Harmonics Transfer Converters
Content
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 12, NO. 2, MARCH 1997
287
General Analysis of Harmonic Transfer Through Converters
˚ Ying Jiang, Member, IEEE and Ake Ekstr¨ om, 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 vectors. The dependency of the relative frequency 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 frequency transformation through converters are also valid for both voltage and current harmonics. The analytical method is validated by digital simulations with EMTP. Index Terms— Current-source converter, harmonic transfer, switching function, voltage-source converter.
transferred through converters. The derived rules are valid for any three-phase converters with symmetrical modulation. Although the derivation is based on three-phase converters 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].
II. SPACE VECTOR I. INTRODUCTION N HIGH-POWER electronics, harmonics generated by converters are often grouped into two categories: characteristic harmonics and noncharacteristic harmonics [1]–[3]. For the calculation of characteristic harmonics, analytical and numerical methods are well established [1]–[3]. Regarding noncharacteristic harmonics, a number of 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 statement about harmonic transferring is not 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 through converters. Based on analyzing basic current and voltage equations of a three-phase voltage-source and current-source converter, simple rules are formulated by using switching functions and the space vector [10]. The rules tell which frequencies will be created on the ac-side of a converter when there is a harmonic on the dc-side and vice-versa. The rules are valid for both current and voltage harmonics. Differing from previous works, the paper gives 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.
I
Three-phase variables, , can be transformed into an orthogonal two-phase variables, , without losing any information of that three phase quantities, if no zero-sequence component exists in the phase components. This condition is automatically fulfilled for voltages/currents transferred through a three-phase transformer for which the winding on at least one side is not grounded. Representing the two orthogonal components , respectively by the real and imaginary components leads to a vector that is often called space vector [10]. The space vector is calculated as follows:
(1)
where (2) It should be noted that the transformation given in (1) results in equal amplitude for the space vector and the threephase quantities. In the following derivation the 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 direction. For instance, a three-phase system quantities 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 sequence component in (3a), whereas the negative
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sign “ ” applies for the negative sequence component
(3a)
(3b) Inversely, the instantaneous values of the three-phase quantities can also be obtained from the corresponding space vector as follows: Re Re Re where “ ” means complex conjugate. III. HARMONIC TRANSFERRED VOLTAGE SOURCE CONVERTER (4a) (4b) (4c)
(a)
BY
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, respectively. The operation of the converter can be illustrated by three binary signals and as shown in Fig. 1(c), which are the so-called switching 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 crossing of the respective ac bus phase voltage. The switching functions can be expressed in Fourier series as follows [1]: (5a)
(b)
(c) Fig. 1. Six-pulse forced-commutated voltage-source converter: (a) basic electric circuit, (b) equivalent circuit, and (c) switching functions.
(5b)
(5c) According to (1), the space vector of the switching function will be as follows: (6)
and etc., i.e., etc. The positive sign “+” in the exponent applies for and the negative sign “ ” for . For simplicity, in the following the fundamental component of will be considered as an approximation of . Therefore, the switching function can be approximately expressed as (7). The influence of the approximation on the result will be discussed later: (7) The basic relationships for the voltage-source converter are derived from the fact that the converter ac voltages are
where
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proportional to the dc-side voltage, which gives (8)
B. Harmonic Transferred from AC-Side to DC-Side According to the principal of superimposition, it is possible to consider each harmonic individually. A simple case that is a harmonic containing only a positive or only a negative sequence component is considered first. Assuming that the angular frequency of the harmonic is , the space vector 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 (17)
(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)
Inserting (8) into (9) and eliminating the equation gives
on both sides of Equation (17) shows that a positive sequence harmonic on the ac-side transferred to the dc-side gives only one of the side-bands, which has a lower frequency, 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 frequency, i.e., ( ). A well-known example is that 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:
(10)
Multiplying (8) by
yields (11), while applying (4) for in (10) and transforming the switching functions into space vector, results in (11) Re (12)
A. Harmonics Transferred from DC-Side to AC-Side Assuming that the dc-component of the dc-side voltage is superimposed with a voltage component having an oscillation angular frequency , the dc-side voltage can be written as follows: (13) Equation (13) can also be written as (14) Substituting (7) and (14) in (11) will give the following equation: (18) Inserting (7) and (18) into (12) gives Re (19) The above equation shows that an asymmetrical harmonic on the ac-side transferred to the dc-side gives two side-bands. The current/voltage amplitudes of the two side-bands may be different depending on the amplitudes of both the positive and negative sequence components. Of the two side-bands on the dc-side, the current amplitude of the lower frequency is proportional to that of the positive sequence component, whereas the current amplitude of the higher frequency is proportional to that of the negative sequence component. A practical case might, for instance, be an unbalanced second harmonic generated by a nearby saturated transformer. The positive sequence component in this case will be more critical as it gives a fundamental component on the dc-side, which in turn gives a second harmonic and the dc-component on the ac-side. The dc-component may saturate the converter transformer, which will further result in additional second harmonic. The procedure aggravates the harmonics continuously until it gets harmonic instability [5].
(15) Equation (15) indicates that a harmonic on the dc-side will be transferred to the ac-side as two side-bands: the harmonic with higher frequency is a positive sequence component and the harmonic with lower frequency is a negative sequence component as ( ) is less than zero as long as is greater than . The two side-bands have the same voltage amplitude.
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(23)
Inserting (22) into (23) and eliminating of the equation yields
on both sides
(24)
(a)
Multiplying (22) by
gives (25), while applying (4) for in (24) and transforming the switching functions into space vector, results in (25) Re (26)
(b) Fig. 2. Six-pulse line-commutated current-source 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 converter, which is often line-commutated in high power applications. Assuming that the overlap is zero, a set of switching functions can be obtained in the same way as for the voltage-source converter, which are shown in Fig. 2(b). Making Fourier analysis on each switching function ( , and ) and applying (1) yields (20) and 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 where (21) The basic relationships for the current-source converter are derived from the fact that the three-phase ac currents are proportional to the dc-side current, which gives (22). The active power balance on the ac-side and dc-side of the converter results in (23), if the losses in the converter bridges are neglected (22)
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 exponent form, and a harmonic 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. These rules are illustrated by Fig. 3. It should be noticed that a single harmonic, which is balanced in 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. VI. EXAMPLES In order to illustrate the presented harmonic transfer rules, two examples are presented in the following. In the first 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 forced-commutated voltage-source converter, which is supposed to be used as a static var compensator [11], [12]. The fundamental frequency of the ac system is 60 Hz.
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Fig. 3. Harmonic transfer rule. Sign “+” denotes positive sequence; sign “ ” denotes negative sequence.
0
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 controller [13], [14]. Fig. 4(a) and (b) shows the 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 result. The other two obvious harmonics are 11th (660 Hz) and 13th (780 Hz) harmonics, which belong to characteristic harmonics. If the approximation had not been made on (6), there should be harmonics with frequencies 660 149 Hz, 780 149 Hz on the ac-side. According 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. Furthermore, according to (14) or Fig. 3 the harmonic with frequency of 89 Hz is of negative sequence 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 driving harmonic. Again, Fig. 4(b) confirms the proposed harmonic transfer rules.
by the spectrum of the internal output voltage of the converter shown in Fig. 5(b). When the second and fourth harmonics are reflected back to the dc-side, they still give the third harmonic, which explains why the dc-side only contains the third harmonic. VII. COMMENTS ON THE INFLUENCE OF ASSUMPTIONS MADE 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 modulation (PWM) may be used in converters, 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 development, only the fundamental component in the switching function has been considered and the higherfrequency components are neglected. It is seen from examples that the higher-frequency components in 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 factor [4]. Therefore, the amplitude of the transferred harmonic may differ from that calculated for the zero overlap angle, but the frequency relationships of harmonic transfer through converters will not be affected. VIII. CONCLUSIONS The presented study clearly shows how harmonics transfer through converters from the dc-side to the ac-side and viceversa. The obtained harmonic transfer rules are valid for
B. Example Two In this example, a second harmonic current of 10% with negative sequence is injected into the ac bus. According to Fig. 3 the second harmonic with negative sequence will give a third harmonic on the dc-side. Fig. 5(a) shows the harmonic spectrum of the dc-side voltage, the third harmonic can be observed on it as expected. The third harmonic will give the second (positive sequence) and fourth (negative sequence) harmonics on the ac-side, this is also confirmed
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(a) (a)
(b)
(b) Fig. 5. Simulation of a forced-commutated VSC at the condition of negative 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.
frequencies ( ) on the ac-side. The method presented in the paper will provide a better understanding on the harmonic coupling of interconnected systems with converters.
REFERENCES
(c) Fig. 4. Simulation of a forced-commutated VSC, which has a natural resonance mode with low damping: (a) dc-side voltage of the converter, (b) harmonic spectrum of the dc-side voltage, and (c) harmonic spectrum of the internal output voltage of the converter. ˚ Ekstr¨ [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. [2] E. W. Kimbark, Direct Current Transmission. New York: Wiley, 1971. [3] J. Arrillaga, High Voltage Direct Current Transmission. Stevenage, U.K.: Peregrinus, 1983. [4] L. Hu and R. Yacamini, “Harmonic transfer through converters and HVDC links,” IEEE Trans. Power Electron., vol. 7, pp. 514–524, July 1992. [5] H. Stemmler, “HVDC back-to-back interties on weak ac systems second harmonic problems analyzes and solutions” in CIGRE Symp., 09-87, Boston, 1987, pp. 1–6. [6] O. Ojo and I. Bhat, “Influence of input supply voltage unbalances on the performance of AC/DC buck rectifiers,” in Proc. 25th Annu. IEEE Power Electronics Specialists Conf. (PESC), 1995, pp. 777–784. [7] M. Gr¨ otzbach 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. Sarshar, M. R. Iravani, and J. Li, “Calculation of HVDC converter noncharacteristic harmonics using digital time-domain simulation method,” in 95 WM 224-6 PWRD. [9] “Draft guide for planning dc links terminating at ac locations having low ´ short-circuit capacities: Part I: ac/dc interaction phenomena,” CIGRE Standard P1204, 1993. [10] P. K. Kovacs, Transient Phenomena in Electrical Machines. Amsterdam: Elsevier Science, 1984. [11] Y. Sumi, Y. Harumoto, T. Hasegawa, and M. Yano, “New static var control using forced-commutated inverters,” IEEE Trans. Power Apparatus Syst., vol. PAS-100, pp. 4216–4224, Sept. 1981.
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 driving harmonic on the dc-side will result in two sideband 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 driving harmonic (angular frequency ) on the ac-side will result in one of the side, bands on the dc-side and one additional frequency ( ) on the ac-side when it is transferred back to the or ac-side. An unbalanced driving harmonic on the ac-side will result in two side-bands on the dc-side and two additional
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[12] G. D. Galanos, C. I. Hatziadoniu, X.-J. Cheng, and D. Maratukulam, “Advance static compensator for flexible AC transmission,” IEEE Trans. Power Syst., vol. 8, pp. 113–121, Feb. 1993. ˚ Ekstr¨ [13] Y. Jiang and A. om, “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] M. M. de Oliveira and Ake Ekstr¨ om, “Transfer functions for voltagesource converter operating as an SVC,” in 2nd Brazilian Power Elec. Conf. COBEP’93, Uberlandia Minas Gerais Brazil, Nov. 29–Dec. 2, 1993, pp. 167–172. [15] W. Rusong, S. B. Dewan, and G. R. Slemon, “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.
˚ Ake Ekstr¨ om received the M.Sc. degree in electrical engineering from Royal Institute of Technology, Sweden, in 1957. He joined ASEA in 1956, where 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 Technology and has been Head of the Department 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¨ om is a member of the Royal Swedish Academy of Engineering Sciences. He has been Chairman of SC 22F, IEC, and has been active for many years in CIGRE. He received the Uno Lamm Award 1995.
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. She received the licentiate degree in 1994 from the Department of Electrical Engineering of the Royal Institute of Technology, 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.
287
General Analysis of Harmonic Transfer Through Converters
˚ Ying Jiang, Member, IEEE and Ake Ekstr¨ om, 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 vectors. The dependency of the relative frequency 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 frequency transformation through converters are also valid for both voltage and current harmonics. The analytical method is validated by digital simulations with EMTP. Index Terms— Current-source converter, harmonic transfer, switching function, voltage-source converter.
transferred through converters. The derived rules are valid for any three-phase converters with symmetrical modulation. Although the derivation is based on three-phase converters 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].
II. SPACE VECTOR I. INTRODUCTION N HIGH-POWER electronics, harmonics generated by converters are often grouped into two categories: characteristic harmonics and noncharacteristic harmonics [1]–[3]. For the calculation of characteristic harmonics, analytical and numerical methods are well established [1]–[3]. Regarding noncharacteristic harmonics, a number of 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 statement about harmonic transferring is not 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 through converters. Based on analyzing basic current and voltage equations of a three-phase voltage-source and current-source converter, simple rules are formulated by using switching functions and the space vector [10]. The rules tell which frequencies will be created on the ac-side of a converter when there is a harmonic on the dc-side and vice-versa. The rules are valid for both current and voltage harmonics. Differing from previous works, the paper gives 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.
I
Three-phase variables, , can be transformed into an orthogonal two-phase variables, , without losing any information of that three phase quantities, if no zero-sequence component exists in the phase components. This condition is automatically fulfilled for voltages/currents transferred through a three-phase transformer for which the winding on at least one side is not grounded. Representing the two orthogonal components , respectively by the real and imaginary components leads to a vector that is often called space vector [10]. The space vector is calculated as follows:
(1)
where (2) It should be noted that the transformation given in (1) results in equal amplitude for the space vector and the threephase quantities. In the following derivation the 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 direction. For instance, a three-phase system quantities 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 sequence component in (3a), whereas the negative
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sign “ ” applies for the negative sequence component
(3a)
(3b) Inversely, the instantaneous values of the three-phase quantities can also be obtained from the corresponding space vector as follows: Re Re Re where “ ” means complex conjugate. III. HARMONIC TRANSFERRED VOLTAGE SOURCE CONVERTER (4a) (4b) (4c)
(a)
BY
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, respectively. The operation of the converter can be illustrated by three binary signals and as shown in Fig. 1(c), which are the so-called switching 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 crossing of the respective ac bus phase voltage. The switching functions can be expressed in Fourier series as follows [1]: (5a)
(b)
(c) Fig. 1. Six-pulse forced-commutated voltage-source converter: (a) basic electric circuit, (b) equivalent circuit, and (c) switching functions.
(5b)
(5c) According to (1), the space vector of the switching function will be as follows: (6)
and etc., i.e., etc. The positive sign “+” in the exponent applies for and the negative sign “ ” for . For simplicity, in the following the fundamental component of will be considered as an approximation of . Therefore, the switching function can be approximately expressed as (7). The influence of the approximation on the result will be discussed later: (7) The basic relationships for the voltage-source converter are derived from the fact that the converter ac voltages are
where
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proportional to the dc-side voltage, which gives (8)
B. Harmonic Transferred from AC-Side to DC-Side According to the principal of superimposition, it is possible to consider each harmonic individually. A simple case that is a harmonic containing only a positive or only a negative sequence component is considered first. Assuming that the angular frequency of the harmonic is , the space vector 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 (17)
(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)
Inserting (8) into (9) and eliminating the equation gives
on both sides of Equation (17) shows that a positive sequence harmonic on the ac-side transferred to the dc-side gives only one of the side-bands, which has a lower frequency, 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 frequency, i.e., ( ). A well-known example is that 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:
(10)
Multiplying (8) by
yields (11), while applying (4) for in (10) and transforming the switching functions into space vector, results in (11) Re (12)
A. Harmonics Transferred from DC-Side to AC-Side Assuming that the dc-component of the dc-side voltage is superimposed with a voltage component having an oscillation angular frequency , the dc-side voltage can be written as follows: (13) Equation (13) can also be written as (14) Substituting (7) and (14) in (11) will give the following equation: (18) Inserting (7) and (18) into (12) gives Re (19) The above equation shows that an asymmetrical harmonic on the ac-side transferred to the dc-side gives two side-bands. The current/voltage amplitudes of the two side-bands may be different depending on the amplitudes of both the positive and negative sequence components. Of the two side-bands on the dc-side, the current amplitude of the lower frequency is proportional to that of the positive sequence component, whereas the current amplitude of the higher frequency is proportional to that of the negative sequence component. A practical case might, for instance, be an unbalanced second harmonic generated by a nearby saturated transformer. The positive sequence component in this case will be more critical as it gives a fundamental component on the dc-side, which in turn gives a second harmonic and the dc-component on the ac-side. The dc-component may saturate the converter transformer, which will further result in additional second harmonic. The procedure aggravates the harmonics continuously until it gets harmonic instability [5].
(15) Equation (15) indicates that a harmonic on the dc-side will be transferred to the ac-side as two side-bands: the harmonic with higher frequency is a positive sequence component and the harmonic with lower frequency is a negative sequence component as ( ) is less than zero as long as is greater than . The two side-bands have the same voltage amplitude.
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(23)
Inserting (22) into (23) and eliminating of the equation yields
on both sides
(24)
(a)
Multiplying (22) by
gives (25), while applying (4) for in (24) and transforming the switching functions into space vector, results in (25) Re (26)
(b) Fig. 2. Six-pulse line-commutated current-source 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 converter, which is often line-commutated in high power applications. Assuming that the overlap is zero, a set of switching functions can be obtained in the same way as for the voltage-source converter, which are shown in Fig. 2(b). Making Fourier analysis on each switching function ( , and ) and applying (1) yields (20) and 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 where (21) The basic relationships for the current-source converter are derived from the fact that the three-phase ac currents are proportional to the dc-side current, which gives (22). The active power balance on the ac-side and dc-side of the converter results in (23), if the losses in the converter bridges are neglected (22)
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 exponent form, and a harmonic 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. These rules are illustrated by Fig. 3. It should be noticed that a single harmonic, which is balanced in 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. VI. EXAMPLES In order to illustrate the presented harmonic transfer rules, two examples are presented in the following. In the first 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 forced-commutated voltage-source converter, which is supposed to be used as a static var compensator [11], [12]. The fundamental frequency of the ac system is 60 Hz.
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Fig. 3. Harmonic transfer rule. Sign “+” denotes positive sequence; sign “ ” denotes negative sequence.
0
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 controller [13], [14]. Fig. 4(a) and (b) shows the 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 result. The other two obvious harmonics are 11th (660 Hz) and 13th (780 Hz) harmonics, which belong to characteristic harmonics. If the approximation had not been made on (6), there should be harmonics with frequencies 660 149 Hz, 780 149 Hz on the ac-side. According 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. Furthermore, according to (14) or Fig. 3 the harmonic with frequency of 89 Hz is of negative sequence 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 driving harmonic. Again, Fig. 4(b) confirms the proposed harmonic transfer rules.
by the spectrum of the internal output voltage of the converter shown in Fig. 5(b). When the second and fourth harmonics are reflected back to the dc-side, they still give the third harmonic, which explains why the dc-side only contains the third harmonic. VII. COMMENTS ON THE INFLUENCE OF ASSUMPTIONS MADE 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 modulation (PWM) may be used in converters, 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 development, only the fundamental component in the switching function has been considered and the higherfrequency components are neglected. It is seen from examples that the higher-frequency components in 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 factor [4]. Therefore, the amplitude of the transferred harmonic may differ from that calculated for the zero overlap angle, but the frequency relationships of harmonic transfer through converters will not be affected. VIII. CONCLUSIONS The presented study clearly shows how harmonics transfer through converters from the dc-side to the ac-side and viceversa. The obtained harmonic transfer rules are valid for
B. Example Two In this example, a second harmonic current of 10% with negative sequence is injected into the ac bus. According to Fig. 3 the second harmonic with negative sequence will give a third harmonic on the dc-side. Fig. 5(a) shows the harmonic spectrum of the dc-side voltage, the third harmonic can be observed on it as expected. The third harmonic will give the second (positive sequence) and fourth (negative sequence) harmonics on the ac-side, this is also confirmed
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(a) (a)
(b)
(b) Fig. 5. Simulation of a forced-commutated VSC at the condition of negative 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.
frequencies ( ) on the ac-side. The method presented in the paper will provide a better understanding on the harmonic coupling of interconnected systems with converters.
REFERENCES
(c) Fig. 4. Simulation of a forced-commutated VSC, which has a natural resonance mode with low damping: (a) dc-side voltage of the converter, (b) harmonic spectrum of the dc-side voltage, and (c) harmonic spectrum of the internal output voltage of the converter. ˚ Ekstr¨ [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. [2] E. W. Kimbark, Direct Current Transmission. New York: Wiley, 1971. [3] J. Arrillaga, High Voltage Direct Current Transmission. Stevenage, U.K.: Peregrinus, 1983. [4] L. Hu and R. Yacamini, “Harmonic transfer through converters and HVDC links,” IEEE Trans. Power Electron., vol. 7, pp. 514–524, July 1992. [5] H. Stemmler, “HVDC back-to-back interties on weak ac systems second harmonic problems analyzes and solutions” in CIGRE Symp., 09-87, Boston, 1987, pp. 1–6. [6] O. Ojo and I. Bhat, “Influence of input supply voltage unbalances on the performance of AC/DC buck rectifiers,” in Proc. 25th Annu. IEEE Power Electronics Specialists Conf. (PESC), 1995, pp. 777–784. [7] M. Gr¨ otzbach 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. Sarshar, M. R. Iravani, and J. Li, “Calculation of HVDC converter noncharacteristic harmonics using digital time-domain simulation method,” in 95 WM 224-6 PWRD. [9] “Draft guide for planning dc links terminating at ac locations having low ´ short-circuit capacities: Part I: ac/dc interaction phenomena,” CIGRE Standard P1204, 1993. [10] P. K. Kovacs, Transient Phenomena in Electrical Machines. Amsterdam: Elsevier Science, 1984. [11] Y. Sumi, Y. Harumoto, T. Hasegawa, and M. Yano, “New static var control using forced-commutated inverters,” IEEE Trans. Power Apparatus Syst., vol. PAS-100, pp. 4216–4224, Sept. 1981.
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 driving harmonic on the dc-side will result in two sideband 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 driving harmonic (angular frequency ) on the ac-side will result in one of the side, bands on the dc-side and one additional frequency ( ) on the ac-side when it is transferred back to the or ac-side. An unbalanced driving harmonic on the ac-side will result in two side-bands on the dc-side and two additional
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[12] G. D. Galanos, C. I. Hatziadoniu, X.-J. Cheng, and D. Maratukulam, “Advance static compensator for flexible AC transmission,” IEEE Trans. Power Syst., vol. 8, pp. 113–121, Feb. 1993. ˚ Ekstr¨ [13] Y. Jiang and A. om, “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] M. M. de Oliveira and Ake Ekstr¨ om, “Transfer functions for voltagesource converter operating as an SVC,” in 2nd Brazilian Power Elec. Conf. COBEP’93, Uberlandia Minas Gerais Brazil, Nov. 29–Dec. 2, 1993, pp. 167–172. [15] W. Rusong, S. B. Dewan, and G. R. Slemon, “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.
˚ Ake Ekstr¨ om received the M.Sc. degree in electrical engineering from Royal Institute of Technology, Sweden, in 1957. He joined ASEA in 1956, where 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 Technology and has been Head of the Department 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¨ om is a member of the Royal Swedish Academy of Engineering Sciences. He has been Chairman of SC 22F, IEC, and has been active for many years in CIGRE. He received the Uno Lamm Award 1995.
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. She received the licentiate degree in 1994 from the Department of Electrical Engineering of the Royal Institute of Technology, 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.
