ZHAO Feng， LI Shute， CHEN Xiaoqiang,2， WANG Ying,2， GAN Yanqi， NIU Xinqiang， ZHANG Fan

(1. School of Automation & Electrical Engineering, Lanzhou Jiaotong University， Lanzhou 730070, China； 2. Key Lab of Opt-Electronic Technology and Intelligent Control of Ministry of Education, Lanzhou Jiaotong University, Lanzhou 730070, China)

Abstract： In predictive direct power control (PDPC) system of three-phase pulse width modulation (PWM) rectifier, grid voltage sensor makes the whole system more complex and costly. Therefore, third-order generalized integrator (TOGI) is used to generate orthogonal signals with the same frequency to estimate the grid voltage. In addition, in view of the deviation between actual and reference power in the three-phase PWM rectifier traditional PDPC strategy, a power correction link is designed to correct the power reference value. The grid voltage sensor free algorithm based on TOGI and the corrected PDPC strategy are applied to three-phase PWM rectifier and simulated on the simulation platform. Simulation results show that the proposed method can effectively eliminate the power tracking deviation and the grid voltage. The effectiveness of the proposed method is verified by comparing the simulation results.

Key words： three-phase PWM rectifier； predictive direct power control； grid voltage sensor free algorithm； third-order generalized integrator； power correction

0 Introduction

Three-phase pulse width modulation (PWM) rectifier has low current distortion rate and adjustable power factor[1-2], which is very popular in active power filter (APF) and renewable energy generation grid connected[3]. In rectifier control, it is necessary to detect the grid voltage as the input. However, the application of voltage sensor is limited under the condition that the sensor volume is strictly required. Therefore, the grid-free voltage sensor algorithm comes into being.

Grid voltage sensor is free controlled based on virtual flux observer, but direct integration of voltage vector will lead to DC bias problem when calculating virtual flux vector[4]. A delay compensation algorithm based on pure integral is proposed, which further avoids the DC bias problem in the calculation of voltage vector integration[5]. However, three PI controllers are used in the system, which makes the process of parameter design tedious. The pure integrator is replaced by low pass filter (LPF) to estimate the virtual flux, the initial phase error is eliminated and the DC bias error is suppressed[6-7]. However, the input current signal directly measured by the sensor has some harmonic components. When filtering the input current harmonics, amplitude and phase errors are introduced by the LPF, which seriously affects the estimation of virtual flux observer, thus affecting the control accuracy of three-phase PWM rectifier. Therefore, the grid voltage sensor free algorithm based on virtual flux observer is easily affected by the harmonic component of input current. Later, the grid voltage sensor free algorithm is applied to the PWM rectifier system[8-9]. However, the used algorithm is still based on the traditional virtual flux observer, so there are many disadvantages. In view of the mature application of second-order generalized integrator (SOGI)[10]in single-phase grid connected phase-locked loop (PLL) system, the improved structure of SOGI is put forward[11], which shows that the phase deviation caused by transformer, filter and other links can be effectively eliminated in SOGI-PLL system. Due to the good filtering characteristics of SOGI and third-order generalized integrator (TOGI) for harmonic components and DC component, it is applied to the grid voltage sensor free algorithm, and good results is achieved combined with the traditional PDPC strategy. SOGI is used to replace pure integrator to estimate virtual flux, which solves the problems of integral drift and DC bias of pure integrator[12]. However, the grid voltage sensor free algorithm is still implemented in the virtual flux observation framework, which leads to complex structure. The principle and structure design of TOGI are analyzed in detail[13]. TOGI is used to construct voltage observer to estimate the grid voltage in current control, and excellent results are achieved[14].

So far, the TOGI-based grid-free voltage sensor algorithm under PDPC strategy has not been fully studied. The voltage observer is constructed to estimate the grid voltage by using the characteristics of TOGI, which can track sinusoidal signal with specific frequency without static error, and a power correction link is designed to correct the inaccurate power reference value in traditional PDPC strategy, so as to eliminate the deviation between actual and reference power. The control structure of the proposed method is simple and almost the same as that of the traditional PDPC strategy. Finally, the proposed method is simulated on MATLAB/Simulink platform.

1 Mathematical model of PWM rectifier

The topology of a three-phase voltage PWM rectifier is shown in Fig.1, whereR,Lare AC side resistance and inductance, respectively.xa,xb,xc(x=e,i) are grid voltage and input current, respectively.Udc,C,RLare DC side voltage, filter capacitor and load resistance, respectively.

Fig.1 Topology of a three-phase voltage PWM rectifier

Inαβframe, the voltage equation of PWM rectifier is expressed as

(1)

wherexα,xβ(x=e,i,u) are the components of grid voltage, input current and AC side voltage of rectifier onαβframe, respectively.

2 Grid voltage sensor free algorithm

2.1 Basic principle of TOGI

The proposed TOGI is used to construct the grid voltage observer[14]. The schematic diagram is shown in Fig.2, whereu(t) is the input singal;ωis the frequency of signalu(t); and the output signals areu1(t),u2(t) andu3(t), respectively;Kis the gain coefficient.

Fig.2 TOGI schematic diagram

The closed-loop transfer function is

(2)

(3)

whereAis the amplitude；φis the initial phase；A0is the DC component； the output signalu1(t) does not contain the DC component；u2(t) andu3(t) both contain the DC componentKA0；u2(t)andu1(t) have the same amplitude；u2(t) phase lagsu1(t) 90°.

Let

(4)

then

(5)

It can be seen from Eq.(5) thatuα(t) anduβ(t) are equal to the amplitude of the input signal.uα(t)is equal to the phase of the input signal.uβ(t) lags the input signal by 90° and does not contain DC component. Therefore, TOGI can not only effectively filter the high-order harmonics in the input signal, but also filter the DC component of the input signal.

2.2 Estimation method of grid voltage

The relationship between grid side voltage and switching function of rectifier is

(6)

whereSa,SbandScare switching functions. Eq.(6) is substituted into Eq.(1), and the grid voltage is estimated according to the input current, DC voltage and switch function. However, high frequency noise will be introduced by directly differentiating the input current in Eq.(1). Since the input current is a sinusoidal signal, its differential value is equal to the amplitude of the original signal, and the phase is 90° ahead of the original signal. So the sine value behind the original signal can be got through TOGI, then the differential value of the original signal is got by taking the inverse. Therefore, the estimated grid voltage is expressed as

(7)

Fig.3 Structure diagram of voltage observer

3 Predictive direct power control

Inαβframe, the instantaneous power is expressed as

(8)

wherepandqare instantaneous active and reactive power.

Its differential equation is expressed as

(9)

3.1 Traditional predictive direct power control

According to the power control objective, the actual value is required to be equal to the reference value at the end ofkTsperiod.Tsis the sampling period. Therefore, letp*(k+1)=p(k+1),q*(k+1)=q(k+1). Assuming that the sampling period is much less than the power frequency of the grid voltage, the grid voltage can be regarded as constant in two adjacent sampling periods, that ise(k+1)=e(k). Therefore, the power difference betweenkTsand (k+1)Tsperiods can be obtained from Eq.(9) as

(10)

wherex(k),x(k+1) (x=e,i) are the actual values of grid voltage and input current ofkTsand (k+1)Tsperiod, respectively.

Because the AC side resistanceRis very small, it has little effect on the system. If the resistance is ignored, it can be obtained from

(11)

From the forward difference, it can be obtained that

(12)

According to Eqs.(10) and (12), the AC side output voltage of the rectifier in the next period is expressed as

(13)

In order to ensure the unit power factor operation of the rectifier in the PDPC system, the instantaneous reactive power commandq*is directly given as 0, and the instantaneous active power commandp*is obtained through the outer voltage loop. In traditional PDPC strategy, the reference value of active power is predicted by linear interpolation method[15-16].

(14)

The AC side output voltage improved by linear interpolation method is expressed as

(15)

whereεp(k) andεq(k) is the difference between the actual and reference power ofkTsperiod.

According to the linear interpolation method, the reference powerp*(k+1) is easy to calculate, but the accuracy is not high. When the system works in steady state, there arep*=p,q*=q, so Eq.(13) can be expressed as

(16)

Substituting Eq.(16) into Eq.(1),iαandiβare all 0. Obviously, after the input information of instantaneous active power deviationεp(k) and instantaneous reactive power deviationεq(k) disappears, the controller can not maintain the output of appropriate control quantity, that is to say, the existence of error difference is necessary to ensure the operation of the controller, so the traditional PDPC strategy has steady-state error.

In order to solve the above problems, the power correction method is introduced in the traditional PDPC strategy to correct the predictive power value.

3.2 Power correction

The purpose of power correction is to eliminate the deviation between the predicted power valuep*(k+1) and the actual power valuep(k+1) at the end of thekTsperiod. In thekTsperiod, the power deviationεp(k) betweenp*(k) andp(k) and the power deviation in previous periods are accumulated to correct the predicted power valuep*(k+1) at the end ofkTsperiod. Therefore, the active power after correction is expressed as

(17)

wherehis the weighted correction coefficient, which can be taken as a constant of 0.05[17], and the calculation of reactive power is the same. The block diagram of power prediction value correction method is shown in Fig.4.

Fig.4 Diagram of correction method for predictive power

Since Eq.(17) can track the step change of power without static error, substituting Eq.(17) into Eq.(13), the AC side voltage of rectifier can be expressed as

(18)

The control block diagram of adding power correction is shown in Fig.5.

Fig.5 Control block diagram of adding power correction

3.3 System control structure

Fig.6 System control structure diagram

4 Simulation analysis

4.1 Description of simulation parameters

A model is built on the MATLAB/Simulink2020a platform to simulate the proposed method and verify its effectiveness. the simulation parameters are shown in Table 1.

Table 1 Simulation parameters

4.2 Simulation model diagrams

The simulation model diagram of the whole system and subsystem simulation model diagram are shown in the Figs.7-15.

Fig.7 TOGI module

Fig.8 Actual power calculation module

Fig.9 Reference power calculation module

Fig.10 Power correction module of predictive direct power controller

Fig.11 Sector judgment module

Fig.12 XYZ calculation module

TxTycalculationprogramis

function[Tx,Ty]=fcn(X,Y,Z,N,Ts)

%#codegen,

Tx=0；Ty=0；

ifN=1

Tx=-Y；Ty=-Z；

elseifN=2

Tx=-Z；Ty=-X；

elseifN=3

Tx=Y；Ty=X；

elseifN=4

Tx=X；Ty=-Y；

elseifN=5

Tx=X；Ty=Z；

elseifN=6

Tx=Z；Ty=Y；

end

ifTx+Ty>Ts

Tx=Tx*Ts/(Tx+Ty)；

Ty=Ty*Ts/(Tx+Ty)；

end

Fig.13 Tabc calculation module

Tcmxcalculationprogram:

function[Tcm1,Tcm2,Tcm3]=fcn(Ta,Tb,Tc,N)

%#codegen,

Tcm1=0；Tcm2=0；Tcm3=0；

ifN=1

Tcm1=Tb；Tcm2=Ta；Tcm3=Tc；

elseifN=2

Tcm1=Ta；Tcm2=Tc；Tcm3=Tb；

elseifN=3

Tcm1=Ta；Tcm2=Tb；Tcm3=Tc；

elseifN=4

Tcm1=Tc；Tcm2=Tb；Tcm3=Ta；

elseifN=5

Tcm1=Tc；Tcm2=Ta；Tcm3=Tb；

elseifN=6

Tcm1=Tb；Tcm2=Tc；Tcm3=Ta；

end

Fig.14 Uabc generation module

Fig.15 abc/αβ coordinate transformation module

4.3 Simulation results

The grid voltage and input current waveforms of phaseAfor the proposed PDPC strategy are shown in Fig.16. When 0.05 s-0.1 s, the DC side is no-load, and when 0.1 s, a sudden load of 50 Ω is applied in the system. It can be seen that the proposed PDPC strategy responds quickly. After 0.1 s, because the reference reactive power remains at 0 throughout the process, the system operates at unity power factor, and the grid voltage and input current waveforms are in phase.

Fig.16 Grid voltage and input current waveforms of phase A

The DC side voltage waveforms is shown in Fig.17.

Fig.17 DC side voltage waveforms

In the initial stage, the DC voltage increases with a faster response speed, and reaches the given voltage value of 750 V in 0.05 s. Due to the sudden load at 0.1 s, the DC voltage drops from 750 V to 700 V, and returns to the given value in a short time. The DC side voltage can accurately track the given value at no-load and load, with smaller voltage drop, shorter regulation time and faster dynamic response. When the simulation reaches the stable state, the load resistance of the DC side is reduced from 50 Ω to 40 Ω att=0.2 s, the DC voltage drops slightly, the amplitude of the input current increases, then returns to the given value, the system remains stable.

The active power waveforms of the traditional PDPC strategy and the proposed method under sudden load is shown in Fig.18. The actual value of the active power of the proposed method always follows the reference value and has almost no deviation. It reaches the steady state after 0.05 s. The actual power of the traditional PDPC strategy also follows the reference power after sudden load, but there is a deviation between them at any time. The steady-state waveform is shown in Fig.19.

(a) Traditional PDPC strategy

(a) Traditional PDPC strategy

The total harmonic distortion (THD) of input current for the traditional PDPC strategy and the proposed method are shown in Fig.20. It can be seen that most of the harmonic current times are between 0 and 600, mainly 100 and 200. Under the same simulation parameters, the THD of the traditional PDPC strategy is 4.16%, and the THD of the proposed method is 3.33%. Both of them meet the national requirements of harmonic distortion rate less than 5%, and use SVPWM to generate on-off signal of rectifier switch. Due to the switching frequency is fixed, so the harmonic number is more concentrated, which is conducive to harmonic filtering.

(a) Traditional PDPC strategy

The waveform of estimated and actual values of α component for grid voltage is shown in Fig.21. It can be seen that there is a deviation between them at the initial time, which is completely consistent after 0.05 s. Therefore, the sensor free algorithm can accurately estimate the grid voltage.

Fig.21 Actual and estimated values of α component for grid voltage

5 Conclusions

1) Compared with the traditional PDPC strategy, this method has the advantages of fast response and good robustness. It eliminates the active power tracking deviation in the traditional PDPC strategy and effectively realizes the power tracking without deviation.

2) The grid voltage sensor free algorithm based on TOGI is not affected by the high-order harmonics and DC components of input current in the process of grid voltage observation, which can accurately estimate the grid voltage. At the same time, it abandons the inconvenience caused by voltage sensor, reduces the cost, and is more flexible and widely used, which provides a new idea for the development of three-phase PWM rectifier grid voltage sensor free algorithm.