Simplified Model and Control Strategy of Three-Phase ... - IEEE Xplore

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IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER ELECTRONICS, VOL. 3, NO. 4, DECEMBER 2015

Simplified Model and Control Strategy of Three-Phase PWM Current Source Rectifiers for DC Voltage Power Supply Applications Yu Zhang, Member, IEEE, Yongxian Yi, Peimeng Dong, Fangrui Liu, and Yong Kang

Abstract— Three-phase pulsewidth modulation (PWM) current source rectifiers (CSRs) that applied as dc voltage power supply include capacitors in the dc bus to obtain dc output voltage, which increase the system complexity and complicate the control strategy. In this paper, a dynamic model of the three-phase PWM CSR is first established in the dq-rotating frame for capacitive load applications. By aligning the vector of the ac capacitor voltage to the d-axis, the model of the CSR can be simplified and decoupled into dc and ac side circuits, which can be controlled separately. Based on this model, a control strategy composed of a state-variable feedback control method for the dc side and an active-damping control method for the ac side is proposed. Moreover, a unity power factor of the CSR is achieved by regulating the q component of the bridge currents. Simulation and experimental results are presented to validate the dynamic and static performance of the proposed control strategy. Index Terms— Active damping, current source rectifier (CSR), dc voltage power supply, dynamic model, state-variable feedback control.

I. I NTRODUCTION

P

HASE controlled rectifiers are widely used for industrial applications largely because of their low cost; however, due to their low-power factor and high current harmonics they are being gradually replaced by pulsewidth modulation (PWM) rectifiers [1]–[3]. There are two main choices of PWM rectifiers: 1) voltage source rectifiers (VSRs) and 2) current source rectifiers (CSRs). VSRs are more popular and can be found in a wide range of applications, from low power to high power, while the CSRs have also been successfully applied to high-power ac motor drives when combined with current

Manuscript received October 20, 2014; revised January 16, 2015; accepted March 31, 2015. Date of publication April 9, 2015; date of current version October 29, 2015. This work was supported in part by the Delta Environmental and Educational Foundation through the Power Electronics Science and Education Development Program under Grant DREG2012007 and in part by the National Natural Science Foundation of China under Award 50877032. Recommended for publication by Associate Editor Sudip K. Mazumder. Y. Zhang, P. Dong, and Y. Kang are with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; [email protected]; [email protected]). Y. Yi was with the State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, Wuhan 430074, China. He is now with the Jiangsu Electric Power Research Institute, Nanjing 211103, China (e-mail: [email protected]). F. Liu is with Power Supply Inc., Toronto, ON M3J 3E5, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JESTPE.2015.2421339

source inverters [3]–[6]. Due to the losses incurred by and the cost of the dc side inductor, CSRs have been ignored in low and medium voltage level circuits. Nonetheless, CSRs have many inherent advantages compared with VSRs. First, in terms of the amplitude of the grid and dc link voltages, CSRs exhibit a step-down conversion. When used as dc voltage source, CSRs can output a lower dc voltage without a bulky, grid-side step-down transformer, as is usually employed in VSRs. Second, because of the inherent current regulation in CSRs, parallel operation and short-circuit protection can be easily achieved [6], [7]. Third, switching losses can be considerably reduced by soft switching method [8], and performance can be improved by methods like tristate operation [9]. Given these advantages, CSRs have excellent potency for use in dc power supply applications. However, contrary to the traditional CSRs that feature only an inductor in the dc bus, CSRs for dc voltage source applications, such as the dc bus of microgrid, modular multilevel converter (MMC), uninterrupted power supply (UPS), and so on, have a bulky capacitor in its output dc bus. A CSR model accounting for this capacitor is of a higher order, which contributes to the complexity of the controller design. The traditional three-phase PWM CSR usually employs an inductor as the dc link and serves as a current source. Their models already exhibit characteristics of high order and nonlinearity, even in the dq-rotating frame [10]–[13]. Common control strategies include PID control, nonlinear control, and predictive control. PID control has a simple structure and high robustness, though it is difficult to optimize the control parameters due to system nonlinearity. Nonlinear methods aim to linearize the system model [11], but they are not robust. Predictive control strategies rely on the accuracy of the system model [12]. Overall, to simplify the CSR model is always an effective way to achieve good performance. In [10], the CSR is considered as a controlled ac current source when viewed from the ac side and treated as a controlled dc voltage source when viewed from the dc side. This helps to achieve separate control of the ac and dc sides. In [13], an effective model in the dq-rotating frame is proposed by setting the q-axis of the ac filter capacitor voltage vector to zero, which also successfully realizes separate control of the dc and ac sides. For applications such as microgrid, MMC, and UPS, CSRs are required to function as dc voltage sources [14]. A dc capacitor is, therefore, connected after the dc link inductor to convert the current source into a voltage source.

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ZHANG et al.: SIMPLIFIED MODEL AND CONTROL STRATEGY OF THREE-PHASE PWM CSRs

Fig. 1.

Circuit of the three-phase CSR as a dc voltage source.

In this way, two LC circuits are present in the ac side and the dc side of the CSR, leading to an even higher order of system dynamic model. In [15] and [16], high-order, small-signal models are established for analysis, but they are complicated for controller design. In [17], the dc voltage is controlled separately while ignoring input filter characteristics. Due to its high order, the CSR model must be simplified before designing the control strategy, which is seldom addressed in the literature. Considering LC resonances, the damping methods are required for both LC circuits. Many damping methods have been proposed for the ac side filter [10], [13], [18]–[28]. The simplest method is to add damping resistors across the capacitor, though unfortunately this approach results in high losses [19]. Active damping can be realized easily in practice by controlling the PWM current from the rectifier bridges to emulate a damping resistor, without any additional power loss [10], [13], [18], [20]–[26]. For the dc side LC filter, damping methods and control strategies are seldom reported. Using the dq-rotating frame transformation, this paper first establishes a dynamic average model of a three-phase PWM CSR. To simplify the model, the d-axis is aligned to the vector of the ac filter capacitor voltage so that the ac side and the dc side model of the CSR can be decoupled. The model enables separate control for the dc and ac sides, and for ac side, resonance damping is performed in the d-axis. Based on this simplified model, we further propose a state-variable feedback control strategy for dc voltage control and an active damping strategy for the ac side. Finally, a unity power factor is achieved by regulating the q-component of the bridge currents. For validation, the proposed modeling strategy and corresponding control methods are applied to a 40-kVA CSR prototype. In addition, simulation results further verify the proposed approach. The remainder of this paper proceeds as follows. In Section II, the dynamic CSR model is described in detail. Next, in Section III, the controller design is presented for both the ac and dc sides. Section IV describes the necessary corrections to achieve unity power factor. In Sections V and VI, simulation and experimental results are given, respectively. Finally, the conclusion is drawn in Section VII. II. DYNAMIC CSR M ODEL AS A DC VOLTAGE S OURCE The CSR as a dc voltage source has the topology shown in Fig. 1, where the load current i o can be treated as a disturbance. Each power switch in the CSR is realized by

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an Insulated Gate Bipolar Transistor (IGBT) and diode in series, and each IGBT has an antiparallel diode. In Fig. 1, the large capacitor Cdc is introduced in the dc bus to provide the dc voltage source. The antiparallel diode D F in the dc side acts as a freewheel diode for the inductor current i dc . In Fig. 1, u sk (k = a, b, c) and i sk (k = a, b, c) are the three-phase grid voltages and currents, respectively, i k (k = a, b, c) are the three-phase input currents of each CSR bridge, u dc is the dc side voltage of the bridge, and u o is the voltage of the dc load. Assuming that the three-phase circuits of the CSR are symmetrical and that the power switches are ideal, their dynamic equations can be given as du o dt di dc L dc ⎡ dt⎤ i sa d ⎣ i sb ⎦ L ac dt i sc ⎤ ⎡ u Ca d ⎣ Cac u Cb ⎦ dt u Cdc

Cc

= i dc − i o

(1)

= u dc − u o ⎡ ⎤ ⎡ ⎤ u sa u Ca = ⎣ u sb ⎦ − ⎣ u Cb ⎦ u sc u Cc ⎡ ⎤ ⎡ ⎤ i sa ia = ⎣ i sb ⎦ − ⎣ i b ⎦. i sc ic

(2) (3)

(4)

The switching variables σk (k = a, b, c) for the CSR bridge are defined as ⎧ ⎪ up-switch ON, down-switch OFF ⎨1 σk = 0 up-switch OFF, down-switch OFF (k = a, b, c) ⎪ ⎩ −1 up-switch OFF, down-switch ON. (5) For σk = 0, both IGBTs in the kth bridge are switched OFF, because the inductor current i dc freewheels through D F . Therefore, the large ON-state losses of the four switches in one bridge (i.e., T1 , D1 , T4 , and D4 ) can be avoided and replaced by the ON-state loss of D F . The relationship between the variables of the dc side and the ac side of the CSR bridges can be expressed as (k = a, b, c) i k = σk · i dc

σk u Ck . u dc =

(6) (7)

k=a,b,c

Based on (1)–(4), (6), and (7), the average model in the abc stationary frame can be obtained by neglecting the high frequency components in the variables and is expressed as [29] ⎧ du o ⎪ Cdc = i dc − i o ⎪ ⎪ ⎪ dt ⎪ ⎪

⎪ di dc ⎪ ⎪ σk u Ck − u o ⎪ ⎨ L dc dt = k=a,b,c (k = a, b, c) (8) di sk ⎪ ⎪ ⎪ = u L − u sk Ck ⎪ ac dt ⎪ ⎪ ⎪ ⎪ du ⎪ ⎪ ⎩Cac Ck = i sk − σk i dc . dt The equivalent circuit of the CSR can be further obtained according to (8), as shown in Fig. 2.

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Fig. 2.


Fig. 3.

Simplified d-axis equivalent circuit of the ac side.

Fig. 4.

Equivalent circuit of the dc side.

Equivalent circuit of three-phase CSR.

Equation (8) is an eighth-order, nonlinear model, making controller design difficult. Hence, it is necessary to simplify the model. Here, the dq-rotating transformation can be performed to simplify the model, which will be derived in the following sections. A. AC Side Model A model of the ac side in the dq frame can be derived from (8) and expressed by ⎧ di sd ⎪ ⎪ L ac = u sd − u Cd + ωs L ac i sq ⎪ ⎪ dt ⎪ ⎪ ⎪ di sq ⎪ ⎪ ⎨ L ac = u sq − u Cq − ωs L ac i sd dt (9) ⎪ ⎪Cac du Cd = i sd + ωs Cac u Cq − σd i dc ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ du ⎪ ⎩Cac Cd = i sq − ωs Cac u Cd − σq i dc dt where u Cd and u Cq are the d and q components of three-phase ac capacitor voltages, σd and σq are the d and q components of three-phase switch variables, i sd and i sq are the d and q components of three-phase grid currents, and u sd and u sq are the d and q components of three-phase grid voltages. Finally, the angular frequency of the grid voltage is given by ωs . By aligning the vector of the ac filter capacitor voltage to the d-axis, the q-axis of the capacitor voltage can be obtained and is given by u Cq = 0.

(10)

Due to the large inductance of inductor L dc in the dc bus, current i dc can be approximated as a constant value of Idc during one grid voltage period. By further neglecting the small coupling component ωs L ac and ωs Cac , the ac side model in the dq frame can be simplified from (9) as ⎧ di sd ⎪ ⎪ L ac = u sd − u Cd ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ di sq ⎨ = u sq L ac (11) dt ⎪ ⎪ du Cd ⎪ ⎪ = i sd − Idc · σd Cac ⎪ ⎪ ⎪ dt ⎪ ⎩0 = i − I · σ . sq dc q In (11), the q-axis equation is first-order with strong stability. In addition, from the d-axis equation an equivalent circuit can be derived and is shown in Fig. 3. Then, u Cd can be further expanded as 1 L ac Idc s u Cd = 2 u sd − 2 σd (12) s s + 1 + 1 ωac ωac

where ωac = 1/(L ac Cac )1/2 , and u sd is considered as a disturbance. Equation (12) is a second-order system containing no damping. Hence, the ac side model has been simplified to a second-order linear system in the d-axis and a first-order system in the q-axis. Therefore, the resonance damping control can be performed only in the d-axis. B. DC Side Model The dc side model can also be derived by the dq-rotating transformation [from (8)] as ⎧ di dc ⎪ ⎨ L dc = 1.5 · (u Cd σd + u Cq σq ) − u o dt (13) ⎪ ⎩Cdc du o = i dc − i o . dt Once the control for the ac side is effective, the symmetrical three-phase ac capacitor voltages can be achieved because they are nearly equal to the grid voltages and do not vary significantly [13]. This implies that u Cd changes quite slowly and can be approximated as a constant u Cd for the dc side. Further, considering (10), (13) can be simplified as ⎧ di dc ⎪ ⎨ L dc = 1.5 · UCd · σd − u o dt (14) ⎪ ⎩Cdc du o = i dc − i o . dt Finally, based on (14), an equivalent circuit of the dc side is shown in Fig. 4, leading to 1.5 · UCd L dc s u o = 2 σd − 2 io s s +1 +1 ωdc ωdc

(15)

where ωdc = 1/(L dc Cdc )1/2 and i o is considered as a disturbance. The above analysis shows that the dc side model is also simplified into a second-order linear system by orienting the vector of the ac capacitor voltage to the d-axis. Thus, the model of the CSR has been divided into two parts, facilitating the system analysis and controller design.


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Fig. 5. Comparison of damping methods. (a) Passive damping. (b) Active damping.

Fig. 6.

Control block diagram of active damping for ac side.

III. C ONTROLLER D ESIGN In this section, the controller design is presented in detail for the ac side and dc side. A. Design of AC Side Controller The simplest way to damp resonance in the ac side LC filter is to add a resistor RDamp across the capacitor, as shown in Fig. 5(a). While these resistors result in high losses, as shown in Fig. 5(b), active damping methods emulate damping resistance—effectively giving a virtual resistance of R H = RDamp —by making the CSR bridges produce an additional PWM current i Damp added to i d [22]. The virtual resistor controller is shown in Fig. 6. It calculates i Damp after measuring u Cd . The resonance of the ac side LC filter will produce harmonics in the capacitor voltage with frequency near ωac , which appears as the ωac − ωs component in u Cd . The fundamental component of capacitor voltage will be dc value in u Cd , which produces dc value in i Damp . However, it is not useful for resonance damping but otherwise may cause overmodulation. Therefore, a high-pass filter (HPF) should be applied to filter out this dc component, which can be expressed as s . (16) HPF(s) = s + ωHP In the dq frame, the design of the HPF is easier than in the abc frame because the dc component is easier to filter out.

Fig. 7. Closed-loop transfer function G ac_1 (s). (a) Bode diagram. (b) Step response.

The cutoff frequency ωHP of the HPF should be much smaller than ωac − ωs , so here, (ωac − ωs )/10 is chosen for ωHP . Finally, the virtual resistor controller composes of HPF and RV (s) = 1/R H . The closed-loop transfer functions for σd and u sd to u Cd can be derived from Fig. 6, as u Cd G ac_1 (s) = u sd 2 (s + ω ) ωac HP = 3 2 s + ω2 ω s + (ωHP + 1/R H Cac )s 2 + ωac ac HP (17) u Cd G ac_2 (s) = ud 2 I L (s 2 + ω s) ωac dc dc HP = 3 . 2 s + ω2 ω s + (ωHP + 1/R H Cac )s 2 + ωac ac HP (18) The Bode diagrams and step responses of G ac_1 (s) and G ac_2 (s) with different values of R H are shown in Figs. 7 and 8, respectively. The resonance in the ac side LC filter excited by harmonics in the grid voltage and the

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Fig. 9.

Control block diagram of the dc side.

Equation (19) has the standard form of a third-order transfer function, with optimal coefficients that can be designed according to the integral of time-weighted absolute value of the error indices for the step response as follows [30]:

2 (20) · Em k1 = ωn3 / ωdc k2 = 1.9ωn L dc /E m

2 − 1 /E m k3 = 2.2ωn2 /ωdc

Fig. 8. Closed-loop transfer function G ac_2 (s). (a) Bode diagram. (b) Step response.

ac side current rectifier is suppressed effectively by R H . In addition, it can be seen that lower values of R H exhibit better damping but slower dynamic response. B. Design of DC Side Controller In the dc side, not only the output voltage u o but also the inductor current i dc need to be controlled. Therefore, a state-variable feedback control is used with voltage u o and dc bus current i dc chosen as the state variables. Since u o is a dc quantity in steady state, an integrator controller G c (S) = k1 /s is introduced to diminish the steady-state error. The diagram of the control is shown in Fig. 9, where k2 and k3 are the state feedback coefficients of i dc and u o , respectively. The closed-loop transfer function can be derived from Fig. 9, and is described by uo G dc_1 (s) = u o_ref 2 k1 E m ωdc = 3 2 · s + k E ω2 s + (k2 E m /L dc ) · s 2 + (k3 E m + 1)ωdc 1 m dc (19) where E m = 1.5 · UCd .

(21) (22)

where ωn is the undamped oscillation frequency chosen for the closed-loop system. The Bode diagrams and step responses of G dc_1 (s) are shown in Fig. 10(a) and (b), respectively, for ωn slightly greater than the resonant frequency ωdc . The system parameters are shown in Table I. The magnitude plot of G dc_1 (s) after design shows a 0 dB gain at low frequency, meaning that the output is able to track the reference without offset. For an LC filter with large time constant, even if the bandwidth of the system is small, the response is still suitable. Equation (15) indicates that the u o is also affected by the output current i o . The transfer function relating i o to u o can be derived from Fig. 9 as G dc_2 (s) uo (s + E m k2 /L dc )/Cdc = = 3 . 2 s + E k ω2 io s + (E m k2 /L dc )s 2 + (E m k3 + 1)ωdc m 1 dc (23) The Bode diagrams and step responses of G dc_2 (s) are shown in Fig. 11(a) and (b), respectively. It can be seen in the magnitude plot of Fig. 11(a) that the gain at low frequency is around −40 dB, which means the output voltage variation generated by a disturbance of the output current is very small. Fig. 11(b) further indicates that the dc side has good dynamic response and current disturbance rejection. C. Control Block for the CSR The proposed control block diagram is shown in Fig. 12. The m dc and m ac are output of the dc side and the ac side controller, respectively. They are summed for space vector modulation (SVM). SVM with single-edge sequences is used to lower the switching losses [31]. With the help of the HPF, m ac has only high frequency components. On the other hand, m dc has mainly low-frequency components due to the slow time constant of the dc LC circuit.


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Fig. 10. Closed-loop transfer function G dc_1 (s). (a) Bode diagram. (b) Step response. Fig. 11.

Transfer function G dc_2 (s). (a) Bode diagram. (b) Step response.

Fig. 12.

Control block diagram of the CSR.

TABLE I S YSTEM PARAMETERS

dc side and the ac side do not affect each other, and therefore, decoupled control of the dc side and ac side is realized. Even if high-frequency fluctuations are introduced by the ac side control, they will be eliminated by the dc side LC filter because of its inherent low-pass property. The control of the

IV. C ORRECTION OF P OWER FACTOR A phasor diagram of the ac side variables of the CSR is shown in Fig. 13.

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Fig. 13.


Phasor diagram of the ac side variables.

The aforementioned control strategy would result in U˙ Ca = 0 and I˙Cd = 0; however, due to the voltage drop across L ac , a phase difference (denoted as θ ) is needed between U˙ s and U˙ Cd to maintain a unity power factor. The current I˙sd is in phase with U˙ Cd and it has phase difference θ with U˙ s . Therefore, a certain amount of current, I˙sq , is necessary to make I˙s in phase with U˙ s . For I˙sq equal to I˙q , due to I˙Cd = 0, the necessary I˙sq can be achieved by controlling I˙q (or σa ) from the CSR bridges. In Fig. 13, current I˙Cq = ωs Cac U˙ Cd can be ignored due to the small value of ωs Cac . Thus, the following equation can be derived based on the triangle theorem: σq Idc ωL ac Is = . Is UCd

(24)

2 , (24) can be Given that Is2 = Id2 + Iq2 = (σd2 + σq2 )Idc given as

σq2 − K σq + σd2 = 0

(25)

where K = UCd /(ωs L ac Idc ). Then, solutions for the q-component of the switching variable σq (the control quantity) from (25) can be expressed as √ √ K + K2 − 4 K − K2 − 4 σd , σq2 = σd . (26) σq1 = 2 2 Equation (26) requires K > 2, hence the solution σq1 > σd is impractical. Moreover, σq2 can be neglected in (25), because it is much smaller than −K σq and σd2 . Finally, the q-component of the switching variable can be derived as σq =

ωs L ac Idc 2 σd . UCd

(27)

By controlling σq according to (27), a unity power factor of the CSR can be achieved, and it will not affect the control of d-axis. V. S IMULATION R ESULTS The performance of the proposed control strategy is first evaluated in MATLAB/Simulink by comparing open-loop control with closed-loop control using the system parameters shown in Table I. The steady-state value of the dc bus voltage (u 0 ) is 432 V. Since the grid voltage can only be measured at the Point of Common Coupling point, a small internal impedance for the grid is assumed in the simulation. Fig. 14(a) shows waveforms for the step-up to full load under open-loop control. During step-up, a large resonance is apparent in the voltage u 0 of the dc side and in the current i sa of the ac side. In addition, on the ac side, the waveform of the grid current i sa is unsatisfactory.

Fig. 14. Simulation results. (a) Under open-loop control. (b) Under closedloop control.

Fig. 14(b) shows the waveforms for step-up to full load under closed-loop control, assuming R H = 1 . During the process, the CSR shows good stability and the maximum drop of the dc voltage is