A Novel Iron Loss Reduction Technique for Distribution Transformers. Based on a Combined Genetic Algorithm - Neural Network Approach

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1 A Novel Iron Loss Reducon Technque for Dsrbuon Transformers Based on a Combned Genec Algorhm - Neural Nework Approach Palvos S. Georglaks Nkolaos D. Doulams Anasasos D. Doulams Nkos D. Hazargyrou and Sefanos D. Kollas Schneder Elecrc AE, Elvm Plan, P.O. Box 59, GR-32011, Inofya Voa, Greece Tel: , Fax: , e-mal: pavlos_georglaks@mal.schneder.fr Dgal Sgnal Processng Laboraory, Deparmen of Elecrcal and Compuer Engneerng Naonal Techncal Unversy of Ahens, Greece Tel: , Fax: , e-mal: ndoulam@mage.ece.nua.gr Elecrc Energy Sysems Laboraory, Deparmen of Elecrcal and Compuer Engneerng Naonal Techncal Unversy of Ahens, Greece Tel: , Fax: , e-mal: nh@power.ece.nua.gr ABSTRACT Ths paper presens an effecve mehod o reduce he ron losses of wound core dsrbuon ransformers based on a combned neural nework - genec algorhm approach. The orgnaly of he work presened n hs paper s ha ackles he ron loss reducon problem durng he ransformer producon phase, whle prevous works were concenraed on he desgn phase. More specfcally, neural neworks effecvely use measuremens aken a he frs sages of core consrucon n order o predc he ron losses of he assembled ransformers, whle genec algorhms are used o mprove he groupng process of he ndvdual cores by reducng ron losses of assembled ransformers. The proposed mehod has been esed on a ransformer manufacurng ndusry. The resuls demonsrae he feasbly and praccaly of hs approach. Sgnfcan reducon of ransformer ron losses s observed n comparson o he curren pracce leadng o mporan economc savngs for he ransformer manufacurer. Keywords: Decson rees, neural neworks, genec algorhms, nellgen core loss modelng, core groupng process, ron loss reducon.

2 1. INTRODUCTION In oday s compeve marke envronmen, here s an urgen need for a ransformer manufacurng ndusry o mprove ransformer effcency and o reduce cos, snce hgh qualy, low cos producs have become he key o survval. Transformer effcency s mproved by reducng load and ron losses. To reduce load losses, he desgner can do one or more of he followng: use lower loss conducor maerals or decrease he curren pah lengh or he curren densy. On he oher hand, he desgner can reduce ron losses by usng lower loss core maerals or reducng core flux densy or flux pah lengh [1]. In general, aemps o reduce load losses cause ncrease of ron losses and vce versa [1]. As a resul, before decdng he opmal desgn mehod, s necessary o deermne whch of he wo losses should be mnmzed. Usually, he ransformer users (e.g., elecrc ules) specfy a desred level of ron losses (guaraneed losses) o deermne he ransformer qualy. Ths s due o he fac ha he accumulaed ron losses n a dsrbuon nework are hgh snce a large amoun of dsrbuon ransformers s nvolved. In addon, ron losses appear 24 hours per day, every day, for a connuously energzed ransformer. Thus, s n general preferable o desgn a ransformer a mnmum ron losses [2] and hs s addressed n hs paper. Inally, ransformers are desgned so ha her ron losses are equal (wh perhaps a safey margn) o he guaraneed ones. In pracce, however, ransformer acual ron losses devae from he desgned (heorecal) ones due o consruconal defecs, whch appear durng he producon phase. Reducon of ransformer acual losses, by mnmzng he effec of consruconal defecs, s a very mporan ask for a manufacurng ndusry. In parcular, (a) ncreases he relably of he manufacurer, (b) reduces he maeral cos, snce smaller safey margn s used durng he ransformer desgn phase and (c) helps he manufacurer no o pay loss penales. The laer occurs n case he acual ransformer losses are greaer (usually 15%) han he guaraneed ones. In general, s clear ha manufacurers, who are able o offer ransformers of beer qualy (lower losses) a he same prce, wll ncrease her marke share. Several works have been proposed n he leraure for he esmaon of ransformer ron losses durng he desgn phase. These approaches can be grouped no wo man caegores. The frs group s based on he arhmec analyss of he elecromagnec feld of he ransformer cores, whle he second group uses ron loss models based on expermenal observaons. In he former approach, fne elemens and fne dfference mehods are manly used. The poenals of he elecromagnec felds are calculaed, by creang mesh models of he ransformer geomery, and usng several feld parameers, such as he magnec flux dsrbuon. Ths analyss s very mporan durng he ransformer desgn phase, when he manufacurer needs o check he correcness of he ransformer drawngs. Key works adopng hs approach are provded nex. In [3], he 3D 2

3 leakage felds are esmaed and n [4] he spaal loss dsrbuon s nvesgaed usng a generc 2D fne dfference mehod. 3D magnec-feld calculaons are performed n [5] o evaluae several ransformer parameers, whle n [6] he effecs of a number of core producon arbues on core loss performance have been examned. Oher works, n hs caegory, model hree-phase ransformers based on he equvalen magnec crcu of her cores [7], [8]. In he second approach, expermenal curves are usually exraced usng a large number of measuremens o nvesgae he effec of several ransformer parameers on ron losses [2]. However, due o he connuous evoluon boh of echncal characerscs of he magnec maerals and he desgn of cores, he expermenal curves should sysemacally be reconsruced when daa change. Alernavely, lnear or smple non-lnear models are used n order o relae ransformer ron losses o he magnec nducon and geomercal properes of he magnec core [9]-[11]. The parameers of hese models are esmaed based on expermenal observaons. However, hese mehods provde sasfacory resuls only for daa (ransformers) or condons on whch hey have been esmaed. Ther performance deeroraes severely n case of new samples, whch are no ncluded n he ranng se. Alhough, all he aforemenoned approaches (heorecal or expermenal) provde a suffcen framework for he calculaon of ransformer ron losses durng he desgn phase, hey do no ake no accoun he effec of consruconal defecs, whch cause he devaon of he acual losses from he heorecal ones. More specfcally, has been found ha he maxmum dvergence beween he heorecal and acual ron losses of a specfc producon bach could as hgh as ±10%. These devaons are o a grea exen arbued o he devaons of he acual core characerscs from he desgned ones. For example, he maxmum devaon of he ron losses of he ndvdual cores can reach up o ±15%, whle he maxmum devaon of he core weghs up o ±1.5% [12]. In hs paper, reducon of ransformer ron losses s acheved durng he ransformer producon phase. In parcular, an opmal mehod s presened o esmae he mos approprae arrangemen of ndvdual cores, whch yelds ransformers of mnmum acual ron losses. Ths s acheved by compensang he consruconal defecs, whch appear n he producon phase. The mehod s reled on a combned neural nework - genec algorhm (GA) scheme. The goal of he neural nework archecure s o predc ransformer acual losses pror o her assembly. For hs reason, several measuremens (arbues) are obaned durng he ransformer producon phase. A decson ree mehodology s adoped nex o selec he mos sgnfcan arbues, whch are fed as npus o he neural nework. A genec algorhm s fnally appled o esmae he opmal arrangemen of ndvdual cores ha assemble a ransformer. In our case, opmaly means ha he ron losses of all consruced ransformers n a producon bach should be as mnmal as possble. The genec algorhm explos nformaon provded by he neural nework archecure o perform he mnmzaon ask. In parcular, 3

4 he nework predcs he ransformer qualy (ron losses) of a gven core arrangemen. The proposed scheme has been appled n a ransformer manufacurng ndusry and he resuls reveal a sgnfcan economc benef. Ths paper s organzed as follows: Secon 2 descrbes he curren pracce for esmang ron losses and for groupng he ndvdual cores. Secon 3 presens a general overvew of he proposed mehod. Secon 4 presens he predcon of ron losses usng neural neworks. In parcular, n hs secon we descrbe he consrucve algorhm used o ran he nework, he mehod appled for arbue selecon and he wegh adapaon algorhm used for mprovng he nework performance. Secon 5 presens he reducon of ron losses usng a genec algorhm (GA). In hs secon, we also dscuss ssues relaed o he GA convergence. Fnally, secon 6 shows he resuls and economc benefs obaned from he applcaon of he proposed echnques n a ransformer ndusry. Secon 7 concludes he paper. 2. CURRENT PRACTICE FOR PREDICTING IRON LOSSES AND GROUPING INDIVIDUAL CORES A hree-phase wound core dsrbuon ransformer s consruced by assemblng wo small and wo large ndvdual cores, accordng o he arrangemen descrbed n Fgure 1. In parcular, he four cores are placed as follows: a small core, followed by wo large cores, followed by anoher small core (from lef o rgh). The wndow wdh of large cores s wce of he wdh of small cores. Based on he prevous arrangemen, N hreephase ransformers are consruced from 2*N small and 2*N large ndvdual cores. Le us denoe as V s ( V l ) he se of all 2*N small (large) cores. A ransformer s represened by a vecor, he elemens of whch corresponds o he four ndvdual cores ha assemble he ransformer, l l r r T = [s l l s ] (1) Varables s l, sr Vs represen he lef and rgh small core of ransformer, whle l l, lr Vl he lef and rgh large core respecvely. Snce only one core (small or large) can be assgned o one ransformer and one poson (lef or rgh), he followng resrcons are held: s l r s, ll l r (2a) { l, r} { l, r} s k s, { l, r} { l, r} l k l wh k, (2b) where { l, r} { l, r} s ( l ) ndcaes he small (large) core n he lef or rgh poson for he ransformer. In he followng subsecon, we analyze how he ron losses are esmaed n curren pracce, whle subsecon 2.2 presens he curren core groupng process, used o assemble a ransformer. 4

5 2.1 Core Loss Esmaon Iron losses consue one of he man parameers for deermnng he ransformer qualy. Usually, cusomers' specfcaons defne an upper lm, say P 0 concernng ransformer ron losses. For hs reason, he ransformer s desgned [13] so ha s heorecal (desgn) ron losses Pd are less or equal o he specfed loss lm P 0, P d ( 1 m) P 0 (3) where m corresponds o he safey margn used durng he ransformer desgn. In curren pracce, he ypcal loss curve s used o esmae he heorecal ron losses Pd of he ransformer. The loss curve expresses he relaonshp beween specfc ron losses S d,.e., losses normalzed per wegh un (n W/Kg) versus magnec nducon B (n Gauss). A ypcal loss curve used n he consdered ndusral envronmen s depced n Fgure 2 as he doed lne. The desgn ron losses Pd of he ransformer are esmaed by mulplyng specfc ron losses nducon, by he heorecal (desgn) oal core wegh, S d, calculaed from Fgure 2 a a gven raed magnec K d, of ransformer Pd = S d * K d (4) The heorecal core wegh of ransformer,.e., ndvdual cores. Tha s, K d, s calculaed from he heorecal weghs of s four K d = 2 * ( K d d s + K l ). (5) where K d s and K d l are he heorecal weghs of small and large cores. The heorecal weghs of ndvdual cores depend on her geomercal dmensons, (.e., wdh and hegh of core wndow, hckness and wdh of core leg), he core space facor and he raed magnec nducon, as descrbed n [14]. The magnec nducon s he same as he one used for he hree-phase ransformer o esmae he specfc losses S d based on he curve of Fgure 2. Based on he above, varous ransformer parameers, whch affec he heorecal ransformer wegh and s specfc ron losses, are examned and he desgn whch sasfes he cusomers' requremens [equaon (3)] a a mnmum cos, s seleced as he mos approprae. 5

6 The oal heorecal (desgn) losses of he four ndvdual cores assembled o consruc he ransformer are gven by, F d = 2 * ( Pd d s + P l ), (6) where P d s, P d l are he heorecal (desgn) ron losses of small and large ndvdual cores, whle F d represens he heorecal (desgn) oal ron losses of he four ndvdual cores of. The heorecal (desgn) ron losses of he four ndvdual cores can be compued based on her loss curve (sold lne of Fgure 2) a he raed magnec nducon used for he hree-phase ransformer. I should be menoned ha he oal ron losses F d are no equal o he ransformer ron losses Pd snce addonal losses n general appear durng he assembly of he four ndvdual cores,.e., F d < Pd. 2.2 Core Groupng Process Alhough all ransformers consruced under he same desgn should presen he same ron losses Pd, her acual losses, say Pa usually dverge from he desgned ones. Ths s due o he fac ha several parameers, nvolved n he consrucon process, such as he formaon of ndvdual cores, he condons of ransformer producon and he qualy of magnec maeral, affec he fnal ransformer qualy. Thus, s possble for he acual ron losses of a ransformer o exceed he upper loss lm P 0. The same happens wh he acual losses of ndvdual cores, whch n general dffer from he desgned ones. In he followng, we denoe as P a l ( P a s ) he acual ron losses of a large (small) ndvdual core from all 2*N avalable large (small) cores. Therefore, random assembly of wo small and wo large cores o form a hree-phase ransformer may resul n ransformers of sgnfcan devaon from her desgned qualy. In parcular, groupng ogeher only cores of low qualy consrucs ransformers of unaccepable qualy. For hs reason, a groupng process of ndvdual cores s performed by assemblng cores of hgh and low qualy ogeher. In hs way, cores of low qualy are compensaed wh cores of hgh qualy o reduce he devaon of ransformer acual losses from he desgned ones. In curren pracce, he followng groupng mehod s used. Inally, ndvdual cores (small or large) are classfed no qualy classes accordng o he devaon of her acual losses from he desgned ones. In parcular, he qualy classes for small/large cores are defned as follows Cs = { s V : (1 + (2k 1) ) Pd < Pa (1 + (2k + 1) ) Pd k s δ s s δ s }, k = (7a) 6

7 Cl k = { l V : (1 + (2k 1) ) Pd < Pa (1 + (2k + 1) ) Pd l δ l l δ l }, k = 3...3, (7b) where n he prevous equaon seven (7) "qualy classes" are assumed. The 2 δ corresponds o he class wdh and P d s, P d l are he heorecal ron losses of a small/large core as has been defned n he prevous subsecon. Posve values of ndex k correspond o cores wh acual ron losses greaer han he desgned ones. On he conrary, negave values ndcae acual losses smaller han he desgned ones. Consequenly, as he ndex k ncreases, he core qualy decreases and vce versa. Cores belongng o he class of zero ndex,.e., or Cl 0, presen acual ron losses close o he heorecal ones whn a devaon of ± δ. s C 0 A grade s assgned o each class ndcang s qualy, so ha all cores of a class are characerzed by he same qualy grade. Snce he class ndex k s nversely proporonal o he core qualy, he negave ndex of he respecve class s defned as s grade, g( s) = k, f Pa s s C k (8a) g ( l) = k, f Pa l C l, (8b) k where we recall ha s Vs, and l Vl s a small/large core from all 2*N small and 2*N large avalable. Based on he qualy grade of each ndvdual core, a groupng process s appled o reduce he devaon of he acual ron losses of he consruced ransformers. In parcular, cores of hgh and low qualy grades are assembled ogeher o preven producon of ransformers wh very low or oo hgh qualy. Ths s accomplshed by selecng he four ndvdual cores, s r l r l, s, l, l, comprsng he ransformer, so ha he sums of he qualy grades of he wo small and wo large cores are close o zero, ha s g ( sr ) + ( l ) = 0, ( r g s g l ) + g( l l ) = 0, sr l, s Vs and l r l, l Vl, (9) or equvalenly s held ha: ( 1 δ ) F d < F a (1 + δ ) F d, (10) where F a represens he oal acual losses of he four ndvdual cores assembled o consruc he ransformer. Equaon (10) ndcaes ha he average acual ron losses of he wo small and wo large ndvdual cores for all ransformers are close o he heorecal ones wh an uncerany nerval of ± δ,.e., he class wdh. In he above mehod, he qualy of ndvdual cores s used o ndcae he qualy of hree-phase ransformers. However, he acual losses of a ransformer are no equal o he losses of s ndvdual cores. Ths 7

8 s due o he fac ha addonal parameers appear durng he ransformer consrucon, lke he exac arrangemen of he four ndvdual cores, whch are no consdered by he above-menoned echnque. For example, reorderng he wo small or he wo large cores of a ransformer, resuls n dfferen acual ron losses hough he average losses of he four cores reman he same. Anoher drawback of he curren groupng process s ha does no provde he opmal arrangemen of he 2*N small and 2*N large cores so ha he ron losses of he N consruced ransformers are as mnmal as possble. 3. PROPOSED METHOD In hs paper, a novel echnque s proposed so ha he 2*N small and 2*N large cores are appropraely arranged o consruc ransformers of opmal qualy. Fgure 3 presens a block dagram of he proposed scheme. Frs, he ransformer desgn s accomplshed based on cusomers' specfcaons and several echnoeconomcal crera as descrbed n subsecon 2.1. In hs phase, several consruconal parameers of he ransformer are specfed, such as he geomerc characerscs of ndvdual cores, he hckness, grade and suppler of magnec maeral, and he raed magnec nducon. Then, he ndvdual cores are consruced and several measuremens are aken for each core o deermne he core performance. Nex, a combned neural nework and genec algorhm approach s used o esmae he opmal core arrangemen whch resuls n hreephase ransformers of mnmum ron losses. More specfcally, he measuremens aken from he core consrucon phase, as well as addonal parameers, affecng he ransformer qualy, are used o predc he acual ron losses of he ransformer. The predcon s accomplshed hrough a neural nework ha relaes all he parameers, called arbues, wh he acual ransformer losses. A new groupng process s hen appled o mnmze he ron losses of all consruced ransformers by he avalable small and large cores. In general, he number of core combnaons s exremely large for a ypcal number of small/large cores. For ha reason a genec algorhm has been adoped o esmae whn a few eraons he opmal arrangemen of he four ndvdual cores so ha ransformers of he bes qualy are consruced. In parcular, a each sep, a populaon of new core arrangemens s generaed and predcon of he acual ron losses of he respecve ransformers s accomplshed by he neural nework model unl mnmal losses are provded for one specfc (opmum) arrangemen. 4. NEURAL NETWORKS FOR PREDICTING IRON LOSSES The neural nework archecure used for predcng he acual ron losses of a hree-phase ransformer s analyzed n hs secon. For each ransformer, several arbues are exraced and gahered n a vecor, say a ( ). Ths vecor s fed as npu o he neural nework. However, for dfferen ypes of magnec maeral and suppler, dfferen relaons beween he exraced arbues and ransformer acual losses are expeced. Ths s 8

9 due o he fac ha each suppler follows a specfc echnology of magnec maeral producon, whle he grade and hckness presen her own characerscs. In he followng, he erm envronmen s used o ndcae a gven suppler, hckness and grade of magnec maeral. Table 1 presens he hree dfferen envronmens used n he consdered ndusry. Le us assume n he followng ha M envronmens are avalable, denoed as Π, =1,2,,M. In hs case, M non-lnear funcons, say h c( ) wh c { Π 1, L, Π M } are defned whch relae he arbues a ( ) of wh he respecve acual specfc ron losses S a. Tha s, S a = h c ( a( )) (11) Snce funcons h c ( ) are acually unknown, feedforward neural neworks are used o esmae hem. The use of feedforward neworks s due o he fac ha hey can approxmae any non-lnear funcon whn any degree of accuracy ( [15], pp , 249). In our case, M feedforward neural neworks are mplemened, each of whch corresponds o a specfc envronmen. A sngle neural nework can be also appled bu usng he envronmen ype as addonal nework npu. However, such an approach provdes greaer generalzaon error han usng M ndependen neworks as s shown n he secon of he expermenal resuls. Le us denoe as h ˆ c( ) an approxmae of funcon h c( ) as s provded by he nework. Then he esmae of specfc ron losses, say Sˆ a, of a ransformer wh arbues a( ) s gven as S ˆ a = hˆ c ( a( )) (12) As can be seen, n equaons (11) and (12), he acual specfc ron losses (n W/Kg) have been used as oupu of he neural nework model, nsead of he acual ron losses (n W). Ths selecon mproves he nework performance (generalzaon) snce normalzaon of he nework oupu s performed per wegh un. Furhermore, neural nework ranng s made more effcen by usng such a normalzaon scheme. Then, he acual ransformer ron losses are calculaed by mulplyng Sˆ a by he sum of he acual weghs of he four ndvdual cores ha assemble he ransformer. Selecon of he mos approprae envronmen s performed durng he desgn phase, where he ype of he magnec maeral and he respecve suppler are deermned. Consequenly, he envronmen ype s known before he ransformer consrucon. 9

10 The neural nework srucure used o approxmae h c( ) s depced n Fgure 4. As s observed, he nework consss of a hdden layer of n neurons, J npu elemens and one oupu neuron. In our case, a lnear oupu un s used, snce he nework approxmaes a connuous valued sgnal,.e., he specfc ron losses of a ransformer. The number of hdden neurons n, as well as he nework weghs are appropraely esmaed based on a consrucve ranng algorhm, whch s descrbed n he followng subsecon. Furhermore, a Decson Tree (DT) mehodology s adoped o selec he mos approprae arbues used as npus o he nework among a large number of canddaes ones (see subsecon 4.2). Fnally, subsecon 4.3 presens a wegh adapaon algorhm used o adap he nework weghs n case ha a slgh modfcaon of he envronmen condons s encounered. 4.1 Nework Tranng and Generalzaon Issues The neural nework sze affecs he predcon accuracy. Parcularly, a small nework s no able o approxmae complcaed non-lnear funcons, snce few neurons are no suffcen o mplemen all possble npu-oupu relaons. On he oher hand, recen sudes on nework learnng versus generalzaon, such as he VC dmenson [16], [17] ndcae ha an unnecessarly large nework heavly deeroraes nework performance. In hs paper, he consrucve algorhm, presened n [18], has been adoped o smulaneously esmae he nework sze and he respecve nework weghs. Usually, consrucve approaches presen a number of advanages over oher mehods used for nework sze selecon. More specfcally, n a consrucve scheme, s sraghforward o esmae an nal sze for he nework. Furhermore, n case ha many neworks of dfferen szes provde accepable soluons, he consrucve approach yelds he smalles possble sze [18]. Le us denoe as h ˆ c, n( ) he funcon, whch mplemens he neural nework of Fgure 4, n case ha n hdden neurons are used. The subscrp c s omed n he followng analyss snce we refer o a specfc envronmen. If we denoe as r j ( ), j=1,2,,n he funcon ha he jh hdden neuron mplemens, hen he nework oupu s gven as n Sˆ a ˆ, n hn ( a( )) = u jrj ( a( )) j 1 (13) = where u j s he wegh, whch connecs he jh hdden neuron o he oupu neuron (see Fgure 4) and S a ˆ, n he esmae of he acual specfc ron losses provded by a nework of n hdden neurons. Based on he neural nework srucure of Fgure 4, funcon r j ( ) s wren as r = J T j ( a ( )) f ( w j a( ) + ϑ j ) (14) k = 1 10

11 where f ( ) denoes he acvaon funcons of hdden neurons (he sgmod n our case), w j he wegh vecor, whch connecs he jh hdden neuron wh he npu layer and as ϑ j he bas of he jh hdden neuron. Le us now assume ha a new un (neuron) s added o he hdden layer of he nework. Le us also denoe as Sˆ a, n + 1 he esmae of specfc acual losses provded by a nework of n+1 hdden uns. Then, based on equaon (13), he followng relaonshp s sasfed Sˆ a ˆ ( ( )) ˆa ( ( )), n h 1 n 1 a S, n u 1 n 1rn 1 a + = = + + = hˆ n( a( )) + un+ 1rn + 1( a( )) (15) In he prevous equaon, r n+1( ) refers o he funcon ha mplemens he new added hdden neuron. As resuls from equaon (14), funcon r n+1( ) s defned by he wegh vecor w n+ 1 and he respecve bas ϑ n+ 1. In he adoped consrucve mehod, only he parameers assocaed o he new hdden un are permed o change,.e., he weghs w n+ 1, he bas ϑ n+ 1 and he wegh oupu u n+ 1. All he oher nework weghs are consdered fxed. In parcular, he new nework weghs are esmaed so ha he error beween he acual specfc ron losses and he ones esmaed by he nework decreases as a new hdden neuron s added. To esmae he new nework weghs, we nally defne he followng quany, 2 Γ < e = n rn + 1 > 2 rn + 1 (16) where en = S a Sˆ a, n = h hˆ n (17) represens he resdual error of he arge non-lnear funcon (acual specfc ron losses) and he one mplemened by a neural nework of n hdden neurons. In equaon (16), he < > corresponds o he nner produc, whle o he norm. Based on funconal analyss, has been proven n [18] ha he error e n ends o zero as he number of n ncreases,.e., lm n e n = 0, f he weghs assocaed o he new hdden added neuron are esmaed by, { w n+ 1, ϑn+ 1} = arg max Γ and (18a) < e, + 1 > + 1 = n r u n n (18b) 2 rn

12 Consequenly, f a neural nework s consruced ncremenally, wh weghs ha sasfy equaon (18) hen srongly convergence o he arge funcon s accomplshed. Maxmzaon of (18) s performed usng he algorhm of [19]. However, n pracce, he exac form of arge funcon h acually s unknown, and hus he error e n canno be drecly calculaed. For hs reason, a ranng se s used, conssng of L ransformers, all belongng o he same envronmen, o provde a conssen esmae of e n. In parcular, le us denoe as S r hs ranng se. Then, an esmae of quany Γ s gven by, ( E + 2 n( ) En) ( rn 1( a( )) rn + 1) Sr Γˆ = ( r + 2 n 1( a( )) rn + 1) S r (19) where E ( ) a ˆa a ˆ n = S S ( ( )), n = S hn a (20) s he absolue dfference beween he acual specfc ron losses and he predced ones for a nework of n hdden neurons n case of a ransformer E n ( ) and r n( ) over all samples of se S r. Sr. In equaon (19), E n and r n+ 1 are he mean values of funcons Equaon (19) expresses he correlaon beween he funcon mplemened by he new added hdden neuron and he prevous resdual error (before he new neuron s added) over all samples (ransformers) of ranng se S r. Ths means ha he new neuron compensaes he resdual error as much as possble and herefore he error over daa of he ranng se decreases as he number of hdden neurons ncreases. The generalzaon performance, however, of he neural nework,.e., he error over daa ousde he ranng se, does no keep on mprovng as more hdden uns are added. Ths s due o he fac ha a large number of hdden uns makes he nework sensve o he daa of S r. Parcularly, wha a nework s learnng beyond a number of hdden neurons s acually nose of daa of he ranng se. As a resul, he generalzaon performance sars o decrease and he ncremenal consrucon of he nework s ermnaed. In our case, hs s accomplshed by applyng he cross valdaon mehod. Accordng o hs mehod, he avalable daa are dvded no wo subses; he frs subse (ranng se) s responsble for esmang he nework parameers, whle he second subse (valdaon se) evaluaes he nework performance. The error on he valdaon se wll normally decrease durng he nal phase of ranng, as does he error on he ranng se. However, when he nework begns o overf he daa, he error on he valdaon se wll ypcally begn o rse and he consrucve ranng algorhm s ermnaed (early soppng). 12

13 4.2 Arbue Selecon Anoher facor, whch affecs he nework performance, s he ype of arbues used as nework npu. For arbue selecon, nally, a large se of canddaes s formed based on exensve research and ransformer desgners experence. Parcularly, n our case, 19 canddae arbues are examned, whch are denoed as I, =1,2,..19 and presened n Table 2. In hs able, x y U s l ( U s l ) denoes he specfc ron losses of magnec maeral a Gauss (17000 Gauss) of he lef small core s l. The specfc ron losses for he oher hree cores are denoed accordngly. denoes he sum of he acual ron losses of he four ndvdual cores ha assemble he ransformer and s defned smlarly o (6) as F a F a = F a + F a + F a + F a, (21) sl ll l r sr where a s l F and F a are he acual (measured) ron losses of he lef and rgh small ndvdual core of. s r Smlarly, a l l F and F a correspond o he acual ron losses of he lef and rgh large ndvdual core. The l r physcal meanng of he oher varables of Table 2 are explaned n secon 2. The Decson Tree (DT) Mehodology A Decson Tree (DT) mehodology [20], [21] has been adoped n hs paper for arbue selecon. Inally, an accepably creron s defned. Le us denoe as and as as follows C a he class, whch conans all accepable ransformers C u he class, whch conans all unaccepable ransformers. In our case, classes C a and C u are defned C { : P a a = < (1 + ξ ) P 0) (22a) C { : P a u = (1 + ξ ) P 0) (22b) where ξ s a consan ndcang he unaccepably hreshold. In order o descrbe he srucure of a DT, we nally presen an example n Fgure 5 creaed from a se of 1680 ransformers of he 1 s envronmen. As observed, he ree consss of wo dfferen ypes of nodes; he ermnal nodes and he non-ermnal nodes. A node s sad o be ermnal f has no chldren. On he conrary, each non-ermnal node has wo chldren and s characerzed by an approprae es (condon) of he followng form 13

14 T : I γ (23) where γ s a hreshold value of arbue I, opmally esmaed durng he DT consrucon. Ths es dchoomzes he non-ermnal node n he sense ha he lef chld conans all ransformer (samples), whch sasfy he es of paren node, whle he rgh chld conans he remanng ransformers. For each node, he number of ransformers (samples) ha hs node conans and he respecve accepably rao s also presened. Based on he accepably rao, a ermnal node s classfed o one of he wo avalable classes. In parcular, n case ha he accepably rao s greaer han 50%, he ermnal node s assgned o class assgned o class C u. The exac noaon used for each DT node of Fgure 5 s explaned n Fgure 6. C a. Oherwse, s A DT s creaed by applyng wo man operaors; he splng operaor and he soppng operaor. The frs esmaes he mos approprae es ha should be appled o a non-ermnal node, whle he second deermnes wheher a node s ermnal or no. For he spng operaor, he opmal splng rule descrbed n [20] s used n our case. More specfcally, he algorhm esmaes he es ha provdes he bes separaon of all ransformers of he examned node no accepable and unaccepable samples. The opmal slng rule s repeaed for each node of he ree, unl a node s labeled as ermnal accordng o he soppng creron. Two dfferen ypes of ermnal nodes are dsngushed; he "LEAF" and "DEADEND" nodes. A node s sad o be "LEAF" f conans ransformers, whch compleely belong (or n pracce almos compleely) n one of he wo classes. On he oher hand, a node s denoed as "DEADEND" f he gan by splng hs node provdes no sgnfcan sascal nformaon. Ths gan s deermned by he rsk level a of he DT [20]- [22]. Implemenaon Issues The rsk level a affecs he srucure of a DT. In parcular, n case a small value of rsk level s used, he ree s grown wh a small number of nodes and vce versa. However, he classfcaon performance of a DT does no keep on mprovng as s sze ncreases. For hs reason, he opmal value of rsk level a s he one ha provdes he maxmum classfcaon accuracy wh he mnmum possble DT complexy (mnmum number of ree nodes). In order o esmae he classfcaon performance of a DT, we use a dfferen evaluaon se. For each sample (ransformer) of hs se, he ess (condons) of he non-ermnal nodes are evaluaed unl a ermnal node s reached. Then, he classfcaon accuracy s compued by comparng he acual class ha hs sample belongs o, wh he class of ermnal node, whch hs sample s assgned o. Fgure 7 llusraes he classfcaon accuracy of he DT of Fgure 5 usng a se of 560 ransformers (samples) of he 1 s envronmen for rsk levels n he range of 0.001% o 10%. As can be seen, he classfcaon accuracy ncreases unl a rsk level smaller han 0.75%. Then, sars o decrease. Furhermore, 14

15 he maxmum accuracy (.e., 95.5%) s reached for rsk levels n he nerval 0.20% %. Fgure 8 llusraes he DT complexy (number of nodes) versus he rsk level. As observed, he DT complexy ncreases wh respec o he rsk level. By combnng Fgures 7 and 8, we can esmae he rsk level value ha provdes he maxmum accuracy a he mnmum possble DT complexy. Ths s acheved usng 13 DT nodes as llusraed n Fgure 5. The DT of Fgure 5 has been creaed by applyng he aforemenoned splng and soppng operaors wh a rsk level equal o 0.25%. As observed, only 5 arbues among he 19 canddae ones are exraced n hs case as he mos approprae, he I 1, I 2, I 5, I 14, and I 15. I has been observed ha he classfcaon accuracy of he DT deeroraes n case s consruced by ransformers belongng o all envronmens [12]. For hs reason, hree dfferen measuremen ses, each of hem correspondng o a specfc envronmen, are used o consruc he DT (2240, 2350 and 1980 samples respecvely). In order o exrac he mos sgnfcan arbues, whch are used as npus o he neural nework, we bul several DTs by ) randomly selecng dfferen ransformer samples of each measuremen se o buld he ree and ) by usng dfferen values of consan ξ. In our case, 30 randomly seleced ses have creaed for each measuremen se (3 30=90 ses for all envronmens), and 5 dfferen values of ξ unformly dsrbued n he nerval 7% - 15%. Then, for each case, he opmal rsk level s esmaed. Ths s performed by examnng 20 dfferen rsk levels n he nerval 0.001% -10% and he one whch maxmzes he DT classfcaon accuracy a he mnmum DT complexy s seleced as he opmal one, as descrbed above. Consequenly 9000 DTs are examned ( =9000), 450 of whch correspond o he opmal rsk level value. The laer (.e., he 450) are used for he arbue selecon. As we have observed, n mos cases, he same arbues are exraced, whereas some of hem are no seleced a all. Furhermore, he same arbues are exraced, even for ransformers belongng o dfferen envronmens. Ths s due o fac ha he envronmen ype deermnes he nfluence of an arbue value on ransformer ron losses bu no he ype of arbues. Takng no accoun all DTs, he arbues wh a probably of appearance greaer han 3% are seleced as nework npus. These arbues are presened n Table 3. I should be menoned ha n hs case we renumbered he seleced arbue ndces of Table 2 as hey are presened n consecuve order n Table 3. A small value of probably has been chosen snce s more preferable o use more arbues as npus o he nework archecure han dscard some (maybe sgnfcan for some suaons) of hem. The selecon of hese arbues s reasonable and expeced. More specfcally, arbue I 1 s he raed magnec nducon, whch s also used n order o calculae ron losses a he desgn phase by usng he loss curve. Arbues I 2 and I 3 express he average specfc losses (W/Kg a Gauss, and Gauss, respecvely) of magnec maeral of he four ndvdual cores used for ransformer consrucon. Arbue I 4 s he rao of acual over heorecal wegh of he four ndvdual cores. Arbue I 5 s equal o he rao of acual 15

16 over heorecal ron losses of he four ndvdual cores. The sgnfcance of he arbue I 5 s ha he ron losses of he hree-phase ransformer depend on he ron losses of s ndvdual cores. In he ndusral envronmen consdered, s observed ha he arrangemen of cores nfluences he assembled ransformer core losses. Ths s refleced hrough he selecon of arbues I 6, I 7, and I 8 by he DT mehodology (see Table 3). 4.3 Wegh Adapaon In some cases, he condons under whch he respecve neural nework has been raned may slghly change over me. For example, s possble ha dfferen baches of magnec maeral, belongng o he same envronmen, presen small varaons n her echncal characerscs. In such cases, he nework performance s mproved by nroducng a wegh adapaon mechansm, whch slghly modfes he nework weghs o he new condons. The wegh adapaon mechansm s acvaed when he nework performance deeroraes. Ths s accomplshed durng he evaluaon phase (see secon 3 and Fgure 3), n whch he predced ron losses are compared wh he acual ones. In case ha he average predcon error s greaer han a pre-deermned hreshold, he wegh adapaon mechansm s acvaed and new nework weghs are esmaed. The hreshold consdered s slghly greaer han he average error over all daa of a es se, whch expresses he generalzaon performance of he nework. The adapve ranng algorhm modfes he weghs so ha he nework appropraely responds o new daa, and also provdes a mnmal degradaon of he old nformaon [23], [24]. Tranng he nework, whou usng he old nformaon, bu only he new daa, would resul n a caasrophc forgeng of he prevous knowledge [25]. In our case, he algorhm proposed n [24] has been adoped o perform he wegh adapaon. 5. GENETIC ALGORITHMS FOR REDUCING IRON LOSSES In hs secon, we descrbe he algorhm used for opmal arrangemen of he ndvdual cores so ha he ron losses of all consruced ransformers are as mnmal as possble. In parcular, n he followng subsecon he problem formulaon s presened whle subsecon 5.2 descrbes he genec algorhm, whch s appled for he opmzaon. Fnally, subsecon 5.3 dscusses ssues relaed o he genec algorhm convergence. 5.1 Opmzaon of Core Groupng Process Le us denoe as c a vecor conanng one possble combnaon of he N hree-phase ransformers, =1,,N, ha can be consruced by he 2*N avalable small / large cores, c = [ T T T T 1 2 L N ], (24) where T ndcaes he ranspose of a vecor. 16

17 Vecor c s of 4 N 1 dmensons snce each ransformer s represened by a 4x1 vecor as equaon (1) ndcaes. A specfc arrangemen (combnaon) of all small and large cores, for consrucng he N hree-phase ransformers, corresponds o a gven value of vecor c. Therefore, any reorderng of he elemens of vecor c resuls n dfferen arrangemen of ndvdual cores,.e., dfferen hree-phase ransformers. Fgure 9 presens an example of vecor c n case ha sx small and sx large cores are avalable. In parcular, he seral numbers from 1 o 6 correspond o small cores, whle he numbers from 7 o 12 o large cores. A randomly seleced arrangemen of hese cores s also presened n Fgure 9 for consrucng hree dfferen ransformers. For example, he frs ransformer consss of he small cores wh labels 5 and 1 and of he large cores wh labels 10 and 12. Ths s represened by he vecor [ ] T n accordance wh equaon (1). Then, vecor c s consruced by concaenang he vecors of he hree ransformers. The core arrangemen for he oher wo ransformers s generaed accordngly and depced n Fgure 9. I s clear ha he esmaon of N ransformers wh opmal qualy (mnmum ron losses) s equvalen o he esmaon of vecor c, whch mnmzes he followng c N = arg mn = arg mn{ a op D P }, (25) = 1 c where c op s a vecor whch conans he opmal arrangemen of all avalable small / large cores so ha he acual losses over all N ransformers are mnmzed. The ransformer acual losses nvolved n (25) are esmaed by he neural nework archecure as has been descrbed n he prevous secon. However, alhough he prevous equaon provdes ransformers of opmal qualy, here s no guaranee ha all he generaed ransformers belong o he accepable class [see equaon (22a)]. For hs reason, a very large value s assgned o a ransformer whose predced ron losses sasfy equaon (22b) (or are slghly smaller o compensae predcon errors). Thus, any unaccepable core arrangemen s rejeced. As observed from equaon (25), esmaon of he opmal core arrangemen resuls n a global combnaoral opmzaon problem. Consequenly, he prevously used feedforward neural nework canno be drecly appled for mnmzng (25). Ths s due o he fac ha a feedforward neural nework s usually suable for funcon approxmaon or classfcaon bu no for funcon mnmzaon. However, oher neural neworks models, lke Hopfeld neworks or Bolzmann machnes, raned based on he smulaed annealng algorhm, can be used o fnd he opmal value c op ([26], pp , ). Genec algorhms (GA) can be also appled [27], [28]. The man advanage of hese GA schemes s ha hey smulaneously proceed mulple sochasc soluon rajecores and hus allow varous neracons among 17

18 dfferen soluons owards one or more search spaces ([29], pp.102, [27]). On he conrary, he neural nework approaches normally follow one rajecory (deermnsc or sochasc) whch s repeaed many mes unl a sasfacory soluon s reached. Furhermore, neural neworks can be appled more appropraely for funcons whose he varables are n produc form, whch s no held n our case. In he followng, a genec algorhm s proposed o perform he aforemenoned mnmzaon. Table 4 summarzes he man seps of he proposed mehod for reducng he ransformer ron losses. 5.2 The Genec Approach In he genec approach, possble soluons of he opmzaon problem are represened by chromosomes whose genec maeral corresponds o a specfc arrangemen of ndvdual cores. Ths means ha vecor c of (24) s represened by a chromosome, whle he seral numbers of ndvdual cores are consdered as he genec maeral of he chromosome. An neger number scheme s adoped for encodng he chromosome elemens (genes) as s llusraed n Fgure 9. Inally, M dfferen chromosomes, say c 1,,c M are creaed o form a populaon. In our case, M possble soluons of he groupng mehod used n he curren pracce are seleced for he nal populaon. Ths s performed so ha he genec maeral of he nal chromosomes s of somehow good qualy and hus fas convergence of he GA s acheved. The performance of each chromosome, represenng a parcular core arrangemen, s evaluaed by he sum of he predced acual ron losses of all ransformers correspondng o hs chromosome. The neural nework model s used as ron loss predcor. For each chromosome, a fness funcon s used o map s performance o a fness value, followng a rank-based normalzaon scheme. In parcular, all chromosomes c =1,2,,M are ranked n ascendng order accordng o her performance,.e., he sum of he predced ransformer losses. Le rank( c ) {1, K, M} be he rank of chromosome c (rank=1 corresponds o he bes chromosome and rank=m o he wors). Defnng an arbrary fness value F 0 for he bes chromosome, he fness F( c ) of he -h chromosome s gven by he lnear funcon F( c ) = F0 [ rank( c ) 1] µ, = 1, K, N, (26) where µ s a decremen rae and s compued n such a way ha he fness funcon F ( c ) akes always posve values, ha s µ < F0 /( M 1). The major advanage of he rank-based normalzaon s ha prevens he generaon of super chromosomes, avodng premaure convergence o local mnma, snce fness values are unformly dsrbued [27], [30]. The paren selecon mechansm hen begns by selecng approprae chromosomes (parens) from he curren populaon. The roulee wheel [28] s used as he paren selecon procedure. Ths s accomplshed by 18

19 assgnng o each chromosome a selecon probably equal o he rao of he fness value of he respecve chromosome over he sum of fness values of all chromosomes,.e., p = M p ( c ) F( c ) / = 1 F( c ) (27) where p p ( c ) s he probably of he chromosome c o be seleced as paren. Equaon (27) means ha chromosomes of hgh qualy presen hgher chance of survval n he nex generaon. Usng hs scheme, M chromosomes are seleced as canddae parens for generang he nex populaon. Obvously, some chromosomes would be seleced more han once whch s n accordance wh he Schema Theorem [28]; he bes chromosomes ge more copes, he "average" say even, whle he wors de off. Consequenly, each chromosome has a growh rae proporonal o s fness value. In he followng sep of he algorhm, couples of chromosomes (wo parens) are randomly seleced from he se of canddae ones, obaned from he paren selecon mechansm. Then, her genec maeral s maed o generae new chromosomes (offsprng). The number of couples seleced depends on a crossover rae. A crossover mechansm s also used o defne how he genes should be exchanged o produce he nex generaon. Several crossover mechansms have been repored n he leraure. In our approach, a modfcaon of he unform crossover operaor [27], [28] has been adoped. As s explaned n he followng secon, hs modfcaon does no spol he GA convergence. In hs case, each paren gene,.e., an ndvdual core, s consdered as a poenal crossover pon. In parcular, a gene s exchanged (undergone crossover), f a random varable, unformly dsrbued n he nerval [0 1], s smaller han a pre-deermned hreshold. Oherwse, he gene remans unchanged. I s possble however for an ndvdual core o appear more han once n he genec maeral of he generaed chromosome. Ths means ha one ndvdual core s placed o more han one ransformer or o more han one poson of he same ransformer, whch corresponds o an unaccepable core arrangemen [equaon (2)]. For hs reason, he followng modfcaon of he unform crossover operaor s adoped. Afer he exchange of one gene beween he wo parens, s hghly possble ha he gene appears wce n he chromosome. In hs case he gene concdng wh he new gene s replaced wh he gene before he exchange. Fgure 10 llusraes an example of he proposed crossover mechansm n case ha sx small and sx large cores are assembled o generae hree ransformers. In hs example, he wo parens exchange her genes only beween he crossover pons 2, 3 and 4 for smplcy. As observed, he genes {10,12,1} of he 1 s paren are exchanged wh he genes {8,12,3} of he 2 nd paren. By applyng hs exchange of genes, n he 1 s chromosome he genes 8 and 3 appear wce, whle genes 10 and 1 dsappear. An equvalen problem occurs n he 2 nd chromosome. For hs reason, n he 1 s chromosome he genes {10,1} are one-by-one exchanged wh genes {8,3} as Fgure 10 depcs. The same happens for he 2 nd chromosome. 19

20 The nex sep s o apply muaon o he newly creaed populaon, nroducng random gene varaons ha are useful for resorng los genec maeral, or for producng new maeral ha corresponds o new search areas [28]. Unform muaon s he mos common muaon operaors and s seleced for our opmzaon problem. In parcular, for each gene a unform number s generaed no he nerval [0 1] and f hs number s smaller han he muaon rae he respecve gene s swapped for oher randomly seleced gene of he same caegory,.e., small or large core. Oherwse, he gene remans unchanged. In our expermen, he muaon rae s seleced o be 5%. Swappng genes of he same caegory s necessary for creang vald core arrangemen. A each eraon, a new populaon s creaed by nserng he new chromosomes, generaed by he crossover mechansm, and deleng her respecve parens, so ha each populaon always consss of M chromosomes. Several GA cycles ncludng fness evaluaon, paren selecon, crossover and muaon are repeaed, unl he populaon converges o an opmal soluon. The GA ermnaes when he bes chromosome fness remans consan for a large number of generaons, ndcang ha furher opmzaon s unlkely. 5.3 GA Convergence The aforemenoned modfcaons of he crossover and muaon operaors do no effec he convergence propery of he GA. To show hs, an analyss s presened n he followng, by modelng he GA as a Markov Chan. In parcular, each sae of he Markov sae corresponds o a possble soluon of he GA,.e., a specfc vecor c. Le us denoe as D a se, whch conans all possble Markov saes. Then, for wo arbrary saes, say,,j D, we denoe as p j he ranson probably from sae o sae j. Gaherng ranson probables for all saes n D, he ranson marx of he chan s formed as P=(p j ). Snce n he GA, ranson from one sae o anoher s obaned by applyng he crossover and muaon operaor, marx P can be decomposed as follows [31], P = C M (28) where marx C ndcaes he effec of crossover operaor and marx M he effec of muaon operaor. Le us denoe as c j he elemens of marx C=( c j ). The c j express he ranson probably from sae D o he sae j D, f only he effec of he crossover operaor s aken no consderaon. Snce he crossover operaor probablscally maps any sae of D o any oher sae of D, marx C s a sochasc marx. More specfcally, a marx s sad o be sochasc f s elemens c j sasfy he followng propery, c j = 1 Marx C=( c j ) s sochasc (29) j 20

21 The prevous equaon means ha from a vald soluon (.e., a sae of D), he crossover operaor produces anoher vald soluon (.e., anoher sae of D). Ths s exacly happened wh he proposed modfcaon of he crossover operaor, snce only vald soluons are permed. On he oher hand, marx M s posve. Ths s due o he fac ha he muaon operaor s appled ndependenly o each gene of a chromosome. Furhermore, each gene can poenally undergo muaon. Consequenly, he elemens m j of marx M, whch express he ranson probables from sae D o sae j D akng no accoun only he effec of he muaon operaor, are srcly posve. m j > 0 Marx M=(m j ) s posve (30) I has been proven n [31] ha f marx C s sochasc and marx M s posve, he ranson marx P = C M [see equaon (28)] of he Markov chan s prmve (.e., here exss k>0: P k s posve). In hs case, has been shown n [31] ha he GA converges o he opmum soluon f he bes soluon s mananed over me. Ths means ha, sarng from any arbrary sae (vald soluon), he algorhm vss any oher sae (vald soluon) whn a fne number of ransons. 6. RESULTS In hs secon, we analyze he resuls obaned by applyng he proposed neural nework-genec algorhm scheme o a manufacurng ndusry followng he wound core echnology. In parcular, subsecon 6.1 presens he performance of he neural nework archecure as an accurae predcor of ransformer ron losses, whle subsecon 6.2 ndcaes he ron loss reducon whch s acheved usng he combned neural-genec scheme. Fnally, subsecon 6.3 dscusses he economc advanages ha arse by he use of he proposed scheme o he examned manufacurng ndusry. 6.1 Iron Loss Predcon To predc ransformer ron losses, we nally consruc hree ndusral measuremen ses (MS), each of whch corresponds o a specfc envronmen. In parcular, he measuremen se of he frs envronmen comprses 2240 acual ndusral samples (ransformers), whle he se of he second and hrd envronmen 2350 and 1980 samples (ransformers) respecvely. Each sample s a par of he egh arbues seleced by he DT, (see Table 3) and he assocaed acual specfc ron losses of he ransformer. The measuremen se of each envronmen s randomly paroned no hree dsjon ses; he ranng se, he valdaon se and he es se. The ranng se s used o esmae he nework parameers (.e., weghs), he valdaon se o ermnae nework ranng (see subsecon 4.1), whle he es se o evaluae he nework accuracy. Table 5 presens he number of daa ncluded n he ranng, valdaon and es se for he hree examned envronmens. 21

22 Fgure 11 llusraes he nework performance versus he number of hdden neurons over all daa n he ranng and valdaon se for he 1 s envronmen. In our case, he nework performance s evaluaed by he average absolue relave predcon error R n, whch s defned as follows 1 Rn = L Sr S a Sˆ a, n *100% S a, (31) where we recall ha S a are he acual (measured) specfc ron losses of ransformer and S a ˆ, n he predced ones n case ha he number of hdden neurons of he nework s equal n. As s observed, he error on ranng se decreases monooncally for an ncreasng number of hdden neurons. Insead, he error on valdaon se decreases unl 8 hdden neurons are added and hen sars o ncrease. Ths s called early soppng pon (8 hdden neurons) and s depced n Fgure 11. When we look a he "ranng" curve (sold lne), appears ha we could mprove he nework performance by usng more han 8 hdden neurons. Ths s due o he fac ha he proposed consrucve algorhm esmaes he new nework weghs so ha he oupu of he new added neuron compensaes he curren resdual error [see equaons (18,19,20)]. As a resul, he error over daa of he ranng se s drven o fall. In realy, however, wha he nework s learnng beyond he early soppng pon s essenally nose conaned n he ranng daa. For hs reason, he "valdaon" curve (doed lne) ncreases beyond hs pon, ndcang ha he generalzaon performance of a nework wh more han 8 hdden neurons begns o deerorae, snce overfng of he ranng daa s accomplshed. As a resul, 8 hdden neurons are seleced for he neural nework assocaed o he frs envronmen. Smlar predcon accuracy s observed for he neworks correspondng o he oher wo envronmens, where he mos approprae number of hdden neurons s esmaed o be 8 and 9 respecvely. The predcon accuracy of he neural nework versus he number of hdden neurons when we mx daa of all examned envronmens n he valdaon se, s depced n Fgure 12 (doed lne). In hs case, he envronmen ype s fed as addonal npu (arbue) o he neural nework. The absolue relave error obaned usng daa only of he frs envronmen s also ploed n hs fgure for comparson purposes. As s observed, smaller predcon error s acheved f he nework has raned usng daa of he same envronmen. Fgure 13 presens he fracle dagram or he Quanle-Quanle plo [32] of he specfc ron losses for he 1 s envronmen. In hs fgure, he real (measured) specfc ron losses are ploed versus he predced (by he loss curve and he neural nework) specfc ron losses. Perfec predcon les on a lne of 45 0 slope. I s observed ha he predcon provded by he neural nework (doed lne n Fgure 13) s closer o he opmal lne of 45 0 han he loss curve predcon (sold lne n Fgure 13). 22