An Evolutionary Programming based Neuro-Fuzzy Technique for Multiobjective Generation Dispatch with Nonsmooth Fuel Cost and Emission Level Functions

Transcription

1 An Evolutionary Programg based Neuro-Fuzzy Technique for Multiobective Generation Dispatch with Nonsmooth Fuel Cost and Emission Level Functions S.K. Dash Department of Electrical Engineering, Gandhi Institute for Technological Advancement,Badaraghunathpur,Madanpur, Bhubaneswar,752054, Orissa, India Abstract A combined approach involving an EP based fuzzy coordination and an ANN(artificial neural network) methods along with a heuristic rule based search algorithm has been propounded in this paper in order to obtain the best fit optimal generation schedules for multiobective generation dispatch problem with non-smooth characteristic functions satisfying various practical constraints. Initially, the economy obective function is imized, followed by imization of emission level obective function. Then, both the obectives are mixed through a fuzzy coordination method to form a fuzzy decision making (FDM) function. Maximizing the FDM function then solves the original two-obective problem. The imization and imization tasks of this optimization problem are solved by the evolutionary programg technique and the results are trained through a radial basis function ANN to reach a preliary generation schedule. Since, some practical constraints may be violated in the preliary stage, a heuristic rule based search algorithm is developed to reach a feasible best compromising generation schedule which satisfies all practical constraints in the final stage. The proposed EP based neuro-fuzzy technique has been applied to standard IEEE-30 bus test system and the results are presented. Simulation results indicate that the accuracy and the capability of very fast computation of generation schedule by this technique seem to be very promising for its suitability for on-line multiobective generation dispatching with any kind of characteristic functions. This technique can be extended to other higher test case systems as well with suitable assumptions. Index Terms Multiobective generation dispatch, Fuzzy coordination method, Evolutionary, Programg (EP), Radial basis function ANN(Artificial Neural Network), Neuro-fuzzy technique, Heuristic Rule. I. INTRODUCTION The purpose of the multiobective generation dispatch is to generate the optimal amount of the generated power for the fossil fuel based generating units in the system by imizing the fuel cost and emission level simultaneously subect to various system constraints. In this procedure both the obectives conflict each other. Therefore, it is difficult to handle them by conventional approaches, which can optimize a single obective function. Some of the optimization techniques for multiobective generation dispatch problem such as goal programg [1], goal-attainment technique [2], classical technique based on coordination equation [3], etc. have been proposed with varying degree of success. These classical approaches need to introduce a compromising factor in order to decide the optimal solution and these results a complicated problem formulation. Further, these methods are not fast enough in terms of execution time. Moreover, they do not have a mechanism to show the vague or fuzzy preference of the human decision-maker in obtaining a compromising solution in presence of such conflicting obectives. Fuzzy systems provide tools for representing and manipulating inexact concepts and the ambiguity prevalent in human interpretation and thought processes. Further, fuzzy sets [4] can be applied for decision making in multiple obectives involving various constraints. Amongst various applications of fuzzy systems, Srinivasan, Chang and Liew [5] have proposed a fuzzy optimal search technique for a multiobective generation scheduling problem. Many interesting applications of fuzzy sets in the power field have been reported in the literature during last decade. Hota et al. [6] have described a simple and efficient technique based on fuzzy set theory for the economic emission load dispatch problem. Further, they have developed an interactive fuzzy satisfying method [7] to solve multiobective generation dispatch problem. The maor advantage of fuzzy technique applied to this kind of problems lies in having a mechanism to show the 134

2 vague/fuzzy preference of the human decision-maker in obtaining a compromising solution in the presence of conflicting obectives. However, the execution time of these fuzzy techniques appears not to be very promising for real time operation where the execution time is crucial. A very fast solution method for multiobective generation dispatch problem is only obtained by using artificial neural network (ANN). Such an approach has been described by Hota et al. [8]. They have developed an ANN based method to obtain well-coordinated economic emission load dispatch solutions suitable in terms of accuracy and speed. However, this method lacks the vague/fuzzy preference of the human decisionmaker in obtaining the compromising solutions. Therefore, an integrated approach combining a fuzzy coordination and an artificial neural network methods along has been developed recently [9]. In this work the authors have proposed a neuro-fuzzy approach, where the conventional quadratic fuel cost and emission level functions of the generators are considered, for which conventional optimization technique has been used. The maor disadvantage of this approach is its incapability of handling non smooth fuel cost and emission level functions. On the other hand, if the non-linear fuel cost and emission level functions are considered, then the use of global optimization techniques such as genetic algorithm, simulated annealing and evolutionary programg may be undertaken. Wong et al [10], have successfully applied simulated annealing to economic dispatch problems. Wong and Fung [11], have demonstrated a method to solve short-term hydrothermal scheduling problem by using simulated annealing optimizaton technique. Hota et al [12], have described a novel method for economic emission load dispatch with nonsmooth fuel cost and emission level functions using a simulated annealing based goalattainment method. Wong et al [13], have proposed a combined genetic algorithm/ simulated annealing/ fuzzy set approach to short-term generation scheduling with take-or-pay fuel contract. In their work, they have considered nonsmooth cost functions of the generating units owing to the effects of valve-point loading. In recent years, another powerful optimization technique called as evolutionary programg (EP) is being continuously applied to various power system optimization problems due to its more powerful ability in finding the global optimum solutions as compared to genetic algorithm or simulated annealing technique. Yang et al [14], have developed an efficient general economic dispatch algorithm for units with nonsmooth fuel cost functions based on EP technique. In this work the authors have compared the results of ED problems when solved by genetic algorithm, simulated annealing and EP. They have shown that the EP method is able to give a cheaper schedule at a less computation time. Hanzheng et al [15], described a solution method for unit commitment using Lagrangian relaxation combined with evolutionary programg. Hota et al [16], have developed an evolutionary programg based algorithm for solution of short-term hydrothermal scheduling problem. They have also shown that when compared to simulated annealing based algorithm for short-term hydrothermal scheduling, EP based algorithm is able to obtain a cheaper hydrothermal schedule at reduced execution time. In this paper, an evolutionary programg (EP) based neuro-fuzzy technique is proposed to solve the multiobective generation dispatch problem with nonsmooth characteristic functions i.e., fuel cost and emission level functions. The developed EP based algorithm is tested on IEEE 30-bus test system and the results are presented. In order to verify the successful working of the proposed EP based algorithm, numerical results obtained from EP based algorithm are compared with those obtained from previous conventional optimization algorithm when the quadratic fuel cost and emission level functions are considered. Simulation results show that the proposed EP based neuro-fuzzy approach is capable of not only solving the multiobective generation dispatch problem with any type of fuel cost and emission level functions, analytical or empirical curves, but also obtaining the very fast global or near global compromising solution. II. PROBLEM FORMULATION The present formulation in this dissertation work treats the multiobective generation dispatch problem as a imization problem which is concerned with the attempt to imize each obective simultaneously. The equality and inequality constraints of the system must meanwhile be satisfied. The multiobective generation dispatch problem has been formulated as the following two-obective optimization problem that deals with the cost of generation and emission level as obective functions. The generating units involved are all thermal units and assumed operating on-line throughout. Equality and inequality system constraints as well as transmission loss have also been included in the problem formulation for completeness of the problem under study. A. Cost of generation Considering a system having N buses and NL lines let the first NG buses have sources for power generation. Taking into account the valve-point effects, the fuel cost function of each generating unit is expressed as the sum of a quadratic and a sinusoidal function Therefore, the 135

3 total cost of generation C in terms of control variable s can be expressed as: f 1 C NG i sin( ( )) R hr ai i bi i ci d i ei i i where, i is the real power output of an i th generator, NG denotes the number of generators and a bi ci d,,, i i ei, are the fuel cost curve coefficients of an ith generator. B. Emission level The combustion of fuel used in fossil based generating units gives rise to four basic forms of emission. Those are oxides of sulphur (SOx), oxides of nitrogen (NOx), carbon dioxides (CO2), and particulates. In the present work, however, all the four forms of emission are treated together as a single emission criterion. The amount of emission from a fossil-based generating unit depends upon the amount of power generated by that unit which is the sum of a quadratic and an exponential function in the present work [13]. Therefore, the total emission level E from all the generating units in the system then can be expressed as: f 2 E where, 2.5 exp( ) lb hr NG 0 i i i i i i i i1,, i, i i i i i, are the emission curve coefficients of the ith generating unit. C. Equality and inequality constraints The following equations and inequalities are satisfied in the present formulation of multiobective generation dispatch problem. Generator load balance The real power balance between generation and the load must be maintained at all time while assug the load at any time as constant. NG i i1 P D P L (3) where, PD is the estimated real power demand and PL is the total transmission system loss of the real power. The total system real power transmission loss is represented as: Ai i NG (4) PL i1 where, Ai are the loss coefficients and are evaluated from base load flow solutions. Evaluation of these coefficients is very fast and simple unlike the evaluation of conventional B-coefficients which is more involved and time consug. The effectiveness and validity of this loss coefficient formulation for generation dispatch problem has been well established. Moreover, these loss coefficients can also be updated on a real-time basis with the change in the system operating condition. Lower and upper limits of generator output Each generating unit is constrained by its lower and upper limits of real power output as shown below to ensure stable operation. i i 2 i (5) where, i and i are the imum and imum real power output of ith unit, respectively. III. EVOLUTIONARY PROGRAMMING TECHNIQUE Evolutionary programg (EP) is a powerful general-purpose technique for solving complex real-world optimization problems. It is also a stochastic optimization technique and can search for global optimum solution. Like genetic algorithm (GA), this technique works on population of trial solutions, imposes random changes to those solutions to create offsprings, and incorporates the use of selection to detere which solutions to maintain into future generations and which are to be removed from the pool of trials [17]. But in contrast to GA, the individual component of a trial solution in EP technique is viewed as a behavioral trait, not as a gene. In other words, EP technique emphasizes the behavioral link between parents and offsprings rather than the genetic link. It is assumed that whatever genetic transformation occurs, the resulting change in each behavioral trait will follow a Gaussian distribution with zero mean difference and some standard deviation. The key feature of EP is in its probabilistic nature of selection by conducting a stochastic tournament for survival at each generation. The probability that a particular trial solution will be maintained is made a function of its rank in the population. The production of an offspring population is called a generation. Many 136

4 such generations are required for the population to converge to an optimum solution, the number increasing according to the problem difficulty. In the EP algorithm the imum number of generations, i.e., imum number of iteration is defined. A. Implementation of the EP Algorithm The implementation of the algorithm is done for following three cases of optimization. a) Minimization of total generation cost, i.e., f1 in (1). b) Minimization of total emission level, i.e., f2 in (2). c) Maximization of fuzzy decision making function, FDM in (18) to be discussed in the following section. a) First select arbitrarily a dependent generating unit from among the committed NG units. Let the unknown generation be the dependent d generation. The can be calculated by d assug that the non-dependent generations, i.e., the for = 1, 2,, NG but, are known. Further, since the power loss is a function of the generation outputs and system topology, to detere the output of dependent unit, d A-loss coefficients are also required as shown in (4). Therefore, is calculated as: d d P D P L NG 1 d (6) In detering the optimal generation schedule for the ELD problem according to the above mentioned problem solving formulation, the main obective is to detere the non-dependent generations which have been assumed to be known by some method. In this work, evolutionary programg algorithm has been applied to detere the non-dependent generations and hence, the global optimal generation schedule with the imization of total generation cost has been obtained. The EP technique implemented to solve the economic load dispatch problem is stated in the following subsections. (i) Representation of trial solution vector According to the formulation for solving the problem, a dependent generation from d committed generator, is randomly selected. The generations from non-dependent generators, i.e., for = 1, 2,, NG, d are together taken as a (NG-1)-dimensional trial vector. Let,,...,,,..., ] [ 1 2 ( d 1) ( d 1 P ) NG be the trial vector designating the ith individual of a population to be evolved. (ii) Initialization of a population of trial vectors (Parents) Taking the population size to be NP, each initial parent trial vector, i = 1, 2,, NP, is selected at random from a feasible range in each dimension. This is done by setting the th component of each parent as: rand[, ] for = 1, 2,, (d-1),(d+1),, NG (7) where, rand [, ] denotes a uniform random variable ranging over [, ]. (iii) Generation of offspring population An offspring is generated according to the relative value of the obective function f() associated with the trial vector. If f() is relatively low, the offspring trial solution is generated near the current parent solution. On the other hand, if the f() is relatively high, the is will be searched within a wider range. To generate an offspring from each parent, a Gaussian random variable with zero mean and standard deviation proportional to the scaled cost values of the parent trial solution is added to the each component of as given by the following 137

5 expressions: (8) [ 1, 2,..., ( d 1), 2 ( d 1),..., and M (0, ) for = 1, 2,, NG, and d (9) where, represents a Gaussian random variable with mean zero and standard deviation. The standard deviation indicates the range of the offspring generated around the parent trial solution and is given by: ( P f f ) i ( ) (10) where, f is the imum cost value among the NP trial solutions and is a scaling factor. (iv) Competition and selection After generation of offspring population, competition and selection procedure is implemented to detere which solutions are to be maintained into the next generation and which are to be removed from the competing pool of trials. The NP parent trial vectors and their corresponding NP offsprings compete with each other in the competing pool for survival. To do this a competitor Pr is selected at random from among the 2NP trial solutions, where r is an integer as given by: N P rand r [ 2 [0,1] 1] (11) 1 In the above equation, rand1[0,1] is an random number ranging over [0,1] and value of r is taken to be the greatest integer less than or equal to the value of the expression in the right hand side. After a stochastic competition, the score for each trail vector is calculated as: N P w wm m1 (12) NG ] and wm = 1, if rand2[0,1] > = 0, otherwise. f ( P f ( r ) ) f ( where, rand2[0,1] is another uniform random number generated between 0 and 1. After the competition is over, the 2NP trial solutions in the competing pool are sorted according to their obtained scores from highest to the lowest. Thereafter, the first NP trial solutions from the sorted pool are selected as the new parent vectors for the next generation. (v) Stopping rule The iterative procedure of generating new trials by selecting those with imum function values from the competing pool consists of equal number of parents and offsprings is terated when there is no significant improvement in the solution. It can also be terated when a preset number of iteration is reached. In the present work, the latter method is employed. The initial values of all components (nondependent generations) of each parent are specified or generated at random before starting the process of evolution. Consequently, the dependent generation is calculated. Thereafter, all the generation levels are checked against their corresponding limits and the generator-load balance is checked. If all the constraints are satisfied, then the current non-dependent generations are taken as the components of the final feasible parent. If otherwise then, using the current value of the dependent generation, all the generation levels are again calculated and all the constraints are checked till all of them are satisfied. The process of generating feasible parent vectors continues till the iteration count equals NP. Similar checking of constraints is performed for generation of each feasible offspring. At the end of the solution process the trial vector with imum function value among the NP trial vectors will be the global optimum solution. (b) In this case same EP procedure is adopted for the emission level function imization as shown in (2). ) 138

6 (c) In many problems, the obective is more naturally stated as the imization of some cost function g(x) rather than the imization of some utility or profit function u(x). Even if the problem is naturally stated in imization form, this alone does not guarantee that the utility function will be nonnegative for all x as we require in fitness function. As a result, it is often necessary to map the underlying natural obective function to a fitness function form through one or more mappings. The duality of cost imization and profit imization is well known. In normal operational research work, to transform a imization problem to a imization problem we simply multiply the cost function by a us one. In evolutionary programg work, this operation alone is insufficient because the measure thus obtained is not guaranteed to be nonnegative in all instances. With evolutionary programg algorithm, the following cost-to-fitness transformation is commonly used: f ( x) C g( x) when g(x) < C, (13) The basic block diagram of the proposed EP-based neuro-fuzzy technique for multiobective generation dispatch with nonsmooth characteristic functions has been shown in Fig. 1. Initially, the economy obective function, i.e., the cost of generation of the multiobective generation dispatch problem is imized, followed by imization of emission level obective function using the global optimization technique namely evolutionary programg. Then, both the obectives are combined through a fuzzy coordination method to form a fuzzy decision making (FDM) function. The original twoobective problem is then solved by imizing the FDM function by using evolutionary programg technique. After this optimization, the results are trained by a radial basis function ANN to reach a preliary generation schedule. Since, some practical constraints may be violated in the preliary schedule, a heuristic rule based search algorithm is developed to reach a feasible best compromising generation schedule which satisfies all practical constraints in the final stage. = 0 otherwise. There are a variety of ways to choose the coefficient C. It may be taken as an input coefficient, as the largest g value observed thus far, as the largest g value in the current population, or the largest of the last k generations. Perhaps more appropriately, C should vary depending on the population variance. The fuzzy decision making imization problem is transformed into a general imization problem by using Equation - 13 which is shown below: f() = C FDM() when FDM < C, (14) = 0 otherwise. where, C is taken as an input coefficient for simplicity. The EP procedure as described for case (a) remains same except the obective function which is replaced by the above function as described in Equation 14. IV. THE PROPOSED EP - BASED NEURO- FUZZY TECHNIQUE Figure 1: Basic block diagram of proposed EP based neuro-fuzzy technique for multiobective generation dispatch with nonsmooth characteristic functions A. Fuzzy coordination method In this method, the fuzzy decision making function [4] is represented by introducing the membership function in the fuzzy set theory. The idea of the membership function is to replace the concept that each variable has a precise value. Rather, each variable is assigned a degree of membership for each possible value of the variable. Fig. 2 depicts the membership function *c for the fuzzy variable signifying total fuel cost fc. This function describes numerically how the decision-maker is satisfied by which level of the index chosen. The decision-maker is fully satisfied with the cost if *c = 1 and not satisfied at all if *c = 0. Therefore, the value of the membership function indicates the adaptability of the economy index. 139

7 1 c Maximize FDM (18) subect to where, : NG-dimensional vector of decision variables, : the set of feasible solutions B. Procedure of neuro-fuzzy approach 0 f f f c cm cd Figure 2: The membership function for fuzzy fuel cost f c The membership function for an ith obective function f i () is defined as: f id fi ( ) i ( ) fid fim (15) where, f im is the imum permissible value of the obective function assumed to be known previously and the parameter f id is the least permissible desired value beyond which the obective is unsatisfactory for the decision-maker. The fuzzy decision making function (FDM) for ith obective is defined as below. 0, i () 0 FDM i ()= i (), 0 i () 1 (16) 1, i () 1 Consequently, FDMi becomes 1 when the ith obective value is most desirable, and it is 0 (zero) when the obective value is most undesirable. The combined fuzzy decision making function (FDM) is obtained as: FDM 2 FDM i ( ) i1 (17) The optimal (best compromising) solution of the multiobective generation dispatch problem is obtained by solving the following optimization: As shown in Fig. 3, the design procedure of the proposed integrated approach consisting of EPbased fuzzy coordination and ANN methods along with a heuristic rule based search algorithm for optimal solutions involves four maor steps, viz. training set creation, training, testing and heuristic search. In the proposed approach, the imum cost of generation and imum emission level are calculated by evolutionary programg technique. Then the optimization of economyemission is done by evolutionary programg based fuzzy coordination method. For deteration of generation dispatches of thermal units, neural networks of supervised learning are needed. This is because, the optimal generation schedule of the thermal units (outputs) for each total system load demand (input) in the training set are required to be known in advance by some suitable method. A radial basis function ANN called as RBANN is employed in the present work for training and testing due to its auto configuring architecture and faster learning ability. The EPbased fuzzy coordination method as described earlier has been applied to create the necessary training set. In the training process, the RBANN is presented with a series of pattern pairs; each pair consists of an input pattern and a target output pattern. The training pattern p is described by: t(p) = {( input (p) ), ( output (p) )} = {( PD (p)), ( 1 (p), 2 (p),..., NG (p) )} (19) The sum of the squared errors (SSE) between the actual and the desired (target) outputs over the entire training sets is used as the measure to find out the convergence of the network. The RBANN 140

8 used is trained by the orthogonal least squares learning algorithm. Training is continued until the given error-goal in terms of SSE is reached. Once the RBANN is trained, there after only the Steps 3 and 4 are used to obtain the optimal solutions of multiobective generation dispatch for any given load P D. In the Step-3 only a preliary generation schedule is obtained since, the practical constraints of lower and upper limits of real power generation outputs of the generators may be violated in the preliary schedule. Therefore, a heuristic rule based search algorithm is developed in the Step-4 to reach a feasible best compromising generation schedule, which satisfies all the practical constraints. C. Heuristic rule based search algorithm for deteration of final schedule In this work the following heuristic rules are applied to refine the preliary schedule and to reach the final best compromising generation schedule. generator exceeds their limits, then also the heuristic rule based search algorithm may be extended in similar ways. i) Removed generation case: For an n-generator case, let the generation level of th generator is such that >. So, the removed generation is - and accordingly, an amount ( - )/(n-1) is added to each generation level of remaining n-1 generators i.e., i for i = 1 to n and i. ii) Added generation case: For an n-generator case, let the generation level of th generator is such that <. So, the added generation is - and accordingly, an amount ( - )/(n-1) is subtracted from each generation level of remaining n-1 generators i.e., i for i = 1 to n and i. Step 1. Training set creation i) Heuristic rule on lower limits of generators Let i, if i i i for i =1,2,,NG (20) ii) Heuristic rule on upper limits of generators Let i, if i i i for i =1,2,,NG (21) When the ANN output for a particular generator either crosses lower limit or upper limit, the generation is fixed at its corresponding limit. The removed generation (in case of exceeding upper limit) or added generation (in case of exceeding lower limit) is so small that even if it is neglected, then also the percentage error is very much well within the acceptable limits for all practical purposes. However, in this work, the removed generation or added generation of a generator is equally shared by remaining generators accordingly as described below. If more than one Figure 3: Design procedure of the proposed EPbased neuro-fuzzy technique for multiobective generation dispatch problem 141

9 V. SYSTEM STUDIES The proposed EP based neuro-fuzzy technique has been applied to the IEEE-30 bus test system. The 30-bus test system consists of three generators in the first three bus and 40 transmission lines. Table 1 summarizes the operating limits of the three generators. The fuel cost function and the emission level function data are given in Table 2 and Table 3, respectively. A base case load of 240 MW has been considered for 30-bus system. The loss coefficients are evaluated from the base case load flows. Table 2 Fuel cost function coefficients A. Simulation studies on 30-bus test system Prior to applying the EP based fuzzy coordination method to the multiobective generation dispatch problem with nonsmooth fuel cost curves of the generators, it has been initially applied to the problem with conventional quadratic fuel cost and emission level functions of the generators to prove its satisfactory working. The control parameters of the EP algorithm are imum iteration number, population size and scaling factor and the most appropriate values of these parameters are set to 500, 50 and 0.001, respectively. These values are obtained after testing and evaluating different combinations. For a load demand of 240 MW the most economical, imum emission and best compromising solution using fuzzy coordination method were computed by sequential quadratic programg approach as described in Reference-9 and the proposed EP approach, and the comparison is shown in Table 4. From this table it is observed that results obtained from both the methods closely match with each other. This indicates that the generation dispatch results can be accurately obtained by the proposed EP approach. In order to explore the converging characteristics of the EP, different random initial solutions were given to the proposed EP algorithm along with the above mentioned control parameters for the most economical solution corresponding to a load demand of 240 MW. The optimal solutions corresponding to each random initial solution (trial) are observed. The total cost variation of the most economical generation schedule obtained from proposed EP approach when executed 10 times with different random initial solutions were observed. It was found that about 90% of the solutions after execution of each trial were converged approximately at the global optimum solution. This further indicates that the EP technique has more powerful ability to achieve the global optimum solution. In this work, all the imization and imization tasks are performed by using the evolutionary programg approach that represents the most powerful tool for global optimization. The obectives f 1() and f 2() are imized separately from 80% to 120% in steps of 5% of base case load of 240 MW to obtain the most economical and imum emission solutions, respectively, considering loss and the results are presented in Tables 5 and 6, respectively. It is observed from Tables 5 and 6 that the generator allocations are not coincident. This may be accounted from the fact that they are optimized based on different performance indices. At base load the total fuel cost is R/h from most economical solution but increases to R/h corresponding to imum emission solution. The total emission level from most economical solution is found to be lb/h but it decreases to lb/h corresponding to the imum emission solution. Thus, Tables 5 and 6 clearly demonstrate the conflicting nature of the two obective functions. The total fuel cost found from imum emission solution, i.e., R/h is set as the imum of the desired value f1d of the obective f1(). The total fuel cost found from most economical solution, i.e., R/h is set as the imum permissible value f1m of the same obective. Similarly, f2d and f2m correspond to the total emission level from the most economical and imum emission solutions, respectively. These solutions are lb/h and lb/h, respectively, for the present case load demand. In this method the range of the obective values are fixed in this way to guarantee the generation of noninferior solutions. However, after setting the fim as described above, the decision-maker may set the value of fid as per his/her satisfaction. It is to be noted here that, fid must be less 142

10 than or equal to fid and certainly more than the fim. The following test case is computed to demonstrate the applicability of the developed algorithm as given in previous section in obtaining the best compromising solutions. In this work no priority of the obectives is assumed, and the desired values of the obectives are set as f1d = f1d and f2d = f2d. The computations were carried out according to the procedure given in section-4.1 for the load values from 80% to 120% of base load in steps of 5%. The best compromising solutions consisting of optimum generations using fuzzy coordination method are presented in Table 7. Performance indices obtained from best compromising solutions by fuzzy coordination method were compared with those obtained from most economical and the imum emission solutions in Tables 5 and 6. It is clear from these tables that the best compromising solutions force both the performance indices to remain in between those obtained from most economical and the imum emission solution procedures as expected. Table 4 Comparison of generation dispatch solutions obtained by Sequential Quadratic Programg and Evolutionary Programg Techniques Table 6 Minimum Emission solutions for 30-bus system Table 7 Best compromising solutions using fuzzy coordination method for 30-bus system Table 5 Most economical solutions for 30-bus system A radial basis function ANN model, namely RBANN is designed for the 30-bus test system. There is only 1 input node (load demand) for the model. The optimal loads of the thermal units in the system, i.e.,,..., are the output nodes. Therefore, there are 3 output nodes for the RBANN. The number of neurons in the single hidden layer is equal to the number of iterations required for training and is set adaptively for RBANN. It is not unusual to get good performance on training data followed by much worse performance on test data. This can be guarded against by ensuring that the training data are uniformly distributed. The cases used to train the networks are as follows: PD is taken as base load i.e., 240MW. The range of load demand is chosen between 80% to 120% of base load in steps of 5% for RBANN. Therefore, 9 different training patterns were generated covering the system load from 192 MW to 288 MW. The training patterns are already given Table 7. The RBANN was trained with its corresponding 9 patterns to reach the error-goal (convergence target) which was SSE = RBANN required only 9 iterations in reaching the convergence target. To achieve the best performance on the test data and good generalization an appropriate value of spread factor (SF) is set. Computations were carried out for different values of SF to find the best value as per the guideline given in Reference-18. For a given set of test patterns the percentage mean absolute error (% MAE) is recorded for each value of SF. Then the value of SF 143

11 corresponding to the imum of the % MAE is taken as the best value of SF. The best SF is found to be 18 for RBANN. For the performance evaluation of the proposed neuro-fuzzy technique, 4 numbers of test cases (load levels other than those in training sets but within 80% to 120% of the base load) are considered. These test cases were generated by fuzzy coordination method. The test cases were computed by the RBANN, which was trained earlier taking the best value of SF i.e., 18. The final optimal generation schedule obtained from the RBANN along with heuristic rule based search algorithm was compared with those obtained from fuzzy coordination method and the %Error= 3 i1 F i 3 NF i i1 i1 3 F i 100 was also computed where F i and NF i are generation schedule obtained from fuzzy coordination method and proposed neuro-fuzzy technique, respectively. The comparison of best compromising generation dispatch solutions between fuzzy coordination method and the neuro-fuzzy technique are shown in Table 8. From this table it is observed that the generation schedule obtained from neuro-fuzzy technique closely matches to that of the fuzzy coordination method. Table 8 Comparison of best compromising generation dispatch solutions between the fuzzy coordination method and EP based neuro-fuzzy technique for 30-bus system The most significant advantage of the proposed neurofuzzy technique is that once the RBANN is trained for a given range of load levels of a multiobective generation dispatch problem then, the computation of best compromising generation schedule corresponding to a new load demand only requires Steps 3 and 4. It may be noted that both Step-1 and Step-2 require relatively lengthy computational effort while that of Steps 3 and 4 require only fraction of a second. However, it is significant to note that the first two steps are simulated off-line only. Present case studies on the test system demonstrates that the absolute % error in scheduling found to be much less than even 1% and when computed on a 2.4 GHz P-IV machine. All the computer programs were implemented using MATLAB 6.1 and run on a Pentium-IV PC with Windows 98 operating system. The proposed EP based fuzzy coordination method required about an average time of 9 utes of total computer time to obtain the best (global optimum) multiobective generation schedule with non-smooth characteristic functions for each load level. But, the average execution time (Steps 3 and 4 of Fig. 3) neurofuzzy method for a given load demand is found to be only 0.1 sec. Therefore, the accuracy and the capability of very fast computation of generation schedule of the proposed EP based neuro-fuzzy technique seem to be very promising for its suitability for on-line multiobective generation dispatching with non-smooth characteristic functions. VI. CONCLUSION An integrated approach combining an evolutionary programg based fuzzy coordination and an artificial neural network methods along with a heuristic rule based search algorithm has been developed in this paper to obtain the best compromising generation schedules for multiobective generation dispatch problem with nonsmooth characteristic functions, satisfying various practical constraints that are suitable both in terms of speed and accuracy, while allowing more flexibility in operation. Initially, the economy obective function is imized, followed by imization of emission level obective function. Then, both the obectives are combined through a fuzzy coordination method to form a fuzzy decision making (FDM) function. Maximizing the FDM function then solves the original two-obective problem. After this optimization, the results are trained by a radial basis function ANN to reach a preliary generation schedule. Since, some practical constraints may be violated in the preliary schedule, a heuristic rule based search algorithm is developed to reach a feasible best compromising generation schedule which satisfies all practical constraints in the final stage. Simulation results indicate that the accuracy and the capability of very fast computation of generation schedule of the proposed EP based neuro-fuzzy technique seem to be very promising for its suitability for on-line multiobective generation dispatching with any kind of characteristic functions. The imization and imization tasks of the optimization problem considered are solved by the evolutionary programg technique. There are two advantages to use EP: first, the output can be represented exactly, secondly, comparing to genetic algorithms the time-consug encodingdecoding manipulations are avoided. Though applied to a moderate size test system, this technique may be applied to large size systems effectively. Further, in the present case studies only two obective functions such as fuel cost and emission level functions are taken. In 144

12 future works obective functions like security, reliability, etc., may be addressed while solving multiobective generation dispatch problems through neuro-fuzzy technique. VII. REFERENCES [1] J.Nanda, D.P.Kothari and K.S.Lingamurthy, Economic Emission Load Dispatch through Goal Programg Technique, IEEE Trans. on Energy Conversion, Vol.3, 1988, pp [2] P.K.Hota, R.Chakrabarti & P.K.Chattopadhyay, Multiobective Generation Dispatch using Goalattainment Method, Journal of Institution of Engineers(India), pt. EL, Vol.82, September-2001, pp [3] J.Nanda, H.Lakshman and M.L.Kothari, Economic Emission Load Dispatch with Line Flow Constraints using a Classical Technique, IEE Proc.- Generation Transmission and Distribution, Vol.141, No.1, 1994, pp [4] H. J. Zimmermann, Fuzzy set theory and its applications, Kluwer-Nihoff publishing, [5] D.Srinivasan, C.S.Chang & A.C.Liew, Multiobective Generation Scheduling using Fuzzy Optimal Search Technique, IEE Proc.- Generation Transmission and Distribution,1 Vol.141, No.3, May- 1994, pp [6] P.K.Hota, R.Chakrabarti & P.K.Chattopadhyay, A Fuzzy-Set Based Optimization Technique for Economic Emission Load Dispatch, Journal of Institution of Engineers(India), pt. EL, Vol.80, November-1999, pp [7] P.K.Hota, R.Chakrabarti & P.K.Chattopadhyay, Economic Emission Load Dispatch through an Interactive Fuzzy Satisfying Method, Electric Power Systems Research, Vol.54(3), 2000, pp [8] P.K.Hota, R.Chakrabarti & P.K.Chattopadhyay, An Integrated Approach to Economic Emission Load Dispatching using Neural Network and Goal-attainment Methods, Electric Machines and Power Systems, Vol.27, No.10, 1999, pp [9] P.K.Hota, R.Chakrabarti & P.K.Chattopadhyay, Multiobective Generation Dispatch through a Neuro- Fuzzy technique, Electric Power Components and Systems, Vol. 32, No. 11, 2004, pp [10] K.P.Wong and C.C.Fung, Simulated annealing based economic dispatch algorithm, IEE Proc.- Generation Transmission and Distribution, Vol.140, No.6, November-1993, pp [11] K.P.Wong and Y.W.Wong, Short-term hydrothermal scheduling Part-I: Simulated annealing approach, IEE Proc.- Generation Transmission and Distribution, Vol.141, 1994, pp [12] P.K.Hota, R.Chakrabarti and P.K.Chattopadhyay, A simulated annealing-based goal-attainment method for economic emission load dispatch with nonsmooth fuel cost and emission level functions, Electric Machines and Power Systems, Vol.28, No.11, 2000, pp [13] K.P.Wong and Suzannah, Y.W.Wong, Combined genetic algorithms/ simulated annealing/ fuzzy set approach to short-term generation scheduling with takeor-pay fuel contract, IEEE Trans. on Power Systems, Vol.11, No.1, 1996, pp [14] H.Yang, P.Yang and C.Huang, Evolutionary programg based economic dispatch for units with non-smooth fuel cost functions, IEEE Trans. on Power Systems, Vol.11, No.1, 1996, pp [15] H.Duo, H.Sasaki, T.Nagata and H.Fuita, A solution for unit commitment using Lagrangian relaxation combined with evolutionary programg, Electric Power Systems Research, Vol.51, 1999, pp [16] P.K.Hota, R.Chakrabarti and P.K.Chattopadhyay, Short-term hydrothermal scheduling through evolutionary programg technique, Electric Power Systems Research, Vol.52, 1999, pp [17] D.B.Fogel, An introduction to simulated evolutionary optimization, IEEE Trans., Neural Networks, Vol.5, No.1, 1994, pp [18] H.Desmuth and M.Beale, Neural Network Tool Box User s Guide, Math Works, Natick, MA,