Total Harmonics Distortion Reduction Using Adaptive, Weiner, and Kalman Filters

Transcription

1 Wester Michiga Uiversity ScholarWorks at WMU Master's Theses Graduate College Total Harmoics Distortio Reductio Usig Adaptive, Weier, ad Kalma Filters Liqaa Alhafadhi Wester Michiga Uiversity, Follow this ad additioal works at: Part of the Electrical ad Computer Egieerig Commos Recommeded Citatio Alhafadhi, Liqaa, "Total Harmoics Distortio Reductio Usig Adaptive, Weier, ad Kalma Filters" 2016). Master's Theses This Masters Thesis-Ope Access is brought to you for free ad ope access by the Graduate College at ScholarWorks at WMU. It has bee accepted for iclusio i Master's Theses by a authorized admiistrator of ScholarWorks at WMU. For more iformatio, please cotact maira.budza@wmich.edu.

2 TOTAL HARMONICS DISTORTION REDUCTION USING ADAPTIVE, WEINER, AND KALMAN FILTERS by Liqaa Alhafadhi A thesis submitted to the Graduate College i partial fulfillmet of the requiremets for the degree of Master of Sciece i Egieerig Electrical) Electrical ad Computer Egieerig Wester Michiga Uiversity Jue 2016 Thesis Committee: Johso Asumadu, Ph.D., Chair Massood Atashbar, Ph.D. Christopher Cho, Ph.D.

3 TOTAL HARMONICS DISTORTION REDUCTION USING ADAPTIVE, WEINER, AND KALMAN FILTERS Liqaa Alhafadhi, M.S.E. Wester Michiga Uiversity, 2016 Total harmoics distortio is oe of the mai problems i power systems due to its effects i geeratig udesirable issues i power quality. These effects iclude heatig i trasformers, capacitors, motors, ad geerators), disoperatio of electroic equipmet, icorrect readigs o meters, disoperatio of protective relays, ad commuicatio iterferece. Besides these problems, harmoics affect the power quality i both trasmissio ad distributio systems. Differet techiques have bee used to mitigate the effects of harmoics. These techiques iclude; passive filters, active power filter, artificial itelliget, ad adaptive selective harmoics reductios. Each method has some advatage ad disadvatage. This thesis preseted ew models of total harmoics distortio reductio usig adaptive, Weier, ad Kalma filters. I order to test the performace of the preseted methods, the output curret of a sigle phase iverter circuit was used as a study case. The ew models reduced the total harmoics distortio from by more tha 50% however Kalma filters give the best performace as compared to adaptive ad Weier filters.

4 Copyright by Liqaa Alhafadhi 2016

5 ACKNOWLEDGEMENTS I would like to express my deepest thaks for my supervisor Dr. Asumadu, for his excellet guidace, carig, patiece, ad providig me with a excellet atmosphere for doig research. Also, I would like to thak my committee members, Dr. Atashbar ad Dr. Cho for their time ad effort. I am heartily thakful to my parets who gave me the moral support I eed. I also would like to express my gratitude to my husbad for his ecouragemet ad support. Liqaa Alhafadhi ii

6 TABLE OF CONTENTS ACKNOWLEDGEMENTS...ii LIST OF FIGURES... vi CHAPTER BACKGROUND AND MOTIVATION Itroductio Thesis Goals Thesis Structure... 3 CHAPTER HARMONICS REDUCTIONS IN POWER AND POWER ELECTRONICS Harmoics Harmoics Reductio Techiques Frequecy Domai Techiques Time Domai Techiques Artificial Itelliget Techiques Curret Cotrol Techiques Adaptive Selective Harmoic Elimiatio Techique CHAPTER ADAPTIVE FILTERING TECHNIQUES AND ALGORITHMS iii

7 Table of Cotets- cotiued 3.1 DC-to-AC Coverters Iverters) Sigle-Phase Iverters Three-Phase Iverters Multi-Level Iverters Adaptive Filterig Algorithms Least Mea Square Algorithm Normalized Least Mea Square Algorithm Leaky Least Mea Square Algorithm Sig LMS ad Sig NLMS Algorithms Weier Filter Kalma Filter CHAPTER SIMULATION AND RESULTS DC-AC Coverter i MATLAB SIMULINK Distortio Reductio Usig New Adaptive Filterig Models Distortio Reductio Usig Weier Filter Distortio Reductio Usig Kalma Filter iv

8 Table of Cotets- cotiued CHAPTER CONCLUSIONS AND FUTURE WORKS Coclusios Future Works BIBLIOGRAPHY v

9 LIST OF FIGURES 2. 1: Sigle Phase Represetatio of Noliear Load ad Passive Shut Filter : Sigle Phase Passive Filter with Shut Cofiguratio : Sigle Phase Passive Filter with Series Cofiguratio : Three Phase, Three Wire Passive Filter for Shut Cofiguratio : Three Phase, Three Wire Passive Filter for Series Cofiguratio : Shut Passive Filter : Series Passive Filter : Sigle Phase Active Filter, Shut Cofiguratio : Sigle Phase Active Filter, Series Cofiguratio : ADALINE Block Diagram by Zouidi : MLP Neural Network Architecture by Che : Structure of a Sigle Frequecy Adaptive Selective Elimiatio techique : Sigle Phase Iverter : Three Phase Iverter : Multi Level iverter : Adaptive filter as a oise caceller : Block Diagram of Wieer Filter : A discrete Time system with Process w ad Measuremet Noise v vi

10 List of Figures- cotiued 3. 7: Block Diagram of Kalma Filter : Sigle Phase Iverter : Simulated Iverter i MATLAB : Iverter s Output Curret ad Voltage for RL Load : Block Diagram of Adaptive filter : Block Diagram of the Proposed Method : Distorted ad Clea Curret Sigal Usig LMS Algorithm Whe µ=1e-8 ad M= : Distorted ad Clea Curret Sigal Usig NLMS Algorithm Whe µ=0.01 ad M= : Distorted ad Clea Curret Sigal Usig Leaky LMS Algorithm Whe µ=2e- 9 ad M= : Distorted ad Clea Curret Sigal Usig Sig LMS Algorithm Whe µ=2e- 9 ad M= : Distorted ad Clea Curret Sigal Usig Sig NLMS Algorithm Whe µ=1 ad M= : MATLAB Code for LMS Algorithm : MATLAB Code for NLMS Algorithm : MATLAB Code for Leaky LMS Algorithm : MATLAB Code For Weier Filter [38] : MATLAB Fuctio For Weier Filter [38] : Distorted ad Clea Curret Sigal Usig Weier Filter Whe M= : MATLAB Code of Kalma Filter : Distorted ad Clea Curret Sigal Usig Kalma Filter : Distorted ad Clea Curret Sigal Usig Kalma Filter whe R=0.5 ad L=1.592e vii

11 CHAPTER 1 BACKGROUND AND MOTIVATION 1.1 Itroductio Harmoics have received great attetio due to their effects i geeratig udesirable issues i power quality. These effects iclude heatig i trasformers, capacitors, motors, ad geerators), disoperatio of electroic equipmet, icorrect readigs o meters, disoperatio of protective relays, ad commuicatio iterferece. Besides these problems, harmoics affect the power quality i both trasmissio ad distributio systems. The effect of harmoics o power quality has received more attetio from both utilities ad customers. The mai factors that make the power quality so importat are; high level of harmoics i power system, the sesitive load equipmet to power variatio, the itercoectio of ay etwork that magify the impact of failure, the awareess of customers about power quality issues. The mai cause of harmoics is the oliear loads which affect the quality of the power i trasmissio ad distributio systems. Differet techiques have bee used to mitigate the effects of harmoics. Passive filters are the most commo power filters that are used for harmoics reductio i power systems. These kid of filters are easy to istalled ad cost effective. However, large ad differet size of filters may be required for differet harmoics orders, which may be cost effective [1]-[2]. Large umber of filters may cause resoace, which leads to ustable system. If additioal iductors or capacitors are used, the circuit becomes bulky. There is 1

12 a eed for flexible ad automated cotrol but flexibility i cotrol caot be achieved usig passive filters. Active power filters APF) techiques are commoly used to mitigate harmoics associated because APFs are more dyamic compared to the passive filters. The mai priciple of APF is to cacel harmoic compoets that are caused by oliear loads ad to cotiuously ad dyamically maitai low or elimiate harmoics to the load. I APF, the system parameters curret ad voltage) are measured ad compared to referece sigals. Power Coverters are used to realize the referece sigals, which are ijected ito the load to elimiate the harmoics [3]. I additio to passive ad active power filters, artificial eural etworks ANN) have bee used also to reduce the harmoics effects ad improve power quality. For example, ANN techiques that have bee used iclude adaptive liear euro ADALINE), back propagatio eural etworks BPNN), ad radial basis eural etworks RBNN). Each oe of these methods has some advatages ad disadvatages as will be show i Chapter 2 [4]-[5]. There are may more methods that have bee developed but ufortuately the log term goal has ot bee satisfied yet. Oe of the methods that has become very commo recetly is the adaptive selective harmoics elimiatio ASHE). The mai cocept of this techique is to attack oe specific harmoic i selected frequecy [6]-[7]. Most of the time the third harmoic is the targeted harmoic for elimiatio. I this thesis, several ew models based o adaptive, Kalma, ad Wier filters will be cosidered for reducig udesirable harmoics i order to improve power quality. I 2

13 these models, all the harmoics cotaied i the distorted sigal, ot specific oe as i other techiques which focused o oe or two harmoics most of the time, will be cosidered. Differet algorithms have bee tested ad all of them give a icredible results i terms of distortio reductios. 1.2 Thesis Goals I this thesis, there are two mai goals. The first oe is to review some of the importat techiques that have bee used to improve the power quality by reducig the effect of harmoics. The secod goal is to ivestigate differet techiques of adaptive, Kalma, ad Weier filters as ways of improvig the power quality. I the secod goal the desire is to reduce ad/or elimiate the effect of harmoics i all the frequecies cotaied i the distorted sigal ad fid out which method gives the best performace i term of adaptatio ad speed, lowest total harmoics distortio THD), ad complexity. 1.3 Thesis Structure The rest of the thesis are orgaized as follow; Chapter 2 covers i detail the curretly used methods to overcome the harmoics problem i power ad power electroics systems, Chapter 3 covers the proposed model, the results ad discussio based o the models will be preseted i Chapter 4, ad fially i Chapter 5 the coclusios ad future works will be preseted. 3

14 CHAPTER 2 HARMONICS REDUCTIONS IN POWER AND POWER ELECTRONICS I the past, harmoic distortio was ot a big issue sice the desigs of power systems were very simple. Nowadays, harmoic distortio has icreased with the use of complex desigs i the idustry. Much emphasis has bee give to harmoic distortio associated with power quality. I a iverter iput DC voltage is coverted ito Ac output. Durig this coversio, harmoics affect the power quality. How harmoic reductio will help to improve the power quality will be explaied i the followig sectios. 2.1 Harmoics Harmoic is defied as a sigal or wave with frequecy that is a iteger multiple of the fudametal frequecy [8]. Besides the fudametal frequecy f, a output sigal may cotai harmoics at frequecies of 2f, 3f, 4f, ad so o. The sigal is a perfect siewave if all eergy is cotaied at the fudametal frequecy. If some eergy is cotaied withi the harmoics, the sigal is ot a pure siusoidal waveform, such as triagular wave, square wave, ad saw tooth wave [8]. The o-siusoidal waveform has odd ad eve harmoics. The odd harmoics are the most dagerous. The mai reaso behid harmoic distortio is oliear loads. A load is cosidered oliear if the curret draw by the load is o-siusoidal eve whe it is coected to a siusoidal voltage [3]. I order to protect the electrical devices from gettig damaged due to harmoic voltage distortio, the harmoic curret is isolated by usig harmoic filters. These filters also help to improve the power quality. Much importace has bee give to power quality because 4

15 the cosumers are worried about power quality issues like iterruptio, saggig, ad switchig trasiets [9]. Also, a etwork has may power systems that are iterally coected. If a failure exists i ay oe of the iteral etworks, whole system is affected due to the itegratio. 2.2 Harmoics Reductio Techiques The umber ad variety of harmoics mitigatio techiques ow available does ot make it easy or a straightforward process to select the best suitable method. Some of these techiques deped o the system coditio i their performace, while the others require extesive system aalysis. I this chapter these techiques will be classified ito several categories Frequecy Domai Techiques I frequecy domai techiques, the harmoic detectio methods deped maily o the Fourier trasform method. These detectio methods iclude three steps: 1) covert the distorted curret or/ad voltage from the time domai ito frequecy domai, 2) isolate the fudametal compoet from the harmoic i the frequecy domai, ad 3) recostruct the compesatig sigal [3]. Girgis et al. [10] ad Mariethoz [11] studied the use of the discrete Fourier trasform DFT) ad the fast Fourier trasform FFT) for harmoic detectio. To use either DFT or FFT, some coditios have to be satisfied. First of all, the sigal to be processed must be periodic. Secod, the samples widow legth must represet a iteger umber of a fudametal cycle. Third, the samplig frequecy has to be equal to or greater tha twice the highest harmoic frequecy cotaied i the sigal based o Nyquist criteria. Fourthly, 5

16 each harmoic icluded i the sigal is a iteger multiple of the fudametal frequecy [10]-[11]. Komrska et al. [12] discussed the possibility of avoidig the high computatio requiremets for the classical DFT. I order to reduce the computatioal complexity, they estimated the fudametal harmoic istead of the whole spectrum of the sigal. This techique is usuitable for olie applicatio because the olie applicatios are way more complex tha the other oes. Ha [13] utilized the FFT algorithm to extract the harmoic compoets i the load curret. The FFT algorithm is more efficiet tha DFT computatioally because FFT requires N/2) log2 N) operatios while DFT eeds N 2 operatios. The mai problem of usig FFT for olie harmoic extractio is that it caot be performed i each samplig period. Istead, a average of magitude ad phase is made to overcome the error due to the samplig. Also, compesatio for phase delay is eeded [13]. Borisov [14] ad Maza- Ortega [15] used Recursive Discrete Fourier Trasform RDFT) which is a versio of FFT suitable for olie applicatio. RDFT is more efficiet tha the DFT ad FFT methods ad it is also differet from the two methods. RDFT is differet by usig a slidig widow; usually this widow is shifted oe sample. For example, the spectrum of the widow at time K is differet from the spectrum of the widow at time K-1 by the first ad last sample. Therefore, there is o eed to calculate the spectrum for the whole widow [14]-[15]. This techique ca be applied i real-time applicatios. Habrouk [16] ad Gree et al. [17] addressed the disadvatages of usig frequecy detectio methods which are the leakage problem, the sychroizatio betwee the 6

17 samplig frequecy ad the fudametal frequecy, the large memory eeded, the large umber of computatios, ad the poor trasiet performace Time Domai Techiques Time domai methods for harmoic detectio ca be applied i power system Filter Based Methods Harmoics i oliear loads are cosidered a major problem i power system because they cause may issues such as overheatig; ad temperature icrease i geerators. These effects may cause damages to the devices [18]. Filters are usig to overcome the issues of harmoics. Filters are used widely for harmoic reductio with the icrease of oliear loads i power system. Istallig a filter helps i reducig harmoic for oliear loads coected i power system. There are two types of filters that are used i order to reduce the harmoic distortio. These are active filters ad passive filters. Those filters are electroic devices that elimiate udesirable harmoics ad distorted sigals i the etwork. Active harmoic filters cosist of active compoets such as IGBTtrasistors, used oly for low voltages. O the other had, passive filters cosist of passive compoets such as resistors, iductors, ad capacitors used for differet voltage levels [18]-[19]. Hybrid filter is also a third type of filter; a combiatio of both passive ad active filters. Usig filters would result i a better quality of power Passive Filters As metioed above, passive filters are used for differet voltage levels. The passive filter is coected as either a parallel or a series resoat circuit with resistor, iductor, ad capacitor. A parallel passive filter is a curret acceptor while a series filter is 7

18 a voltage acceptor. Also, the filter provides maximum atteuatio ear the resoat frequecy. The passive filter reduces or elimiates harmoic by atteuatig the frequecy of the harmoic. The reduced harmoic frequecy must be equal to the resoat frequecy of the circuit, thus the impedace of the etwork ad the low impedace of the filter elimiate the harmoic curret [18], [20]. Passive filters are used to reduce harmoic currets i the power system by miimizig the harmoics due to oliear loads. That is why the passive filters produce better results i reducig harmoics effect. Figure 2.1 shows a sigle phase represetatio of distributio system with oliear load ad passive shut filter. A passive filter is oly capable of elimiatig oe targeted harmoic [9]. For example, a passive filter desig for elimiatig 3 rd harmoic will oly elimiate this harmoic order. The resoace frequecy of the passive filter must be lower tha the frequecy of the harmoic to be elimiated i order to avoid shift i frequecy of the filter due to chages i the parameters of the filter [18]. Figure 2.1: Sigle Phase Represetatio of Noliear Load ad Passive Shut Filter. 8

19 Types of Passive Filters There are two types of passive filters Shut passive filters Series passive filters Both shut ad series passive filters are used for sigle phase ad three phase power system. Also, more tha oe shut ad series passive filters ca be used with ad without each other i a system. There are some differeces betwee the shut ad series passive filters: a. The shut passive filters carry oly part of the total load curret while the series passive filters carries the full load curret [18]. b. Compared to the series passive filters, shut passive filter are cheaper so they are used more tha the series oes [18]. Figures 2.2 ad 2.3 show sigle phase passive filter with shut ad series cofiguratios, respectively. 9

20 Figure 2.2: Sigle Phase Passive Filter with Shut Cofiguratio. Figure 2.3: Sigle Phase Passive Filter with Series Cofiguratio. Figures 2.4 ad 2.5 show three phase three wire passive filter for shut ad series cofiguratios, respectively. 10

21 Figure 2.4: Three Phase, Three Wire Passive Filter for Shut Cofiguratio. Figure 2.5: Three Phase, Three Wire Passive Filter for Series Cofiguratio. 11

22 Three filters or more are usually coected i a system to reduce harmoics. To reduce the effect of harmoics which are less effective, the first two filters are coected ad the a high pass filter is used. Figures 2.6 ad 2.7 show shut ad series coected passive filters respectively. Figure 2.6: Shut Passive Filter. Figure 2.7: Series Passive Filter Active Filters Where reactive power requiremet is low, active filters are used for low voltages as explaied earlier. These filters work as follow: the output load with the voltage 12

23 waveform is obtaied by boostig the voltage throughout each half cycle by the filter. The power, which is produced teds to rectifiers i the power supply to gai curret. The power factor ad the duty cycle are thus improved. Also, the harmoics filters moitor the curret that is produced due to load ad geerate a waveform which coicides with the exact shape of the oliear portio of the load curret [18]-[20]. The output distortio is reduced, depedig o the active harmoic filter used. Active filters are used i order to provide dyamic compesatio where the harmoic orders chage i terms of magitudes ad phase. They ca be used i oliear load coditios where the harmoics depedet o the time. Depedig o the source type which create harmoics i the power system, active filters ca be coected i either series or parallel. Figure 2.8 ad 2.9 show sigle phase active filters i shut ad series cofiguratio respectively. Figure 2.8: Sigle Phase Active Filter, Shut Cofiguratio. 13

24 Figure 2.9: Sigle Phase Active Filter, Series Cofiguratio. The advatages of active filters icludes the reductio of harmoics ad useful for lowerig flickerig. But active filters are costly, caot be used for small loads [18], ad caot be used i certai applicatios where harmoics are preset i both the curret ad voltage. More complex filters are eeded i case where both curret ad voltage are leadig to a directio i power system. These filters are made up of combiatio of active ad passive filters which called as hybrid filters [18] Artificial Itelliget Techiques To overcome the disadvatages of time ad frequecy domai filters, the artificial itelliget filters have bee used. There are three mai techiques that have bee used i artificial itelliget filterig i) adaptive liear euro ADALINE), ii) Feed Forward Multiplayer Perceptio MLP), ad iii) radial basis fuctio eural etwork RBFNN). 14

25 Adaptive Liear Neuro The most popular eural etwork techique used i active power filter is the ADALINE algorithm. The mai reasos behid the popularity of this algorithm are the simplicity ad the ease of hardware implatatio. The ADALINE has the followig features: i) it has oe euro, ii) it is used as olie idetifier, iii) the relatioship betwee the iput ad the output is liear, ad iv) it uses least mea square method LMS) to adjust its weights. Figure 2.5 shows the block diagram of the ADALINE algorithm [21]. Figure 2.10: ADALINE Block Diagram by Zouidi. The harmoics preset i the polluted curret represet the iputs of this etwork. The weights of this etwork represet the Fourier expasio cofficiets. The weighted sum of the iputs multiply by a costat is the output. The error betwee the estimated load curret ad the actual load curret is used to update the weights, based o Widro_ Hoff algorithm [21]. This was used by Zouidi et al to idetify the discret Fourier expasio 15

26 cofficiets of harmoic polluted waveform. Based o these coffeciets, the magitude ad the phase of the fudemetal harmoic ca be estimated Feed Forward Multilayer Perceptro MLP) To estimate the total harmoic cotets for three phase oliear, Che et al [22] used multi output MLP eural etwork with back propagatio algorithm. Figure 2.6 illustrates the eural etwork structure used by Che. Accordig to Che s proposed scheme, the firig agel i geeratig traiig data was costat. Some remarks ca be oticed o Che s proposed scheme: the architecture eed to be tested for a wide rage, the umber of hidde euros is large, ad the traiig time is ot metioed. Figure 2.11: MLP Neural Network Architecture by Che. 16

27 Radial Basis Neural Networks Chag et al. [23] itroduced radial basis fuctio eural etwork RBFNN) to extract selective harmoics. Figure 2.11 explais the eural etwork structure used by Chag. This algorithm uses similar traiig algorithm of back propagatio. The iput is a delayed vector of the polluted sigal. The output is the value of the give harmoic order first, third, fifth, or seveth). Just like the back propagatio algorithm used i MLP, the error betwee the RBNN output ad the desired iput is used to update the weights ad ceters based o traiig algorithm [23] Curret Cotrol Techiques The choice of the cotrol techique affects the performace of the active power filter. To resolve this issue, several curret cotrol techiques have bee used for active power filter: such as space vector cotrol, Proportioal-Itegral PI) cotrol, predictive cotrol, deadbeat cotrol, resoat cotrol, slidig mode cotrol, carrier phase shift, hysteresis curret cotrol, fuzzy cotrol, ad artificial eural etwork cotrol [24]-[26]. Hysteresis curret cotrol techique is the most commo oe used i active power applicatio because it shows distict performace over the other cotrol methods. The hysteresis curret cotrol has some advatages which are: simplicity, ease to implemet, accuracy, fast respose, ad stability. However, it has the followig disadvatages: variable switchig frequecy ad iterferece i three phase case [24]. The mai techiques used to improve the basic hysteresis techiques are: Space vector based hysteresis curret cotroller. Adaptive hysteresis bad curret cotrol techique Fuzzy based hysteresis curret cotroller 17

28 Neural etwork based hysteresis cotroller More details about these techiques ca be foud i [27]-[30] Adaptive Selective Harmoic Elimiatio Techique Adaptive selective harmoic elimiatio ASHE) algorithm has bee used as a permaet solutio i a wide rage of power electroics applicatios. I power applicatios, cotrol with objectives like fast respose of system o referece ad disturbace chage is a primary objective. However, the algorithm of adaptive cacellatio of selected harmoic compoets has bee well developed i digital sigal processig ad filterig [6]-[7]. ASHE is used to elimiate udesirable higher harmoic compoets from selected variable curret or voltage. This task requires oly kowledge of the frequecy of the compoets to be elimiated [31]. Figure 2.12 shows the structure of a sigle frequecy adaptive selective harmoic elimiatio filter. The ASHE algorithm or filter, as show i Figure 12.2, cosists of combier, least mea square LMS) algorithm, ad a summig poit. 18

29 Figure 2.12: Structure of a Sigle Frequecy Adaptive Selective Elimiatio techique. The way this filter operates as followig: the referece sigal has two orthogoal compoets cosie ad sie) which is sampled at f s. The frequecy of the sigal should be elimiated from the primary iput sigal. Multiply the referece iput by the correspodig weights. The, added the weighted sie ad cosie compoets together to match the amplitude ad phase agle of the primary iput. The adaptio process adjust the weights to match the amplitude ad phase agle of the iterferig siusoid i the primary iput [30]. The sigal created by combier is subtracted from the primary iput to compute the error sigal, which is also the output of the filter. 19

30 CHAPTER 3 ADAPTIVE FILTERING TECHNIQUES AND ALGORITHMS This chapter presets the study case used i this thesis which is the iverter ad what are the solutios to improve the quality of the iverters outputs. 3.1 DC-to-AC Coverters Iverters) There is widespread use of power electroics iverters i filter applicatios. A iverter coverts DC to AC with the desired amplitude ad frequecy [32]. The output AC voltage ca be either sigle phase or three phase. Also, the magitude of the AC voltage rage varies ad the frequecies are typically 50 HZ, 60 HZ, or 400 HZ. The output curret waveform of a ideal iverter is expected to be siusoidal waveform, but i real applicatios they have distortios due to harmoics presece i the system which leads to distorted waveform [32]. The output sigal will cotai harmoics at frequecies of 2f0, 3f0, 4f0, ad so o besides the fudametal frequecy f0. The odd harmoics are the most domiat oes [33]. There are may applicatios of iverters; but the basic oe would be a uiterruptible power supply UPS). The other applicatios of a iverter is the variable frequecy drives which cotrols the frequecy ad voltage of power supplied to the motor, thus cotrollig the speed of AC motor. Also, iverter is used to regulate the speed by chagig the frequecy of AC output [34]-[35]. There are three types of iverters: Sigle Phase Iverters 20

31 Three Phase Iverters Multi-Level Iverters Sigle-Phase Iverters Sigle phase iverters have differet topology. Figure 3.1 shows a sigle phase iverter. It is also called a H bridge iverter based o the way the switches ad the load are coected. This sigle phase iverter cosists of four choppers for example, IGBT trasistors with diodes i parallel). The loads are coected betwee the two IGBTs devices ad the diodes. The way these diodes work is: whe trasistors S1 ad S2 are o the trasistors S3 ad S4 will be off ad the positive voltage appears o the load. Whereas whe trasistors S1 ad S2 are off the trasistors S3 ad S4 will be o ad the egative voltage appears o the load. I order to cotrol the turig o ad off of the IGBT circuits, differet kids of cotrollers ca be used [32]. Figure 3.1: Sigle Phase Iverter. 21

32 3.1.2 Three-Phase Iverters Three-phase iverters also have differet topologies which ca be used similar to the sigle phase iverter. Figure 3.2 shows a three phase iverter circuit. Three sigle phase iverters are coected to form the cofiguratio of three phase iverter. There is a phase shift of 180 degrees betwee differet legs i case of sigle phase iverter while i three phase iverter the phase shift is 120 degrees. The 120-degree phase shift helps i elimiatig the odd harmoics from the three legs of the iverter. Besides the odd harmoics, eve harmoics ca be elimiated as well if the output is pure AC wave form [35]. The way the three phase iverter works is almost similar to that of sigle phase iverter. Figure 3.2: Three Phase Iverter. 22

33 3.1.3 Multi-Level Iverters The costructio of multi-level iverters is similar to the sigle ad three phase iverters as explaied earlier. Figure 3.3 shows a multi-level iverter circuit. It is a extesio of sigle- ad three-phase iverter circuits. As show i the figure, four IGBT circuits are coected i three differet legs ad the diodes are coected i parallel to each legs i opposite directio. The loads are coected betwee two IGBT circuits for each leg. The mai advatages of usig multi-level iverters istead of sigle- or three-phase iverters are as follow [9], [35] Multi-level iverters are used for high power applicatios. They have higher capability of reducig harmoics because of multiple DC levels. 23

34 Figure 3.3: Multi Level iverter. 3.2 Adaptive Filterig Algorithms It is very importat to produce siusoidal waveform sice the output of DC to AC iverter is alteratig. To do so, filters are implemeted which reduce the harmoics effect by removig the third ad higher order harmoics from the system. These filters cosist of large umber of iductors ad capacitors which leads to more costly filters to remove the harmoics of higher order [34]-[35]. To avoid the cost of such expesive ad complex filters, adaptive filter algorithms are applied. Adaptive filterig is a liear filter with ability 24

35 to chage its weights iteratively i order to get the optimum solutio of the cost fuctio. Figure 3.4 describes the mai cocept of adaptive filter. The geerated referece sigal is the iput to the adaptive filter ad the output of the filter will be subtracted from the iput sigal to compute the error which is cotrollig the filter weights i order to reduce the error as much as possible. Based o the defiitio of the cost fuctio ad the kow ad ukow system parameters, there are may ways to reach the optimizatio goal. Sayed ad Hayki metioed differet ways for approachig the optimizatio goals i their textbooks [36]-[37]. Adaptive filterig has bee used widely for oise cacellatio. Figure 3.4 illustrates the process i simple pictures. Least mea square LMS) ad ormalized least mea square NLMS) are the most popular algorithms that have bee used successfully to reduce harmoics. Next sectios will describe these algorithms i detail. Figure 3.4: Adaptive filter as a oise caceller. 25

36 3.2.1 Least Mea Square Algorithm LMS is the most commo algorithm that has bee used to reduce harmoics because of its distiguished properties. It has bee used i differet applicatios. It is simple, stable, ad easy to implemet. Its iterative process icludes the followig three steps: 1) compute the output of adaptive filter which, produce by a group of tap iput, 2) geerate a estimated error by comparig the output of the filter to a desired sigal, ad 3) use the error sigal to adjust the tap weights [38]. Choosig the right value of step size is very importat to cotrol the stability ad covergece speed of LMS algorithm. I this algorithm, the cost fuctio is defied as the mea square error betwee the estimated output ad the actual oe. Usig steepest descet i LMS algorithm to fid the filter weights which miimize the cost fuctio. The weights, error, ad output equatio of LMS algorithm ca be derived as follow [36]. For a give adaptive filterig system with iput matrix x ), which is M*M matrix: x ) [ x ) x 1).. x M 1)] 3.1) where M is the filter taps ad is the umber of curret iput samples. The output of the adaptive filter will be y ) which is: T T y ) [ y ) y 1).. y M 1)] x ) w ) 3.2) where T refers to the traspose of the matrix ad w ) is the weight vector which is: w ) [ w0) w1) w2).. w M 1) ] T 3.3) 26

37 27 The mai idea of LMS ad other adaptive algorithms is to use steepest decet method to fid the filter weight which miimize the cost fuctio. The cost fuctio C ) is defied as the expected value of the square of the error sigal ) e. The error sigal is the differece betwee the desired output ) d ad the actual oe ) y : T M e e e e )] 1.. 1) ) [ ) 3.4) T M d d d d )] 1.. 1) ) [ ) 3.5) Based o the defiitio of the cost fuctio, 2 ) ) e E C 3.6) where e ) is the error at the curret sample ad E is the expected value. If we apply the steepest decet method the this meas takig the partial derivatives with respect to the idividual filter weights: w e e e E C w w ) ) 2 ) ) 2 3.7) But T w x d e ) ) ) ) 3.8) Hece ) ) x w e 3.9)

38 To take the miimum of the cost fuctio, take a step i the opposite directio of C). To express this poit mathematically: w w 1) w ) C ) w ) w 1) w ).2. e ). x ) 2 w 1) w ). x ). e ) 3.11) where is the step size. Equatio 3.11) is the weight update which is the relatioship betwee the curret weight vector ad the ext step Normalized Least Mea Square Algorithm The filter legth ad the power of the sigal affect o the stability, covergece, ad steady state behavior of the LMS algorithm. To solve this problem, NLMS which is a variat of the LMS has bee used widely. It solves the problem by ormalizig the power of the iput. I the NLMS algorithm, all the equatios metioed i the previous sectio stay the same except the weight update equatio will chage as follow [36]: w 1) w ) * x )* e ) 2 x ) 3.12) where x ) 2 H x ) * x ) is the ormalizig factor, H deotes the complex cojugate trasposed, ad is the regularizatio size. 28

39 3.2.3 Leaky Least Mea Square Algorithm It is very hard to approach the stability of LMS algorithm without choosig a learig rate of the step size that guaratee the stability. To deal with this problem a variat of LMS used, which is leaky LMS algorithm. This algorithm also desiged to overcome the slow covergece of LMS [39]. I leaky LMS, the iput, the output, ad the error sigals stay the same as i LMS. The chage will happe i the weight update equatio as follow: w 1) 1 * ) w ) * e )* x ) 3.13) where is the leaky factor, which is a very small positive umber <<1) Sig LMS ad Sig NLMS Algorithms These two algorithms are exactly similar to LMS ad NLMS. The oly differece is the weight update equatios. I LMS ad NLMS, the equatios depeds o the error sigal. Whereas i sig LMS ad sig NLMS use the sig of the error sigal oly to update the weight equatios as show i 3.14) ad 3.15) respectively: w 1) w ) *sg e ))* x ) 3.14) w 1) w ) *sg e ))* x ) 2 x ) 3.15) where sg e )) refers to the sig of the sigal e ). 29

40 3.3 Weier Filter Wieer filter is a class of liear optimum discrete time filters. The mai goal of usig these filters is to miimize a appropriate fuctio of error which is kow as the cost fuctio [38]. The most commoly cost fuctio that used i optimum filters is the mea square error MSE). Miimizig the MSE icludes the secod order statistics oly ad leads to a theory of liear filter that is useful i may applicatios. The Wieer filters ca be used i differet applicatios such as image processig, sigal processig, cotrol system, ad digital commuicatio. Figure 3.5 explais the mai cocept of Wieer filter. Figure 3.5: Block Diagram of Wieer Filter. The basic idea is that Wieer filter was desiged to produce a estimate sigal d ) of the desired sigal d ) usig a liear combiatio of the data x ) such that the mea square error fuctio is miimized. This process icludes filterig, smoothig, predicatio, ad decovolutio. 30

41 31 Assumig the processes x ), d ), etc., have zero mea value ad the filter coefficiets do ot chage with time. The output is equal to the covolutio of the iput ad the filter coefficiet. Hece [38] ) ) ) 1 0 x m x d w w T M m m 3.16) where M is the umber of filter coefficiets, w T is the filter weights as i adaptive filter, ad T M x x x x )] 1 1)... ), [ ) The cost fuctio J is defied as the miimum mea square of the differece betwee the actual output ad desired oe. )} { } )) ) { 2 2 e E d d E J 3.17) With some simplificatio: w R w w w x x E w x d E w d E w x x w w x d d x w d E x d x w d E x w d E e E w J x T dx T d T T T T T T T T T T T P w 2 )} ) { )} ) { 2 ) { } ) ) ) ) ) ) ) { ))] ) )][ ) {[ } )) ) {[ )} { ) where 2 d is the variace of the desired sigal d ), ad Pdx is the cross correlatio vector betwee d ) ad x ). T dx dx dx dx M P P P P )] _1 1),... 0), [ 3.18)

42 P dx r P r 0) 0), 1) 1)..., M 1) M 1) dx dx dx P dx r dx x ) x 1) E. x ), x 1)... x M 1) Rx. x M 1) E{ x ) x )} E{ x ) x 1)}... E{ x ) x M 1)} E{ x 1) x )} E{ x 1) x 1)}... E{ x 1) x M 1)} E{ x M 1) x )} E{ x M 1) x 1)}... E{ x M 1) x M 1)} 3.19) r x r x r x 0) 1) M 1) r r r 1) x 0) x M 2) x r... M 1) x... r M 2) x... r 0) x The Rx matrix is the correlatio matrix of the iput data, ad it is symmetric because the radom process is assumed to be statioary. Updatig the filter weights i Weier filter is ot that easy as i adaptive filter, eve if they follow the same path. A detailed process o how to update these weights are available i [37]-[38]. 3.4 Kalma Filter Kalma filter has become very popular i differet applicatios such as trackig objects, i ecoomics, avigatio, computer visio, ad may more. It has widely used to cotrol oisy system. The mai idea of Kalma filter is to reduce the oise by extractig the useful data from the oisy oe. For this approach, it uses immediate measured power [40]. I additio to oise reductio, Kalma filter ca be used to estimate the states of the system. Power egieers used this filter to estimate failure probability which is cosidered 32

43 very helpful [41]. By usig Kalma filter, it is possible to achieve good results. This sectio will preset some of Kalma filter s applicatios. Kalma filter has bee widely applied to the Global Positioig System GPS) which is satellite based avigatio system that provides users with the proper equipmet to useful ad accurate positioig iformatio aywhere o the globe. The oly problem could be due to the modelig error usig Kalma filter i avigatio process is divergece. To overcome this problem Jow, et al [42] proposed three strategies. First, applyig adaptive filter algorithms to prevet divergece of exteded Kalma filter EKF) i case the precise kowledge o the system are ot available. Secod, derivig a better oliear dyamic process model ad the utilizig a appropriate oliear estimate approach. Fially, iteractig multiple model IMM) algorithms takes ito accout for differet maeuverig coditios. IMM has bee used to allow the possibility of usig high dyamic model i the problem of divergece ad the improve the estimatio accuracy. Kalma filter was also used successfully as a motio estimator i video codig [43]. It is cosidered to be oe of the most commo tools that is extremely used i sigal processig field especially speech ehacemet ad speaker trackig [44]. Dikoomou et al [45] focused o the applicatio of Kalma filter i the Electroecephalogram EEG) processig. They preseted the differece betwee the Kalma filter KF) ad the Kalma smoother KS) based o how they use the observatios to perform estimatio. I the last few years Kalma filter techiques have bee used successfully for oise cacellatio to approach high precisio ad fast covergece. The work of Kazimi et al 33

44 [46] focused o two areas which are audio oise cacellatio ad electrocardiogram ECG) oise filterig. Based o the previous sigificat applicatios of Kalma filter, a ew model of harmoic reductio usig Kalma filter is itroduced. I additio to all applicatios of Kalma filter metioed earlier, this filter was used perfectly to reduce harmoics. It was applied to attack all harmoics cotaied i the distorted sigal ad good results were obtaied. The techique of Kalma filter works very well to get rid of harmoics that exist i the output curret sigal. The strategy is similar to oise cacellatio usig Kalma filter. The simplicity of derived equatios ad the other properties make it easy to use Kalma filter for harmoic reductio. I liear dyamic system, Kalma filters are discretized i the time domai. The state of the system is represeted as a vector of real umbers. Figure 3.6 shows a block diagram of a discrete time system with processed ad some oise mixed i it. Kalma filter was used i order to estimate the iteral state of the process give oly a sequeces of oisy observatios. The process must be modelled i accordace with the framework of Kalma filter as follow: x j A x j B u j w 3.20) 1 j z j H x j v j 3.21) where A is the state matrix gai, B is the iput matrix gai, Wj is the process oise, H is the output matrix gai, vj is the measuremet oise, ad j is the time variable. 34

45 At each step the iitial state ad the oise vectors are assumed to be mutually idepedet {x0, w1.wj, v1.vj}. Figure 3.6 is a correspodig block diagram of the system described i 3.20) ad 3.21). Figure 3.6: A Discrete Time system with Process w ad Measuremet Noise v. Sice Kalma filter is a recursive estimator, the estimated states from previous time step ad curret measuremet are eeded to estimate for the curret state. The state of the filter is represeted by two variables X, ad P. The block diagram of Kalma filter is show i Figure 3.7. Kalma filter is most ofte writte as two distict phases: predict ad update. The predict phase uses the state estimate from previous time step to produce a estimate of the state at the curret time step, which is also kow as the a priori state estimate. I the update phase, the curret a priori predictio is combied with curret observatio iformatio to refie the state estimate. This update state estimate is also kow as the posteriori state estimate. The predictor equatio is give by ˆ 3.22) j j1 j x Axˆ Bu 35

46 where x j ˆ 1 ˆ ad x j The update equatio is give by are the curret ad the previous estimated states respectively. xˆ j xˆ j K j z Hx j ˆ j ) 3.23) where K j is the Kalma gai. The priori P j ad a posteriori P j covariace matrix are give by: T P E { } { ˆ ) ˆ ) j e je j x j x j x j x j T P E { } { ˆ ) ˆ ) j e je j x j x j x j x j T T } } 3.24) 2.25) where E is the expected value, ej is the differece betwee the estimated ad the true value of the state, ad is the matrix traspose. Figure 3.7: Block Diagram of Kalma Filter. 36

47 CHAPTER 4 SIMULATION AND RESULTS This chapter presets the simulatios ad results of the study case described i this thesis ad what are the suggested solutios ad how they perform. 4.1 DC-AC Coverter i MATLAB SIMULINK A sigle-phase iverter, show i Figure 4.1, was simulated i MATLAB/SIMULINK [47] to study its performace i terms of power quality. The simulated circuit is show i Figure 4.2. As show i the figure, there are four choppers trasistors with diodes i parallel). Whe trasistors S1 ad S2 are tured o ad S3 ad S4 are off, positive voltage appears at the load. Whe trasistors S3 ad S4 are tured o ad S1 ad S2 are off, egative voltage appears at the load [32]. The amplitude of the output voltage is equal to the value of the DC voltage source. The simulatio parameters of the iverter circuit were selected as follow: Vdc = 600 V, the load was RL load with R = 0.5 Ohm ad L= mille Hery. PWM is a commo method for cotrollig the switches of the iverters [32]. For the simulatio of sigle phase switches cotroller, we use PWM geerator block i MATLAB/SIMULINK with the followig parameters: two arms geerator mode, 1000 Hz carrier frequecy, 0.6 modulatio idex, 60 Hz frequecy of the output voltage, ad 0 degree phase shift of the output voltage. The output curret ad voltage of this iverter will be distorted, as show i Figure

48 Figure 4.1: Sigle Phase Iverter. Figure 4.2: Simulated Iverter i MATLAB. 38

49 Figure 4.3: Iverter s Output Curret ad Voltage for RL Load. The mai cause of the distortio is the harmoics. To study the possible techiques that we suggested i chapter III for harmoics reductio, we chose to focus o the curret sigal oly. 4.2 Distortio Reductio Usig New Adaptive Filterig Models The distorted curret sigal of sigle phase iverter was used i this thesis. LMS, NLMS, Leaky LMS, sig LMS, ad sig NLMS algorithms are used as adaptive filterig techiques to clea the sigal. Istead of geeratig referece sigal as i Figure 4.4, a delayed versio of the sigal was used as iput sigal for the adaptive filter as show i Figure 4.5. The output sigal of the adaptive filter was subtracted from the distorted iput sigal to obtai the error sigal, which is resposible for cotrollig the weights of the adaptive filter. The error ad filter weights are computed i each iteratio based o the 39

50 equatios described i Chapter 3 for each algorithm. Istead of focusig o specific harmoics as i ASHE, all harmoics cotaied i the distorted sigal are targeted for elimiatio. For certai values of step size µ) ad filter legth M), Adaptive filterig algorithms gave a very clea sigal compared to the distorted oe as show i Figures These algorithms are very powerful i reducig the distortio despite the time required for the adaptatio process. Durig this time the output curret seems to be differet from the rated values but there is o overshoot i the sigal. I LMS algorithm, the step size has to be very small example 1e-8 rage) to reach the adaptio goal while i NLMS the goal is reached with higher values of step size example 0.01). Based o Figures , we ca say that NLMS ad sig NLMS are better tha other algorithms i term of step size. If we igored this term, the all algorithms give a good performace. The MATLAB code for LMS algorithm is show i Figure For NLMS, sig LMS, sig NLMS, ad Leaky LMS, the weight equatios are updated ad the LMS code is modified, as show i Figures Siged LMS ad siged NLMS use the same code as i LMS ad NLMS, respectively. The oly chage is to use the fuctio sg e )) i the weight update equatios. The total harmoic distortio THD) of the distorted curret sigal before processig was 4.9%. With adaptive filters, the THD is reduced to 2.5-3%. Leaky LMS gives the best performace i term of THD reductio. The THD reductio usig adaptive filters seem to be less effective eve if the output sigal looks very clea after the processig. The mai cause is the short time that adaptive filters eed to reach their optimizatio goal, which explais why the output sigal is ot stable i the begiig. If the load was chaged i a certai way so the THD got icrease adaptive filters are still be able to reduce it by more tha 50%. 40

51 Figure 4.4: Block Diagram of Adaptive filter. Figure 4.5: Block Diagram of the Proposed Method. 41

52 distorted cleaed Figure 4.6: Distorted ad Clea Curret Sigal Usig LMS Algorithm Whe µ=1e-8 ad M= distorted cleaed Figure 4.7: Distorted ad Clea Curret Sigal Usig NLMS Algorithm Whe µ=0.01 ad M=

53 Figure 4.8: Distorted ad Clea Curret Sigal Usig Leaky LMS Algorithm Whe µ=2e-9 ad M=120. Figure 4.9: Distorted ad Clea Curret Sigal Usig Sig LMS Algorithm Whe µ=2e- 9 ad M=

54 Figure 4.10: Distorted ad Clea Curret Sigal Usig Sig NLMS Algorithm Whe µ=1 ad M=120. clc; close all; primary=v:,1); t=0:0.5/legthprimary)): /legthprimary)); =1:legtht); d)=primary;%%%%%%%%%%d=primary is the distorted sigal x=zeros1,legtht)); x2:legtht))=d1:legtht)-1); %x is the delayed versio of the distorted sigal mu= ; M=120; N=legthx); y=zeros1,n);% iitialized output of the filter w=zeros1,m)% iitialized filter coefficiet vector e=zeros1,n);% iitialized output error of the filter for =M:N x1=x:-1:-m+1);% for each the vector x1 is of legth M withe elemets from x i reverse order y)=w*x1'; e)=d)-y); w=w+mu*e)*x1; % some books use other eq. which is w=w+2*mu*e)*x1; 44

55 ed; subplot2,1,1); plott,d); ylabel'distorted'); grid o; subplot2,1,2); plott,y); ylabel'cleaed'); grid o; Figure 4. 1: MATLAB Code for LMS Algorithm. mu=0.1; M=120; %%%%%%%%%%%%%%%%%%%%%%%%%%% epsilo = 1e-2; N=legthx); y=zeros1,n);% iitialized output of the filter w=zeros1,m)% iitialized filter coefficiet vector e=zeros1,n);% iitialized output of the filter for =M:N x1=x:-1:-m+1);% for each the vector x1 is of legth M withe elemets from x i reverse order y)=w*x1'; e)=d)-y); factor = epsilo + ormx1)^2; w = w + mu/factor)*e)*x1; % NLMS update % some books use other eq. Figure 4.12: MATLAB Code for NLMS Algorithm. 45

56 mu=0.01/ ; M=120; gamma=0.1; % gamma is the leakig factor which is greater tha zero but very very smaller tha 10<gamma<<1) N=legthx); y=zeros1,n);% iitialized output of the filter w=zeros1,m)% iitialized filter coefficiet vector e=zeros1,n);% iitialized output of the filter for =M:N x1=x:-1:-m+1);% for each the vector x1 is of legth M withe elemets from x i reverse order y)=w*x1'; e)=d)-y); w=1-mu*gamma)*w+mu*e)*x1; % some books use other eq. with 2 multiplied by mu; ed; Figure 4.13: MATLAB Code for Leaky LMS Algorithm. 4.3 Distortio Reductio Usig Weier Filter Weier filter ca be used to clea the distorted curret sigal from the iverter output. The steps metioed i Poularikas ad Ramada textbook [38] were exactly followed. I their textbook, they provide detail method o how to use Weier filter for oise cacellatio. The same steps they described ca be used for harmoics reductio sice harmoics are othig other tha sum of oise. The book also provides MATLAB code that ca be modified i order to fit our problem. Weier filter gives better performace if ormalized sigal is used istaed of the traditioal sigal. The MATLAB codes for this sectio are show i Figures The distorted ad clea sigal usig Weier filter are show i Figure The THD of the output curret sigal usig Weier filter is 46