Abstract—This paper presents the application of a space vectormodulation (SVM) strategy for a multimodule converter systemthat 1) enables low switching frequency, 2) eliminates/minimizesac-side voltage harmonics, particularly low-order harmonics, and3) provides maximum ac-side fundamental voltage component.The SVM strategy is based on a sequential sampling technique. TheSVM provides harmonic cancellation/minimization by introducingappropriate phase shift for the corresponding voltage harmonics ofthe converter modules, while maintaining the fundamental voltagecomponents of modules in-phase to obtain maximum ac-sidevoltage. The SVM eliminates the need for complicated transformerarrangement for harmonic reduction, and thus provides highdegree of modularity by utilization of identical transformers forconverter modules. The proposed SVM strategy is developed fora back-to-back HVDC converter station in which each convertersystem is composed of four identical modules. Based on a dynamicmodel of the system, converter controls are designed and perfor-mance of the SVM strategy in terms of converter harmonics anddynamic performance are presented. The studies are performedin time-domain, using the PSCAD/EMTDC software tool.
Index Terms—Dynamic model, harmonics, HVDC, multimoduleconverter, space vector modulation (SVM).
I. INTRODUCTION
SIGNIFICANT developments in semiconductor technologyand commercial availability of high-power switches [e.g.,
insulated-gate bipolar transistors (IGBTs)], have resulted inwide acceptance of the switch-mode voltage-source converter(VSC) as the building block for HVDC converters [1]–[6].The requirement to meet high voltage levels, both at ac anddc sides, of an HVDC converter system is met by the use ofmultilevel [7] and/or multimodule [8] VSC configurations. Tomaintain HVDC converter loss within an acceptable limit, theswitching frequency of the VSC modules of a converter systemhas to be at the lowest possible value. Low switching frequencygenerates low-order ac-side harmonics which also have to belimited within permissible values.
Manuscript received February 28, 2006; revised June 8, 2006. Paper no.TPWRD-00076-2006.
M. Saeedifard and R. Iravani are with the Center for Applied Power Elec-tronics (CAPE), Department of Electrical and Computer Engineering, Univer-sity of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]; [email protected]).
H. Nikkhajoei is with the Wisconsin Power Electronics Research Center(WisPERC), Department of Electrical and Computer Engineering, Universityof Wisconsin-Madison, Madison, WI 53706 USA (e-mail: [email protected]).
A. Bakhshai is with the Department of Electrical and Computer Engineering,Queen’s University, Kingston, ON K7L 3N6, Canada (e-mail: [email protected]).
Digital Object Identifier 10.1109/TPWRD.2006.886777
If low switching frequency is adopted for a multimoduleHVDC converter system, harmonic cancellation/minimizationis conventionally achieved by the following approaches.
• The first approach is based on selective harmonic elimina-tion pulse-width modulation (SHEPWM) techniqueswhich allows certain harmonics, usually low-ordernon-triplen harmonics, to be eliminated/minimized byproper selection of switching instants [9]–[11]. Due to thelow switching frequency and thus low switching powerloss, and good harmonic performance, SHEPWM tech-niques have been used for series-connected half-bridgeconverters [11]. The main drawbacks of SHEPWM tech-niques are 1) implementation difficulty, i.e., requirement tosolve a set of nonlinear equations and calculate switchingangles, and 2) the need for additional memory and hard-ware to store the preprogrammed switching angles.
• The second approach is based on a transformers windingsarrangement that provides appropriate phase shift for (a setof) harmonics, based on the concept of 12-, 24-, or even48-pulse operation [12], [13]. The advantage of this ap-proach is that it can utilize very low switching frequency.Its drawbacks are 1) the need for complicated transformerswindings and 2) lack of modularity since the transformersare not identical.
• The third approach is based on carrier phase-shifted sinu-soidal PWM (SPWM) techniques [8], [14]. The main fea-ture of this method is that it does not require complicatedtransformer configurations. Its main drawbacks are that 1)it practically can be used if the SPWM frequency modu-lation index is larger than (or at least equal to) 9, and 2)the SPWM inherent per-phase switching nature results inredundant switchings that partially offset the low loss as aresult of low switching frequency.
Space vector modulation (SVM) is the preferred PWMstrategy for low- and medium-power three-phase VSC units,particularly in view of its inherent property for digital imple-mentation [15], [16]. However, the conventional SVM methodsrequire relatively high switching frequency, and thus are not thebest option for high power applications. This paper proposesand investigates the use of a low switching frequency SVMtechnique for a multimodule converter system (e.g., an HVDCconverter system [17]). Salient features of the proposed SVMstrategy are the following.
• It can operate at switching frequencies lower than that of aphase-shifted carrier SPWM.
• It can provide appropriate phase shift among the cor-responding harmonics of VSC modules of a converter
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Fig. 1. Schematic representation of a back-to-back HVDC converter station.
system. This results in harmonic cancellation/minimiza-tion through the use of identical transformers for the VSCmodules, and thus eliminates the need for complicatedtransformer windings and enhances modularity of theconverter structure. The proposed SVM guarantees max-imum possible ac-side voltage by keeping the fundamentalvoltage components of all VSC modules inphase.
This paper also develops a comprehensive mathematicalmodel for the overall multimodule-based HVDC system. Thedeveloped model provides a tool for system studies and sys-tematic controller design of the HVDC system.
The rest of the paper is organized as follows. Section II intro-duces the HVDC study system. Section III provides an overviewof the SVM-based switching strategy. Then, principles of se-quential sampling technique for SVM is explained. The tech-nique is then employed in an SVM strategy to eliminate low-order harmonics of the ac-side voltage of a four-module con-verter system. Section IV develops a fundamental-frequencymodel of the multimodule-based HVDC system. In Section V,the model is used to design controllers for the HVDC system.Performance of the HVDC system, in terms of harmonics andtransient responses, is evaluated based on digital time-domainsimulation studies in the PSCAD/EMTDC environment and re-ported in Section VI. Conclusions are stated in Section VII.
II. HVDC SYSTEM
Fig. 1 shows a schematic representation of a four-moduleback-to-back HVDC converter station. Each converter system(i.e., either VSC-1 or VSC-2), is composed of four VSC moduleswhich are connected in series at both ac- and dc-side. Each VSCmodule is shown in Fig. 2. The configuration of Fig. 1 providesa high degree of modularity since all VSC modules and the cor-responding dc-side capacitors and the ac-side transformers areidentical. In the system of Fig. 1, the corresponding VSC mod-ules of VSC-1 and VSC-2 share the same dc-side capacitor, andnodes to are common between VSC-1 and VSC-2. It shouldbe noted that the proposed SVM switching, modelling and con-clusions of the paper are equally valid, if VSC-1 and VSC-2 haveseparate dc-side capacitor arrangements. The number of VSCmodules in each converter system can be increased dependingon power and voltage ratings as required by the application.
Fig. 2. Schematic diagram of each VSC module.
To maximize dc to ac voltage transfer ratio of each four-module converter, fundamental components of terminal volt-ages of the corresponding VSC modules are generated at thesame phase-angle. However, each low-order, ac-side, voltageharmonic is appropriately phase shifted to be cancelled/elim-inated when ac-side voltages of modules are added up. Thispaper proposes an SVM-based switching strategy, based on asequential sampling technique, to provide the required phaseshifts for minimization/cancellation of low-order, ac-side har-monics [18]. In comparison with a SPWM switching strategy,the proposed SVM switching strategy: 1) generates a larger fun-damental component of ac-side voltage and 2) requires a lowerswitching frequency for each VSC module that, in turn, resultsin a lower switching loss.
III. SVM SWITCHING STRATEGY
This section briefly reviews the conventional SVM strategyand introduces the proposed sequential sampling SVM tech-nique adopted for each VSC module of Fig. 1.
A. Conventional SVM Switching Strategy
A conventional SVM is a digital modulation technique inwhich a sampled reference vector is synthesized by time-aver-aging of a number of appropriate switching state vectors. Thereference and the switching state vectors are represented in acomplex plane by a transformation from to coordinates[15], [16]
(1)
SAEEDIFARD et al.: SPACE VECTOR MODULATION APPROACH 1645
Fig. 3. Representation of the switching state vectors and the reference vectorin the �� plane.
When applied to the eight permitted switching states of the VSCmodule of Fig. 2, this transformation generates six nonzerovoltage space vectors ( ) that form a hexagon,centered at the origin of the plane, and two zero switchingvectors , located at the origin of the plane, Fig. 3. Theac-side voltage of the VSC module is synthesized by [15]
(2)
where is the switching period, is the reference voltagevector, and , and are the respective on-duration time in-tervals of switching vectors, , and , Fig. 3. The on-du-ration time intervals are calculated from [15]
(3)
(4)
(5)
where is the angle between and , Fig. 3, modulationindex is
(6)
and is the dc voltage of the VSC module.Reference voltage vector of an SVM strategy can be syn-
thesized based on different combinations of switching vectors.Thus, there exist multiple switching patterns, each with a dif-ferent switching frequency and harmonic spectrum. Similar toa SPWM switching strategy, the switching pattern of an SVMstrategy has a lower impact on the corresponding frequencyspectrum when its switching frequency is higher.
For a high-power converter (e.g., the four-module convertersystem of Fig. 1), low switching frequency is more desirablesince it reduces switching loss. Therefore, to meet the loss andharmonic spectrum targets, both the SVM switching frequencyand the switching pattern are to be simultaneously considered,as further discussed in the following sections.
Fig. 4. Schematic representation of the sequential sampling technique.
Fig. 5. Sequential sampling-based SVM for the four VSC modules in Sector I.As the reference voltage vector rotates in the�� plane, each modulator samplesat a specified instant.
B. Sequential Sampling SVM Technique
Sequential sampling SVM technique is an approach to mini-mize/reduce low-order ac-side harmonics of a multimodule con-verter system, e.g., that of Fig. 1, by means of a low switchingfrequency SVM strategy [17]. Fig. 4 shows a schematic blockdiagram for implementation of the proposed sequential sam-pling technique for the four-module converter system of Fig. 1.The SVM strategy adopts the same reference vector, , for allof the SVM modulators of each converter system, e.g., VSC-1,of Fig. 4. All of the SVM modulators use the same samplingrate; but at sampling instants that are delayed by s cor-responding to 15 electrical degrees, Fig. 5. Since is thesame for all VSC modules of VSC-1, the corresponding ac-sidefundamental-frequency voltage components are in-phase. How-ever, as a result of the sequential sampling technique, harmoniccomponents are phase-shifted. The phase shifts, as shown in thenext section, are used for harmonic cancellation at the ac-sideof each four-module converter system.
In contrast to a phase-shifted carrier SPWM technique [8],[14], the proposed sequential sampling SVM generates theswitching patterns on a three-phase basis which permits alower switching frequency. Also, since can be synthe-sized based on different combinations of switching vectors, as
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Fig. 6. Proposed space vector switching pattern in Sector I and I.
compared with a carrier SPWM technique, the proposed SVMstrategy provides higher degree of flexibility to 1) minimizethe switching frequency and 2) minimize/cancel low-frequencyharmonics.
C. SVM Switching Pattern
In a conventional SVM switching strategy, two factorsdetermine the switching frequency: 1) sampling rate and2) switching pattern. In a low switching frequency SVM whichis desirable in high-power applications, the sampling rate mustbe an integer multiple of 6 to synchronize the PWM and preventsub-harmonics. Therefore, to attain a low switching frequencybased on the proposed sequential sampling technique, eachSVM modulator samples the reference vector at the lowest pos-sible rate of once per sector (every 60 ). Since the switchingpattern also affects the switching frequency, it should 1) havethe lowest possible switching frequency to minimize switchinglosses and 2) provide an appropriate phase shift for harmonicsup to the order of the first dominant harmonic to minimize themwhen superimposed by the interface transformer. Fig. 6 showsa space vector pattern that satisfies these features [17]. As Fig. 6shows, each sampling period is divided into two intervals andthe switching pattern of the first-half period is repeated at thesecond-half period. This results in a switching frequency of
Hz for each VSC module.Based on the space vector pattern of Fig. 6, the ac-side fun-
damental components of all modules remain in-phase, however,the harmonic components have almost opposite phase-angles.Magnitudes and phase-angles of the fundamental, the fifth andthe seventh components, as functions of sampling angle , areshown in Fig. 7. Based on the sequential sampling SVM, thesampling angles of the four corresponding VSC modules are
, , 1, 2, 3. Fig. 7 shows that harmonicphases sharply change from negative to positive in the vicinity of
. With four VSC modules, the corresponding harmonicsappear at the ac-sides of modules are in almost opposite phases.Thus, superposition of low-order voltage harmonics, throughthe system sides of the transformers, Fig. 1, cancels/minimizesthe harmonics. The proposed SVM switching pattern has thefollowing salient features.
• The selected switching frequency is (or6 p.u.), since there is one sample per each sector for eachVSC module. Effective modulation frequency ratio of afour-module converter system is . Thus, the firstdominant harmonic component at the system-side of eachfour-module converter system is of order 23, as illustratedin Fig. 8.
Fig. 7. Magnitude and phase angle of harmonics versus a sampling angle form = 1 for VSC modules of the four-module converter system of Fig. 1: (a) mag-nitude and (b) phase angle.
Fig. 8. AC-side voltage spectra of the four-module converter system of Fig. 1versus modulation index m.
Based on Fig. 7 and the harmonic analysis presented inAppendix A, 1) the fundamental component of the ac-sidevoltage varies linearly with the modulation index and 2) forthe whole linear range of modulation, the first dominantharmonic is of order 23. However, as shown in Fig. 7(a),at the sampling angle of the fundamental compo-nent of the corresponding module is reduced. At samplinginstant (i.e., and for ), the on-du-ration time of zero voltage vector, based on (3) to (5), be-comes zero. This only applies when and results in
SAEEDIFARD et al.: SPACE VECTOR MODULATION APPROACH 1647
the amplitude of fundamental component to decrease andthe amplitudes of the fifth and seventh harmonics to in-crease at .
• In general, for an -module converter system that em-ploys the sequential sampling SVM technique, the effec-tive switching frequency is . The mathematicalproof is presented in Appendix A.
• For the whole range of voltage control, magnitudes ofharmonics up to order 23 are significantly minimized,although not completely cancelled out. The reason is thatin the sequential sampling SVM for which the number ofmodules is not too large, the phase-angles of the low-orderharmonics are not exactly opposite of each other and donot cancel out completely. Therefore, as shown in Fig. 8,residual low-order harmonics appear in the ac-side voltagespectra of the four-module converter system of Fig. 1.Increasing the number of VSC modules while keeping theswitching frequency constant, practically eliminates thelow-order harmonics and reduces higher-order harmonicsin the vicinity of frequency .
• In comparison with a three-level converter that has beenalso investigated for HVDC systems, the proposed four-module converter provides better harmonic performance.Details are presented in Appendix B.
IV. SYSTEM MODEL
This section presents a fundamental-frequency model for thefour-module converter system of Fig. 1 to design the convertersystem controllers and evaluate harmonics and dynamic perfor-mance of the HVDC system under the proposed SVM switchingstrategy.
First, a fundamental-frequency model of the system, in theabc frame, is developed. Then, the model is transferred to the
frame to deduce a fundamental-frequency-based modelfor the system.
A. System Model in abc Frame
Phase-a voltage equation corresponding to ac System-1 ofFig. 1 is
(7)
where is the operator, and are the interface re-sistance and inductance between the VSC-1 and ac System-1,and is the net terminal voltage of VSC-1, Fig. 1. Phase-aterminal voltage of VSC module of VSC-1, Fig. 1, is [18]
(8)
where is the switching function of the switch that connectsphase-a terminal of the VSC module to its corresponding pos-itive dc-capacitor terminal, and is the dc-capacitor voltageof module . Substituting for in terms of its Fourier seriescomponents [17] in (8)
(9)
where and are respectively the amplitude modulationindex and the angle modulation index which are the same forall VSC modules. is the angular frequency of the switchingfunctions of module , and and (in degrees), which areindependent of , are
(10)
(11)
The net terminal voltage of VSC-1, Fig. 1, is the sum of terminalvoltages of the corresponding four VSC modules, through theinterface transformers, i.e.,
(12)
Substituting for from (9) in (12) and neglecting har-monics of order 23 and higher, we deduce
(13)
It should be noted that harmonics up to order 23 are cancelled/minimized based on the proposed sequential sampling SVMtechnique. To deduce (13), it is assumed that the dc-capacitorvoltages of VSC modules are the same and equal to . Sub-stituting for from (13) in (7), and also considering phases“b” and “c,” we obtain
(14)
where , .Analogous to (14), the voltage equations of VSC-2 of Fig. 1 are
(15)
where , .and are the interface resistance and inductance between
VSC-2 and ac System-2, and , , and are, respectively,angular frequency of the switching functions and amplitude andangle modulation indices of the VSC modules.
For the dc-link circuit of VSC module of VSC-2, Fig. 1, wededuce
(16)
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where
(17)
(18)
and are the dc-link currents of module , and ,and represent switching functions of the ac sides of
VSC module . For the dc-link arrangement of Fig. 1, we have
(19)
Equations (14), (15). and (19) represent a fundamental-fre-quency model of VSC-1 of Fig. 1 in the abc frame.
B. Converter Model in dq Frame
Since the dynamic model of an electrical power system istraditionally developed in the frame, it is desirable to obtainthe model of the system of Fig. 1 in the dq frame.
To transfer the variables of ac System-1 to a frame, atransformation matrix is selected such that the and currentcomponents of ac System-1 are proportional to its real and reac-tive power components, respectively. Thus, the control of eachcurrent component regulates the corresponding power compo-nent. The ac System-1 variables are transferred to the frameby [20]
(20)
where transformation matrix is
(21)
(22)
is the frequency of ac System-1, and is the phase angleof , Fig. 1. Transforming the variables of ac System-1, asgiven by (14), based on (20), we deduce
(23)
where is equal to the amplitude of . Analogous to (23),for the ac System-2 of Fig. 1, we can deduce
(24)
In (24), and are the amplitude and angle of .
Fig. 9. Control systems of VSC-1 and VSC-2 of Fig. 1.
Fig. 10. Schematic representation of the system of Fig. 1 including power andcontrol subsystems.
If switching functions , and are represented bythe corresponding fundamental components as
(25)
by substituting for from (25) in (17), we deduce
(26)
In (26), and are the and current components of theac System-1 current. Analogous to (26), for the VSC-2 dc-linkcurrent of module , we obtain
(27)
where and are the and current components of theac System-2 current. Equations (19), (23), and (24) represent afundamental-frequency model of the HVDC system of Fig. 1.
SAEEDIFARD et al.: SPACE VECTOR MODULATION APPROACH 1649
Fig. 11. Steady-state current and voltage waveforms of the four-module converter system of Fig. 1 : (a,b) line voltage of VSC-1 and its spectrum. (c,d) Line voltageof one VSC module of VSC-1 and its spectrum. (e,f) AC System-1 current and its spectrum. (g) Net dc-capacitor terminal voltage. (h) Per-module dc-capacitorvoltage.
V. CONVERTER CONTROL
The control parameters of each VSC module of Fig. 1 are themagnitude and angle modulation indices. The and compo-nents of terminal voltage of VSC-1 based on (23) are
(28)
(29)
where
(30)
(31)
Similarly, for VSC-2, from (24), we deduce
(32)
(33)
where
(34)
(35)
Based on (28), (29), (32), and (33), block diagrams of VSC-1and VSC-2 control systems are deduced as shown in Fig. 9.Inputs are the and current components of ac System-1 andac System-2. Outputs of the control systems are the magnitudeand angle modulation indices of VSC-1 and VSC-2.
The control system of Fig. 9 consists of two subsystems,where each one is a two-input two-output controller and eachcan be substituted with two single-input single-output (SISO)controllers [18] through intermediate variables [18], [20]. ,
, and are selected as intermediate variables of thecontrol systems of Fig. 9. Relationships between the interme-diate variables and the system variables to be controlled (i.e.,
, , , and are given by (30), (31), (34), and (35). Theserelationships are used to deduce SISO controllers from the con-trol systems of Fig. 9 [21], and design the controllers based onMATLAB SISO tools.
VI. STUDY RESULTS
This section evaluates performance of the HVDC systemof Fig. 1 when operating under the control system of Fig. 9.Fig. 10 shows a schematic block representation of the powersystem and the controls of Figs. 1 and 9. The reported studiesin this section are carried out based on time-domain simulationin the PSCAD/EMTDC environment. The system parametersare given in Appendix C.
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Fig. 12. Dynamic response of the system of Fig. 10 to a step change in the dcvoltage reference: (a) DC-link voltage. (b,c) Real and reactive components ofac System-1 currents.
A. Harmonics
Initially, the system operates under a steady-state conditionat the rated load. Since both VSC-1 and VSC-2 operate basedon the same sequential sampling technique and similar voltageand current values, only the waveforms corresponding toVSC-1 are presented. Fig. 11(a) and (c) shows instantaneoushigh-voltage side voltages of VSC-1 and one of its convertermodules. A comparison of the corresponding harmonic spectra[i.e., Fig. 11(b) and (d)] shows that all harmonic components ofthe VSC-1 and, in particular, the low-frequency components,are effectively minimized by the proposed SVM strategy. Thefirst dominant harmonic of four-module converter system isof order of 23, as shown in Fig. 11(b). However, as explainedin Section III-C, due to imperfections, residual low-orderharmonics do exist in the vicinity of the dominant harmonicof order 23. Fig. 11(e) shows line current of phase ’a’ of acSystem-1, and Fig. 11(f) shows the corresponding currentharmonics. Fig. 11(f) indicates that even without any filter,harmonic content of the current is well within the acceptablelimits [22]. The low-order harmonic components are almostcancelled out and negligible. Fig. 11(g) and (h) shows thenet and per module dc-voltage waveforms. Fig. 11(g) and (h)indicates that dc-voltage ripple is within the limits.
B. Reference Tracking
DC voltage control: Fig. 12 shows the system response to astep change in the net dc-voltage reference from 83% to 100%rated value at . Fig. 12(a) shows that the controlsystem of Fig. 9 effectively regulates the dc voltage at the newreference value. Fig. 12(b) and (c) shows the dynamics of realand reactive current components of ac System-1 in response tothe change in the dc-voltage reference signal.
Real/reactive power control: The system is initially under asteady-state operating condition. Reactive and real power de-mands of ac System-2 change from 0.6 to p.u. at
, and from 0 to 0.6 p.u. at , respectively. Fig. 13shows dynamic response of the system to the step changes inpower demand. Fig. 13(a) and (b) shows step changes in ref-erence signals and corresponding to a power demandchange in ac System-2, and the corresponding changes inand imposed by the controllers. Fig. 13(a) and (b) showsthat and faithfully follow the corresponding references.
Fig. 13(c) and (d) shows real and reactive power exchangedwith ac System-2. Fig. 13(a) and (b) demonstrates that the - and-axis current components of the ac System-2 are well decou-
pled. However, due to the weakness of both ac Systems and theleakage inductances of the transformers, real and reactive powercomponents are not fully decoupled and, thus, glitches are vis-ible during transients. Fig. 13(e) shows the net dc-link voltageresponse. Fig. 13(e) demonstrates that the net dc-link voltage iswell regulated subsequent to the disturbances.
Fig. 13(f) and (g) shows changes in amplitude and angle mod-ulation indices of VSC-2 to meet the power demand in responseto changes in and . Fig. 13 shows that the control systemproperly tracks the specified signals. Fig. 13 also illustrates thatthe control system effectively regulates the system operatingconditions in response to step changes in real and reactive powerdemands.
C. Disturbance Rejection Capability
The system is initially under a steady-state operating condi-tion when subjected to a single-phase to ground fault, with thefault resistance of , at the middle of the line in acSystem-2. The fault occurs at and is self-cleared at
. Fig. 14 shows transient behavior of the system duringand subsequent to the fault. Fig. 14(a) shows current waveformof the faulty line of ac System-2. Fig. 14(b) shows that the netdc voltage is maintained at the corresponding reference duringand subsequent to the fault. Fig. 14(c) and (d) shows that realand reactive current components of ac System-2 contain doublefrequency oscillations due to the unbalanced fault condition.Fig. 14(e) and (f) shows real and reactive current componentsof ac System-1, and illustrate that VSC-1 reacts to maintain thesystem power balanced in response to the fault. The study re-sults of Fig. 14 illustrate disturbance rejection capability of thecontrol system. The control system effectively maintains the op-erating conditions of the system during and subsequent to thefault.
VII. CONCLUSION
This paper proposes and investigates the use of an SVM-based switching strategy, based on a sequential sampling tech-nique for a back-to-back four-module HVDC converter system.The proposed technique minimizes each low-order, ac-side har-monic by introducing an appropriate phase shift for the corre-sponding harmonic components of each module. A mathemat-ical model is also developed for the HVDC system to designand to evaluate performance of the system based on the pro-posed switching strategy. Time-domain simulation results of
SAEEDIFARD et al.: SPACE VECTOR MODULATION APPROACH 1651
Fig. 13. Dynamic response of the system of Fig. 10 to step changes in the ac System-2 real and reactive power demands: (a,b) Real and reactive components ofac System-2 currents. (c,d) Real and reactive power components of ac System-2. (e) Net dc-link voltage. (f,g) Amplitude and angle modulation indices of VSC-2.
the system, in the PSCAD/EMTDC environment conclude thefollowing.
• While the switching frequency of each VSC module is keptas low as 6 p.u. (360 Hz), the effective switching frequencyof the converter in terms of harmonics is equivalent to
, and the first dominant harmonic is of order 23.• All harmonic components of the converter ac-side, and
in particular low-order harmonics up to order of 23, areeffectively cancelled/minimized by the proposed sequen-tial sampling SVM technique. Furthermore, For the wholerange of linear modulation, the magnitude of each low-order harmonic does not exceed 3%.
• Current harmonic content is well within acceptable limits,even without the use of filters. Harmonics up to order 23are effectively cancelled out.
• The HVDC control based on the proposed strategy can en-sure proper reference tracking, and disturbance rejectioncapability.
APPENDIX AHARMONIC ANALYSIS OF AN -MODULE CONVERTER BASED
ON A SEQUENTIAL SAMPLING SVM TECHNIQUE
This section develops an analytical model to determine har-monic components of a -module converter that employs se-quential sampling SVM technique.
The analysis utilizes the existing relationship betweenSPWM and SVM techniques [23]. This relationship es-tablishes a common mathematical basis between the twotechniques. Adding an offset signal to the three-phase refer-ences of an arbitrary SPWM modulator, it can be transformedinto an equivalent SVM modulator [23]. For an SVM modu-lator, distribution of zero voltage vectors within a samplinginterval plays the same role as an offset signal added to aSPWM modulator [10], [23]. The offset signal for the SPWMmodulator depends on the distribution of zero voltage vectorswithin a sampling interval.
Analysis of One VSC Module: For the SVM pattern usedin this paper and with respect to the distribution of zerovoltage vectors in Fig. 6, the equivalent reference waveformof phase-a SPWM modulator of one VSC module within oneperiod is [23]
(36)
The mathematical model of SVM modulator can be de-rived based on its corresponding SPWM equivalent of (36).
1652 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
Fig. 14. Transient response of the system of Fig. 10 to a single-phase to groundfault: (a) ac System-2 phase-a current. (b) DC-link voltage. (c,d) Real and reac-tive components of ac System-2 currents. (e,f) Real and reactive components ofac System-1 currents.
According to phase-a waveform function, the modulated wave-form is constructed based on a double Fourier series [24]. Themodulation function of phase-a is shown in Fig. 15, where
, and . and are the carrier andfundamental frequencies, respectively. The modulated phase-a
Fig. 15. Modulation function of SVM-based VSC module.
waveform, i.e., modulation function, is defined by ,where is equal to in the shaded region of Fig. 15 and zeroout of the region. The boundary is expressed by
(37)
The complex Fourier harmonic component form can be devel-oped for the double variable controlled waveform as
(38)
where
(39)
For the line voltage, we deduce
(40)
can be expressed in the real form as
(41)
SAEEDIFARD et al.: SPACE VECTOR MODULATION APPROACH 1653
where in terms of the Bessel function is expressed as
(42)
Analysis of -Module Converter: In a -module con-verter that utilizes the sequential sampling technique, the corre-sponding sampling instants of contiguous VSC modules shouldbe delayed by
(43)
This delay corresponds to shifting the carrier phase by in aphase-shifted carrier SPWM technique. Based on (40), phase-avoltage of VSC module is
(44)
For an -module converter, the phase-a voltage is
(45)
and the line voltage is
otherwise(46)
Fig. 16. AC-side voltage spectra of an NPC converter.
where is an integer. Considering (41), (46) is simplified to
(47)
as shown in (48), at the bottom of the page. Based on (47) and(48), we conclude the following.
• The net amplitude of the fundamental component of themultimodule converter is times that of one VSC module.
• The lowest side-band harmonic is around , i.e., theequivalent switching frequency is .
APPENDIX BHARMONIC SPECTRA OF A THREE-LEVEL CONVERTER
Fig. 16 shows the ac-side voltage of a HVDC converter whichis composed of a neutral point diode clamped (NPC) converter[25]. The NPC converter utilizes an SVM switching strategywith the switching frequency of The NPC con-verter ratings are the same as the four-module converter unit ofFig. 1. A comparison of the corresponding harmonics of Figs. 16and 8 concludes the following.
• The four-module converter offers a superior performancesince the amplitudes of low-order harmonics in the vicinityof frequency are much smaller that those of theNPC converter.
• In both cases, the first dominant harmonic is of order 23.However, the number of on-off state of switches of the NPCconverter is half of the switching frequency, i.e., 12 p.u.,while for the four-module converter system is 6 p.u.
APPENDIX CSYSTEM PARAMETERS
Please see Tables I and II.
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1654 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
TABLE IPARAMETERS OF THE SYSTEM OF FIG. 1
TABLE IIPARAMETERS OF PI CONTROLLERS OF FIG. 9
REFERENCES
[1] Y. H. Liu, J. Arrillaga, and N. R. Watson, “A new high-pulse voltage-sourced converter for HVDC transmission,” IEEE Trans. Power Del.,vol. 18, no. 4, pp. 1388–1393, Oct. 2003.
[2] X.-P. Zhang, “Multiterminal voltage-sourced converter-based HVDCmodels for power flow analysis,” IEEE Trans. Power Syst., vol. 19, no.4, pp. 1877–1884, Nov. 2004.
[3] Z. Huang, B. T. Ooi, L. A. Dessaint, and F. D. Galiana, “Exploitingvoltage support of voltage-source HVDC,” Proc. Inst. Elect. Eng.,Gen., Transm. Distrib., vol. 150, no. 2, pp. 252–256, Mar. 2003.
[4] B. Anderson and C. Barker, “A new era in HVDC?,” Inst. Elect. Eng.Rev., vol. 46, no. 2, pp. 33–39, Mar. 2000.
[5] N. Hingorani and L. Gyugyi, Understanding FACTS. Piscataway, NJ:IEEE, 2000.
[6] B. R. Andersen, L. Xu, P. J. Horton, and P. Cartwright, “Topologiesfor VSC transmission,” Power Eng. J., vol. 16, no. 3, pp. 142–150, Jun.2002.
[7] J. S. Lai and F. Z. Peng, “Multilevel converters-A new breed of powerconverters,” IEEE Trans. Ind. Appl., vol. 32, no. 3, pp. 509–517, May/Jun. 1996.
[8] J. Kuang and B. T. Ooi, “Series connected voltage-source convertermodules for force-commutated SVC and DC-transmission,” IEEETrans. Power Del., vol. 9, no. 2, pp. 977–983, Apr. 1994.
[9] F. C. Zach, R. J. Berthold, and K. H. Kaiser, “General purpose mi-croprocessor modulator for a wide range of PWM techniques for ACmotor control,” in Proc. IEEE Ind. Appl. Soc., Oct. 1982, pp. 446–451.
[10] D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Con-verters. Piscataway, NJ: IEEE Press, 2003.
[11] L. Czarkowski, L. Yaguang, and P. Pillay, “Multilevel selectiveharmonic elimination PWM technique in series-connected voltageinverters,” IEEE Trans. Ind. Appl., vol. 36, no. 1, pp. 160–170, Jan.2000.
[12] C. Schauder, M. Gernhardt, E. Stacey, T. Lemak, L. Gyugyi, T. W.Cease, and A. Edris, “Operation of 100 MVAr TVA STATCON,” IEEETrans. Power Del., vol. 12, no. 4, pp. 1805–1811, Oct. 1997.
[13] A. R. Bakhshai, G. Joos, and H. Jin, “EMTP simulation of multi-pulseunified power flow controllers,” in Proc. IEEE CCECE, May 1996, vol.2, pp. 847–850.
[14] B. Mwinyiwiwa, Z. Wolanski, and B. T. Ooi, “Microprocessor-imple-mented SPWM for multiconverters with phase-shifted triangle carriers,”IEEE Trans. Ind. Appl., vol. 34, no. 3, pp. 487–494, May/Jun. 1998.
[15] H. W. Vander Brocek, H. C. Skudelny, and G. Stank, “Analysis andrealization of a pulse width modulator based on voltage space vectors,”in Proc. IEEE Ind. Appl. Soc., Jan. 1992, vol. 1, pp. 12–17.
[16] J. Holtz, “Pulsewidth modulation-A survey,” in Proc. IEEE PowerElectron. Specialists Conf., Apr. 1994, vol. 1, pp. 11–18.
[17] M. Saeedifard, A. Bakhshai, and G. Joos, “Low switching frequencyspace vector modulators for high powermulti-module converters,” IEEETrans. Power Electron., vol. 20, no. 6, pp. 1310–1318, Nov. 2005.
[18] H. Nikkhajoei and R. Iravani, “Electromagnetic transients of a micro-turbine based distributed generation system,” in Proc. Int. Conf. PowerSystems Transients, Montreal, QC, Canada, Jun. 2005, IPST05-070-8b.
[19] P. C. Krause, Analysis of Electric Machinery. New York: McGraw-Hill, 1986.
[20] J. M. Maciejowski, Multivariable Feedback Design. Reading, MA:Addison-Wesley, 1989.
[21] H. Nikkhajoei, A. Tabesh, and R. Iravani, “Dynamic model of a matrixconverter for controller design and system studies,” IEEE Trans. PowerDel., vol. 21, no. 2, pp. 744–754, Apr. 2006.
[22] IEEE Recommended Practices and Requirements for Harmonic Con-trol in Electrical Power Systems, , Apr. 1993, IEEE Std. 519.
[23] K. Zhou and D. Wang, “Relationship between space-vector modula-tion and three-phase carrier-based PWM: A comprehensive analysis[three-phase inverters],” IEEE Trans. Ind. Electron., vol. 49, no. 1, pp.186–196, Feb. 2002.
[24] S. R. Bowes and B. M. Bird, “Novel approach to the analysis and syn-thesis of modulation processes in power converters,” Proc. Inst. Elect.Eng. , vol. 122, no. 5, pp. 507–513, May 1975.
[25] A. Yazdani and R. Iravani, “Dynamic model and control of the NPC-based back-to-back HVDC system,” IEEE Trans. Power Del., vol. 21,no. 1, pp. 414–424, Jan. 2006.
Maryam Saeedifard (S’06) received the B.Sc. andM.Sc. degrees in electrical engineering from IsfahanUniversity of Technology, Isfahan, Iran, in 1998 and2002, respectively. She is currently pursuing thePh.D. degree in the Department of Electrical andComputer Engineering at the University of Toronto,Toronto, ON, Canada.
Her research interests include power electronics,application of power electronics in power systems,and control systems.
Hassan Nikkhajoei (M’05) received the B.Sc. andM.Sc. degrees from Isfahan University of Tech-nology, Isfahan, Iran, in 1992 and 1995, respectively,and the Ph.D. degree from the University of Toronto,Toronto, ON, Canada in 2004, all in electricalengineering.
He was a Postdoctoral Fellow in the University ofToronto from 2004 to 2005, and a Faculty Memberwith Isfahan University of Technology from 1995 to1997. Currently, he is a Research Associate in theDepartment of Electrical and Computer Engineering,
University of Wisconsin-Madison. His research interests include power elec-tronics, distributed generation systems, and electric machinery.
Reza Iravani (F’03) received the B.Sc. degree fromTehran Polytechnic University, Tehran, Iran, in 1976,and the M.Sc. and Ph.D. degrees from the Universityof Manitoba, Winnipeg, MB, Canada, in 1981 and1985, respectively, all in electrical engineering.
Currently, he is a Professor at the University ofToronto, Toronto, ON, Canada. His research inter-ests include power electronics and power system dy-namics and control.
Alireza Bakhshai (M’04) received the B.Sc. andM.Sc. degrees from the Isfahan University of Tech-nology, Isfahan, Iran, in 1984 and 1986, respectively,and the Ph.D. degree from Concordia University,Montreal, QC, Canada, in 1997.
From 1986 to 1993 and from 1998 to 2004, hewas on the faculty of the Department of Electricaland Computer Engineering, Isfahan University ofTechnology. He was a Postdoctoral Fellow from1997 to 1998 at Concordia University. Currently, heis with the Department of Electrical and Computer
Engineering, Queen’s University, Kingston, ON. His research interests includehigh-power electronics and applications, control systems, and flexible actransmission services (FACTS).
