Geeralizatio of Selective Harmoic Cotrol/Elimiatio J.R. Wells, P.L. Chapma, P.T. rei Graiger Ceter for Electric Machiery ad Electromechaics Departmet of Electrical ad Computer Egieerig Uiversity of Illiois at Urbaa-Champaig Urbaa, IL U.S.A. wells@uiuc.edu, plchapma@uiuc.edu, krei@uiuc.edu Abstract Previous work o selective harmoic elimiatio/cotrol has made fudametal assumptios that eforce output waveform quarter- or half- wave symmetry, presumably i order to reduce the complexity of the resultig equatios. However, the quarter- or half-wave symmetric assumptio is ot required ad it restricts the solutio space, which ca result i sub-optimal solutios with regard to the ucotrolled harmoic distributio. More geeral formulatios ca be proposed which have varyig degrees of additioal complexity. I order to uderstad how these more geeral formulatios ca be obtaied, a qualitative descriptio of the waveform costructio process for the two-level waveform case will be discussed followed by presetatio of the resultig system of equatios. This two-level case is the geeralized to the m- level, -harmoic cotrol problem. Fially, this geeralizatio is used to aalyze three-level waveforms. All solutios preseted i this paper are uattaiable utilizig previous techiques. I. INTRODUCTION Harmoic elimiatio has bee a research topic sice the early 96 s, first examied i [] ad developed ito a mature form i [-4] durig the 97 s. Harmoic elimiatio, a reduced switchig techique for iverters, provides direct cotrol over output waveform harmoics. This makes it a viable alterative to stadard voltage sourced iversio (quasi-square wave switchig sigals) or low frequecy PWM i high power coversio applicatios. I the past, the problem has bee formulated from a few differet perspectives, all assumig quarter-wave symmetry (except [5] which eforces half-wave symmetry). The most familiar formulatio costructs a output waveform, two-level or three-level, by otchig a pre-existig square wave with each otch represetig a harmoic cotrolled [-3]. Aother approach uses a double pulse waveform as a basis fuctio to create the desired two- or three-level output waveform [4]. Both formulatios result i the same Fourier series represetatio for a two-level or three-level waveform. The resultig equatios based o the Fourier series for the twolevel waveform are described by si k i f m ( -) cos θ = = + θi N () 4 where is the th elemet i a set of cotrolled harmoics N havig k elemets, θ is a vector of legth k, ad q i is the i th switchig agle. The magitude of the harmoic cotet i th the harmoic is m. The three-level waveform equatios are si k i+ f ( θ ) = m (-) cos = θi N () 4 The set of all equatios geerated from () ad () have multiple solutios, which ca be obtaied usig iterative methods [6, 7], elimiatio theory [8], homotopy methods [5] or optimizatio theory [9, ]. Aother formulatio [, ], which maitais quarter-wave symmetry, uses Walsh fuctios as a basis to create the desired harmoic elimiatio waveform. This method results i a set of algebraic matrix equatios. The quarter-wave symmetry assumptio guaratees that the eve harmoics will be zero ad that all harmoics will be either i phase or ati-phase with the fudametal [4]. Although this is coveiet, the quarter-wave symmetry costrait limits the solutio space. If the quarter-wave symmetry costrait is relaxed to a half-wave symmetry costrait as preseted i [6], the eve harmoics are still zero but ow the harmoic phasig is free to vary. The method proposed i this paper advaces the work doe i [3] ad uses geeral periodic switchig fuctios (GPSF) [3] as a basis for the harmoic elimiatio waveform s creatio. Ulike the quasi-square wave basis, this basis fuctio gives freedom to place every switchig edge i the waveform idepedetly. It is show that this more geeral problem formulatio results i a ifiite umber of solutios due to the uder-costraied ature of the resultig system of equatios. The o-half-wave symmetric two-level waveforms have uique ucotrolled harmoic profiles which vary sigificatly from traditioal results [6, 8, 3]. As such, some solutios may have merit relative to others with respect to system losses, ripple characteristics, or some other system aspect. For example, these uique harmoic profiles ca be chose such that they place sigificat eergy i triple harmoics, a beefit for applicatios with balaced threephase wye-coected loads. -783-933-4/5/$. 5 IEEE. 358
This paper presets the two-level problem formulatio ad aalyzes several solutios for cases where triple harmoics are cotrolled ad ot cotrolled. The problem formulatio is the geeralized to the m-level, -harmoic case. Fially, this geeralizatio is used to aalyze three-level waveforms. Several solutios to the harmoic cotrol problem are preseted which are uobtaiable with previous techiques icludig the geeralizatio of [5]. II. TWO-LEVEL WAVEFORMS A. Problem Formulatio As discussed i the itroductio, the restrictio of quarterwave or half-wave symmetry costrais the selective harmoic cotrol problem. Although this reduces the complexity of the trascedetal equatios, it reduces the solutio space which may result i a suboptimal solutio for a give applicatio. To relax the symmetry costraits, geeral periodic switchig fuctios [3] are summed to obtai the desired switchig waveform as depicted i Figure. This basis fuctio gives freedom to place every switchig evet i the etire waveform idepedetly. The resultig system of equatios to be solved is ow posed by (3) ad (4), where is the th elemet i a set of cotrolled harmoics N, D is a vector of legth with each elemet related to the duty ratio of the k th switchig fuctio as described by [4], φ is a vector of legth with each elemet describig the phase shift of the k th switchig fuctio, ad m ad m are the desired th ad iary compoets of the harmoic. ( D ) si f m N i i ( D, φ ) = = cos φ (3) ( D ) si f m N i i ( D, φ ) = = si φ (4) Note that the harmoic cotet ca also be described i polar coordiates such that m = m cos( γ ) (5) m = msi ( γ ) (6) where m is the magitude ad g is the phase of the th harmoic i the set N. Due to the lack of half-wave symmetry i this formulatio, eve harmoics are o loger guarateed to be zero. Thus, i additio to the equatios described by (3) ad (4), it is ow ecessary to cotrol the dc compoet of the waveform by eforcig the additioal equatio f ( D, φ ) = = Di (7). I the geeralizatio preseted i [5], cotrollig all harmoics out to a particular harmoic required at least switch max + = 4 + switchig evets where max is the maximum harmoic to be cotrolled ad is always odd. This is the same umber of switchig evets required i the traditioal case preseted i (). Usig the geeral switchig fuctio approach, for the same harmoic cotrol, oly switch max + = 4 + switchig evets are eeded. Sice each geeral periodic switchig fuctio provides two switchig evets, this leaves a additioal degree of freedom. The best utilizatio of this additioal degree of freedom remais a ope questio. Possibilities iclude simply fixig a switchig evet at a (8) (9).5.5.5.5.5.5 5 5 5 3 35 5 5 5 3 35 5 5 5 3 35 a) Geeral Periodic Switchig Fuctio (GPSF ) b) GPSF c) GPSF 3.5.5.5.5.5 -.5-5 5 5 3 35 5 5 5 3 35 5 5 5 3 35 d) GPSF 4 e) GPSF 5 f) Figure. Waveform costructio, fudametal, d, 3 rd, ad 4 th cotrolled 5 GPSFi 359
coveiet place such as the zero crossig of the wave form, imposig a additioal optimizatio costrait equatio, or elimiatig oe GPSF per every two fudametal periods. The problem posed by (3) ad (4) ca be solved usig ay of the techiques that have bee previously discussed i the cotext of the more restrictive quarter-wave symmetric problem icludig iterative approaches [6, 7], elimiatio theory [8], miimizatio techiques [9, ], homotopy methods [5] or geetic algorithms [5]. Ay solutio to the quarter-wave symmetric problem will also be a solutio to the more complete formulatio posed i this paper. B. Example Solutios Figure presets several plots that illustrate oe set of solutios to the two-level problem with the st through the th harmoics cotrolled. This requires at least geeral switchig fuctios to solve 3 equatios ( for the compoets, for the iary compoets, ad for the dc compoet). Sice each switchig fuctio itroduces two degrees of freedom (D i ad f i ), this results i 4 degrees of freedom ad the problem is thus uder-costraied. A iterative based solver was used to obtai solutios for D ad φ across a rage of modulatio depth ad the results are show i a ad b. Figure c shows a example waveform ad harmoic cotet of the solutio with a modulatio depth of.. Note that the waveform o loger possesses quarter- or half-wave symmetry ad eergy exists i the eve harmoics of the ucotrolled spectrum. Figure 3 presets similar iformatio for the case where triple harmoics are ot cotrolled as is typical whe usig harmoic elimiatio i motor drive applicatios. III. GENERAL MULTI-LEVEL WAVEFORMS A. Problem Formulatio The previous sectio detailed a more geeral problem formulatio ad selected solutios for the two-level harmoic cotrol problem which is typically ecoutered. This problem is actually a special case of the more geeral m-level, - harmoic cotrol problem defied by ( D ) si f m c N i i i ( D, φ ) = =, cos φ () ( D ) si f m c N i i i ( D, φ ) = =, si φ () f ( D, φ ) = = ci, Di + ci, (). where is the th elemet i a set of cotrolled harmoics N, D is a vector of legth with each elemet related to the duty ratio of the k th quasi-square wave as described by [4], φ is a vector of legth with each elemet describig the phase shift of the k th switchig fuctio, m ad m are the desired ad iary compoets of the th harmoic, ad c i, describes the magitude ( = ) ad offset ( = ) of the i th switchig fuctio. The magitude ad offset of the switchig fuctios, c i,, take discrete values from a set C which is defied by the available voltage levels i the..5..4.6.8 35 3 5 5 5..4.6.8 Switchig State Harmoic Magitude - 5 5 5 3 35 Agular Time Scale (degrees).5 5 5 5 3 35 4 Harmoic Number a) Duty ratio vs. modulatio depth b) Phasig of the switchig fuctios vs. modulatio depth c) Example switchig waveform ad the at a modulatio depth of. (see Appedix) Figure. Example solutios for cotrollig the st through the th harmoics for a two-level waveform.5..5..5..4.6.8 35 3 5 5 5..4.6.8 Switchig State Harmoic Magitude - 5 5 5 3 35 Agular Time Scale (degrees).5 5 5 5 3 35 4 Harmoic Number a) Duty ratio vs. modulatio depth b) Phasig of the switchig fuctios vs. modulatio depth c) Example switchig waveform ad the at a modulatio depth of. (see Appedix) Figure 3. Example Solutios for Cotrollig the st through 3 th No-triple Harmoics of a Two Level Waveform 36
coverter as l l C = m m i (3) = li l i where m is the umber of levels i the coverter ad l i ad l are elemets from the set L which cotais all possible coverter level magitudes. B. Example Solutios for a Three-Level Waveform The three-level waveform problem which is ofte discussed i harmoic elimiatio papers ca be obtaied usig the geeralizatio of the previous sectio by defiig the set L as L = {,, } (4). The by (3), the set C is defied as C,, = (5). I order to solve (-), it is ecessary to choose a appropriate umber of each type of GPSW from C. Although C icludes three possible types of GPSFs for mathematical completeess, topology cosideratios will ofte limit which choices should be used. I this example, the choice of c = (6) would require switchig all four devices i a stadard H- bridge iverter cotributig uecessarily to icreased switchig losses. Figures 4 ad 5 preset solutios to (-) for the threelevel harmoic elimiatio problem with triples cotrolled ad ucotrolled respectively. For both examples, at least GPSFs are required to solve 9 equatios (9 for the compoets, 9 for the iary compoets, ad for the dc compoet). I these examples, 5 GPSFs were chose with ad 5 were chose with c = c = (7) (8) Agai, ote that the solutios do ot possess the quarter- or half-wave symmetry that is foud i previously published solutios. IV. CONCLUSION This paper has preseted a more complete formulatio for the selective harmoic cotrol problem geeralizig the basic theory set forth by [-4] ad extedig the geeralizatio of [6]. It was show that several solutios exist which have ot bee previously idetified despite claims of completeess i several papers (although these papers did likely fid all solutios to the problem they posed). I particular, these solutios differ by ot imposig half- or quarter-wave symmetry costraits that previous solutios must have. The work preseted i this paper may have several implicatios regardig traditioal harmoic cotrol applicatios. First, as suggested by [3], there exists a optimal switchig waveform for a give harmoic cotrol goal with respect to the ucotrolled harmoic cotet. This problem formulatio gives the desiger much more flexibility i selectig the ucotrolled harmoic cotet ad may result i improvemets i a system cost fuctio such as system efficiecy, electromagetic compatibility, acoustic properties,.5.4.3....4.6.8 35 3 5 5 5..4.6.8 Switchig State Harmoic Magitude - 5 5 5 3 35 Agular Time Scale (degrees).5 5 5 5 3 35 Harmoic Number a) Duty ratio vs. modulatio depth b) Phasig of the switchig fuctios vs. modulatio depth c) Example switchig waveform ad the at a modulatio depth of. (see Appedix) Figure 4. Example Solutios for Cotrollig the st through 9 th Harmoics of a Three Level Waveform.5.4.3....4.6.8 35 3 5 5 5..4.6.8 Switchig State Harmoic Magitude - 5 5 5 3 35 Agular Time Scale (degrees).5 5 5 5 3 35 4 Harmoic Number a) Duty ratio vs. modulatio depth b) Phasig of the switchig fuctios vs. modulatio depth c) Example switchig waveform ad the at a modulatio depth of. (see Appedix) Figure 5. Example Solutios for Cotrollig the st through 3 th No-triple Harmoics of a Three Level Waveform 36
etc. Ivestigatio of such optimizatios is a subect for future research. I fact, the additioal degree of freedom may allow direct optimizatio of such a cost fuctio i the solutio process. As a fial remark, it is importat to ize whe searchig for harmoic elimiatio waveforms that multiple solutios ca arise as result of two pheomea. First, the problem itself is fudametally uder-costraied. As such, a free parameter exists which ca be varied idepedetly givig rise to a cotiuum of solutios. Secod, eve if this free parameter is fixed, the system of trascedetal equatios ca be rewritte as a system of polyomial equatios usig multiple agle idetities [8, 6-8] which iheretly has multiple solutios. APPENDIX Experimetal verificatio of example waveforms displayed i paper (Note all waveform fudametal frequecies are 6Hz): Figure 4c Figure c Figure 5c ACNOWLEDGMENT The authors would like to thak the Graiger Ceter for Electric Machiery ad Electromechaics ad Motorola Commuicatios Ceter for providig fudig ad/or facilities required for this work. Figure 3c REFERENCES [] F. G. Turbull, "Selected harmoic reductio i static dc-ac iverters," IEEE Trasactios o Commuicatio ad Electroics, vol. 83, pp. 374-378, 964. [] H.S.Patel ad R.G.Hoft, "Geeralized techiques of harmoic elimiatio ad voltage cotrol i thyristor iverters: Part II-Voltage cotrol Techiques," IEEE Trasactios o Idustry Applicatios, vol. IA-, pp. 666-673, 974. [3] H.S.Patel ad R.G.Hoft, "Geeralized techiques of harmoic elimiatio ad voltage cotrol i thyristor iverters: Part I-Harmoic elimiatio," IEEE Trasactios o Idustry Applicatios, vol. IA-9, pp. 3-37, 973. 36
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