIMULATION RESULTS OF 15 GENERATOR SYSTEMS Fig 2.Convergence characteristics of 15 generator system. The convergence characteristics of the total generation cost using CSAPSO shown in Fig.2.From the graph, it can be notice that the value of the cost function converges smoothly to the optimum solution at reasonable time. c) Test system III The system consists of 40 units. Simulation results are carried out for 10500 MW load demand. Owing to the limits of space, effect on population size and effect of change in acceleration coefficients cannot be listed. TABLE V OPTIMAL OUTPUT FOR 40 GENERATOR SYSTEMS Optimization Maximum Minimum Average Generation cost($/hr) 122678.61 121686.43 122022.58 Evaluation time(s) 15.52 3.60 5.47 TABLE VI SIMULATION RESULTS OF 40 GENERATOR SYSTEMS Unit Individual Generation (MW) Unit Individual Generation (MW) P1 112. 9815 P21 444.2734 P2 113. 1549 P22 434.7422 P3 120 P23 434.3485 P4 180 P24 453.7839 P5 97 P25 494.2260 P6 140 P26 433.7495 P7 300 P27 11.8074 P8 300 P28 11.7527 P9 289. 1003 P29 09.3051 P10 129. 989 P30 97 P11 244. 284 P31 190 P12 319.9531 P32 173. 2620 P13 393.8832 P33 190 P14 395.0087 P34 200 P15 306. 3063 P35 199.0023 P16 395.3322 P36 200 P17 489. 6725 P37 110 P18 487.6419 P38 110 P19 499.8845 P39 110 P20 455.517 P40 423.0379 0 50 100 150 200 250 300 350 400 3.25 3.255 3.26 3.265 3.27 3.275 3.28 3.285 3.29 x 10 4 FINDING OPTIMAL GENERATION COST-15 generator No of Iteration G en er at io n C o st ( $/ h r) Unit (MW) GA[19] CEP[5] CSAPSO Pg1 451.4 449.946 455.000 Pg2 455.0 450.00 455.000 Pg3 130.0 130.000 130.000 Pg4 129.1 130.000 130.000 Pg5 337.1 335.007 260.000 Pg6 429.5 455.039 460.000 Pg7 464.4 015.000 015.000 Pg8 060.0 060.000 060.360 Pg9 026.6 025.000 025.823 Pg10 027.1 020.001 020.000 Pg11 025.7 020.000 058.878 Pg12 059.0 055.004 075.000 Pg13 025.0 025.000 025.000 Pg14 015.0 015.000 015.000 Pg15 015.0 465.000 464.939 Generation cost ($/hr) 32517.0 32507.553 32506.892 2013 International Conference on Power, Energy and Control (ICPEC) 410 The optimal generation cost and optimal power output of each generator are listed in Table V and Table VI respectively.The results given above demonstrate the effectiveness and applicability of the proposed method. The results represent the average of 40 trials. VI. CONCLUSION An approach for the determination of the most appropriate generation dispatch solution, which best meets the load demand in power system operation has been proposed. The implementation of the approach is based on global optimization technique. The CSAPSO method has been utilized successfully in the present work to solve ED problem. A chaotic local search operator is introduced in the proposed algorithm to avoid the premature convergence.The test results show that the proposed method can lead to better solution than any other method. Many non- linear characteristics of the generator such as ramp rate limits, valve point effect and prohibited operating zones are considered for practical generator operation. The solution gives the most appropriate generation dispatch scheme. This method converges to the global or near global point, irrespective of the shape of the cost function. The better computation efficiency and excellent convergence characteristics of the CSAPSO approach shows that it can be applied to a wide range of optimization problem for practical applications. ACKNOWLEDGMENT The authors would like to thank the management of Vellore Institute of Technology, VIT University for supporting our research work. REFERENCES [1] Farag, A.. AL-BaiyatS., and Cheng T.C.: ‘ Economic dispatch multi objective optimisation procedures using linear programmingtechniques’IEEETrans. Power Syst., 1995, 10, (2), pp. 731-738. [2] Wong, K.P and Fung, C.C, ‘Simulated annealing based algorithm for minimum emission dispatch’, Proc.International Power Engineering conference, March 1993. [3] D.P.Kothari, j.S.Dhillon “ power system optimisation” Prentice-Hall of India, 2004. [4] Rani C., Rajesh Kumar M. K..Pavan., ‘Multi-objective GenerationDispatch using Particle Swarm Optimization’ IEEE International Conference on power Electronics, Dec 18-19, 2006. 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