Global transformer design optimization using deterministic and non-deterministic algorithms

Transcription

1 Global transformer desgn optmzaton usng determnstc and non-determnstc algorthms Eleftheros I. Amorals Member, IEEE Natonal Techncal Unversty of Athens 9 Iroon Polytechnou Street, 5780 Athens, Greece eamr@tee.gr Marna A. Tsl Natonal Techncal Unversty of Athens 9 Iroon Polytechnou Street, 5780 Athens, Greece marna.tsl@gmal.com Dmtros G. Papargas Independent Electrcal/Electronc Manufacturng Professonal dpapar@tee.gr Abstract -- The present paper compares the applcaton of two determnstc and three non-determnstc optmzaton algorthms to global transformer desgn optmzaton. Two determnstc optmzaton algorthms (Mxed Integer Nonlnear Programmng and Heurstc Algorthm), are compared to three non-determnstc approaches (Harmony Search, Dfferental Evoluton and Genetc Algorthm). All these algorthms are ntegrated n a desgn optmzaton software appled and verfed n the manufacturng ndustry. The comparson yelds sgnfcant conclusons on the effcency of the algorthms and the selecton of the most sutable for the transformer desgn optmzaton problem. Index Terms-- Transformers, Power transformers, Desgn optmzaton, Optmzaton methods, Algorthms, Artfcal ntellgence, Genetc algorthms, Heurstc algorthms, Software packages, Desgn methodology, Desgn for manufacture. I. INTRODUCTION In today s compettve market envronment, there s an urgent need for the transformer manufacturng ndustry to mprove transformer effcency and to reduce costs, snce hgh-qualty, low-cost products and processes have become the key to survval n the global economy. In optmum desgn of transformers, the man target s to mnmze the manufacturng cost. Therefore, the objectve functon s a cost functon wth many terms, ncludng materal costs, labor costs, and overhead costs. These component costs, as well as the constrant functons, must be expressed n terms of a basc set of desgn varables []. Determnstc methods provde robust solutons to the transformer desgn optmzaton problem. In ths context, the determnstc method of geometrc programmng has been proposed n [2] n order to deal wth the desgn optmzaton problem of both low frequency and hgh frequency transformers. Furthermore, the complex optmum overall transformer desgn problem, whch s formulated as a mxednteger nonlnear programmng problem, by ntroducng an ntegrated desgn optmzaton methodology based on evolutonary algorthms and numercal electromagnetc and thermal feld computatons, s addressed n [3]. However, the overall manufacturng cost mnmzaton s scarcely addressed n the techncal lterature, and the man approaches deal wth the cost mnmzaton of specfc components such as the magnetc materal [4], the no-load loss mnmzaton [5] or the load loss mnmzaton [6]. Technques that nclude mathematcal models employng analytcal formulas, based on desgn constants and approxmatons for the calculaton of the transformer parameters are often the base of the desgn process adopted by transformer manufacturers [7]. Apart from determnstc methods, Artfcal Intellgence technques have been extensvely used n order to cope wth the complex problem of transformer desgn optmzaton, such as genetc algorthms (s) that have been used for transformer constructon cost mnmzaton [8] and constructon and operatng cost mnmzaton [9][0], performance optmzaton of cast-resn dstrbuton transformers wth stack core technology [], torodal core transformers [2], furnace transformers [3], small low-loss low frequency transformers [4] and hgh frequency transformers [5]. s also employed for the optmzaton of dstrbuton transformers coolng system desgn n [6]. Neural network technques are also employed as a means of desgn optmzaton as n [7] and [8], where they are used for wndng materal selecton and predcton of transformer losses and reactance, respectvely. The comparson of determnstc and non-determnstc optmzaton algorthms s scarcely encountered n the relevant lterature, as n [9] where and Smulated Annealng are compared to Geometrc Programmng for hgh-frequency power transformer optmzaton. It s therefore clear that global transformer optmzaton remans an actve research area, snce several approaches for ts mplementaton have not yet been nvestgated. It must be noted that there s no sngle best optmzaton algorthm for all problems, ths s called no free lunch theorem [20]. Therefore, the purpose of the paper s to ndcate a sutable optmzaton algorthm dedcated to ths problem as well as to meet the demandng requrements of the ndustry. The present paper compares the applcaton of two determnstc and three non-determnstc optmzaton algorthms to global transformer desgn optmzaton. The appled determnstc optmzaton algorthms are the Mxed Integer Nonlnear Programmng () and Heurstc Algorthm (), whle

2 the three non-determnstc algorthms are Harmony Search (), Dfferental Evoluton () (the use of both and for transformer desgn optmzaton s ntroduced n ths paper) and Genetc Algorthm. The paper s organzed as follows: Secton II descrbes the mathematcal formulaton of the transformer desgn optmzaton problem and the software developed to mplement t. Secton III provdes a bref theoretcal background for the determnstc and non-determnstc optmzaton algorthms. Secton IV presents the results of the applcaton of the fve algorthms to four dfferent dstrbuton transformer ratngs. Fnally, Secton V concludes the paper. II. TRANSFORMER SIGN OPTIMIZATION The am of transformer desgn optmzaton s the detaled calculaton of the characterstcs of all transformer components based on prescrbed specfcatons, usng avalable materals economcally to acheve lower cost, lower weght, reduced sze, and better operatng performance. The dervaton of transformer desgns s mplemented wth the use of a software platform (TDO Transformer Desgn Optmzaton) developed to provde an ntegrated desgn, smulaton and vsualzaton envronment (Fg. ) [2]. The software package s based on advanced optmzaton technques, enablng computaton of the optmal transformer actve and mechancal part confguraton, analyss and optmzaton of coolng system, mechancal desgn of tanks, optmal selecton between dfferent core and wndng materals, losses and short-crcut mpedance analyss as well as economc evaluaton n transformer management. The ntegraton of the software n an automated desgn envronment results to sgnfcant economc benefts for transformer manufacturers, acheved wthout compromses to the qualty of the optmum desgns, as far as conformty to the desgn specfcatons and the rest of the performance parameters are concerned. TDO s used for: Techncal calculatons of wound core dstrbuton transformers (alternatve core confguratons such as stack core or wndng materals such as alumnum can be consdered) wth fully parameterzed nput data and specfcatons. Manufacturng Cost Mnmzaton ensurng satsfacton of all mposed constrants on the techncal characterstcs Optmal confguraton of coolng system based on the thermal calculatons Total Owng Cost Mnmzaton, ncorporatng the loss cost to the desgn optmzaton Sze Constraned Applcatons Customzed Desgns A number of dfferent Standards such as NEMA-TP / Department of Energy (DOE) Standard Effcency, IEEE and CENELEC can be used for the techncal constrants. Economc analyss of the optmum desgns Detaled Losses and Short-Crcut Impedance Evaluaton usng advanced feld analyss technques (FEM-Fnte Element Method) Detaled thermal performance evaluaton based on FEM Lnk to automated AutoCad routnes n order to provde a complete lst of ndustral drawngs ready to be nterpreted by the producton lne engneers or to be embedded n a complete study folder. A. Transformer Input Data In order to perform the desgn optmzaton process, the user must nsert the man sxteen nput parameters (usng the respectve felds of the man TDO software screen) concernng the transformer techncal characterstcs. These parameters are: Sngle-phase or three-phase transformer nomnal power (n kva): the rated power of the transformer guaranteed short-crcut mpedance (n %): the guaranteed value of short-crcut mpedance. prmary and secondary wndng materal: two choces are provded, copper or alumnum prmary and secondary lne-to-lne voltage (n V): the nomnal voltage of the prmary and secondary wndng prmary and secondary wndng connecton type: three choces are provded, delta, star or zg-zag (coverng all the possble connecton types) prmary and secondary wndng conductor type: fve choces are provded: sngle or double crcular or rectangular wre and sheet. operatng frequency (n Hz): two choces are provded, namely 50 and 60 Hz type of magnetc materal: three types of materals are ncluded n the software, namely M4, HB and Fe-based amorphous. The user can also nsert ther own materal through txt fles. method for the determnaton of the wndngs cross-secton: the cross-secton calculaton s based on the current densty, whch can be defned by three dfferent methods, explaned n detal the next paragraph. guaranteed no-load and load losses: they can be defned accordng to CENELEC standard or upon user selecton. The above sxteen nput parameters n the Transformer Input Parameters frame are suffcent for the dervaton of the rest of the transformer characterstcs, snce the software mplements calculatons that defne a sgnfcant number of other electrcal and mechancal data (e.g. guaranteed shortcrcut mpedance, basc nsulaton level, wndngs nsulaton type, number of coolng ducts, detals of the tank and ts corrugated panels, etc.), whle a certan number of desgn constants are predefned based on the experence of transformer desgn engneers n the manufacturng ndustry as well as expermental data on a large amount of produced and tested transformers. However, t must be noted that the possblty to access and modfy these data s provded by the software, enablng more

3 expert users on transformer engneerng to examne more specalzed desgns. In total, the transformer desgner has the opportunty to change around one hundred nput parameters, based on the desgn specfcatons, n order to nvestgate and fnally reach the best possble result. Fg.. TDO man screen [2]. B. Selecton of the Current Densty Input Method One of the crucal desgn varables durng the transformer desgn optmzaton s the calculaton of the conductors cross-secton. The conductors cross-secton derves from the current densty of the hgh voltage (HV) and low voltage (LV) wndng, whch consst crucal desgn parameters, dependent on the transformer ratng and loss category. The Current Densty Method comprses three flexble approaches for the successful defnton of the values of the HV and LV wndng current densty (n A/mm 2 ), denoted as WCD HV and WCD LV, respectvely. At the frst approach (Method a), the transformer desgner can defne drectly the value of the WCD HV and WCD LV. The man drawback of ths approach s that the transformer desgner should be qute experenced n order to correctly set ths value and drect the method to the optmal soluton. At the second approach (Method b), an nterval wth a set of dscrete c LV and c HV values for the LV and HV wndng, respectvely, can be defned. In ths case, the optmzaton algorthm wll be executed for all possble c LV.c HV combnatons of current densty. Although ths approach s more tme-consumng, t assures a global optmum desgn. At the thrd approach (Method c), the desgner can ncrease the vector of the four desgn varables (explaned at the next subsecton) nto sx. In partcular, the correct defnton of the current densty value s under the rules (supervson) of the optmzaton method. In ths way, the transformer desgner defnes the ntal, the upper and the lower value of the WCD HV and WCD LV and the proposed method fnds an optmum transformer desgn, desgnatng the values of the sx varables of the desgn vector. Method b s recommended n general snce t provdes the easest method to examne a wde range of current denstes and result to the most effcent one. Ths feature s useful to both nexperenced and experenced users, and s qute mportant n case of desgns wth dffcult specfcatons

4 (e.g. low guaranteed losses) as well as specal desgns. C. Objectve Functon By default, the objectve of the transformer desgn optmzaton s to mnmze the cost of the transformer man materals, accordng to the followng equaton: 8 mn Z( x) = mn c f ( x) () x j= where c j and f j are the unt cost ( /kg) and the weght (kg) of each component j of the eght man materals, namely: the prmary and secondary wndng materal (Fg. 2) magnetc materal (Fg. 2) nsulatng lqud nsulatng paper (Fg.2) duct strps tank sheet steel (Fg. 3) corrugated panels materal (Fg. 3) and x s the vector of the four desgn varables,.e. the number of low voltage turns, the magnetc nducton magntude (B), the wdth of core leg (D) and the core wndow heght (G) (Fg. 2). j j where DNLL denotes the desgned no-load loss (W), DLL denotes the desgned load loss (W), CRM denotes the cost of the transformer remanng materals ( ), LC denotes the labor cost ( ), M denotes the transformer sales margn (%), A denotes the equvalent no-load loss cost rate ( /W), and B denotes the equvalent load loss cost rate ( /W). Another strong pont of the proposed software s that the desgner can defne the loss evaluaton factors (A and B), ether drectly or accordng to the IEEE Standard C57.20 [22]. III. OPTIMIZATION METHODOLOGIES The structure of TDO software enables the combnaton of the detaled desgn calculatons descrbed n Secton II wth fve dfferent optmzaton methodologes and the comparson of the respectve results, as depcted n Fg. 4. Fg. 4. Optmzaton Methodologes used n TDO. A. Determnstc Methodologes For the producton of the optmum transformer desgn, two determnstc methodologes can be used n TDO, as descrbed n the followngs. Fg. 2. Actve part confguraton Fg. 3. Transformer tank. However, upon user selecton, the transformer loss cost can also be ntegrated nto the objectve functon enablng to seek for the optmum desgn based on the total ownng cost (TOC),.e., the transformer purchasng cost plus the transformer operatng cost: 8 c x + CRM+ LC j j= mntoc= mn ( + A DNLL+ B DLL) (2) x j x j j ( M) ) Methodology The mxed nteger nonlnear programmng algorthm seeks an optmum for the transformer desgn, defned by a set of nteger varables lnked to a set of contnuous varables that mnmze the objectve functon and meet the restrctons mposed on the transformer desgn problem. These restrctons are desgnated by the tolerances n the devaton between the desgned and guaranteed values of losses and short crcut mpedance [23], as well as manufacturng constrants. The objectve functon varables,.e. the desgn varables are: the number of secondary wndng turns, the magnetc nducton magntude (B), the wdth of core leg (D) and the core wndow heght (G). Snce the wndngs crosssecton s a major factor affectng the overall transformer desgn, and t s lnked to the wndngs current densty, the possblty to nsert the prmary and secondary wndng current densty to the vector of the desgn varables s provded by the software, ncreasng the number of desgn varables from four to sx. method enables non-expert users to desgn an optmum transformer wth the least possble knowledge, by provdng default nput values for the desgn vector (ntal

5 value, lower and upper value) and the soluton space accordng to the transformer nomnal ratng and voltage [24][25]. These values are based on the constructonal experence on a wde spectrum of varous dstrbuton transformer ratngs. Ths methodology s recommended for: non-expert users defnton of the range of nput varables of the desgn vector (refnement of the soluton space) desgns wth specfc techncal requrements 2) Heurstc Methodology Ths algorthm s based on mplementaton of the desgn calculatons for dscrete values of the desgn varables (n contrast to the algorthm, where the desgn varables can varate among a contnuous range of values). Each combnaton of the dscrete values of the desgn vector corresponds to a canddate soluton [26]. For each one of the canddate solutons, t s checked f all the specfcatons (lmts) are satsfed, and f they are satsfed, the manufacturng cost s estmated and the soluton s characterzed as acceptable. On the other hand, the canddate solutons that volate the specfcaton are characterzed as non-acceptable solutons and are rejected by the algorthm. Fnally, among the acceptable solutons, the transformer wth the mnmum manufacturng cost s selected, whch s the optmum transformer. However, all of the acceptable solutons are stored and lsted by manufacturng cost, provdng the user the ablty to select anyone of them, dsplay t to the man form and nvestgate ts characterstcs. Gvng n LV dfferent values for the turns of the low voltage (LV) col, n D values for the core s dmenson D, n FD tres for the magnetc nducton (flux densty), n G dfferent values for the core s dmenson G, c LV dfferent values for the LV wndng current densty and c HV dfferent values for the HV wndng current densty, the total canddate solutons, N teratons, are calculated from the followng equaton: N teratons = n LV n D n FD n G c LV c HV (3) Ths methodology s recommended for: expert users refnement of the optmum solutons provded by drect transformer desgn based on gven nput values of the desgn vector It must be noted that the defnton of the dscrete values of the desgn varables often requres pror experence n transformer desgn. Selectng a small number of teratons and ncorrect values mght result to canddate solutons that do not satsfy the mposed constrants. In case of non-expert users, t s recommended that the algorthm s used for an ntal optmzaton run and the optmum value of the desgn vector s used n order to decde the dscrete values of the desgn varables of the heurstc algorthm. B. Non-Determnstc Methodologes Apart from the aforementoned determnstc methodologes, three non-determnstc methodologes can also be used n TDO, as descrbed n the followngs [27]. ) Harmony Search Algorthm The Harmony Search algorthm () s a new metaheurstc populaton search algorthm proposed by Geem et al. [28]. was derved from the natural phenomena of muscans behavor when they collectvely play ther muscal nstruments (populaton members) to come up wth a pleasng harmony (global optmal soluton). Ths state s determned by an aesthetc standard (ftness functon). The s smple n concept, less n parameters, and easy n mplementaton. It has been successfully appled to varous benchmarkng, and real-world problems lke travelng salesman problem [29]. The man steps of are as follows [28][30]. Step ) Intalze the algorthm parameters. Step 2) Intalze the harmony memory. Step 3) Improvse a new harmony. Step 4) Update the harmony memory. Step 5) Check the termnaton crteron. These steps are descrbed n the next subsectons. a) Intalzaton of algorthm parameters The algorthm parameters are: the harmony memory sze (HMS), or the number of soluton vectors n the harmony memory; harmony memory consderng rate (HMCR); ptch adjustng rate (PAR); and the number of mprovsatons (NI), or stoppng crteron. The harmony memory s a memory locaton where all the soluton vectors (sets of decson varables) are stored. Here HMCR and PAR are parameters that are used to mprove the soluton vector, whch are defned n Step 3. b) Intalzaton of Harmony Memory In ths step, the HM matrx wth as many randomly generated soluton vectors as the HMS: x x2... xn x N x x2... xn x N..... HM =..... (4)..... HMS HMS HMS HMS x x2... xn xn HMS HMS HMS HMS x x2... xn x N Statc penalty functons are used to calculate the penalty cost for an nfeasble soluton. The total cost for each soluton vector s evaluated usng M 2 ftness( X ) = f ( X ) + α mn[0, g ( x)] + = P j= β ( ) 2 ( mn[0, hj ( x)] ) (5)

6 c) Improvsaton of a new harmony I A new harmony vector x = ( x, x 2,..., x N ) s generated, based on three crtera: ) memory consderaton, 2) ptch adjustment, and 3) random selecton. Generatng a new harmony s called mprovsaton. Accordng to memory consderaton, -th varable x = ( x x ) The HMCR, I HMS whch vares between 0 and, s the rate of choosng one value from the hstorcal values stored n the HM, whle (- HMCR) s the rate of randomly selectng one value from the possble range of values, as shown n [28]: f ( rand() < HMCR) x x x x x 2 HMS {,,..., } else x x X end where rand () s a unformly dstrbuted random number between 0 and and X s the set of the possble range of values for each decson varable. For example, an HMCR of 0.85 ndcates that A wll choose decson varable value from hstorcally stored values n HM wth 85% probablty or from the entre possble range wth 5% probablty. Every component obtaned wth memory consderaton s examned to determne f ptch s to be adjusted. Ths operaton uses the rate of ptch adjustment as a parameter as shown n the followng: f ( rand() < PAR) x = x ± rand() bw else x = x end where bw s an arbtrary dstance bandwdth for the contnuous desgn varable and rand () s unform dstrbuton between and. d) Update of harmony vector I If the new harmony vector x = ( x, x 2,..., x N ) has better ftness functon than the worst harmony n the HM, the new harmony s ncluded n the HM and the exstng worst harmony s excluded from the HM. e) Check of the termnaton crteron The A s termnated when the termnaton crteron (e.g., maxmum number of mprovsatons) has been met. Otherwse, steps 3 and 4 are repeated. Ths methodology s recommended for: non-expert users defnton of the range of nput varables of the desgn vector (refnement of the soluton space) desgns wth specfc techncal requrements (6) (7) 2) Dfferental Evoluton Algorthm Dfferental Evoluton () s a parallel drect search method whch utlzes NP D-dmensonal parameter vectors x, G, =,2,3,..., NP (8) as a populaton for each generaton G. NP does not change durng the mnmzaton process. The ntal vector populaton s chosen randomly and should cover the entre parameter space. As a rule, we wll assume a unform probablty dstrbuton for all random decsons unless otherwse stated. In case a prelmnary soluton s avalable, the ntal populaton mght be generated by addng normally dstrbuted random devatons to the nomnal soluton x nom,0. generates new parameter vectors by addng the weghted dfference between two populaton vectors to a thrd vector. Let ths operaton be called mutaton. The mutated vector s parameters are then mxed wth the parameters of another predetermned vector, the target vector, to yeld the so-called tral vector. Parameter mxng s often referred to as crossover. If the tral vector yelds a lower cost functon value than the target vector, the tral vector replaces the target vector n the followng generaton. Ths last operaton s called selecton. Each populaton vector has to serve once as the target vector so that NP compettons take place n one generaton. More specfcally s basc strategy can be descrbed as follows [3][32]: a) Mutaton For each target vector x, G, a mutant vector s generated accordng to v =, G x + r, G F ( x + r, G xr, G ) (9) 2 3 wth random ndexes r, r 2, r 3 {, 2,3,..., NP}, nteger, mutually dfferent and F>0. The randomly chosen ntegers r, r 2, and r 3 are also chosen to be dfferent from the runnng ndex, so that NP must be greater or equal to four to allow for ths condton. F s a real and constant factor [0, 2] whch controls the amplfcaton of the dfferental varaton ( x x ). r2, G r3, G b) Crossover In order to ncrease the dversty of the perturbed parameter vectors, crossover s ntroduced. To ths end, the tral vector: u = ( u, u,..., u ) (0), G+, G+ 2, G+ D, G+ s formed where, u j, G+ f ( randb( j) CR) or j= rnbr( ) u j, G+ = x j, G f ( randb( j) > CR) and j rnbr( ) () j=, 2,..., D. In () randb( j) s the jth evaluaton of a unform random number generator wth outcome [0, ]. CR s the crossover constant [0, ] whch has to be determned by the user. rnbr( ) s a randomly chosen ndex,2,,d whch ensures that u, G + gets at least one parameter from v, G+. c) Selecton To decde whether or not t should become a member of

7 generaton G+, the tral vector u, G + s compared to the target vector x, G usng the greedy crteron. If vector u, G + yelds a smaller cost functon value than x, G, then x, G + s set to u, G+ ; otherwse, the old value x, G s retaned. Ths methodology s recommended for: expert users defnton of the range of nput varables of the desgn vector (refnement of the soluton space) 3) Genetc Algorthm The Genetc Algorthm metaheurstc s tradtonally appled to dscrete optmzaton problems. Indvduals n the populaton are vectors, coded to represent potental solutons to the optmzaton problem. Each ndvdual s ranked accordng to a ftness crteron (typcally just the objectve functon value assocated wth that ndvdual). A new populaton s then formed as chldren of the prevous populaton. Ths s often the result of cross-over and mutaton operatons appled to the fttest ndvduals [33]. In our case, 30 runs of the algorthm are performed and the best soluton s chosen as the optmum one. The populaton type s bt strng of sze equal to 20. A random ntal populaton s created, that satsfes the bounds and lnear constrants of the optmzaton problem. Rank ftness scalng s employed, scalng the raw scores based on the rank of each ndvdual, rather than ts score. Stochastc unform selecton functon s used, whch lays out a lne n whch each parent corresponds to a secton of the lne of length proportonal to ts expectaton. The algorthm moves along the lne n steps of equal sze, one step for each parent. At each step, the algorthm allocates a parent from the secton t lands on. The frst step s a unform random number less than the step sze. As far as mutaton and crossover functons are concerned, the frst one s adaptve feasble (t randomly generates drectons that are adaptve wth respect to the last successful or unsuccessful generaton - a step length s chosen along each drecton so that lnear constrants and bounds are satsfed) and the second one s scattered (t creates a random bnary vector, selects the genes where the vector s a from the frst parent, and the genes where the vector s a 0 from the second parent, and combnes the genes to form the chld.). The lmt for the ftness functon s set to 0.5, whle the maxmum number of generatons (teratons) s equal to 500. Ths methodology s recommended for: expert users defnton of the range of nput varables of the desgn vector (refnement of the soluton space) IV. RESULTS AND DISCUSSION It s essental to fnd an optmum transformer that satsfes the techncal specfcatons and the purchaser needs wth the mnmum manufacturng cost. The, (determnstc group) and,, (non-determnstc group) optmzaton algorthms are appled for the desgn optmzaton of four kV three-phase dstrbuton transformers, of 60kVA, 400kVA, 630 kva and 000 kva ratng. Tables I, II, III and IV compare the respectve optmzaton results. In addton, Fgs 5-8 llustrate the comparson of the guaranteed versus desgned short-crcut mpedance, as well as the total guaranteed versus desgned losses for each examned transformer ratng based on the fve dfferent optmzaton method results. To be more precse, spder charts are used to compare and evaluate each algorthm performance for each transformer desgn, based on two mportant characterstcs: short-crcut mpedance and total losses. In each spder graph, the blue polygon represents the guaranteed values and the red straght dotted lne polygon shows the desgned values (fnal results from each optmzaton method). It must be noted that the maxmum permssble devaton between the guaranteed and desgned values s equal to ± 0% n the case of short-crcut mpedance and +0% n the case of total losses. Tables I-IV show the results of the fve optmzaton algorthms based on the rated power. In partcular, the frst four lnes of each Table show the optmum values of the desgn vector, the next four lnes depct the guaranteed losses and short-crcut mpedance, the next four present the desgned losses and short-crcut mpedance, and fnally the last two lnes refer to the cost analyss. Regardng Table I, shows to have the best performance n comparson wth the other algorthms n terms of cost. However, durng the decson makng process, values of techncal specfcatons can nfluence our fnal choce. In ths case, and have qute good behavor concernng the total losses and the short-crcut mpedance (Fg. 5), and the frst algorthm () domnates to the second one () due to the lower total losses and lower cost of the respectve optmal transformer. As a result, seems to be the best possble selecton. In the case of the 400 kva transformer (Table II) s the lowest cost soluton, however the results of or are more effcent n terms of techncal performance (better total losses) (Fg. 6). In the case of the 630 kva transformer (Table III) provdes the most effcent soluton n terms of cost. As far as losses are concerned, the soluton exhbts slghtly hgher losses compared to but an mproved short-crcut mpedance value (Fg. 7). The non-determnstc algorthms correspond to optmal desgns of hgher cost and losses but better short-crcut mpedance results (especally the algorthm). Fnally, n the case of 000 kva transformer (Table IV) and provde the best solutons whch are very close n cost and performance characterstcs. TABLE I Comparson of the Optmzaton Algorthms for the 60 KVA transformer. Under the greedy crteron, a new parameter vector s accepted f and only f t reduces the value of the cost functon. Characterstcs 60kVA

8 of the optmum transformer desgn Low voltage turns D (mm) G (mm) B (Gauss) Guaranteed Fe Guaranteed Cu Total guaranteed Guaranteed Short Crcut Impedance (%) Desgned Fe Losses (W) Cu Losses (W) Total desgned Short-Crcut Impedance (%) Cost ( ) Cost Classfcaton Guaranteed Usc (%) Desgned Usc (%) (a) Total guaranteed Total desgned (b) Fg. 5. Comparson of the guaranteed and desgned short-crcut mpedance (a), and total guaranteed and desgned losses (b) for each 60 KVA transformer desgn based on the fve dfferent optmzaton results. TABLE II Comparson of the Optmzaton Algorthms for the 400 KVA transformer. Characterstcs of the optmum transformer desgn 400kVA Low voltage turns D (mm) G (mm) B (Gauss) Guaranteed Fe Guaranteed Cu Total guaranteed Guaranteed Short Crcut Impedance (%) Desgned Fe Losses (W) Cu Losses (W) Total desgned Short-Crcut Impedance (%) Cost ( ) Cost Classfcaton

9 Total desgned Short-Crcut Impedance (%) Cost ( ) Cost Classfcaton Guaranteed Usc (%) Desgned Usc (%) (a) Guaranteed Usc (%) Desgned Usc (%) (a) Total guaranteed Total desgned (b) Fg. 6. Comparson of the guaranteed and desgned short-crcut mpedance (a), and total guaranteed and desgned losses (b) for each 400 KVA transformer desgn based on the fve dfferent optmzaton results TABLE III Comparson of the Optmzaton Algorthms for the 630 KVA transformer. Characterstcs of the optmum transformer desgn 630kVA Low voltage turns D (mm) G (mm) B (Gauss) Guaranteed Fe Guaranteed Cu Total guaranteed Guaranteed Short Crcut Impedance (%) Desgned Fe Losses (W) Cu Losses (W) Total guaranteed Total desgned (b) Fg. 7. Comparson of the guaranteed and desgned short-crcut mpedance (a), and total guaranteed and desgned losses (b) for each 630 KVA transformer desgn based on the fve dfferent optmzaton results. TABLE IV Comparson of the Optmzaton Algorthms for the 000 KVA transformer. Characterstcs of the optmum transformer desgn 000kVA Low voltage turns D (mm) G (mm) B (Gauss) Guaranteed Fe

10 Guaranteed Cu Total guaranteed Guaranteed Short Crcut Impedance (%) Desgned Fe Losses (W) Cu Losses (W) Total desgned Short-Crcut Impedance (%) Cost ( ) Cost Classfcaton Guaranteed Usc (%) Desgned Usc (%) (a) Total guaranteed Total desgned (b) Fg. 8. Comparson of the guaranteed and desgned short-crcut mpedance (a), and total guaranteed and desgned losses (b) for each 000 KVA transformer desgn based on the fve dfferent optmzaton results. In non-determnstc methodologes, as well as n determnstc methodologes, the desgn vector defnton s crucal n order to meet some desred performance objectve. For example, the explorng of the geometrcal core parameters can be ensured through careful plannng, and thus the qualty of transformer desgn can be establshed durng the defnton of ntal desgn vector values. However, the nput data of the desgn vector are deployed randomly, n the case of the non-determnstc methods. As a result, there s possblty of lttle control over nvestgatng the entre soluton space. Therefore, determnstc methods are often pursued for only a selected subset of the desgn vector wth the am of n-depth searchng of the soluton space. Accordng to the above results, provdes the best soluton both n terms of cost and operatng performance (especally on total losses). Heurstc algorthm does not guarantee optmal, or even feasble, soluton and s often used wth no theoretcal guarantee. Despte ths man dsadvantage, heurstc evaluatons stll perform an mportant role n the transformer desgn, and f mplemented properly can provde powerful results. Based on the case studes that have been carred out, t should be noted that snce the non-determnstc methods or stochastc methods use random processes, an algorthm run at dfferent tmes can generate dfferent transformer desgns. Therefore, a partcular transformer study needs to be run several tmes before the soluton s accepted as the global optmum. On the contrary, and belong to determnstc methods whch are able to fnd the global mnmum, but by an exhaustve search. In ths case, n order to avod huge calculatons, stochastc methods can provde us wth the frst suboptmal soluton, and afterwards, can be used n order to fnalze our decson. Snce no other method gves an absolute guarantee of fndng the global mnmum n a fnte number of steps, technque becomes mportant. Despte the fact that fve dfferent optmzaton technques were nvestgated n order to fnd the most economc transformer desgn wth respect to a sequence of mechancal and electrcal constrants, the transformer manufacturng factores declare that a relatve near-optmal soluton (to the optmal one) s often preferred and fnally s chosen to be constructed. Under these condtons, t s obvous that the crteron of cost s not the only factor whch should be taken nto account at the fnal decson but also the transformer specfcatons of each optmum desgn, such as the no-load and load losses and the short crcut mpedance, are vtal aspects. Based on the above-mentoned fact, and become mportant methods, snce they can store a wde range of several optmum solutons wth dfferent techncal specfcatons, especally the. It must be ponted out that the tunng of non-determnstc algorthms has derved through comparson of varous combnatons and choce of the best one between them (n order to exclude the possblty that they are not properly tuned, thus they cannot converge to the global optmum as the determnstc algorthm). The methods of stochastc nature fal to fnd the global optmum due to the fact that the optmalty of the soluton provded by them cannot be guaranteed and multple runs may result to dfferent suboptmal solutons [33], wth a sgnfcant dfference between the worst and the best one. On the other hand, determnstc methods provde more robust solutons to the

11 transformer desgn optmzaton problem and are more sutable for the search of global optmum. V. CONCLUSION In the present paper, comparson of determnstc and nondetermnstc optmzaton methods has been carred out n order to acheve optmal global transformer desgn. The desgn optmzaton has been carred out wth the use of an ntegrated software platform, whch has been expermentally verfed and ntegrated n the automated desgn process of several transformer manufacturng ndustres. The combnaton of the proposed methods s very effectve because of ts robustness, ts hgh executon speed and ts ablty to effectvely search the large soluton space. The ablty to locate the global optmum s llustrated by the applcaton to a wde spectrum of actual transformers, of dfferent power ratngs. The development of user-frendly software based on the combnaton of these methods provdes sgnfcant mprovements n the desgn process of the manufacturng ndustry. Accordng to the results, provdes the best soluton both n terms of cost and operatng performance. The methods of stochastc nature fal to fnd the global optmum due to the fact that the optmalty of the soluton provded by them cannot be guaranteed and multple runs may result to dfferent suboptmal solutons, wth a sgnfcant dfference between the worst and the best one. On the contrary, and belong to determnstc methods whch are able to fnd the global mnmum, but by an exhaustve search. It s however ponted out that the goal s not only to fnd the most economc transformer, but a desgn that meets the techncal specfcatons wth the less possble devaton from the guaranteed values. In ths context, the crteron of cost s not the only factor whch should be taken nto account at the fnal decson but also the transformer specfcatons of each optmum desgn. REFERENCES [] E. I. Amorals, M. A. Tsl, A. G. Kladas, Transformer desgn and optmzaton: a lterature survey, IEEE Trans. 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