A Comprehensive Study of Neutral-Point V oltage Balancing Problem in Three-Level Neutral-Point-Clamped V oltage Source PWMInvertersNikola Celanovic,Student Member,IEEE,and Dushan Boroyevich,Member,IEEEAbstract—This paper explores the fundamental limitations of neutral-point voltage balancing problem for different loading con-ditions of three-level voltage source inverters.A new model in DQ coordinate frame utilizing current switching functions is developed as a means to investigate theoretical limitations and to offer a more intuitive insight into the problem.The low-frequency ripple of the neutral point caused by certain loading conditions is reported and quantified.Index Terms—Neutral-point voltage balancing,space vector modulation,three-level converter.I.I NTRODUCTIONS INCE it’s introduction in1981[1],the three-level neutral-point-clamped(NPC)voltage source inverter(VSI),Fig.1, has been shown to provide significant advantages over the con-ventional two-level VSI for high-power applications.The main advantages are as follows.1)V oltage across the switches is only half the dc bus voltage.This feature effectively doubles the power rating of VSI’s for a given power semiconductor device.Moreover,this is achieved without additional,often cumbersome,hard-ware for voltage and current sharing.2)The first group of voltage harmonics is centered aroundtwice the switching frequency[1],[7].This feature en-ables further reduction in size,weight,and cost of passive components while at the same time improving the quality of output waveforms.On the other hand this topology also has its disadvantages.1)Three-level VSI’s require a high number of devices.2)The complexity of the controller is significantly in-creased.3)The balance of the neutral-point has to be assured.The three-level VSI was first considered with respect to high-capacity high-performance ac drive applications[1].To this day, it remains the area where this topology is most widely used [2]–[4],[7]–[9],[15],and[16].Other interesting applications ofManuscript received March10,1999;revised September22,1999.Recom-mended by Associate Editor,F.Z.Peng.The authors are with the Department of Electrical and Computer Engi-neering,Virginia Polytechnic Institute and State University,Blacksburg,V A, 24061-0111USA.Publisher Item Identifier S0885-8993(00)02327-9.Fig.1.Circuit schematic of a three-level VSI.this technology include static V AR compensation systems[11],[12],HVDC transmission systems[18],active filtering applica-tions,as well as applications in power conditioning systems forsuperconductive magnetic energy storage(SMES)[13].The neutral-point(NP)voltage balancing problem ofthree-level NPC VSI’s has been widely recognized in litera-ture.Various strategies have been presented,and successfuloperation has been demonstrated with a dc-link voltage balancemaintained.In addition,some of the proposed algorithms avoidthe narrow pulse problem[5],[9],minimize losses by notswitching the highest current[10],or share the balancing taskwith front-end converters as in[2].NP control for the carrier-based PWM has been studiedin[15]–[17].In[15],the switching frequency optimal PWMmethod is introduced.This method controls the NP by,essen-tially,adding the zero sequence voltage to the inverter output.This work was extended in[16],where the authors propose ananalytical method for analysis of the NP potential variation,show some limitations of the NP control,and also deal with thedc-link capacitors design issues.In[17],the authors analyzethe stability of the NP control based on an insightful dynamicmodel of the NP control they developed.This paper discusses the issues of NP control from the spacevector modulation(SVM)point of view.In addition,the broaderrange of inverter operating conditions is addressed,and a newmathematical formulation of NP balancing problem is given.Furthermore,low-frequency NP voltage ripple,normalized withthe output current and the size of the dc-link capacitors,is givenfor all operating conditions.0885-8993/00$10.00©2000IEEEII.P RINCIPLE OF O PERATIONAll available voltage space vectors for three-level VSI’s are shown in Fig.2.These vectors,called switching-state vectors,represent inverter output line voltages in two-dimensional(,,or),n e g a t i v e ()p o i n t o f t h e d c l i n k .S h o w n i n F i g .2,s w i t c h i n g s t a t e p o n ,f o r e x a m p l e ,i s p r o d u c i n g l i n e v o l t a g eso u t p u t i s c o n n e c t e d t o t h en e u t r a l p o i n t ,w h i c h r e s u l t s i n t h e c u r r e n t d i s t u r b i n g t h e N Pv o l t a g e b a l a n c e .N o t a l l t h e v e c t o r s a f f e c t t h e N P b a l a n c e .T h e o n e s t h a t d o a r e s u m m a r i z e d i n T a b l e I .L a r g e v e c t o r s d o n o t a f f e c t t h e N P b a l -a n c e b e c a u s e t h e y c o n n e c t t h e p h a s e c u r r e n t s t o e i t h e r t h e p o s i -t i v e o r n e g a t i v e d c r a i l ,a n d t h e N P r e m a i n s u n a f f e c t e d .M e d i u m v e c t o r s c o n n e c t o n e o f t h e p h a s e c u r r e n t s t o t h e N P m a k i n g t h e N P p o t e n t i a l d e p e n d e n t i n p a r t o n t h e l o a d i n g c o n d i t i o n s .T h e ya r e t h e m o s t i m p o r t a n t s o u r c e o f t h e N P p o t e n t i a l u nb a l a nc e .S m a l l v e c t o r s c o m e i n p a i r s .E a c h v e c t o r i n a p a i r g e n e r a t e s t h e s a m e l i n e -t o -l i n e v o l t a g e s .A s m a l l v e c t o r t h a t c o n n e c t s a p h a s e c u r r e n t t o N P p o i n t w i t h o u t c h a n g i n g t h e s i g n o f t h e c u r -r e n t w i l l b e r e f e r r ed t o a s a p o s i t i ve s m a l l v e c t o r.T h e o t h e r o n e ,c o n n e c t i n g t h e p h a s e c u r r e n t w i t h t h e n e g a t i v e s i g n ,w i l l b e c a l l e d a n e g a t i v e s m a l l v e c t o r.T h e m a j o r i t y of t h e N P v o l t ag e b a l a n c i n g s ch e m e s u s e di n S V M r e l i e s o n s o m e f o r m o f m a n i p -u l a t i o n o f s m a l l v e c t o r s i n a p a i r ,w h e r e t h e r e l a t i v e d u r a t i o n o f p o s i t i v e a n d n e g a t i v e s m a l l v e c t o r s i n a p a i r i s u s u a l l y a dj u s t e d i n o r d e r t o c o m p e n s a t e f o r t h e e r r o r i n N P.I I I .NE U T R A L P O I N T C U R R E N T M O D U L A T I O N G e n e r a l l y ,t h e t a s k o f t h r e e -l e v e l V S I ’s i s t o s y n t h e s i z e t h ed e s i r e d o u t p u t p h a s e v o l t a g es(1)w h e r e t h e m o d u l a t i o n i n d ex,asshown in Fig.3.The reference vector may be synthesized using the space vector modulation (SVM)of the three switching state vectors that are nearest to the reference vector at every sampling instant.The nearest three vectors are selected by locating the reference vector in one of the four small triangles illustrated in Fig.3.For the outer small triangle shaded in Fig.3,the reference vector is synthesizedas(2)(3)Fig.2.Switching state vectorsof three-level VSI.TABLE IN EUTRAL P OINT C URRENTi FORD IFFERENT S PACE V ECTORSFig.3.Synthesis ofVi ii i irelative duration of the positive()smallvectorswithin.In further text,the relative duration of posi-tive and negative small vectors will be called current modulationindex(()is(7)T h e e x p r e s s i o n s f o r t h e o t h e r o u t e r s m a l l t r i a n g l e (2)–(7)a r e s y m m e t r i c .F r o m (7)i t i s o b v i o u s t h a t t h e N P c u r r e n t c o n s i s t s o f t h e n o n c o n t r o l l a b l e c o m p o n e n t,,b u t a l s o o n t h e l o a d c u r r e n t a n d s m a l l v e c t o rd u t y c y c le .T h e s e a d d i t i o n a l c o n s t r a i n t s s i g n if i c a n t l y l i m i t t h e c o n t r o l a u t h o r i t y o v e r t h e N P c u r r e n t i n t h i s s m a l l t r i a ng l e r e -g i o n .T h e m i d d l e s m a l l t r i a n g l e r e g i o n ,s h o w n i n F i g .4,i s m o r ef a v o r a b l e f o r b a l a n c i ng th e N P v o l t a g e si n c e t w o s m a l l v e c t o r s a r e a v a i l a b l e .T h e r e f e r e n c e v e c t o r i s s y n t h e s i z e d i n t h e r e g i o nas(9)(11)(13)w h e r e a n da r e t h e N P -c u r r e n t m o d u l a t i o n i n d e x e s f o r t h e s m a l l v e c t o rs(16)(18)T h e i n n e r s m a l l t r i a n g l e r e g i o n i s t h e m o s t a d v a n t a g e o u s f o rt h e N P v o l t a g e b a l a n c i n g b e c a u s e o n l y t h e s ma l l v e c t o r s ,t h o s e t h a t a l l o w f u l l c o n t r o l o f t h e N P c u r r e n t (18),a r e u s e d .U n f o r t u -n a t e l y ,i n t h i s r e g i o n t h e d c -l i n k v o l t a g e i s p o o r l y u t i l i z e d ,a n di t i s r e a s o n a b l e t o e x p e c t i n v e r t e r t o o p e r a t e i n t h i s r e g i o n o n l y d u r i n g s t a r t u p a n d /o r t r a n s i e n t s.F i g .4.S y n t h e s i s o fV i n i n n e r s m a l l t r i a n g l e .I V .L OW F REQUENCY R IPPLE IN N EUTRAL P OINT C URRENT Steady-state low-frequency ripple in the NP-current is caused by periodic variation of the components in (7),(13),and (18)over the output voltage line cycle.In a steady state,the voltagereference vector (1)has constantamplitude,,,,and,as shownin Fig.6.Although the duty cycle functions in Fig.6.are continuous,they are the duty cycles of different switching state vectorsin different sectors.One way to represent how different switching state vectors,used in different sectors,affect the NP current,is to introduce current switching functions.Current switching functions define a mapping between the duty cycle of the vectors,and the NP current that those vectors produce.For example,using this representation the NP current produced by medium vectors can be representedas,is the dutycycle of the medium vector pon,andfor,andfor ,the neutral point current contribution from the medium vector isFig.6.Duty cycles of SVM for modulation index m =0:8.TABLE IIC URRENT S WITCHING F UNCTION FOR M EDIUM VECTORSBy extending this reasoning to small vectors and allsix sectors a set of current switching functions can be defined for small vectors as well.Small vectors’switching functions are shown in Tables III and IV .Finally,all these pieces can be com-bined into a single expression valid for the NP current over the entire linecycleS MALL VECTORSTABLE IVC URRENT S WITCHING F UNCTION FOR SFig.7.Weighing factors for medium vectors for m=0:8.active and reactive components of the load current,respectively. By substituting(21)into(20),the NP current can be expressedas,and so on.Note that(22)is essentially the composite expression com-bining(7),(13),and(18)into one matrix equation valid overthe full line cycle of the output voltage.The NP current in thisformulation still consists of noncontrollable current producedby the application of the medium vector,and the controllablecurrent produced by the small vectors.NP current,resulting from application of medium vectors,can be found by multiplying the direct current by the directweighingfactor.It is apparentthat quadrature component of the current will be weighed muchmore heavily,and will produce much larger low-frequency(LF)ripple than the direct component current.Similarly,NP current resulting from application of small vec-tors depends on“controllable”direct and quadrature weighingfactors multiplying the direct,,and quadrature,,load cur-rent.These factors are given in Fig.8for.These four weighing factors depend not only on small vec-tors duty cycles and the current switching functions that are de-termined by particular type of SVM used,but also on the con-trolinputsand as defined earlier.Two distinct sets ofweighing factors are given in Fig.8.One set of weighing factorswhen only positive small vectors are used(i.e.,)is indicated by a dashed line.Between these two extremecases the weighing factors can be controlled by adjusting cur-rent modulation indexes.The weighing factors for medium vectors are periodic func-tions with zero average value over a line cycle.This means thatin the ideal steady-state case,and currents are constant andthe NP current from medium vectors naturally balances over aline cycle.Finding the size of the LF ripple under these con-ditions will be used to help determine the size of the dc-linkcapacitor for a given NP voltage ripple.Note that the ratio of active and reactive weighing factors isopposite for medium and small rge means largecontrol authority over the NP current through the manipula-tion of current modulation indexes of small vectors,and smallmeans small disturbance from middle vectors.On the otherhand,large means large disturbance from middle vectors,andsmall means small control authority over NP from small vec-tors.This confirms the fact that it is much easier to suppress theLF ripple in the NP when the load has a high power factor.V.NP B ALANCE C ONTROLThere seems to be equivalence in the NP balance controlmechanism between carrier-based,and SVM-based PWMschemes.For carrier-based PWM modulation,all the controlschemes appear to be based on the same concept:they all usesome form of manipulation of output zero sequence voltage.Similarly,all the NP control schemes for SVM-based PWMschemes appear to use some form of manipulation of theredundant small vectors.Note that the difference between thephase voltages of two small vectors in a pair is,in fact,the zerosequence voltage.This seems to be another proof of the dualityof the two PWM methods.Regarding NP balancing control for SVM,and with the re-striction to NTV,three distinctive approaches to the control ofNP might be as follows.1)Passive“control,”where the positive and negative smallvector is selected alternatively in each new switchingFig.8.Weighing factors for small vectors for m=0:8.cycle.This method can work only in the case of perfectly balanced load and perfectly balanced PWM scheme, which is unlikely to happen in practice.This method would have difficulties to recover from line or load transients[15].Still,this“control”method can be used to establish a benchmark for NP controller performance.This benchmark can then be used to evaluate the perfor-mance of other NP control methods.2)Hysteresis type control is perhaps the simplest and mostpopular closed loop NP control scheme.This method requires the knowledge of the current direction in each phase.Based on that information,the small vectors that will move the NP voltage in the direction opposite from the direction of unbalance can be selected.The downside of this method is the current ripple at half the switching frequency.3)Active control schemes that control the current modula-tion indexes m and mFig.10.Normalized amplitude of the LF chargeripple.Fig.11.Capacitor sizes for specified NP ripple with and without NP control.From the analysis in the previous section,it should be ob-vious that,regardless of control scheme,the control authority over the NP current is limited,and the region where exact bal-ancing can be achieved in each switching cycle must exist.This region is given as a shaded area in Fig.9.Note that the graph is symmetrical,and that the unity power factor load represents the most favorable case.For that case,the NP voltage can be bal-anced in every switching cycle for a modulation index as highas,and load powerfactor(23)This should provide sufficient guidelines to size the dc link capacitors for any expected operation mode and desired neutral point voltage ripple value.Consider an example with 1800V dc-link voltageandA,peak phase current,and allow 1%voltage ripple(,the comparison of capacitor sizes for the feedback NP control and the case with passive NP control is summarized in Fig.11.The greatest savings in the size of capacitor can be achieved when the inverter is predominantly supplying active power,while for the operation with purely reactive power the benefits of feedback NP control diminish.VI.C ONCLUSIONIn this paper,NP balancing was investigated for all possible operating conditions of a three-level VSI.A new and general model in the DQ coordinate frame was introduced as a way to investigate the theoretical and practical limitations of NP bal-ancing problem regardless of the type of SVM used.Addition-ally,the low-frequency ripple of the neutral point voltage caused by all possible loading conditions was reported and quantified.Results presented in this study should clarify the tradeoffs be-tween the size of the dc-link capacitor,size of the NP voltage ripple,and the NP balancing method.Based on the investigations reported in this paper and the re-sults reported by other researchers,it can be concluded that the NP balancing problem in three-level NPC VSI topology does not limit the usefulness of this topology for practical applica-tions.This problem can be solved in a satisfactory way using various techniques,depending on the particular system,and its operating point constraints.R EFERENCES[1] A.Nabae,I.Takahashi,and H.Akagi,“A new neutral-point clampedPWM inverter,”IEEE Trans.Ind.Applicat.,vol.IA-17,no.5,pp.518–523,Sept./Oct.1981.[2]M.C.Klabunde,Y.Zhao,and T.A.Lipo,“Current control of a3-levelrectifier/inverter drive system,”IEEE Trans.Power Electron.,vol.11, no.1,pp.57–65,Jan.1996.[3]R.Rojas,T.Ohnishi,and T.Suzuki,“An improved voltage vector controlmethod for neutral-point-clamped inverters,”IEEE Trans.Power Elec-tron.,vol.10,no.6,pp.666–672,Nov.1995.[4]H.L.Liu,N.S.Choi,and G.H.Cho,“DSP based space vector PWMfor three-level inverter with DC-link voltage 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[12]G.H.Jung,G.C.Cho,S.W.Hong,and G.H.Cho,“DSP based controlof high power static V AR compensator using novel vector product phase locked loop,”in IEEE Annu.Power Electron.Spec.Conf.Rec.,vol.1, 1996,pp.238–243.[13]H.Mao,D.H.Lee,H.Dai,F.C.Lee,and D.Boroyevich,“Evaluationand development of new power electronic technologies for supercon-ductive magnetic energy storage(SMES)using PEBB,”in Proc.VPEC Sem.Proc.,Sept.1997,pp.129–134.[14]V.H.Prasad,S.Dubovsky,N.Celanovic,R.Zhang,and D.Boroyevich,“DSP based implementation of a power electronics control system,”in Proc.VPEC Sem.,Sept.1997,pp.61–67.[15]J.Steinke,“Switching frequency optimal PWM control of a three-levelinverter,”IEEE Trans.Power Electron.,vol.7,July1992.[16]S.Ogasawara and H.Akagi,“Analysis of variation of neutral point po-tential in neutral-point-clamped voltage source PWM inverters,”in Proc.IEEE Ind.Applicat.Soc.Conf.Rec.,1993,pp.965–970.[17] C.Newton and M.Summer,“Neutral point control for multi-level in-verters:Theory,design and operation limitations,”in IEEE Ind.Ap-plicat.Soc.Conf.Rec.,1997,pp.1336–1343.[18]G.Lipphardt,“Using a three-level GTO voltage source inverter in aHVDC transmission system,”in Proc.EPE Conf.Rec.,Sept.1993,pp.151–155.Nikola Celanovic(S’95)received the B.S.degreein electrical engineering from the University ofNovi Sad,Yugoslavia,in1994,the M.S.degree inmechanical engineering from Vanderbilt University,Nashville,TN,in1996,and is currently pursuingthe Ph.D.degree at the Virginia Polytechnic Instituteand State University(Virginia Tech),Blacksburg.He is a Graduate Research Assistant with theCenter for Power Electronics Systems,VirginiaTech.During the summer of1999,he was a summerintern with the General Electric CR&D Center, Schenectady,NY,where he was working on modeling and control of multilevel three-phase drive systems.His research interests include modeling,control design,and applications of high-power,high-frequency power electronicssystems.Dushan Boroyevich(M’99)received the B.S.degreefrom the University of Belgrade,Yugoslavia,in1976,the M.S.degree from the University of Novi Sad,Yu-goslavia,in1982,and the Ph.D.degree from the Vir-ginia Polytechnic Institute and State University(Vir-ginia Tech),Blacksburg,in1986.From1986to1990,he was an Assistant Professorand Director of the Power and Industrial ElectronicsResearch Program,Institute for Power and Elec-tronic Engineering,University of Novi Sad,andlater,Acting Head of the Institute.In1990,he joined the Bradley Department of Electrical and Computer Engineering,Virginia Tech,as an Associate Professor.From1996to1998,he was an Associate Director with the Virginia Power Electronics Center,and since1998,has been the Deputy Director of the NSF Engineering Research Center for Power Electronics Systems,where he is now a Full Professor.His research interests include multiphase power conversion,high-power PWM converters,modeling and control of power converters,applied digital control,and electrical drives. He has published over100technical papers,has three patents,and has been involved in numerous government and industry-sponsored projects in the areas of power and industrial electronics.Dr.Boroyevich is a member of the IEEE Power Electronics Society AdCom, IEEE Industry Applications Society Industrial Power Converter Committees, and Phi Kappa Phi.。
