A PSO with Quantum Infusion Algorithm for Training Simultaneous Recurrent eural etworks

Transcription

1 A PO wth Quantum Infuson Algorthm for Tranng multaneous Recurrent eural etworks Bpul utel and Ganesh Kumar Venayagamoorthy Abstract multaneous Recurrent eural etwork (R ) s one of the most powerful neural network archtectures well suted for estmaton and control of complex tme varyng nonlnear dynamc systems. R tranng s a dffcult problem especally f multple nputs and multple outputs (MIMO) are nvolved. Partcle swarm optmzaton wth quantum nfuson (PO-QI) s ntroduced n ths paper for tranng such R s. In order to llustrate the capablty of the PO-QI tranng algorthm, a wde area montor (WAM) for a power system s developed usng a multple nputs multple outputs Elman R. The R estmates speed devatons of four generators n a multmachne power system. nce MIMO structured R s are hard to tran, a two step approach for tranng s presented wth PO-QI. The performance of PO-QI s compared to that of the standard PO algorthm. Results demonstrate that the R traned wth the PO-QI n the two step approach tracks the speed devatons of the generators wth the mnmum error. I. INTRODUCTION IMUTANEOU Recurrent Neural Network (RN) are known to be a powerful class of neural network archtectures. As the name sgnfes, the recurrence s nstantaneous.e. many tmes wthn a samplng perod []. The RN provdes response of a dynamc nonlnear system even when the weghts are fxed and therefore s more approprate for approxmatng more complex nonlnear systems wth less number of neurons. The RNs have the capablty of approxmatng non-smooth functons whch cannot be approxmated by conventonal Multlayer Perceptrons (MPs) []. Because of the nherent recursve calculaton nvolved n RN, they are hard to tran usng tradtonal tranng algorthms such as backpropagaton through tme, whch suffer from local mnma [3]. Computatonal Intellgence (CI) based algorthms have got popularty n tranng of neural networks because of ther ablty to fnd global soluton n mult-dmensonal search space. warm and evolutonary based algorthms lke Partcle warm Optmzaton (PO) have shown promses n tranng of RNs. A hybrd of PO and Evolutonary Algorthm (EA) called the PO-EA s used n engne data classfcaton n [3]. By combnng the best features of the partcpatng ndvdual algorthms, hybrd algorthms are The fundng provded by the Natonal cence Foundaton, UA under the CAREER grant EC #0348 and EFRI # s gratefully acknowledged. Bpul utel and Ganesh K. Venayagamoorthy are wth the Real-Tme Power and Intellgent ystems aboratory, Mssour Unversty of cence and Technology, Rolla, MO 6540 UA (e-mal: bl7f3@mst.edu and gkumar@eee.org). more robust and have been used n varous knds of optmzaton problems. An electrc power grd s a geographcally large nterconnected network of generators, transmsson lnes, real and reactve power compensators, loads etc. Power sources and generators are wdely dspersed n a modern power system confguraton. For the stablty and securty of the power system, dstrbuted control agents are employed to provde reactve control at several places on the power network through Power ystem tablzers (Ps), Automatc Voltage Regulators (AVRs), Flexble AC Transmsson ystems (FACT) devces, etc. Although local optmzaton s acheved by the control agents such as Ps and AVRs, the lack of coordnaton among the local agents may cause serous problems such as system oscllatons (nter-area) n the power network. Wde Area Control ystem (WAC) scheme s proposed n [4, 5] n order to mnmze the problems encountered n a dstrbuted power network. The ncreasng complexty and hghly nonlnear nature of the power system today requres a Wde Area Montor (WAM) for fast and accurate montorng, for effectve control of power networks wth an adaptve WAC. Ths s mportant for dfferent purposes such as renforcement of power system based on accurate feedback sgnals obtaned durng analyss of system dynamcs, coordnated approach for the executon of fast stablzng actons n case of sever network dsturbances etc. [6] The WAM provdes nformaton to the WAC whch then sends approprate control/feedback sgnals to the dstrbuted agents n the power network based on some predefned objectve functons. In ths study, quantum prncple obtaned from Quantum PO (QPO) has been combned wth tradtonal PO to form a new hybrd algorthm called as PO wth Quantum Infuson (PO-QI). A multple nputs multple outputs (MIMO) RN s used to mplement the WAM for a two area multmachne system. nce tranng of a MIMO RN s computatonally complex, a two step tranng approach s suggested. It s shown through results that PO-type algorthms can be used to tran RNs. It has been shown n lteratures that hybrd algorthms perform better n tranng of complex neural network archtectures [6]. Hence, ths study focuses on mprovng the hybrd technque for accuracy. To mprove the tranng accuracy n mplementng a MIMO WAM, PO-QI s used to tran the network usng the two step approach. The followng sectons of the paper are

2 arranged as follows: Multmachne power system s descrbed n ecton. In ecton 3, local and wde area montors are descrbed. PO-QI algorthm s descrbed n ecton 4. Results and dscusson are gven n ecton 5 and concluson n ecton 6. Σ Σ V ref II. MUTIMACHINE POWER YTEM The practcal power system s a complex system wth thousands of buses, several hundreds of generators and nteractons between multple areas wth several nter-area modes of oscllatons. The two-area power system [7] of Fg. s a test system whch s commonly used to show the effectveness of controllers n dampng slow mode oscllatons. The two-area system wth the WAM (Fg. ) conssts of two fully symmetrcal areas lnked together by two transmsson lnes. Each area s equpped wth two dentcal synchronous generators rated 0kV/900 MVA. All the generators are equpped wth dentcal speed governors and turbnes and AVRs and excters (Fg. ). Generators G, G, G3 and G4 are all equpped wth Ps. The swtch (Fg. ) can be used to provde tranng and auxlary control sgnals to the generators. The swtch (Fg. ) s used to add the P sgnal to the exctaton system. The loads are represented as constant mpedances and splt between the areas n such a way that Area s transferrng about 43 MW to Area. Three electromechancal modes of oscllaton are present n ths system; two nter-plant/ntra-area modes, one n each area, and one nter-area low frequency mode [8]. The nonlnear behavor of the complete power system n Fg. s smulated n detal n the PCAD/EMTDC envronment (PCAD, 004) for ths study. The parameters of the two area system are gven n [3]. ω ( ω ( ω 4 ( ω 3 ( Fg. : Block dagram of the AVR-excter model and showng the nterface to the WAC. III. WIDE AREA MONITOR A WAM mplemented n ths work conssts of a three layered Elman RN. An Elman RN has ts feedback from the hdden layer output to the context layer nputs. It has an nput layer wth 8 nput nodes, a hdden layer node wth 5 hdden nodes and an output layer wth 4 output nodes. Beng an Elman network, t also has a context layer wth 5 nodes whose nputs are the outputs of the correspondng hdden layer nodes. Fg. 3 shows the Elman RN used as WAM. Vref ( Vref ( Vref 3( Vref 4( ω ( t ) ω ( t ) ω 3 ( t ) ω 4 ( t ) ω ( ω ( ω 3 ( ω 4 ( Vref ( ω (t-) Vref ( ω (t-) Vref 4( ω 4(t-) Vref 3( ω 3(t-) H(k-) Fg. 3: Elman RN used n the WAM. Fg. : Two-area power system wth a WAM predctng the speed devatons of generators G, G, G3 and G4. Then nputs to the RN are the current devatons n reference voltage V ref (Fg. ), caused by the PRB exctaton, of the four generators. The prevous step speed devatons of these generators are also the other set of nputs. The RN WAM receves these nputs every 0 ms (00 Hz), whch s possble wth today s phasor measurement unt technology [3]. The hdden nodes have sgmod actvaton functon and the output nodes are lnear. In vector notaton,

3 an Elman RN s mplemented as follows [9]: H ( t, k) f ( A* I( t, k) + B * H ( t, k ) + K) = () O ( g( C * H ( t, k) + K') = When k = R () where I = [ Vref, ω ] for = to 4 s the set of nputs, H s the set of outputs from the hdden nodes and O = O ] = [ ω, ω, ω 3, ω 4 ] s the set of [, O, O3, O4 outputs. A contans weghts from nput layer to the hdden layer, B contans weghts from context layer to the hdden layer, C contans weghts from hdden layer to the output layer, k s the tme ndex of nternal recurrence, t s the tme ndex of the nput sample, R s the total number of nternal recurrences, K and K are the bases, f and g are the two actvaton functons. The WAM under consderatons s a MIMO system and dentfcaton of such system s very dffcult. Hence a two step procedure for tranng the MIMO RN s mplemented. In tep, RN as shown n Fg. 3 s used and s traned usng PO and PO-QI to obtan the nput and output weghts. In tep, the nput weghts obtaned n tep are kept fxed, and the same RN s traned to obtan the output weghts, wth only one output at a tme. The output of the WAM s the predcted values of the speed devaton for the current sample. IV. PARTICE WARM OPTIMIZATION WITH QUANTUM INFUION Partcle swarm optmzaton wth quantum nfuson s a new approach to hybrdzaton of PO and Quantum Partcle warm Optmzaton (QPO) [0]. Here, the quantum prncple n QPO s used to create a new offsprng. After the poston and velocty of the partcles are updated usng standard PO equatons, a randomly chosen partcle from PO s pbest populaton s utlzed to carry out the quantum operaton; and thus, create an offsprng by mutatng the gbest. The ftness of the offsprng s evaluated and the offsprng replaces the gbest partcle of PO only f t has a better ftness. Ths ensures that the ftness of the gbest partcle s equal to or better than ts ftness n the prevous teraton. Thus, t s mproved and pulled towards the best soluton over teratons. By nfusng the quantum theory to the standard PO, a new hybrd algorthm s evolved whch ncorporates the best features of the respectve ndvdual algorthms and thus a better ftness s acheved. In PO-QI, fast convergence property obtaned by PO n the frst few teratons, and the convergence to a lower average error property obtaned by QPO, have been combned and hence the performance s sgnfcantly mproved, as s shown n the results and fgures below. It s descrbed below n detal. PO s an evolutonary-lke algorthm developed by Eberhart and Kennedy n 995 []. It s a populaton based search algorthm and s nspred by the observaton of natural habts of brd flockng and fsh schoolng. In PO, a swarm of partcles moves through a D dmensonal search space. The partcles n the search process are the potental solutons, whch move around a defned search space wth some velocty untl the error s mnmzed or the soluton s reached, whch s decded by the ftness functon. The partcles reach to the desred soluton by updatng ther poston and velocty accordng to the PO equatons. In PO, each ndvdual s treated as a volume-less partcle n the D-dmensonal space, wth the poston and velocty of the th partcle represented as: x = x, x,..., x ) (3) ( D v = v, v,..., v ) (4) v d ( D ( t + ) = wv + c rand ( P x gd d ( + c rand ( x d ( ) ( t + ) = x ( + v ( t + ) ( P d ( x d ( ) (5) d d d (6) These partcles are randomly ntalzed over the search space wth ntal postons and veloctes. They change ther postons and veloctes accordng to (5) and (6) where c and c are cogntve and socal acceleraton constants respectvely, rand () and rand () are two random functons unformly dstrbuted n the range of [0,] and w s the nerta weght ntroduced to accelerate the convergence speed of PO []. Vector P = (P, P,...,P D ) s the best prevous poston (the poston gvng the best ftness value) of partcle called the pbest, and vector P g = (P g, P g,..., P gd ) s the poston of the best partcle among all the partcles n the swarm and s called the gbest. x d, v d, P d are the d th dmenson of vector of x, v, P. PO has been shown n the flowchart n Fg. 3. QPO was ntroduced by un n 004 []. Accordng to the uncertanty prncple, poston and velocty of a partcle n quantum world cannot be determned smultaneously. Thus QPO dffers from standard PO manly n the fact that exact values of x and v cannot be determned. In quantum mechancs, a partcle, nstead of havng poston and velocty, has a wavefuncton gven by: ( r, ψ (7) whch has no physcal meanng but ts ampltude squared gves the probablty measure of ts poston n any one dmenson r at tme t. The governng equaton of quantum mechancs s the chrodnger s equaton gven by: ψ ( r, = H ( r) ψ ( r, t jh (8)

4 where H s a tme-ndependent Hamltonan operator gven by: acheved: u y / = e (4) h H ( r) = + V ( r) m (9) y ± ln(/ u) = (5) where h s Planck s constant, m s the mass of the partcle and V p (r) s the potental energy dstrbuton [3]. Based on the probablty densty functon, a partcle s probablty of appearng n poston x can be determned. Therefore n QPO, a Delta-potental-well based probablty densty functon has been used wth center of the well at pont J = (j, j,.., j D ) n order to avod exploson and help the partcles n PO to converge [4]. Assumng a partcle n onedmensonal space havng ts center of potental at J, normalzed probablty densty functon Q and dstrbuton functon D f can be obtaned [5]. et y=x-j, then the form of ths probablty densty functon s gven as follows and depends on the potental feld the partcle les n: Q( y) D f y / = e (0) = y y / Q( y) dy = e ( y) () where the parameter s the length of the potental feld whch depends on the energy ntensty and s called the creatvty or magnaton of the partcle that determnes ts search scope [4]. can be evaluated as the dstance between the partcles current poston and pont J as follows: = β J x () The parameter β s the only parameter of the algorthm. It s called the creatvty coeffcent and s responsble for the convergence speed of the partcle. In QPO, search and soluton spaces are two unque spaces of dfferent qualty. o a mechansm s necessary to map the poston of a partcle n the search space to the soluton space. Ths s called collapsng and s acheved by applyng the Monte Carlo smulaton. Ths has been explaned n [] as follows. et s be any random number unformly dstrbuted between 0 and /. For a unform random number u n the nterval [0, ], s s defned as: s u = (3) Now, equatng (0) and (3), the followng relaton s The poston equaton s gven as follows: x J ± ln(/ u) = (6) where the partcle s local attractor pont J has coordnates gven by the followng equaton: J d ( gd d t = α P ( + α P ( ) (7) where α = a/(a + b) and α = b/ (a + b), and a and b are two unformly dstrbuted random numbers. From () and (5), the new poston of the partcle s calculated as: x( t + ) = J ( ± β J ( x( ln(/ u) (8) Ths Delta-Potental-well based quantum PO s called the QDPO n []. Ths has been mproved further by defnng a manstream thought [5] or the Mean Best Poston, mbest, as: mbest( = = = = P ( P (,..., = P D ( (9) where s the sze of the populaton, D s the number of dmensons and P s the pbest poston of each partcle. Now the poston update equaton n (8) s gven as (0), where the addton or subtracton s carred out wth 50% probablty: x( t + ) = J ( ± β mbest( x( ln(/ u) (0) By usng (7) ths can also be wrtten as follows to show the mutaton on gbest: x( t + ) = αpgd ( + α P ( ± β mbest( x( ln(/ u) d () PO-QI s comparable to Estmaton of Dstrbuton PO (EDPO) [6] where new partcle s created based on the

5 probablstc models of the search space. Hence the PO-QI mutatons are more lkely to produce better offsprng than other random mutaton technques. V. TUDIE AND REUT An Elman RN based wde area montor s developed n ths paper. A forced tranng s carred out n whch all four generators are subjected to a PRB exctaton and ther correspondng speed devatons are measured. One thousand data samples obtaned n ten seconds are used for the tranng. Prevous samples of each generator s speed devaton are fed as nput to the RN along wth the devatons n the reference voltage of the generators due to the PRB exctaton. The RN outputs are one step ahead predcted values of speed devatons. The RN s traned usng PO and PO-QI. In tep of the two step tranng process, RN has been traned usng all the nputs and outputs. In tep, the nput weghts obtaned from the frst approach have been kept fxed and the RN s traned only for the output weghts, one at a tme for each output. After the network has been traned, t has been tested on the same dataset. For both tranng algorthms, Mean quared Error (ME) between the output of the RN and the actual output of the generator has been used as the measure of ftness. For each generator, ME for tep can be wrtten as (). 4 ME T = ME 4 where, ME = = k= ( ω ( k) ω ( k)) () (3) where ω s the actual output of the generator and s the predcted output from the RN at sample k. Eq. (3) gves the ME for tep. Fg. 4 shows the dstncton between the two steps of tranng RN. The outputs are evaluated and compared n terms of ther mean squared error as well as the absolute relatve error. The absolute relatve error (ARE) s defned as (4). ARE ω ( k) ω ( k) = ω ( k) where the symbols have the same meanng as (3). (4) ω Fg. 4: Two step tranng of RN. In [6], t s shown that a thrd order model s enough to represent the power system under consderaton. nce four of such systems are beng consdered here, the network should be able to model four thrd order systems n order to correctly dentfy the whole system. From tral and experence, a network wth the followng parameters s used, but s not clamed to be optmal. Input Nodes (n) = 8 (4 PRB, 4 speed devatons) Hdden Nodes (m) = 5 Output Nodes (r) = 4 (tep ), (tep ) Number of samples ( ): 000 The followng parameters are used for PO and PO-QI: c, c = w = lnearly decreasng from 0.9 to 0.4 Populaton ze: 30 Number of teratons: 0 β = lnearly ncreasng from 0.5 to Dmenson (D) = 405 (tep ), 5 (tep ) The speed devaton output obtaned whle testng s plotted along wth the actual output. The plots shown are for any random tral. Fg. 5 shows the PRB nput to the

6 generator G. mlar nputs are appled to the other generators smultaneously. The speed devaton predcton of G obtaned from the RN traned usng tep s shown n Fg. 6. The fgure shows the ablty of PO-QI to better tran the neural network and hence t predcts the output more accurately. The same output after the RN s further traned n tep s shown n Fg. 7. Ths fgure clearly shows a sgnfcant amount of mprovement n the predcton by both the algorthms. However, the output of PO-QI s more close to the actual output than PO. mlar comparson of the outputs of generators G, G3, and G4 for the two step tranng process are shown n Fgs. 8 to 3. The numercal values of ME averaged over 0 trals are compared n Table. These results show that PO-QI performs better than PO n both steps. It also confrms that RN traned n teps and s able to predct the speed devatons much better than the RN traned n tep alone. Although tranng of MIMO neural network s dffcult and computatonally complex, usng the proposed two step tranng process and PO-QI algorthm, better accuracy n tranng s acheved. Fg. 7: Testng plot for G n step. Fg. 8: Testng plot for G n step. Fg. 5: PRB nput to G. Fg. 9: Testng plot for G n step. Fg. 6: Testng plot for G n step.

7 Fg. 0: Testng plot for G3 n step. Fg. 3: Testng plot for G4 n step. Fg. : Testng plot for G3 n step. G I II III IV Avg. TABE I COMPARION OF REUT OBTAINED IN TWO TEP Mean ARE ME (0-5 ) Algorthm tep tep tep tep One Two One Two PO PO-QI PO PO-QI PO PO-QI PO PO-QI PO PO-QI Fg. : Testng plot for G4 n step. VI. CONCUION A new algorthm PO wth quantum nfuson and a two step approach for tranng MIMO RNs has been presented n ths paper. By mplementng quantum mechancal concept n mutaton of the gbest partcle, PO-QI produces offsprng more ntellgently, than other evolutonary technques wth random mutaton, n the vcnty of the soluton and thus ncreasng the speed of convergence. The performance of the PO-QI algorthm was compared to that of PO n terms of the mean squared error between the actual and the predcted outputs and the absolute relatve error at each sample. Results show that a MIMO RN performance s mproved sgnfcantly wth PO-QI and the two step tranng approach. These sgnfcant mprovements n RN performance are at the cost of more tranng tme. It has been shown that a MIMO RN can be effectvely used as a wde area montor n multmachne power systems to predct the speed devatons of the generators. For further research, t s mportant to study f any mprovement n reducng the number of teratons requred for tranng n tep to that of tep. The applcaton of such tranng

8 approach n other MIMO problems also needs to be explored. Along the lnes of power system, comparson of multple local montorng unts wth a sngle wde area montor and consderaton of transmsson delays n the predcton tme are also topcs of future research. REFERENCE [] J. A. Geb, G. erpen, Computatonal Promse of multaneous Recurrent Network wth a tochastc earch Mechansm, n Proceedngs of the Internatonal Jont Conference on eural etworks, vol. 3, Jul. 004, pp [] M. aka, N. Homma, K. Abe, tatstcal earnng Method of Dscontnuous Functons usng multaneous Recurrent Networks, n Proceedngs of the ICE Annual Conference, vol. 5, Aug. 00, pp [3] X. Ca, D. C. Wunsch II, Engne Data Classfcaton wth multaneous Recurrent Network usng a Hybrd PO-EA algorthm, n Proceedngs of the Internatonal Jont Conference on eural etworks, vol. 4, Jul. 005, pp [4] H. N, G. T. Heydt, Power ystem tablty Agents usng Robust Wde Area Control, IEEE Transactons on Power ystems, vol. 7, Nov. 00, pp. 3-3 [5] C. W. Taylor, D. C. Erckson, R. E. Wlson, V. Venkatasubramanan, WAC-Wde Area tablty and Voltage Control ystem: R & D and Onlne Demonstraton, n Proceedngs of the IEEE, vol. 93, 005, pp [6] G. K. Venayagamoorthy, Onlne Desgn of an Echo tate Network based Wde Area Montor for a Multmachne Power ystem, eural etworks, 007, pp [7] P. Kundur, Power ystem tablty and Control, 994, pp. 83, McGraw-Hll. [8] M. Klen, G. J. Rogers, P. Kundur, A Fundamental tudy of Interarea Oscllatons n Power ystems, IEEE Transactons on Power ystems, vol. 6-3, Aug. 99, pp [9] M. Wang, W. u, Y. Zhong,, mple Recurrent Network for Chnese Word Predcton, n Proceedngs of the Internatonal Jont Conference on eural etworks, vol., Oct. 993, pp [0] B. utel, G. K. Venayagamoorthy, Partcle warm Optmzaton wth Quantum Infuson for the Desgn of Dgtal Flters, n Proceedngs of warm Intellgence ymposum (I), ept. 008, pp. -8 [] Y. del Valle, G. K. Venayagamoorthy,. Mohaghegh, J. C. Hernandez, R. G. Harley, Partcle warm Optmzaton: Basc Concepts, Varants and Applcatons n Power ystems, IEEE Transactons on Evolutonary Computaton, vol., Apr 008, pp [] J. un, B. Feng, W. Xu, Partcle warm Optmzaton wth Partcles havng Quantum Behavor, n Congress on Evolutonary Computaton, vol., June 004, pp [3] R. C. Eberhart, Y. h, Comparng nerta weghts and constrcton factors n partcle swarm optmzaton, n Proceedngs of the Congress on Evolutonary Computaton, vol., Jul. 000, pp [4] J. un, W. Xu, B. Feng, Global earch trategy of Quantum-behaved Partcle warm Optmzaton, n IEEE Conference on Cybernetcs and Intellgent ystems, vol., Dec 004, pp. -6 [5] J. un, W. Xu, B. Feng, Adaptve Parameter Control for Quantumbehaved Partcle warm Optmzaton on Indvdual evel, n IEEE Internatonal Conference on ystems, Man and Cybernetcs, vol. 4, Oct. 005, pp [6] M. El-Abd, M.. Kamel, Partcle warm Optmzaton wth Varyng Bounds, n IEEE Congress on Evolutonary Computaton, ep. 007, pp