A space vector based rectifier regulator for ac dc ac conver


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A space vector based rectifier regulator for ac dc ac conver

  1. 1. ~ IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993 30 A Space Vector-Based Rectifier Regulator for AC/DC/AC Converters Thomas G. Habetler, Senior Member, IEEE Abstruct- A voltage-sourced rectifier control scheme for use with ac/dc/ac variable speed drives is presented in this paper. A control scheme is derived that directly calculates the duration of time spent on the zero state and on each switching state adjacent to the reference vector, over a constant switching interval, in order to drive the line current vector to the reference vector. In addition, under transient conditions, when deadbeat control is not possible, a control scheme is presented that ensures that the line current vector is driven in the direction of the reference current vector. The current reference for the rectifier controller is derived from the bus voltage error and a feedforward term based on the estimated converter output power. The proposed space vectorbased rectifier regulator is shown to exhibit improved harmonic and transient performance over existing per-phase duty cycle prediction methods, especially at modulation indices near unity. The deadbeat control of the rectifier input current is accomplished every half-cycle with constant switching frequency while still symmetrically distributing the zero state within the halfcycle period. In this way, satisfactory performance under various operating conditions is achieved with relatively low switching frequencies and high bus voltage ripple. Fig. 1. Schematic of ac/dc/ac converter with controlled rectifier. Rectifier Inverter I. INTRODUCTION 8 C ONTROLLED rectifiers offer distinct advantages over typically used uncontrolled diode, or phase-controlled thyristor rectifiers in ac/dc/ac converters for variable speed drive applications. These advantages include unity input power factor and greatly reduced input line current harmonic distortion due to the nearly sinusoidal input line current attainable with controlled rectifiers. Reversible power flow is also possible. Fig. 1 shows a schematic of the transistor-diode implementation of the voltage-sourced controlled rectifierinverter drive system that is discussed in this paper. It has been shown, [l], [2], that in order to control the dc link voltage U&, the input line currents must be regulated. In typical rectifier controllers that have been presented to date, such as those in [3]-[5], the dc bus voltage error is used to synthesize a line current reference. Specifically, the line current reference is derived through multiplication of a term proportional to the bus voltage error by a template sinusoidal waveform. The sinusoidal template is directly proportional to the input voltage, thus resulting in unity input power factor. The line current is then controlled to track this reference using a suitable current regulator. The use of load feedforward [ 121 eliminates the steady-state error that exists when using only proportional bus voltage Manuscript received November 5 , 1991; revised August 7, 1992. The author is with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332. IEEE Log Number 9204294. 9 J Magktude of ia Feedforward Fig. 2. Block diagram of rectifier control including output power estimator for load feedforward. feedback. This is accomplished by deriving the line current magnitude reference from the average load current, found from an output power estimator, plus the proportional-integral (PI) controlled bus voltage error. The PI controller then eliminates the steady-state error in the bus voltage while still maintaining proper current regulation. In addition, the dynamic response of the bus voltage to changes in load is significantly increased due to the load feedforward. A block diagram of this type of control scheme is shown in Fig. 2. The task of current regulation can be accomplished through the use of any number of existing current control schemes 0885-8993/93$03.00 0 1993 IEEE
  2. 2. 31 HABETLER: A SPACE VECTOR-BASED RECTIFIER REGULATOR for use in voltage-sourced inverters. These include hysteresis regulators [ 131 and synchronous reference frame regulators [ 141, which use a proportional-integral (PI) controller and sine-triangle pulsewidth modulator (PWM). The sine-triangle PWM-based schemes have the advantage of fixed switching frequency, greatly reducing device stresses in comparison with the hysteresis regulator. The problem of current control in ac/dc converters differs from dc/ac converters, even though the topology is unchanged. This is because the circuit quantities (input voltage, source impedance, and dc link voltage) that determine the input line current are known in ac/dc converters. Therefore, the current control scheme can be formulated that derive the rectifier switching function by estimating or predicting the line current trajectory. One such scheme, Predicted Current Control with Fixed Switching Frequency (PCFF), was presented in [6] and [7] for use in rectifier control. This scheme calculates the duty cycle required for each inverter leg that drives the respective line current to the reference value in one switching period. This scheme is based on symmetrical uniform-sampled PWM where the reference voltage is found from a calculation of the rectifier input (ac side) voltage required to drive the associated line current to the reference value in a deadbeat fashion. In this paper a rectifier current regulation scheme is presented that relies on calculation of a duty cycle for each switching state (space vector) of the rectifier. Calculation of a space vector-based duty cycle was shown to be effective method for inverter voltage regulation [8]. This fundamental concept is extended to a deadbeat, predictive, rectifier current regulator in this paper. Direct control of the current space ( d q ) vectors has the advantages of improved harmonic performance, especially at low dc bus voltages (modulation index near unity), and improved dynamic response to a transient in the load power or dc bus voltage at any operating condition in comparison to per-phase PWM techniques. 11. PRELIMINARY CONSIDERATIONS FOR RECTIFIER CONTROL where the e superscript denotes a rotating reference frame quantity. The stationary and rotating reference frame representations of the input line currents are (4) and I" = (Id cos wt = I; + Iq sin wt) + j(-ld sin wt + Iq cos wt) +jI; (5) respectively. B . Formulation of the Current Reference In order to obtain unity power factor, it is desired that 1 : be zero since I t is the component of I in quadrature with E. Therefore, I,"* = 0 (6) where the * superscript denotes a reference or command value. With I: controlled to zero, I: corresponds to the magnitude of the input line current. As depicted in Fig. 2, the magnitude of the input current is formulated from a combination of the bus voltage error and a load feedforward term. The d-axis reference current then, is, is where Pout an estimate of the output power delivered to the motor load and R is a parameter that determines the trade-off between steady-state and transient performance. Quantity R is a user-definable control parameter, not a circuit resistance. It can be noted that if R is small, the bus regulation term is dominant, and steady-state bus voltage ripple is low. If R is large, the transient performance is improved. The derivation of the output power is given in [2] and is found from an estimate of the motor torque and the stator frequency. A . Reference Frame Transformations It is assumed that the source is a balanced, sinusoidal threephase voltage supply with frequency w. Because unity input power factor is desired, it is convenient for this analysis to take the angle of the a-phase input voltage as the reference angle. That is, E, = E cos wt Eb = E cos(wt - 2 ~ / 3 ) E, = E cos(wt + 2 ~ / 3 ) . I* = I: +jI,* = I:* coswt (1) The input voltage in a stationary dq reference frame is given by ]E = Ed+jEq = E,+j =E where V I is the dq vector of the stator voltage, A, is the stator flux, and IT is the component of stator current in quadrature with A,. The current reference is then transformed back to the stationary reference frame by cos wt+jE sin wt. /? + jId* sinwt. (9) The stationary reference frame current I* is then used by the space vector controller described in the next section. 111. DEADBEAT STEADY-STATE SPACE VECTORCURRENT REGULATION AI The input voltage can then be transformed to a rotating reference frame dq quantity aligned with E,. This is given by E"= Edcoswt+jEqsinwt =E (3) The current control method proposed in this paper is accomplished by estimating the appropriate duty cycles for the dq current vectors adjacent to the reference vector in such a way that the line current is driven to the reference value at the end
  3. 3. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993 32 T s 'a I I '.W Fig. 3 . Rectifier switching states and input voltage vectors. Fig. 4. Switching function generation for space vector-based regulator. of a constant, known period of time. In the case of steadystate operation, this is equivalent to calculating a reference voltage vector that accomplishes the deadbeat control and calculating the duty cycles for the appropriate states using space vector PWM as described in [8]. In the case where the voltage vector necessary to drive the current vector to the reference value cannot be synthesized in half the switching period (i.e., transient operation), an altemative scheme is used to ensure the current is driven in the direction of the reference vector while still maintaining a constant switching frequency. This is described in detail in Section IV. Let T, denote half the switching period. That is, T, = 1/(2fsw), where f s w is the rectifier switching frequency. If T,, is assumed to be small in comparison with the frequency of E, then IE can be assumed constant over the period T,. Let t, denote the time at the beginning of one T, period. Then the change in line current over one period (neglecting the source resistance) is where Tk and Tk+l are the time durations spent on states k and k 1, respectively. The remainder of the period is spent on the zero state. That is, A I = I(t, + T,) - I(t,) = L i - T, (10) where V is the average value of the ac side rectifier voltage over T,. In order to control the line current in a deadbeat manner, A I must be AI = I* - I(t,). (1 1) The value of average rectifier input voltage V *, which satisfies (ll), is The rectifier ac side voltage dq vector that corresponds to the kth rectifier switching state ( k = 0 , . . . ,7) is given by The seven rectifier switching states and the v k vectors are shown in Fig. 3. If VI, and v k + l are the two voltage vectors adjacent to V * ,then V * can be synthesized using the space vector PWM by 1 v* = -(VkT T, k f Vk+lTk+l) (14) + To = T , - Tk - Tk+1. (15) In order to minimize the ripple content of the line current, the time spent on the zero state is equally distributed at the beginning and the end of the T, period. The rectifier leg switching functions and the definitions of To,Tk, and Tk+1, are illustrated in Fig. 4. With space vector current regulation, the line currents are controlled to the reference value over the period T,, and each of the three inverter legs has been exercised once. Therefore, it is clear that with this scheme the current is controlled on a twice-per-switching cycle basis. This is not the case with per-phase duty cycle control as in [6]. In the PCFF scheme, the voltage is updated only once per switching period. The ability of the space vector current regulator to control the line current twice per switching cycle is important several reasons. The first is, clearly, that the ripple current is reduced since the current (and voltage) references are updated twice as fast with the space vector control. This is particularly important at lower switching frequencies where error is introduced by assuming that IE is constant over the switching period. Another important reason is that in most current regulator applications, including the one described here, the current reference I* typically cannot be predicted at the end of the switching cycle. The reference current is known only in real time. Therefore, replacing I* with I*(t,) in (12), results in the line current following the reference current with one period of delay. Therefore, the steady-state line current error is significantly reduced with the space vector current regulation. PWM PREDICTIVE CONTROL IV. ASYMMETRICAL The harmonic current distortion of the PCFF duty cycle control scheme presented in [6] can be improved, especially at lower switching frequencies (4 kHz), by updating the duty cycle twice per switching period, as is the case with space vector current regulation and conventional asymmetrical uniform sampled PWM [ 151. This is accomplished by updating the duty cycle and the control voltage U* for each phase, at the beginning of each straight line segment of the triangle carrier. This is illustrated in Fig. 5. The control voltage that
  4. 4. HABETLER: A SPACE VECTOR-BASED RECTIFIER REGULATOR ,4-TS L - I V - Fig. 7. Diagram illustrating the change in line current associated with each rectifier switching state. - +. I r. 1 1 AI. I vc f 4- Ts + I I va 33 s Rectifier Inverter Harmonic Distortion Comparison Fig. 8. Block schematic of ac/dc/ac converter V. HARMONIC PERFORMANCE COMPARISON 0 0.2 0.4 0.6 0.8 1 1.2 Modulation Index Fig. 6. Line current harmonic distortion as a function of modulation index. accomplishes the deadbeat current regulation is The total harmonic distortion in steady-state operation can be compared for the three control schemes discussed above. The simulation results are illustrated in Fig. 6, with a switching frequency of 2 kHz. The operating conditions for each switching scheme are identical. It is very important to note that the space vector current control provides superior harmonic performance to per-phase duty cycle generation since the space vector control results in a switching pattern wherein the zero state is symmetrically distributed at the beginning and end of each half cycle. This ensures minimum current ripple. The PCFF scheme also symmetrically distributes the zero state but only does so over the entire switching period, 2T, = l/fsw. VI. TRANSIENT OPERATION where Vtri is the peak of the triangle carrier as indicated in Fig. 5, and 2,. is the reference current for phase j a, b, c. This scheme will be referred to as “asymmetrical PWM predictive control.” In comparing (12) with (16), it is very important to note that the actual duty cycle of each pole of the rectifier is identical for the asymmetrical PWM predictive control and the space vector current regulator. This is to be expected since they both accomplish the same deadbeat control. The switching pattems as a function of time, however, are not the same since the space vector current regulator distributes the time spent on the zero state symmetrically within the period and the asymmetrical PWM predictive control does not. This accounts for the difference in harmonic performance as discussed in the next section. Maximizing transient performance is very important in rectifier control in order to minimize the magnitude of transients in the dc bus voltage and, subsequently, to minimize the size of the dc bus capacitance required. The semiconductor switches in the boost-type rectifier discussed in this paper must block a higher voltage than the devices used in conjunction with a conventional diode rectifier. Operation at modulation index near unity is therefore desirable. Therefore, in addition to considering the response of the current regulator to load transients, it is also important to consider the performance of the current regulator at modulation index near unity. These two topics are addressed in this section. Transient operation can be defined as an operation condition where the solution of (14) results in the sum TI, Tk+1 being +
  5. 5. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993 34 25 . I I 25 . I I I I I I I I I 1 1.25 - & 1.25 a v v + $ 0 c ( $ 0 2 3 -1.25 -1.25 -2.5 0 I I 1 I I I I 1 2 3 4 5 6 -2.5 -2.5 7 Q -1.25 1 . U 25 . Ids (pu) wt (rad) (b) 35 . & 3 G 2 $ 2 04 . M z 5 0.8 e 25 . v 54 15 . 0 4 1 0 'E .- 2 -0.4 3 = 05 . -08 0 0 1 2 3 4 ,5 6 7 wt (rad) (c) 3 A & - 2 I g 1 U .B d 0 Y B -1 g c -2 2 -3 O i 2 3 4 wt (rad) 5 6 7 (e) Fig. 9. Converter waveforms under steady-state operating conditions. (a) Input line current. (b) Input line current dq vector. (c) DC bus voltage. (d) Line-line switching function. (e) Inverter line current. larger than T,. In other words, the value of V* calculated in (12), cannot be synthesized, in an average sense, from VI, and Vk+l over the period T,. Under this condition, deadbeat control of the current is no longer possible. The best transient performance that can be achieved is to switch the rectifier such that the line current is driven in the direction of the reference value. That is, it is desired that the angle of AI be control to be equal to the angle of AI*. Controlling the rectifier so that the actual average voltage at the input to the rectifier is proportional to V* does not cause the current to be driven in the correct direction. This can easily be seen from (12). The angle of AI is equal to the angle of V* - E. Therefore, an actual input voltage vector proportional to V* (but not equal to V*) results in an incorrect angle for AI. Let AI, be the change in line current associated with switching state m. That is, AI, = E-V, Li ~ T, m= 1,2,...6 . (17)
  6. 6. 35 HABETLER: A SPACE VECTOR-BASED RECTIFIER REGULATOR The two adjacent states, m = k and m = k + l , which drive the line current in the correct direction, can be found by calculating AI, for each of the six nonzero switching states. Then k and k 1 correspond to the two AI, vectors that are adjacent to AI*. Note that the value of T, is not needed to determine k and IC 1. Referring to Fig. 7, states 2 and 3 are adjacent to the reference current vector. 1 are known, then the values of T and k Once k and k Tk+1, which would drive the current to the reference value, can be found from + + + LiAI* = E(Tk f Tk+1)- VkTk - Vk+1Tk+1- 4.5 3.6 a Y 2.7 % $ d 1.8 n 4 0.9 (18) 0 0 Equation (18) represents two equations with Tk and Tk+1 as k Tk+1 will be larger than T, in the unknowns. The sum T transient condition. Therefore, in order to maintain a constant switching frequency, T and Tk+1 can be multiplied by a k constant without changing the angle of AI*. That is, 1 2 3 4 5 6 7 wt (rad) + (a) 2 - 1.5 Y where T and TL+l are the actual switching intervals for states L IC and k + 1, respectively. Note from (19) that T + L = T,, and therefore the zero state is not used under transient i! 0.5 conditions. This is to be expected since the current is to be driven to the reference value as quickly as possible. -1.5 VII. RESULTS A block schematic of the complete rectifier control system including the space vector current regulator is shown in Fig. 8. Simulation results for the ac/dc/ac converter with the proposed space vector current regulator is shown in Figs. 9 and 10. In all the simulation results given, including those in Fig. 6, E = 1.0 pu, I , = 0.7 PU, fSw = 2.1 kHz, V,., = 3.0 Vl pu, input frequency = 60 Hz, output frequency = 90 Hz, 1 / ( 2 ~ 6 0 C )= 3 PU, ( 2 ~ 6 0 ) L i= 0.2 PU, ( 2 ~ 6 0 ) L ,= 0.2 pu, R = 0.05 pu, from (7) K = 12. In order to interpret the rectifier regulator performance in terms of actual units, for a 20-kW converter, with a 300-Vdc bus, the value of C in the simulations is 100pF. Given these actual units, from Fig. 9(c), the peak-peak ripple on the dc bus would be 35 V. Fig. 9 illustrates steady-state operating conditions. The inverter is controlled by a conventional synchronous frame current regulator with a PI controller. Note that the use of load feedforward eliminates steady-state error in the dc bus voltage. Also, the steady-state input line current error caused by the half-cycle delay in current regulation is compensated by the bus voltage controller. No addition compensation is required. Fig. 10 shows the response of the rectifier control system to a step change in inverter current command. Note the quick response of the dc bus voltage and the input line current resulting from the space vector current control scheme described above. It should by noted that in examining Fig. 10(a), the bus capacitance is very small, as mentioned in the previous paragraph. Therefore, the peak voltage under the transient condition could certainly be reduced by increasing the value of C. The important point here is that the line current 0 1 2 3 4 5 6 7 5 6 7 wt (rad) (b) 3 2 - 1 & Y 4- 8 0 5 1 -2 0 1 2 3 4 wt (rad) (C) Fig. 10. Converter waveforms illustrating transient response to a step change in inverter current reference. (a) DC bus voltage. (b) Rectifier line current. (c) Inverter line current and reference current. responds very quickly in order to control the bus voltage. Also, the time constant of the bus voltage response depends of the value of R in (7). As R becomes smaller (or K becomes larger) the transient response time is reduced. This, of course, comes at the expense of steady-state performance. VIII. CONCLUSIONS A new method for current regulation in for use in controlled rectifiers has been presented in this paper. The proposed control scheme is based on a deadbeat or predictive control
  7. 7. 36 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 8, NO. 1, JANUARY 1993 of the input line current. This is accomplished by calculating the duration of time spent on the appropriate rectifier states in order to drive the line current to the reference value at the end of the cycle. The current is controlled on a half-cycle basis (i.e., the switching times are calculated twice per switching cycle). This results in superior harmonic performance. The space vector current regulator was shown to result in lower harmonic current distortion in comparison to existing predictive schemes, especially at a high modulation index, where the rectifier is typically operated. This is because space vector control results in the zero state being symmetrically distributed each half cycle. In addition, a method of controlling the current under transient conditions was introduced, wherein the line current is always driven in the direction of the reference voltage, reducing the response time of the current regulator. The space vector current control, when used in conjunction with load feedforward and proportional- integral control of the dc bus voltage, was shown to provide unity input power factor, low-bus voltage ripple, and excellent transient response, even with low values of dc link capacitance. [8] H. W. van der Broeck, H.-Ch. Skudelny, and G. Stanke, “Analysis and realization of a pulse width modulator based on voltage space vectors,” IEEE Trans. Industry Applications, vol. IA-24, no. 1, pp. 142-150, 1988. [9] K. P. Gokhale, A. Kawamura, and R. G. Hoft, “Dead beat control of PWM inverter for sinusoidal output waveform synthesis,” IEEE Trans. Industry Applications, vol. IA-23, no. 5, pp. 901-910, 1987. [IO] J. Holtz and S. Stadtfeld, “A PWM inverter drive system with on- line optimized pulse pattems,” European Conf. Power Electron. Applications, Brussels, Belgium, 1985, pp. 321-325. [ l I] M. P. Kazmierkowski, M. A. Dzieniakowski, W. Sulkowski, “Novel space vector based current controllers for PWM inverters,” IEEE Trans. Power Electron., vol. 6, no. 1, pp. 158-166, 1991. [I21 S. K. SUI and T. A. Lipo, “Design and performance of a high frequency link induction motor drive operating at unity power factor,” IEEE Trans. Industry Applications, vol. 26, no. 3, pp. 4344L0, 1990. 1131 D. M. Brod and D. W. Novotny, “Current control of VSI-PWM inverters,” IEEE Trans. Industry Applications, vol. IA-21, no. 4, pp. 769-775, 1984. [I41 T. M. Rowan and R. J. Kerkman, “A new synchronous current regulator and an analysis of current regulated PWM inverters,” IEEE Trans. Industry Applications, vol. IA-22, no. 4, pp. 678490, July/Aug. 1986. [I51 G. Franzo, M. Mazzucchelli, L. Puglisi, and G. Sciutto, “Analysis of PWM techniques using uniform sampling in variable-speed electrical drives with large speed range,” in IEEE-IAS Annual Meeting Conf. Rec., 1984, pp. 568-575. REFERENCES [ I ] R. Mahadevan, “Problems in analysis, control, and design of switching inverters and rectifiers,” Ph.D. dissertation, Califomia Institute of Technology, Pasadena, CA, 1986. [2] T. G. Habetler and D. M. Divan, “Angle controlled current regulated rectifier for ac/ac converters,” IEEE Trans. Power Electron., vol. 6, no. 3, pp. 463469, July 1991. [3] J. W. Dixon, A. B. Kulkami, M. Nishimoto, and B. T. Ooi, “Characteristics of a controlled current PWM rectifier-inverter link,” IEEE Trans. Industry Applications, vol. IA-23, no. 6, pp. 1022-1028, 1987. [4] M. Nishimoto, J. W. Dixon, A. B. Kulkami, and B. T. Ooi, “An integrated controlled-current PWM rectifier-chopper link for sliding mode position control,” in IEEE-IAS Annual Meeting Conf. Rec., 1986, pp. 685-691. [5] B. T. Ooi, J. W. Dixon, A. B. Kulkami, and M. Nishimoto, “An integrated ac drive system using a controlled-current PWM rectifier inverter link,” IEEE Trans. Power Electron., vol. PE-3, no. 1, pp. 64-71, 1988. [6] R. Wu, S. B. Dewan, and G. R. Slemon, “A PWM ac to dc converter with fixed switching frequency,” IEEE Trans. Industry Applications, vol. 26, no. 5, pp. 88G885, 1990. [7] R. Wu, S. B. Dewan, and G. R. Slemon, “Analysis of a PWM ac to dc voltage source converter under predicted current control with fixed switching frequency,” IEEE Trans. Industry Applications, vol. 27, no. 4,pp. 756764, 1991. Thomas G. Habetler (S’82-M’83-S’SS-M’89) received the B.S.E.E. and M.S. degrees in electncal engineering from Marquette University, Milwaukee, WI, and the Ph.D. degree from the University of Wisconsin-Madison, in 1981, 1984, and 1989, respectively. From 1983 to 1985 he was employed by the Electro-Motive Division of General Motors as a Project Engineer He was involved in the design of switching power supplies and voltage regulators for locomotive applications. In 1985 he was awarded the General Motors Fellowship to attend the University of Wisconsin-Madison. He is currently an Assistant Professor of Electncal Engmeenng at the Georgia Institute of Technology, Atlanta. His research interests are in switching converter technology and electnc machine dnves. Dr. Habetler was corecipient of the 1989 first- and the 1991 secondpnze paper awards of the Industnal Dnves Committee of the IEEE Industry Applications Society. He is chair of the Membership and Publicity Committee of the IEEE Power Electronics Society and is a member of the IEEE-IAS Industnal Power Converter Committee and Industnal Drives Committee.