Multi-objective Emission constrained Economic Power Dispatch Using Differential Evolution Algorithm

Transcription

1 Multi-objective Emission constrained Economic Power Dispatch Using Differential Evolution Algorithm Sunil Kumar Soni, Vijay Bhuria Abstract The main aim of power utilities is to provide high quality power supply to the consumer at lowest possible cost and minimum emission while operating to meet the limits and constraints imposed on the generating units. The economic emission dispatch (EED) problem is a sub problem of an optimal power flow. In this paper, Differential Evolution (DE) based optimization technique is presented to solve the economic emission dispatch (EED) problem which is a nonlinear function of generated power. DE is a population-based stochastic search technique that works in the general framework of Evolutionary Algorithms. The design principles of DE are simplicity, efficiency and use of real coding. It starts to explore the search space by randomly choosing the initial candidate solutions within the boundary. Then the algorithm tries to locate the global optimum solution for the problem by iterated refining of the population through reproduction and selection. In this paper differential evolution technique is presented to solve the economic load dispatch problem which is a nonlinear function of generated power. In the experimental study, Differential Evolution (DE) is analysed and demonstrated on standard IEEE 30 bus system consisting of six generating units. Key words- Economic Dispatch, Emission Dispatch, Multi-Objective Function, Price Penalty Factor, Differential Evolution (DE). I. INTRODUCTION Economic emission dispatch (EED) has become an essential function in operation and control of modern power system and it is a sub problem of an optimal power flow. The main aim of the economic emission dispatch problem is to find an optimal combination of the output power of all the online generating units that minimize the total fuel cost and reduction of pollution level up to a safe limit of the generation and supplies the power demand while satisfying unit constraints, equality and inequality constraints [1]. The major part of the power is generated due to the fossil fired plants and hence their emission contribution cannot be ignored. Fossil fired electric power plants use coal, oil, gas as primary energy resources and produce atmospheric emission whose nature and quantity depend upon fuel type and its quality. The particulate matter such as ash and gaseous pollutant i.e. CO 2, NO X (oxides of nitrogen) etc. are produced due to coal. Hence it is a needed to reduce the emission from there fossil fired plants either by design or by operational strategies [2]. In classical Economic Load Dispatch (ELD) problem, mathematical model of fuel cost function has been approximated as a single quadratic cost function [3]. A number of conventional optimization techniques have been applied to solve the EED problem such as linear Programming (LP) [4], nonlinear programming (NLP) [5], quadratic programming (QP) [6], and interior point methods [7]. Although these conventional techniques offer good results but when the search space is non-linear and has discontinuities they become very complicated with a slow convergence ratio and sometimes unable to find the optimal solution. To overcome these difficulties new numerical methods are needed which have high speed search to the global optima and does not suffer from the problem of local minima [8]. In the recent years the soft computing techniques such as Genetic algorithm (GA) [9], Evolutionary programming (EP), Simulated annealing (SA), Tabu search (TS) technique, Particle Swarm Optimization (PSO) [ ] etc. may prove to be very effective in solving nonlinear EED problems. In 1995 Storn and Price [13] have proposed one of the most prominent new generation evolutionary algorithm i.e. Differential Evolution (DE) to exhibit consistent and reliable performance in nonlinear and multimodal environment [14] and proven effective for constrained optimization problems. The advantages of DE over other EAs, like simple and compact structure, few control parameters, high convergence characteristics, have made it a popular stochastic optimizer. In this paper differential evolution technique is presented to solve the economic emission dispatch problem which is a nonlinear function of generated power. In the experimental study, Differential Evolution (DE) is analyzed and demonstrated on standard IEEE 30 bus system consisting of six generating units. II. PROBLEM FORMULATION The primary objective of EED problem is to determine the most economic loading of the generating units such that the load demand in the power system can be met. The EED planning must be performed satisfying different equality and inequality constraints. Consider a power system having N generating units each loaded to. The generating units should be loaded in such a way that minimizes the total fuel cost while satisfying the power balance and other constraints. The economic emission dispatch problem is an optimization problem that determines the power output of 120

2 each online generator that will result in a least cost system operating state with minimum emission. The EED problem can then be written in the following form: Minimize f(x) Subject to: g(x) = 0 h(x) 0 f(x) is the objective function, g(x) and h(x) are respectively the set of equality and inequality constraints. x is the vector of control and state variables. A. Objective Function (1) Cost- The objective of the ELD is to minimize the total system cost by adjusting the power output of each of the generators connected to the grid. The total system cost is modelled as the sum of the cost function of each generator (1). The generator cost curves are modelled with smooth quadratic functions and is given by: total system loads. Equilibrium is only met when the total system generation equals the total system load (P D ) plus system losses (P L ) The exact value of the system losses can only be determined by means of a power flow solution. The most popular approach for finding an approximate value of the losses is by way of Kron s loss formula (6), which approximates the losses as a function of the output level of the system generators. D. Inequality Constraints Generating units have lower and upper ( ) production limits of power output of the unit. These bounds can be defined as a pair of inequality constraints, as follows: (i=1, 2, 3, NG) Where N is the number of online thermal units, is the active power generation at unit i and, and are the Fuel cost coefficients of the generating units. (2) Emission- The atmospheric pollution such as sulphur oxide (SO X ), nitrogen oxide (NO X ) and carbon dioxide (CO 2 ) caused by fossil fuel generator can be modelled separately. The emission equation of generation can be expressed as Where, and are emission coefficients of the generating unit. B. Multi-Objective Economic Dispatch EED is a multi-objective problem, which is combination of both economic and environmental dispatches that individually make up different single problems. At this point, this multi-objective problem needs to be converted in to single-objective form in order to fulfil optimization [15]. The conversion process can be done by using price penalty factor. However, the single-objective EED can be formulated as shown in below equation Where h i is the price penalty factor, and is formulated as follows Where is the maximum power generation of the unit in MW. C. Equality Constraints The equality constraint is represented by the power balance constraint that reduces the power system to a basic principle of equilibrium between total system generation and III. DIFFERENTIAL EVOLUTION OPTIMIZATION PROCESS Differential Evolution is a simple and efficient adaptive scheme for global optimization over continuous spaces. In this section, we review the differential evolution (DE) algorithm that was used for searching the optimum solution of ELD problems. Differential evolution solves real valued problems based on the principles of natural evolution [16] using a population P. the population size remains constant throughout the optimization process Differential Evolution (DE) is a parallel direct search method which utilizes NP, D-dimensional parameter vectors, In DE, a population of NP solution vectors is randomly created at the start this population is successfully improved over G generations by applying This optimization process is carried out with three basic operations mutation, crossover and selection operators, to reach an optimal solution [17]. The main steps of the DE algorithm are given bellow: Initialization Mutation Crossover Evaluation Selection Until (Termination criteria are met) (1) as a population for each generation G. NP does not change during the minimization process. The initial vector population is chosen randomly and should cover the entire parameter space. As a rule, we will assume a uniform probability distribution for all random decisions unless otherwise stated. In case a preliminary solution is available, the initial population might be generated by adding normally distributed random deviations to the nominal solution. DE generates new parameter vectors by 121

3 adding the weighted difference between two population vectors to a third vector. Let this operation be called mutation. The mutated vector s parameters are then mixed with the parameters of another predetermined vector, the target vector, to yield the so-called trial vector. Parameter mixing is often referred to as crossover in the ES-community and will be explained later in more detail. If the trial vector yields a lower cost function value than the target vector, the trial vector replaces the target vector in the following generation. This last operation is called selection. Each population vector has to serve once as the target vector so that NP competitions take place in one generation. More specifically DE s basic strategy can be described as follows: A. Initialization The first step in the DE optimization process is to create an initial population of candidate solutions by assigning random values to each decision parameter of each individual of the population. Such values must lie inside the feasible bounds of the decision variable. B. Mutation After the population is initialized, The mutation operator creates mutant vectors by perturbing a randomly selected vector xa with the difference of two other randomly selected vectors xb and xc, Where xa, xb and xc are randomly chosen vectors among the NP population, and a b c. xa, xb and xc are selected anew for each parent vector. The scaling constant F is an algorithm control parameter used to adjust the perturbation size in the mutation operator and improve algorithm convergence. For each target vector a mutant vector is generated using a differential mutation operation according to following equation (2) Where F is a real and constant factor, commonly known as scaling factor or amplification factor, is a positive real number, typically less than 1.0, that controls the amplification of the differential variation with random indexes integer. C. Crossover DE uses binomial crossover operation this crossover constant CR is an algorithm parameter that controls the diversity of the population The crossover operator creates the trial vectors, which are used in the selection process. In order to increase the diversity of the perturbed parameter vectors, crossover is introduced. To this end, the trial vector: is formed, where J = 1, 2,..,D (4) In (4) is the evaluation of a uniform random number generator with outcome. CR is the crossover (3) constant which has to be determined by the user. Is a randomly chosen index which ensures that gets at least one parameter from D. Selection The selection operator forms the population by choosing between the trial vectors and their predecessors (target vectors) those individuals that present a better fitness or are more optimal. This optimization process is repeated for several generations allowing individuals to improve their fitness as they explore the solution space in search of optimal values.de has three essential control parameters: the scaling factor (F), the crossover constant (CR) and the population size (NP). The scaling factor is a value in the range [0, 2] that controls the amount of perturbation in the mutation process. The crossover constant is a value in the range [0, 1] that controls the diversity of the population. The population size determines the number of individuals in the population and provides the algorithm enough diversity to search the solution space. To decide whether or not it should become a member of generation G+1 the trial vector is compared to the target vector using the greedy criterion. If vector yields a smaller cost function value than then is set to otherwise, the old value is retained. IV. DESCRIPTION OF THE TEST SYSTEM In the study of experiment, DE algorithm is tested over standard IEEE 30-bus power system with six generating units as shown in figure 1. Fig 1: Single Line Diagram of IEEE 30-Bus Test System The algorithm is tested for load demand 500MW, 700MW and 900MW. The coefficient EED problem and transmission loss coefficients matrix are taken from Rughooputh and King (2003). The transmission loss coefficients matrix is given by equation 122

4 Table3. Best Fuel Cost Solution for the Test Power System. V. RESULT AND DISCUSSION Here the Multi-objective EED problem is solved by the DE algorithm for the standard IEEE 30-bus power system. In this case study problem, the variable given in Table 1. A test system having six thermal units is considered. The proposed method is applied for EED with load demands 500 MW, 700MW and 900 MW and it is compared with FSGA and NSGA-II. Minimum fuel cost solution with all load demands are considered in Table 2, Table 3 and Table 4. Table1. Coefficient of Fuel Cost, Emission and Capacities of the Six Generating Units. Table4. Best Fuel Cost Solution for the Test Power System. Table2. Best Fuel Cost Solution for the Test Power System. Minimum NO X emission effect solution for EED problem with all load demands are considered in Table 5, Table 6 and Table 7. Table5. Best Emission Effects (NO X) For the Test Power System. 123

5 Table6. Best Emission Effects (NO X) For the Test Power System. Table9. Best Compromise Solution for the Test Power System with 700 MW. Table7. Best Emission Effects (NO X) For the Test Power System. Table10. Best Compromise Solution for the Test Power System with 900 MW. Table 8, Table 9 and Table 10 give the best compromise solution for EED problem using DE, FCGA and NSGA-II with load demand 500 MW, 700 MW and 900 MW for six generator system. Table8. Best Compromise Solution for the Test Power System with 500 MW. VI. CONCLUSION In this paper the difficult optimization problem is solved by using DE algorithm. In order to prove the effectiveness of algorithm it is applied to three different cases with six generating units. Cases 1, 2 & 3 are 500 MW, 700 MW & 900 MW. The multi-objective problem is converted in to single objective form by means of price penalty factor with the consideration of problem constraints. After comparing the results with other algorithm it is observed that DE is well suited for obtaining the best solution, so that both fuel cost and emission effect are reduced for different load demands. ACKNOWLEDGEMENT The authors are thankful to Director, Madhav Institute of Technology & Science, Gwalior (M.P) India for support and facilities to carry out this research work. 124

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