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- 1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME43MULTI OBJECTIVE ECONOMIC LOAD DISPATCH USING HYBRIDFUZZY, BACTERIAL FORAGING-NELDER–MEAD ALGORITHMBharathkumar S1, Arul Vineeth A D2, Ashokkumar K3, Vijay Anand K41,4II Year ME Power System, EEE Department, Anna University Regional centre ,Coimbatore2II Year ME Information Technology, IT Department, Anna University Regional centre,Coimbatore,Tamilnadu,India3II Year ME Control and Instrumentation, EEE Department, Anna University Regionalcentre, CoimbatoreABSTRACTIn this paper, a new approach is proposed to solve the economic load dispatch (ELD)problem. Power generation, spinning reserve and emission costs are simultaneouslyconsidered in the objective function of the proposed ELD problem. In this condition, if thevalve-point effects of thermal units are considered in the proposed emission, reserve andeconomic load dispatch (ERELD) problem, a non-smooth and non-convex cost function willbe obtained. Frequency deviation, minimum frequency limits and other practical constraintsare also considered in this problem. For this purpose, ramp rate limit, transmission line losses,maximum emission limit for specific power plants or total power system, prohibitedoperating zones and frequency constraints are considered in the optimization problem. Ahybrid method that combines the bacterial foraging (BF) algorithm with the Nelder–Mead(NM) method (called BF–NM algorithm) is used to solve the problem. In this paper, theperformance of the proposed BF–NM algorithm is compared with the performance of otherclassic (non-linear programming) and intelligent algorithms such as particle swarmoptimization (PSO) as well as genetic algorithm (GA), differential evolution (DE) and BFalgorithms. The simulation results show the advantages of the proposed method for reducingthe total cost of the system.Index Terms- Economic Dispatch, Differential Evolution, Evolutionary Algorithms, ValvePoint Loading Effects, Prohibited Operating Zones, Piecewise Quadratic Cost Functions.INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING& TECHNOLOGY (IJEET)ISSN 0976 – 6545(Print)ISSN 0976 – 6553(Online)Volume 4, Issue 3, May - June (2013), pp. 43-52© IAEME: www.iaeme.com/ijeet.aspJournal Impact Factor (2013): 5.5028 (Calculated by GISI)www.jifactor.comIJEET© I A E M E
- 2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME44I. INTRODUCTIONThe economic load dispatch of power plants is one of the most important problems inpower system operations. In this regard, the ELD minimizes the generation cost of powerplants so that the generated power satisfies the load demand by considering practical systemconstraints [1–3]. This is an extremely important problem in restructured power systems.Due to increasing sensitivity regarding power plant emissions, the ELD must beperformed such that the environmental emissions of power plants are minimized [4].Furthermore, during a specific period of time, the emission constraint is considered in [5–7]to solve the ELD problem. The prohibited power generation zone is another constraint thatcan be considered in the ELD problem [8–10]. In addition, economic dispatch can be solvedby considering frequency constraints [1].To develop a complete model of the ELD problem, the effect of the spinning reserveconstraint [11–13] as well as the valve-point effect [14–21] and transmission line losses[22,24] can also be taken into account. In most studies, the generation cost function isconsidered to be quadratic function, but a cubic cost function more closely conforms to thegeneration cost [14]. Therefore, the use of a cubic cost function leads to more accuratemodelling of power plant costs.The ELD problem is an optimization problem; thus, a large number of methods areavailable to solve this problem. Recently, stochastic search algorithms such as PSO, GA,direct search, and DE algorithms [17] have been successfully used to solve the ELD problem.Each of these algorithms has its own advantages and disadvantages. For example, the directsearch method and GA have slow execution speeds, and the PSO algorithm requires theexecution of many repeated stages. The above-mentioned search methods determine the localoptimal point but cannot find that optimal solution.The BF algorithm is a new optimization algorithm that has recently been consideredto solve the real world optimization problem. It covers a wide search region but has lowconvergence speed. In this respect, a hybrid method combines BF algorithm and NM method(BF–NM algorithm) with the combination of the fuzzy logic is used. By combining thesethree methods, the search power of intelligent methods and the precision of conventionalmethods are simultaneously employed [27]. Addition, the transmission losses, maximumemission limit, and practical constraints of the power plants are considered in the problem.The frequency deviation, minimum frequency limit and maximum permissible environmentalemission constraints are also used in the problem to assure the power system security. Thesimulation results validate the performance and accuracy of the proposed method for solvingthe ERELD problem by placing practical constraints in power system.II. ECONOMIC LOAD DISPATCHA. Problem formulationThe proposed ERELD problem consists of an objective function and practical constraints.The objective function and constraints are introduced in following subsections.Objective Functions: The objective of the classical economic dispatch is to minimize the totalsystem cost (1) 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.
- 3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME45(1))(....1321 iNiinCost PFFFFFF ∑==+++=where Pi and Fi are the output power and the generation cost of ithgenerating units and Nis the number of power plants.The cost function of each generator establishes the relationship between the powerinjected to the system by the generator and the cost incurred to load the machine to thatcapacity. Generators are typically modelled by smooth quadratic functions such as (2), inorder to simplify the corresponding optimization problem, as well as to facilitate theapplication of proposed technique. The cost function is generally considered to be a squarecost function [29,30]. However, a cubic cost function is more appropriate and accurate. So,the proposed total generation cost can be expressed as follows:(2))PPP(min 3i2i1iCost iNiiii dcbaF +++= ∑=Valve-Point Effect: If the power output of a generator with multi-valve steam turbines isincreased to meet the increased demand, various steam valves should to be opened insequence. As shown in Fig. 1, the valve-point effect can be considered by adding the absolutevalue.Fig. 1. A cost function of a unit with valve-point effect and prohibited operating zones.of a sinusoidal function with a cubic cost function [14–18]. Thus, the cost function ismodified as follows:(3)))]P-(Psin(f.(e)PPP[(miniminiii3i2i1iCost ++++= ∑=iNiiii dcbaFPower Plant Spinning Reserve Cost Function: Plants should have enough spinning reserve toprovide energy without interruption for customers. This reserve provides cost for the system[12]. Thus,(4))(....1321iNiinCostRFRFRFRFRFRF∑==+++=
- 4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME46where FRcost is the total reserve cost of the whole system and Ri is the reserve for the ithunit.The determination of spinning reserve values to minimize the FRcost function is one of themain objectives in power system operations. Therefore,(5))RR(min 2i1iCost ∑=++=Niribiri cbaFRwhere ari, bri and cri are the coefficients of the reserve cost of the ithgenerator.Multiple Fuel types: Some generating units are capable of operating using different types offuels. The use of multiple fuel types may result in multiple cost curves that are notnecessarily parallel or continuous. The lower region of the resulting cost curve determineswhich fuel type is most economical to burn.Fig. 2 Fuel cost function of a thermal generation unitsupplied with multiple fuel typesThis cost function can be represented by a piecewise curve (see Fig. 2), and the segmentsare defined by the range in which each fuel is used (6). The ED problem with piecewisequadratic cost curves is very difficult to solve by standard techniques. Piecewise quadraticcost functions have as many segments as fuel types.F୧൫Fୋ൯ ൌەۖ۔ۖۓa୧,ଵ b୧,ଵPୋ c୧,ଵPୋଶ, Pୋଵ൏ Pୋ൏ Pୋഠଵതതതതa୧,ଶ b୧,ଶPୋ c୧,ଶPୋଶ, Pୋଶ൏ Pୋ൏ Pୋഠଶതതതതڭa୧,୩ b୧,୩Pୋ c୧,୩PୋଶڭPୋ୩൏ Pୋ൏ Pୋഠ୩തതതതሺ6ሻwhere Pୋ୩and Pୋഠଵതതതത are the lower and upper bound respectively of the kthfuel of unit i, and ai,kbi,k ci,k are the kthfuel cost coefficients of unit i.Prohibited Operating Zones: Generating units may have certain regions where operation iseither undesired or impossible due to physical limitations of
- 5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME47the machine components or issues related to instability. These regions produce discontinuitiesin the cost curve since the unit must operate under or over certain specified limits. This typeof cost functions results in non-convex sets of feasible solution points, which are modelled asfollows:൞ܲ, ܲ ܲ,ଵܲ,ିଵ௨ ܲ ܲ,ܲ,௭௨ ܲ ܲ,௫(7)where Pli,k and Pui,k are the lower and upper limits of the kthof the prohibited zone of ithgenerating unit and z is the number of the prohibited zones of the ithunit, respectively.The Proposed Objective Function: Fuel, spinning reserve and emission costs are in conflictwith each other. In other words, as the minimum generation and reserve costs and minimumemission do not occur at a single point, it is necessary to optimize them, simultaneously.Multi-objective optimization methods can be used to solve this optimization problem. Togenerate the non-inferior solutions of a multi-objective optimization problem, the weightingmethod can be used [7,31]. This approach aggregates all objective functions in a weightedcombination, producing a single one [31]. Therefore, the ERELD problem can be convertedinto a scalar optimization problem as follows:min ߶ ൌ ቂݓଵ ቀܨ௦௧, ܴܨ௦௧, ቚ݁. sin ቀ݂൫ܲ, െ ܲ൯ቁቚቁேୀଵ ݓଶ൫ܳ,. ܨா௦௦,൯ቃ ሺ8ሻwhere w1 and w2 are non-negative weights, such that w1 + w2 = 1. w1 and w2 are used tomake a trade-off between emission and total cost (energy and reserve costs). So theseweighting factors vary between w1 = 1.0, w2 = 0.0 and w1 = 0.0, w2 = 1.0. It means that, if w1= 1.0, w2 = 0.0, the economic and reserve dispatch will be performed instead of ERELD.Also, if w1 = 0.0, w2 = 1.0, the emission dispatch will be performed instead of ERELD. If w1= w2, emission and total cost (energy and reserve cost) have similar importance. Many studiesuse this weight setting to convert a multi-objective problem into a single-objective.III. THE HYBRID BF–NM ALGORITHM WITH FUZZY LOGICIn order to solve the proposed OPF problem, the hybrid bacterial foraging (BF)algorithm and the Nelder–Mead (NM) method are employed to minimize the cost function ofthe problem. The BF algorithm is a stochastic optimization algorithm. It covers a wide searchregion, but it has low convergence speed. In this respect, the BF algorithm and the NMmethod can be combined [27]. By combining these two methods, the search power of theintelligent methods and the precision of conventional methods are simultaneously exploited.Therefore, in this section, the BF algorithm and the NM method are first introduced. The BF–NM combinational algorithm is then presented to solve the proposed ERELD problem.
- 6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME48AlgorithmStep 1. Form the objective function for the ERELD problem.Step 2. Set the initial count of the bacteria.Step 3. Determine the P0 for tthhour.Step 4. Check the constrains are satisfied the Update limits for tthhour.Step 5. Random selection of each bacteria.Step 6. Find the localized optimized value for the each bacteria by Nelder Meadalgorithm.Step 7. Find the localized optimized value satisfy the objective function if not thebacteria are been places locally but the fuzzy logic according to the membershipfunction and rule based.Step 8. Check for the Optimized feasible solution for the problem.Fig. 3. Convergence properties of different optimization algorithmsThe bacteria movement is been given bySwimming: (9)(i)(i)(i))(l)k,j,(),,1(T∆∆∆+=+ iClkj iiθθTumble: (10)(i)(i)(i))(l)k,1,j(),,1(T∆∆∆++=+ iClkj iiθθWhere θi(j,k,l) represents the position of ithbacterium at jthchemotaxis, kthreproduction, and lthelimination and dispersal, respectively. Also, C(i) and ∆(i) are themovement length and direction random vector, respectively. If the value of the cost functionin ithchemotactic step is smaller than the value of the cost function in i-1thchemotactic step,the moving direction will be correct and the bacterium swims in the same direction.Otherwise, the moving direction will be incorrect. In this case, a new random direction(tumble) is set for this bacterium.
- 7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME49In this A simple method has been developed for finding a local minimum point from afunction of several variables by Nelder and Mead [28]. The Nelder–Mead method is used tocompare the objective function values in n + 1 of the vertex for the solution of the ndimensional optimization problem. At each stage, one new vertex is generated.Then, if the new vertex has less objective function value relative to the previousvertices, this new vertex is replaced with the worst vertex of the previous level. As anexample, the NM method, which is a pattern search for a problem with 2 variables, comparesthe values of the objective function at the three vertices of a triangle. The Nelder–Mead directsearch method is shown in Fig. 4 for the minimization of a non-linear function in a two-dimensional space is shown in the diagram. By moving toward the minimum point in thismethod, the size of the triangle becomes increasingly small.Fig. 4. The Nelder–Mead algorithm.So that the bacteria need not search for food in every area the Nelder Mead simplexalgorithm would search locally for the food so that the speed is enhanced if the values are notsatisfied the constrain then the fuzzy logic is used to relocate the bacterial search in the newarea which are been modelled mathematically according to the ERELD problem.The Convergence of the optimal value is very fast and accuracy more than any othermodel which would lead to solve the problem more effective than any other optimizationtechnique for any real world problem.IV. CONCLUSIONIn this paper, by considering spinning reserve, emission, and the valve-point effects, anew ELD problem was presented and solved using the BF–NM algorithm. In this problem,the frequency constraints the practical constraints of power plants and the maximum emissionlimit were also considered. By investigating the simulation results, it was found that if thefrequency constraints are inserted in the proposed problem, it can be solved by controlling thefrequency within the permissible limit. Thus, the frequency-constrained ERELD effectivelyincreases social welfare for consumers and GENCOs. The simulation results confirm thevalidity of proposed FC-ERELD problem solved by BF–NM algorithm with fuzzy incomparison with conventional method and other optimization algorithms.
- 8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME50V. REFERENCES[1] Doherty R, Lalor G, Malley MO. Frequency control in competitive electricity marketdispatch. IEEE Trans Power System 2005;20(3):1588–96.[2] Bai J, Gooi HB, Xia LM, Strbac G, Venkatesh B. A probabilistic reserve marketincorporating interruptible load. IEEE Trans Power Syst 2006;21(3):1079–87.[3] Kumar A, Gao W. Pattern of secure bilateral transactions ensuring power economicdispatch in hybrid electricity markets. Applied Energy 2009;86 (7–8):1000–10.[4] Vahidinasab V, Jadid S. Multi objective environmental/techno-economic approach forstrategic bidding in energy markets. Applied Energy 2009;86(4):496–504.[5] Xuebin L. Study of multi-objective optimization and multi-attribute decision makingfor economic and environmental power dispatch. Electric Power Syst Res 2009;79(5):789–95.[6] Abido MA. Multiobjective particle swarm optimization for environmental/economicdispatch problem. Electric Power System Res 2009;79(7):1105–13.[7] Dhillon JS, Kothari DP. Economic-emission load dispatch (EELD) using binarysuccessive approximation-based evolutionary search. IET Gener Transm Distrib 2009;3(1):1–16.[8] Adhinarayanan T, Sydulu M. Reserve constrained economic dispatch with prohibitedoperating zones using ‘‘k-logic’’ based algorithm. IEEE power engg society general meeting;2007. p. 4244–53.[9] Adhinarayanan T, Sydulu, M. A new optimising concept to ramp-rate constrainedeconomic dispatch with prohibited operation zones. IEEE power eng society general meeting;2007. p. 1–6.[10] Coelho LS, Lee CS. Solving economic load dispatch problems in power systemsusing chaotic and Gaussian particle swarm optimization approaches. Int J Electric PowerEnergy Syst 2008;30(5):297–307.[11] Wu J, Liu J, Duan D, Niu H, Xie L, Li W. Research on operation reserve capacity inpower market environment. IEEE Int Conf Electric Util Derg Restr Power Technol(DRPT2004) 2004.[12] Xin JQ, Bompard E, Napoli R. Security coordinated economic dispatch for jointenergy and reserve markets. Int Conf Power Syst Technol 2006;3(4):1324–9.[13] Misraji J, Conejo AJ, Morales JM. Reserve-constrained economic dispatch: cost andpayment allocations. Electric Power Syst Res 2008;78(5):919–25.[14] Al-sumait JS, Sykulski JK, Al-othman AK. Solution of different types of economicload dispatch problems using a pattern search method. Electric Power Comp Syst2008;36(3):250–65.[15] Victoire TAA, Jeyakumar AE. Hybrid PSO–SQP for economic dispatch with valve-point effect. Electric Power Syst Res 2004;71(1):51–9.[16] Park JB, Lee KS, Shin JR, Lee KY. A particle swarm optimization for economicdispatch with nonsmooth cost function. IEEE Trans Power System 2005;20(1):34–42.[17] Wang SK, Chiou JP, Liu CW. Non-smooth/non-convex economic dispatch by anovel hybrid differential evolution algorithm. IET Gener Transm Distrib 2007;1(5):793–803.[18] Alsumait JS, Sykulski JK, Al-Othman AK. A hybrid GA–PS–SQP method to solvepower system valve-point economic dispatch problems. Appl Energy 2010;87(5):1773–81.
- 9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME51[19] Al-Othman AK, El-Naggar KM. Application of pattern search method to powersystem security constrained economic dispatch (SCED) with non-smooth cost function.Electric Power Syst Res 2008;78(4):667–75.[20] Da-kuo H, Fu-li W, Zhi-zhong M. Hybrid genetic algorithm for economic dispatchwith valve-point effect. Electric Power Syst Res 2008;78(4):623–33.[21] He D, Wang F, Mao Zh. A hybrid genetic algorithm approach based on differentialevolution for economic dispatch ED with valve-point effect. Int J Electric Power Energy Syst2008;30(1):31–8.[22] Chen CL, Chen N. Direct search method for solving economic dispatch problemconsidering transmission capacity constraints. IEEE Trans Power System 2001;16(4):764–9.[23] Gaing ZL. Particle swarm optimization to solving the economic dispatch consideringthe generator constraints. IEEE Trans Power System 2003;18(3):1324–32.[24] Niknam T. A new fuzzy adaptive hybrid particle swarm optimization algorithm fornon-linear, non-smooth and non-convex economic dispatch problem. Applied Energy2010;87(1):327–39.[25] Vlachogiannis JG, Lee KY. Economic load dispatch – a comparative study onheuristic optimization techniques with an improved coordinated aggregation based PSO.IEEE Trans Power Syst 2009;24(2):991–1001.[26] Passino K. Biomimicry of bacterial foraging for distributed optimization and control.IEEE Control Syst Mag 2002;16(4):52–67.[27] Panigrahi BK, Pandi VR. Bacterial foraging optimization: Nelder–Mead hybridalgorithm for economic load dispatch. IET Gener Transm Distrib 2008;2(4):556–65.[28] Nelder JA, Mead R. A simplex method for function minimization. Comput J1965;7:308–13.[29] Georgopoulou CA, Giannakoglou KC. Two-level, two-objective evolutionaryalgorithms for solving unit commitment problems. Appl Energy 2009;86 (7-8):1229–39.[30] Subbaraj P, Rengaraj R, Salivahanan S. Enhancement of combined heat and powereconomic dispatch using self adaptive real-coded genetic algorithm. Appl Energy2009;86(6):915–21.[31] Parsopoulos KE, Vrahatis MN. Particle swarm optimization and intelligence:advances and applications. New York: Information Science Reference; 2010.[32] Chaturvedi KT, Pandit M, Srivastava L. Modified neo-fuzzy neuron-based approachfor economic and environmental optimal power dispatch. Appl Soft Comput 2008;8(4):1428–38.[33] Venkatesh P, Gnanadass R, Padhy NP. Comparison and application of evolutionaryprogramming techniques to combined economic emission dispatch with line flow constraints.IEEE Trans Power Syst 2003;18(2):668–97.[34] Kyoto protocol to the United Nation framework convention on climate changeEnglish conference on the parties third session Kyoto; 1997. p. 1–10.[35] Vijay Kumar, Dr.Jagdev Singh, Dr.Yaduvir Singh and Dr.Sanjay Sood, “Design &Development of Genetic Algorithms for Economic Load Dispatch of Thermal GeneratingUnits”, International Journal of Computer Engineering & Technology (IJCET), Volume 3,Issue 1, 2012, pp. 59 - 75, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.[36] D. Pattanayaka , M. Basub and R.N. Chakrabartic, “Multi-Objective DifferentialEvolution for Optimal Power Flow”, International Journal of Electrical Engineering &Technology (IJEET), Volume 3, Issue 1, 2012, pp. 31 - 43, ISSN Print : 0976-6545,ISSN Online: 0976-6553
- 10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME52AUTHOR’S PROFILES.Bharathkumar received the B.E. degree from the Department of Electricaland Electronics Engineering, Anna University Tirunelveli from VinsChristian College of Engineering Nagercoil, in 2011, and is currentlypursuing the M.E degree in the Department of Electrical and ElectronicsEngineering in Power Systems Engineering, Anna University RegionalCentre Coimbatore. His research interests include Real time Optimization,Power Systems Optimization, Non linear Controlling Techniques, Fuzzy improvisingSystems and Digital Image Processing.A.D.Arul Vineeth received the B.E. degree from the Department ofComputer Science Engineering, Anna University Tirunelveli from SunEngineering Nagercoil, in 2011, and is currently pursuing the M.Techdegree in the Department of Information Technology, Anna UniversityRegional Centre Coimbatore. His research interests include Real timeOptimization, Cloud Computing, Soft Data Computing, and NetworkSecurity and in Fuzzy improvising Systems.K.Ashokkumar received the B.E. degree from the Department of Electricaland Electronics Engineering, Anna University Chennai from OdaiyappaCollege of Engineering and Technology Theni, in 2009, and is currentlypursuing the M.E degree in the Department of Electrical and ElectronicsEngineering Control and Instrumentation, Anna University Regional CentreCoimbatore. His research interests include Real time Control Systems, Nonlinear Controlling Techniques.K. Vijay Anand received the B.E. degree from the Department of Electricaland Electronics Engineering, Madras University from IFET EngineeringCollege Villuppuram, in 2004, and completed MBA (HRM), AnnamalaiUniversity Chidambaram in 2011 and is currently pursuing the M.E degreein the Department of Electrical and Electronics Engineering in PowerSystems Engineering, Anna University Regional Centre Coimbatore. Hisresearch interests include Real time Optimization and in Power Systems Optimization.

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