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