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General Analysis of Harmonic Transfer Through Converters ˚ Ying Jiang, Member, IEEE and Ake Ekstrom, Ekstr¨ o¨ m, Fellow Member, IEEE

Abstract—Harmonic transfer from dc-side to ac-side and viceversa through a three-phase bridge converter is treated by using space spa ce vec vector tors. s. The depen dependen dency cy of the re relat lative ive fr frequ equenc ency y of a harmonic and its sequence (positive/negative) is illustrated. It is shown that the derived relationships are valid for both voltagesource and current-source converters. Thereby, the relationships for the freq frequency uency transform transformation ation through conve converter rterss are also valid vali d for both voltag voltagee and current harmoni harmonics. cs. The analyt analytical ical method is validated by digital simulations with EMTP.

transferred transfer red thro through ugh converter converters. s. The deri derived ved rule ruless are valid valid for any three-phase converters with symmetrical modulation. Although Alth ough the derivati derivation on is based on thre three-pha e-phase se conve converter rterss with fundamental switching frequency, the result is also valid for converters with higher switching frequency. Examples are given for two purposes, i.e., to verify the derived rules and to show how they are used. The validation of the rules can also be veriﬁed by the studied cases presented in [4]–[6], and [8].

Current-source converter, harmonic transfer, Index Terms— Current-source switching function, voltage-source converter.

II. SPACE VECTOR I. INTRODUCTION

I

N HIGH-POWER electronics, harmonics generated by converters vert ers are often often grouped grouped into two categories categories:: chara character cter--

istic harmonics and noncharacteristic harmonics [1]–[3]. For the calculati calculation on of character characteristi isticc harm harmonic onics, s, analytica analyticall and numerical numer ical meth methods ods are well established established [1]–[3]. [1]–[3]. Regar Regarding ding noncharac nonch aracteri teristic stic harmoni harmonics, cs, a number number of papers papers have been published [4]–[8]. However, the treatments in the published works are either limited to a certain frequency [5], [6], or too complicated [4], [7], [8] to draw a simple rule that can be used to judge or predict which frequencies will be resulted on one side of a converter when there is a harmonic on the other side. It has, for instant, been stated that frequencies that are present on the ac-side of a converter will be transferred to the dc-side as side-bands and vice-versa [9]. It is easy to prove that this statemen statementt about harmon harmonic ic transfer transferring ring is not quite quite true. A well-known example is that a balanced fundamental component on the ac-side does not necessarily give a second harmonic on the dc-side besides the dc component. This paper presents a general analysis of harmonic transfer thr throug ough h conver converter ters. s. Based Based on analyz analyzing ing basic basic curren currentt and voltage voltage equations equations of a three three-phas -phasee volt voltage-s age-source ource and current-source converter, simple rules are formulated by using switch swi tching ing functi functions ons and the spa space ce vector vector [10]. [10]. The rul rules es te tell ll whic which h fr freq eque uenc ncie iess wi will ll be cr crea eate ted d on th thee ac ac-s -sid idee of a conv conver erte terr when when ther theree is a ha harm rmon onic ic on th thee dc dc-s -sid idee and and vice-versa. The rules are valid for both current and voltage harmonics harm onics.. Dif Differi fering ng from previou previouss works works,, the paper paper give givess not only harmonic amplitude and frequency relationship when a current/voltage harmonic is transferred from the dc-side to ac-side of a converter, or vice-versa, but also the frequencyand sequence-dependent relationship when the harmonics are Manuscript received May 8, 1995; revised July 30, 1996. The authors are with the Department of Electric Power Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden. Publisher Item Identiﬁer S 0885-8993(97)01852-8.

Three-phase variables, , can be transfor orme med d into nto an or orth thog ogo ona nall two-p wo-pha hasse varia ariabl blees, , without losing any information of that three phase quantities, if no zero-sequence component exists in thefor phase components. This condition is automatically fulﬁlled voltages/currents transfer tran sferred red through through a three-pha three-phase se tran transfor sformer mer for which the winding on at least one side is not grounded. Representing the two ortho rthog gon onaal compo ompone nen nts , respe especctive tively ly by the the real and imaginary components leads to a vector that is often cal called led space space vec vector tor [10]. [10]. The space space vec vector tor is calcul calculate ated d as follows:

(1)

where (2) It should be noted that the transformation given in (1) results in equa equall ampl amplit itud udee fo forr the the spac spacee vect vector or and and the the thre threeephasee quantiti phas quantities. es. In tthe he fol followin lowing g deri derivati vation on the subscript subscript “ ” in the space vector notation will be dropped out in order to make the notation simpler. One advantage of using space vector is that it reduces a three-phase system to a two-phase system. The other advantage of the space vector is that it gives the sequence information of sinusoidal three-phase variables simply by its exponent, as a space vector for different sequence will rotate in different directio dire ction. n. For instance instance,, a three three-phas -phasee syst system em quant quantitie itiess as expressed in (3a) have the space vector form as given in (3b). The positive sign “+” in the exponent in (3b) applies for the positive posi tive sequenc sequencee comp component onent in (3a), whereas the negat negative ive

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sign “ ” applies applies for the negative negative sequence sequence component component

(3a)

(3b) Inversely, the instantaneous values of the three-phase quantities can also be obtained from the corresponding space vector as foll follows: ows: Re

(4a)

Re

(4b)

Re

(4c)

(a)

where “ ” means complex conjugate. conjugate. II III. I. HARMONIC TRANSFERRED BY VOLTAGE SOURCE CONVERTER The derivation of the harmonic transfer is ﬁrst made for a six-pulse bridge forced-commutated voltage-source converter with fundamental switching frequency. Fig. 1(a) and (b) show the basic electric circuit conﬁguration and its equivalent circuit, cuit, respec respectiv tively ely.. The operat operation ion of the conver converter ter can be illustrated by three binary signals a nd as shown shown in Fig. Fig. 1(c), 1(c), which which are the soso-cal called led swi switch tching ing functions. A logic “1” means that the corresponding ac phase is connected to the positive terminal of the dc-side. A logic “0” means that the ac phase is connected to the zero terminal of the dc-side. It is assumed that the switching occur at equal time intervals, i.e., at constant delay (or advance) angle measured from the zero crossi crossing ng of the respecti respective ve ac bus phase voltage. The switching functions can be expressed in Fourier series as follows foll ows [1]:

(b)

(5a)

(c) Fig. 1. Six-pulse Six-pulse forc forced-co ed-commuta mmutated ted voltag voltage-sou e-source rce conv converter: erter: (a) basic electric circuit, electric circuit, (b) equivalen equivalentt circui circuit, t, and (c) switch switching ing functions functions..

where

(5c)

and etc., i.e., etc. etc. Th Thee po posi siti tive ve sign sign “+ “+”” in the the expo expone nent nt applies for and the negative sign “ ” for . For simplicity, in the following the fundamental compon com ponent ent of will will be con consid sidere ered d as an app approx roxim imati ation on of . Therefore, the switching function can be approximately expressed as (7). The inﬂuence of the approximation on the result will be discussed later:

According to (1), the space vector of the switching function will be as follows:

(7)

(5b)

(6)

The basic basic relation relationships ships for the volt voltageage-sourc sourcee conve converter rter are de deri rive ved d fr from om the the fact fact that that the the conv conver erte terr ac vo volt ltag ages es are are

¨ M: GENERAL ANALYSIS OF HARMONIC TRANSFER THROUGH CONVERTERS JIANG AND EKSTRO

proportional to the dc-side voltage, which gives (8)

289

B. Harmonic Tran Transferred sferred from AC-Side to DC-Side

(8)

Neglecting the losses in the converter bridges, the requirement of active power balance on both the dc-side and ac-side of a converter yields

(9)

According to the principal of superimposition, it is possible to consi consider der each harmonic harmonic individuall individually. y. A simple simple case that is a harmonic containing only a positive or only a negative sequence seque nce component component is consider considered ed ﬁrst. Assumin Assuming g that the angular angul ar frequenc frequency y of the harm harmonic onic is , the the sp space ace vec vector tor of that harmonic will be as follows: (16) Inserting (7) and (16) into (12) gives the corresponding harmonic on the dc side, as expressed in Re

In Inse sert rtin ing g (8 (8)) into into (9 (9)) and and elim elimin inat atin ing g the equation gives

on bo both th si side dess of

(10)

Mult Mu ltip iply lyin ing g (8) by

(17)

yiel yields ds (11) (11),, while while app apply lyin ing g (4) for for in (10) and transfor transforming ming the switching switching

functions into space vector, results in (11) Re

(12)

Equation (17) shows that a positive Equation positive sequence harmo harmonic nic on the ac-side transferred to the dc-side gives only one of the sid idee-b -ban and ds, whi which has has a lowe lowerr fr freq equ uen enccy, i.e. i.e.,, ( ), and that a negative sequence harmonic transferred to the dcside also gives only one of the side-bands, but it has a higher fr freq eque uenc ncy, y, i.e. i.e.,, ( ). A we well ll-k -kno nown wn exam exampl plee is that that the the seventh characteristic harmonic, which is of positive sequence, gives the sixth harmonic on the dc-side, whereas the negative sequence ﬁfth harmonic also gives the sixth harmonic on the dc-side. A more general case is an unbalanced harmonic with both positive and negative sequence components. The space vector form for that harmonic will be as follows:

A. Harmonics Trans Transferred ferred ffrom rom DC-Side to AC-Side

Assuming that the dc-component of the dc-side voltage is superimposed with a voltage component having an oscillation angula ang ularr fr frequ equenc ency y , the dc-sid dc-sidee voltage voltage can be wri writte tten n as follows: (13)

(18) Inserting (7) and (18) into (12) gives Re (19)

Equation (13) can also be written as

Equation (15) indicates that a harmonic on the dc-side will

The above equation shows that an asymmetrical harmonic on the ac-side transferred transferred to the dc-side gives two side-band side-bands. s. The current/voltage amplitudes of the two side-bands may be dif differe ferent nt depending depending on the amplitu amplitudes des of both the posit positive ive and negative sequence sequence comp component onents. s. Of the two side-band side-bandss on the dc-side, the current amplitude of the lower frequency is proportio proportional nal to that of the positive sequence sequence component, component, wherea whe reass the cur curren rentt amp amplit litude ude of the hig higher her fre freque quency ncy is proportional to that of the negative sequence component. A practical case might, for instance, be an unbalanced second harmonic harm onic generated generated by a nearb nearby y saturate saturated d trans transform former. er. The positive sequence component in this case will be more critical as it gives a fundamental component on the dc-side, which

be transferred to the ac-side as two side-bands: the harmonic with higher frequency is a positive sequence component and the harmo harmonic nic with with low lower er fre freque quency ncy is a negati negative ve sequen sequence ce component as ( ) is less less tha than n zer zero o as long long as is gre greate aterr than . The two side-bands have the same voltage voltage amplitude. amplitude.

in tur turn n gives gives a sec second ond harmon harmonic ic and the dcdc-com compon ponent ent on the ac-side. The dc-c dc-compon omponent ent may saturate the conv converte erterr transformer, which will further result in additional second harmonic. The procedure aggravates the harmonics continuously until it gets harmonic instability [5].

(14) Substitut Substi tuting ing (7) and (14) (14) in (11 (11)) wil willl giv givee the followin following g equation:

(15)

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(23)

In Inse sert rtin ing g (2 (22) 2) into into (2 (23) 3) and and elim elimin inat atin ing g of the equation yields

on bo both th side sidess

(24) (a)

Mult Mu ltip iply lyin ing g (22 (22)) by

give givess (25), (25), whi while le appl applyi ying ng (4 (4)) for in (24) and transfor transforming ming the switching switching functions into space vector, results in (25) Re

(b) Fig. 2. Six-pulse Six-pulse line-commuta line-commutated ted current-sou current-source rce converter: converter: (a) electric circuit diagram and (b) switching functions.

IV. HARMONIC TRANSFERRED BY CURRENT SOURCE CONVERTERS Fig. 2(a) illustrates the basic electric circuit for a currentsource sou rce con conver verter ter,, which which is often often line-c line-comm ommuta utated ted in high high power applications. Assuming that the overlap is zero, a set of switching functions can be obtained in the same way as for the volt voltageage-sourc sourcee converter converter,, which are shown in Fig. 2( 2(b). b). Maki Ma king ng Fo Four urie ierr an anal alys ysis is on ea each ch sw swit itch chin ing g fu func ncti tion on ( , and and ) and and app apply lyin ing g (1) (1) yiel yields ds (20) where

a nd etc., i.e., etc. The positive sign in the exponent applies for and the negative sign for . Furthermore, if a similar approximation is made for (20) as made for (6), we obtain

(26)

Obviously, (25) and (26), respectively, are the dual form of (11) and (12). If a harmonic current on the dc-side is expressed in the complex complex exponent exponent form form,, and a harmonic harmonic voltage voltage on the ac-side is described as a space vector, the same conclusion can be obtained for the harmonic transfer though current-source converters, i.e., a current harmonic on the dc-side transferred to the ac-side gives two side-bands with equal current amplitudes. One of the side-band has a positive sequence and the other has a negative sequence. On the other hand, a harmonic voltage on the ac-side transferred to the dc-side gives one side-band of higher frequency if it is of negative sequence, one side-band of lower frequency if it is of positive sequence and two sidebands if it is an unbalanced harmonic (the voltage amplitudes of the two side-bands depend on the amplitudes of positive and negative sequence components, respectively). V. HARMONIC TRANSFER RULES Summarizing above results for the voltage-source converter as well as for the current-source converter leads to simple rules regarding voltage/current harmonic transfer through converters. rules are illustrated Fig. 3. Itinshould be noticed that These a single harmonic, which isbybalanced three phases, on the ac-side generates only one of the side-bands on the dc-side and not both side-bands as sometimes stated in the literatures [9]. It is important to notice this relationship. Otherwise, it is easy to lead to the wrong conclusion that converters not only transfer harmonics from the ac-side to dc-side, and vice-versa, but also proliferate the number of harmonics.

(21) The basic basic relation relationships ships for the curre current-s nt-source ource convert converter er are derive der ived d fro from m the fac factt tha thatt the thr threeee-pha phase se ac curren currents ts are propor pro portio tional nal to the dc-sid dc-sidee curren current, t, which which gives gives (22). (22). The active power balance on the ac-side and dc-side of the converterr results verte results in (23), (23), if the loss losses es in the converter converter bridges are neglected (22)

VI. VI. EXAMPLES In order to illustrate the presented harmonic transfer rules, two exa examp mples les are prese presente nted d in the fol follow lowing ing.. In the ﬁrs ﬁrstt example, the driving harmonic source is on the dc-side of the converter, while in the second example the driving harmonic is on the ac-side. Considering that 12-pulse converters are mostly used in high-power applications, the EMTP simulation is based on a 12-pulse 12-pulse forc forced-c ed-commu ommutate tated d volt voltageage-sourc sourcee converte converter, r, which is supposed to be used as a static var compensator [11], [12]. The fundamental frequency of the ac system is 60 Hz.

¨ M: GENERAL ANALYSIS OF HARMONIC TRANSFER THROUGH CONVERTERS JIANG AND EKSTRO

Fig. 3.

291

Harmonic Harmonic transfer transfer rule. Sign “+” denotes pos positive itive sequence; sequence; sign “ ” denotes negative sequen sequence. ce.

A. Example One

A dominant harmonic with frequency around 149 Hz on the dc-side is excited during a transient, due to the low damping on the system interaction resonance and high gain over the controll cont roller er [13], [13], [14]. Fig. 4( 4(a) a) and (b) shows the measured measured dc-side voltage and its harmonic spectrum respectively. The dc-side harmonic voltage transferred to the ac-side gives two side-bands, which are 89 and 209 Hz, as can be seen from the harmonic spectrum of the internal output phase voltage of the converter shown in Fig. 4(c). Checking closely, it is found that the ratio between the amplitudes of 149-Hz voltage harmonic on the dc-side and that of 209 (or 89) Hz on the ac-side is approximately equal to 1.27, which corresponds to a factor of for a 12-pulse converter and is in good agreement with the analytical analytical result. The othe otherr two obvious harmonic harmonicss are 11th (660 Hz) and 13th (780 Hz) harmonics, which belong to characte char acterist ristic ic harm harmonic onics. s. If the appr approxim oximatio ation n had not been made made on (6), (6), there there should should be harmon harmonics ics wit with h fre freque quenci ncies es 660 149 149 Hz Hz,, 780 780 149 149 Hz on the the ac ac-s -sid ide. e. Acc Accor ordi ding ng to Fig. 4(c), these harmonic components are very small and negligible in this case. This shows that the approximation made on the switching function is reasonable. Furthe Fur therm rmore ore,, accord according ing to (14 (14)) or Fig. Fig. 3 the harmon harmonic ic with wit h fre freque quency ncy of 89 Hz is of neg negati ative ve sequen sequence ce and the harmonic with 209 Hz is of positive sequence. Both harmonics transferred back to the dc-side will give the same frequency on the dc-side, i.e., 149 Hz, which is the same as the original dri drivin ving g har harmon monic. ic. Again, Again, Fig. Fig. 4( 4(b) b) con conﬁrm ﬁrmss the pro propos posed ed harmonic transfer rules.

B. Example Two

In this example, a second harmonic current of 10% with negative negati ve seq sequen uence ce is inject injected ed int into o the ac bus bus.. Acc Accord ording ing to Fig. Fig. 3 the second second harmon harmonic ic with with neg negati ative ve sequen sequence ce will will give a third give third har harmo monic nic on the dc-sid dc-side. e. Fig. Fig. 5( 5(a) a) shows shows the harmonic spectrum of the dc-side voltage, the third harmonic can be observed on it as expected. The third harmonic will givee the second giv second (posit (positive ive sequen sequence) ce) and fourth fourth (negat (negative ive sequence) sequ ence) harmon harmonics ics on the ac-side, this is also conﬁrmed conﬁrmed

0

by the spectrum of the internal output voltage of the converter shown in Fig. 5(b). When the second and fourth harmonics are reﬂected reﬂected back to the dc-side, dc-side, they still still giv givee the third harmonic, which explains why the dc-side only contains the third harmonic.

VI VII. I. COMMENTS ON THE I NFLUENCE OF A SSUMPTIONS M ADE In the preceding derivation, it was assumed that the converters were comprised of a six-pulse bridge and commutated at fundamental frequency. However, in most applications pulsewidth wid th mod modula ulatio tion n (PW (PWM) M) may may be used used in conver converter ters, s, or converters consist of more than one six-pulse bridge. In these cases, the relative magnitudes of the fundamental component and the other higher-frequency components in the switching functions will be modiﬁed. However, the dominant component is still the fundamental component. Therefore, the harmonic transfer through converters will still follow the proposed rules. Examples can be found in [5] and [6]. In the developme development, nt, only the fundament fundamental al comp component onent in the switching switching function function has been considered considered and the higherfrequency components are components neglected. It in is seen from examples that the higher-frequency the switching functions give almost invisible effect on the harmonic transfer. It has also been shown in [15] that the fundamental component is the main factor in inﬂuencing harmonic transfer. The other assumption made previously is that the overlap angle in the line-commutated current-source converter is equal to zero. This assumption is not valid in most applications. If the overlap angle is taken into consideration, the magnitude of each component in the switching function will be modiﬁed by a fact factor or [4]. There Therefore fore,, the amplitu amplitude de of the transfer transferred red harmonic may differ from that calculated for the zero overlap angle, angl e, but the frequency frequency relationship relationshipss of harmonic harmonic transfer through converters will not be affected. VIII VIII.. CONCLUSIONS The presented study clearly shows how harmonics transfer through converters from the dc-side to the ac-side and viceversa. ver sa. The obt obtain ained ed harmon harmonic ic transf transfer er rules rules are val valid id for

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(a) (a)

(b)

(b) Fig. 5. Simula Simulation tion of a force forced-co d-commuta mmutated ted VSC at the cond condition ition of neg negative ative sequence second harmonic injection on the ac-side: (a) harmonic spectrum of the dc-side voltage and (b) harmonic spectrum of the internal output phase voltage of the converter.

fr freq eque uenc ncie iess ( ) on the the ac-s ac-sid ide. e. The The me meth thod od pr pres esen ente ted d in the paper will provide a better understanding on the harmonic coupling of interconnected systems with converters.

REFERENCES (c) Fig. 4. Simulation Simulation of a forced-comm forced-commutated utated VS VSC, C, which has a natural resresonance mode with low damping: (a) dc-side voltage of the converter, onance converter, (b) harmonic spectrum of the dc-side voltage, and (c) harmonic spectrum of the internal output voltage of the converter.

both current and voltage harmonics with any frequencies. The harmonic transfer rules have also be illustrated with Fig. 3. It gives a complete harmonic frequency pattern that can be used to judge or predict which frequencies can be created on one side of a converter if there is a driving harmonic source on the other side. The rules are further summarized as follows. A dri drivin ving g har harmo monic nic on the dc-si dc-side de wil willl res result ult in two sidesideband frequencies on the ac-side, but no additional harmonic on the dc-side when the harmonics in the ac-side are transferred back to the dc-side. A balanced balanced driving harmon harmonic ic (angu (angular lar fr freq eque uenc on-sid the ac-si acdee wil will l tion re resu sult ineque on one e of side-, band ba nds s ncy ony the the) dc-s dc ide e and an dside one on ad addi diti onal allt fr freq uenc ncy y ( the sideor ) on the the ac-s ac-sid idee wh when en it is tr tran ansf sfer erre red d back back to th thee ac-side. An unbalanced driving harmonic on the ac-side will result result in two sid side-b e-band andss on the dc-sid dc-sidee and two add additi itiona onall

˚ Ekstr¨om, [1] A. om, “Calculation of transfer function for a forced-commutated voltage-source converter,” in Proc. 22nd Annu. IEEE Power Electronics Specialists Conf. (PESC), Boston, MA, June 1991, pp. 330–337. Transmission. New Y [2] E. W. Kimbark, Kimbark, Direct Current Transmission. York: ork: Wiley, 1971. [3] J. Arrillaga, Arrillaga, High Voltage Direct Current Transmission. Stevenage, U.K.: Peregrinus, 1983. [4] L. Hu and R. Yacamin acamini, i, “Harm “Harmonic onic transfer through through converter converterss and HVDC links,” IEEE Trans. Power Electron., vol. 7, pp. 514–524, July 1992. [5] H. Stemmler, “HVDC back-to-back back-to-back interties on weak weak ac systems second harmonic harmo nic prob problems lems analy analyzes zes and solut solutions” ions” in CIGRE Symp., 09-87, Boston, 1987, pp. 1–6. [6] O. Ojo and I. Bhat, “Inﬂu “Inﬂuence ence of input supply volta voltage ge unbalan unbalances ces on the performance of AC/DC buck rectiﬁers,” in Proc. 25th Annu. IEEE Power Electronics Specialists Conf. (PESC), 1995, pp. 777–784. [7 [7]] M. Gr Grotzbach o¨ tzbach and J. Xu, “Noncharacteristic line current harmonics in diode rectiﬁer bridges produced network asymmetries,” in Proc. Fourth European Conf. Power Electronics and Appl. (EPE’93), pp. 64–69. [8] A. Sar Sarsha shar, r, M. R. Ira Iravan vani, i, and J. Li, “Calcu “Calculati lation on of HVD HVDC C conconverter noncharacteristic harmonics using digital time-domain simulation method,” in 95 WM 224-6 PWRD. [9] “Draft guide for planning planning dc links terminating at ac locations locations having low ´ short-circuit capacities: Part I: ac/dc interaction phenomena,” CIGR E Standard P1204, 1993. [10] P. K. Kovacs, Kovacs, Transient Phenomena in Electrical Machines. Amsterdam: Elsevier Science, 1984. [11] [11] Y. Sum Sumi, i, Y. Harumo Harumoto, to, T. Hasega Hasegawa, wa, and M. Yano, ano, “Ne “New w static static var control control using force forced-co d-commutat mmutated ed inver inverters,” ters,” IEEE Tr Trans. ans. Power Apparatus Syst., vol. PAS-100, pp. 4216–4224, Sept. 1981.

¨ M: GENERAL ANALYSIS OF HARMONIC TRANSFER THROUGH CONVERTERS JIANG AND EKSTRO

[12] G. D. Galanos, Galanos, C. I. Hatziadoniu, Hatziadoniu, X.-J. Cheng, and D. Maratukul Maratukulam, am, “Advance static compensator for ﬂexible AC transmission,” IEEE Trans. Power Syst., vol. 8, pp. 113–121, Feb. 1993. ˚ Ekstr¨ [13] Y. Jiang Jiang and A. Ekstrom, o¨ m, “Study of the interaction oscillation between the AC system and the forced-commutated voltage-source converter,” in Proc. First Int. Power Electronics and Motion Control Conf. (IPEMC),

Beijing, China, June 1994, pp. 1050–1055. ˚ [14] [14] M. M. de Oliv Oliveir eiraa and Ake Ekstr¨o om, m, “Transfer functions for voltagesource converter operating as an SVC,” in 2nd Brazilian Power Elec. Conf. COBEP’93, Uberlandi Uberlandiaa Minas Gerais Brazil, Nov. 29–Dec. 29–Dec. 2, 1993, pp. 167–172. [15] W. Rusong, Rusong, S. B. Dewan, Dewan, and G. R. Slemo Slemon, n, “Analysis “Analysis of an ac-to-dc voltage source converter using PWM with phase and amplitude control,” IEEE Trans. Ind. Applicat., vol. 27, pp. 355–364, Mar./Apr., 1991.

Ying Jiang received the B.Sc. and M.Sc. degrees in electrical engineering from Huazhong University of Science and Technology, China, in 1984 and 1987, respectively respe ctively.. She received the licentiate licentiate degree degree in 1994 from the Department of Electrical Engineering of the Royal Institute of Techno Technology, logy, Stockholm, Stockholm, Sweden, where she is presently working toward the Ph.D. degree. She was with the Department of Electrical Engineering of Huaihai University from 1987 to 1991 as an Assistant Professor. Her areas of interest are FACTS devices with time-domain simulation and mathematical analysis and harmonic interactions in the power system with converters.

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˚ Ake Ekstr ¨o Ekstr ¨ om m received the M.Sc M.Sc.. degre degreee in electrical engineering from Royal Institute of Technology, Sweden, in 1957. He joined joined ASEA in 1956, 1956, wher wheree has been the Chief Engineer in the HVDC Design Department. Presently, he is Senior Vice President at the Power Transmission and Distribution Segment ABB. He is also the Professor at the Royal Institute of Technolog nology y and has been been Hea Head d of the Depar Departme tment nt of High Power Electronics there since 1986. He has published and presented more than 40 papers and holds several patents in HVDC design. Mr. Ekstr¨ o om m is abeen member of the of Royal Swedish of Engineering Sciences. He has Chairman SC 22F, IEC,Academy and has been active for many years in CIGRE. He received the Uno Lamm Award 1995.