Proceedngs of the 28 Internatonal Conference on Electrcal Machnes Paper ID 157 Optmzaton of Bg Power Low Voltage Inducton Motor usng Hybrd Optmzaton Algorthm Sebastan Łachecńsk, Mara Dems Techncal Unversty of Lodz, Insttute of Mechatroncs and Informaton Systems, 18/22 Stefanowskego Street, 9-924 Lodz, Poland, Tel: (+48)-42-6312571, fax: (+48)-42-636239 E-mal: slachecnsk@wshe.lodz.pl, mdems@p.lodz.pl Abstract- Ths paper presents results of optmzaton calculaton of the bg power low voltage nducton motor. The results receved by new three hybrd algorthms:, ESµ -R, were compared. These hybrd algorthms were created by connectng three algorthms of global optmzaton,.e. genetc algorthm GA, evolutonary strategy ES and partcle swarm optmzaton PSO wth a sutable modfed Rosenbrock s method. I. INTRODUCTION The optmzaton of the bg power low voltage nducton motors s a more complcated problem than the optmzaton of the small nducton motors and requres the soluton of the optmzaton problem wth lmtatons and contnuous, dscontnuous nteger and fractonal varables [2,3]. A bg dffculty n the optmzaton of ths object s the occurrence of the addtonal constructon and explotaton lmtatons and a very flat objectve functon. In result, the global optmum s on the edge of the permssble area or often outsde ths area. The use of the global optmzaton algorthms, n partcular genetc and evolutonary algorthms, gves good results n ths area. We can obtan better results usng the hybrd optmzaton algorthms whch combne methods of global optmzaton wth selected methods of classcal optmzaton. [1,5] II. METHODS OF THE OPTIMIZATION For the frst tme n ths optmzaton three algorthms of global optmzaton,.e. genetc algorthm GA, evolutonary strategy ES and partcle swarm optmzaton PSO were used. In these algorthms the contnuous, dscontnuous nteger and fractonal varables were taken nto account. The consderable number of dscontnuous varables.e. number of stator slots Q s, number of rotor slots Q r, and number of parallel paths of the stator wndng a sq were used. In result of dscreet changes, a number of seres turns of stator wndng N s, stator wndng ptch y s and number of parallel wres of the stator wndng a sd were obtaned. For the second tme, the values of the dscontnuous varables were stable and the results of the soluton obtaned by the algorthm of global optmzaton were the startng pont of the modfed determnstc Rosenbrock s algorthm. In the Rosenbrock s algorthm only 8 contnuous varables were used [7,8]. The collecton of 15 lmtatons was taken nto consderaton by external penalty functon. In the calculaton addtonal lmtaton of the maxmal values of the stator external dameter Dse, admssble for the assumpton mechancal values of the motor were taken nto account. Objectve functon F c n the form F c = K m + (K p + K q )* n (1) was mnmzed, where: n =1 or, K m, K e = K p + K q - materals and operatng unt costs rato to the prce of 1 kwh [5]. The lmtatons were taken nto account by ntroducng the external penalty functon. In result, the functon n the form s mnmzed: F p (x) = F c + P(x) (2) where external penalty functon P(x) s n the form: n P( x) w *( p ( x)) (3) 1 w penalty coeffcent p ( x) g ( x) g g f g ( x) f g ( x) g set value of functon of the lmtaton, g (x) - value of functon of the lmtaton for the varable n the x ndvdual p (x) - value of functon of the lmtaton rato to the value of the lmtaton. In the optmzed calculaton mathematcal crcut model of the nducton motor wth varyng parameters ncorporatng saturaton effect and skn effect as well was used [4]. The presented model wth the mult-crcut representaton of the rotor cage takes nto account the varaton of the leakage reactance due to the magnetc saturaton of the magnetc crcut. Ths model was verfed by the feld-crcut calculaton as well as the result of measures made for many nducton motors [6]. All algorthms of the optmzaton were mplemented n the computer program OPTYM [12] n Matlab v. 7. n whch for the calculaton of the successve verson of the motor the computer program SPOS1 [11] was used. (4) 978-1-4244-1736-/8/$25. 28 IEEE 1
Proceedngs of the 28 Internatonal Conference on Electrcal Machnes Paper ID 157 At the begnnng the populaton of basc soluton s generated. For every soluton, the full calculaton of the motor s performed. If durng these calculatons the constructed lmtatons wll be overdraft, the process of calculaton s stopped, whch shortens the tme of calculaton. After makng of the calculatons for all solutons of the basc populaton the results were transformed by use of the crossng or mutaton, dependng on the type of the global optmzaton algorthm. In result new populatons of the soluton are generated. III. THE RESULT OF THE CALCULATIONS The object of the nvestgaton was a three-phase, two-pole, low voltage nducton squrrel-cage motor of 38 V (trangle connected) rated output power 25 kw. For the frame sze of ths motor equal to 355 mm, the maxmal external dameter of the stator core s Dse = 6 mm. The optmzaton varables are as follows [9,1]: Ds [mm] nsde dameter of the stator core, dsw- nsde dameter of the stator yoke rato to nsde dameter of the stator core lsw- length of the stator core rato to nsde dameter of the stator core hsyw heght of the stator yoke rato to nsde dameter of the stator yoke bsdw wdth of the stator tooth on the half of the stator tooth heght rato to the slot ptch of the stator brdw- wdth of the rotor tooth on the half of the rotor tooth heght of the workng cage rato to the slot ptch of the rotor hrdx - heght of the rotor tooth of the startng cage rato to heght of the rotor tooth of the workng cage hrdpw - heght of the rotor tooth of the workng cage rato to wdth of the rotor tooth on the half of the rotor tooth heght of the workng cage The dscontnuous varables are: Qs, Qr - the number of stator and rotor slots asq the number of parallel paths of stator wndng. The lmtatons are as follows: dtets steady temperature ncrease (dtets = 115 C), Mb1m, Mk1m mnmal values of the pll-out and startng torque (Mb1m = 1.7, Mk1m=.9), Sk1m maxmal value of the startng complex power (Sk1m=1), etam - mnmal value of the motor effcency (etam=.92), Bpmax, Bymax, Bdmax - maxmal values of the flux densty n the core of the motor, bs2mn, bs3mn, hs3mn, br2mn, br3mn, hrsmn - mnmal values of the dmensons stator and rotor slots, kqs11max - maxmal value of the fllng factor of the stator slot (kqs11max=.8). Only the solutons where the external dameter of the stator core was smaller than admssble value of motor frame sze were taken nto account (Dse <= 6 mm). Fg.1 shows the results of the calculatons for the best solutons of the global optmzaton algorthms, for 1 populatons and 1 ndvduals n each of them. objectve functon Fc,48,46,44,42,4,38,36 genetc algorthm GA evolutonary strategy ES partcle swarm optmzaton PSO number of populatons 1 2 3 4 5 6 7 8 9 1 Fg. 1 The values of the objectve functon Fc vs. number of populatons for dfferent algorthms of the global optmzaton, for the lmtatons Sk1m=1, Mk1m=.9. Table I shows the values of the components of the objectve functon (for n = 1) calculated usng all optmzaton algorthms. TABLE I THE VALUES OF THE COMPONENTS OF OBJECTIVE FUNCTION FOR THE LIMITATIONS SK1M=1, MK1M=.9. Km Kp Kq Fc GA.638.2265.972.3875 ESµ.585.2135.1162.3882 PSO.67.2149.922.3678.638.2259.972.3869 ESµ -R.585.2135.1162.3882.67.2149.922.3678 Exstng motor.654.265.1555.4813 The values of the optmzaton varables and the objectve functon calculated usng all hybrd algorthms are shown n Table II. TABLE II THE VALUES OF THE OPTIMIZATION VARIABLES AND OBJECTIVE FUNCTION FOR THE LIMITATIONS SK1M=1, MK1M=.9. ESµ -R Exstng dsw 1.216 1.196 1.224 1,271 lsw 1.47 1.336 1.336 1,129 hsyw.188.198.17,219 bsdw.49.41.418,418 brdw.491.468.457,386 hrdx.175.157.114,71 hrdp 3.999 2.98 4. 3,195 Ds [mm] 32.9 3. 3. 31, asq 2 2 1 2 Qs 48 36 48 36 Qr 52 28 52 28 Fc.3869.3882.3678,4813 Table III shows the flux densty n the motor core, the magnetzng current and the basc electromagnetc parameters 978-1-4244-1736-/8/$25. 28 IEEE 2
Proceedngs of the 28 Internatonal Conference on Electrcal Machnes Paper ID 157 of the optmzed motor calculated usng all optmzaton algorthms. Addtonally, n these tables the values calculated for the exstng motor are presented. Comparng values shown n Table I and Table II, we can state that the lowest value of the total unt costs we can obtan usng algorthm partcle swarm optmzaton PSO, for receved values of the lmtatons. In ths case the total costs of the optmzed motor are 23.6% lower than for the exstng motor. TABLE III THE FLUX DENSITY, MAGNETIZING CURRENT AND THE BASIC ELECTROMAGNETIC PARAMETERS OF THE OPTIMIZED MOTOR FOR THE LIMITATIONS SK1M=1, MK1M=.9. ESµ -R Exstng Flux densty n the stator 1.612 1.597 1.589 1.795 yoke [T] Flux densty n the stator 1.746 1.778 1.589 1.683 Flux densty n the rotor 1.485 1.62 1.431 1.695 Magnetzng current [A] 8.5 91.1 69.4 142.1 Core losses [kw] 2.52 2.569 2.271 4.177 Stator wndng losses [kw] 2.817 3.75 3.23 2.668 Rotor wndng losses [kw] 1.482 1.288 1.722 2.89 Total losses [kw] 12.549 11.864 11.94 14.47 Stator current [A] 418.7 425.9 415.9 45.4 Power factor.9522.9341.957.8922 Effcency [%] 95.27 95.47 95.44 94.53 Startng current [A] 3761 3723 3495 2939 Startng Torque [Nm] 941 137 935 1131 Maxmal Torque [Nm] 398 258 2891 235 In ths case, the optmzed motor has better operatng parameters,.e. lower magnetzng current and n result lower total current and hgher power factor than the exstng motor. The optmzed motor has also lower total losses and hgher effcency than the exstng motor. The optmzed motor has worse startng parameters, but t s the result of the receved lmtaton of the maxmal value of the startng complex power Sk1m = 1 and startng torque Mk1m=.9. objectve functon Fc,66,62,58,54,5,46,42,38 genetc algorthm GA evolutonary strategy ES partcle swarm optmzaton PSO number of populatons 1 2 3 4 5 6 7 8 9 1 Fg. 2 The values of the objectve functon Fc vs. number of populatons for dfferent algorthms of the global optmzaton, for the lmtatons Sk1m=8, Mk1m=1.2. The next calculatons were made for the maxmal value of the startng complex power Sk1m = 8 and startng torque Mk1m=1.2. For these new lmtatons, the values of the objectve functon Fc calculated usng all algorthms of the global optmzaton, for 1 populatons and 1 ndvduals n each of them, are shown on the Fgures 2. Fg.3 shows the results of these calculatons for 2 numbers of the teratons for modfed Rosenbrock s method, used n the second step of the optmzaton of the motor. objectve functon Fc,42,415,41,45,4,395,39,385,38,375 genetc algorthm GA evolutonary strategy ES partcle swarm optmzaton PSO number of teratons 2 4 6 8 1 12 14 16 18 2 Fg. 3 The values of the objectve functon Fc vs. number of the teratons for modfed Rosenbrock s method, for the lmtatons Sk1m=8, Mk1m=1.2. The values of the optmzaton varables and the objectve functon calculated usng all hybrd algorthms, for the lmtatons Sk1m=8, Mk1m=1.2, are shown n Table IV. TABLE IV THE VALUES OF THE OPTIMIZATION VARIABLES AND OBJECTIVE FUNCTION FOR THE LIMITATIONS SK1M=8, MK1M=1.2. ESµ -R Exstng dsw 1.252 1.27 1.229 1,271 lsw 1.79 1.336 1.336 1,129 hsyw.18.19.17,219 bsdw.58.43.41,418 brdw.455.381.442,386 hrdx.93.6.81,71 hrdp 2.381 3.932 3.42 3,195 Ds [mm] 317.9 3. 3. 31, asq 2 1 1 2 Qs 48 48 48 36 Qr 34 4 34 28 Fc.483.383.3797,4813 Comparng values of the objectve functon for both maxmal values of the startng parameters (Table II and Table IV) we can state that the lowest value of the objectve functon gves the hybrd optmzaton algorthm. Ths algorthm gves also good results very fast - after about 1 populatons n the frst step, and 5 teratons n the second step of the optmzaton process. Table V shows the flux densty n the stator yoke, stator and rotor tooth, the magnetzng current, core and wndng losses and the basc electromagnetc parameters of the optmzed 978-1-4244-1736-/8/$25. 28 IEEE 3
Proceedngs of the 28 Internatonal Conference on Electrcal Machnes Paper ID 157 motor calculated usng all optmzaton algorthms, for the new lmtatons Sk1m=8, Mk1m=1.2. TABLE V THE FLUX DENSITY, MAGNETIZING CURRENT AND THE BASIC ELECTROMAGNETIC PARAMETERS OF THE OPTIMIZED MOTOR FOR THE LIMITATIONS SK1M=8, MK1M=1.2. ESµ -R Exstng Flux densty n the stator 1.614 1.597 1.585 1.795 yoke [T] Flux densty n the stator 1.446 1.778 1.659 1.683 Flux densty n the rotor 1.586 1.62 1.483 1.695 Magnetzng current [A] 76.6 91.1 69.4 142.1 Core losses [kw] 2.518 2.12 2.294 4.177 Stator wndng losses [kw] 2.789 3.56 3.13 2.668 Rotor wndng losses [kw] 1.839 1.723 1.644 2.89 Total losses [kw] 13.13 12.282 11.816 14.47 Stator current [A] 423.5 419.1 42.7 45.4 Power factor.9435.959.9455.8922 Effcency [%] 95.5 95.32 95.49 94.53 Startng current [A] 338 339 338 2939 Startng Torque [Nm] 963 143 11 1131 Maxmal Torque [Nm] 2458 2433 2327 235 Comparng the results obtaned for dfferent values of the lmtatons of the startng parameters we can state that more restrcted lmtatons gve a lttle bgger values of the objectve functon Fc (about 5% 7 % for the algorthms GA,, PSO,, and about 1.4% for algorthm ESµ For algorthm ESµ -R for more restrcted lmtatons the value of the objectve functon s about.4% lower than for the lmtatons Sk1m=1, Mk1m=.9. The decrease of the total unt costs (Fc) rato to the total costs of the exstng motor, calculated usng dfferent optmzaton algorthms for lmtatons of the startng parameters Sk1m=8, Mk1m=1.2 s shown n the Fg.4. 25% 2% 15% 1% 5% % 13,19% 15,17% 18,18% 2,98% 21,11% 19,11% 1 2 3 GA ES PSO Fg. 4 Decrease of the total costs of the optmzed motor rato to the total cost of the exstng motor, for the lmtatons Sk1m=8, Mk1m=1.2. In ths case the bgger decrease of the total unt costs of the optmzed motor rato to the total costs of the exstng motor we obtan usng hybrd optmzaton algorthms. Fgure 5 presents the decrease of the total unt costs (Fc) of the optmzed motor obtaned by usng hybrd algorthms rato to the total costs calculated usng only global optmzaton algorthms 4,% 3,5% 3,% 2,5% 2,% 1,5% 1,%,5%,% 2,27% 3,43% 2,47% GA/ ES/ PSO/ Fg. 5 Decrease of the total costs of the optmzed motor calculated usng hybrd algorthms rato to the total costs calculated usng only global optmzaton algorthms, for the lmtatons Sk1m=8, Mk1m=1.2. Fgures 6, 7, 8, 9 present the curves of the stator current, power factor, effcency and rotor speed versus output power of the optmzed motor, calculated usng all hybrd algorthms. current [A] 6 5 4 3 2 1 exstng motor,, power output [kw] 5 1 15 2 25 3 35 Fg. 6 Curves of the stator current of the optmzed motor calculated usng hybrd algorthms vs. output power, for the lmtatons Sk1m=8, Mk1m=1.2. power factor 1,,95,9,85,8,75,7,65,6 exstng motor power output [kw] 5 1 15 2 25 3 35 Fg. 7 Curves of the power factor of the optmzed motor calculated usng hybrd algorthms vs. output power, for the lmtatons Sk1m=8, Mk1m=1.2. 978-1-4244-1736-/8/$25. 28 IEEE 4
Proceedngs of the 28 Internatonal Conference on Electrcal Machnes Paper ID 157 effcency 1,,98,96,94,92,9,88,86 exstng motor power output [kw] 5 1 15 2 25 3 35 Fg. 8 Curves of the effcency of the optmzed motor calculated usng hybrd algorthms vs. output power, for the lmtatons Sk1m=8, Mk1m=1.2. rotor speed [rpm] 3 2995 299 2985 298 2975 297 2965 exstng motor power outpout [kw] 5 1 15 2 25 3 35 Fg. 9 Curves of the rotor speed of the optmzed motor calculated usng hybrd algorthms vs. output power, for the lmtatons Sk1m=8, Mk1m=1.2. Comparng curves shown n Fgures 6 to 9 we can state that the optmzed motor has better operatng characterstcs,.e. lower stator current and hgher power factor and effcency, for all range of the output power. The results obtaned by use of all hybrd algorthms are smlar. Fgures 1 and 11 present the curves of the stator current and torque vs. slp of the rotor. current [A] 35 3 25 2 15 1 5, exstng motor slp of the rotor,,1,2,3,4,5,6,7,8,9 1, Fg. 1 Curves of the stator current of the optmzed motor calculated usng hybrd algorthms vs. slp of the rotor, for the lmtatons Sk1m=8, Mk1m=1.2. torque [Nm] 3 25 2 15 1 5 exstng motor slp of the rotor,,1,2,3,4,5,6,7,8,9 1, Fg. 11 Curves of the startng torque of the optmzed motor calculated usng hybrd algorthms vs. slp of the rotor, for the lmtatons Sk1m=8, Mk1m=1.2. For the lmtaton Sk1m = 8 and Mk1m = 1.2 the characterstcs of the optmzed motor and exstng motor are smlar for all values of the slp of the rotor. IV. CONCLUSIONS The use of the hybrd optmzaton algorthms created by connectng dfferent algorthms of global optmzaton wth a sutable modfed Rosenbrock s method gve the soluton stuated nearby global optmum. The use n the second step of the optmzng process Rosenbrock s modfed method decreases the total unt costs of the optmzed motor about 3% rato to the results obtaned by the use of only global optmzaton algorthms. The average decrease of the total unt costs of the optmzed motor rato to the total costs of the exstng motor, calculated usng dfferent hybrd optmzaton algorthms amounts to about 2%. All hybrd algorthms gve smlar results. The operatng parameters of the optmzed motor are better than those of the exstng motor. ACKNOWLEDGMENT Mnstry of Scence and Hgher Educaton Project 3T1A 62 28 sponsors the work. REFERENCES [1] M. Dabrowsk, Desgn of Electrcal Machnes, WNT,Warszawa 1988 [n polsh]. [2] M. Dabrowsk, A. Rudeńsk, Hybrd optmzaton method n desgnng of the three-phase nducton motors wth ncreased numbers of optmzaton varables, Proceedngs of XL Internatonal Symposum on Electrcal Machnes SME 26, 3-6 July, Cracow, Poland, 26, pp.131-134 [n polsh]. [3] M. Dabrowsk, A. Rudeńsk, Dscrete varables n nondetermnstc optmzaton of the electrcal machne, Proceedngs of XL Internatonal Symposum on Electrcal Machnes SME 25, 14-17 June, Jarnołtówek, Poland, 25, pp.731-737 [n polsh]. [4] M. Dems M., Z. Rutkowsk, The calculaton of the magnetzng current and startng parameters of the nducton motors, Scentfc Bulletn Techncal Unverstety of Poznan, Elektryka, no. 4, 1992, pp.63-73, [n polsh]. [5] J. Fontchastagner, F. Messne, A. Lefevery, A new ratonal way combnng analytcal and numercal models wth a determnstc global optmzaton algorthm for the desgn of electrcal rotatng machnes 978-1-4244-1736-/8/$25. 28 IEEE 5
Proceedngs of the 28 Internatonal Conference on Electrcal Machnes Paper ID 157 Proceedngs of Internatonal Conference of Electrcal Machne ICEM 26, Greece, pp. 555, 26 [6] K. Komęza, M. Dems, S. Wak, Analyss of the nfluence of the assumpton of equvalent saturaton on startng currents n nducton motor, COMPEL, The Internatonal Journal for Computaton and Mathematcs n Electrcal and Electronc Engneerng, vol 19, No. 2, 2, pp. 463-468. [7] S. Łachecńsk, M. Dems, Hybrd optmzaton algorthms n desgnng of the systems wth occurrence contnuous and dscontnuous varables, Proceedngs of 1 st PD FCCS 25 Polsh and Internatonal PD Forum- Conference on Computer Scence, Łódź-Bronsławów, 11-14 Aprl, Poland, 25 [n polsh]. [8] S. Łachecńsk, M. Dems, Usng genetc algorthm n optmzaton of the nducton motor, Proceedngs of 2 nd PD FCCS 26 Polsh and Internatonal PD Forum-Conference on Computer Scence, Łódź- Smardzewce, 16-19 October, Poland, 26 [n polsh]. [9] S. Łachecńsk, M. Dems, S. Wak, Hybrd optmzaton algorthms n desgnng of electromechancal converters, Polsh Journal of Envronmental Studes, vol. 17, No. 2A, 28, pp. 9-94.Poland, 28. [1] T. Ślwńsk, The optmal synthess of the nducton motor, Scentfc Bulletn Techncal Unverstety of Poznan, No 4, 1992, Poland, [n polsh]. [11] M. Dems, Z. Rutkowsk, Computer program SPOS 1 Optmal desgnng of low-voltage nducton motor Computer programs lbrary of Insttute of Mechatroncs and Informaton Systems, Techncal Unversty of Lodz, Poland, 1994 [12] M. Dems, S. Łachecńsk, Z. Rutkowsk, Computer program OPTYM v.2, Optmal desgnng of low-voltage nducton motor usng hybrd optmzaton algorthms, Computer programs lbrary of Insttute of Mechatroncs and Informaton Systems, Techncal Unversty of Lodz, Poland, 27. 978-1-4244-1736-/8/$25. 28 IEEE 6