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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 52, NO. 2, APRIL 2005

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Steady-State and Dynamic Study of One-Cycle-Controlled Three-Phase Power-Factor Correction Guozhu Chen, Member, IEEE, and Keyue M. Smedley, Senior Member, IEEE

Abstract—One-cycle control power-factor correction (OCCPFC) with vector operation is a promising rectification method that eliminates harmonics and improves the power factor. It features great simplicity, high performance, and excellent stability. This paper performs analysis and design of OCC-PFC in both the steady-state and dynamic transients. The sufficient stability condition for three-phase OCC-PFC is derived. Some typical large-signal perturbations in practice are then used to verify the theoretical predictions. The paper also provides some guidelines for the selection of the circuit parameters in practical application. All analysis results were verified by simulation or experiments based on a three-phase 1-kW pre-industrial OCC-PFC prototype. Index Terms—One-cycle control (OCC), power-factor correction (PFC), power quality control, rectifier.

I. INTRODUCTION

E

LECTRONIC equipment, such as computers, communication equipment, appliances, etc., are widely used in homes, industry, space, and the military. In many cases, a rectifier is required to convert ac electricity to dc for the use of electronic equipment. Unfortunately, the traditional rectifiers of diode or thyristor type draw pulsational current from the utility lines, which adds low power factor (PF) to the system due to both the phase displacement and harmonic distortion factors. The harmonics not only lower the transmission efficiency, but also harm the transformers and other equipment connected to the line. To limit the low PF and total harmonic distortion (THD) of electronic equipment, the international standards such as IEC 1000-3-2, EN61000-3-2, and IEEE 519 are in place for electronic products of 75 W or higher. Many active powerfactor-correction (PFC) techniques have been developed for rectifications including single phase for low to medium power, and three phase for high power [1]–[4]. Among the three-phase rectifiers, the six-switch bridge rectifier is one of the commonly used topologies, in which the low

Manuscript received December 30, 2003; revised March 18, 2004. Abstract published on the Internet January 13, 2005. This work was supported by the California Energy Commission under Grant 52100A/00-27. G. Chen was with the Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697 USA, on leave from the College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China. K. M. Smedley is with the Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TIE.2005.843920

PF is mainly caused by the harmonic distortion. In this topology, (1) where and are the three-phase real power and apparent power respectively; the and THD are the displacement angle between the fundamental voltage and current and the THD and relative to the fundamental component, respectively; are the displacement factor (narrow sense PF) and distortion factor, respectively; as a combination, the is defined as PF (in a broad expression sense). Therefore, the modern rectifier should have the function of harmonics suppression. In most of the reported PFC rectifiers, hysteresis control [5] and – [6] transformation methods were used to control six-switch bridge boosts. Hysteresis control results in variable switching frequency, which causes difficulties for filter design. Constant switching frequency modulation also was provided in [4]. However, many sensors for the voltages and multipliers are required to form the three-phase current references, which results in a complex circuit, high cost, and low stability. The one-cycle control (OCC) technique [7] established a large-signal nonlinear pulsewidth-modulation (PWM) scheme that features a simple circuit, high performance, and universal applications. OCC has been successfully implemented in many sectors of power electronics including dc/dc converters, amplifiers, dc/ac inverters, PFC, and active power filter (APF) in both single phase and three phase [8]–[11], [14]. The OCC controller usually uses an integrator with reset to force the controlled variables to meet with the control goal in each switching cycle. It has the advantages of no reference calculation, fast response, and high precision. A three-phase vector-operated OCC-PFC was first reported in the literature [12]. It achieves three-phase unity PF and low THD without complex reference calculation. The OCC control circuit is very simple: one integrator with reset along with a few linear and digital components. The OCC-PFC features constant switching frequency (desirable for magnetic design) and vector operation (lower switching losses compared to PWM operation). In order for a new concept to be industrially and commercially adoptable, it is necessary to perform a rigid evaluation for all possible practical scenarios and to develop solid engineering design guidelines. This paper studies the performance

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 52, NO. 2, APRIL 2005

Fig. 1. Circuits of the three-phase OCC-PFC with vector operation. (a) Main circuit of the OCC-PFC. (b) OCC controller.

of both steady-state and dynamic states. All analyses are verified by the experiments based on a three-phase 1-kW pre-industrial OCC-PFC prototype. II. REVIEW OF OCC-PFC WITH VECTOR OPERATION The OCC-PFC shown in this paper is composed of a threephase bridge as the main circuit and a one-cycle controller as shown in Fig. 1(a) and (b), respectively. The main circuit is a is realized three-phase bridge rectifier. The dc-bus voltage by a capacitor, which acts as an energy storage bank for the dc load. The power line provides symmetric sinusoidal three-phase and currents in normal operation. voltages The bridge rectifier contains three inductors and six switches on the up-leg (when ) or the ) for phase respectively. The low-leg (when two switches in each phase are operated in a complementary fashion. The converter operates in continuous conduction mode (CCM) with high switching frequency and proper design of the inductors [12].

The control circuit contains an OCC core, some vector operation logic, and a feedback circuit, as shown in Fig. 1(b). The OCC core includes three adders, two comparators, and an integrator with reset as well as two flip-flops; the vector operation logic contains a vector region selection circuit that divides a line cycle into six regions according to the power line voltage , from a multiplex circuit that selects the vector currents and directs two drive sigthe three phase currents nals to the right switches, and the feedback loop that regulates the dc-bus voltage (slow loop) and feeds the OCC fast loop. The so dc-bus voltage is regulated against a reference voltage that it keeps constant during steady-state operation. The error to signal passes a proportional–integral (PI) controller . The OCC control core takes the form the modulation signal together with the vector currents modulation signal and to produce switch trigger signals and at the output of the flip/flops, whose status are (1: ON; 0: OFF). The duty ratios are associated with switch . For vector operation, a line cycle is equally divided into six regions. Only the switches of two phases are controlled using the

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TABLE I VECTOR OPERATION MECHANISM OF THE OCC-PFC

duty ratios given by the OCC core at each vector region while the switches in the remaining phase are kept ON or OFF during the whole selected region, which results in minimum switching losses. The vector turns to a new region in each 60 of one line cycle. This mechanism is shown in Table I. With the circuit in Fig. 1(b) operating according to the rotation sequence given in Table I, the control core (2) for PFC are can be obtained, which means that the duty ratios and controlled by the selected phase currents and the dc-bus voltage error (2) and

defined as (3)

where and are the duty ratios of the switches and current-sensing resistance

is

(4) is the emulated resistance that reflects the real power where of the load. Equation (4) is the control goal of OCC-PFC, i.e., the line currents are sinusoidal and in phase with the line voltages and therefore, unity power factor and low THD are achieved. III. DYNAMIC ANALYSIS OF OCC-PFC A. Poincare Mapping Analysis for OCC-PFC The OCC-PFC is a large-signal nonlinear system. Using the Poincare mapping technique [13], the OCC modulation is convergent when the condition given in (5) is satisfied

Fig. 2. Operating waveforms of the OCC controller. (a) Zoom-in waveforms of the switching operation (T : switching period). (b) Zoom-out waveforms of the region rotation (T : line period).

can be considered constant at the conjoint switching cycles in Fig. 2(a) and their long-term outlines vary in a low frequency and can be as shown in Fig. 2(b). Therefore, the slopes calculated from the switch states as in Table II, which is verified by simulation. In Table II, and and, therefore, their linear combinations in region I (0–60 ) are calculated and listed based on different and are the line–line voltages of phases switching states; and to phase , respectively; is the inductance of the switching filter. It should be noted that only two switching sequences are possible—I, II, IV or I, III, IV, therefore, the slopes of are as follows: (6)

(5) For the control waveforms shown in Fig. 2(a), and are the rising and falling slope of the current , or ), respectively; is the equivalent slope generated by the integrator with reset. of the PFC is usually Since the switching frequency thousands of times higher than that of the vector currents, the and ripples of the control signals are dominated by the switching actions, while their averages

Combination (5) and (6) yields the local stability condition, (7) for a three-phase three-wire Considering rectifier system, (8) can be obtained – –

(8)

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TABLE II EQUIVALENTS AND CURRENT SLEW RATE OF i

When the control goal in (4) is satisfied, and are in and , respectively, in region I as shown in phase with Fig. 2(b), which has the physical meaning that the system has unity PF.

•

It should be noted that the result derived from or from other regions is essentially the same as that shown in (7) and (8). This is because of the symmetry of the three-phase voltage and the rotational regions. where , the worst case of (7) occurs when reaches the minimum and light load, since reflects the load level referring to (3). From (7), the following can be concluded, Using

•

Equation (7) is a sufficient condition for global convergence of the OCC algorithm. However, it is not necessary and over stringent for circuit parameter selection. For example, the dc-bus voltage must be large enough to keep the system operating in the boost mode. Usually, the and small control stability desires large inductance according to the equation. current sensor resistance However, the maximal inductance and minimal reare restricted by other factors and, therefore, sistance the sufficient condition is hard to be satisfied from the viewpoint of parameter selection in practice.

•

AND i

IN

REGION I

The OCC-PFC shall function stably even when (7) is only partially satisfied. As shown in Fig. 2(b) in region I – , for example, varies between and , therefore, (7) is always satisfied when approaches its high end of the region if is designed. Supposing (7) is not satisfied when is at its low end of the region, the divergence will be limited locally, i.e., the current will not grow unboundedly before the convergence condition is caught again. The global stability for OCC-PFC is guaranteed when , which gives another sufficient condition for stability and a much greater freedom for circuit parameter selection.

B. Stability Verified With Large-Signal Perturbations To verify the global stability condition of OCC-PFC, some typical dynamic states of the circuit in practical operation are investigated, which include the following: • changes of operation state, e.g., start and/or stop of the circuit; • fast load change, e.g., load power varies in a step; • large random perturbations from fault or environment. The first two groups of perturbations were simulated as shown in Fig. 3. Referring to Fig. 1(b), the signals applied to the comparators should have proper values so that the duty ratios and

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Fig. 4. Large random signal perturbations to the OCC modulation.

Fig. 3. Simulation for some large-signal perturbations of OCC-PFC. TABLE III FOUR TYPES OF PERTURBATION OF OCC MODULATION

locate in a practical range, e.g., and are the maximal and minimal duty ratios, respectively. Here, one of the two signals is investigated in one of the six regions (e.g., region I). The result is valid for all regions due to the symmetric operation (9) The physical meaning of this equation is that the values of must locate within the envelope of . In OCC-PFC, is adjusted according to the power level with a PI controller according to (3) and the variable amplitude carrier is implemented as shown in Fig. 3. have Any random perturbations adding to signal the equivalent effect as changing the relative position of signal to the carrier. There are four common types in practice: I, II, III, and IV, which are positive or negative perturbation added to the rising or falling edges, respectively, as shown in Table III and Fig. 4. Since the OCC-PFC is globally stable according to the convergence condition analyzed above, the system is stable with the

Fig. 5.

1-kW pre-industrial prototype of the OCC-PFC with vector operation.

large random perturbations. However, maximal duty ratio must be used to prevent short-through. This is because the perturbation of type III/IV may cause the comparator’s never turning over during the whole switching cycle. IV. THREE-PHASE OCC-PFC PROTOTYPE AND EXPERIMENTS A 1-kW pre-industrial prototype three-phase APF (PFC) with OCC and vector operation was built as shown in Fig. 5. The prototype has built-in housekeeping power supplies, overvoltage, overcurrent, and overheat protections, stop–start preset sequence, and isolation, which are usually required in industrial applications. The experiment conditions and parameters are as follows: power line 120 Vrms/60 Hz/three phase; output power 1.0 kW (varies from 10% to 110%);

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Fig. 6. Steady-state experimental waveforms. (a) Line voltage (ch1) and current (ch2) before OCC-PFC started (phase A). (b) Line voltage (ch1) and current (ch2) after OCC-PFC started (phase A), (c) Three-phase line current (ch2 and ch3 are measured with transducer, while ch4 with current probe). TABLE IV EXPERIMENTAL CURRENT COMPONENTS OF OCC-PFC (PHASE A)

dc-bus voltage/capacitance 365 V /90 F; sensor resistor (loop gain) ; sensor bandwidth 12.5 kHz; kHz; switching frequency switching filter (adjustable) 0.4–2.2 mH.

The steady-state waveforms of the load, the power line, and the OCC-PFC with vector operation are shown in Fig. 6(a)–(c), respectively. The spectrum of the phase current is obtained using the fast Fourier transform (FFT) analyzer of the TDS3202 oscilloscope.

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Fig. 7. Dynamic waveforms of OCC-PFC. (a) Start transient: line voltage (ch1) and current (ch4) (Phase A). (b) Start transient: dc-bus voltage. (c) Load transient: power changes from 50% to 100%.

The data of the current in the frequency domain are given in Table IV. Fig. 6 and Table IV demonstrate that the designed OCC-PFC improves the current THD caused by the nonlinear load from about 94% to about 3% and, thus, it improves the PF from 0.73 to 0.99, i.e., the PFC controls the three-phase boost rectifier so that the system draws sinusoidal current from the power line. To demonstrate the dynamic performance of the prototype, both the startup and load step transients were investigated. Fig. 7(a) and (b) shows the dynamic waveforms when the OCC-PFC started. The response of the line current and the dc-bus voltage together with the in-phase voltage are shown, respectively. It can be seen that it is stable, with no dangerous overshoot, and the response speed is proper after the PFC is started. The dynamic period shown in Fig. 7(a) is 3–5 line cycles, which is

acceptable in practice and can even be shortened with adjustment of the PI parameters of the voltage loop. Fig. 7(c) shows the dynamic waveforms when the nonlinear load power changed from 50% to 100% of 1 kW in step. The response of the line current (phase ) referring to the line voltage is shown in Fig. 7(c) (ch1 and ch4, respectively). It can be seen that the load transient state is also smooth, fast, and stable with the OCC-PFC. From Figs. 6 and 7, it can be summarized that the OCC-PFC has the following: • excellent steady-state performance of harmonics suppression; • no overshoot occurs in transient states including the circuit start and power changed in step; • fast dynamic current-tracking ability; • stable and reliable operation.

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V. CONCLUSION OCC-PFC with vector operation reported in the literature [12] provides a simple, effective, and reliable rectification method. The OCC-PFC rectifier use a three-phase bridge as its power stage and an integrator with reset along with a few logic and linear components as its control core to realize three-phase input current PFC. As shown in [15], the OCC-PFC rectifier has rigid adaptability to the power system whether it is three-phase balanced or unbalanced. While this method is being adopted for industrial and aerospace applications, it is important to map out its global stability. This paper has studied the performance both in steady-state and dynamic transients. A convergence condition for the OCC algorithm is derived using the Poincare method. It is found that the global stability can be guaranteed by partial satisfaction of the convergence function, because the divergence occurs only in a confined region of the line cycle. Soon the operation proceeds outside the region, and the convergence condition is caught, which traps the system motion within the bounded region. In addition, as the divergence occurs during the lower portion of the line current, the spikes caused by the divergence are negligible. Some typical large-signal perturbations in practice were investigated using the stability condition. Guidelines were provided for system design with stable operation and high performance. Experimental results based on a three-phase 1-kW pre-industrial PFC prototype verified that the OCC-PFC achieves excellent performance in both the steady-state and dynamic transients. REFERENCES [1] P. D. Ziogas, “An active power factor correction technique for threephase diode rectifiers,” IEEE Trans. Power Electron., vol. 6, no. 1, pp. 83–92, Jan. 1991. [2] J. W. Kolar, H. Ertl, and F. C. Zach, “Space vector-based analytical analysis of the input current distortion of a three-phase discontinuous-mode boost rectifier system,” IEEE Trans. Power Electron., vol. 10, no. 6, pp. 733–745, Nov. 1995. [3] N. Mohan and R. Naik, “Analysis of a new power electronics interface with approximately sinusoidal 3-phase utility currents and a regulated DC output voltage,” IEEE Trans. Power Delivery, vol. 8, no. 2, pp. 540–546, Apr. 1993. [4] H. Mao, D. Boroyevich, and F. C. Lee, “Analysis and design of high frequency three-phase boost rectifiers,” in Proc. IEEE APEC’96, vol. 2, San Jose, CA, Mar. 3–7, 1996, pp. 538–544. [5] M. S. Dawande, V. R. Kanetkar, and G. K. Dubey, “Three-phase switch mode rectifier with hysteresis current control,” IEEE Trans. Power Electron., vol. 11, no. 3, pp. 466–471, May 1996. [6] F. Z. Peng and J. S. Lai, “Generalized instantaneous reactive power theory for three-phase power system,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 293–297, Feb. 1996. [7] K. Smedley and S. Cuk, “One-cycle control of switching converters,” in Proc. IEEE PESC’91, Cambridge, MA, 1991, pp. 888–896. [8] K. M. Smedley, L. Zhou, and C. Qiao, “Unified constant-frequency integration control of active power filters-steady-state and dynamics,” IEEE Trans. Power Electron., vol. 16, no. 3, pp. 428–436, May 2001. [9] K. M. Smedley and S. Cuk, “Dynamics of one-cycle controlled Cuk converters,” IEEE Trans. Power Electron., vol. 10, no. 6, pp. 634–639, Nov. 1995.

[10] C. Qiao, T. Jin, and K. Smedley, “Unified constant-frequency integration control of three-phase active-power-filter with vector operation,” in Proc. IEEE PESC’01, vol. 3, Jun. 2001, pp. 1608–1614. [11] C. Qiao, K. M. Smedley, and F. Maddaleno, “A comprehensive analysis and design of a single phase active power filter with unified constant-frequency integration control,” in Proc. IEEE PESC’01, vol. 3, Jun. 17–21, 2001, pp. 1619–1625. [12] C. Qiao and K. M. Smedley, “A general three-phase PFC controller for rectifiers with a parallel-connected dual boost topology,” IEEE Trans. Power Electron., vol. 17, no. 6, pp. 925–934, Nov. 2002. [13] K. Smedley, “Poincare stability analysis of switching converters with nonlinear control,” IEEE Power Electron. Soc. Newslett., vol. 14, no. 1, pp. 3–4, Jan. 2002. [14] G. Chen and K. M. Smedley, “Steady-state and dynamic study of onecycle controlled three-phase active power filter,” in Conf. Rec. IEEE-IAS Annu. Meeting, vol. 2, Oct. 12–16, 2003, pp. 1075–1081. [15] T. Jin and K. Smedley, “Operation of unified constant-frequency integration controlled three-phase power factor correction in unbalanced system,” in IASTED PES’02, 2002, pp. 441–447.

Guozhu Chen (M’03) received the B.S. degree in electrical engineering from Hangzhou Commerce University, Hangzhou, China, in 1988, and the M.S. and Ph.D. degrees in power electronics from Zhejiang University, Hangzhou, China, in 1992 and 2001, respectively. In 1992, he joined the faculty of the College of Electrical Engineering, Zhejiang University, where, since 2000, he has been an Associate Professor. He visited the Department of Electrical Engineering and Computer Science, University of California, Irvine, as a Postdoctoral Researcher during 2002–2004. He has authored more than 40 technical papers. His professional experience and research interests are power electronics equipment and their digital control, which includes single-phase and three-phase active power filters, power-factor correction, inverters, dc–dc converters, chargers, and motor driving.

Keyue Ma Smedley (SM’97) received the B.S. and M.S. degrees from Zhejiang University, Hangzhou, China, in 1982 and 1985, respectively, and the M.S. and Ph.D. degrees from the California Institute of Technology, Pasadena, in 1987 and 1991, respectively, all in electrical engineering. From 1990 to 1992, she was an Engineer at the Superconducting Super Collider, where she was responsible for the design and specification of ac–dc conversion systems for all accelerator rings. She is currently a Full Professor in the Department of Electrical Engineering and Computer Science, University of California, Irvine (UCI). She is also the Director of the UCI Power Electronics Laboratory. Her research interests include topologies, control, and integration of high-efficiency dc–dc converters, high-fidelity class-D power amplifiers, active and passive soft-switching techniques, single-phase and three-phase power-factor-corrected rectifiers, active power filters, and grid-connected inverters for alternative energy sources. She has published numerous technical articles and is the holder of eight U.S. patents. Dr. Smedley is an At-Large AdCom Member of the IEEE Power Electronics Society, an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS, a Co-Chair of the Industry/Education Committee of the Power Sources Manufacturer’s Association, and the Chair of the IASTED and IEEE Power Electronics Society co-sponsored 2003 International Conference on Power and Energy Systems. She was also the Chair of the 2004 Industrial Conference on Power Electronics for Distributed and Cogeneration.