 Energy and Power En gi neering, 2011, 3, 113-119 doi:10.4236/epe.2011.32015 Published Online May 2011 (http://www.SciRP.org/journal/epe) Copyright © 2011 SciRes. EPE Economic Dispatch with Multiple Fuel Options Using CCF Radhakrishnan Anandhakumar, Srikrishna Subramanian Department of Electrical Engineering, FEAT, Annamalai University, Annamalainagar, India E-mail: anand_r1979@yahoo.com, dr_smani@yahoo.co.in Received February 6, 2011; revised March 24, 20 1 1; accepted April 2, 2011 Abstract This paper presents an efficient analytical approach using Composite Cost Function (CCF) for solving the Economic Dispatch problem with Multiple Fuel Options (EDMFO). The solution methodology comprises two stages. Firstly, the CCF of the plant is developed and the most economical fuel of each set can be easily identified for any load demand. In the next stage, for the selected fuels, CCF is evaluated and the optimal scheduling is obtained. The Proposed Method (PM) has been tested on the standard ten-generation set system; each set consists of two or three fuel options. The total fuel cost obtained by the PM is compared with earlier reports in order to validate its effectiveness. The comparison clears that this approach is a promising alterna-tive for solving EDMFO problems in practical power system. Keywords: Economic Load Dispatch, Composite Cost Function, Multiple Fuel Options, Piecewise Quadratic Function, Mathematical Model 1. Introduction The economic dispatch problem in a power system is to determine the optimal combination of power outputs for all generating units which will minimize the total cost while satisfying the load and operational constraints . The economic dispatch problem is very complex to solve because of its colossal dimension, a non-linear objective function, and a large number of constraints. Conven-tional techniques lambda iteration method and quadratic programming offer good results, but when the search space is nonlinear and has discontinuities, they become very complicated with a slow convergence ratio and do not always seek the optimal solution. New numerical methods are needed to cope with these difficulties, espe-cially those with high speed search for the optimal and not being trapped in local minima . The stochastic search algorithms such as Simulated Annealing (SA) and Genetic Algorithm (GA) have been applied to determine the optimal generation schedule for economic dispatch problem in a power system . SA is applied in many power system problems, but setting the control parameters of the SA algorithm is a difficult task, and the convergence speed is slow when applied to a real system. The GA methods have been employed to suc-cessfully to solve complex optimization problem, recent research has identified some deficiencies in GA per-formance. Particle swarm optimization method (PSO) has been applied for solving economic dispatch problems with various operating constraints . A novel optimization approach, Artificial Immune System (AIS) has been applied to solve constrained eco-nomic load dispatch problem . This approach utilizes the clonal selection principle and evolutionary approach wherein cloning of antibodies is performed followed by hyper mutation. A novel coding scheme for practical economic dispatch by modified particle swarm optimiza-tion approach has also been proposed to solve economic dispatch problem . The heuristic search technique, Differential Evolution has been suggested for solving economic dispatch problems . Bacterial Foraging- Nelder Mead method has been applied for the solution of economic dispatch problems . In certain fossil fire systems, the generation cost func-tion is represented as a segmented piecewise quadratic function. The generating unit, supplied with multi-fuel sources like coal, natural gas or oil suffers with the problem of determining the most economic fuel to burn. Such a problem has been solved using the Hierarchical Method (HM) of Lagrangian multipliers method to find the incremental fuel cost for subsystems comprising sets of units . The solution searches for the optimal for various choices of fuel and generation range of the units iteratively. A Hopfield neural network approaches to 114 R. ANANDHAKUMAR ET AL. economic dispatch problems has been proposed . An improved adaptive Hopfield Neural Network (HNN) approach has been proposed for finding the solution for economic dispatch with multiple fuel options . The HNN suffers with slow convergence rate and normally takes a large number of iterations. A hybrid real coded GA method has been presented for solving the economic dispatch problem with multip le fuel options [12 ]. An enhanced Lagrangian neural network has been ap-plied to solve the economic load dispatch problems with piecewise quadratic cost functions . In this method the convergence speeds are enhanced by employing by momentum technique and providing criteria for choosing the learning rate. Economic dispatch solutions with piecewise quadratic cost functions has been solved by using improved genetic algorithm . In order to im-prove the effectiveness of genetic algorithm multi-stage algorithm and directional crossover methods are pro-posed and projection method is introduced to satisfy a linear equality constraint from power balance. The heu-ristic search techniques such as PSO  ,Taguchi method (TM) , Evolutionary Programming (EP)  and its improved version are also been applied to solve the economic dispatch problems with multiple fuel op-tions [18,1 9] . The CCF is a non-iterative direct method, gives the most economic dispatches of the online units with less computation time. In th is paper, the CCF is used to solve the economic dispatch problem with multiple fuel op-tions. 2. Problem Formulation 2.1. Nomenclature ai, bi, ci Fuel cost coefficients of the units asj, bsj, csj Fuel cost coefficients of the set S with k fuel options A, B, C Composite cost coefficients AS, BS, CS Composite cost coefficients of the set S AP, BP, CP Composite cost coefficients of the plant FCSj Fuel cost function of the set S with k fuel options, in \$/h FCS Fuel cost function of the set S, in \$/h FCP Fuel cost function of the plant, in \$/h k Number of fuel options in a plant N Number of generation units minSP Minimum power generation of set S, in MW maxSP Maximum power generation of set S, in MW PS Economic dispatch of the set S, in MW PG Power generation of the plant, in MW PD Power demand, in MW S Set of generating units in a plant λ Incremental production cost, in \$/MWh 2.2. Economic Dispatch Problem with Multiple Fuels The main objective of economic dispatch is to find the optimal combination of power generation that minimizes the total generation cost while satisfying equality and inequality constraints. A piecewise quadratic function is used to represent the input-output curve of a generator with multiple fuel options. For a generator with k fuel options, the cost curve is divided into k discrete regions between lower and upper boun d s. The economic dispatch problem with piecewise quadratic function is defined as Minimize 1NiiiFP 2m11122221 22m1,1,,2,,,iiiiiiiiiiii iiiiiiik iik iikikiiaPbP cfuelPPPaPbPcfuelPPPFPaPbP cfuelkPPP   in1ax (1) where Fi(Pi) is the fuel cost function of the ith unit, Pi is the power output of the ith unit, N is the number of gen-erating units in the system and aik, bik and cik are cost coefficients of the ith unit using fuel type k. Minimization of the generation cost is subjected to the following constraints: 1) The power balance constraints 1NiDiPP (2) where DP is the total system demand in MW. 2) Generating capacity constraints min maxiiiPPP (3) where, and are the minimum and maxi-mum power outputs of the ith unit. miniPmaxiP 3. Composite Cost Function (CCF) The composite cost coefficients were reported in the lit-erature . 1211 11NAaa a (4) 112 2NNBbaba baA (5) 2;GAP B where GDPP2iGiibPaN where (6) 1, 2,,iCopyright © 2011 SciRes. EPE R. ANANDHAKUMAR ET AL. 11512222 2112244 4NNNCcc cbababa BA 4 (7) 2PPGPG PFCAPBPC (8) The A, B and C are the composite cost coefficients and can be easily calculated by using Equations (4), (5) and (7) respectively. For a particular load demand, the opti-mal generation of units is directly computed using Equa-tion (6). 4. Proposed Approach for Economic Dispatch with MFO The proposed methodology consist two stages of the most economic fuel identification and economic sched-uling. In the first stage, the composite cost function of the plant is developed as de tailed in the previous section. The incremental cost of the plant for a particular load demand is calculated and the generation dispatch for each unit is determined. The dispatch of each set directly indicates the most economical fuel and feasible operating region. In the second stage, the dispatch of the generating units is refined within the feasible operating region. The composite cost function is developed with the selected fuels and is solved to obtain the most economic dispatch of generating units. 4.1. Identification of the Most Economic Fuel Consider a plant consists of ‘S’ set of generation, each set consists of ‘k’ fuel options. The fuel cost function of the set ‘S’ with ‘k’ fuel op-tions is, 2min max,;1, 2,,;1,2,,SjSj SSj SSjSSSFCaPb PcPPPSNjk (9) The composite cost function of set ‘S’ is calculated by using Equa tions (4), (5) and (7). 2SSSSS SFCAPBPC (10) 1211 11SkAaa a (11) 112 2SkBbaba baAkS (12) 12222 21122444 4Skkk SCcccbababa BA S (13) In this manner, the CCF for the plant is calculated and is given as, 2PPGPG PFCAPBPC (14) The composite coefficients are constant for any load demands. The incremental production cost of the plant for the demand is calculated by 2;wherePGP GDAPBP P (15) The economic dispatch of the set ‘S’ is calculated as, 2SSSBPA (16) The above equation provides the dispatch of each set to meet the load demand. Based on this dispatch, the most economical fuel and the feasible operating region of each set can be easily identified. 4.2. Economic Dispatch of the Selected Fuels The composite cost function is developed using the most economic fuels. The generation dispatch is refined within the feasible limits as detailed in Section 3. The computational flow of the proposed methodology for solving economic dispatch problem with multiple fuel options is presented as a flow chart in Figure 1. Start Read the cost coefficients, power Ge neration limits and load Compute composite cost coefficients A, B, and C for each set and evaluate the composite cost function for the plant Compute optimum power generation required for each set to meet the load demand Ident ify t he most econ omical f uel of each set Compute composite coefficients A, B, C for the selected fuels Obtain the economic schedule and ca lculate total fu el c ost Stop Figure 1. Flow chart of the proposed method. Copyright © 2011 SciRes. EPE R. ANANDHAKUMAR ET AL. 116 5. Numerical Simulation Results and Discussion The proposed technique has been implemented in MATLAB on a 2.10 GHz core2Duo processor PC. The simulation studies have been carried out on ten-generat-ing unit system with multiple fuel options . This problem includes one objective function with ten vari-able parameters (P1, P2,, P10), one equality and twenty inequality constraints, i.e. power balance con-straint and maximum and minimum limits of each gen-erating unit. By the proposed strategy, the economic dispatch solu-tion for the given system can be obtained in two stages: 1) evaluate the composite fuel cost function for the plant and is solved to identify the most economic fuel and fea-sible operating region of each set, and 2) the optimal dispatches are calculated by solving composite fuel cost function with selected fuels. The implementation of the proposed strategy for the given system is detailed as follows. The equivalent cost function of set ‘S’ is calculated using the Equations (11), (12) and (13). In the selected sample system the number of sets in the plant is ten and the number of units is twenty nine. For example, the equivalent cost function of the set 5 is, 25550.0000448070.0493 91.5841FCP P (17) Similarly, for set 8, the equivalent cost function is, 28880.0000673690.04404 126.6098FCPP (18) In this manner, the equivalent cost function of each set is determined. By combining these cost coefficients, the equivalent cost function of the plant is calculated. The equivalent cost function of the plant is, 06 27.32870.3933 1518PGGFCe PP (19) The incremental cost of the plant is obtained by using Equation (15) . For a load demand of 2400 MW, the λ is 0.4285 and the dispatch of each set is determined by using Equation (16). Then, the dispatches of the set 5 and 8 are 278 MW and 265 MW respectively. In this manner, the dispatch of each set is identified and it indicates the most economical fuel and the feasible operating region. For the set 5, the dispatch is 278 MW, then the most economical fuel is 1 and the feasible operating region is 190 MW to 338 MW. For the set 8, the dispatch is 265 MW, then the most economical fuel is 3 and the feasible operating region is 200 MW to 265 MW. Similarly, the economic dispatch is performed to de-termine the optimal dispatches within the feasible oper-ating region by using the composite cost function of the selected fuels. During the calculation of dispatch, if the generation of any unit violates the effective limits, its generation are fixed at the violated limit. Then that unit is eliminated from the dispatch procedure. The genera-tion of all the units except the violated unit is recalcu-lated using the above procedure with total generation equal to the load demand minus the generation of the limit violated unit. The simulation is performed for various load demands of 2400 MW, 2500 MW, 2600 MW and 2700 MW. The optimal dispatches obtained by the proposed methodol-ogy and HM  for the above mentioned load demands are compared and the comparison is presented in Table 1. These results signify that the proposed CCF always pro-vides better solution than HM . Though the HM method is an iterative mathematical approach, suffers with the assumption of initial la mbda values. In addition , two operating points having same incremental cost also exist and it requires valid assumption to choose the op-timal fuel for a particular demand, improper selection will lead infeasible solution. Additionally, the proposed CCF is a direct or non iterative method, it does not de-mand any initial guess values for economic dispatch of units for the given loa d demand. The comparison of total fuel cost obtained by pro-posed methodology, HM , HNN , AHNN , HGA , Modified PSO (MPSO) , TM , Im-proved Fast EP (IFEP) , Fast EP (FEP) , Classi-cal EP (CEP) , PSO , and Improved EP (IEP)  is presented in Table 2. As seen the comparison, the gen-eration costs obtained by CCF are lowest among the re-sults. Moreover, the optimal fuel cost obtained through the CCF method is exactly same as the HGA , except the load demand of 2600 MW. However, the proposed method directly provides the optimal schedule and it utilizes the CCF for fuel selection and economic sched-uling. These comparison results confirm that the CCF provides better solution quality. The salient features of proposed approach over exist-ing methods are:  The approach is conceptually simple.  It is a non iterative method.  The simplified generalized expression directly gives the most economical fuel and the feasible operating region for a particular load demand.  It also provides the most economic schedule of gen-eration with less computational effort.  It requires negligible computational time, hence it is suitable for on line applications.  The performance of the proposed is independent of system size; hence it is suitable for system of any size. Copyright © 2011 SciRes. EPE R. ANANDHAKUMAR ET AL. Copyright © 2011 SciRes. EPE 117 Table 1. Comparison of simulation results between proposed method and HM. LOAD DEMAND = 2400 MW LOAD DEMAND = 2500 MW PM HM  PM HM  UNIT FTGEN (MW) FT GEN (MW)FT GEN (MW)FTGEN (MW) 1 1 189.7405 1 193.2 2 206.5190 2 206.6 2 1 202.3427 1 204.1 1 206.4573 1 206.5 3 1 253.8953 1 259.1 1 265.7391 1 265.9 4 3 233.0456 3 234.3 3 235.9531 3 236.0 5 1 241.8297 1 249.0 1 258.0177 1 258.2 6 3 233.0456 1 195.5 3 235.9531 3 236.0 7 1 253.2750 1 260.1 1 268.8635 1 269.0 8 3 233.0456 3 234.3 3 235.9531 3 236.0 9 1 320.3832 1 325.3 1 331.4877 1 331.6 10 1 239.3969 1 246.3 1 255.0562 1 255.2 LOAD DEMAND = 2600 MW LOAD DEMAND = 2700 MW PM HM  PM HM  UNIT FTGEN (MW) FT GEN (MW)FT GEN (MW)FTGEN (MW) 1 2 216.5442 2 216.4 2 218.2499 2 218.4 2 1 210.6058 1 210.9 1 211.6626 1 211.8 3 1 278.1441 1 278.5 1 280.7228 1 281.0 4 3 239.0967 3 239.1 3 239.6315 3 239.7 5 1 275.5154 1 275.4 1 278.4973 1 279.0 6 3 239.0967 3 239.1 3 239.6315 3 239.7 7 1 285.7585 1 285.6 1 288.5845 1 289.0 8 3 239.0967 3 239.1 3 239.6315 3 239.7 9 1 343.8134 1 343.3 3 428.5216 3 429.2 10 1 271.5861 1 271.9 1 274.9667 1 275.2 Table 2. Comparison of the total cost with different techniques. TOTAL FUEL COST (\$/h) TECHNIQUES 2400 MW 2500 MW 2600 MW 2700 MW HM  488.500 526.700 574.030 625.180 HNN  487.870 526.130 574.260 626.120 AHNN  481.700 526.230 574.370 626.240 HGA  481.7226 526.2388 574.3808 623.8092 MPSO  481.723 526.239 574.381 623.809 TM  481.6 ----- ----- 623.7 IFEP  ----- 526.25 ----- ----- FEP  ----- 526.26 ----- ----- CEP  ----- 526.25 ----- ----- PSO  ----- ----- ----- 623.88 IEP  481.779 526.304 574.473 623.851 PROPOSED METHOD 481.7226 526.2338 574.0105 623.8092 R. ANANDHAKUMAR ET AL. Copyright © 2011 SciRes. EPE 118 6. Conclusions The economic dispatch problem with multiple fuel op-tions is a complex optimization problem whose impor-tance may increase as competition in power generation intensifies. This paper presents economic dispatch prob-lem with multiple fuel options using composite cost function. The proposed CCF based solution for economic dispatch with MFO offers a best contribution in the area of economic dispatch. In contrast to the HM , this approach fully explores the cost coefficients and gives a promising value of power for providing improved eco-nomic dispatch. The HM method requires the valid as-sumptions such as initial value of lambda and it itera-tively solves the problem. The proposed methodology is a non-iterative method that directly gives the optimal generation schedule of the generating set and it does not require any assumptions. The numerical results demon-strate that the proposed approach offers a better conver-gence rate, minimum cost to be achieved and better solu-tion than the existing methods. As power systems are usually large scale systems, the proposed meth od may be suggested for the solution of economic load dispatch problems and it is also suitable for online applicatio ns. 7. 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