An Artificial Neural Network Method For Optimal Generation Dispatch With Multiple Fuel Options

Transcription

1 An Artfcal Neural Network Method For Optmal Generaton Dspatch Wth Multple Fuel Optons S.K. Dash Department of Electrcal Engneerng, Gandh Insttute for Technologcal Advancement,Badaraghunathpur,Madanpur, Bhubaneswar,752054, Orssa, Inda Emal: Abstract Ths paper presents an artfcal neural network method to solve the optmal generaton dspatch problem wth multple fuel optons. Tradtonally, n the optmal generaton dspatch problem, the cost functon for each generator s approxmately represented by a sngle quadratc functon. However, t s more realstc to represent the generaton cost functon for fossl fred plants as a segmented pecewse quadratc functons, as n the case of valve pont loadng. Some generaton unts especally, those unts whch are suppled wth multple fuel sources (gas and ol) are faced wth the problem of determnng whch s the most economcal fuel to burn. The proposed method has applcatons to fossl fred generatng unts capable of burnng gas and ol, etc., as well as other problems whch result n multple ntersectng cost curves for a partcular unt. An advantage of ths method s the capablty to optmze over a greater varety of operatng condtons. The smulaton results show that the soluton method s practcal and vald for real tme operaton. Index Terms Optmal generaton dspatch, Multple fuel optons, Hybrd cost functon, Radal bass functon network, Hopfeld neural network. I. INTRODUCTION Tradtonally n economc load dspatch problem, the cost functon for each generatng unt has been approxmately represented by sngle quadratc functon [1]. It s more realstc, however, f the cost curve of each generatng fossl-fred unt can be represented as a segmented pece-wse quadratc functons [2], as n the case of valve pont loadng. For pece-wse quadratc cost functon, the generatng unts are suppled wth multple fuel sources such as (1) gases or gases wth dfferent heat content, (2) coal or dfferent heat content of coal and (3) ol or dfferent heat content of ol. Thus, the cost functon of any fossl-fred unt can be parttoned nto dfferent segments for multple fuels. Each segment for mult-fuel cost functon s beng assocated wth dfferent types of fuel. For ths reason any or sngle unt for mult-fuel opton can be burnt wth at least two types of fuels. Some generaton unts especally, those unts whch are suppled wth multple fuel sources (gas/ ol/ coal, etc.) are faced wth the problem of determnng whch s the most economcal fuel to burn. As fossl fuel costs ncrease, t becomes even more mportant to have a good model for the producton cost of each generator. In ths work, the pece-wse quadratc functon s used to represent multfuel whch s avalable to each generatng unt. For any gven unt wth multple cost curves, these curves can be supermposed as shown n Fg.1. The resultng cost functon s known as hybrd cost functon [3]. The hybrd cost functon s known as pece-wse cost functon also. The hybrd ncremental cost functon can be obtaned from hybrd cost functon whch s ndcated n the Fg.1. The economc load dspatch for multplefuel generaton schedule of any unt s done n such a way that the fuel cost s at mnmum level,.e., burnng of fuel of each unt s done economcally. Therefore, an effcent method whch s to be used to obtan the generaton schedule of each unt for such type of ELD problem s needed to be developed. There has been a growng nterest n neural network models wth massvely parallel structures, whch mmc to resemble the human bran [4]. Owng to the powerful capabltes of neural networks such as learnng, optmzaton and fault-tolerance, neural networks have been appled to the varous felds of complex, non-lnear and large-scale power systems [5-7]. Novak [8] has descrbed the varous felds of power system where the Radal bass neural network can be appled successfully. J.Park and I.W.Sandberg [9] have descrbed the radal bass functon networks as unversal tool for functon 126 ISSN (Prnt) : , Volume-2, Issue-1, 2013

2 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) approxmaton. The most promsng advantage of ths network over back propagaton neural network s ts auto-confgurng archtecture. Besdes, the tranng tme for a practcal szed problem n case of Radal bass functon network s sgnfcantly less as compared to that of the back-propagaton network. On the other hand, the Hopfeld neural network [10-11] has been appled to varous felds snce Hopfeld proposed the model n In the problem of optmzaton, the Hopfeld neural network has a well demonstrated capablty of fndng solutons to dffcult optmzaton problems. The TSM (travelng salesman problem)[12], typcal problems of NP (nondetermnstc polynomal)- complete class, A/D converson, lnear programmng and job-shoppng schedule are good examples [13-14] whch the Hopfeld network provdes wth solutons. In the feld of power systems, the Hopfeld network has been appled to unt commtment [15], and economc load dspatch problems [16-18]. Ths paper partcularly presents a new method to solve the problem of optmal generaton dspatch wth multple fuel optons usng a Radal bass functon neural network along wth a heurstc rule based search algorthm and a Hopfeld neural network. The smulaton results show that the soluton method s practcal and vald for real-tme operaton. An advantage of the proposed method s ts capablty to optmze over a greater varety of operatng condtons. The proposed ANN method has applcatons to fossl fueled generaton unts capable of burnng coal, gas, and/or ol as well as other problems whch result n multple ntersectng cost curves for a partcular generatng unt. dfferent heat contents. In general, these lead to hybrd cost functons and hybrd ncremental cost (IC) functons as gven below [18]. The notatons of the followng hybrd cost functons and hybrd ncremental cost functons are: Pmn s the mnmum power generaton lmt of jth unt wth fuel (1). P1 s the maxmum power generaton lmt of jth unt wth fuel (1).... Pn-1 s the mnmum power generaton lmt wth fuel (n), and Pmax s the maxmum power generaton lmt of jth unt wth fuel (n). Subscrpt j ndcates generatng unt number and subscrpt k ndcates type of fuel. ajk, bjk and cjk are cost curve coeffcents of jth generatng unt wth fuel (k). Cost (Pj) s the hybrd cost functon and IC(Pj) s the hybrd ncremental cost functon of jth generatng unt. Fgure 1: Multple cost curves II. PROBLEM FORMULATION The modern power system experences that the cost functons of fossl-fuel fred generatng unts are found to be pece-wse quadratc functons. The pece-wse quadratc cost functons are generally equpped wth ether the provson of mult-fuels or ol/ gases wth III. NUMERICAL COMPUTATIONAL ALGORITHM The followng steps are requred for developng an algorthm to solve the economc load dspatch problem wth multple fuel optons. Step-1. Read the values of hybrd cost coeffcents ajk, bjk and cjk of each unt along wth ther maxmum and mnmum generatng capacty correspondng to each fuel and also total load demand (PD), where j denotes unt number and k denotes type of fuel opton. 127 ISSN (Prnt) : , Volume-2, Issue-1, 2013

3 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) Step-2. Intally assume fuel-1 opton for all unts,.e., k =1.Step-3. Compute jk and jk usng equaton (5),.e., 11=1 and 11= 0 2k= c1k/c2k and 2k= (b1k-b2k)/2c2k 3k= c1k/c3k and 3k= (b1k-b3k)/2c3k : : : : : : mk= c1k/cmk and mk= (b1k-bmk)/2cmk Step-4. Compute the generaton of unt-1 usng equaton (11),.e., and also compute generaton of other unts n terms of P1k usng equaton (8),.e.,. Step-5. Check generaton of each unt Pjk to reman wthn ts operatng lmts,.e. and If generaton of all unts are wthn ther operatng lmts of correspondng fuel optons then go to Step-6. Otherwse, for each unt whose nequalty s volated, fnd out the new fuel opton k, so that the nequalty s satsfed and then aj1, bj1 and cj1 are replaced by ajk, bjk and cjk, respectvely and then repeat the Steps 3 to 5. Step-6. Prnt the fuel opton and generaton schedule of each unt. IV. ANN BASED METHOD The basc block dagram of the proposed ANN method to solve the optmal generaton dspatch problem wth multple fuel optons has been shown n Fg. 2. Intally, the proposed numercal technque s appled to fnd out the economc fuel opton for each unt correspondng to a partcular load demand. The dotted box shows ths for generaton of tranng patterns for the Radal bass functon ANN. Once ths ANN s traned, then for a gven load demand, a prelmnary economc fuel opton for each generatng unt s obtaned. In fact the fuel optons are nteger values lke 1, 2 or 3, etc. But, n the prelmnary economc fuel opton results, we may get fractonal values lke 1.008, 0.987, etc. Therefore, a smple heurstc rule based algorthm s developed to reach a correct economc fuel opton,.e., an nteger value for each output. After obtanng the economc fuel opton for each generatng unt correspondng to a gven load demand, the optmal generaton dspatch result s obtaned by a Hopfeld neural network whch satsfes the operatonal constrants. As shown n Fg. 3, the desgn procedure of the proposed ANN technque consstng of Radal bass functon ANN, Hopfeld ANN methods along wth a heurstc rule based search algorthm for optmal generaton dspatch solutons wth multple fuel optons nvolves fve major steps, vz. tranng set creaton, tranng, testng of Radal bass functon ANN, heurstc search and Hopfeld ANN. In the proposed ANN technque, the economc fuel optons are obtaned by proposed numercal technque. For determnaton of economc fuel opton for each generatng unt, neural network of supervsed learnng s needed. Ths s because, the economc fuel opton for each generatng unt (outputs) for each total system load demand (nput) n the tranng set are requred to be known n advance by some sutable method. A radal bass functon ANN called as RBANN s employed n the present work for tranng and testng due to ts auto confgurng archtecture and faster learnng ablty. In the tranng process, the RBANN s presented wth a seres of pattern pars; each par conssts of an nput pattern and a target output pattern. The tranng pattern 'p' s descrbed by: t(p) = {( nput (p) ), ( output (p) )} = {( PD (p)), ( F1(p), F2 (p),..., FNG (p) )} (16) Here, Fj(p) ndcates the fuel opton of jth generatng unt correspondng to pth tranng pattern. The sum of the squared errors (SSE) between the actual and the desred (target) outputs over the entre tranng sets s used as the measure to fnd out the convergence of the network. The RBANN used s traned by the orthogonal least squares learnng algorthm. Tranng s contnued untl the gven error-goal n terms of SSE s reached. Once the RBANN s traned, there after only the Steps 3 and 4 are used to obtan the economcal fuel opton for each generatng unt for any gven load PD. In the Step- 3 only a prelmnary (non-nteger) economcal fuel optons may be obtaned. Therefore, a heurstc rule based search algorthm s developed n the Step-4 to reach a correct (nteger) economcal fuel opton for each unt. Once the economcal fuel optons are found out, then the optmal generaton dspatch s merely an ELD problem whch s solved by a Hopfeld neural network n fnal step (Step-5). Fgure 2: Block dagram of proposed ANN technque for optmal generaton dspatch wth multple fuel optons 128 ISSN (Prnt) : , Volume-2, Issue-1, 2013

4 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) A. Heurstc rule based search algorthm for determnaton of fnal economcal fuel optons In ths work the followng heurstc rules are appled to transform the prelmnary fuel opton results (nonnteger values) nto nteger values. neuron has the range and the nput-output functon s a contnuous and monotoncally ncreasng functon of the nput U to neuron. The typcal nput-output functon g (U) s a sgmodal functon. The dynamcs of the neuron s defned by Fgure 3: Desgn procedure of the proposed ANN technque for optmal generaton dspatch problem wth multple fuel optons V. HOPFIELD NEURAL NETWORK Hopfeld neural network model s a sngle layer recursve neural network, where the output of each neuron s connected to the nput of every other neuron. There s an external nput to the each neuron. In a Hopfeld network all connectve weght values are calculated ntally from system data. Then as patterns or nput values are appled, the network goes through a seres of teratons untl t stablzes on a fnal output. Thus, the values of neuron nputs and the outputs change wth tme and form a dynamc system. It s mportant to ensure that the system wll converge to a stable soluton. Ths requres fndng a bounded functon (a Lyapunov or energy functon) of the state varables such that all state changes result n a decrease n energy. There are two types of Hopfeld neuron model. The orgnal model of Hopfeld neural network used a bnary neuron model. The contnuous and determnstc model of the Hopfeld neural network [19] s based on contnuous varables and responses but retans all of the sgnfcant behavors of the orgnal model and hence, used n the present work. The output varable V for From ths, t can be seen that de/dt s always less than zero because g s a monotonc ncreasng functon. Therefore, the network soluton moves n the same drecton as the decrease n energy. The soluton seeks out a mnmum of E and comes to a stop at such pont.c. A. The Economc Load Dspatch Problem The ELD problem s to fnd the optmal combnaton of power generaton whch mnmzes the total fuel cost whle satsfyng the total requred demand. In ths dssertaton work, the cost functon s as follows: C ( ) where, a C: total cost b P c P 2. a, b, c: cost coeffcents of generator P: the generated power of generated In mnmzng total fuel cost, the followng constrants should be satsfed. a) Power balance: PD + L = P (20) where, PD: total load L: transmsson loss. 129 ISSN (Prnt) : , Volume-2, Issue-1, 2013

5 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) The transmsson loss can be represented as D P L NG 1 where, NG: number of generators D: transmsson loss coeffcents 2 (21) b) Maxmum and Mnmum lmts of Power: The generator power of each generator should be lad between maxmum and mnmum lmt. That s, P mn where, P P max (22) Pmn and P max power output of th unt, respectvely. are the mnmum and maxmum real B. Mappng of the ELD nto the Hopfeld Neural Network In order to solve the ELD problem, the followng energy functon s defned by combnng the objectve functon wth the constrant as defned by Eqn. (19) and Eqn. (20), respectvely. 2 2 E A( PD L P) (23) / 2 B ( a b P c P ) / 2 where, A ( 0) and ( 0) B are weghtng factors. The synaptc strength and the external nput are obtaned by mappng the above energy functon nto the Hopfeld energy functon as descrbed by Eqn. (23) nto the Hopfeld energy functon, Eqn.(17). Frst by assumng that the loss L s constant, the Eqn. (23) s expanded and compared to Eqn. (17) n whch V and Vj correspond to P and Pj, respectvely: E A[ ( P L) 2( P L)( P D P ) ( B ) ]/ 2 ( a b P c P ) / 2 D = 2 A ( PD L) / 2 [ A( P L) B D b / 2] ( A B ) / 2 A / 2 B j c P P j P P a P / 2. (24) Thus, by comparng Eqn. (17) wth Eqn. (24), the synaptc strength and external nput of neuron n the Hopfeld network are gven by T = -A-Bc Tj = -A (25) I = A(PD +L)-Bb/2. The dfferental synchronous transton model [18] used n the computaton for ths Hopfeld neural network s as follows: U U ( k ) ( k 1) ( k) j T j V j I (26) Accordngly, the output value P can be obtaned by ths Hopfeld neural network and the transmsson loss can be calculated by the loss formula as descrbed n Eqn. (21). Agan the calculated loss s assumed as a constant value and thereafter the above process s repeated. In representng a large value wth the neural network, the bnary number representaton requres a large number of neurons whch s a clear-cut dsadvantage. Therefore, n ths work a modfed sgmodal functon as suggested n Reference-18 s used, whch s gven below as: V V ( k 1) [ ( k)]. g U g U max mn mn ( ) ( P P )(1/(1 exp( U / u ))) (27) 0 P VI. SYSTEM STUDIES In ths secton a neural network procedure comprsng of a radal bass functon network along wth a heurstc rule based search algorthm and a Hopfeld neural network has been proposed to solve the economc load dspatch problem wth multple fuel optons. The test system consstng of four generatng unts wth unt- 1 suppled wth two types of fuels and other remanng unts suppled wth three types of fuels, has been consdered for the performance evaluaton of the proposed algorthm. The hybrd cost coeffcents,.e., the cost curve coeffcents of each unt correspondng to dfferent types of fuel are gven n Table 1. The mnmum and maxmum generaton capacty of each unt correspondng to each type of fuel optons are summarzed n Table 2. For determnaton of economc fuel opton of each thermal unt, neural networks of supervsed learnng are 130 ISSN (Prnt) : , Volume-2, Issue-1, 2013

6 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) needed. Ths s because, the optmal fuel opton of the thermal unts (outputs) for each total system load demand (nput) n the tranng set are requred to be known n advance by some sutable method. A radal bass functon ANN called as RBANN s employed n the present work for tranng and testng due to ts auto confgurng archtecture and faster learnng ablty. The proposed numercal technque as descrbed earler has been appled to create the necessary tranng set. A radal bass functon ANN model, namely RBANN s desgned for the purpose. There s only 1 nput node (load demand) for the model. The optmal fuel optons of the thermal unts n the system,.e.,,..., are the output nodes. Therefore, there are 4 output nodes for the RBANN. The number of neurons n the sngle hdden layer s equal to the number of teratons requred for tranng and s set adaptvely for RBANN. It s not unusual to get good performance on tranng data followed by much worse performance on test data. Ths can be guarded aganst by ensurng that the tranng data are unformly dstrbuted. Two dfferent cases were computed by the proposed ANN technque,.e., (1) Optmal generaton dspatch wth multple fuel optons wthout consderng losses and (2) Optmal generaton dspatch wth multple fuel optons consderng losses. For both the test cases, to tran the networks PD was vared n the range 850 MW to 950 MW n steps of 5 MW. Therefore, 21 dfferent tranng patterns were generated coverng the system load from 850 MW to 950 MW for each test case. Two dfferent radal bass functon networks namely, the RBANN1 for test case 1 and RBANN2 for test case 2 were traned wth ther correspondng 21 patterns to reach the error-goal (convergence target) whch was SSE = Two dfferent neural networks are used as the outputs.e., the optmal fuel optons for same load demand PD for loss ncluson case and wthout loss case may be dfferent. However, ther archtecture remans same. RBANN1 and RBANN2 requred 19 and 20 teratons (epoch), respectvely, n reachng the convergence target. To acheve the best performance on the test data and good generalzaton an approprate value of spread factor (SF) s set. Computatons were carred out for dfferent values of SF to fnd the best value of SF. For a gven set of test patterns the percentage mean absolute error (% MAE) s recorded for each value of SF. Then the value of SF correspondng to the mnmum of the % MAE s taken as the best value of SF. Accordngly, the best SF s found to be 5 for both RBANN1 and RBANN2. For the performance evaluaton of the RBANN, 4 dfferent load levels other than those n tranng sets but wthn 850 MW to 950 MW are consdered. These test cases were generated by proposed numercal technque. The test cases were computed by the RBANN1, whch was traned earler takng the best value of SF.e., 5. The fnal optmal fuel optons obtaned from the RBANN1 along wth heurstc rule based search algorthm was compared wth those obtaned from proposed numercal method and the comparson s shown n Table 3. From ths table t s observed that the optmal fuel optons obtaned from RBANN1 along wth heurstc rule-based search algorthm matches to that of the proposed numercal method. Smlarly, Table 4 shows the comparson of fnal optmal fuel optons obtaned from the RBANN2 along wth heurstc rule based search algorthm and those obtaned from proposed numercal method. The RBANN2 results matches to that of the proposed numercal method. A. Smulaton of Hopfeld Neural Network After obtanng the economc fuel opton for each generatng unt correspondng to a gven load demand, the optmal generaton dspatch result s obtaned by a Hopfeld neural network whch satsfes the operatonal constrants. Durng smulaton, t was found that the assumed ntal solutons dd not affect the results for dfferent cases snce they are convex problems. Table 3 Comparson of optmal fuel optons obtaned from RBANN1 along wth heurstc rule based search algorthm and proposed numercal method. 131 ISSN (Prnt) : , Volume-2, Issue-1, 2013

7 Generated Power Generated Power Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) B. Determnaton of weghtng factors Determnaton of weghtng factors n case of Hopfeld neural network s very crucal n achevng the optmal generaton schedules. A s the penalty factor to the constrant of the total load demand and B s the penalty factor to the constrant of the objectve functon. It was observed that when A was bgger than 0.5 regardless of B values, the network oscllated. Usually, when there s self-feedback (T 0), the solutons can be n oscllaton as reported n Reference-. Through smple tral and error method, t was found that A = 0.5 and B = 0.06 were approprate values. The nequalty constrants of maxmum-mnmum lmts are dealt by the sgmodal functon varaton as shown n Eqn.(27). The results of dfferent case studes are shown n Table 5 and Table 6, respectvely, for wthout loss and loss ncluson cases and compared wth those of proposed numercal technque. The results of the Hopfeld network method shows small error n power balance (the msmatch power s MW n wthout loss case and MW n loss ncluson case). When ths error s converted nto the fuel cost of a power plant wth the hghest cost functon, the total cost ncrease s extremely small compared wth the total cost of numercal method. The convergence characterstcs for both the cases have been observed and shown n Fgs. 4 (a) and (b). In second case, where the transmsson loss s consdered, the Hopfeld neural network method also shows good results. Table 5. Comparson of optmal generaton dspatch results obtaned from Hopfeld Neural Network and Proposed Numercal Method wthout consderng loss Table 6 Comparson of optmal generaton dspatch results obtaned from Hopfeld Neural Network and Proposed Numercal Method consderng loss (a) Wthout loss case P3 (b) Wth loss case P P4 P1 P P1 P4 P Iteraton Iteraton Fgure 4: Convergence characterstcs of Hopfeld neural network VII. CONCLUSION Ths paper presents an ANN method consstng of Radal bass functon ANN, Hopfeld ANN methods along wth a heurstc rule based search algorthm for optmal generaton dspatch solutons wth multple fuel optons. Intally, a numercal technque as proposed n ths paper s appled to fnd out the economc fuel opton 132 ISSN (Prnt) : , Volume-2, Issue-1, 2013

8 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) for each unt correspondng to a partcular load demand whch s used for generaton of tranng patterns for the Radal bass functon ANN. Once ths ANN s traned, then for a gven load demand, a prelmnary economc fuel opton for each generatng unt s obtaned. In fact the fuel optons are nteger values lke 1, 2 or 3, etc. But, n the prelmnary economc fuel opton results, we may get fractonal values lke 1.008, 0.987, etc. Therefore, a smple heurstc rule based algorthm s developed to reach a correct economc fuel opton,.e., an nteger value for each output. After obtanng the economc fuel opton for each generatng unt correspondng to a gven load demand, the optmal generaton dspatch result s obtaned by a Hopfeld neural network whch satsfes the operatonal constrants. The smulaton results show that the soluton method s practcal and vald for realtme operaton. An advantage of the proposed method s ts capablty to optmze over a greater varety of operatng condtons. The proposed ANN method has applcatons to fossl fueled generaton unts capable of burnng coal, gas, and/or ol as well as other problems whch result n multple ntersectng cost curves for a partcular generatng unt. Ths neural network method has the specal advantage of solvng the ELD problem wthout calculatng ncremental fuel costs and ncremental losses requred by conventonal numercal methods. The proposed ANN technque for solvng optmal generaton dspatch problem wth multple fuel optons was mplemented n MATLAB language whch was run on a 2.4 GHz Pentum-IV machne. The average computaton tme for each load demand n case of ANN method was found to be about 40 second whle the average computaton tme of the proposed numercal technque s about 1 mnute. Ths ndcates that there s not much dfference n computaton tme. But, when mplemented n hardware, the proposed neutral network technque can acheve much faster real tme response than the proposed numercal technque. Therefore, the proposed ANN based technque promses to have a good mert n ts applcatons. VIII. REFERENCES [1]. B.H.Chowdhury and S.Rahman, A revew of recent advances n economc dspatch, IEEE Trans., Power Systems, Vol.5, No.4, 1990, pp [2]. D.C.Walters and G.B.Sheble, Genetc algorthm soluton of economc dspatch wth valve pont loadng, IEEE Trans. on Power Systems, Vol.8, No.3, 1993, pp [3]. M.E.El-Hawary and G.S.Chrstensen, Optmal Economc Operaton of Electrc Power Systems, Academc Press, [4]. J.Freeman and D.Skapura, Neural Networks Algorthms, Applcatons and Programmng technques, Addson-Wesley Publshng Company, Inc., [5]. P.K.Kalra and A.Srvastava, Possble applcatons of neural nets to power system operaton and control, Electrc Power Systems Research, Vol.25, 1992, pp [6]. M.El-Sharkaw, R.Marks and S.Weerasoorya, Neural Networks and ther Applcaton to Power Engneerng, Academc Press, [7]. V.S.S.Vankayala and N.D.Rao, Artfcal neural networks and ther applcatons to power systems-a bblographcal survey, Electrc Power Systems Research, Vol.28, 1993, pp [8]. B.Novak, Superfast autoconfgurng artfcal neural networks and ther applcaton to power systems, Electrc Power Systems Research, Vol.35, 1995, pp [9]. J.Park and I.W.Sandberg, Unversal approxmaton usng radal bass functon networks, Neural computaton, Vol.3(2), 1991, pp [10].J.J.Hopfeld, Neural networks and physcal systems wth emergent collectve computatonal abltes, Proceedngs of the Natonal Academy of Scence, USA, Vol.79, Aprl 1982, pp [11].J.J.Hopfeld, Neurons wth graded response have collectve computatonal propertes lke those of two-state neurons, Proceedngs of the Natonal Academy of Scence, USA, Vol.81, May 1984, pp [12].J.J.Hopfeld and D.W.Tank, Neural computaton of decsons n optmzaton problems, Bologcal Cybernetcs, Vol.52, 1985, pp [13].D.W.Tank and J.J.Hopfeld, Smple neural optmzaton networks: an A/D converter, sgnal decson crcut and a lnear programmng crcut, IEEE Trans., Crcuts and Systems, Vol. CAS-33, No.5, 1986, pp [14].M.P.Kennedy and L.O.Chua, Unfyng the Tank and Hopfeld lnear programmng crcut and the canoncal nonlnear programmng crcut of Chua and Ln, IEEE Trans., Crcuts and Systems, Vol. CAS-34, No.2, 1987, pp [15].H.Sasak, M.Watanabe and R.Yokoyama, A soluton method of unt commtment by artfcal neural networks, IEEE Trans., Power Systems, Vol.7, No.3, 1992, pp [16].T.D.Kng, M.E.El-Hawary and F.El-Hawary, Optmal envronmental dspatchng of electrc power systems va an mproved Hopfeld neural network model, IEEE Trans., Power Systems, Vol.10, No.3, 1995, pp ISSN (Prnt) : , Volume-2, Issue-1, 2013

9 Internatonal Journal of Advanced Electrcal and Electroncs Engneerng (IJAEEE) [17].Y.Uek and Y.Fukuyama, An applcaton of artfcal neural network to dynamc economc load dspatchng, Proceedngs of the Frst Internatonal Forum on the Applcaton of Neural Networks to Power Systems, Scattle, July 1991, pp [18].J.H.Park, Y.S.Km, I.K.Eom and K.Y.Lee, Economc load dspatch for pecewse quadratc cost functon usng Hopfeld neural network, IEEE Trans. on Power Systems, Vol.8, No.3, 1993, pp [19].J.J.Hopfeld and D.W.Tank, Computng wth neural crcuts: A model, Scence, Vol.233, 1986, pp ISSN (Prnt) : , Volume-2, Issue-1, 2013