Line Voltages THD Optimization on a Multilevel Inverter for PV Systems

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1 Lie Voltages THD Optimizatio o a Multilevel Iverter for PV Systems LUIS DAVID PABON, JORGE LUIS DIAZ RODRIGUEZ, ALDO PARDO GARCIA. Departmet of Electrical Egieerig ad Mechatroics Uiversity of Pamploa Ciudadela Uiversitaria, Km via Bucaramaga, Pamploa COLOMBIA davidpabo@uipamploa.edu.co, jdiazcu@uipamploa.edu.co, apardo3@uipamploa.edu.co Abstract: - This paper presets the optimizatio of total harmoic distortio (THD) applied directly to the lie voltages output from a multilevel power iverter, with H-bridges cascaded commo source of ie steps per phase applied to a photovoltaic system. The mathematical model of the THD of lie voltages is preseted i terms of commutatio agles of the PWM phase modulatio followig the IEEE-59 stadard. This equatio gives a ew ad purely umerical optimizatio character, allowig to obtai excellet results usig geetic algorithms reachig a THD of 0.3% i lie voltages. To validate the results obtaied by mathematical modelig ad optimizatio, a algorithm usig the FPGA Virtex5 XUPV5-LX0T is implemeted. Key-Words: - Three-phase multilevel power iverter, multilevel PWM modulatio, H-bridges cascaded commo source, THD, Geetic algorithm, Optimizatio.. Itroductio Curretly there is a proouced iterest at geeratig high quality electricity, through alterative eergy sources, this is due to evirometal disasters caused by may factors, oe of them is the covetioal sources of eergy, i this cotext solar eergy has bee widely accepted because of its may advatages as a source of reewable eergy ad has preseted a cotiuous growth i recet years [], settlig large umber of photovoltaic systems aroud the world []. The productio of electricity from solar paels is give i direct compoet, usig power iverters is a complete solutio to covert AC voltage output [3] for the use of eergy or coectig to the power etwork []. However, covetioal coverters used i this applicatio have certai problems i terms of power quality, due to harmoic distortio [4], [5]. Sice i photovoltaic (PV) systems ito alteratig the power quality depeds o the power iverter [6], today there have bee umerous ivestigatios aimed o optimizig the harmoic cotet of these [7], [8], presetig multilevel coverters as a solutio to the problem, because these iverters have may advatages compared with covetioal two levels: waveforms of high quality eergy, low loss switchig capacity of high voltage, low electromagetic iterferece, etc. [9]- []. I this way exclusive area harmoic cotet optimizatio of photovoltaic systems through multilevel coverters works are preseted [] - [4]. This research through usig geetic algorithms aims to optimize the harmoic cotet of the three phase photovoltaic systems, miimizig THD lie directly through a cascaded asymmetric H-bridge commo source multilevel power coverter [5] ie levels phase adopt the modulatio foud by the optimizatio algorithm.. State of The Art The multilevel power coverter appears i the year 975 with Baker ad Lawrece patet [6], performig the first implemetatio i 98 [7], from these works, umerous ivestigatios have come to get a variety of coverters [] some of which are cosidered as classics [3]: fixig coverters diodes, capacitors ad floatig H-bridges cascade. I the same way recetly topologies that are cosidered iovative ad seek to reduce the umber of compoets, icrease efficiecy or improve power quality have appeared [8]-[0]. Iside the classic topologies, the cascade H- bridges is cosidered advatageous i terms of quality of the waveform ad umber of compoets [3], withi this a coveiet sub-topology appears i photovoltaic systems usig a accumulator block, because it uses a sigle voltage source directly, accompaied by a trasformer at the output of the ISSN:

2 H-bridge, this topology is amed as a multilevel coverter cascaded H-bridge commo source []. This cofiguratio is very suitable for low cost reewable eergy applicatios, because it feeds the etire system from a sigle source [9] however, usig a trasformer to the output, creates problems i terms of the perturbatio of the waveform ad makes the system eve more complex [9], []. Problems have bee studied ad have bee solved i works like [6] ad [3]. I the field of optimizatio of the harmoic cotet of the modulatios of multilevel coverters there have bee umerous proposed techiques depedig o the topology used, the specific objective ad the shape of the search for a optimum [4], [5], however there are very promisig strategies i the lie of evolutioary algorithms ad particle swarm (PSO) ad geetic algorithms (GA) [8], some of these studies coclude that outperforms geetic algorithms [8] ad [6]. Similarly these optimizatios mostly made i the phase voltages of the coverters [5], [8], [9] ad very few are made directly i the lie voltages [30] reachig results ot so ear from zero. For these reasos i this paper a multilevel coverter H- bridges commo source cascade that optimizes THD of lie voltages directly, usig geetic algorithms, as proposed to improve power quality three-phase PV systems is preseted. but the triplicatio of the sigle phase preseted i Figure. As show, the three phases share the DC bus. Fig.. Coverter topology. The waveform geerated by the phase coverter show i Figure 3. This voltage waveform is achieved if phase-multilevel PWM modulatio is geerated. It is expected that the lie voltage coverter has more steps ad more pulses due to the subtractio betwee phase voltages, this will be show i later sectios. 4. Mathematical modelig I this study obtaied a expressio that quatifies the total harmoic distortio of lie voltages i terms of commutatio agles of each step of the phase voltage, both o ad off agles. 3. Cascade Multilevel Coverter Phase of Commo source of Nie Steps for Phase The topology of multilevel iverter selected for this work is the iverter H-bridge asymmetrical cascade commo source with : 3 ratio of stages, which geerates 9 levels of output voltage of each phase, this topology is show i Figure. Fig.. Asymmetry Cascade Iverter i commo source of two stages with 9 levels at the output, asymmetry. : 3 Topology. b) Vo. This coverter allows a miimum of stages of H- bridges ( steps) to achieve the possible maximum umber of output voltage steps (9 steps). The Figure shows the three-phase topology, which is othig Fig. 3. PWM modulatio waveform of 9 steps per phase. I order to determie this equatio it was modeled usig Fourier phase voltages A ad B series of a ABC phase system of ie steps PWM modulatios, ad sice the lie voltage is the subtractio of the phase voltages (o a coectio ad for the coverter), the Fourier series of the lie voltage is obtaied by the respective differece. Gettig the Fourier series of lie voltage THD is calculated i terms of the first 50 harmoics followig the IEEE 59 stadard [3]. Phase A As the shape of the modulatio show i a quarter (/4) guardig symmetry wave, it is oly ecessary to defie the firig agles i the first quarter wave (Fig. 4), the remaiig parts of the modulatio are costructed usig trigoometric relatioships. Thus a vector L = [a b c d] represetig the total umber of ISSN:

3 switch o ad switch off agles at each step is established. The Fourier series for periodic waveforms states: α a0 v( t) = + ( acos ω0t+ bsi ω0t) () = Fig. 4. A quarter of a modulatio waveform i terms of switch o ad switch off of each step agles. Where is the harmoic umber, w o the fudametal frequecy wave, time t, a 0 / is the DC compoet, which is calculated by the expressio: π a0 = v( ωt) d( ωt) π () 0 a coefficiet of the Fourier series is calculated π with (3): a = v( ωt) cos ( ω0t) d( ωt) π (3) 0 b coefficiet of the Fourier series, calculated by: π b = v( ωt) si ( ω0t) d( ωt) π (4) 0 The waveform (Fig. 4) presets ueve symmetry ad the positive part is equal to the egative applyig thereby, symmetries as the theory of Fourier series. a o = 0 ad a = 0 (5) Oly the coefficiet related withi the sius will remai, therefore the waveform i terms of the Fourier series, will be expressed as follows: α ( ) si ω0 v t = b t (6) = b i terms of L vector for waveform: 4 L 4v i cd j b = ( ) for odd ; (7) π i= j= y b = 0 for eve. Where i is the umber of the step (hece the sum is from to 4), L i the vector's i compoet L ad α the j agle of the i step, this ca be switch o or switch off. For a ladder vector compoets L must be odd. If the peak magitude of harmoic, i the Fourier series is defied as: h = a + b (8) Substitutig (5) ad (7) (8) the peak magitude of each harmoic, where oly odd harmoics will exist because b =0, if is eve therefore obtaied as: 4 L 4v i cd h = π i= j= The expressio i the form of Fourier series phasors of a waveform i terms of the fudametal ad harmoic compoet: j ( ) for =, 3,..., (9) α v t a h t ( ) = + si ( ω + ϕ ) 0 (0) = Where a 0 is the direct compoet, h the amplitude for each harmoic calculated i (9) ad ϕ the phase of each harmoic calculated i (): a ϕ = ta () b For this case ϕ ad ull a 0 ad therefore the associated terms for the Fourier series for phase A, i the form of equatio (0), are: 0 for eve 4 L h 4 i = vdc j () ( ) for odd π i= j= ϕ = 0 (3) Phase B Determiig the Fourier series of phase B of the ABC system, the A waveform from Fig. 3 was offset. 0 i arrears, this correspods to each agle preseted i Fig. 4 should be shifted π 3 rad, ad due to this displacemet the odd symmetry is ot applicable to this waveform, thus followig the same process of A phase it is possible to calculate the Fourier series of the phase B but i this case should be calculated o the a ad b full cycle. The shifted phase wave form still has symmetry about the x-axis compoet thus direct voltage remais ull Phase B: a o = 0 (4) The terms associated for the Fourier series for phase B, i the form of equatio (0), are those show i (5) ad (6): 0 for eve 4 Li 4v dc j C = ( ) for odd (multiple of 3) π i= j= 4 Li 4vdc j ( ) for odd (o multiple of 3) π i= j= (5) k represets the umber of odd harmoic ot multiple of 3, for example k=, =5, k= to =7, k=3 to =, so o. Lie Voltage For calculatig the Fourier series i the lie voltages set out from the Fourier series of the voltages phase differece: ISSN:

4 ( ) ( ) ( ) v t = v t v t (6) AB A B By makig the respective differeces of the Fourier series of Phases A ad B, i the forms of equatio (0) with the amplitudes ad offsets give by (), (3), (5) ad (6) the Fourier series for v AB lie voltage is obtaied, but i the article oly the harmoic amplitude is preseted, sice the objective is to miimize THD ad for this mismatch harmoics are required, therefore the harmoic amplitudes of the lie voltages shall be expressed by equatio (7) = 4 Li 4 3vdc j ( ) α π i= j= h 0 for eve 0 for odd (multiple of 3) cos for odd (o multiple of 3) (7) THD of lie voltage The IEEE 59 stadard i 99 o page 63 [3] defies the total harmoic distortio as equatio0: 50 h = THD = 00 (8) h Where h is the fudametal harmoic compoet ad h the peak of the harmoic. Replacig (7) i (8): THD 50 4 Li j ( ) i j = = = = 4 Li j ( ) i= j= 00% (9) Where takes ot odd values o multiple of 3, i.e. 5, 7,, 3, 7, etc. ad L i are the compoets of the vector L = [a b c d]. Thus (9) defies the objective fuctio to be miimized by the optimizatio algorithm. 5. Optimizatio Algorithm With the help of Matlab ad GA commad (Geetic Algorithm) algorithms for the mathematical model of the fitess fuctio (Equatio 9) ad its respective optimizatio usig geetic algorithms were programmed. The size of the populatio for the algorithm is take from 0 idividuals, each idividual (X) cosists of the total of shootig agles i the first wave quarter: X = αα αxαα α yα3 α3zα α w (0) Accompaied by the L vector idicatig the program resposible for evaluatig the fitess fuctio agles correspodig to each step. The flowchart of the algorithm show i Fig. 5. Fig. 5. Flowchart of the optimizatio algorithm (GA). 6. Results L = [ ] is the vector which defies the umber of firig agles i each step idicatig that the first ad secod step have a pulse ad a half, i the third step two pulses ad a half, i the last step 4 pulses ad a half i the first quarter wave voltage phase. Firig agles of phase are summarized i Table I TABLE I: ANGLES IN DEGREES OF THE BEST INDIVIDUAL. a=3 b=3 c=5 d= The THDv of lie voltages of a balaced threephase system which uses a ie steps phase modulatio with shootig previous fire agles theoretically will be of %. THDv itroduced a phase of 0.863%. I Figure 6 the v A waveform voltage phase is show, ad i Figure 6.b its harmoic spectrum. Figure 6, b shows that the harmoics appearig i the phase modulatio are triplig, because of this the THD phase is high (0.863%), these harmoics are deleted i the lie voltage at withdrawig the two positive sequece phases. The waveforms of threephase lie modulatios obtaied by subtractig the phases of the positive sequece waveforms is show ISSN:

5 i Figure 6, show i Figure 7.a ad its spectrum is show i Figure 7.b. As show i Figure 7.a, there i the lie voltage appear additioal steps due to the subtractio doe betwee the phases, phase modulatio has ie steps, while the lie has 5 steps, similarly more pulses occur, ad this makes the waveform get closer to the siusoidal ad therefore the harmoic cotet decreases reachig a value of %. the 60 Hz omial frequecy, ad a omial output voltage of 7 Vrms ad 0 Vrms phase lie. The cotrol of coverter is performed oly with twelve sigals, ecessary to cotrol the upper MOSFETs of the bridges, the lower is cotrolled by the deial of the sigs of major cotrol ad allocatio of dowtime was made alog with the deial through hardware. The cotrol device i the experimetal stage was based o the use of FPGAXUPV5-LX0T. The desig of trasformers, critical devices i achievig the waveform, raised accordig to the methodology proposed i [3]. The implemeted prototype is show i Figure 8. Fig. 6. Phase modulatio. a) Waveform. b) Harmoic spectrum. 8. Experimetal Test Figure 8 shows assembly for the experimetal tests carried out, here is possible to observe the basic compoets of the iverter: H-bridges, trasformers, driver stage ad FPGA. The aalyzer power quality Fluke 434 ad the 5 Fluke oscilloscope: the accumulator block, represeted by laboratory sources from which the drive ad measurig equipmet are fed is also observed. Fig. 7. Lie Modulatio. a) Waveform. b) Harmoic spectrum. 7. Implemetatio I order to validate, a three-phase iverter prototype was implemeted accordig to the topology selected i sectio III, based o the use of MOSFETs as switchig devices. The power coverter is rated 00 VA, the iput voltage 48 V dc, aimed to be applicable to low cost photovoltaic systems i which oly a accumulator block for this voltage is take, Fig. 8. Moutig experimetal tests with FPGA. With the above assembly waveforms, RMS values, THD, phasor diagrams of the phase voltages ad lie, as the output currets of the iverter, for testig i vacuum, resistive load (resistive bak 50 W) were observed, iductive load (iductio motor Hp) ad capacitive load (50W ad 50 VAR), i order to validate the harmoic cotet of the implemeted modulatios. Figure 9.a shows the waveform of the lie voltages empty coverter, with the spectrum give by the aalyzer (figure 9.b This figure was expaded showig the ordiate axis with 0% ecouragemet of evidece that there is o harmoic), also, the spectrum of the phase voltage is preseted to show the presece of harmoics i the phase ad demostrate effectively that optimizatio is doe directly olie. I Figure 9.a. all pulses are ot see due to the display resolutio of the aalyzer, Figure 0.a shows ISSN:

6 the shape of the waveform of the lie voltage captured o the oscilloscope i which the presece of all pulses is observed ad i figure 0.b the phasor diagram of the voltages ad currets for iductive load test shows the accuracy gaps ad the system frequecy are evidet. I Figure 0.c spectrum is show to the three lie voltages, which shows the replica of behavior i the three lie voltages. b) a) b) c) Fig. 9. a) Lie voltage waveform. b) Spectrum. c) Spectrum of the phase voltage. A cosolidate of all the tests is show i Table II, factor imbalace ad percet cotrol are calculated as preseted i [3]. This table shows that the THD of the phase voltages is quite high i all tests, but the THD of lie that is optimized i that work oscillates at very low values of %. c) Fig. 0. a) Waveform lie voltage. b) Phasor diagram. c) Lie voltages spectrum i the iductive test. TABLE II: Cosolidated tests results. Variable Parameter Phase Voltage Lie Voltage. Lie Curret. Power Test No load Resist. Iduc. Capac. Mea Vrms (V) % Regulatio % Ubalace factor %THDv 0, Mea Vrms (V) % Regulatio % Ubalace factor %THDv Mea Irms (A) % Ubalace factor %THD I Output Power (W) Power Factor 0 0. L 0.88 C Reactive Power(VA) Likewise, it is observed that the imbalaces i the phase voltages ad lie meets EN5060 which establishes a maximum of %, as percetages of regulatio which is maitaied withi the rage established by the IEEE 59 which states that the ormal bad operatio voltage must be withi 0% fluctuatio. a) 9. Coclusios The phase voltage THD i this work is quite high because the proposed optimizatio is directly i lie voltages. The value of THDv phase raged from 0.9% i the vacuum test ad.6% i the test resistive load. This shows that the load disturbs the behavior of the coverter; however variatios are ot very represetative. As for the measured harmoic spectra phase voltages each of the tests ISSN:

7 demostrated that the compoets preset were triples because as deduced i mathematical modelig these harmoics would be elimiated i the lie voltages by the potetial differece betwee the two phases. THD of lie voltages evideces good optimizatio performed by the algorithm, sice all THD tests were below the limit set by the IEEE 59 which is 5%. It presetig the highest THD at a value of 0.6%, which is a very low value. The results regardig the THDv voltage lie reflects the good desig of the coverter, as this reproduces the waveforms calculated i a accurate maer, although the coverter uses trasformers; the same desig miimizes perturbatios they could geerate, is for this reaso that the THD is very low despite the presece thereof. The optimizatio algorithm developed will be able to override all defied harmoic cotet betwee ad 50 if a sufficiet umber of shootig agles is give, however, modulatios that are ot possible to implemet ca occur. The optimized theoreticia THD of this project is defied as 0.07% that this is the harmoic cotet of the modulatio optimized to the 50 th harmoic, beig the upper boud of evaluatio established by the IEEE 59 o page 63. From the experimetal poit of view the optimized THD for lie voltage waveform is defied as 0.6% because this is the largest THD preset i the tests. The coverter meets the quality stadards of eergy; this is why this coverter is a good proposal to improve the quality of three-phase PV eergy systems. Refereces []. Mohammad Barghi Latra, Ahmet Teke, Ivestigatio of multilevel multifuctioal grid coected iverter topologies ad cotrol strategies used i photovoltaic systems, Reewable ad Sustaiable Eergy Reviews, Volume 4, February 05, Pages , ISSN: []. P. Veea, V. Idragadhi, R. Jeyabharath, V. Subramaiyaswamy, Review of grid itegratio schemes for reewable power geeratio system, Reewable ad Sustaiable Eergy Reviews, Volume 34, Jue 04, Pages 68-64, ISSN: [3]. [3] Abdelaziz Fri, Rachid El Bachtiri, Abdelaziz El Ghzizal, A Comparative Study of Three Topologies of Three-phase (5L) Iverter for a PV System, Eergy Procedia, Volume 4, 03, Pages , ISSN [4]. Mohammad, S. Z.; Omar, A. N. ad Ibrahim, I. R. A Review of Sigle-Phase Sigle Stage Iverter Topologies for Photovoltaic System. 4 th Cotrol ad System Graduate Research Colloquium, Shah Alam, Malaysia, Vol. 9, N. 0, Aug. 03, pp [5]. Mehmet, Y.; Seydi, V.; Seci, V. ad Hasa, C. Compariso of Output Curret Harmoics of Voltage Source Iverter used Differet PWM Cotrol Techiques. WSEAS Trasactios o Power Systems Vol. 3, 008. pp [6]. Diaz Rodríguez, J. L., Pabo Ferádez, L.D: y Pardo García, A. THD Improvemet of a PWM Cascade Multilevel Power Iverters Usig Geetic Algorithms as Optimizatio Method, Wseas Trasactios O Power Systems ISSN: 4-350X, 05 voumel:0 fasc: Grecia pages: [7]. T. Sudhakar Babu, K. Priya, D. Maheswara, K. Sathish Kumar, N. Rajasekar, Selective voltage harmoic elimiatio i PWM iverter usig bacterial foragig algorithm, Swarm ad Evolutioary Computatio, Volume 0, February 05, Pages 74-8, ISSN [8]. Abdul Moeed Amjad, Zaial Salam, A review of soft computig methods for harmoics elimiatio PWM for iverters i reewable eergy coversio systems, Reewable ad Sustaiable Eergy Reviews, Volume 33, May 04, Pages 4-53, ISSN , [9]. Aup Kumar Pada, Yellasiri Suresh, Research o cascade multilevel iverter with sigle DC source by usig three-phase trasformers, Iteratioal Joural of Electrical Power & Eergy Systems, Volume 40, Issue, September 0, Pages 9-0, ISSN [0]. Maliowski, M.; Gopakumar, K.; Rodriguez, J. ad Pérez, M. A Survey o Cascaded Multilevel Iverters, IEEE Tras. o Id. Elect., Vol. 57, N. 7, July 00, pp []. Ilhami Colak, Ersa Kabalci, Ramaza Bayidir, Review of multilevel voltage source iverter topologies ad cotrol schemes, Eergy Coversio ad Maagemet, Volume 5, Issue, February 0, Pages 4-8. []. Naya Kumar, Tapas Kumar Saha, Jayati Dey, Modelig, cotrol ad aalysis of cascaded iverter based grid-coected photovoltaic system, Iteratioal Joural of Electrical Power & Eergy Systems, Volume 78, Jue 06, Pages 65-73, ISSN [3]. Vekatachalam Kumar Chiaiya, Jovitha Jerome, J. Karpagam, A experimetal ISSN:

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