IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER

Transcription

1 TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER A Harmonically Coupled Admittance Matrix Model for AC/DC Converters Yuanyuan Sun, Guibin Zhang, Wilsun Xu, Fellow,, Julio G. Mayordomo, Member, Abstract This paper proposes a new method to model the harmonic generating characteristics of ac/dc converters. The model transforms the time domain nonlinear characteristic of the converter into a frequency domain linear admittance matrix. The matrix represents the coupling among the converter ac side harmonic voltages currents accurately it does not vary with the harmonic conditions of the system. The proposed model opens up a new way to conduct harmonic power flow studies. This paper presents the theoretical foundation analytical derivation of the admittance model for both single-phase three-phase bridge rectifiers. Potential applications of the model are discussed. Index Terms Converter, harmonic analysis, harmonic model, harmonics. I. INTRODUCTION THE voltage current waveforms in power systems are often distorted by harmonics due to the presence of harmonic-producing loads such as power electronic converters. Over the past 20 plus years, various models have been proposed to represent the harmonic sources to support network-wide harmonic analysis. The predetermined harmonic current sources are the most commonly used model for harmonic source representation [1]. The magnitudes phases of the sources can be calculated from the typical harmonic current spectrum of the harmonicproducing device. The model cannot provide adequate results when the voltage distortions at the harmonic source buses are high. In order to improve its accuracy, harmonic-voltage-dependent current source models have been proposed, which leads to the well-known iterative harmonic analysis (IHA) method [2] [5]. To determine harmonic currents generated by a device, the current source model is first determined from the device analytical equations or time domain simulation based on estimated terminal harmonic voltages. The results are then used to compute bus harmonic voltages. The voltages are in turn used to calculate more accurate harmonic current sources. This method has gained industry acceptance. The IHA can be viewed as a Gaussian iteration method to the harmonic power flow problem. Manuscript received January 29, 2007; revised April 23, This work was supported by the China Scholarship Council Program under Grant [2006]3037, the China Cheung Kong Scholars Program, the Natural Sciences Engineering Council of Canada. Paper no. TPWRS Y. Sun is with the school of Electrical Engineering, Shong University, Jinan, China ( sunyy@mail.sdu.edu.cn). G. Zhang is with GP Technologies, Inc., Edmonton, AB T6G 2V4, Canada. W. Xu is with the University of Alberta, Edmonton, AB T6G 2V4, Canada ( wxu@ualberta.ca). J. G. Mayordomo is with the Department of Electrical Engineering, Universidad Politécnica of Madrid, Madrid, Spain. Digital Object Identifier /TPWRS Fig. 1. Single-phase bridge converter-schematic circuit for analysis. The Newton iteration method has also been proposed [6], [7]. In order to calculate the Jacobian elements for the Newton method, this approach generally requires knowing the analytical equations representing the harmonic generating characteristics of the device. The resulting solution techniques can become very complicated. In recent years, some researchers have shown that the relationship between the ac side harmonic voltages currents of a rectifier can be further clarified through a transformation matrix [8] or through an admittance relationship [9]. The significance of these findings is that they reveal a new direction to establish elegant harmonic models for power electronic devices analytically. Building on the work of [8] [9], we have managed to derive such a model for the bridge rectifier devices. The result is that the time-domain nonlinear characteristic of a converter is transformed into a frequency-domain linear admittance matrix with little approximation. Furthermore, it was found that elements of the matrix are independent of the supply system harmonic voltages, i.e., the matrix is constant as long as the converter firing angle commutation angle do not vary. The model therefore opens up a new way to conduct harmonic power flow studies. II. MODEL DEVELOPMENT FOR SINGLE-PHASE BRIDGE CONVERTER Since the proposed model is entirely analytical, the derivation of the model equations can be quite complicated. The model for single-phase converter shown in Fig. 1 is developed first to highlight the main steps. A. Switching Function Modeling of the Rectifying Process It is well known that the operating principle of an ideal converter without commutation process can be understood as the process of modulating the ac side voltage with a switching function to obtain the dc link voltage [10] [12], as follows: (1) /$

2 2 TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007 where is the rms value of th harmonic voltage is its phase angle, is the highest harmonic order available or of interest. Based on the modulation theory, the dc side voltage can be determined by multiplying with in frequency domain. The result is a Fourier expression of. For simplicity, the result is shown in the following symbolic form: Fig. 2. Modulation process of dc side voltage for single-phase converter. Fig. 2 shows this process where the dc voltage can be determined by multiplying the ac side voltage with the square waveform switching function. If the converter firing angle is, the switching function shifts right by degree, but the modulation process is still valid. As a result, a general expression, assuming no commutation, for the voltage switching (or modulating) function is Equation (2) shows that thyristors 1 4 conduct between the time interval of, whereas thyristors 2 3 are on between. Similarly, the ac current can be determined by modulating the dc current with the current switching function as follows: For the ideal converter,. Since the analytical expression of the switching function is known, we can find its frequency-domain expression analytically. The result is shown in (4) where, is the fundamental angular frequency of the system. (2) (3) (4) (6) where is the dc component of the converter dc voltage, is the rms value of th harmonic voltage is its phase angle. Analytical expressions for these variables are available but are omitted here to save space. The dc current of the converter can be determined from the above voltage the dc side impedance, in frequency domain, as follows: where is the th harmonic component of. Note that (7) is based on assumption that the dc motor voltage is zero. If is not zero, can also be derived but its form is slightly more complex. The ac current can be derived by the multiplication of Fourier series of dc current the current switching function as follows: C. Analytical Expression for the Converter AC Current Equations (5) (8) can be manipulated analytically. The final result that relates the converter ac harmonic current to its terminal voltage has the following form: (7) (8) B. Analytical Determination of the Converter AC Current The converter ac side voltage generally contains harmonics. The general expression of this voltage can be written as (5)

3 SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS 3 The formulas to determine each of the elements in, are shown in (12) (15) at the bottom of the page, where,,, A compact form for the above equation can be written as (9) (10) where is the ac current phasor matrix, is the ac voltage phasor matrix is its conjugate matrix. When the dc voltage source is taken into consideration, a new independent vector matrix needs to be added to the above equation. The result is (11) D. Characteristics of the Admittance Matrices Despite its complexity in derivation, the final converter model (11) can be constructed easily based on formulas for the matrix elements. Equation (11) is the harmonically coupled matrix model for the single-phase converter. It can be seen that there is no approximation in the equations. The model is, therefore, an exact frequency-domain representation of the nonlinear single-phase bridge converter. It effectively transforms a nonlinear element into harmonically-coupled admittance element. It is interesting to note that this model has two distinct features in comparison with common linear elements. The first one is that it couples all harmonic voltages current components. Each harmonic voltage will contribute to the generation of harmonic currents at different frequencies. The second one is that the current is not only a function of the supply voltage; it is also a function of the voltage conjugate. The most important characteristic of the admittance matrix model, however, is its constant nature with respect to the supply harmonic voltages: the elements of the matrix are independent of the harmonic voltages applied to the device. This implies that the model represents a linear component in the frequency domain. Because of the linearity, the matrices can be computed (12) (13) (14) (15)

4 4 TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007 before the network harmonic condition is known. As will be discussed in Section VI, this feature leads to a new elegant way to conduct harmonic power flow analysis. III. MODEL DEVELOPMENT FOR THREE-PHASE BRIDGE CONVERTER The model deduction process for three-phase bridge converter is similar to that of the single-phase bridge converter. For the component shown in Fig. 3 assuming a balanced operating condition no commutation, the switching functions have the following form: Fig. 3. Three-phase bridge converter-schematic circuit for analysis. The dc voltage can thus be determined as where (16) (17) (19) are the three-phase ac voltages. The equation for dc current is similar to that of (7). Due to the balanced condition, we only need to determine the phase-a ac current, which has the following form: (18) After extensive mathematical operations, the admittance matrix model for phase A is derived it has the following form: The compact form for (19) can also be written as (20) It can be seen that there is no triple order harmonics in the model. If the dc voltage source is included, the model has the following form:

5 SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS 5 The formulas to determine each of the elements in, are given in (22) (25) when (22) (23) (24) when ; when ; when ; when (25) when where,,,, are integers. It can be seen that the matrix model for three-phase converter has the same characteristics as that for the single-phase converter. As a result, conclusions drawn for the single-phase converter model are also applicable to the three-phase converter. IV. MODEL ENHANCEMENT In this section, the proposed admittance matrix model will be extended to take into account more realistic converter operating Fig. 4. Actual switching function for single-phase three-phase converters. conditions such as commutation. A complete model will be developed after the following analysis. A. Phase Angle of Fundamental Frequency Voltage If the formulas for the matrix elements are examined in detail, one can find that the elements are functions of, the phase angle of the fundamental frequency voltage. This phenomenon is expected since essentially represents the shift of the entire ac voltage current waveforms by degrees. From this perspective, the model is not strictly linear. However, this complication can be overcome due to the following consideration: the fundamental frequency phase angles of network bus voltages are mainly if not exclusively determined by the fundamental frequency power flow conditions of the system. It is theoretically possible but practically rare that the harmonic currents produced by a converter are strong enough to change the voltage phase angles at the fundamental frequency. As a result, if we know the fundamental frequency load flow results, angles for the converter buses become known quantities. The matrices can be computed according to the formulas. The problem then becomes how to compute the fundamental frequency power flow when a network contains converter loads. The simplest approach is to treat the converters as constant power loads, which results in a conventional power flow solution process. A more sophisticated approach is discussed in Section VI. B. Modeling of the Thyristor Commutation Process In reality, the change of the thyristor current from one steady state value to another can not happen instantaneously due to the existence of an inductance in the supply side of the converter. The converter current is commutated from one phase to another. Because of the commutation process, the switching functions for the voltage current modulation are no longer rectangular waves the function for voltage is different from that for the current. Fig. 4 shows the actual switching functions needed to take into account the commutation process accurately for the single-phase three-phase converter respectively, where in the figure is the commutation angle for the thyristors. It can be seen that the voltage switching function is close to rectangular waveform. However, there are intervals of zero

6 6 TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007 It is worth pointing out that the proposed current switching function is an improvement over that used in [11]. Simulation results have shown that the current waveforms will experience large errors without this improvement. The commutation reactance also affects the dc side current during the thyristor conducting period. The total circuit impedance experienced by the current becomes (28) Fig. 5. Adopted switching function for single-phase three-phase converters. values. The current switching function is no longer a rectangular wave. The method proposed in this paper to deal with the complex switching functions is to approximate them with more sophisticated but manageable waveforms [10] [12]. In this work, we proposed to model the voltage switching function as a rectangular function shown in Fig. 5, which is similar with the switching function used in [11] [12]. As a result, the Fourier expression for the switching function is the same as before, but the Fourier coefficients are changed to the following: (26a) (26b) (26c) where,,, is the Fourier coefficient for single-phase three-phase, respectively. If, the above equation becomes original switching functions (4) (16). The switching function for the current modulation is more complicated. The ac current during commutation is a function of system impedance, firing angle, the dc current. To simplify the analysis without introducing significant errors, a trapezoidal function is proposed to approximate the actual switching function as illustrated in Fig. 5. As a result, the Fourier expressions of the switching functions in (4) (16) also remain the same. The only changes are the Fourier coefficients, which have the following forms: (27a) (27b) (27c) where is the commutation inductance. A multiplier of 2 is used since the ac current passes through phase A commutation inductance first, goes to the dc side, returns through the phase B (or C) commutation inductance. is the value that should be used to construct the matrix. In summary, the commutation process has been included through the modified Fourier coefficients of the switching functions augmented circuit impedance. The deduction process of the matrix model becomes the same as that presented earlier. As a result, the formulas for the matrix their elements are the same. The only difference is that the elements will be functions of the commutation angel inductance as well. According to [13], the commutation angles can be computed approximately using the fundamental frequency parameters as follows: (29a) (29b) where is the thyristor firing angle, is the ac side inductance, is the dc current mean value, is the ac side phase voltage. V. CASE STUDY RESULTS The proposed model has been compared with the time domain simulation results. Waveform comparison is more desirable than harmonic spectrum comparison because the former ensures that both the harmonic magnitudes phase angles are verified. The converter is supplied by harmonic voltage sources. In the single-phase converter, only the odd order harmonics are considered since the even order harmonics in the supply voltage source are usually of much smaller amplitude. For the threephase converter, no triple order harmonic is presented in the supply side. A lot of simulation work has been done the following are some of the representative results. The system data is shown in Table I. A. No Commutation Process Simulation results are shown in Figs. 6 7 for single-phase converter. The waveforms of converter dc voltage, dc current ac current from the matrix method are compared with the

7 SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS 7 TABLE I PARAMETERS FOR BRIDGE CONVERTERS Fig. 6. Waveforms for V (t) S (t) V (t) (single-phase converter). Fig. 7. Waveforms for I (t) S (t) I (t) (single-phase converter). Matlab Simulink results. The figure shows that the results essentially overlap each other. This is expected since there is no approximation for the matrix method in this case. Figs. 8 9 show the results for the three-phase converter. The three-phase switching functions for the ideal converter are first illustrated in Fig. 8. Under ideal conditions, voltage current switching function are the same for each phase. The converter dc voltage, dc current along with the ac current of phase A is shown in Fig. 9. Again, the results have a good agreement. Fig. 8. Waveforms for three-phase converter switching function. Fig. 9. Waveforms for V (t) I (t) I (t) (three-phase converter). B. Impact of Commutation Sample waveform comparison is shown in Figs for the cases where there is a commutation overlap. As expected, the waveforms do not match exactly since the matrix model uses approximate switching functions. To further verify the model accuracy at different commutation angles, an error index is defined as follows: (30) where is the harmonic current phasor from Simulation is that from the matrix model. This index measures the relative error between the results from the matrix method those from the time domain simulation. Such comparative results are shown in Table II. It can be seen that the error increases with the overlap angle but it is small from the practical application perspective. If it is desirable to reduce the error further, more sophisticated approximation to the switching functions could be used.

8 8 TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 4, NOVEMBER 2007 is the power factor of the three-phase converter load is the specified converter active power at the fundamental frequency. They are given data. is the converter fundamental frequency line to line voltage obtained from the power flow results. If commutation process is considered, angles are determined by solving the following two nonlinear equations [14]: (32a) (32b) Fig. 10. AC current waveform for single-phase converter ( =30 ). Fig. 11. AC current (Phase A) waveform for three-phase converter ( =30 ). TABLE II RELATIVE ERROR FOR DIFFERENT COMMUTATION ANGLES VI. POTENTIAL APPLICATIONS OF THE MATRIX MODEL The proposed admittance model for converters has some significant applications. The most direct one is a new harmonic power flow method that is noniterative. This method is explained briefly here. The method first computes the fundamental frequency power flow by treating the converters as constant power loads. From the power flow results, the converter firing angles dc voltage sources, assuming no commutation, can be determined using the following equations [13]: where (31a) (31b) (31c) where is the commutating reactance. can be solved from the above equations using, for example, the Newton algorithm. The dc magnitudes can then be obtained from (33a) (33b) Accordingly, the matrix models for the converters can be established as follows: (34) where is the converter number in the network. is a known current vector that has included the impact of the fundamental frequency voltage the converter dc voltage source as follows: (35) The unknowns in (34) are the converter ac side harmonic voltage current vectors. The rest of the network is linear can be modeled as a harmonically decoupled admittance matrix as follows: (36) where is the harmonic order interested. Equations (34) (36) are both linear equations. They can be solved together without iteration. The results are harmonic voltages at each bus that have included the impact of harmonic currents from all harmonic sources. This technique has the potential

9 SUN et al.: A HARMONICALLY COUPLED ADMITTANCE MATRIX MODEL FOR AC/DC CONVERTERS 9 to analyze systems with distributed harmonic sources. For such cases, the impact of background harmonics in the form of diversity attenuation effects must be considered [15]. The above noniterative solution method relies on some approximations. The main one is that the harmonic voltages will not affect the converter firing angle ( overlap angle). If one must model such an effect, an iterative solution for the firing angle can be implemented. This iterative method has firing angles overlap angles of the various converters as variables so it leads to a new iterative harmonic power flow method. This is the second potential application of the model. The third application is that the model provides an excellent platform to analyze the mutual couplings among the various harmonic components. The result will help us to underst the accuracy characteristics of the harmonic current source model. It is important to note that the current source model is the foundation for a number of harmonic-related subjects besides assisting harmonic power flow calculations. One example is the harmonic source detection problem which is formed on the assumption that the harmonic sources can be approximated as harmonic current sources. VII. CONCLUSION This paper has presented a new elegant model for analyzing the harmonic characteristics of the ac/dc converters. The model is in the form of harmonically-coupled admittance matrices that relate the ac side harmonic voltages currents. Despite the complexity of derivation, the final matrix model can be computed easily according to a set of formulas. The proposed model has the following useful characteristics. The model is linear, which leads to the formulation of alternative harmonic power flow solution algorithms. The model is analytical. It will enable one to investigate the harmonic coupling characteristics of the converters much more thoroughly. The model is developed for a wide range of operating conditions. Unlike the linearized model, the proposed model is operating point independent. Our subsequent effort is to investigate the new harmonic power flow algorithms described in Section VI, with the distributed harmonic sources as the main target of application. Research is also underway to establish a similar model for the thyristor-controlled reactors. REFERENCES [1] Task force on harmonics modeling simulation, Modeling simulation of the propagation of harmonics in electric power networks Part I: Concepts, models simulation techniques, Trans. Power Del., vol. 11, no. 1, pp , Jan [2] H. W. Dommel, A. Yan, W. Shi, Harmonics from transformer saturation, Trans. Power Del., vol. PWRD-1, no. 2, pp , Apr [3] W. Xu, J. E. Drakos, Y. Mansour, A. Chang, A three-phase converter model for harmonic analysis of HVDC systems, Trans. Power Del., vol. 9, no. 3, pp , Jul [4] W. Xu, J. R. Marti, H. W. Dommel, A multiphase harmonic load flow solution technique, Trans. Power Syst., vol. 6, no. 1, pp , Feb [5] V. Sharma, R. J. Fleming, L. Niekamp, An iterative approach for analysis of harmonic penetration in the power transmission networks, Trans. Power Del., vol. 6, no. 4, pp , Oct [6] D. Xia G. T. Heydt, Harmonic power flow studies, Part I: Formulation solution, Trans. Power App. Syst., vol. 101, pp , Jun [7] L. T. G. Lima, A. Semlyen, M. R. Iravani, Harmonic domain periodic steady state modeling of power electronics apparatus: SVC TCSC, Trans. Power Del., vol. 18, no. 3, pp , Jul [8] S. G. Jalali R. H. Lasseter, Harmonic interaction of power system with static switching circuits, in Proc Power Electronics Specialist Conf., pp [9] J. G. Mayordomo, L. F. Beites, R. Asensi, F. Orzaez, M. Izzeddine, L. Zabala, A contribution for modeling controlled uncontrolled ac/dc converters in harmonic power flows, Trans. Power Del., vol. 13, no. 4, pp , Oct [10] J. Arrillaga N. R. Watson, Power System Harmonics. New York: Wiley, 1997, p [11] L. Hu R. Yacamini, Harmonic transfer through converters HVDC links, Trans. Power Electron., vol. 7, no. 3, pp , Jul [12] R. Carbone, F. D. Rosa, R. Langella, A. Testa, A new approach to model AC/DC/AC conversion systems, in Proc Power Eng. Soc. Summer Meeting, pp [13] M. H. Rashid, Power Electronics: Circuits, Devices Application. Englewood Cliffs, NJ: Prentice-Hall, 1993, vol. 2. [14] E. W. Kimbark, Direct Current Transmission. New York: Wiley, 1971, vol. 1, p. 80. [15] E. E. Ahmed, W. Xu, G. Zhang, Analyzing systems with distributed harmonic sources including the attenuation diversity effects, Trans. Power Del., vol. 20, no. 4, pp , Oct Yuanyuan Sun received the B.Sc. degree in electrical engineering from Shong University, Jinan, China, in She is currently pursuing the Ph.D degree in the School of Electrical Engineering, Shong University. Her main research interest is harmonics. Guibin Zhang received the B.Sc. M.Sc. degrees in electrical engineering from Shong University, Jinan, China, in , respectively, the Ph.D. degree from Zhejiang University, Hangzhou, China, in He was a Postdoctoral Fellow with the University of Alberta, Edmonton, AB, Canada, from 2003 to He is currently with GP Technology, Inc., Edmonton. Wilsun Xu (M 90 SM 95 F 05) received the Ph.D. degree from the University of British Columbia, Vancouver, BC, Canada, in He was an Electrical Engineer with BC Hydro from 1990 to He is presently a Professor with the University of Alberta, Edmonton, AB, Canada, an Adjunct Professor at Shong University, Jinan, China. His main research interests are harmonics power quality. Julio G. Mayordomo (M 97) received the Dipl.-Eng. Ph.D. degrees from the Universidad Politécnica of Madrid (UPM), Madrid, Spain, in , respectively. Currently, he is a Professor at UPM. His main research interest is low frequency disturbances.