To the Graduate Council:

Transcription

1 o the Graduate Counil: I am submitting herewith a dissertation written by Yan Xu entitled A Generalized Instantaneous Nonative Power heory for Parallel Nonative Power Compensation. I have examined the final eletronis opy of this dissertation for form and ontent and reommend that it be aepted in partial fulfillment of the requirements for the degree of Dotor of Philosophy, with a major in Eletrial Engineering. John N. Chiasson Major Professor We have read this dissertation and reommend its aeptane: Jak S. Lawler Suzanne M. Lenhart Seddik M. Djouadi Aepted for the Counil: Anne Mayhew Vie Chanellor and Dean of Graduate Studies (Original signatures are on file with offiial student reords.)

2 A Generalized Instantaneous Nonative Power heory for Parallel Nonative Power Compensation A Dissertation Presented for the Dotor of Philosophy Degree he University of ennessee Yan Xu May 2006

3 Copyright 2006 by Yan Xu All rights reserved. ii

4 Aknowledgments I would like to thank many people for their kind help with my study and researh so that I ould finish my dissertation. I sinerely appreiate my advisor, Dr. John N. Chiasson, for his instrution, his supervision, and his selfless support. I am very thankful to my ommittee members, Dr. Jak S. Lawler, Dr. Suzanne M. Lenhart, and Dr. Seddik M. Djouadi, for being on my ommittee and for their support and enouragement. Speial thanks go to Dr. Fang Z. Peng and Dr. Leon M. olbert, for their inspirational instrution and valuable diretions. I would like to thank all the people in the Power Engineering Laboratory at the University of ennessee, espeially Dr. Zhong Du, Dr. Kaiyu Wang, and Surin Khomfoi. I am also very thankful to Jeremy B. Campbell at Oak Ridge National Laboratory who has worked together with me on the experiments. I also thank Dr. Burak Ozpinei and Madhu Sudhan Chinthavali at Oak Ridge National Laboratory for their advie and help. Finally I would like to express my appreiation and love to my parents and sisters. hey have always loved me, supported me, and enouraged me unonditionally. iii

5 Abstrat Although the definitions of ative power and nonative power in a three-phase sinusoidal power system have been aepted as a standard, it has been an issue on how to define instantaneous nonative power in power systems with non-sinusoidal or even nonperiodi voltage and/or urrent. his dissertation summarizes these nonative power theories, and a generalized instantaneous nonative power theory is presented. By hanging the parameters in the instantaneous ative urrent and nonative urrent definitions, this theory is valid for power systems with different harateristis. Furthermore, other nonative power theories an be derived from this generalized theory by hanging parameters. his generalized nonative power theory is implemented in a shunt nonative power ompensation system. A three-phase four-wire nonative ompensation system onfiguration is presented, and a ontrol sheme is developed. he ompensator provides the nonative urrent omponent, whih is determined by the generalized nonative power theory. Unity power fator, and/or fundamental sinusoidal utility soure urrent an be ahieved despite the harateristis of the nonlinear load. Both the simulation and experiment results verify the validity of the theory to different systems and the apability in a shunt nonative ompensation system. iv

6 Contents Chapter 1 Introdution Standard Definitions in Power Systems Single-Phase Fundamental Nonative Power hree-phase Fundamental Nonative Power Nonlinear Loads and Distortion in Power Systems...10 Chapter 2 Literature Survey Nonative Power heories Instantaneous Nonative Power heories in ime Domain Instantaneous Nonative Power heories in Frequeny Domain Nonative Power Compensation Series-Conneted Compensators Parallel-Conneted Compensators Stati Synhronous Compensator (SACOM) Summary...28 Chapter 3 Generalized Nonative Power heory Generalized Nonative Power heory Averaging Interval = is a finite value Referene Voltage v p (t)...35 v

7 3.2 Charateristis of the Generalized Nonative Power heory hree-phase Fundamental Systems Periodi Systems Sinusoidal Systems Periodi Systems with Harmonis Periodi Systems with Sub-Harmonis Non-Periodi Systems Unbalane Generalized heory Summary...61 Chapter 4 Implementation in Shunt Compensation System Configuration of Shunt Nonative Power Compensation Control Sheme and Pratial Issues DC Link Voltage Control Nonative Current Control Simulations hree-phase Periodi System hree-phase Balaned RL or RC Load hree-phase Harmoni Load and Sinusoidal Voltage hree-phase Harmoni Load and Distorted System Voltage hree-phase Harmoni Load and Sinusoidal Referene Voltage hree-phase Diode Retifier Load hree-phase Load with Sub-Harmonis hree-phase Unbalaned RL Load Single-Phase Load Single-Phase RL Load Single-Phase Pulse Load Non-Periodi Load Disussion...98 vi

8 4.4.1 Averaging Interval, DC Link Voltage v d Coupling Indutane, L DC Link Capaitane Rating, C Summary Chapter 5 Experimental Verifiation System Configuration dspace System he Utility Loads Inverter Experimental Results hree-phase Balaned RL Load hree-phase Unbalaned RL Load Sudden Load Change and Dynami Response hree-phase Diode Retifier Load Single-Phase Load Conlusions and Summary Chapter 6 Conlusions and Future Work Conlusions Future Work Referenes Vita vii

9 List of ables able Page able 2.1. Comparison of nonative power theories in the time domain...21 able 3.1. Definitions of the generalized nonative power theory...37 able 3.2. Definitions of the generalized nonative power theory in sinusoidal systems able 3.3. Summary of the parameters v p and in the nonative power theory...60 able 4.1. Parameters of the three-phase RL load ompensation...72 able 4.2. Parameters of the three-phase harmonis ompensation with fundamental sinusoidal v s...79 able 4.3. Parameters of the three-phase harmonis ompensation with distorted v s...81 able 4.4. Parameters of the three-phase harmonis ompensation with sinusoidal v p...83 able 5.1. Ratings of the ommerial inverter able 5.2. Components of the nonative power ompensation system able 5.3. Parameters of the experiments able 5.4. Parameters of the three-phase balaned RL load ompensation able 5.5. Powers of the three-phase balaned RL load ompensation able 5.6. Parameters of the three-phase unbalaned RL load ompensation able 5.7. RMS urrent values of the unbalaned load ompensation viii

10 able 5.8. Parameters of the three-phase sudden load hange ompensation able 5.9. Parameters of the three-phase retifier ompensation able HD of the urrents in the retifier load ompensation able Parameters of the single-phase RL load ompensation able RMS urrent values of the single-phase load ompensation ix

11 List of Figures Figure...Page Figure 1.1. A two-port power network...2 Figure 1.2. Single-phase RL load in an a system...5 Figure 1.3. Ative power and nonative power in a single-phase RL load...6 Figure 1.4. hree-phase RL load in an a system...8 Figure 1.5. Ative power and reative power in a three-phase RL load...8 Figure 2.1. opologies of series and parallel ompensators...26 Figure 2.2. A three-phase inverter...27 Figure 3.1. Sequene analysis of a three-phase waveform...58 Figure 4.1. System onfiguration of shunt nonative power ompensation...64 Figure 4.2. DC link voltage ontrol diagram...66 Figure 4.3. Nonative urrent ontrol diagram...67 Figure 4.4. Control diagram of the shunt ompensation system...68 Figure 4.5. Simulation model diagrams of the nonative power ompensation system...69 Figure 4.6. hree-phase RL load simulation...71 Figure 4.7. hree-phase harmoni load with fundamental v s...78 Figure 4.8. hree-phase harmoni load ompensation with distorted v s...82 Figure 4.9. hree-phase harmoni load ompensation with sinusoidal v p...84 x

12 Figure Simulation of retifier load ompensation...85 Figure Simulation of sub-harmonis load ompensation...88 Figure hree-phase unbalaned RL load...90 Figure Simulation of single-phase load in a single-phase system...93 Figure Simulation of single-phase pulse load, = / Figure Simulation of single-phase pulse load, = Figure Compensator urrent normalized with respet to load urrent for different ompensation times and load urrent phase angles...96 Figure Compensator energy storage requirement and power rating requirement...97 Figure Simulation of non-periodi load ompensation...99 Figure 5.1. Experimental onfiguration of the nonative power ompensator Figure 5.2. dspace system onfiguration Figure 5.3. Experiment setup of the nonative power ompensation system Figure 5.4. Shemati of the interfae between dspace and the inverter Figure 5.5. Control diagram of the experimental setup Figure 5.6. hree-phase balaned RL load ompensation Figure 5.7. Regulation of DC link voltage in three-phase balaned RL load ompensation Figure 5.8. hree-phase unbalaned RL load ompensation Figure 5.9. Dynami response of three-phase RL load ompensation Figure A three-phase diode retifier load Figure Diode retifier load ompensation Figure Single-phase load ompensation xi

13 CHAPER 1 Introdution Alternating urrent, three-phase eletri power systems were developed in the late 1800s and have been adopted as the standard for more than 100 years. A typial power system onsists of generation, transmission, and distribution. In today s power systems, most of the ative power is generated by synhronous generators, and then transmitted to load enters by long distane transmission lines. In an ideal power system, the voltage and urrent are sinusoidal at a onstant frequeny (50 Hz or 60 Hz), and balaned (the same magnitude for eah of the three phases, and the phase angle between onseutive phases being 2/3 radians). A system is onsidered ideal if the generation is ideal (a sinusoidal waveform with onstant magnitude and onstant fundamental frequeny), and all the loads are linear, i.e., the loads only onsist of resistane R, indutane L, and/or apaitane C. here are single-phase loads (two-wire), three-phase three-wire loads (three-phase without neutral), and three-phase four-wire loads (three-phase with neutral) in a power systems. 1.1 Standard Definitions in Power Systems A power network N has two ports as shown in Figure 1.1. he voltage between the two ports is v(t), and the urrent is i(t) with the diretion flowing into the network N. 1

14 Figure 1.1. A two-port power network. he network an be an individual devie, suh as a generator, or a load. It also an be a network whih onsists of multiple devies. For an ideal eletri power system as defined above, both the voltage and the urrent are sinusoidal waveforms. For a single-phase system, let the voltage v(t) and the urrent i(t) be, respetively, v( t) 2V os( t), (1.1) and i ( t) 2I os( t ). (1.2) where 2 f is the frequeny in radians/seond, and f if the frequeny in Hz. 1/ f is the period in seond. is the phase angle between v(t) and i(t), and (, ]. V and I are the root mean square (rms) values of the voltage v(t) and the urrent i(t) over the period, respetively, V 1 2 v () tdt, (1.3) 0 and I 1 0 i 2 ( t) dt. (1.4) he instantaneous power p(t) is defined by 2

15 p() t v()() t i t. (1.5) In this dissertation, the term instantaneous means that the voltage/urrent/power is defined for eah time t rather than as an average value or an rms value. he instantaneous power p(t) is the rate of hange at whih the eletri energy is being transmitted into or out of the network N [1]. In the ase of a sinusoidal voltage and urrent as given in (1.1) and (1.2), the instantaneous power is pt () VI[os(2 t) os ]. (1.6) he instantaneous power has a onstant term, VIos, and a sinusoidal term, VIos(2t+), with a frequeny whih is twie of the fundamental frequeny. he average ative power P over the time period is the average value of the instantaneous power over one period of the waveform; that is, P 1 ptdt (). (1.7) 0 In the ase of a sinusoidal voltage and urrent as given in (1.1) and (1.2), respetively, it follows that 1 P p() t dt VI os. (1.8) 0 he average power is equal to the onstant term in the expression for instantaneous power given in (1.6). P is positive if ( /2, /2), whih indiates that the network onsumes ative power; P is negative if (, /2) or ( /2, ), whih shows that the network generates ative power; and P is zero if /2 or /2, whih means that the network does not onsume or generate any ative power. he apparent power is defined by 3

16 S VI. (1.9) S is the apaity that a devie or a network to generate, to onsume, or to transmit power, inluding both ative power and nonative power. herefore, S is often used to indiate the power rating of a devie. he average nonative power is defined by 2 Q S P 2. (1.10) In the ase of a sinusoidal voltage and urrent as given in (1.1) and (1.2), respetively, it follows that Q VIsin (1.12) Q is positive if (0, ), that is, the urrent i(t) is leading the voltage v(t), and it indiates that the network generates nonative power; Q is negative if (,0), i.e., the urrent is lagging the voltage, and it indiates that the network onsumes nonative power. Q is zero if 0, and in this situation, the urrent is in phase with the voltage and no nonative power is generated or onsumed by the network. he power fator pf is defined by pf P. (1.13) S If the voltage and urrent are as given in (1.1) and (1.2), respetively, pf os. For a multi-phase system with the phase number m, define the voltage vetor v(t) by v ( t) [ v ( t), v ( t),..., v ( t)]. (1.14) 1 2 m he urrent vetor i(t) is defined by i ( t) [ i ( t), i ( t),..., i ( t)]. (1.15) 1 2 m 4

17 Load i(t) L v(t) R Figure 1.2. Single-phase RL load in an a system. he instantaneous power p(t) is defined by m p() t v() t i() t vk() t ik() t. (1.16) k 1 hese definitions of instantaneous power in (1.5) and (1.16) are also true for voltages and urrents with arbitrary waveforms. However, other definitions are restrited to fundamental sinusoidal power systems. Many modifiations have been made to these standard definitions to extend them to non-sinusoidal, non-periodi, and other ases. hese power theories will be disussed in Chapter 2. In the following two subsetions, a single-phase RL load and a three-phase RL load will be used to illustrate the standard definitions in this subsetion. 1.2 Single-Phase Fundamental Nonative Power Figure 1.2 shows a resistive and indutive load (RL load) onneted to an a power soure. Let v(t) denote the a power soure given by v( t) 2V os( t). (1.17) he impedane of the load is 5

18 (a) pf < 1 (b) Unity power fator Figure 1.3. Ative power and nonative power in a single-phase RL load. j Z R j L Z e, (1.18) where 2 Z R L ( ) 2, (1.19) and L R 1 tan ( ). (1.20) he instantaneous urrent i(t) in this system is it () 2Ios( t ) (1.21) where I is the rms value of i(t) and may be written as I V. (1.22) Z he waveforms of v(t), i(t), p(t), as well as P, Q, and S are plotted in Figure 1.3. In Figure 1.3a, as L 0, it follows that 0 and i(t) is lagging v(t) by. When p(t) > 0, v(t) and i(t) have the same polarity, whih indiates that the power is flowing (on average) from the power soure to the load; when p(t) < 0 (the red areas in Figure 1.3a), the 6

19 polarity of i(t) is opposite to the polarity of v(t), whih indiates that the power is flowing (on average) from the load to the power soure. he power that flows bak and forth between the power soure and the load is referred to as the nonative power. his nonative power has zero average value (over one period), but is atual power flowing in the power network. he average nonative power Q indiates the amount of nonative power that flows in this network. In Figure 1.3b, the load has only resistane (L = 0), so that = 0 and i(t) is in phase with v(t). he instantaneous power p(t) is greater than or equal to zero at all times, whih indiates that power is delivered from the soure to the load at all times and no power is flowing bak and forth. In both ases, the same amount of power is delivered to the load (average power P over one period), but the magnitude of i(t) in Figure 1.3b is smaller than the urrent in Figure 1.3a. If i(t) is in phase with v(t) (i.e., unity power fator), i(t) is the minimum urrent required to transmit any given amount of power. he average nonative power Q is zero, whih indiates that there is no nonative power flowing in this network, so that the apparent power S is equal to the average power P. 1.3 hree-phase Fundamental Nonative Power A three-phase RL load in an a system is shown in Figure 1.4. he voltage vetor v(t) and urrent vetor i(t) are shown in Figures 1.5a and 1.5b. v( t) 2 V[os( t),os( t2 / 3),os( t 2 / 3)], (1.23) i( t) 2 I[os( t),os( t2 / 3 ),os( t2 / 3 )] (1.24) he instantaneous power p(t) is pt () 3VIos. (1.25) 7

20 i(t) R R R Load L L L v(t) Figure 1.4. hree-phase RL load in an a system. (a) Voltage v(t) (b) Current i(t) () pf < 1 (d) Unity power fator Figure 1.5. Ative power and reative power in a three-phase RL load. 8

21 he average ative power P is P 3VIos. (1.26) he apparent power S is S 3VI. (1.27) he average nonative power Q is Q 3VIsin. (1.28) he phase angle between the voltage and the urrent is negative in a RL load system, i.e., the urrent is lagging the voltage. Figure 1.5 shows the phase angle between the voltage and the urrent in phase a. he instantaneous power p(t), the average ative power P, the average nonative power Q, and the apparent power S are also plotted in Figure 1.5. he instantaneous power, whih is the sum of the instantaneous powers of three phases, is equal to the average ative power at all times, although the instantaneous power of eah phase is a sinusoidal waveform similar to the instantaneous power in the single-phase RL load ase whih was disussed in the previous subsetion. Q is negative, whih indiates that the load onsumes nonative power. In Figure 1.5d, the same amount of average ative power, whih is also equal to the instantaneous power, is onsumed by the load. However, the urrent is in phase with the voltage and the amplitude of the urrent is smaller than the one in Figure 1.5. he apparent power is equal to the average ative power and the average nonative power is zero. If the urrent is in phase with the voltage, the required amount of urrent to arry a given amount of ative power is minimum. Moreover, the average nonative power is zero and the apparent is also the minimum value, whih is equal to the average ative power. 9

22 Loads of unity power fator are desired from the system point of view. If loads draw a large amount of nonative power from the synhronous generators in a power system, the ative power generation apability of the generators will be weakened. Furthermore, the urrent to arry the nonative power inreases system losses and demands higher urrent ratings of the transmission lines. In a power system, the generation and onsumption of ative power must essentially be equal at any time as there is very little energy storage in a power system. In today s power systems, most ative power is generated by synhronous generators and transmitted to loads. Furthermore, the generation and onsumption of nonative power must also be equal. he transmission lines and most loads onsume nonative power (indutive). he shortage of nonative power an ause voltage instability and even voltage ollapse whih may lead to a blakout at a weak end of the system whih is far away from the generator. he nonative power ould be generated by synhronous generators; however, it is not an optimal hoie. One reason is beause the generation of nonative power by synhronous generators is at the ost of less ative power generation, and it is usually at a higher prie; and the other reason is the transmission of the nonative power generated by synhronous generators to loads results in inreased losses. herefore, nonative power should be ompensated and balaned loally for the reason of reduing transmission losses and providing voltage support as well. 1.4 Nonlinear Loads and Distortion in Power Systems Due to the nonlinear loads in a power system, there is distortion in the voltage and/or urrent waveforms, whih deteriorates the quality of the eletri power. For example, the 10

23 harmonis generated in the system may be amplified beause of parallel and series resonane in the system. he existene of harmonis redues the energy effiieny of a power system at the generation, transmission, and onsumption points of the system. Harmonis an ause extra heating of the ore of transformers and eletrial mahines. Furthermore, the harmonis may expedite the aging of the insulation of the omponents in the system and shorten their useful lifetime. A nonative ompensator based on power eletronis devies an be installed loally to ompensate the nonlinear omponents. Speifially, a nonative power ompensator installed in the distribution level is an effetive method to provide the fundamental nonative omponent of load urrent so that it does not need to be transmitted all the way from the generator to the load. his redues the losses of the transmission lines, and ompensates the distortion in the system due to nonlinear loads. In a system with distortion in the urrent and voltage waveforms, the standard definitions given in subsetion 1.1 for nonative power and nonative urrent are no longer appliable to defining, measuring, or desribing the power quality, whih is a preondition to improve the power quality as well as to eliminate the distortion in the system. Different nonative power theories have been proposed in the literature [2]-[37], and different nonative power ompensation methods have been implemented to ompensate the nonative power omponent in a power system [38]-[46]. In this dissertation, a new approah is presented on instantaneous nonative power definitions, whih an be used in a nonative power ompensation system for fundamental nonative power and other nonative power as well. 11

24 he following is a brief summary of the hapters in this dissertation. In Chapter 2, a literature survey is made of nonative power theories and nonative power ompensation. In Chapter 3, a generalized instantaneous nonative power theory is proposed. In Chapter 4, an implementation of the nonative power theory in a shunt nonative power ompensator is presented and simulated. Chapter 5 presents an experimental study of the proposed method. In the last hapter, the main results of this generalized instantaneous nonative power theory are summarized, and further researh is proposed. 12

25 CHAPER 2 Literature Survey In this hapter, instantaneous nonative power theories that have been proposed in the literature are reviewed, and their advantages as well as disadvantages are summarized. Also, various nonative power ompensation topologies are introdued and their shunt ompensation is then disussed. 2.1 Nonative Power heories As early as in the 1920s and 1930s, Fryze [2] and Budeanu [3] proposed instantaneous nonative theories for periodi (but non-sinusoidal) waveforms in the time domain and the frequeny domain, respetively. Different theories have been formulated in the last 30 years to desribe the inreasingly omplex phenomena in power systems. Due to the existene of nonlinear loads, urrents as well as voltages are distorted so that they are no longer pure sinusoids and sometimes not even periodi. hey an be divided into the following ategories: 1. Single-phase or multi-phase. 2. Periodi or non-periodi. 13

26 3. Balaned or unbalaned. For multi-phase voltages or urrents, if the magnitude of eah phase is equal and the phase angles between onseutive phases are also equal, then it is balaned, otherwise; it is unbalaned Instantaneous Nonative Power heories in ime Domain Various nonative power theories in the time domain have been disussed [4], and most of them an be divided into two ategories. he first ategory is Fryze s theory [2] and its extensions, and the other is p-q theory [11]-[12] and its extensions. Fryze s heory In 1931, Fryze introdued his definition of nonative power in [2] for sinusoidal power systems with harmonis by deomposing the soure urrent i(t) into an ative omponent i a (t), whih had the same waveform shape as (i.e., in phase with) the soure voltage v(t), and a nonative omponent i n (t) by the following definitions: P ia () t v(), t i () () () (2.1) 2 n t i t ia t v where v is the root mean square (rms) value of v(t) given by 1 2 v v t () dt (2.2) 0 and P is the average value of p(t) = v(t)i(t) over one period. If the soure voltage is a pure sinusoid as in (1.1) (without harmonis), then v V as in (1.3). he urrents i a (t) and i n (t) are mutually orthogonal; that is, ia() t in() t dt 0 (2.3) 0 he rms values i a and i n satisfy 14

27 2 2 a i i i n 2. (2.4) By multiplying (2.4) by v 2, Fryze defined nonative power Q F as Q v i n ( 2.5) F Q F is the same as the standard definition of Q in Chapter 1 if the voltage and urrent are steady-state fundamental sinusoidal waveforms. he apparent power S is defined as (2.6) S v i he apparent power, ative power, and nonative power satisfy S P Q F. (2.7) In an ideal system with fundamental sinusoidal voltage and urrent, Fryze s theory is onsistent with the standard definitions in Chapter 1. his theory is valid for systems with harmonis. Based on average and rms values, the nonative power Q F is an average value, while the ative urrent i a (t) and nonative urrent i n (t) are instantaneous. Fryze s definition is the first definition of instantaneous nonative power in the time domain given in the literature. It has been extended to multiple phases, and some ontemporary definitions of ative urrent are modifiations of his idea. he theories disussed below are based on Fryze s theory. Indutive and Capaitive Power In 1979, Kusters and Moore presented a theory for periodi non-sinusoidal waveforms in [5]-[6]. he ative urrent i a is defined the same as Fryze s theory in [2], but the nonative urrent is subdivided into two omponents the indutive (apaitive) reative urrent and a residual urrent. he indutive (apaitive) 15

28 urrent an be ompletely eliminated by adding a apaitor (indutor) in parallel with the load. heir theory is appliable to passive ompensation in single-phase systems. Instantaneous Multi-Phase Currents Willems [7] instantaneous multi-phase urrent theory, Rosssetto and enti s [8] instantaneous orthogonal urrent, Peng and Lai s [9] generalized instantaneous reative power theory for three-phase system, and the instantaneous reative quantity for multi-phase systems by Dai et al. [10] are essentially the same. hey are an extension of Fryze s theory to multi-phase systems and use instantaneous values instead of average values. he voltage and the urrent are vetors in a multi-phase system, i.e., v() [,,..., ] t v1 v2 v m i() [,,..., ] t i1 i2 i m (2.8) (2.9) where m is the number of phases. he instantaneous ative urrent is defined by vi ip() t v (2.10) vv he instantaneous nonative urrent is then defined as i () t i() t i () t q p (2.11) i p (t) and i q (t) are orthogonal. he differene between the theory in [9] and [10] from that given in [7] and [8] is the definition of instantaneous nonative power, whih in [9] and [10] is the ross-produt of the voltage vetor and the urrent vetor. he theory in [9] is only appliable to three-phase systems with or without neutral line. hese theories assume that the voltage is purely sinusoidal and balaned. he urrent is instantaneous and not limited to periodi waveforms. However, if i(t) has harmonis, this theory generates harmonis with frequenies different from the harmonis in i(t) due to the 16

29 multipliation between the fundamental omponent and a harmoni or two harmonis with different frequenies. p-q heory In 1984, Akagi et al. proposed an instantaneous reative power theory known as p-q theory for three-phase systems [11]-[12]. Park s transformation is applied to transform the a-b- quantities of the three-phase system to the - two-phase equivalent quantities aording to va v 2 1 1/2 1/2 v b v 3 0 3/2 3/2 (2.12) v and ia i 2 1 1/2 1/2 i b i 3 0 3/2 3/2. (2.13) i he instantaneous ative power p and reative power q are then defined as p v v i q v v i. (2.14) A Fourier series expansion of p onsists of a DC omponent p, whih is referred to as the fundamental omponent, and the higher order harmoni remaining terms denoted by p ; that is, p p p (2.15) Similarly, q may be written as q q q. (2.16) 17

30 he urrents i and i are divided into two omponents. he -axis omponent of the instantaneous ative urrent is defined by v v v i p p p p. (2.17) v v v v v v he -axis omponent of the reative urrent is defined by v v v i q q q q. (2.18) v v v v v v Similarly, the -axis omponent of the ative urrent is defined by v v v i p p p p (2.19) v v v v v v and the -axis omponent of the reative urrent is defined by v v v i q q q q. (2.20) v v v v v v Let the reative urrent be defined by v v i p ( q q ), v v v v (2.21) v v i ( ) p q q. (2.22) v v v v If i and i an be supplied by a ompensator, then the power utility only needs to provide the following ative urrents i s v 2 2 v v p (2.23) i s v v 2 2 v p (2.24) 18

31 and therefore only the ative power vi vi p. s s he p-q theory has the following limitations: 1. he original p-q theory was developed for three-phase, three-wire systems, and later was used in three-phase, four-wire systems by introduing the zero-phase sequene omponents. However, it annot be used in single-phase systems. 2. he assumption of this theory is that the voltage is purely sinusoidal and the three phases are balaned. However, in real power systems, there is distortion in the voltage, and the p-q theory will introdue errors if it is applied to a nonative power ompensation system [13]. 3. Average values are still used ( p, q, p ~, and q ~ ) in the theory, so it is not an atual instantaneous definition of real and reative omponents, i.e., data from at least half of a yle of the fundamental period are needed to alulate the average values. p-q-r heory Kim et al. [14]-[15] proposed a theory based on two sets of transfo rmations. he voltages and urrents are transformed first from Cartesian a-b- oordinates to Cartesian 0-- oordinates, and then from Cartesian 0-- oordinates to Cartesian p-q-r oordinates. It is valid in three-phase four-wire systems and ompletely eliminates the neutral urrent even if the system voltage is unbalaned or distorted by harmonis. In addition to these theories, the Fryze-Buhholz-Depenbrok (FBD) method by Depenbrok [16] is valid in multi-phase periodi systems. he definitions of the ative urrent and the nonative urrent are an extension of Fryze s idea to multi-phase systems. 19

32 Beause average values are used, omplete ompensation of nonative urrent is only possible under steady-state onditions. Enslin and Van Wyk [17]-[20] modified Fryze s theory to single-phase non-periodi systems. hey introdued a time interval for a non-periodi system and the ative urrent and the nonative urrent are defined based on this time interval. herefore, the definitions depend on the interval and the time at whih the interval starts. Ferrero and Superti-Furga s Park power theory [21]-[22] is the same as p-q theory [11]-[12], exept the definition of Park imaginary power (instantaneous nonative power). In addition, they also extended Fryze s theory [2] to three-phase three-wire systems. Nabae and anaka [23] defined the voltage and urrent spae vetors in polar oordinates. It is restrited to three-phase systems. All the definitions of urrents and powers are originated from fundamental frequeny and balaned systems. Systems with harmonis or unbalaned are not disussed in the papers. able 2.1 summarizes the nonative power theories in the time domain Instantaneous Nonative Power heories in Frequeny Domain Instantaneous nonative power theories developed in the frequeny domain share a key idea that the voltage and the urrent are deomposed into a fundamental omponent and harmonis by using the Fourier series deomposition. Of ourse, this is appliable to periodi systems only. Furthermore, it requires at least one half yle of the fundamental period to ompute the average values and thus is not instantaneous. Beause of the ross multipliation between harmonis, the definition of power is more omplex than in the time domain. o meet the standard definitions in [1], some theories have to define several 20

33 able 2.1. Comparison of nonative power theories in the time domain. heory [Referene] Phases ime Period Current Waveforms Voltage m Average Instantaneous Fundamental Harmonis Non- Periodi Fryze s heory [2] a Indutive and Capaitive Power [5]-[6] Instantaneous Multi-Phase Currents [7]-[10] p-q heory [11]-[12] p-q-r heory [14]-[15] a b b FBD Method [16] b Single-Phase Non- Periodi [17]-[20] Park Power [21]-[22] Nabae and anaka [23] d b b Notes: 1. he assumptions of the system voltage, whih are a. Fundamental sine wave b. Fundamental sine wave and balaned. Periodi, but non-sinusoidal d. No assumption 21

34 nonative power omponents, whih do not have expliit physial meanings or provide a good foundation for nonative power ompensation. Budeanu s Approah he first definition of nonative power in the frequeny domain was proposed in 1929 by Budeanu [3]. He extended the notion of nonative power from systems with sinusoidal voltage and urrent to those with periodi nonsinusoidal waveforms using the Fourier series deomposition. he voltage v(t) and urrent i(t) are deomposed as vt () 2V n os( nt ), (225). n n1 it () 2Inos( nt n). (2.26) n1 he ative power P is given by 1 P v()() t i t dt VnInos( n n). (2.27) 0 n1 he rms values of v(t) and i(t) are, respetively, V V (2.28) n1 2 n and I I. (2.29) n1 2 n he apparent power S is given by S VI. (2.30) he total reative power Q B B is defined as 22

35 B Q V I sin( ). (2.31) B n n n n n1 However, S P Q B, whih is not onsistent with the standard definitions in [1]. herefore, the distortion power D B is introdued so that S P Q D (2.32) B B hey do not have physial meanings and the minimization of Q BB does not neessarily improve the power fator [24]. Hilbert Spae ehniques [25] An n-phase system is onsidered, in whih the timeby a summable Fourier series, domain vetor waveforms v(t) and i(t) are periodi with a fundamental period. Suh waveforms an be represented i.e., vt () l Ve l jlt (2.33) and jlt it () Ie. n l (2.34) V l and I l are defined as the Fourier oeffiients, that is, 1 jlt Vl vte ( ) dt vt ( ), where l,..., l (2.35) and 1 jlt Il i( t) e dt i( t), whe l re l,...,. (2.36) he instantaneous ative urrent i a(t) and the instantaneous inative urrent i x(t) are defined by 23

36 i() t v() t p() t ia () t v() t v() t v() t v() t v() t v() t (2.37) i () t i () t i () t (2.38) x a he definitions of ative urrent and nonative urrent in this theory are the same as the theories in [7]-[10], and their problems are: 1. he ative urrent i a (t) may have harmoni omponents with frequenies different from the harmonis in i( t). As shown in (2.37), the new harmoni omponents are generated due to the multipliation between the fundamental omponent and a harmoni or two harmonis with different frequenies. he goal of nonative power ompensation is to provide the nonative omponent in the urrent, inluding fundamental nonative power, harmonis, and other nonative omponents. Although the aim of this definition is to minimize the soure urrent ( in rms) that supplies the same power as the original load urrent, this definition inreases the harmoni ontent in the urrent instead of improving it. 2. If n = 1 (a single-phase system), i a (t) i(t), and i x (t) 0. his theory is not able to separate the ative power and nonative power in a single-phase system. In addition to the theories whih have been disussed in the frequeny domain, referenes [26]-[36] deompose the nonative power or nonative urrent into several omponents. Some of the definitions of ative power in these papers are onsistent with the physial meaning of ative power, while others are not. he definitions of nonative power in some theories have ative power omponents. All the theories in the frequeny domain assume that the Fourier series of the voltage and urrent are known, whih is not pratially useful for nonative power ompensation appliations. 24

37 2.2 Nonative Power Compensation Linear loads whih do not have unity power fator draw fundamental nonative power. Nonlinear loads ause problems suh as harmonis, unbalane, and distortion in power systems, in addition to the problem of fundamental nonative power. Some nonlinear loads draw irregular urrents, whih are neither sinusoidal nor periodi. Variable-speed motor drives, ar furnaes [37], omputer power supplies, and single- (unbalaned in a three-phase system) are the most ommon nonlinear loads phase loads in power systems. Both linear and nonlinear loads ause pollution in power systems, i.e., low power fator, exess transmission losses, voltage sags, voltage surges, and eletromagneti interferene (EMI), whih deteriorate the power quality, and utility or onsumer equipment is damaged in some extreme ases. Nonative power ompensation is an effetive method to lean up the pollution in power systems. Synhronous generators, apaitor banks, and parallel- or series- onneted reators an ompensate the fundamental reative power and improve the power fator, and passive L-C filters an redue harmonis. However, these are all steady-state devies and have little instantaneous apability. With the development of power eletronis tehnology, solid-state swithing devies suh as thyristors, gate turnoff thyristors (GOs) and insulated gate bipolar transistors (IGBs), new approahes are possible for nonative power ompensation [38]-[40] Series-Conneted Compensators Depending on the harateristis of a nonlinear load, it an be represented as a voltage-soure load or as a urrent-soure load. For a voltage-soure load, a ompensator 25

38 v s Z s v + v _ Load _ + v L v s i s i L Z Load i (a) Series-Conneted Compensator (b) Parallel-Conneted Compensator Figure 2.1. opologies of series and parallel ompensators. is onneted in series with the load (Figure 2.1a). Most loads are represented as urrentsoure loads [39], and the ompensator ats as a nonative power soure in parallel (shunt) with the utility and injets a ertain amount of urrent (the nonative omponent of the load urrent) to the system so that the utility only provides the ative omponent to the load Parallel-Conneted Compensators As mentioned above, most loads an be represented as urrent soures. o ompensate the nonative omponent in the load urrent, a parallel ompensator is needed whih injets the same amount of nonative urrent (i in Figure 2.1b) as the nonative omponent in the load urrent so that the utility provides ative urrent only (i s in Figure 2.1b) Stati Synhronous Compensator (SACOM) here are different topologies of nonative power ompensators, suh as the thyristor ontrolled reator (CR), the thyristor swithed apaitor (SC), and the stati synhronous ompensator (SACOM). Among them, SACOM is a shunt-onneted 26

39 + V d D 1+ D D _ D 1- D 2- D 3- Figure 2.2. A three-phase inverter. stati var ompensator whose output urrent an be ontrolled independent of the a system voltage. A SACOM uses IGBs or GOs, whih have turn-on and turn-off apabilities; it does not require an energy soure, and requires only very limited energy storage if it performs nonative power ompensation only. Also referred to as an ative filter, the SACOM has lent itself to real-time nonative power/urrent ompensation appliations whih is independent of the system voltage beause of its flexibility in the ontrol of swithes. Figure 2.2 shows a typial topology of a three-phase inverter, whih is widely used in SACOMs. his is a voltage-soure inverter whih has apaitors on the d side as it provides nonative urrent only so that no energy soure is needed. he a side is onneted to the system in parallel with the utility to provide demanded urrent to the load. he system onfiguration of a SACOM will be disussed in more detail in Chapter 4. Nonative power theories are implemented for a shunt nonative ompensation, and the ontrol strategies and pratial issues are disussed in [41]-[46]. 27

40 2.3 Summary A survey on the nonative power/urrent theories in both the time domain and the frequeny domain was provided. As explained, the theories in the time domain are divided into two ategories, Fryze s approah and the p-q approah. Both approahes were haraterized aording to the number of phases they are appliable to, the time period whih is used to define the quantities (instantaneous or average), and their ability to handle urrent waveforms (fundamental, distorted with harmonis, or non-periodi). heir advantages and disadvantages were disussed as well as their appliability to the ompensation of nonative power. he nonative power/urrent theories developed in the frequeny domain were also briefly disussed. It was stated that they are restrited to periodi systems and result in the definitions of two or more nonative power omponents whih do not have any physial meanings. Some of the proposed definitions of ative power were not onsistent with the real (physis-based) meaning of ative power. It is assumed that both the voltage and the urrent are in steady state so that they an be presented by the Fourier series, so the ability of these theories to handle a transient state is poor. Furthermore, at least one yle of the fundamental period is needed to perform the Fourier deomposition. herefore, these theories annot be applied to instantaneous nonative power ompensation. 28

41 CHAPER 3 Generalized Nonative Power heory Many definitions of nonative power/urrent have been disussed in the previous hapter. here is little debate about the definitions of ative power or nonative power in sinusoidal ases, but the theories differ in the ase of non-sinusoidal and non-periodi waveforms. Some definitions work well in three-phase, three-wire systems, but are not able to handle the single-phase ase, while others an only be applied to periodi waveforms. Furthermore, some definitions are related to physial quantities, while other do not have any physial meaning. he theory proposed by Peng and Lai [9] is a generalized theory of instantaneous ative power/urrent and instantaneous nonative power/urrent for a three-phase system. Ative power is the time rate of energy generation, transmission, or onsumption. he urrent that arries ative power is the ative urrent, whih is the omponent of urrent in phase with the voltage. Nonative power is the time rate of the energy that irulates bak and forth between the soure and load. In a multi-phase iruit, it also inludes the power that irulates among phases. 29

42 3.1 Generalized Nonative Power heory he generalized nonative power theory proposed in this dissertation is based on Fryze s idea of nonative power/urrent disussed in setion and is an extension of the theory proposed in [47]. Let a voltage vetor v(t) and a urrent vetor i(t) be given by v () t [ v (), t v (),..., t v ()] t, (3.1) 1 2 m and i ( t) [ i ( t), i ( t),..., i ( t)], (3.2) 1 2 m respetively, where m is the number of phases. he instantaneous power p(t) is defined by m p() t v ()() t i t vk() t ik() t. (3.3) k 1 he average power P(t) is defined as the average value of the instantaneous power p(t) over the averaging interval [t-, t], i.e., t 1 Pt () p( ) d. (3.4) t he instantaneous ative urrent i a (t) = [i a1 (t), i a2 (t),, i am (t)] and instantaneous nonative urrent i n (t) = [i n1 (t), i n2 (t),, i nm (t)] are, respetively, ia Pt () () t v p () t, (3.5) V () t 2 p and i () t i() t i () t. (3.6) n a he rms value V(t) of a voltage vetor v(t) is defined by 30

43 1 t Vt ( ) ( ) ( ) d v v. (3.7) t In (3.5), the voltage v p (t) is the referene voltage, whih is hosen based on the harateristis of the system and the desired ompensation results. It will be elaborated in subsetion V p (t) is the orresponding rms value of the referene voltage v p (t), i.e., t 1 Vp ( t) ( ) ( ) d vp v. (3.8) p t Based on the above definitions for i a (t) and i n (t), the instantaneous ative power p a (t) and instantaneous nonative power p n (t) are defined as m pa () t v () t ia () t vk() t iak() t, (3.9) k 1 and m pn () t v () t i n() t vk() t ink() t. (3.10) k 1 he rms values of the ative urrent i a (t), nonative urrent i n (t), and urrent i(t) are, respetively, 1 t Ia () t ( ) ( ) d i t a i, (3.11) a 1 t In ( t) ( ) ( ) d i t n i, (3.12) n and 1 t It () ( )( ) d i i. (3.13) t 31

44 he average ative power P a (t) is defined as the average value of the instantaneous ative power p a (t) over the averaging interval [t-, t], i.e., t 1 Pa() t pa( ) d. (3.14) t he average nonative power P n (t) is defined as the average value of the instantaneous nonative power p n (t) over the averaging interval [t-, t], i.e., t 1 Pn() t pn( ) d. (3.15) t Based on the rms values defined above, the apparent power S(t) is defined by St () VtIt () (). (3.16) he apparent ative power P p (t) is defined by P () t V() t I () t. (3.17) p a he apparent nonative power Q(t) is defined by Qt () V() ti() t. (3.18) n Averaging Interval he standard definitions for an ideal three-phase, sinusoidal power system use the fundamental period to define the rms values and average ative power and nonative power. In the generalized nonative power theory, the averaging time interval an be hosen arbitrarily from zero to infinity, and for different, the resulting ative urrent and nonative urrent will have different harateristis. he flexibility of hoosing different as well as the referene voltage results in this theory being appliable for defining nonative power for a larger lass of systems than in the urrent literature. For 32

45 eah ase, a speifi value of an be hosen to fit the appliation or to ahieve an optimal result. he hoie of will be disussed in the following subsetions = 0 In this ase, the definitions of average powers are the same as the instantaneous powers, and the rms definitions have different forms, i.e., Vt () v () tv () t, (3.19) I() t i ()() t i t, (3.20) and V () t v () t v () t. (3.21) p p p Further, the definitions of instantaneous ative urrent i a (t) and instantaneous nonative urrent i n (t) are, respetively, i a pt () () t v p () t, (3.22) V () t 2 p and i () t i() t i () t. (3.23) n a If v p (t) = v(t), the instantaneous ative power p a (t) is equal to the instantaneous power p(t), and the instantaneous nonative power p n (t) is identially zero, that is, p () t v () t i () t p() t, (3.24) a a p () t v () t i () t 0. (3.25) n n More speifially, in a single-phase system, the instantaneous ative urrent i a (t) is always equal to the urrent i(t), and the instantaneous nonative urrent i n (t) is always 33

46 zero, therefore, = 0 is not suitable for nonative power/urrent definitions in singlephase systems is a finite value For most appliations, will be hosen as a finite value. For a periodi system with fundamental period, is hosen as = /2. If v p (t) is hosen as a periodi waveform with period, then the average power P(t) and the rms value V p (t) are both onstant numbers, i.e., P(t) = P, and V p (t) = V p. P i a () t vp () t. (3.26) V 2 p he instantaneous ative urrent is proportional to and has the same shape as the referene voltage. herefore, by hoosing different referene voltages, the instantaneous ative urrent an have different waveforms. he generalized nonative power theory does not speify the harateristis of the voltage v(t) and urrent i(t), i.e., they an theoretially be any waveforms. However in a power system, the voltage is usually sinusoidal with/without harmoni distortion, and the distortion of the voltage is usually lower than that of the urrents (the total harmoni distortion (HD) of the voltage is usually less than 5%.). herefore in this dissertation, the voltage is assumed to be periodi for all ases. A non-periodi system is referred to as a system with periodi voltage and a non-periodi urrent. In a non-periodi system, the instantaneous urrent varies with different averaging interval, whih is different from the periodi ases. In appliations suh as nonative power ompensation, is usually hosen to be 1-10 times that of the fundamental period based on the tradeoff between 34

47 aeptable ompensation results and reasonable apital osts. his will be illustrated in Chapter his is a theoretial analysis of a non-periodi system, in whih a finite an not ompletely aount for the entire nonative omponent in the urrent. It will be elaborated in Subsetion However, is not pratial in an atual power system appliation, and a finite will be used instead Referene Voltage v p (t) If P(t) and V p (t) are onstant, whih an be ahieved with = /2 for a periodi system, or with for a non-periodi system, the ative urrent i a (t) is in phase with v p (t) (as shown in (3.26)), and the waveforms of i a (t) and v p (t) have the same shape and they differ only by a sale fator. heoretially, v p (t) an be arbitrarily hosen, but in pratie, it is hosen based on the voltage v(t), the urrent i(t), and the desired ative urrent i a (t). Choies for v p (t) inlude 1. v p (t) = v(t). If v(t) is a pure sinusoid; or the ative urrent i a (t) is preferred to have the same waveform as v(t). 2. v p (t) = v f (t), where v f (t) is the fundamental positive sequene omponent of v(t). In power systems, if v(t) is distorted or even unbalaned, and a purely sinusoidal i a (t) is desired, then v p (t) is hosen as the fundamental positive sequene omponent of v(t). his ensures that i a (t) is balaned and does not ontain any harmonis. 3. Other referenes to eliminate ertain omponents in urrent i(t). For example, in a hybrid nonative power ompensation system with a SACOM and a passive 35

48 LC filter, the lower order harmonis in the urrent are ompensated by the SACOM, and the higher order harmonis will be filtered by the passive filter due to the limit of the swithing frequeny of the SACOM. In this ase, a referene voltage with the higher order harmonis is hosen so that the resulting nonative urrent whih will be ompensated by the SACOM does not ontain these harmonis. By hoosing different referene voltages, i a (t) an have various desired waveforms and the unwanted omponents in i(t) an be eliminated. Furthermore, the elimination of eah omponent is independent of eah other. he definitions of the generalized nonative power theory are onsistent with the standard definitions for ideal sinusoidal ases. he theory also extends the definitions to other situations. able 3.1 summarizes the definitions of the generalized nonative power theory. All the definitions are funtions of time. Column 1 ontains the definitions of the instantaneous voltage, urrents, and powers. he voltage and urrents are vetors, while the powers are salars. Column 2 ontains the rms values of the instantaneous voltages and urrents in olumn 1. hey are the root mean square values of the orresponding instantaneous definitions from olumn 4 over the time interval [t-, t]. Column 3 ontains the average values of the instantaneous powers in olumn 1 over the time interval [t-, t]. Finally, olumn 4 gives the definitions of apparent powers, whih are derived from the rms values of the voltages and urrents. he harateristis and the physial meaning of the definitions and the relationship of these definitions will be disussed in more detail in the following subsetion. 36

49 able 3.1. Definitions of the generalized nonative power theory. Instantaneous Definitions Root Mean Square (rms) Definitions Average Powers Apparent Powers v(t) = 1 t [v 1 (t),, v m (t)] Vt ( ) ( ) ( ) d v v t i(t) = 1 t I() t ()() d [i 1 (t),, i m (t)] i i t Pt () ia() t vp() t V () t 2 p 1 t Ia() t () () d i t a ia i () t i() t i () t n a 1 t In() t ( ) ( ) d i t n in p(t) = v (t)i(t) p a (t) = v (t)i a (t) p n (t) = v (t)i n (t) 1 Pt () p( ) d t t 1 Pa() t pa( ) d t t 1 Pn() t pn( ) d t t S(t) = V(t)I(t) P p (t) = V(t)I a (t) Q(t) = V(t)I n (t) 37

50 3.2 Charateristis of the Generalized Nonative Power heory In general, the nonative power theory has the following harateristis. 1. i a (t) and i n (t) are orthogonal. he instantaneous ative urrent i a (t) and the instantaneous nonative urrent i n (t) are orthogonal, that is, t t i ( ) i ( ) d 0, (3.27) a so that I 2 n 2 2 ( t) I ( t) I ( t). (3.28) a n he instantaneous ative urrent is in phase with the voltage v(t), while the instantaneous nonative urrent is 90 out of phase with v(t). he physial meaning of this harateristi is that the ative urrent arries ative power and the nonative urrent arries nonative power. 2. he instantaneous ative power p a (t) and nonative power p n (t). he instantaneous power p(t) is deomposed into two omponents, the instantaneous ative power p a (t) and the instantaneous nonative power p n (t), whih satisfy p( t) p ( t) p ( t). (3.29) a n Similar to the instantaneous ative urrent and nonative urrent, at any moment, the power flowing in the system has two omponents, i.e., the ative power omponent and the nonative power omponent. 3. he apparent powers. If v p (t) = v(t), the apparent powers S(t), P p (t), and P n (t) satisfy 38

51 S () t P () t Q () t. (3.30) p he generalized theory has different harateristis when it is applied to different systems. hree-phase systems will be first disussed in subsetion 3.2.1; periodi and non-periodi systems will be disussed in subsetions and he unbalaned system will be disussed in subsetion hree-phase Fundamental Systems In partiular, a three-phase fundamental system will be disussed in this subsetion sine most of the power systems are three-phase. he voltage v(t) and urrent i(t) are, respetively v () t v () t v () t v () t, (3.31) and i () t i () t i () t i () t. (3.32) he instantaneous ative urrent and the instantaneous nonative urrent are, respetively, ia1() t vp 1() t Pt () () t i () t v () t, (3.33) i () t v () t ia a2 2 p2 Vp () t a3 p3 and in1() t i1() t ia1() t i () t i n2() t i2() t n ia 2() t. (3.34) in3() t i3() t ia3() t where 39

52 t t 1 Pt ( ) v( ) i( ) v( ) i ( ) v( ) i( ) d, (3.35) and t p p1 p2 p3 t V ( t) v ( ) v ( ) v ( ) d. (3.36) For a three-phase four-wire system, the neutral urrent i 0 (t) is the sum of the other three phases, i.e., i () t i () t i () t i () t. (3.37) he instantaneous ative power p a (t) and the instantaneous nonative power p n (t) are, respetively, ia1() t pa() t v1() t v2() t v3() t ia2() t v1() t ia 1() t v2() t ia2() t v3() t ia3() t, (3.38) ia 3() t and in1() t pn() t v1() t v2() t v3() t in2() t v1() t in 1() t v2() t in2() t v3() t in3() t. in 3() t (3.39) In a three-phase balaned sinusoidal system, both the voltage and the urrent are sinusoidal and have the same frequeny. he voltage and urrent vetors are represented as v( t) 2Vos( t) 2Vos( t2 /3 ) 2Vos( t2 /3 ), (3.40) and i() t 2Ios( t) 2Ios( t2 /3 ) 2Ios( t2 /3 ). (3.41) 40

53 he phase angle between the voltage and the urrent is given by -. Let the averaging interval be half of the system period, i.e.,. (3.42) 2 he instantaneous power p(t) and average power P(t) are pt () v ()() ti t 3VIos( ), (3.43) and 1 t Pt () ( ) ( ) d 3VIos( ) v i. (3.44) t P(t) and p(t) are equal in this three-phase system. Furthermore, P(t) is equal to p(t) in any multi-phase balaned sinusoidal system. Let the referene voltage be the system voltage v(t), that is, v () t v () t. (3.45) p he instantaneous ative urrent i a (t) and the instantaneous nonative urrent i n (t) are, respetively, given by 2Ios( )os( t) Pt () ia() t v () t 2Ios( )os( t 2 /3 ) 2, (3.46) p Vp () t 2Ios( )os( t2 /3 ) and 2Isin( )sin( t) in() t i() t i () t 2Isin( )sin( t2 /3 ). (3.47) a 2Isin( )sin( t2 / 3 ) 41

54 he ative urrent vetor i a (t) is in phase with the system voltage vetor v(t) and is the ative omponent of i(t), and i n (t) is the nonative omponent. he two urrents i a (t) and i n (t) are orthogonal, that is, 1 t t a n i ( ) i ( ) d 0. (3.48) he instantaneous ative power p a (t) and instantaneous nonative power p n (t) are, respetively, p () t v () t i () t 3VIos( ) p() t, (3.49) a and a p () t v () t i () t 0. (3.50) n n he average ative power P a (t) and the average nonative power P n (t) are, respetively, 1 t Pa ( t) ( ) ( ) d 3VIos( ) P( t) v i t a, (3.51) and 1 t Pn ( t) ( ) ( ) d 0 v i t n. (3.52) he apparent power S(t), apparent ative power P p (t), and the apparent nonative power Q(t) are, respetively St () 3VI, (3.53) P () t 3VIos( ), (3.54) p and 42

55 Qt () 3VIsin( ). (3.55) he sum of the instantaneous ative power of the three phases is equal to the sum of the instantaneous power (3.49), and the sum of the nonative power is zero (3.50). his indiates that there is no nonative power at any time for all three phases, but there is nonative power/urrent flowing in eah phase, as shown in (3.47). he amplitude of the urrent in eah phase of i(t) is 2 I, in whih only the ative omponent (the amplitude of the urrent in eah phase of i a (t), whih is given by 2I os( ) ) is the effetive part to arry ative power, and the rest of it (the amplitude of the urrent in eah phase of i n (t), whih is 2I sin( ) ) is the urrent flowing bak and forth between the soure and the load whih arries nonative power. he average powers P(t), P a (t), and P n (t) indiate the energy utilization in the system. P(t) is always equal to P a (t), and P n (t) is equal to zero. he apparent powers S(t), P p (t), and Q(t) indiate the power flow in the system, whih inludes the nonative power flowing bak and forth between the soure and the load or among the phases Periodi Systems he three-phase, sinusoidal ase disussed in the previous subsetion is generalized to a periodi system with m phases. For a periodi system with periodi, if = k/2, k = 1, 2,, and v p (t) = v(t), P(t) and V p (t) are onstant values, i.e., P(t) = P, and V p (t) = V(t) = V. Speifially, t t 1 P P 1 2 a( ) ( ) ( ) ( ) 2 p 2 V V t t, (3.56) P t v v d v d P 43

56 t t 1 P P 1 2 n( ) ( ) ( ) ( ) ( ) ( ) 0 2 p 2 V V t t P t v i v d P t v d. (3.57) hat is, P a (t) = P and P n (t) = 0. Over the time interval [t-, t], P a (t), the average value of p a (t), is equal to P(t) (the average value of p(t)); and P n (t), the average value of p n (t), is zero. It indiates that instantaneously p(t) has both ative and nonative omponents, but on average, P(t) has only the ative power, whih is equal to P a (t). he average value of p n (t) is zero, whih indiates that the nonative power p n (t) flows bak and forth, and over the time interval, there is no net energy utilization. his is onsistent with the onventional definition of ative power and nonative power. For a periodi system, the apparent power S(t), apparent ative power P p (t), apparent nonative power Q(t) satisfy S () t P () t Q () t. (3.58) p A periodi funtion f(t) of fundamental period satisfies f ( t) f ( t k). (3.59) f(t) an be expressed as a Fourier series given by f ( t) A 0 Ak os( k 1t k ) k1, where 2 1. (3.60) As to periodi waveforms in a power system, if the frequenies are all integral multiples of the system fundamental frequeny, it is said to have only harmonis (Subsetion ); if the waveform is pure sine wave with the fundamental frequeny, it is a sinusoidal ase (Subsetion ); and if the frequeny ontent of the waveforms ontains frequenies whih are not integral multiples of the system fundamental frequeny (50/60 Hz), then it is said to ontain sub-harmonis (Subsetion ). 44

57 Sinusoidal Systems A sinusoidal system with voltage vetor v(t) and urrent vetor i(t) and period, v ( t) [ v ( t), v ( t),..., v ( t)], (3.61) 1 2 m where v ( t) 2V os( t2 k / m), k 0,1,..., m 1, and 2 /. k i () t [ i (), t i (),..., t i ()] t, (3.62) 1 2 m where i ( t) 2Ios( t2 k / m), k 0,1,..., m 1. k For both the voltage and urrent, the magnitude of eah phase is equal, and the phase angle between eah two ontiguous phases is also equal, i.e., the system is balaned. he average power P(t) is 1 t Pt () p( ) d mvios( ) P. (3.63) t Let v p (t) = v(t), the rms value of v p (t) is V () p t mv V p. (3.64) herefore, the instantaneous ative urrent i a (t) and the nonative urrent i n (t) are, respetively P 2 2( m1) i ( t) ( t) 2Ios( )[os( t),os( t ),...,os( t )] a v, V m m 2 p i n (3.65) ( t) 2 2( m 1) 2Isin( )[sin( t),sin( t ),...,sin( t )]. m m (3.66) he instantaneous ative power p a (t) and the instantaneous nonative power p n (t) are, respetively 45

58 p ( ) os( ) a t mvi, (3.67) and p () 0 n t. (3.68) he rms values of i(t), i a (t), and i n (t) are, respetively I( t) mi, (3.69) I a ( t) mi os( ), (3.70) and I n ( t) mi sin( ). (3.71) he apparent power S(t), the apparent ative power P p (t), and the apparent nonative power Q(t) are, respetively St () VtIt ()() mvi, (3.72) P () t V() t I () t mvios( ), (3.73) p and a Qt () VtI () () t mvisin( ). (3.74) n he average ative power P a (t), and the average nonative power P n (t) are, respetively P ( t) mvi os( ) P( t) P ( t), (3.75) a and p P n ( t) 0. (3.76) he instantaneous ative urrent is in phase with the voltage, and magnitude is redued to 2I os( ). he instantaneous nonative urrent is 90 out of phase with 46

59 the voltage, and the magnitude is 2I sin( ). Both the instantaneous ative urrent and the nonative urrent are independent of the number of phases m. Other definitions are listed in able 3.2. here are some interesting harateristis for a balaned sinusoidal system. 1. he instantaneous ative urrent is in phase with the voltage, and the instantaneous nonative urrent is 90 out of phase with the voltage. 2. he instantaneous power, ative power, and nonative power are all onstant, and the instantaneous ative power is equal to the instantaneous power; the instantaneous nonative power is zero. 3. All the rms values and the average powers are onstant. he average powers are equal to their orrespondent instantaneous powers. 4. he apparent ative power is equal to the average ative power, but the apparent nonative power is not equal to the average nonative power. he average powers indiate the power utilization in a system; therefore, the average ative power is equal to the average power, while the nonative power is zero. he apparent powers indiate the power arried by the urrents, therefore, if there is nonative urrent flowing in the system, there is apparent nonative power, whih is proportional to the rms value of the instantaneous nonative urrent. he apparent powers satisfy S P Q 2. (3.77) 2 2 p 5. he definitions are independent of the hoie of, as long as is zero or = k/2, k = 1, 2, Usually smaller is preferred, whih will be illustrated in the next 47

60 able 3.2. Definitions of the generalized nonative power theory in sinusoidal systems. Instantaneous Definitions rms Definitions Average Powers Apparent Powers v () t Vt () mv i () t It () mi i () t I () t mi os( ) i a a i () t I () t mi sin( ) n n pt () mvi os( ) Pt () mvi os( ) p () t mvi os( ) Pt () mvios( ) a a p n n(t) = 0 P n n(t) = 0 S(t) = mvi Pt () mvi os( ) p Qt () mvi sin( ) 48

61 hapter when the theory is implemented in a shunt nonative power system. In this ase, is hosen as / Periodi Systems with Harmonis It is assumed that both the voltage and urrent are balaned, i.e., the magnitude of eah frequeny omponent is equal, and the phase angle between two ontiguous phases is also equal. herefore, the voltage v(t) and urrent i(t) an be expressed as v () t [ v (), t v (),... t v ()] t, (3.78) 1 2 m 2k 2k where vk( t) 2V1os( 1t 1) 2Vhos( h1t h), m hm k = 0,, m-1. h2 i () t [ i (), t i (),... t i ()] t, (3.79) 1 2 m 2k 2k where ik( t) 2I1os( 1t 1) 2Ihos( h1t h), m hm k = 0,, m-1. he averaging interval is hosen to be half of the fundamental period, i.e., h2. (3.80) 2 1 he average power P(t) is t t h2 Pt ( ) v ( ) i( ) d mvios( ) m VI os( ). (3.81) h h h h he first term in (3.81) is the power ontributed by the fundamental omponent, and the rest is the ontribution from the higher-order harmonis. A harmoni omponent ontributes to average real power only when both the voltage and the urrent have the 49

62 same frequeny omponent. his is beause sinusoids of different frequenies are orthogonal to eah other. he rms value of v(t) and i(t) are, respetively, Vt () mv ( V 2 ), (3.82) 2 1 h2 h and It () mi ( I 2 ). (3.83) 2 1 h2 h In this ase, P(t), V(t), and I(t) are all onstant if is a multiple of half of the fundamental period. When the voltage is not sinusoidal, it is very important to hoose the referene voltage. For a voltage with harmonis, the referene an be hosen as the voltage itself or the fundamental omponent of the voltage. First, let v(t) itself be the referene voltage, i.e., v () t v () t. (3.84) p hen the instantaneous ative urrent and the nonative urrent are, respetively VI os( ) VI os( ) Pt () i v t h h h h h2 a() t () () 2 p t v Vp () t 2 2 V1 Vh h2, (3.85) and i () t i() t i () t. (3.86) n a 50

63 Both P(t) and V p (t) are onstant, therefore the instantaneous ative urrent i a (t) has the same shape as the voltage v(t) and is in phase with it. he rms values of the omponents of i a (t) are I ak VI os( ) VI os( ) V, where k = 1, 2,,. (3.87) h h h h h2 2 2 V1 Vh h2 k Let i a (t) be i () t [ i (), t i (),..., t i ()] t, (3.88) a a1 a2 am 2k 2k where iak ( t) 2Ia 1os( 1t 1) 2Iah os( h1t h), k = 0,, m-1. m hm h2 he rms values of i a (t) and i n (t) are, respetively, VI os( ) VI os( ) I () t m( I I ) V() t, (3.89) a h h h h 2 2 h2 a1 ah h2 2 2 V1 Vh h2 and I () t m( I I 2 nh ). (3.90) n 2 n1 h2 he apparent power S(t) is h 1 h. (3.91) h2 h2 2 St () VtIt () () m ( V V )( I I) he apparent ative power P p (t) is p a 1 h a1 h2 h2 2 P () t V() t I () t m ( V V )( I I ah ). (3.92) 51

64 he apparent nonative power Q(t) is n 1 h n1 nh. (3.93) h2 h2 2 Qt () V() ti() t m ( V V )( I I ) he average ative power P a (t) and the nonative power P n (t) are, respetively t t t Pt a a 2 () t t t ( ) Pa( t) p ( ) d v ( ) i ( ) d v ( ) v( ) d P( t), V t (3.94) and t 1 Pn() t pn( ) d 0. (3.95) t If a sinusoidal waveform is preferred for i a (t), the referene voltage an be hosen as the fundamental omponent of v(t), i.e., v ( t) [ v ( t), v ( t),..., v ( t)], (3.96) p p1 p2 pm 2k where vpk ( t) 2V1os( 1t 1), k = 0,, m-1. m he instantaneous ative urrent i a (t) is Pt () ia() t vp() t V () t 2 p V h 2( m 1) 2I1os( 11) 2 Ihos( h h) [os( 1t1),...,os( 1t 1)] h2 V1 m (3.97) i a (t) arries the ative power from both the ative urrent of the fundamental omponent and the ative urrents of the harmonis. 52

65 In (3.97), the first omponent of i a (t) represents the ative power of the fundamental omponent, and the remaining omponents represent the ative power in eah of the harmoni omponents. Now the ative urrent i a (t) is a pure sinusoid and in phase with the referene voltage. he ative urrent i a (t) arries all the ative power in urrent i(t), and all the nonative power is arried by the nonative urrent i n (t) = i(t) - i a (t), speifially, i n ( t ) [ in1( t), in2 ( t),..., inm ( t)], (3.98) where 2k ink () t 2I1sin( 11)sin( 1t 1) m Vh 2k Ihos( h h)os( 1t 1) V m h2 1 2k 2Ihos( h1t h) hm h2 k = 0, 1,, m-1. (3.99) he first term in i nk (t) is the fundamental nonative omponent in the load urrent; the seond and the third terms are the nonative omponents in the harmonis. he rms value of i a (t) is I a ( t) V m( I1 os( 1 1) I h os( h h )). (3.100) h h2 V1 he average ative power P a (t) is 1 t P ( t) v( ) i ( ) d VI os( ) V I os( ) Pt ( ). a a h h h h t h2 he average nonative power over is zero as (3.101) 53

66 1 t 1 t Pn( t) v( ) in( ) d v( )( i( ) ia( )) d 0 t t. (3.102) Periodi Systems with Sub-Harmonis Sub-harmonis are the omponents in a waveform whose frequenies are not an integral multiple of the fundamental frequeny. For simpliity of exposition, let there be only one sub-harmoni in a single-phase system with the voltage v(t) and the urrent i(t). v t) 2V os( t ) 2V s os( t ), (3.103) ( s1 s and i t) 2I os( t ) 2I s os( t ), (3.104) ( s2 s where 1 2 f1, s 1 2 fs 1, and s2 2 fs2. he instantaneous power p(t) is pt () vtit ()() VIos( ) VIos(2 t ) VI os[( ) t ] VI os[( ) t ] 1 s 1 s2 1 s 1 s 1 s2 1 VI os[( ) t ] VI os[( ) t ] s 1 1 s1 s 1 s 1 1 s1 s 1 VI s s s 1 s2 t s s VI s s s 1 s2 t s s os[( ) ] os[( ) ] (3.105) he instantaneous power p(t) ontains seven frequenies due to the modulation (multipliation) between the fundamental and the sub-harmonis, whih are 2f 1, f 1 + f s1, f 1 - f s1, f 1 + f s2, f 1 - f s2, f s1 + f s2, and f s1 - f s2. Let v p (t) be hosen as the fundamental omponent of v(t), i.e., s vp t V1 1t 1 () 2 os( ). (3.106) 54

67 he rms value V p (t) is then t 1 2 p() p( ) t V t v d V, (3.107) 1 and the ative urrent i a (t) is Pt () i t V t. (3.108) ( ) 2 os( a 2 1 ) 1 1 V1 If is hosen as an integral multiple of the periods of all the frequenies in p(t), the average value P(t) is a onstant. herefore, i a (t) is purely sinusoidal and in phase with the fundamental omponent of v(t). is the ommon multiple of the periods of all these frequenies. Now let v p (t) be hosen as v(t) itself, i.e., vp( t) 2V1os( 1t1) 2Vsos( s 1t s ). (3.109) here are three new frequenies 2f s1, f 1 +f s1, and f 1 -f s1 in v 2 () p t. If is hosen as an integral multiple of the periods of all the harmonis in p(t) and v 2 () t, both the average p value P(t) and rms value of the referene voltage Vp(t) are onstants. herefore, the waveform of i a (t) has the same shape as the waveform of v(t). In the periodi system with harmonis ase desribed in subsetion , where all frequenies were integral multiples of the fundamental frequeny, hoosing as an integral multiple of a half yle of the fundamental frequeny will guarantee that is the ommon multiple of the periods of all the frequenies in p(t) and v 2 p() t, i.e., ia(t) is proportional to v p (t). 55

68 In the sub-harmonis ase disussed in this subsetion, if is the ommon multiple of the periods of all the harmonis in p(t) and v 2 p() t (with sub-harmonis present, is larger than the fundamental period), then the waveform of i a (t) has the same shape as the waveform of v p (t). If is not hosen this way, there are still sub-harmoni omponents in i a (t), i.e., the nonative omponent is not ompletely eliminated Non-Periodi Systems For a non-periodi system, hoosing = t, the average power P(t), rms value of the referene voltage v p (t), and the ative urrent i a (t) are defined by t 1 1 P( t) v ( ) i( ) d v ( ) i( ) d, (3.110) t t 0 t t 0 t t 1 1 V p ( t) v p ( ) v p ( ) d v p ( ) v p ( ) d, (3.111) t and P( t) ia ( t) v ( t) 2 p. (3.112) V ( t) p In a power system, the voltage and the urrent are finite power waveforms, i.e., t 0 1 Pt ( ) lim v ( ) i( ) d P, (3.113) t t t 0 1 Vp( t) lim vp( ) vp( ) d V p, (3.114) t t and Pt () P i () t v () t v (), t as t. (3.115) a 2 p 2 p Vp () t Vp 56

69 he generalized nonative power theory is valid for voltage and urrent of any waveform, and the nonative urrent an only be ompletely eliminated when = t and t (i a (t) has the shape as and is in phase with v p (t) so that unity power fator is ahieved). However, this is not pratial in a power system, and is hosen to have a finite value. he voltage in an a power system usually is a sine wave with fundamental frequeny f, and for a non-periodi voltage, it is usually a sum of the fundamental omponent and the non-periodi omponent. is usually hosen as a few multiples of the fundamental period, and this finite will mitigate the distortion. his will be disussed further in Chapter Unbalane Let the voltage and urrent in an unbalaned sinusoidal system be, respetively, v ( t) 2V1os( t1) 2V2os( t2) 2V3os( t ) 3, (3.116) and i ( t) 2I1os( t1) 2I2os( t2) 2I3os( t ) 3. (3.117) If the referene voltage is hosen as the voltage itself, then the definitions are similar to the definitions in subsetion In this ase, the ative urrent is still unbalaned beause of the unbalaned voltage. If a balaned ative urrent is preferred, a balaned referene voltage is needed. Usually, the referened voltage is hosen as the positive sequene omponent of the system voltage. 57

70 V 3 V 1 V 2 V 3 + V 2 + V 1 + V 3 - V 1 - V 2 - V 1 0 V 2 0 V 3 0 positive sequene negative sequene zero sequene Figure 3.1. Sequene analysis of a three-phase waveform. Using omplex rotating vetors V () t, V () t, and 1 2 V 3() t to represent v () t Re{ V ()} t 1, v () t Re{ V ()} t, and v () t Re{ V 3()} t : 3 V () t 2Ve j( t 1), 1 1 V () t 2V e j( t2 ), (3.118) 2 2 V () t 2Ve j( t 3 ). 3 3 he unbalaned voltages of the three phases are deomposed into three sequenes: a positive sequene, a negative sequene, and a zero sequene (as shown in Figure 3.1). Speifially, let j( t ) V e j( t ) V t a a V 2 t V e 3, (3.119) ( ) () V 3 ( t ) j t V t 2Ve V () t 2 1 a a V 2 1( t) 1 2 () 1 () 2 2 j 3 where a e. 2V, 2V, and 0 2V are the amplitudes of the positive sequene, negative sequene, and zero sequene omponents, respetively, and the angles,, and 0 58

71 are the phase angles of phase 1 of these three sequenes. herefore, the positive sequene omponent v () t of v(t) is v 2 os( ) 1 () t V t v () t v2 () t 2V os( t 2 /3). (3.120) v3 () t 2V os( t 2 / 3) he negative sequene omponent v () t is v 2 os( ) 1 () t V t v () t v2 () t 2V os( t 2 /3). (3.121) v3 () t 2V os( t 2 / 3) he zero sequene omponent v 0 () t is v. (3.122) v () t 2V os( t ) v1 () t 2V os( t ) () t v2 () t 0 2V os( t ) If the system is balaned, the voltage has only the positive sequene omponent, and both the negative sequene omponent and the zero sequene omponent are zero. In an unbalaned system, usually the positive sequene omponent v () t is the dominant omponent, whih is sinusoidal and balaned by design. herefore, a sinusoidal and balaned ative urrent an be ahieved by hoosing v () t as the referene voltage. 3.3 Generalized heory able 3.3 shows the different ombinations of v p and for different loads, and the orresponding ompensation results are shown in the last olumn. he hoie of the 59

72 able 3.3. Summary of the parameters v p and in the nonative power theory. Load Current i(t) v p Ative Current i a (t) hree-phase fundamental nonative urrent v 0 /2 Unity pf and pure fundamental sine wave Single-phase or multi-phase fundamental nonative urrent and harmoni urrent v /2 Unity pf and same shape of v v f /2 v f Pure sine wave and in phase with v f+. f+. v 0 Instantaneous urrent Sub-harmoni urrent v f n Non-periodi urrent v f n Non-periodi urrent v Unbalaned voltage v f+ /2 v f v f v f+ Pure fundamental sine wave or smoothed sine wave Smoothed and near sine wave In phase with v with unity pf Balaned and sinusoidal urrent Notes: Both v(t) and i(t) are distorted exept in Case 1. v f is the fundamental omponent of v. v f+ is the positive sequene of the fundamental omponent of v. 60

73 referene voltage influenes the ative urrent. A pure fundamental sinusoidal ative urrent an be ahieved by hoosing the positive sequene omponent of the fundamental voltage. For a system with fundamental omponent and/or harmonis, the nonative urrent omponent an be ompletely eliminated by hoosing the averaging interval to be /2. he above disussion shows that this theory is valid in different situations, and for eah situation, it is speified by the parameters v p and. his theory overs many of the other nonative power theories presented in Chapter 2. For p-q theory by Akagi et al. [11]-[12], hoosing a three-phase system with v p = v (fundamental only and balaned using the method of p-q theory), and =, the theory proposed here will redue to p-q theory. For omplete ompensation of the nonative power, p-q theory requires the d omponent of the instantaneous power, whih requires at least half of the fundamental yle to determine it. herefore, it has the same result as this generalized theory when is hosen to be /2 or. For the Hilbert spae tehnique based theory in the frequeny domain [25], the definitions of ative urrent i a (t) and inative urrent i x (t) are exatly equal to the generalized theory of this dissertation if 0. However, this is not reommended as the ative urrent is neither sinusoidal nor in phase with the voltage. 3.4 Summary In this hapter, a generalized nonative power theory was presented. he instantaneous ative urrent i a (t), the instantaneous nonative urrent i n (t), the instantaneous ative power p a (t), and the instantaneous nonative power p n (t) were 61

74 defined in a system with a voltage v(t) and a urrent i(t). he average ative power P a (t) and average nonative power P n (t) are defined by averaging the instantaneous powers over time interval [t-, t]. he apparent power S(t), apparent ative power P p (t), and the apparent nonative power Q(t) are also defined based on the rms values of the voltage and urrents. he generalized nonative theory presented in this dissertation did not have any limitations suh as the number of the phases; the voltage and the urrent were sinusoidal or non-sinusoidal, periodi or non-periodi. hat is, the theory in this dissertation is valid for 1. Single-phase or multi-phase systems 2. Sinusoidal or non-sinusoidal systems 3. Periodi or non-periodi systems 4. Balaned or unbalaned systems he referene voltage v p (t) and the averaging interval were two important fators. By hanging v p (t) and, this theory had the flexibility to define nonative urrent and nonative power in different systems. Several ases were disussed in details ombining the system harateristis and the hoies of v p (t) and. It was a generalized theory that other nonative power theories disussed in Chapter 2 ould be derived from this theory by hanging the referene voltage and the averaging interval. he flexibility was illustrated by applying the theory to different ases suh as a sinusoidal system, a periodi system with harmonis, a periodi system with subharmonis, and a system with non-periodi urrents. 62

75 CHAPER 4 Implementation in Shunt Compensation System A SACOM is a shunt nonative power ompensator, whih utilizes a d-a inverter to ompensate the nonative urrent omponent in a load urrent without any energy soures. It provides power fator orretion, harmonis elimination, peak urrent mitigation, and urrent regulation. he generalized nonative power theory proposed in Chapter 3 is implemented in a SACOM to determine the nonative urrent that the SACOM needs to ompensate. A ompensation system onfiguration for three-phase, four-wire system ompensation is also proposed. Control shemes of regulating the DC link voltage and ontrolling the ompensator urrent are disussed. he ompensation system is simulated, and results of several ases are analyzed. he averaging interval and the referene voltage v p (t) are disussed as well as some pratial issues, suh as the DC link apaitane, the DC link voltage, and the oupling indutane required for a SACOM installation. 4.1 Configuration of Shunt Nonative Power Compensation In a shunt nonative power ompensation system, the ompensator is onneted to the utility at the point of ommon oupling (PCC) and in parallel with the load. It injets a ertain amount of urrent, whih is usually the nonative omponent in the load urrent 63

76 i s v s i l Utility Load i L i v d v d * v s i l v Inverter swithing signals Controller i * Nonative urrent alulation v d Figure 4.1. System onfiguration of shunt nonative power ompensation. i n (t), into the system so that the utility only needs to provide the ative omponent of the load urrent i a (t). In Figure 4.1, i l is the load urrent, i s is the soure urrent, and i is the ompensator urrent. herefore, if the nonative urrent is ompletely produed by the ompensator, then i s = i a (t), and i = i n (t). he theory an be implemented in a three-phase three-wire, three-phase four-wire, or a single-phase system. he system onfiguration shown in Figure 4.1 is a three-phase four-wire system. he ompensator onsists of three parts, i.e., the nonative power/urrent alulator, the ontroller, and the inverter. In the nonative power/urrent alulator, the system voltage v s and load urrent i l are measured and used to alulate the nonative omponent of i l as explained in Chapter 3. his is the urrent the shunt ompensator needs to injet to the power system ( * i in Figure 4.1). In the theory disussed in Chapter 3, the ompensation system was assumed to be lossless; however in an atual power system, 64

77 losses exist in the swithes, apaitors, and indutors. If the ompensator does not have any energy soure, the energy stored in the DC link apaitor is drawn to ompensate these losses ausing v d to drop. he ative power required to replae the losses is drawn * from the soure by regulating the DC link voltage v d to the referene V d. 4.2 Control Sheme and Pratial Issues here are two omponents in the ompensator urrent i. he first is the nonative omponent i n to ompensate the nonative omponent of the load urrent, and the seond is the ative omponent i a to meet the ompensator s losses by regulating the DC link voltage v d, then i i i a (4.1) n DC Link Voltage Control A PI ontroller is used to regulate the DC link voltage v d, as shown in Figure 4.2 [48]. he ative urrent required to meet the losses is in phase with v s (In Figure 4.2, has a minus sign beause its diretion is opposite to the onvention of i in Figure 4.1). he amplitude of the ative urrent is ontrolled by the differene between the referene * * voltage V and the atual DC link voltage vd. is alulated by the amplitude d modulation of vs as follows t * * * ia vs KP1( Vd vd ) KI1 ( Vd vd) dt. (4.2) 0 i a * i a 65

78 V d * + - PI -i a * v d v s Figure 4.2. DC link voltage ontrol diagram Nonative Current Control he equivalent iruit of the shunt ompensation system is shown in Figure 4.3a, where v s is the system voltage, v is the output voltage of the inverter, L is the oupling indutane, and i is the ompensator urrent. he ative urrent of the ompensator i a is a small fration of the whole ompensator urrent i, so the ative omponent is negleted at present for simpliity. he relationship between the ompensator s urrent i and the system voltage is di L v v s. (4.3) dt he referene of the ompensator nonative urrent ompensator output voltage * v satisfy * i and the referene of the * di * L v v s (4.4) dt * where is the nonative urrent alulated based on the nonative power/urrent theory i presented in Chapter 3. he referene ompensator output voltage * v is di v v. (4.5) * * L dt s 66

79 Utility i s v s i i l Load d dt L v s v * v L i * PI v Compensator i (a) Equivalent iruit (b) Feedforward ontroller Figure 4.3. Nonative urrent ontrol diagram. Subtrat (4.3) from (4.4) to obtain L * d( i i) dt * v v. (4.6) Using a PI ontroller and substituting (4.5) into (4.6), the output voltage of the ompensator v is alulated by * t di * * s L KP2 KI2 dt 0 v v ( i i ) ( i i ) dt (4.7) A feedforward ontroller based on (4.7) is shown in Figure 4.3b. Figure 4.4 is the omplete ontrol diagram of the shunt ompensation system. he inner loop is the ompensator urrent ontrol whih ontrols the output voltage of the ompensator aording the required nonative urrent * i. he outer loop ontrols the ative urrent drawn by the ompensator by regulating the DC link voltage v * * ombines the ative urrent together with the nonative urrent. i a i n d, and 67

80 v s d dt L v s i l Nonative Current Calulation i n * i * PI v v d PI -i a * i V d * v s Figure 4.4. Control diagram of the shunt ompensation system Simulations he ompensation system presented in subsetion 4.1 is simulated in Matlab Simulink using the Power System Blokset. Figure 4.5 shows the main diagram of the simulation model. he power system and the ontroller are separated as shown in Figure 4.5a. he simulation in the power blok is faster than the simulation in the ontrol blok. here are an inverter and a pulse width modulation (PWM) generator in the power blok, whih require a small simulation stepsize for aurate simulation, while the ontrol blok requires a larger simulation stepsize to simulate how the ontroller would exeute in real time. Speifially, the simulation stepsize in the power blok is 1 s, while the simulation stepsize in the ontrol blok is 50 s, exept for the ase of high-order harmoni ompensation, whih requires a smaller stepsize. he power blok diagram is shown in Figure 4.5b. he utility, the load, and the inverter with a DC link apaitor are simulated. he system voltage, load urrent, ompensator urrent, and the DC link 68

81 (a) Main diagram, the power blok (fast) and the ontrol blok (slow) (b) Power blok () Control blok Figure 4.5. Simulation model diagrams of the nonative power ompensation system. 69

82 voltage are measured and output to the ontrol blok; while the referene inverter output voltage is provided by the ontrol blok. he diagram in Figure 4.5 is the ontrol blok, whih takes the measurements from the power blok, alulates the nonative urrent omponent of the load urrent, ontrols the DC link voltage by drawing a ertain amount of ative urrent from the utility, and generates a referene voltage for the inverter output. his referene voltage is ompared to a triangle waveform with the frequeny of the inverter swithing frequeny to generate PWM signals for the inverter hree-phase Periodi System he simulation starts with a ommon ase, i.e., a three-phase RL load. Compensation of harmonis is simulated in three different onditions, harmoni urrent and fundamental sinusoidal voltage, and harmonis in both urrent and voltage where the referene voltage is hosen as the distorted system voltage itself or the fundamental omponent. he requirements of the DC link apaitane C, the DC link voltage v d, and the oupling indutane L are determined in the RL load ase, and these requirements in other systems are analyzed based on the RL load system hree-phase Balaned RL or RC Load In this ase, the load onsists of resistors, and/or indutors, and/or apaitors, and the values of R, L, and C in eah phase are equal. Moreover, the transmission lines and most loads in power systems are indutive, therefore, a three-phase balaned load with resistors and indutors are simulated. he load urrent is a fundamental sine wave, and the phase angle between the urrent and the voltage is lagging. As shown in Figure 4.6, the line-to-neutral voltage is 120V (rms) (Figure 4.6a), the load urrent is 20A (rms) 70

83 (a) System voltage v s (V) (b) Load urrent i l (A) () Phase a voltage v sa and load urrent i la (d) Phase a voltage v sa and soure urrent i sa (e) Soure urrent i s (f) Compensator urrent i (g) Phase a alulated soure urrent and (h) DC link voltage v d real soure urrent Figure 4.6. hree-phase RL load simulation. 71

84 able 4.1. Parameters of the three-phase RL load ompensation. rms of the system voltage v s (line-to-neutral) (V) 120 rms of the fundamental load urrent i l (A) 20 Fundamental frequeny (Hz) 60 DC link apaitane C (F) 100 DC link voltage v d (V) 350 Coupling indutane L (mh) 1 Averaging interval /2 Swithing frequeny (khz) 20 (Figure 4.6b), and the phase angle of the urrent is /6 radians lagging (Figure 4.6). he parameters are shown in able he nonative power and urrent are alulated aording to the generalized nonative power theory presented in Chapter 3. In eah phase, the resistor R = 5.196, and the indutor L = 7.96mH. he voltage and the load urrent are, respetively, v( t) [120 2 os( t),120 2 os( t2 / 3),120 2 os( t 2 / 3)], (4.8) and i l ( t) [20 2 os( t / 6),20 2 os( t5 / 6), 20 2 os( t / 2)].(4.9) Aording to the generalized nonative power theory presented in Chapter 3, this load urrent i l (t) has the ative urrent omponent i a (t) and the nonative urrent omponent i n (t). he ative urrent is provided by the soure, and is denoted as i s (t) and is alulated aording to (3.5) (with v p (t) = v(t), P(t) = 3VI = W, and V p (t) = 3V = 207.8V), i ( t) [ 2I os( t), 2I os( t2 / 3), 2I os( t 2 / 3)], (4.10) s s s s 72

85 where I s = he nonative urrent i n (t), i.e., i (t) is alulated aording to (3.6), i ( t) [ 2I sin( t), 2I sin( t2 / 3), 2I sin( t 2 / 3)], where I = 10. (4.11) he soure urrent and ompensator urrent before and after the ompensator start to run are shown in Figures 4.6d - 4.6f, where the time interval A is before the ompensator starts, and the time interval B is after it starts. he ompensator starts to run at t = 0.017s, Figure 4.6d shows the phase a soure urrent i sa (t) (blue waveform) is in phase with the phase a voltage and its amplitude is smaller than the amplitude before ompensation. Figure 4.6e shows all the three soure urrents, whih are sinusoidal and balaned. Figure 4.6f shows the ompensator urrent, whose amplitude is equal to the value alulated from the generalized nonative power theory in (4.11). After the ompensator starts to run, the i (t) is the nonative omponent of the load urrent, whih is out of phase with the voltage. As disussed in subsetion 3.2.2, the average power P(t) and the rms value of v(t) are onstants, and they do not hange whether the averaging interval is hosen as 0 or integer multiples of /2. herefore, the instantaneous nonative power ompensation without average values (average power P(t) and rms value of v(t)) an be ahieved in this ase. he following shows how to determine the requirements of the DC link voltage V d and the oupling indutane L. 73

86 DC link voltage V d Based on the equivalent iruit of the nonative power ompensation system in Figure 4.3a, the ompensator urrent i (t), the ompensator output voltage v (t), and the system voltage v s (t) satisfy L di () t dt v () t v () t (4.12) s he fundamental omponent of v (t) is in phase with v s (t), and the amplitude is larger than v s (t) when the ompensator is providing nonative power to the system. If the high frequeny omponents in v (t) are negleted, v (t) an be written as v ( t) [ 2V os( t), 2V os( t2 /3), 2V os( t 2 /3)]. (4.13) Solve i (t) from (4.12), i t os tdt ia (0) 0 sin (0) t t ia V V s V Vs ( t) 2 os( t2 / 3) dtib (0) 2 sin( t2 / 3) ib (0) L L 0 sin( t 2 / 3) i (0) t os( t2 / 3) dti (0) 0 (4.14) he initial values i a (0), i b (0), and i (0) are zero beause the urrents flowing through the indutors annot hange instantaneously. herefore, the nonative urrent is V V i ( t) 2 s [sin( t),sin( t 2 / 3),sin( t 2 / 3)]. (4.15) L he amplitude 2I of the nonative urrent is alulated in the generalized nonative power theory. he rms value of the ompensator output voltage V is 74

87 V LI Vs. (4.16) his is the rms value of the ompensator output voltage for fundamental nonative urrent ompensation. Aording to [49], the DC link voltage V d is V 2 2V 2 2( L I V ). (4.17) d s A smaller oupling indutor is preferred so that the ompensator output voltage is lower; however, a ertain amount of indutane is needed to filter the ripple in the nonative ompensator urrent i (t). For ompensation of harmoni load urrent, V d should inlude the harmonis voltage drop on the oupling indutor, whih is proportional to the harmonis frequenies. Coupling indutane L In (4.7), the differential of the referene urrent (the seond term on the right) an be negleted, and the integral gain K I2 is muh smaller than the proportional gain K P2. herefore, (4.7) an be approximately written as K ( * v v P2 i i ). (4.18) s If rms values are used, it an be written as * V ( ) Vs KP2 I I Vs KP2 I. (4.19) Combine (4.16) and (4.19), LI K I. (4.20) P2 herefore, the proportional gain K P2 an be determined by I I K L X I P2 I. (4.21) 75

88 K P2 is proportional to the oupling impedane X, and inversely proportional to the ompensator urrent error I. A smaller error requires a larger gain; however in a real system, there is noise from the measurement, EMI from the swithing and other interferene, whih may be amplified to an intolerable level if a large gain is used. his is another reason a smaller L is preferred. he indutane L between the power system and the ompensator ould be the indutane of a step-up transformer or a oupling reator. It also ats as a filter for the ompensation urrent i, whih an have high ripple ontent if a PWM inverter is used. If L is too small, it annot filter the ripple in the ompensation urrent i ; on the other hand, if L is too large, as mentioned above, a higher DC link voltage is required, espeially for higher-order harmonis ompensation. If the ompensator urrent is I, then the voltage drop on the oupling indutor L is V L. (4.22) LI his voltage drop V L is usually hosen as 3% - 9% of the system voltage V s for fundamental omponent. his number is hosen for two reasons. A smaller L is preferred from the point of dereasing the DC link voltage V d and the proportional gain K P2. However, distortion will be introdued to the system voltage v s if L is too small. he oupling indutane an be determined from (4.22), L Vs (3% 9%). (4.23) I he oupling indutane L is maintained small to redue the DC link voltage, and aordingly the gains of the PI ontroller are kept small also, as long as the desired ompensation results are obtained. 76

89 hree-phase Harmoni Load and Sinusoidal Voltage Harmonis are a ommon ause of distortion in power systems, existing in both voltage and urrent. Harmonis ompensation is independent of the averaging interval if is an integer multiple of /2. Aording to the definition of ative urrent given in (3.5): ia Pt () () t v p() t. (4.24) V () t 2 p For harmonis ompensation, P(t) and V p (t) are onstant, i.e., i a (t) is in phase with v p (t) and has the same waveform shape as v p (t) if is an integer multiple of /2. If the system voltage is not distorted, and the load urrent has harmonis, then the system voltage itself an be used as the referene voltage to ahieve sinusoidal, balaned, and unity power fator soure urrent after ompensation. his ase is simulated with the parameters in able 4.2 (exept as noted below), and the simulation results are illustrated in Figure 4.7. he load urrent in Figure 4.7b ontains harmoni omponents from the 5 th order to the 23 rd order. Beause of the high order harmonis, the swithing frequeny is inreased to 50 khz (20 khz in the previous ase). he DC link voltage is inreased to 420V, and other parameters are shown in able 4.2. he fundamental system voltage and the distorted load urrent are shown in Figures 4.7a and 4.7b, respetively. Figure 4.7 shows the phase a load urrent is distorted and lags the voltage, while in Figure 4.7d, the soure urrent is a fundamental sine wave and in phase with the voltage when the ompensator is operating (after t = 0.017s in Figure 4.7). he soure urrent in Figure 4.7e now ontains the ative omponent of the load urrent, and the ompensator urrent in Figure 77

90 (a) System voltage v s (V) (b) Load urrent i l (A) () Phase a voltage v sa and load urrent i la (d) Phase a voltage v sa and soure urrent i sa (e) Soure urrent i s (f) Compensator urrent i (g) Phase a alulated soure urrent (h) DC link voltage v d and real soure urrent Figure 4.7. hree-phase harmoni load with fundamental v s. 78

91 able 4.2. Parameters of the three-phase harmonis ompensation with fundamental sinusoidal v s (t). rms of the system voltage v s (t) (line-to-neutral) (V) 120 rms of the fundamental load urrent i l (t) (A) 20 Fundamental frequeny (Hz) 60 DC link apaitane C (F) 1000 DC link voltage v d (V) 420 Coupling indutane L (mh) 1.5 Averaging interval /2 Swithing frequeny (khz) f has the nonative omponent. In Figure 4.7g, the phase a alulated ative load urrent (magenta) and the atual soure urrent are ompared. he atual soure urrent is slightly larger than the alulated ative load urrent, beause of the ative urrent drawn by the ompensator to meet the losses. he DC link voltage v d (t) is shown in Figure 4.7g. he variation of v d is inversely proportional to the DC link apaitane. he output voltage of the inverter satisfies di () t v () t L v s () t. (4.25) dt Phase a of the system voltage v s (t) is v () t 2V os( t). (4.26) sa s Phase a of the ompensator urrent is i ( t) 2I os( t ) 2I os( kt ). (4.27) a k k 2 k 2 79

92 he plus sign indiates that the ompensator provides apaitive nonative power, and the minus sign indiates that the ompensator provides indutive nonative power. Phase a of the ompensator voltage is v () t 2( V L I )os( t) 2kL I os( kt ). (4.28) a s k k k2 For a ompensator urrent with harmonis, the magnitudes of the harmonis in v (t) are k times of the harmonis in the urrent. For the worst ase, in whih the ompensator is providing indutive fundamental nonative urrent and all the harmonis reah the peak value at the same time, the DC link voltage v d should be twie the peak line-to-line voltage v (t) to be apable of ompletely ompensating the nonative omponents in the load urrent. Compared to the fundamental nonative power ompensation, a larger inverter output voltage, i.e., a larger DC link voltage v d is required, and the magnitude is proportional to the frequeny of the harmonis hree-phase Harmoni Load and Distorted System Voltage If the system voltage is also distorted, hoosing a different referene voltage will result in a different soure urrent. he ompensation with the system voltage itself is disussed in this subsetion, and the ompensation by hoosing the fundamental omponent of the system voltage as the referene voltage will be disussed in the next subsetion. In both of these ases, the load urrent and the system voltage ontain 5 th and 7 th harmonis. A ompensator with a swithing frequeny of 20 khz will be used in these two simulations. he parameters of the simulation are listed in able 4.3. he DC link voltage is higher than for the RL load ase, sine there are harmonis in the ompensator 80

93 able 4.3. Parameters of the three-phase harmonis ompensation with distorted v s. rms of the system voltage v s (line-to-neutral) (V) 120 rms of the fundamental load urrent i l (A) 20 Fundamental frequeny (Hz) 60 DC link apaitane C (F) 500 DC link voltage v d (V) 370 Coupling indutane L (mh) 0.5 Averaging interval /2 Swithing frequeny (khz) 20 urrent. he DC link apaitane is larger than for the RL load ase beause the instantaneous power is not zero in this ase whih requires the DC link apaitor to provide and to onsume power from time to time. he system soure voltage v s (t), the load urrent i l (t), the soure urrent i s (t), and the ompensator urrent i (t) are shown in Figures 4.8a 4.8d, respetively. he soure urrent is still distorted after ompensation. More speifially, the system voltage and the soure urrent of eah phase are ompared in Figures 4.8e 4.8g, whih show that the waveform of the soure urrent has the same shape as the system voltage, therefore, a unity power fator is ahieved in this ase hree-phase Harmoni Load and Sinusoidal Referene Voltage If a fundamental sinusoidal soure urrent is preferred, another referene voltage should be used other than the system voltage when it is distorted, whih is usually the fundamental omponent of the system voltage v s (t). he soure urrent will be sinusoidal and in phase with the referene voltage v p (t). he simulation parameters are listed in 81

94 (a) System voltage v s (V) (b) Load urrent i l (A) () Soure urrent i s (A) (d) Compensator urrent i (A) (e) Phase a voltage v sa and soure urrent i sa (f) Phase b voltage v sb and soure urrent i sb (g) Phase voltage v s and (h) DC link voltage v d soure urrent i s Figure 4.8. hree-phase harmoni load ompensation with distorted v s. 82

95 able 4.4. Parameters of the three-phase harmonis ompensation with sinusoidal v p. rms of the system voltage v s (line-to-neutral) (V) 120 rms of the fundamental load urrent i l (A) 20 Fundamental frequeny (Hz) 60 DC link apaitane C (F) 500 DC link voltage v d (V) 400 Coupling indutane L (mh) 1 Averaging interval /2 Swithing frequeny (khz) 20 able 4.4 and the results are illustrated in Figure 4.9. he DC link voltage is higher than the previous ase, beause all the nonative omponents in the load urrent are provided by the ompensator this time. he system voltage and the load urrent are shown in Figures 4.9a and 4.9b. he referene voltage, whih is the fundamental omponent of the system voltage, is shown in Figure 4.9. he phase a soure urrent and referene voltage are ompared in Figure 4.9d, whih shows that the soure urrent is fundamental sinusoidal and in phase with the referene voltage hree-phase Diode Retifier Load A three-phase diode retifier is a ommon nonlinear load. he urrent of a diode retifier is shown in Figure 4.10a (phase a). It is a periodi waveform with the same period as the fundamental period of the system voltage, thus the urrent an be deomposed into a fundamental omponent and harmonis. However, different from the harmonis load disussed in the previous subsetion, the urrent drawn by a diode-based retifier is not ontinuous, and the variation of the urrent with time di/dt is very large 83

96 (a) System voltage v s (V) (b) Load urrent i l (A) () Referene voltage v p (V) (d) Phase a referene voltage v pa and soure urrent i sa (e) Soure urrent i s (t) (A) (f) Compensator urrent i (A) (g) Phase a alulated soure urrent and real soure urrent (h) DC link voltage v d (V) Figure 4.9. hree-phase harmoni load ompensation with sinusoidal v p. 84

97 (a) Load urrent of phase a, i la (b) Soure urrent of phase a, i sa, V d = 2.6V s () Compensator urrent of phase a, i a, (d) DC link voltage v d, V d = 2.6V s Vd = 2.6V s (e) Soure urrent of phase a, i sa,, V d = 5.2V s Figure Simulation of retifier load ompensation. 85

98 (di/dt = A/s in this example). herefore, there are speial issues in the ompensation of a retifier load. At t = 0.04 s, there is a sudden load hange whih auses the load urrent to inrease (the seond dash line in Figure 4.10a). In Figure 4.10b, during the time interval A, i s (t) is supplying the full load urrent while the ompensator is off. he ompensator is on during the time interval B and i s (t) is lose to a sinusoid, and in the time interval C, i s (t) is shown after the sudden load hange. Figure 4.10 shows the nonative urrent whih is provided by the ompensator, whih is highly distorted. he load hange has an impat on the DC link voltage, whih drops about 50 volts when the load hanges. o regulate v d, some ative urrent is drawn. he sudden load hange requires a larger apaitane than a steady load. he soure urrent i sa (t) in Figure 4.10b has some spikes, whih is aused by the large variation of the urrent with time. Aording to (4.3), the ompensator output voltage v is given by di v v. s L dt A large di /dt requires a large v ; however, if the DC link voltage does not meet this requirement, the ompensator is unable to provide the nonative urrent as required by the nonative power theory, and spikes our in the soure urrent aordingly. Figure 4.10b and Figure 4.10e show the soure urrent when the DC link voltage V d is 2.6V s and 5.2V s, respetively, where V s is the amplitude of the system line-to-neutral voltage. he spikes in Figure 4.10e are smaller than those in Figure 4.10b, whih shows that a soure urrent with less spikes an be ahieved by inreasing the DC link voltage. 86

99 However, a higher DC link voltage requires a higher voltage rating of the ompensator and results in higher ompensator losses. In pratie, the DC link voltage an be hosen between 2V s to 3V s hree-phase Load with Sub-Harmonis he frequenies of sub-harmonis are not a multiple integer of the fundamental frequeny. In this simulation, the voltage is fundamental pure sine wave, and the load urrent has a sub-harmoni omponent with frequeny f s = 2f/3, where f is the fundamental frequeny. he nonative urrent is ompletely ompensated if the averaging interval is a multiple integer of the ommon multiple of the fundamental omponent and the sub-harmoni omponent, whih is 3 in this ase. he simulation results are shown in Figure he system voltage v s (t) and load urrent i l (t) are shown in Figures 4.11a and 4.11b respetively. Figures 4.11 and 4.11d are the soure urrent and the ompensator urrent when = /2. Figures 4.11e and 4.11f are the soure urrent and the ompensator urrent when =. he fundamental nonative omponent is ompletely ompensated, but the sub-harmonis omponent is not ompletely eliminated. As illustrated, there are still sub-harmonis in the soure urrent. he sub-harmonis are ompletely ompensated if = 3, as shown in Figures 4.11g and 4.11h hree-phase Unbalaned RL Load In a three-phase four-wire system, i.e., a neutral line is present, and the load or the soure voltage is unbalaned, there will be a neutral urrent flowing in the neutral line. A 87

100 (a) Soure voltage v s (V) (b) Load urrent i l (A) () Soure urrent i s, = /2 (d) Compensator urrent i, = /2 (e) Soure urrent i s, = (f) Compensator urrent i, = (g) Soure urrent i s, = 3 (h) Compensator urrent i, = 3 Figure Simulation of sub-harmonis load ompensation. 88

101 three-phase four-wire system is balaned if the phase angle between eah two phases is 120, and all the phases have the same magnitude; otherwise it is unbalaned. An unbalaned load urrent with both unequal amplitudes and unbalaned phaseangles is first simulated. In this ase, the voltage is balaned. he rms values of the threephase urrents are 16A (20% lower), 20A, and 18A (10% lower), and the phase angles are 30, 27, and 33 lagging to the voltage, respetively (Figure 4.12b). he voltage is fundamental pure sine wave and balaned as shown in Figure 4.12a. he phase a soure urrent before ompensation (time interval A) and after ompensation (time interval B) is shown in Figure 4.12d (red waveform). After ompensation, the soure urrent is in phase with the phase a voltage (blue waveform). Figure 4.12e shows the three-phase soure urrent before and after ompensation. he three-phase urrents are now balaned, i.e., equal amplitude and equal phase angle between eah other. Figure 4.12f shows the neutral urrent of the utility before and after ompensation. After ompensation, the soure neutral urrent is near zero (theoretially it is zero, but it is near zero beause of the ontrol traking error), and the load neutral urrent now is provided by the ompensator Single-Phase Load Single-Phase RL Load Based on the equivalent iruit of the nonative power ompensation system in Figure 4.3a, the ompensator urrent i (t), the ompensator output voltage v (t), and the system voltage v s (t) satisfy di () t L v() t v s () t. (4.29) dt 89

102 (a) System voltage v s (V) (b) Load urrent i l (t) () Phase a voltage v sa and load urrent i la (d) Phase a voltage v sa and soure urrent i sa (e) Soure urrent i s (A) (f) Soure neutral urrent i sn (A) Figure hree-phase unbalaned RL load. 90

103 In the fundamental nonative power ompensation ase, let v () t 2V os( t). (4.30) s s he fundamental omponent of the v (t) is in phase with v s (t), and the amplitude is larger than v s (t) when the ompensator is providing indutive power to the system. If the high frequeny omponents in v (t) are negleted, v (t) an be written as v () t 2V os( t). (4.31) Apply the Laplae transformation to (4.29), V V I () 2 s s L s (4.32) herefore, the nonative urrent in the time domain is V Vs i () t 2 sin( t). (4.33) L he amplitude of the nonative urrent is alulated aording to the generalized nonative power theory presented in Chapter 3, and denoted as 2I. he rms value of the ompensator output voltage V is then V LI Vs. (4.34) his is the rms value of the ompensator output voltage for fundamental nonative urrent ompensation, whih is also valid in three-phase systems. A smaller oupling indutor is preferred so that the ompensator output voltage is lower; however, a ertain amount of indutane is needed to filter the ripples in the nonative ompensator urrent i (t). he DC link voltage V d is equal to the peak value of V. he energy stored in the DC link apaitor E d (t) is 91

104 1 2 Ed ( t) Cvd ( t). (4.35) 2 he maximum energy variation in C is Ed C( vd ( tmax ) vd ( tmin )), (4.36) 2 where v d (t max ) is the maximum value of the DC link voltage and v d (t min ) is the minimum value. he energy variation E d an also be written as tmax E v ( t) i ( t) dt, (4.37) d d tmin d where v d (t) and i d (t) are the voltage and urrent of the DC link apaitor, respetively. Usually the DC link urrent i d (t) is not measured, therefore, the energy variation on the DC link apaitor E d is approximately equal to the energy variation of the nonative ompensator, i.e., t max E d vs ( t) i ( t) dt Vs I. (4.38) 4 tmin Combine (4.36) and (4.38), and it is assumed that the DC link voltage varies around V d, i.e., d C(( V 2 Vd ) ( Vd Vd ) ) Vs I. (4.39) 4 herefore, the DC link apaitane C is C V I V V s. (4.40) 2 d d he DC link voltage variation V d is inversely proportional to the apaitane, therefore for a nonative power ompensator, the apaitane requirement an be 92

105 (a) Voltage v s (t) and load urrent i l (b) Voltage v s (t) and soure urrent i s () Voltage v s (t) and ompensator urrent i (d) DC link voltage v d Figure Simulation of single-phase load in a single-phase system. determined if the DC link voltage variation is speified. he apaitane requirement in a three-phase fundamental nonative power system is muh smaller than in a single-phase system with the same amount of nonative urrent, beause the instantaneous nonative power is zero in a three-phase system and not zero in a single-phase system, whih requires more DC link energy storage, i.e. higher DC link apaitane for a given system. A single-phase RL load is simulated and the simulations results are shown in Figure Figure 4.13a shows the system voltage v s (t) (red waveform) and the load urrent i l (t) (blue waveform), whih is lagging. Figure 4.13b shows the soure urrent before (time interval A) and after ompensation (time interval B). After ompensation, the soure urrent is in phase with the voltage. Figure 4.13 shows the ompensator urrent and the system voltage. he ompensator urrent is 90 out of phase with the voltage. Figure 4.13d shows the DC link voltage v d (t), whih has a ripple, but the average value is 93

106 kept onstant by drawing a ertain amount of ative urrent to make up for ompensator losses Single-Phase Pulse Load Some loads draw a large magnitude urrent for a short period (from half of a yle to a few yles), whih may ause voltage sag in a weak power system. Singlephase loads in three-phase power systems an ause the system to beome unbalaned (either unbalaned magnitude or unbalaned phase angle). his may result in a high neutral urrent, or a distortion in voltage. Single-phase loads also result in harmonis at all odd multiples of the fundamental, but the most substantial of them are typially the triplen harmonis. riplens add together in the neutral line and result in a high neutral urrent. A single-phase pulse load is studied in this setion. A pulse urrent is represented by a triangle waveform. he duration of this urrent may be a fration of the line frequeny yle, or up to several yles. he urrent may be zero or a sine wave of a muh smaller magnitude ompared to the irregular pulse urrent; therefore, it is simplified to zero at all the times other than the pulse urrent interval (i l is shown in Figure 4.14a). Figures 4.14 and 4.15 show the simulation results for = /2 (Figure 4.14) and = 2 (Figure 4.15) with v p = v f for both ases. o simplify the simulation, the voltage waveform is assumed to be a sine wave. However, note that the results also apply to the ase of a soure voltage that ontains harmonis or is nonsinusoidal. Figure 4.16 shows the peak soure urrent normalized with respet to load urrent for various ompensation intervals and phase angles between the urrent pulse and soure voltage for a pulse waveform suh as those shown in Figure 4.14 and Figure With the averaging interval hanging from /2 to 2, i s dereases from 0.6 p.u. 94

107 (a) System voltage v s and load urrent i l (b) Instantaneous load power p l () Average load power P l (d) Soure urrent i s (e) Compensator urrent i (f) DC link voltage v d Figure Simulation of single-phase pulse load, = /2. (a) System voltage v s and load urrent i l (b) Instantaneous load power p l () Average load power P l (d) Soure urrent i s (e) Compensator urrent i (f) DC link voltage v d Figure Simulation of single-phase pulse load, = 2. 95

108 (a) Peak soure urrent (b) Peak ompensator urrent Figure Compensator urrent normalized with respet to load urrent for different ompensation times and load urrent phase angles. to 0.2 p.u. of the load urrent while i inreases modestly from 0.6 to 0.9 p.u. of the load urrent. If the phase angle between the soure voltage and pulse urrent is small, whih is ommon for many systems, a small inrease in energy storage apaity will result in a muh better ompensation (i.e., muh smaller soure urrent). In the ase of a large phase angle differene with a ompensation period of 2, i inreases 10% (from 0.9 to 1.0 p.u. of the load urrent); while in the ase of a ompensation period of /2, i inreases approximately 70% (from 0.6 to 1.0 p.u.). his is beause if the ompensator uses = 2, it must have a larger energy storage apaity (relative to hoosing = /2) to aommodate not only a longer ompensation interval, but also a larger instantaneous reative omponent. hus, a ompensator with larger energy storage an provide better ompensation and support a load that has a large reative omponent. 96

109 (a) Peak soure urrent plotted as a funtion of the ompensator s energy storage requirement (b) Power requirement for different and load urrent phase angles Figure Compensator energy storage requirement and power rating requirement. Figure 4.17 shows the peak soure urrent plotted as a funtion of the ompensator s energy storage requirement and the instantaneous power requirement for various ompensation times and load urrent phase angles. At longer averaging intervals, the power drawn from the soure is distributed during so that it has a smaller peak value instead of a short duration, high power pulse. he ompensator s instantaneous power requirement is more onentrated so that a higher energy rating is required for short time durations. Over the omplete ompensation period, the ompensator provides only reative power and does not onsume or generate any ative power beause the load energy is provided by the soure. hus, a trade-off between a smoother soure urrent waveform with lower amplitude and the size of the ompensator omponents must be onsidered when hoosing the value of. 97

110 4.3.4 Non-Periodi Load heoretially, the period of a non-periodi load is infinite (a period muh larger than the fundamental period of the utility). he nonative omponents in a load annot be ompletely ompensated by hoosing as /2 or, or even several multiples of. Figure 4.18a shows a three-phase non-periodi load urrent waveforms (red waveform) and the system voltage waveforms (blue waveform). Figure 4.18b shows the soure urrents after ompensation, with = /2, 2, and 10, respetively. With = /2, there is still signifiant nonative omponent in i s with variable peak values and a nonsinusoidal waveform. With = 2, the variation of the amplitude of i s is smaller, and with = 10, i s is lose to a sine wave with less nonative omponent. A longer smoothes the soure urrent waveforms. heoretially, i s ould be a pure sine wave if goes to infinity, but in pratie, suh a annot be implemented nor is it neessary. If is large enough, inreasing further will not typially improve the ompensation results signifiantly. For example, in the ase shown in Figure 4.18, the total harmoni distortion (HD) of i s is 6% with = 2, and with = 10, the HD is only slightly smaller (4%). ypially, there is no need to inrease to a larger value as the small derease in HD is often not worth the larger apital osts (higher ratings of the ompensator omponents and therefore higher apital expenses). 4.4 Disussion In nonative power ompensation, there are several fators whih have signifiant influene on the hoie of ompensator type, the power rating of the ompensator, the energy storage requirement of the ompensator, and the ompensation results. Some of 98

111 (a) Load urrent i l and system voltage v s (b) Phase a of soure urrent i sa after ompensation = /2, 2, and 10 Figure Simulation of non-periodi load ompensation. 99

112 these fators are related to the nonative power theory itself, whih inlude the averaging interval, and the referene voltage v p, while others are pratial issues related to the implementation of the ompensation system. In this dissertation, the oupling indutane L, the power rating of the ompensator, and the apaitane of the DC link have been taken into onsideration Averaging Interval, If there are only harmonis in the load urrent, as in subsetions , does not hange the ompensation results as long as it is an integral multiple of /2, where is the fundamental period of the system. Here, the nonative urrent is ompletely ompensated, and a purely sinusoidal soure urrent with a unity power fator is ahieved. However, in other ases, suh as a three-phase load with sub-harmonis (see subsetion ), a single-phase pulse load (see subsetion ), or a non-periodi load (see subsetion 4.3.4), has signifiant influene on the ompensation results, and the power and energy storage rating of the ompensator s omponents. With longer, a better soure urrent will result, but at the ost of higher power rating for the swithes and apaitane. here is a tradeoff between better ompensation and higher system ratings (i.e., osts). On the other hand, a longer does not neessarily yield a signifiantly better soure urrent waveform. For a speifi system, there is an appropriate with whih the ompensation an be ahieved. For example, in subsetion , depends on the frequeny of the sub-harmonis. 100

113 4.4.2 DC Link Voltage v d Generally, the ompensator output voltage v is the sum of the system voltage and the voltage drop on the oupling indutor. he rms value of the required ompensator output voltage V at the worst situation, i.e., all the nonative omponents reah the peak value at the same time is V XI V s. (4.41) he ompensator urrent I inludes all the nonative urrent omponents that are ompensated, therefore, the reatane X has different values for nonative omponents of different frequenies. he higher order harmonis will have larger voltage drop on the indutane. he required DC link voltage V d for a three-phase system is V 2 2V. (4.42) d he required DC link voltage V d for a single-phase system is V 2V. (4.43) d Coupling Indutane, L he oupling indutane L between the power system and the ompensator ould be the indutane of a step-up transformer or a oupling reator. It ats as the filter of the ompensation urrent i, whih has high ripple ontent due to the ompensator s PWM ontrol of the swithes. Generally a smaller L is preferred, beause a large L requires a higher ompensator output voltage, i.e., a higher DC link voltage. In pratie, L is hosen so that the fundamental voltage drop on this indutor is 3% - 9% of the system voltage V s. he indutane L is 101

114 L Vs (3% 9%). (4.44) I DC Link Capaitane Rating, C he DC link apaitane rating C is proportional to the maximum energy storage variation of the apaitor. Aording to the disussion in Setion 3.1, the average power of the ompensator P (t) over is zero. Energy is neither generated nor onsumed by the ompensator, therefore, the energy stored in the apaitor is a onstant at rated DC link voltage V d given by 2 E 1 CV 2 d. (4.45) However, the instantaneous power is not neessarily zero. he ompensator generally has a apaitor for energy storage, and this apaitor operates in two modes, i.e., harge and disharge. Different apaitane values are required to fulfill different ompensation tasks. he maximum energy variation in the apaitor is the integral of v s (t)i (t) between time t max when the apaitor goes from disharge to harge, and t min when the apaitor goes from harge to disharge, or vie versa. E t t max min v () t i () t dt. (4.46) s he energy variation on the DC link apaitor auses the voltage variation, that is E E( tmax ) E( tmin ) C( Vd( tmax ) Vd( tmin )). (4.47) 2 he required DC link apaitane C is 102

115 tmax 2E 2 vs ( t) i( t) dt tmin d max d min d max d min C. (4.48) V ( t ) V ( t ) V ( t ) V ( t ) For different appliations, the energy variation E is different, whih determines the apaitane rating, for a given DC link voltage variation. In a three-phase RL load ase, beause the instantaneous nonative power is zero at all times, the urrent flowing into or out of the DC link apaitor is zero. herefore, a small apaitor an meet the requirement of this ase. A large energy storage variation happens when the load is unbalaned, or there is a sudden load hange, or a single-phase system, and a large apaitane is required. 4.5 Summary he generalized nonative power theory was implemented in a shunt ompensation system, whih was onfigured to perform the nonative power ompensation for a threephase four-wire system. he urrent that the ompensator was required to provide was alulated based on the generalized nonative power theory presented in Chapter 3. A ontrol sheme was developed to regulate the DC link voltage of the inverter, and to generate the swithing signals for the inverter based on the required nonative urrent. he ompensation system was simulated, and different ases were studied. he simulation results showed that the generalized nonative power theory proposed in this dissertation was appliable to the nonative power ompensation in three-phase four-wire systems, single-phase systems, load urrents with harmonis, and non-periodi load urrents. his theory was adapted to different ompensation objetives by hanging the 103

116 referene voltage v p (t) and the averaging interval. he pratial issues suh as the DC apaitane rating, the DC link voltage, and the oupling indutane were also disussed. 104

117 CHAPER 5 Experimental Verifiation In the previous hapter, the generalized nonative power theory was implemented in a parallel nonative power ompensator, and different ases were simulated. A ompensator experimental setup was built, whih is desribed in this hapter. Four different types of load, a three-phase balaned RL load, a three-phase diode retifier load, a three-phase unbalaned RL load, and a single-phase load are tested. he experimental results are analyzed, and the generalized instantaneous nonative power theory is verified. he experimental system onfiguration and the equipment will be shown in subsetion 5.1, and the experimental results are elaborated and analyzed in subsetion 5.2. Subsetion 5.3 ontains onlusions and summary of this hapter. 5.1 System Configuration he system onfiguration simulated in Chapter 4 is adopted in the experimental system, as shown in Figure 5.1. he atual power omponents are used in the experiments, while the alulation and ontrol is performed in Simulink. 105

118 Figure 5.1. Experimental onfiguration of the nonative power ompensator. 106

119 Host PC Expansion box panels Figure 5.2. dspace system onfiguration dspace System dspace, a real-time ontrol platform, is used to implement the Simulink ontroller on hardware to perform the real-time ontrol of the nonative power ompensator. In this experiment, as shown in Figure 5.2, the dspace system uses a personal omputer (PC) as the host omputer. An expansion box is onneted to the host omputer via an optial fiber. wo panels are onneted to the expansion box, one is an analog-to-digital board, and the other is a PWM signal output board. he voltages and urrents are measured by voltage potential transduers (Ps) and urrent transduers (Cs), and the measurement signals are input to the analog-to-digital board, where these analog signals are onverted to digital signals for alulation and ontrol purposes. he PWM gate signals used to ontrol the inverter swithes are generated and output from the dspace PWM output board. 107

120 he ontrol system of the parallel nonative power ompensator onsists of two parts, i.e., MALAB/Simulink and dspace. MALAB is a platform for numeri alulations, analysis, and visualization. Simulink is an interative environment for modeling and offline simulation with blok diagrams. he state diagrams in Simulink an be integrated with Stateflow and Real-ime Workshop to automatially generate C ode from Simulink blok diagrams and Stateflow systems. ogether with dspace s Real-ime Interfae, these tools provide a seamless transition from a blok diagram to dspace s real-time hardware. he ompensation is operating in a high EMI environment mainly beause of the inverter swithing on and off high voltage and/or high urrent. he measurement devies and the ontrol system are vulnerable to the high EMI, and errors are introdued to the system. he inverter was put into a metal box to isolate the strong EMI soure from other equipment. Some filtering apaitors are used in the ompensation system to mitigate the ripple in the ompensator urrent. A piture of the experiment setup is shown in Figure 5.3. he omponents of the setup will be presented in the following subsetions he Utility he utility is a three-phase, four-wire power system. In the simulations presented in Setion 4.3.1, the system voltage is highly distorted, whih will not be the ase in the experiments, sine the voltage distortion in the real power system is fairly small. 108

121 1. Resistor load bank 2. Interfae of dspace and the inverter 3. Power supplies for the interfae and the inverter drive board 4. Voltage transduers 5. Coupling indutors and load indutors 6. DC power supply 7. Inverter 8. Current transduers 9. dspace host PC Figure 5.3. Experiment setup of the nonative power ompensation system. 109

122 able 5.1. Ratings of the ommerial inverter. DC link voltage DC link apaitane IGB urrent Swithing frequeny 800 V 3300 F 75 A 20 khz Loads A three-phase balaned RL load, a three-phase diode retifier load, a three-phase unbalaned RL load, and a single-phase load will be tested to verify the validity of the nonative power theory. A three-phase diode retifier is a typial nonlinear load whih draws a highly-distorted urrent onsisting of a fundamental omponent and high-order harmonis Inverter A Powerex POW-R-PAK M onfigurable IGB based 3-phase inverter is used in the experiment. he topology of the inverter is the same as shown in Figure 2.2. he ratings of this inverter are shown in able 5.1. In Figure 5.3, the DC power supply is used to harge the DC link apaitor to the rating value. When the ompensator is operating, the DC power supply is swithed off. he ompensator does not use a power soure and the DC link voltage is self-regulated by obtaining ative power from the utility soure. 110

123 An interfae is needed to onvert the PWM output signals from the dspace, whih are 5V; to 15V PWM signals to drive the swith gate drives on the inverter. An interfae was built, and its shemati is shown in Figure 5.4. hree apaitors C f are onneted between phases between the oupling indutors and the PCC, as shown in Figure 5.1. hese apaitors filter the ripple in the ompensator urrent. able 5.2 is a list of the omponents in the experimental setup, and their ratings are also shown. 5.2 Experimental Results In the hosen set of experiments, the nonative urrent required by the load is provided by the ompensator so that the soure only needs to provide ative power. In the experiments, as explained in Chapter 4 (see Figure 4.4), the three-phase urrent produed by the ompensator is given by i * () t i * () t i * () t. (5.1) n a Reall that i * () t v ()[ t K ( V * v ) K ( V * v ) dt], (5.2) a s P1 d d I1 d d 0 t where v s (t) = [v sa (t), v sb (t), v s (t)] is the three-phase system voltage. his is the urrent required to keep the ompensator apaitor harged against losses in the ompensator. Also, reall that * i n is speified as * i () t i () t i () t, (5.3) n l s 111

124 Figure 5.4. Shemati of the interfae between dspace and the inverter. able 5.2. Components of the nonative power ompensation system. Devie Inverter hree-phase resistor bank Load indutor Coupling indutor Voltage transduer Current transduer DC power supply Manufaturer /model Powerex M POW-R-PAK PP75120 Avtron K595 riad-utrad C-59U Amveo L1510 LEM CV Danfysik ULRASAB 866 Magna-power Current (A) Ratings DC voltage (V) Power (kw) Line-to-line voltage (V) /500 Indutane (mh) Current (A) DC Indutane (mh) Current (A) DC Primary voltage (V) Bandwidth 500 DC to 300 khz Primary urrent (A) Bandwidth 600 DC to 100 khz Current (A) DC voltage (V)

125 i s i () () sa t vpa t () () () Pt t isb t () 2 () vpb t Vp t, (5.4) i () () s t vp t t 1 Pt () p( ) d, (5.5) t p() t v () ti () t v ti() t v ti() t, (5.6) sa la sb lb s l and t 1 Vp() t vp( ) v p( ) d. (5.7) t where, in all the experiments, v p (t) = v s (t). he ontrol diagram of the experimental setup is shown in Figure 5.5. able 5.3 lists the parameters that do not hange throughout the experiments. Other parameters that do hange with the experiments will be listed in eah experiment hree-phase Balaned RL Load A three-phase balaned RL load ompensation is performed under the onditions listed in able 5.4. he three phase system voltage waveforms [v sa (t), v sb (t), v s (t)] are shown in Figure 5.6a. he load urrent waveforms [i la (t), i lb (t), i l (t)] are plotted in Figure 5.6b, together with the phase a system voltage v sa (t), to show the phase angle between the system voltage and the load urrent. he load urrent lags the system voltage in a RL load. Figure 5.6 shows the soure urrent i s (t) after ompensation, together with phase a voltage. he amplitude of the soure urrent is smaller than the load urrent, and the 113

126 Figure 5.5. Control diagram of the experimental setup. able 5.3. Parameters of the experiments. System voltage (line-to-neutral rms) System frequeny f = 1/ Coupling indutane DC link apaitane Control stepsize Swithing frequeny 120 V 60 Hz 10 mh 3300 F 80 s 12.5 khz DC link voltage PI ontroller KP = 0.2, KI = 0 able 5.4. Parameters of the three-phase balaned RL load ompensation. Load resistane Load indutane DC link voltage mh 400 V Averaging interval /2 Nonative urrent PI ontroller KP = 40, KI = 0 114

127 (a) System voltage v s s(t) (b) Load urrent i l (t) () Soure urrent i s s(t) (d) Compensator urrent i (t) (e) Instantaneous powers (f) Average powers (g) Apparent powers (h) power fators Figure 5.6. hree-phase balaned RL load ompensation. 115

128 able 5.5. Powers of the three-phase balaned RL load ompensation. heoretial powers (VA) Experimental powers (VA) Average power P(t) 2426 Average load power P l (t) 2426 Average ative power P a (t) 2426 Average soure power P s (t) 2533 Average nonative power P n (t) 0 Average ompensator power P (t) -107 Apparent power S(t) 2933 Apparent load power S l (t) 2934 Apparent ative power P p (t) 2426 Apparent soure power S s (t) 2537 Apparent nonative power Q(t) 1649 Apparent ompensator power S (t) 1584 soure urrent is in phase with the system voltage. he ompensator urrent i (t) is illustrated in Figure 5.6d. his urrent, whih is about 90º out of phase with the system voltage, ontains mostly nonative urrent. o further analyze the ompensation results, the instantaneous powers (instantaneous power p(t), instantaneous ative power p a (t), and instantaneous nonative power p n (t)), the average powers (average power P(t), average ative power P a (t), and average nonative power P n (t)), and the apparent powers (apparent power S(t), apparent ative power P p (t), and apparent nonative Q(t)) are alulated based on the system voltage and the load urrent. he average powers and apparent powers are shown in the seond olumn in able 5.5. P(t) = P a (t) = P p (t), P n (t) = 0, and S () t P () t Q () t. hey are p onsistent with the disussion in Chapter 3. he powers of the experimental results are listed in the fourth olumn of able 5.5. he instantaneous load power is 116

129 p () t v () t i () t. (5.7) l s l he instantaneous soure power is p () t v () t i () t. (5.8) s s s he instantaneous ompensator power is p () t v () t i () t. (5.9) s he average load power is t 1 Pl () t ( ) ( ) d vs i. (5.10) l t he average soure power is t 1 Ps ( t) vs ( ) is( ) d. (5.11) t he average ompensator power is t 1 P ( t) vs ( ) i( ) d. (5.12) t he apparent load power is S () t V () t I () t. (5.13) l s l he apparent soure power is S () t V () t I () t. (5.14) s s s And the apparent ompensator power is S () t V () t I () t. (5.15) s he instantaneous load power p l (t), instantaneous soure power p s (t), and the instantaneous ompensator power p (t) are shown in Figure 5.6e. Aording to the 117

130 analysis in Chapter 3 (able 3.2), the instantaneous ative power is equal to the instantaneous power, and the instantaneous nonative power is zero (see the instantaneous powers in the first and seond olumns of able 5.5). In Figure 5.6e, the instantaneous soure power is slightly larger than the instantaneous load power, and the instantaneous ompensator power is negative. his is beause the ompensator draws a small amount of ative urrent from the utility to meet the losses. herefore, the ompensator urrent ontains a small amount of ative urrent, and the instantaneous ompensator power ontains a small amount of ative power omponent. he average load power P l (t), average soure power P s (t), and the average ompensator power P (t) are shown in Figure 5.6f. hey have similar harateristis as the instantaneous powers exept that these waveforms are smoother. he apparent load power S l (t), apparent soure power S s (t), and the apparent ompensator power S (t) are shown in Figure 5.6g. heoretially, these nine powers are all onstant (able 3.2). he average values of the waveforms in Figures 5.6f and 5.5g are alulated and listed in the fourth olumn of able 5.5. he differene between the seond olumn and the fourth olumn of able 5.5 is mainly beause of the DC link voltage regulation whih is not onsidered in the generalized nonative power theory. In the experiment, instead of zero, the ompensator power has some ative power omponent drawn from the utility. Figure 5.6h shows the power fators before and after ompensation. pf 1 is the power fator before the ompensation, whih is P(t)/S(t) = ; after ompensation, the power fator of the soure urrent (pf 2 ) is P a (t)/p p (t) = herefore, the nonative power ompensator an improve the load power fator to nearly unity power fator. 118

131 (a) DC link voltage hange (b) DC link voltage hange (short period) () Soure urrent i s (t) (d) Compensator urrent i (t) Figure 5.7. Regulation of DC link voltage in three-phase balaned RL load ompensation. Figure 5.7 shows the DC link voltage ontrol. In Figure 5.7a, the red waveform is the referene DC voltage, whih is 400V at the beginning, and a step hange is made to the referene to 450V at t = 2.66s. he blue waveform is the atual DC link voltage measured by the P. Figure 5.7b is a short period around the step hange. o regulate the DC voltage from 400V to 450V, the ompensator draws a larger amount of ative urrent than usual to harge the apaitor. his is ontrolled by the DC link voltage PI ontroller. he dynami proess of the soure urrent and the ompensator urrent are shown in Figures 5.7 and 5.7d. Beause of the onstraints of the system, for instane, the urrent and/or power ratings of the omponents in the ompensator, only a limited urrent an be drawn from the utility, whih auses the atual DC link voltage to reah the new 119

132 referene value in about 0.1s. A higher urrent rating would allow a faster dynami response hree-phase Unbalaned RL Load For this test, the indutors of the RL load are not equal in eah phase, therefore, the three phase load urrents are not balaned any more, as shown in Figure 5.8b. he load indutors are 30mH, 10mH, and 10mH, respetively. he DC link voltage is 450V. Other parameters are listed in able 5.6. he three phase system voltages are balaned, whih is shown in Figure 5.8a. he rms values of the three phase load urrent is shown in the seond olumn of able 5.7. Figure 5.8 shows the soure urrents after ompensation, whih are nearly balaned ompared to the load urrent. he ompensation urrent is shown in Figure 5.8d. he three phase soure urrent rms values are listed in the third olumn of able 5.7. he unbalane of the three-phase urrents is alulated as I where unbalane max{ Ia Ib, Ib I, I Ia }, (5.16) Avg( I, I, I ) a b Avg( I, I, I ) ( I I I )/3. (5.17) a b a b he unbalane of the load urrent is 38.97%, and the unbalane of the soure urrent is improved to 4.92% after ompensation Sudden Load Change and Dynami Response In this ase, a sudden hange is applied to the three-phase RL load to test the dynami response of the nonative power ompensator. he parameters are listed in able 5.8. he waveforms in Figure 5.9a are the three phase system voltage. In Figure 5.9b, the load 120

133 (a) System voltage v s s(t) (b) Load urrent i l (t) () Soure urrent i s s(t) (d) Compensation urrent i (t) Figure 5.8. hree-phase unbalaned RL load ompensation. able 5.6. Parameters of the three-phase unbalaned RL load ompensation. Load resistane Load indutane DC link voltage , 10, 10 mh 450 V Averaging interval /2 Nonative urrent PI ontroller KP = 40, KI = 0 able 5.7. RMS urrent values of the unbalaned load ompensation. I l (A) I s (A) Phase a Phase b Phase I unbalane 38.97% 4.92% I unbalane 121

134 able 5.8. Parameters of the three-phase sudden load hange ompensation. Load resistane Load indutane DC link voltage Averaging interval Nonative urrent PI ontroller mh 400 V 5 KP = 40, KI = 0 (a) System voltage v s s(t) (b) Load urrent i l (t) () Soure urrent i s s(t) (d) Compensator urrent i (t) (e) Phase a voltage and load urrent (f) Phase a voltage and soure urrent Figure 5.9. Dynami response of three-phase RL load ompensation. 122

135 urrent is zero at the beginning, and it is hanged to 10A (peak value) at t = 3.366s. he soure urrent and ompensator urrent are shown in Figures 5.9 and 5.9d, respetively. here is a transient in both the soure urrent and the ompensator urrent. here are two fators that influene the dynami response of the nonative ompensation, one is the averaging interval, and the other one is the DC link voltage regulation. In this ase, = 5, therefore, the soure urrent slowly inreases from zero to the steady-state value. Before the soure urrent reahes the steady state, it inreases to a maximum value and then dereases to the steady state value, this is beause of the DC link voltage regulation, whih draws some ative urrent to regulate the DC link voltage. For a faster dynami response, a shorter is preferred. Figures 5.9e and 5.9f are the phase a load urrent and soure urrent with the system voltage, to show the phase angle between the voltage and the urrent. he load urrent lags the voltage, while the soure urrent is in phase with the voltage; therefore, a unity power fator is ahieved hree-phase Diode Retifier Load A three-phase diode retifier load is a typial nonlinear load in the power system. he topology of a three-phase diode retifier is shown in Figure 5.10a, and the typial load urrents are shown in Figure 5.10b. A resistive load is used on the DC side of the retifier. he parameters are shown in able 5.9. he DC link voltage is set to 500V. he measured load urrent is shown in Figure 5.11b, whih is highly distorted. Figures 5.11 and 5.11d show the soure urrent and ompensator urrent when the DC link voltage is 500V. he soure urrent is still distorted, whih is mainly beause of the large di/dt in the load urrent. As disussed in Chapter 4, a large di/dt requires a large DC link voltage for full ompensation, i.e., a large inverter output voltage, so that the ompensator an 123

136 able 5.9. Parameters of the three-phase retifier ompensation. DC link voltage 500, 600, 700 V Averaging interval /2 Nonative urrent PI ontroller KP = 60, 100, 120, KI = i l D 1 D 3 D 5 C d v d R L D 4 D 6 D 2 _ (a) hree-phase diode retifier (b) Current waveforms of a diode retifier Figure A three-phase diode retifier load. 124

137 (a) System voltage v s s(t) (b) Load urrent i l (t) () Soure urrent, V d = 500 V (d) Compensator urrent, V d = 500 V (e) Soure urrent, V d = 600 V (f) Soure urrent, V d = 700 V Figure Diode retifier load ompensation. 125

138 able HD of the urrents in the retifier load ompensation. HD of i l (%) V d = 500V V d = 600V V d = 700V HD of i s (%) HD of i l (%) HD of i s (%) HD of i l (%) HD of i s (%) phase a phase b phase average injet a large amount of urrent in a very short period. However, it is often not pratial to have very high DC link voltage. Inreasing the DC link voltage to 600V and 700V, the resulting soure urrents are shown in Figures 5.11e and 5.11f, respetively. he red sinusoidal waveforms in these figures are the phase a voltage, to show that the urrents are in phase with the voltage. By inreasing the DC link voltage from 500V to 600V and then 700V, the waveforms of the soure urrent are improving, i.e., loser to a sinusoid. able 5.10 lists the HD of the load urrents and soure urrents in the three ases. he HD of the load urrent is about 30%. he HD of the soure urrent is 25.32% (average value of the three phases) when the DC link voltage is 500V, and it dereases to 21.24% and 18.84% when the DC link voltage is 600V and 700V, respetively Single-Phase Load An RL load is onneted between phase a and phase b in the three-phase system while phase is left open (unloaded). he resistor is 29.2, and the indutor is 10mH. he DC link voltage is 500V. Other parameters are shown in able he voltage and 126

139 able Parameters of the single-phase RL load ompensation. Load resistane 29.2 Load indutane 10 mh DC link voltage 500 V Averaging interval /2 Nonative urrent PI ontroller KP = 40, KI = 0 the load urrent are shown in Figures 5.12a and 5.12b, respetively. he load urrent in phase a and phase b are equal in magnitude and opposite in phase, and the urrent in phase is zero. he rms values of the three phase load urrents are listed in the seond olumn of able he rms value of the phase load urrent is slightly more than zero beause of the measurement error as a result of IGB swithing. he soure urrent and the ompensator urrent are shown in Figures 5.12 and 5.12d, respetively. he magnitudes of phase a and phase b soure urrents are redued and there is a urrent in phase. he rms values of the three phase soure urrents are shown in the third olumn of able he values of phase a and phase b are redued and the three phases are more balaned after ompensation. 5.3 Conlusions and Summary he generalized nonative power theory proposed in this dissertation is implemented in a parallel nonative power ompensation system. A three-phase balaned RL load, a three-phase unbalaned RL load, a three-phase diode retifier load, and a single-phase RL load are tested. he experimental results verify that the generalized nonative power theory is suitable for parallel nonative ompensation. he experimental results are 127

140 (a) System voltage v s s(t) (b) Load urrent i l (t) () Soure urrent i s s(t) (d) Compensator urrent i (t) Figure Single-phase load ompensation. able RMS urrent values of the single-phase load ompensation. I l (A) I s (A) Phase a Phase b Phase I unbalane % 22.42% I unbalane 128

141 onsistent with the theoretial analysis in Chapter 3 as well as the simulation results in Chapter he experimental results show that this theory is valid in three-phase and single phase systems. he single-phase load is onneted between two phases in a three-phase system. After ompensation, the soure urrent has a unity power fator and is nearly balaned. 2. he theory is appliable to both sinusoidal and non-sinusoidal systems. An RL load (fundamental nonative power ompensation) and a diode retifier load (harmonis load) are tested. In the ase of fundamental nonative power ompensation, the fundamental nonative urrent omponent is provided by the ompensator, and nearly unity power fator is ahieved. In the diode retifier ase, a redution in the harmonis ontent of the soure urrent is ahieved. 3. he theory is appliable to both balaned and unbalaned systems. In the experiments, there is no onnetion between the inverter s DC link middle point and the power system s neutral point. he ompensator topology and ontrol sheme are simpler than for a three-phase four-wire system. he results show that the three-phase unbalaned load urrents are balaned and the power fator is unity after ompensation. 4. he experimental results are onsistent with the disussion in Chapter 4 on the DC link voltage requirement and the DC link apaitane. 5. he definitions of ative and nonative urrents/powers are instantaneous, whih allows a fast dynami response by the ompensator. he dynami 129

142 response is related to the averaging interval, the power rating and/or urrent rating of the ompensator, and the ontrol system speed. 130

143 CHAPER 6 Conlusions and Future Work he dissertation is summarized in subsetion 6.1, and the future work is proposed in subsetion Conlusions Despite that mature power systems have adopted the form of three-phase a power for over 100 years, modern power systems have been ompliated by a variety of nonlinear loads. With the existene of harmonis, sub-harmonis, non-periodi waveforms, et., the standard definitions of average ative power and reative power of three-phase, sinusoidal a power systems are no longer able to express these physial phenomena. Definitions of nonative urrent and the instantaneous nonative power are required to desribe, measure, and ompensate the nonative urrent and power in power systems. Several theories have been proposed sine the 1920s; however, none of them provide an expliit definition of instantaneous nonative power/urrent whih is appliable to all of the different ases. heir disadvantages inlude 1. No generalized theory. hey are valid only in some speifi situations. Some are valid only in three-phase, three-wire systems, some are valid only in periodi systems, and some are valid only in steady-state systems. 131

144 2. Not stritly instantaneous definition. At lease one yle of the fundamental period is required to alulate the harmoni omponents. In addition, some definitions do not have any physial meanings, and some definitions annot be used in a nonative power ompensation appliation. A generalized nonative power theory was presented in this dissertation for nonative power ompensation. he instantaneous ative urrent, the instantaneous nonative urrent, the instantaneous ative power, and the instantaneous nonative power were defined in a system whih did not have any limitations suh as the number of the phases, the voltage and the urrent were sinusoidal or non-sinusoidal, periodi or non-periodi. By hanging the referene voltage and the averaging interval, this theory had the flexibility to define nonative urrent and nonative power in different ases. It was shown that other nonative power theories disussed in Chapter 2 ould be derived from this more general theory by hanging the referene voltage and the averaging interval. he flexibility was illustrated by applying the theory to different ases suh as a sinusoidal system, a periodi system with harmonis, a periodi system with subharmonis, and a system with non-periodi urrents. his theory was implemented using a shunt ompensator. he urrent that the ompensator was required to provide was alulated based on the generalized nonative power theory. A ontrol sheme was developed to regulate the DC link voltage of the inverter, and to generate the swithing signals for the inverter based on the required nonative urrent. he ompensation system for different ases was studied. he simulation and experimental results showed that the theory proposed in this dissertation was appliable to nonative power ompensation in three-phase four-wire systems, 132

145 single-phase systems, load urrents with harmonis, and non-periodi load urrents. he DC link apaitane rating, the DC link voltage requirement, and the oupling indutane rating are determined in a three-phase RL load ase, and these requirements for other ases are also disussed. 6.2 Future Work he generalized nonative power theory was presented, and its appliation in a shunt nonative power ompensation system was simulated and implemented in experiments. 1. More researh on the eonomi analysis. he generalized nonative power theory was used in different systems, and the averaging interval, the referene voltage, the DC link apaitane, the DC link voltage, and the oupling indutane were disussed and the requirements were determined based on the ompensation objetives. More researh needs to be done on the analysis of the apital ost by hoosing different ratings and the system and/or ustomer benefit by installing a nonative power ompensator. 2. he implementation of the generalized nonative power theory in other flexible a transmission systems (FACS) appliations. he theory is appliable for nonative power ompensators with various onfigurations, suh as a hybrid nonative ompensator, a multilevel-inverter-based nonative ompensator, an interline power flow ontroller, or a unified power flow ontroller. hese FACS devies ontrol both the ative and the nonative power flow between the soure and the load and between different transmission lines. hey provide nonative power ompensation, as 133

146 well as anillary servies suh as voltage stability, transient stability, and power osillation damping. 3. he utilization of the nonative power theory in distributed energy resoures. he power eletroni interfaes of the distributed energy resoures provide the onnetion to the power system, by onverting d power to a power, and/or adjusting the power level, and/or synhronizing with the power system. he interfaes an also be used to ontrol both the instantaneous ative power and the nonative power of the distributed energy resoures themselves and the power systems as well. herefore, the distributed energy resoures an simultaneously be ative power soures and nonative power ompensation systems that an perform multi-purpose tasks. 4. System-wide ative and/or nonative power ontrol and operation. System-wide, or at least the nearby distribution system is taken into onsideration by the nonative power ompensation system, instead of only the system/load operation information at the loal site. Network ommuniation tehnology will be needed to perform the system-wide monitoring, analysis, and ontrol. he individual nonative power ompensation system and other FACs will have brains to exhange information with the entral ontrol and other intelligent devies, reeive remote ommands from the entral ontrol, and perform loal ative and/or nonative power flow ontrol tasks based on the loal situation and the system-wide operation requirements. 134

147 Referenes 135

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152 Vita Yan Xu reeived her B.S. in July 1995 from Shanghai Jiaotong University, China majoring in Eletri Power Engineering. She reeived her M.S. in April 2001 from North China Eletri Power University, China majoring in Eletri Power Engineering. She worked at Anhui Provine Eletri Power Company, Maanshan Branh from August 1995 to August 1998 as an eletrial engineer. She also worked at Beijing Pengfa Xingguang Power Eletronis Co. Ltd. from August 2001 to Deember 2001 as a researh and development engineer. Yan Xu started her Ph.D. program at the Department of Eletrial and Computer Engineering, he University of ennessee in January At the same time, she joined the Power Eletronis Laboratory at he University of ennessee as a graduate researh assistant, working on reative power theory and SACOM ontrol issues. She has worked at Oak Ridge National Laboratory sine 2005 by helping to install and setup a SACOM in the Reative Power Laboratory. She graduated with a Dotor of Philosophy degree in Eletrial Engineering from he University of ennessee in May