Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi


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1 Journal of Engneerng and Natural Scences Mühendslk ve Fen Blmler Dergs Sgma 005/1 A NEW APPROACH BASED ON HOPFIELD NEURAL NETWORK TO ECONOMIC LOAD DISPATCH Naser Mahdav TABATABAEI 1, Ahmet NAYIR *, Gholam AHMADI 3 1 Electrcal Engneerng Department, Azarbaan Unversty of Tarbat Moallem, TabrzIRAN Insttute of Energy, Istanbul Techncal Unversty, Ayazağa Kampüsü,MaslakISTANBUL 3 Electrcal Engneerng Department, Unversty of Tabrz, TabrzIRAN Gelş/Receved: Kabul/Accepted: ABSTRACT The Economc Load Dspatch (ELD) problem s how to real power output of each controlled generatng unt n an area s selected to meet a gven load and to mnmze the total operatng cost n the area. Ths s one of the mportant problems n a power system. The Hopfeld Neural Network (HNN) has a good capablty to solve optmzaton problems. Recently, the economc load dspatch problem solved by usng the Hopfeld neural network approach and good result has obtaned. Ths paper presents a new approach for solvng ELD problem consderng the returnng cost usng HNN model. In ths approach two energy functons are ntroduced. The frst energy functon consst of msmatch power, total fuel cost and transmsson lne losses. Each term of ths functon s multpled by a weghtng factor whch represents the relatve mportance of those terms. The other energy functon composed of total fuel cost and losses power cost. Our purpose s to mnmze these two functon and the results shows that solvng ELD problem wth ths approach yeld more savng cost. Keywords: HNN, ELD, Power system, Lagrangan method, Transmsson lne losses. EKONOMİK YÜK RAPORUNDA HOPFIELD SİNİR AĞINA DAYALI YENİ BİR YAKLAŞIM ÖZET Ekonomk yük raporu (ELD) problem bu alandak toplam en düşük şletm malyet ve verlen br yük le karşıldığında br alanda kontrol edlen her br üretm brmnn gerçek güç çıkışıdır. Bu güç sstemndek öneml br problemdr. Hopfeld snr ağı (HNN) en y kullanım problemlernn çözümünde y br kapasteye sahptr. Son zamanlardak ekonomk yük raporu problem Hopfeld snr ağı yaklaşımı kullanılarak çözülmüş ve y sonuç elde edlmştr. Bu makale, HNN model kullanılarak gerletlen malyet göz önüne alınarak ELD problemnn çözümünde yen br yaklaşımı arz etmektedr. Bu yaklaşımda k ener fonsyonu arz edlmektedr. İlk ener fonksyonu, toplam yakıt malyet ve taşıma hattı kayıbı olan, msmatch gücünden barettr. Bu fonksyonun her br term bu termlern brbrne göre önemn arz eden br faktörle çarpılır. Ötek ener fonksyonu toplam yakıt malyetnden ve güç malyet kayıbından barettr. Amacımız bu k fonksyonu mnmze etmektr ve sonuçlar malyetn daha çok korunduğu bu yaklaşım vermyle ELD problemnn çözüldüğünü göstermştr. Anahtar Sözcükler: HNN, ELD, Güç sstem, Lagrangan metodu, İletm hattı kaybı. * Sorumlu Yazar/Correspondng Author; eposta: tel: (01)
2 N. M. Tabatabae, A. Nayır, G. Ahmad Sgma 005/1 1. INTRODUCTION Economc load dspatch s one of the most mportant problems n a power system. The goal of solvng ths problem s obtanng optmum generatng power for generaton unts n an electrc power system to meet a gven load and to mnmze the total operatng cost. For solvng ths problem, many approaches have presented. One of them s Lagrangan method. In ths method a Lagrangan augmented functon s frst formulated [13]. The optmal condtons are obtaned by partal dervaton of ths functon. Calculaton of the penalty factors and ncremental losses s always the key ponts n the soluton algorthm. Incremental losses and thus the penalty factors are determned by the Bcoeffcent method whch states that the transmsson losses can be expressed n quadratc forms of the generaton powers. Recently, the ELD problem has been solved by usng the Hopfeld Neural Network and genetc algorthm. The HNN has a good capablty to solve optmzaton problems. In ths method the obectve functon of ELD problem s transformed n to a Hopfeld energy functon and numercal teraton are appled to mnmze the energy functon. Gee and Prager ntroduced methods to mprove the HNN approach by ntroducng slack varables for handlng nequalty constrants [4]. J.H. Park et. al have proposed a method to use the HNN to solve the ELD problem wth a pecewse quadratc cost functons [5]. HNN converges very slowly whch n the advantage methods have been used to update the slops or bas of the network to speed the convergence. The pecewse quadratc cost functons n ELD are used to represent multple fuels whch are avalable to each generaton unt. In these unts t may be more economcal to burn a certan fuel for some MW outputs and another knd of fuel for other outputs [6]. Reducng of transmsson lne losses s another parameter whch must be taken nto account n ths problem, because transmsson losses are the energes that the customers don t pay drectly ther costs. In ths paper a new approach and mappng technque s presented for solvng ELD problem n a power system, consderng the cost that customer pay t, by usng HNN. The proposed method has also acheved effcent and accurate solutons for dfferent szes of power systems.. ECONOMIC LOAD DISPATCH MODEL The ELD problem s to fnd the optmal soluton of power generaton that mnmzes the total cost whle satsfyng the system constrants. Mathematcally, ths problem can be expressed as [1]: C = ( a + b P + cp ) (1) P : The power output of th generator a, : Cost coeffcents of th generator b, c C : The generaton cost of the th plant Subect to satsfyng the followng constrants: (a) The actve power balance equaton: P = PL + PD () P = PB P (3) L P D : Total demand load 46
3 A New Approach Based on Hopfeld Neural P L : Transmsson loss B : Transmsson loss coeffcents (b) Maxmum and mnmum lmt of power: P = P = P (4), mn, max P : The mnmum generaton lmt of unt, mn P, max : The maxmum generaton lmt of unt The well known soluton method to ths problem usng the coordnaton equaton s df( P) PF df ( P ) df( P) 1 1 k k 1 =... = PFk =... = PF (5) df1 dpk dp PF k k PF k s the penalty factor of unt k gven by 1 = 1 P / P = 1,,..., L k, and PL / Pk s the ncremental loss of unt k. The penalty factors can be computed from losses formula (3). 3. THE STANDARD HOPFIELD NEURAL NETWORK The Hopfeld neural network conssts of a set of neurons and a correspondng set of unt delays, formng a multple loop feedback system. The number of feedback loops s equal to the number of neurons. The nput of neuron s suppled by two dfferent sources, e.g., the output of other neurons and the external nput. The nputoutput relaton s descrbed generally by a sgmod functon gven below []: V = g ( U) (7) V s a contnuous varable n the nterval 0 to 1, and g ( U ) s a ncreasng functon whch constrants V to ths nterval. 1 g( U) = (8) U 1+ exp( ) u 0 U : The total nput of neuron V : The output of neuron u 0 : The shape constant of sgmod functon The dynamc characterstc equaton of the system can be descrbed by: du dt = T V + I I : The nput bas current to neuron (6) (9) 47
4 N. M. Tabatabae, A. Nayır, G. Ahmad Sgma 005/1 T : Interconnecton conductance from the output of neuron to the nput of neuron T : Self connecton conductance of neuron The energy functon of the Hopfeld neural network s defned as [4]: E = ( 1/ ) T V V I V (1 The tme dervatve of the energy functon can be proven to be negatve: de / dt < 0 (11) So the model state always moves n such a way that the energy functon gradually reduces and converges to a mnmum. 4. TRANSFORM OBJECTIVE FUNCTION INTO HOPFIELD ENERGY FUNCTION To solve the ELD problem usng the Hopfeld method wth consderng returnng cost, two energy functons are defned as follows [3]: E1 = ( A/ )[( PD ) P ] + ( B / ) ( a + b P + cp ) + ( C / ) P (1) and E ( a + b P + cp ) + 90PL = (13) The frst energy functon, E 1, s composed of power msmatch, total fuel cost and transmsson lne losses. The postve weghtng factors A, B and C ntroduce the relatve mportance for ther respectve assocated terms and they are determned by tral and error. The other energy functon, E, s consst of total generatng cost and transmsson losses cost whch the cost doesn t return to the system and our goal s to reduce that [5, 6]. Consderng $0.09 for each KWh energy ($90 for each MWh), the cost of transmsson losses wll be 90 PL whch taken nto account n E. If the generaton output of unt changes from P 0 to P then the transmsson losses wll change from P L0 to P L, whch may be expressed as P L PL 0 + IL0( P P (14) I L0 s the ncremental loss of unt at power generaton of P 0. Substtutng (1 nto (9) yelds E1 = ( A/ )[( PD ) P ] + ( B / ) ( a + b P + cp ) + ( C / )( PL 0 + I L0 ( P P 0 )) so E1 ( A/ )( PD + ( B / ) a + ( C / )( PL 0 I L0P [ A( PD 0 ) ( Bb / ) ( C / ) I L0] P + ( A + Bc ) P / + AP P / then E 1 H [ A( PD ( Bb / ) ( C / ) I L0] P + ( A + Bc ) P / + ( AP P / ) (15) Where H s a constant and s equal 48
5 A New Approach Based on Hopfeld Neural a + ( C / )( PL 0 H = ( A/ )( PD 0 ) + ( B/ ) IL0P (16) and E = a + b P + c P ) + 90 PB P = G + ( b ) P + ( c + 90B) P + 90 BP P 1 ( (17) Where G s a constant and G = a (18) The power output value, P, n Hopfeld model can be expressed as follows: P = g( U) = ( P, max P, mn)/(1 + exp( U / u) + P, mn (19) By comparng (1 wth (1) the weght parameters and external nput of neuron n the network are gven T = A Bc T = A ( I = A( PD Bb / CIL0 / By usng (9) and ( we have U = { A( PD 0 P ) ( B / )( b + c P ) CI L 0 / } t, (1) P = g ( U ) For computng energy functon, E, by comparng (1 and (13) the followng equaton are gven T = ( c + 90B ) / T = 90B () I = b / By usng equatons (9) and () we have U = {(( 90B )/ ) V ) ( c / ) V b / )} t, P = g ( U ) Usng teraton methods, equatons (1) and (3) can be solved separately. In each teraton, the output of each neuron must be updated untl the soluton converges gradually to ther feasble local mnmum. 5. SIMULATION RESULTS To llustrate the applcaton of the proposed method, an example system s employed. The example system has 0 generatng unts to supply a total load demand of 0 MW [3, 6]. Table 1 gves fuel cost coeffcent and generaton lmts for each unt. Weghtng factors A, B and C are obtaned from tral and error. The computaton results whch are obtaned by usng teraton method and proposed approach are gven n Table. Accordng to Table we fnd that the generatng value for each unt, whch are obtaned from proposed method, has a lttle dfference wth ts value, whch are obtaned from teraton method, whle transmsson losses s smaller n the proposed method. (3) 49
6 N. M. Tabatabae, A. Nayır, G. Ahmad Sgma 005/1 Fgure 1 shows the lttle dfference between two methods. The cost that consume for generatng power and transmsson losses, related to two methods s calculated as follows: C 1 = = $/h and C = = $/h C 1 s the cost n λ method and C s ths cost n the proposed method. From the values of C 1 and C t s obvous that the consumed cost n the proposed method s small and the dfference s equal C 1 C = $/h = $/year In the other words, f we use the proposed method that s presented n ths paper, more cost wll be saved n one year. 6. CONCLUSIONS Ths paper presents a new method based on Hopfeld model for solvng economc load dspatch problem. The proposed method essentally obeys the equalncrementalcost crteron followed by conventonal economc dspatch methods. The Hopfeld neural network has a nonherarchcal structure and ts connectve conductances and external nput can be determned by employng the system data. Thus, the proposed model unlke other neural networks requres no tranng. Usng the Hopfeld neural network for solvng economc load dspatch decreases computaton tme. Table 1. The coeffcent of total fuel cost functon and generatng power lmt unts of test system Unt a ($/h) b ($/MWh) c ($/MWh) P, mn (MW) P, max MW)
7 A New Approach Based on Hopfeld Neural Table. Computaton results of two methods n the test system Unt Generaton (MW) P 1 P P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 P 1 P 13 P 14 P 15 P 16 P 17 P 18 P 19 P 0 P LOSS λ Iteraton Method The Proposed Method Fgure 1. The λ teraton method values and the proposed method values for the test system 51
8 N. M. Tabatabae, A. Nayır, G. Ahmad Sgma 005/1 REFERENCES [1] Happ H.H., "Optmal Power Dspatch," IEEE Transactons PAS, vol. PAS93, pp , [] Ln C.E., Chen S.T. and Huang C.L., "A Drect NewtonRophson Economc Dspatch," IEEE Transactons on Power System, vol. 7, no. 3, pp , 199. [3] Su C.T. and Chou G.J., "A Fast Computaton Hopfeld Method to Economc Dspatch of Power System," IEEE Transactons on Power Systems, vol. 1, no. 4, pp , [4] Gee A.H. and Prager R.W., "Polyhedral Combnatorcs and Neural Networks," Neural Computng, vol. 6, pp , [5] Park J.H., Km Y.S., Eom I. K. and Lee K. Y., "Economc Load Dspatch for Pecewse Quadratc Cost Functon Usng Hopfeld Neural Network," IEEE Transactons on Power System, vol. 8, no. 3, pp , [6] Lee K.Y., SodeYome A. and Park J.H., "Adaptve Hopfeld Neural Networks for Economc Load Dspatch", IEEE Tansactons on Power Systems, vol. 13, no., pp , May
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