Volume 114 No. 8 2017, 143-153 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Multi Objective Economic Load Dispatch problem using A-Loss Coefficients D.Poornima 1, Sishaj P. Simon 2 B.Sonia 3,T.Sunita 4 1,3,4 Vignan s Institute of Information Technology, Duvvada. 2 NIT, Tiruchirapalli, 1poornima.dumpa@gmail.com 2sishajpsimon@gmail.com 3sonia.888b@gmail.com 4suni.sunitha26@gmail.com May17,2017 Abstract Multi Objective Economic Load Dispatch (MOELD) is one of the main objectives of power system operation while dispatching the output power of various generating units. The main objective of this problem is to minimize the Generation Cost, Emissions of fossil fuel plants and Transmission losses in the network. This problem is an extension of Economic-Emission dispatch problem which also includes the minimization of transmission losses. In this paper, the transmission losses are evaluated using nominal A-loss coefficients which can be derived for any transmission network from the knowledge of load flow analysis at few operating conditions using perturbation method. As the evaluation of these loss coefficients involves more than one operating conditions of the network in contrast to that for conventional B-loss coefficients, 143 1
these are proven to be accurate in calculating transmission losses. So, these A-loss coefficients are used in this paper for solving MOELD problem. Conventional Weighed Sum (WS) approach and Strength Pareto Genetic Algorithms (SPGA) are used to solve the problem of MOELD and the effectiveness of the algorithms are compared based on the results obtained for IEEE 30 bus system with 6 generating units. Key Words : Multi Objective Economic Load Dispatch, A-Loss coefficients, Economic-Emission dispatch, Weighed sum approach, Strength Pareto Genetic Algorithm 1 Introduction: Economic Load dispatch problem plays a key role in load dispatch process which minimize the cost of generation by suitable scheduling of committed generating units, while satisfying different operational constraints. Due to the U.S, Clean Air Act amendments 1990, the power generation companies using fossil fuels are enforced to revise their strategies such that atmospheric emissions are reduced. So, the emissions are included as one of the objectives to be minimized. As the average transmission losses can also be minimized by properly distributing the generation among various power plants, losses are also considered as one of the objectives to be minimized in this paper. By considering these three objectives at a time, the problem is converted into a Multi Objective Optimization Problem (MOOP). The transmission losses of power system network are conventionally calculated using B-loss coefficients which are derived at a particular operating condition of the network by making some assumptions. So, the calculation of transmission losses is not so accurate with these loss coefficients. In literature, A-loss coefficients have been proposed [1], [2] to be effective in calculating transmission losses. These coefficients are evaluated using perturbation method by considering more than one operating conditions of the power system network which makes it effective and accurate in calculating losses. So, A-loss coefficients are used in solving MOELD problem in this paper, which are proven to be very effective. The Multi Objective Optimization Problems (MOOP)have been solved using different techniques in literature. Blaze Gjorgiev& Marko Cepin.,[3] proposed Weighted Sum approach to solve the Economic-Environmental power dispatch and the constraints are handled using a penalty function. M.S. Osmana, M.A. Abo-Sinnab, A.A. Mousab., [4] 144 2
presented a novel Multi Objective Genetic Algorithm for solving Economic-Emission load dispatch problem in which the nondominated solutions iteratively updated based on the concept of ε- dominance. M. A. Abido, [5]presented a new methodology based on Strength Pareto Evolutionary Algorithm (SPEA) to solve the Economic -Environmental power dispatch problem which uses a clustering algorithm to bring about Pareto-optimal solution. A best compromise non dominated solution is extracted using Fuzzy set theory. In this paper, the three objective MOELD problems is solved by using A-loss coefficients with Weighed Sum approach and Strength Pareto Genetic Algorithms which are proven to be effective in literature and results are compared to justify the effective method. The IEEE 30 bus system with 6 generating units is chosen as the test system to carry out the simulations 2 Formulation of Multi Objective Economic Load dispatch (MOELD) The MOELD problem minimizes three competitive objectives while satisfying different operational constraints and is formulated as below. A. Objective functions: a) Cost of generation: The cost of generation of thermal plants can be expressed as a quadratic function of its real power output (Pgi) and is given by Eq (1) 2. F ( P ) a ( P ) b ( P ) c $ / h...(1) 1 nt gi i gi i gi i i 1 where, nt is number of generating units, ai, bi, ci are cost coefficients of i th generating unit. b) Emissions of pollutants: The amount of pollutants released into atmosphere can be expressed as a function of the output power of the plant (Pgi) with the help of emission coefficients (α, β, ) as given by Eq (2). F P P P ton h 2 ( ) ( ) ( ) /...(2) 2 nt gi i i gi i gi i 1 145 3
c)transmission losses: The transmission losses can be expressed as a function of output power(pgi) of generating units with help of A-loss coefficients as given in Eq (3). nt 2 F ( P ) ( )...(3) 3 AP MW gi i gi i 1 B. Constraints: a) Equality constraint: This is also calledpower balance constraint which sets the total power generation to be equal to t sum of load demand and transmission losses in the network. It is expressed as in Eq (4). nt i 1 P P P gi load loss...(4) where,p load is the total demand and P loss is the transmission losses calculated using Eq (3). 3 Solution Methodologies The Multi Objective Optimization Problem can be solved using various methods like conventional weighed sum approach in which a single objective function is defined which is a weighed combination of the objectives of the problem and evolutionary algorithms with pareto set approach. These methodologies are explained below in brief. A. Weighed Sum Approach: In this approach, the MOOP problem can be solved by converting it to a one objective optimization problem by using the linear combination of all objectives as a weighed sum such that m i=1 w i = 1 and wi 0 (i=1,2,,m) where m is the number of objective functions and wi is the weighing coefficient of i th objective function. This technique requires well known domain knowledge to assign appropriate weighing coefficients to each objective function. So this problem is solved using different 146 4
weightage combinations as shown in Table1 to locate non-inferior solution set. The best compromise weightage combination is determined with a fuzzy mechanism known as membership function. In this paper, Newton-Raphson method and Real coded Genetic algorithms are chosen to solve the problem using this approach. a) Newton - Raphson method: In this method, the constrained MOELD problem which is formed as a single objective function F is altered into unconstrained scalar optimization problem using Lagrangian multiplier function as shown in Eq (6). The optimality conditions are derived by taking the partial derivatives of this augmented objective function with respect to Pgi, λ. The augmented objective function is given by, ng P gi j = ).. (6) L = m i=1 w i F i + λ(p load + P loss b) Real coded Genetic Algorithm (GA): It is a randomised search algorithm which is guided by the principle of natural genetic systems. This algorithm is robust and requires no auxiliary information and can offer significant advantages in solution methodologies. The process of GA is explained as follows to solve this MOELD problem. 1) Generate a random feasible solution set which is known as population 2) Assign fitness value to each member of the population based on its evaluation. 3) Select solutions with lowest fitness value (value of F which is the weighed combination of three objective functions) to be parent the new solutions during reproduction process. 4) The new solution set replaces the less fitted old solutions based on selection rate. 147 5
5) Continue the process from step 2 till the convergence criterion is satisfied. Non Pareto solution set is formed for different combinations of weightages among which the optimum solution is extracted based on the membership value. c) Membership function: This is one of the effective method in Fuzzy logic which derives Pareto-optimal solution from a group of non-inferior solutions. The membership function of i th objective function of a solution is given by Eq (7), F i,max F i F ii,max F i,min μ F i = 1 F i F i,min F i,min < F i < F i,max (7) 0 F i F i,max Where F i,max and F i,min are the maximum and minimum values of i th objective function. The normalized membership function corresponds to k th non-dominated solution is given by μ k D = m i= 1 K m k= 1 i= 1 μ(f k i μ(f The solution with maximum optimal solution. ) k i ) (8) B) Strength Pareto Genetic Algorithm (SPGA): k μ D value is chosen as the Pareto It is one of the potential algorithm for Multi Objective Optimization Problems which works based on the Pareto set approach. In this approach, non-dominated solution set is determined using Pareto dominance principle which is defined for a minimization problem as, j {1,2,.m}: fi(x1)<fi(x2) i {1,2,.m} : fi(x1) fi(x2) Here x1 is known as non-dominated solution within the set 148 6
{x1, x2}.the non dominatedsolution within the entire search space is known as Pareto-optimalsolutions. A value known as strength is assigned to each solution within the range [0,1) for evaluation. This strength is defined to be proportional to the number of solutions covered by it. The fitness of an individual is calculated asthe sum of the strengths of all external Pareto solutions by which it is covered. The steps to be followed to solve MOELD problem using SPGA method are explained below. 1) An initial population of random feasible solutions is generated. 2) The non-dominated solutions are identified using dominance principle to update the archive set. 3) Assign fitness value to each member and sort out the population from maximum to minimum fitness value. 4) The generation values correspond to maximum fitness value is considered as pareto optimal solution. C) Steps to form initial feasible solution set using A-loss coefficients: 1. Create the population of real power outputs of generators except for slack bus power (Pg1) 2. Assume Pg1=0. 3. Calculate losses by using Eq (3). 4. For each combination of chromosomes, calculate the generation of the slack bus (Pg1) by using the Eq (10). P P P nt P...(10) g1 load losses gi i 2 5. Calculate the losses by using Eq (3). 6. Calculate the difference in losses evaluated in steps 3 and 5. If it is more than 0.0001, go to step 3 and repeat the steps. Otherwise stop the process. 4. Simulation Results: The Multi Objective Economic Load Dispatch problem is solved 149 7
Cost ($/h), Emission s(ton/h) Losses (MW) International Journal of Pure and Applied Mathematics for IEEE-30 bus system at a total load of 283.4MW, using SCILAB-5.4.1 software. The transmission losses are calculated using nominal A loss coefficients. Newton-Raphson method (NR method), Genetic algorithms are used to solve the problem with Weighed Sum approach and Strength Pareto Genetic Algorithm is also used to solve the problem and results are compared with these algorithms. Table 1 Different weightage combinations Combination number w1 w2 w3 1 1 0 0 2 0 1 0 3 0 0 1 4 0.85 0.15 0 5 0.7 0.3 0 6 0.55 0.45 0 7 0.4 0.6 0 8 0.25 0.75 0 9 0.1 0.9 0 10 0.85 0 0.15 Non dominated solutions 1000 800 600 400 200 0 Cost of generation Emissions Losses 10 8 6 4 2 0 Objective considered to be minimized By applying membership technique, the most optimal solution obtained by these three methods is determined. The respective normalized membership values. 150 8
Cost ($/h), Emissions(ton/hs Losses, MW International Journal of Pure and Applied Mathematics 1000 800 600 NR method 15 10 400 200 5 0 1 4 7 10 13 16 19 22 25 28 Weightage combination number 0 Fig 1: Non-inferior solutions with NR method and WS approach From the membership values, we can justify the effectiveness of Strength Pareto Genetic algorithm in solving Multi Objective Optimization problems. 5. Conclusion: In this paper, the formulation and implementation of Multi objective optimization problems have been explained. The Multi Objective Economic Load Dispatch problem is solved in this paper which optimizes three objectives, Cost of generation, Emissions and the Transmission losses of power system operation. The losses are calculated using nominal A loss coefficients. Both conventional and evolutionary algorithms are used to solve the problem with Weighed Sum approach and Strength Pareto Genetic Algorithm. The use of A loss coefficients in calculating transmission losses, while solving Multi objective ELD is attempted successfully. The feasibility of the above algorithms has been checked by validating them in IEEE 30 bus system. It is observed that Strength Pareto Genetic algorithm gives better result when compared with Weighed Sum approach. 6. References: [01] J.Nanda., L.L. Lai, A novel approach to computationally efficient algorithms for transmission loss and line flow formulations, Elsevier, Electrical Power and Energy Systems 21 (1999) 555 560. [02] C. H. Ram Jethmalani, PoornimaDumpa, Sishaj P. Simon and K. Sundareswaran, Transmission Loss Calculation using A and B Loss Coefficients in Dynamic Economic Dispatch Problem, Int. J. Emerg. Electr. Power Syst. DOI 10.1515/ijeeps-2015-01812016 151 9
[03] Blaze Gjorgiev& Marko Cepin., A multi-objective optimization based solution for the combined economic-environmental power dispatch problem, Engineering Applications of Artificial Intelligence 26 (2013) 417 429. [04] M.S. Osmana, M.A. Abo-Sinnab, A.A. Mousab, An ε-dominancebased multi objective genetic algorithm for economic emission load dispatch optimization problem, Electric Power Systems Research 79 (2009) 1561 1567. [05] M. A. Abido., Environmental/Economic power dispatch using multi objective evolutionary algorithms, IEEE transactions on power systems, Vol. 18, No. 4, November 2003 [06] M. A. Abido., Multi objective evolutionary algorithms for Electric power dispatch problem, IEEE transactions on evolutionary computation,vol. 10, No. 3, June 2006 [07] Bin Shi a, Lie-Xiang Yan a, WeiWub, Multi-objective optimization for combined heat and power economic dispatch with power transmission loss and emission reduction, Energy 56 (2013) 135e143 [08] B.Venkateswara Rao, G.V.Nagesh Kumar, M.Ramya Priya, and P.V.S.Sobhan, " Implementation of Static VAR Compensator for Improvement of Power System Stability ", International Conference on Advances in Computing, Control, and Telecommunication Technologies, ACT 2009 organized by ACEEE and CPS, Trivandrum, Kerala, India,, 28-29 December, 2009, Pages: 453-457. [09] M. A. Abido, A new multi objective evolutionary algorithm for environmental/economic power dispatch IEEE Power Eng. Soc. Summer Meeting, Vancouver, BC, Canada, Jul. 15 19, 2001., 1263 1268. 152 10
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