Avalable onlne at www.scencedrect.com ScenceDrect Proceda Computer Scence 85 (206 ) 976 986 Internatonal Conference on Computatonal Modelng and Securty (CMS 206) Comparatve Analyss of NR and TBLO Algorthms n Control of Cascaded MLI at Low Swtchng Frequency V.Josh Manohar a, M.Trnad b, K.Venkata Ramana c * a&b Dept. of EEE, Guntur Engneerng College, Guntur,52209,Inda c Dept. of CSE, R.V.R&J.C College of Engneerng, Guntur,52203, Inda Abstract To control multlevel nverters n medum voltage drves applcatons whch operate at a power level of mega watts, modulaton control strateges at hgh swtchng frequency are not preferable due to hgh thermal losses and poor converter effcency. Instead, fundamental swtchng frequency technque such as selectve harmonc elmnaton technque s one of the wdely used methods. The non lnear transcendental trgonometrc SHE equatons set whch are formed n control of multlevel nverter are hghly non lnear n nature, hence t has been a challengng task for researchers to obtan feasble soluton at desred value of modulaton ndex. So far, n lterature several technques have been proposed to obtan the feasble swtchng angles n control of MLI wth less %THD. All the proposed stochastc optmzaton technques need the declaraton of algorthmc specfc parameters. Ths paper presents, comparatve analyss of NR method wth random ntal guess and novel algorthm such as teachng learnng based optmzaton technque to solve SHE equaton set at dfferent values of modulaton ndex. To valdate the proposed technque, a three phase CHB seven level nverter has been consdered and the developed algorthms are tested at dfferent modulaton ndces and comparatve analyss have been carred out. MATLAB programmng and SIMULINK envronment has been consdered to valdate proposed method. The %THD produced at dfferent values of modulaton ndces are comples wth IEEE 59-992 harmonc gudelnes too. Keywords: cascaded multlevel nverter; NR method; selectve harmonc elmnaton technque; teacher learnng based optmsaton; THD. Introducton Multlevel nverter technology s an emergng area n the past few decades as t posses the advantages of * Correspondng author. Tel.: +9-99376009 E-mal address:vjoshmanohar@gmal.com 877-0509 206 The Authors. Publshed by Elsever B.V. Ths s an open access artcle under the CC BY-NC-ND lcense (http://creatvecommons.org/lcenses/by-nc-nd/4.0/). Peer-revew under responsblty of the Organzng Commttee of CMS 206 do:0.06/j.procs.206.05.290
V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 977 syntheszng hgher, voltage levels wth lower ratng of sem conductor swtch ratngs for applcaton n medum voltage and hgh power area. Such as medum voltage tracton, medum voltage drves and hybrd electrc vehcles, power qualty. The commercal avalablty of mult level nverters n present market are classfed to three topologes such as dode clamped MLI(DC MLI), Cascaded H-Brdge MLI(CHB MLI) and Flyng capactor MLI(FC MLI). Among all these topologes, Cascaded H-brdge MLI s generally preferred topologes for many ndustral applcatons. The man lmtatons n the control of adjustable speed drves are poor converter effcency and motor bearng falure. The man Reasons for the problems above are presence of delectrc stress, crculatng currents, voltage transents, hgh, common mode voltage and swtchng frequency. Sem conductor Technology has advanced to levels where 600V can be swtched for from on to off wthn mll seconds whch n turn causes electromagnetc nference (EMI)[]. The broad classfcaton of modulaton strateges for control of varable speed drves are low swtchng frequency and hgh swtchng frequency. The hgh frequency swtchng technques as space vector pulse wdth modulaton technque and carrer based pulse wdth modulaton technques can cause the followng phenomenon. Increased thermal loss reducng system effcency Damage of motor bearng & Insulaton due hgh Sde Bands of carrer n power of order 0-30KHZ causes electromagnetc nterference Rpples n voltage & current wave forms. Instead, low swtchng frequency modulaton technques such as selectve harmonc elmnaton (SHE) has nherent advantages such as low swtchng losses, control over output voltages harmoncs and good harmoncs profle [2]. SHE Technque suffers from the drawbacks of solvng non lnear transcendental SHE equatons to provde a feasble soluton at a partcular modulaton ndex to produce low %THD to comply wth IEEE 59-992 harmonc gudelnes. Ths topc has attracted researches n past few decades and n recent tmes s use of varous soft computng technques for solvng SHE equatons to get better soluton wth less computatonal effort and tme. In lterature several technques have been proposed such as Newton Raphson Method, WALSH, Resultant theory, theory of systematc polynomals, all these technques suffers long computatonal tme, tedous calculatons and unable to provde feasble solutons durng complete range of modulaton ndex from 0 to [3]-[5]. Stochastc optmsaton technques lke, genetc algorthm, modfed speces based optmsaton technque and fre fly algorthms are used to provde the feasble solutons for SHE equaton set. All these optmsaton technques requre ntalzaton parameters such as mutaton rate, socal parameter, cogntve parameter, constrcton factor etc[6]-[8]. Recently, a novel learnng based Optmzaton Technque algorthm was proposed by Rao et.al., amed to solve machne desgn problems whch s effectve, roboust and compact n terms of codng compared to bo nspred algorthms[9]. Ths paper compares the effectveness of TBLO algorthm to that of conventonal NR method n solvng SHE equatons set whch are formed n control of three phase CHB 7-level nverter. 2. SHE Technque and Problem Formaton The selectve harmonc elmnaton method s also known as fundamental swtchng frequency method based on the harmonc elmnaton theory. It s a modulaton strategy whose goal s to determne the proper swtchng angles to elmnate specfc harmoncs especally lower harmoncs whch cause to mnmze the output waveform THD. SHE technque has several features such as drect control over the output voltage harmoncs, low swtchng frequency modulaton technque whch results n low swtchng losses and also the ablty to leave trple harmoncs whch s the multple of thrd order harmoncs. SHE method can also called as programmed PWM method and has been wdely used n applcatons such as mprovement of power qualty, proper functonng of MV drves, also HVDC systems and dstrbuton phenomenon. Hgh swtchng frequency modulaton technques lke carrer based PWM technques and space vector modulaton technques produces sde bands around carrer frequency whch results n hgh %THD. However, n control of multlevel nverter operatng at above MW power level, devce swtchng frequency above 500 Hz wll results n hgh swtchng losses and thermal losses. Hence, low swtchng frequency technque such as selectve
978 V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 harmonc elmnaton technque s one of the preferred technques at MW power levels. However, provdng the feasble solutons (swtchng angles) for the non lnear transcendental SHE set s a challengng task for the researchers over decades. As voltage levels ncreases, the number of varables to be found n SHE equatons set also ncreases, thus the computng burden also ncreases[0]. One of the preferred topology n varable speed applcatons s three phase CHB 7-level to -level nverter. Hence, three phase CHB 7-level nverter s chosen for analyss. Fg. shows the crcut confguratons of a three phase cascaded H-brdge nverter and whle Fg.2 presents the per phase output voltage waveform for a seven level nverter wth ts swtchng pattern. Applyng Fourer seres for the starcase output voltage waveform of CHB 7 level nverter, as shown n Fg., t s gven by 4vdc Van( t) (cos( k)... cos( k ))sn( k t) k,3,5.. s (2) k -brdges are connected per phase and cascaded wth each other. V DC s nput voltage of each H -brdge and total output voltage n each phase s gven by: V V V V (3) RN H H 2 H 3 S S 3 R S 4 S 2 S 2 S 32 S 42 S 22 S 3 S 33 S 43 S 23 N Fg.. Sngle phase CHB 7-level nverter In SHE technque the constrant of the swtchng angles s consdered n between 0 o and 90 o o ). Accordng to Fourer seres, when the square wave gets decomposed nto ther Fourer sne wave components, t wll not have even harmoncs because of symmetry and the trpled harmoncs can be elmnated by adjustng modulaton ndex to multples of three. Consderably the fundamental output voltage (V ) n terms of swtchng angles can be gven as:
V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 979 4vdc (cos( ) cos( 2)...cos( s)) V M I s defned as the rato of fundamental output voltage (V ) to maxmum obtanable fundamental voltage Vmax.Vmax can be acheved by makng all swtchng angles to zero degrees. V max 4s M I can be expressed as: M V 4 sv (0M ) I dc I Snce there are three H-brdges per phase for a CHB 7-level nverter t conssts wth three degrees of freedom. Among those three degrees of freedom, one degree of freedom s to produce fundamental voltage and remanng two degrees are used to elmnate lower order harmoncs.e., 5 th and 7 th. The modulaton methods used n multlevel nverters can be classfed accordng to swtchng frequency. The man am s to attan the maxmum fundamental voltage wth mnmzaton n lower order harmonc content. So wth the Combnaton of equatons () and (5), SHE equatons can be presented as [cos( ) cos( ) cos( )]/ 3 M cos(5 ) cos(5 ) cos(5 ) 0 cos(7 ) cos(7 ) cos(7 ) 0 2 2 2 3 3 3 I (4) (5) (6) (7) 3 2 - -2-3 2 3 /2 t 3-3 2-2 - Fg. 2. Per phase Voltage waveform of CHB 7-level nverter representng seven steps n output volatge.
980 V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 Cost functon Ftness functon or cost functon s formed by addng orgnal objectve functon to penalty factor. The ftness value s a measure of the exactness of a soluton wth respect to the orgnal objectve. Here, the man am s to get a set of swtchng angles so that the magntude of fundamental harmonc reaches a desred value. For each soluton the ftness functon s calculated as follows [6]. 4 2 * V V s Vh n f mn 00 50 ; * n2 (8) V hn V =, 2,...s Subject to: 0 (9) 2 3. Newton-Raphson Method NR method s one of the wdely used and well establshed technque to solve non lnear equatons.h. S. Patel & R. G. Hoft n 973[], frst appled NR method to solve SHE equatons to obtan feasble swtchng angles to elmnate harmoncs. Though, ths technque s extremely powerful one, t suffers from the drawback of requrement of good ntal guess. Snce the search space s unknown, provdng good ntal guess s so complex. Hence, n order to overcome the above mentoned drawback, an approach of any random ntal guess has been consdered for developng algorthm [2]. The steps nvolved n development of the algorthm are Step : Assume any random Intal Guess (Say ) Step 2: Set Step 3: Calculate F( )),B( ) and Jacoban matrx J( )) Step 4: Calculate error Step 5: Update the swtchng angles.e Step 6: Repeat the steps (3) to (5) for suffcent number of teratons to attan error goal. Step 7: Increment by a fxed step Step 9: Repeat steps (2) to (7) for complete range of. The proposed algorthm s developed on MATLAB programmng envronment and the effectveness of proposed algorthm s tested n solvng non lnear transcendental SHE equaton set. 4. Teachng Learnng Based Optmsaton Technque TLBO s a teachng-learnng based optmzaton algorthm based on the effect of nfluence of a teacher on the two basc modes of the learnng (known as teacher phase) and nteracton wth the other learners wthn the class (known as learner phase). Ths optmzaton algorthm consst a group of learners that are consdered as populaton and dfferent desgn varables are consdered as dfferent subjects. Among entre populaton the best soluton s teacher. The output of the TLBO algorthm s consdered among the terms of results or grades of the learners that lterally depends on the qualty of teacher. Here hghly learned person s the teacher who trans the learners to get better results among the categores of ther marks or grades. Moreover, learners gets an opportunty to learn a lot from the nteractons among themselves so, t helps wth the mprovement of ther results or grades [9]. The algorthm descrbes two basc modes of the learnng: () Through teacher (known as teacher phase) and
V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 98 () Interactng wth the other learners (known as learner phase). Implementaton of TLBO algorthm for optmzaton s as follows: Defne the optmzaton problem and ntalze the optmzaton parameters. Intalze populaton sze (P n ), number of generatons (G n ), number of desgn varables (D n ) and lmts of desgn varables (U L, L L ). Generate a random populaton accordng to the populaton sze and number of desgn varables. For TLBO, the populaton sze ndcates the number of learners and the desgn varables ndcates the subjects offered. Ths populaton s expressed as follows: x Populaton= x2... xp n, x x x 2... 22 Pn,2............ x D x 2D... xp n, D Teacher phase: Calculate the mean of populaton column-wse, whch wll gve the mean for the partcular subject as M D m, *, m2,..., m D. () The best soluton wll be act as a teacher for that teraton X (2) teacher X f ( X ) mn The teacher wll try to shft the mean from M *,D towards X *,teacher whch wll act as a new mean for the teraton. The dfference between two means s expressed as Dfference r M T M ), (3) *, D ( new, D F *, D where the value of T F s selected as or 2. The obtaned dfference s added to the current soluton to update ts values usng X Dfference (4) X new, D old, D *, D Accept X new f t gves better functon value. Learners phase: Learners ncrease ther knowledge wth the help of ther mutual nteracton. The mathematcal expresson s as follows: X X new, new, X X old, old, r ( X r ( X X ) j j X ) Termnaton crtera stop f the maxmum generaton s acheved; otherwse repeat from Teacher phase. 5. Smulaton Results and Dscussons The proposed technques lke NR method and TLBO algorthms are used to solve non lnear transcendental SHE equatons set whch are formed n control of three phase CHB 7-level nverter and comparatve analyss have been carred out. The MATLAB programmng envronment has been consdered to developed the code for the proposed technques and SIMULINK model s also developed to observe %THD & FFT analyss. (0) (5)
982 V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 5. Newton Raphson method The proposed algorthm s developed n MATLAB programmng envronment and run from modulaton ndex of 2 3 whch produce less %THD at each value of modulaton ndex are tabulated n Table I. 2 3 at varous values of modulaton ndex. It s observed that there s no soluton for SHE equatons set below the value 0.4 and above the value of 0.8. Multple solutons are attaned durng the range of 0.5 to 0.62 M I. Sngle soluton s obtaned durng the range of 0.4 to 0.49 M I and from 0.63 to 0.79 M I. The phase voltage waveform of CHB 7-level nerter whch represents seven steps of output voltage s represented n Fg. 4 and lne to lne voltage waveform s represented n Fg. 5 respetvely. Average computatonal tme requred to run the algorthm at selected values of modulaton ndces are 07.00 sec only. It s also observed the proposed NR method could not provde the feasble soluton at the modulaton ndces of values 0.85, 0.9, 0.95 and.0. Fg. 3. Swtchng angles Vs Modulaton Index Table. Swtchng angles n degrees & %THD S.No M I 2 3 %THD 0.5 39.42 56.25 80.09 3.07 2 0.55 7.9 50.39 86.50 7.3 3 0.6.82 4.7 85.75 3.26 4 0.65 25.62 52.2 64.25 0.57 5 0.7 8.30 44. 64.36 2.29 6 0.75 3.52 36.6 6.63 9.02 7 0.8.50 28.7 57.0 8.86 8 0.85 No soluton 9 0.9 No soluton 0 0.95 No soluton No soluton
V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 983 From Table I, t s observed that %THD value gradually decreases as the modulaton ndex ncreases and mnmum %THD of value 8.86% has obtaned at a modulaton ndex of value 0.8. FFT analyss at the value of value of 0.8 M I s shown n Fg. 6, t s observed that the targeted order of harmoncs say 5 th and 7 th are mnmzed to that greater extant. The magntudes of harmonc voltages are below 5%. The %THD produced has comples wth IEEE 59-992 harmonc gudelnes too. The %THDs obtaned at the values of modulaton ndex below 0.8 are hgh and needs an addtonal flters to brng the values nearer to IEEE 59-992 harmonc gudelnes too. Fg. 4. Phase voltage waveform of three phase CHB 7-level nverter. Fg. 5. Lne to lne output volatge of three phase CHB 7-level nverter
984 V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 Fg. 6. FFT analyss representng harmonc order at the M I of value 0.8. 5.2 Teachng Learnng based Optmzaton Technque: The code for Proposed TLBO algorthm s developed n MATLAB programmng envronment and the program s run at dfferent modulaton ndces from 0.5 to.0 n steps of 0.5 and at each step the obtaned swtchng angles of SHE equaton set and %THDs are tabulated n Table 2. The values of %THD obtaned at dfferent steps of modulaton ndces are decreased gradually as the modulaton ndex s ncreased. Table 2 Swtchng angles n degrees and %THD S.No M 2 3 %THD 0.5 2.59 54.8 84.09 9.42 2 0.55 8.36 47.50 85.06 7.85 3 0.6 9.8 43.70 84.95 8.37 4 0.65 7.33 30.6 85.37 8.28 5 0.7 5.26 37.35 72.70 5.27 6 0.75.96 36.3 63.48 5.8 7 0.8 9.38 3.56 39.8 2.5 8 0.85 4.02 37.07 45.8 2.9 9 0.9 6.5 3.25 43.62 0.98 0 0.95 2.70 6.2 32.25 6.95 5.23 7.05 30.98 7.30 The value of %THD obtaned at the modulaton ndex of value 0.95 s 6.95% and completely comples wth IEEE 59-992 harmonc gudelnes too. FFT analyss at 0.95 modulaton ndex s shown n Fg. 7. It s observed that all targeted order of harmoncs are mnmzed and below 5% n magntude only. It s further observed that the average computatonal tme requred for runnng the TLBO code s 6.6 sec. The potental of TLBO algorthm s, t has successfully solved SHE equaton set at the modulaton ndces of value 0.85,0.9, 0.95 and.0.
V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 985 Fg. 7. FFT analyss representng harmonc order at the M I of value 0.95. Comparatve analyss of %THD at dfferent values of modulaton ndces are shown n Fg. 8. It s observed that though the %THDs obtaned by NR method s mnmum but t could not provde the feasble solutons at the modulaton ndex of values 0.85 and above. The TLBO algorthm has successfully solved the SHE equaton set where NR method could not solve t. The TLBO algorthm has solved the SHE equaton set at hghest magntude of the fundamental voltage wth mnmum %THD of value 6.95%. The computatonal tme requred for both the algorthms are less but NR method s stll less wth the average value of 07 sec. THD NR Method %THD TBLO Method %THD Modulaton ndex Fg. 8. Comparatve analyss of %THDs at dfferent modulaton NR method and TLBO method
986 V. Josh Manohar et al. / Proceda Computer Scence 85 ( 206 ) 976 986 Conclusons Ths paper manly focused on comparatve analyss of varous algorthms to solve non lnear transcendental SHE equatons set to obtan the feasble swtchng angles n control three phase CHB 7-level nverter. It s observed that TLBO algorthm s rugged, effcent and wth less computatonal effort t has solved SHE equaton set form 0.5 to.0 values of modulaton ndex where NR method could not provde the feasble soluton at the values of modulaton ndces 0.85, 0.9, 0.95 and.0. Though, the %THDs obtaned by TLBO algorthm s hgh at dfferent values of M I are hgh but at.0 M I the %THD obtaned by TLBO algorthm s less n value say 6.95% whch comples wth IEEE 59-992 harmonc gudelnes too. The computatonal tme for NR method s very less say 07 sec but t has a problem of struck at global mnma. The TLBO algorthm takes somewhat more computatonal tme say 7 sec but t can fnd feasble solutons at dfferent values of M I where NR method could not fnd. However, t s observed that, TLBO algorthm needs several runs to fnd the near global values for ths optmzaton problem. Future research may focus on workng on Eltst TLBO optmzaton and some other optmsaton technques to overcome the above problem [4]. References. J. Rodríguez, J. S. La, and F. Z. Peng, Multlevel nverters: A survey of topologes, controls, and applcatons, IEEE Transacton on Industral electroncs, vol. 49, no. 4, pp. 724738, Aug. 2002. 2. J. Napoles, A. J. Watson, J. J. Padlla, J. I. Leon, L. G. Franquelo, P. W. Wheeler and M. A. Agurre, Selectve Harmonc Mtgaton Technque for Cascaded H-Brdge Converters wth Non-Equal DC Lnk Voltages, IEEE Transactons on power electroncs, pp. -9, 202. 3. John N. Chasson, Leon M. Tolbert, Keth J. McKenze and Zhong Du, A Complete soluton to the harmonc elmnaton problem, IEEE transactons on power electroncs, vol. 9, no. 2, pp. 49-498,March 2004. 4. J.Sun and H.Grotstollen, Solvng Nonlnear equatons for Selectve harmonc elmnated PWM usng predcted ntal values, n Proc.Int.Conf.Industral Electroncs, Control, Instrumentaton, Automaton, 992, pp. 259-264 5. J.Chasson, L. Tolbert, K. McKenZe, and Z. Du, Elmnatng harmoncs n a mult-level converter usng resultant theory of symmetrc polynomals and resultants, IEEE trans. Control Syst. Technol., Vol.3,No 2, pp. 26-223, Mar. 2005 6. Reza S, Naeem F, Mehrdad A and Syed Hamd F, Elmnaton of Low order harmoncs n Multlevel Inverter usng Genetc Algorthm, Journal of Power Electron, vol., no. 2, pp. 3239, 20. 7. Ozpnec B, Tolbert LM and Chasson JN, Harmonc Optmzaton of Multlevel Converters Usng Genetc Algorthms, IEEE Power Electron Lett, vol. 3, no. 3, pp. 9295, 2005 8. Tarafdar M, Taghzadeh H and Raz K, Harmonc Mnmzaton n Multlevel Inverters Usng Modfed Speces- Based Partcle Swarm Optmzaton, IEEE Trans Power Electron, vol.24, no. 0, pp. 22592266, 2009. 9. Venkata Rao R and Vvek P, A teachng-learnng based optmzaton algorthm for solvng complex constraned optmzaton problems, Int J Ind Eng Comput, vol. 3, pp. 535-560, 202. 0. Samr Kouro, S. La Rocca, B. Cortes, P. Alepuz, S. Bn Wu Rodrguez, J. Predctve control based selectve harmonc elmnaton wth low swtchng frequency for multlevel converters Energy Converson Congress and Exposton, 2009. ECCE 2009. IEEE, pp- 330 336.. H. S. Patel and R. G. Hoft, Generalzed harmonc elmnaton and voltage control n thyrstor nverters: Part IHarmonc elmnaton,ieee Trans.Ind. Appl., vol. IA-9, no. 3, pp. 3037, May/Jun. 973. 2. Jagdsh Kumar, Bswarup Das and Pramod Agarwal, Selectve harmonc elmnaton technque for a multlevel nverter, Ffteenth Natonal power Systems Conference (NPSC), IIT Bombay, December 2008. 3. Venkata Rao R and Vvek P, An eltst teachng-learnng based optmzaton algorthm for solvng complex constraned optmzaton problems, Int. J Ind Eng Comput., vol. 3, pp. 535560, 202.