# Current PIλ Control of the Single-Phase Grid Inverter

Authors: Xueqin Yang; Xingyu Liu; Jichao Li; Binbin Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In a grid-connected power generation system, the grid-connected current of the inverter is sensitive to nonlinear factors such as periodic disturbance of grid voltage, which results in grid-connected current waveform distortion. By establishing a single-phase photovoltaic grid-connected inverter control system model, designing an inverse current fractional-order PI (PIλ or FO-PI) controller and the dynamic and steady-state performance, antidisturbance and grid connection inversion characteristics of the system are simulated and compared under the action of the integer-order PI controller and fractional-order PI controller. The quality of the inverter grid-connected current is analyzed by using the fast Fourier transform (FFT). The simulation results show that the fractional-order control system can reduce the total harmonic distortion (THD) of the grid-connected current and dynamic performance and antidisturbance ability of the improving system while satisfying the steady-state performance indexes.


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## Body

## 1. Introduction

Solar grid-connected photovoltaic (PV) system has great strategic significance to alleviate the current energy crisis and environmental pollution. Inverter as the core of the grid-connected PV system, the output of the grid-connected current directly affects the power quality of the power grid. Therefore, it is of great practical significance to study an effective grid-connected current control strategy so that the photovoltaic system can realize the grid-connected operation while the total harmonic distortion (THD) of the grid-connected current is as small as possible.The common current digital control strategies of the grid-connected current are PI control, voltage and current double-closed loop control, repetitive control, no beat control, and so on. The PI control strategy is widely used in engineering because of its simple control method, easy implementation, and good dynamic performance. In literature [1], the parameters of the PI controller are self-tuned by using the principle of fuzzy control, which effectively reduces the current tracking error and improves the dynamic response performance of the system. In this method, the parameter of the controlled object is relatively rough, and the fuzzy inference is more dependent on the rich engineering experience. Literature [2] proposes a current-tracking control strategy combining repetitive control and PI control, which can effectively improve the grid-connected current waveform and ensure that the inverter output current and grid voltage are with the same frequency and phase, but their dynamic response is poor. In literature [3], different duty ratio control functions are generated by the controller to control the output current of the grid-connected inverter. The output power quality of the inverter is relatively stable, but the response speed of the system is slow, and the application scope is small. In literature [4], combined with repetitive control and H∞ control, an inverter current controller is designed to improve the tracking performance of the system and reduce the total harmonic distortion (THD). However, the method needs to solve the Riccati equation, and the operation is more complex, so the method cannot be widely used. In literature [5], a composite control strategy based on internal model control and repetitive control is proposed by using the internal model control to simplify the parameters of the controller in order to achieve the purpose of improving the dynamic performance of the system and the repetitive control to suppress the disturbance of the power grid in order to achieve the purpose of improving the steady-state performance of the system. This method requires that the filter has low-pass characteristics, but in practice, any high-frequency signals cannot achieve the desired results.In view of the above problems, this paper based on the analysis of the current waveform distortion causes of the single-phase PV grid-connected inverter, combined with the PIλ controller flexible control structure; when it acts on the controlled object, the system has better dynamic, steady-state performance, and robustness [6, 7]; the PIλ controller is applied to the single-phase PV grid-connected power generation system. Starting from the control structure of the grid-connected inverter and taking the inverter output current as the control object, the inverter control model is established, PIλ controller is designed, and the MATLAB/Simulink simulation model is set up; the performance of the control system is simulated and compared with the PIλ controller and the integer-order PI controller, and the effectiveness of the control strategy is verified.


## 2. Single-Phase PV Grid-Connected Inverter Control Strategy

The output of the grid-connected inverter adopts the current control mode. Actually, the grid-connected system and the grid are AC sources and voltage sources in parallel. The output voltage of the inverter is automatically clamped to the grid voltage. Therefore, it is only needed to control the output current of the inverter that tracks the grid voltage to achieve the purpose of the grid-connected operation. This paper adopts the control strategy of current instantaneous value feedback and triangular wave comparison, and the controller uses the fractional-order proportional-integral (FO-PIλ) control method.


### 2.1. Grid-Connected Inverter Control Structure

In the single-phase PV grid-connected inverter control structure as shown in Figure1, the main circuit uses the two-level topology: boost type DC/DC step-up chopper circuit is formed by pre-energy storage inductor L1, IGBT switch tube VT1, and rectifier diode D0, mainly to achieve the PV terminal voltage to the required voltage of the grid. The latter stage is composed of four IGBT switches to form a DC/AC full-bridge inverter circuit, which can realize the grid-connected current and grid voltage with the same frequency and phase control and DC bus voltage stability. Cpv for the PV output side of the storage capacitor is used to stabilize the PV module output DC voltage and Cdc is the storage capacitor of the inverter DC side.Figure 1
Main circuit inverter control structure.The specific control process is as follows: compared with the difference between grid-connected current reference valueiref and the actual value of the grid-connected current is adjusted by the PIλ controller and compared with the triangular modulation wave, the SPWM control signal is generated to drive the switching tube of the inverter, and inverter output is filtered through the inductor L and fed into the same frequency and phase with the grid voltage sine wave current is.


### 2.2. The System Control Model

During the grid operation, switch frequency (10 kHz) is much higher than the grid voltage rated frequency (50 Hz); therefore, it can ignore the switch delay and switch off action effects on the system, and the full-bridge inverter is approximated as a link gain of K. Design grid-connected current control block diagram [8, 9], as shown in Figure 2.Figure 2
Block diagram of grid-connected current control.In Figure2, Gc(S) is the transfer function of the controller, L is the filter inductor, and R is the equivalent resistance of the filter inductor. Assuming the power switch as an idealized switch, the transfer function of the full-bridge inverter in the SPWM control mode can be approximated as a small inertia link [10], that is, Kinv/(sTinv + 1), where Kinv is the gain of the inverter and Tinv is the switching time period. From the analysis of literature [11], it can be seen that the influence of grid voltage disturbance on the system can be neglected when the controller parameters are reasonable, and the mathematical model of the control object can be obtained as(1)Gss=KinvsTinv+1⋅1sL+R=KinvsTinv+11/RsL/R+1.


## 2.1. Grid-Connected Inverter Control Structure

In the single-phase PV grid-connected inverter control structure as shown in Figure1, the main circuit uses the two-level topology: boost type DC/DC step-up chopper circuit is formed by pre-energy storage inductor L1, IGBT switch tube VT1, and rectifier diode D0, mainly to achieve the PV terminal voltage to the required voltage of the grid. The latter stage is composed of four IGBT switches to form a DC/AC full-bridge inverter circuit, which can realize the grid-connected current and grid voltage with the same frequency and phase control and DC bus voltage stability. Cpv for the PV output side of the storage capacitor is used to stabilize the PV module output DC voltage and Cdc is the storage capacitor of the inverter DC side.Figure 1
Main circuit inverter control structure.The specific control process is as follows: compared with the difference between grid-connected current reference valueiref and the actual value of the grid-connected current is adjusted by the PIλ controller and compared with the triangular modulation wave, the SPWM control signal is generated to drive the switching tube of the inverter, and inverter output is filtered through the inductor L and fed into the same frequency and phase with the grid voltage sine wave current is.


## 2.2. The System Control Model

During the grid operation, switch frequency (10 kHz) is much higher than the grid voltage rated frequency (50 Hz); therefore, it can ignore the switch delay and switch off action effects on the system, and the full-bridge inverter is approximated as a link gain of K. Design grid-connected current control block diagram [8, 9], as shown in Figure 2.Figure 2
Block diagram of grid-connected current control.In Figure2, Gc(S) is the transfer function of the controller, L is the filter inductor, and R is the equivalent resistance of the filter inductor. Assuming the power switch as an idealized switch, the transfer function of the full-bridge inverter in the SPWM control mode can be approximated as a small inertia link [10], that is, Kinv/(sTinv + 1), where Kinv is the gain of the inverter and Tinv is the switching time period. From the analysis of literature [11], it can be seen that the influence of grid voltage disturbance on the system can be neglected when the controller parameters are reasonable, and the mathematical model of the control object can be obtained as(1)Gss=KinvsTinv+1⋅1sL+R=KinvsTinv+11/RsL/R+1.


## 3. PIλ Controller Design

### 3.1. PIλDμ Controller

PIλDμ controller is proposed by Professor I. Podlubny [12], and the transfer function is(2)Gcs=Kp+Kisλ+Kdsμ.PIλDμ controller includes an integral order λ and a differential order μ, where λ and μ can be arbitrary real numbers. In the design of a fractional-order control system, the controller parameters Kp, Ki, Kd, λ, and μ must be optimized to meet the requirements according to the performance index of the system.From formula (2), when λ and μ take different combinations, respectively, (λ, μ) = {(0, 0), (0, 1), (1, 0), (1, 1)}, it can be obtained as a conventional proportional (P) controller, a PD controller, a PI controller, and a PID controller. The value plane of λ and μ is shown in Figure 3.Figure 3
P-I-D order value plane.As can be seen from the figure, the conventional PID controller is a special case of the PIλDμ controller. Since λ and μ can vary continuously, the PIλDμ controller is more flexible than the integer-order PID controller [13, 14]. By selecting the values of λ and μ reasonably, the dynamic performance of the fractional-order control system can be better adjusted.


### 3.2. PIλ Controller Design

The PIλDμ controller transfer function is expressed by formula (2), which makes μ = 0, and then the transfer function of the PIλ controller can be obtained:(3)Gcs=Kp+Kisλ.In order to facilitate the analysis, formula (3) can be rewritten as(4)Gcs=Kp1+Ki′sλKi=KpKi′.For the specific control objectGs(s), the parameters of the PIλ controller are optimized by using the frequency characteristic of the control system and the given cut-off frequency ωc and the phase margin φm. The open-loop transfer function Gk(s) can satisfy the following performance index by designing the PIλ controller Gc(s) [15]:(1) Phase margin:(5)ArgGkjωc=ArgGcjωcGsjωc=−π+ϕm.(2) Gain robustness:(6)dArgGcjωGsjωdωω=ωc=0.(3) Amplitude criterion:(7)Gkjωc=GcjωcGsjωc=1.The frequency response of the PIλ controller is obtained by formula (4) combined with the Euler’s formula:(8)Gcjω=Kp1+Ki′ω−λcosλπ2−jKi′ω−λsinλπ2.Its phase and amplitude are(9)ArgGcjω=−arctanKi′ω−λsinλπ/21+Ki′ω−λcosλπ/2,(10)Gcjω=KpKi′ω−λsinλπ22+1+Ki′ω−λcosλπ22.For the controlled objectGs(s), the inverter gains Kinv and 1/R can be transferred to the proportional gain Kp of the controller without affecting the gain of the whole inverter control link [16]. Therefore, for the sake of generality, the gain and 1/R in formula (1) are normalized to 1. At this point, there is(11)Kp′=1RKpKinv.The frequency response of the controlled object is(12)Gsjω=1jωTinv+1⋅1jωL/R+1.Its phase and amplitude are(13)ArgGsjω=−arctanωTinv−arctanωLR,(14)Gsjω=11+ωTinv2⋅1+ωL/R2.The open-loop transfer function is(15)GkS=GcS⋅GsS.From formulas (9), (13), and (15), the phase angle of the open-loop transfer function is(16)ArgGkjω=ArgGcjω+ArgGsjω=−arctanKiω−λsinλπ/21+Kiω−λcosλπ/2−arctanωTinv−arctanωLR.From formulas (10), (14), and (15), the amplitude of the open-loop transfer function is(17)Gkjω=GcjωGsjω=Kp1+Kiω−λcosλπ/22+Kiω−λsinλπ/221+ωTinv21+ωL/R2.From formula (16) and the phase angle margin criterion, the relationship between K′I and λ can be obtained as(18)Ki′=−tanarctanωcTinv+arctanωcL/R+ϕm−πωc−λsinλπ/2+ωc−λcosλπ/2tanarctanωcTinv+arctanωcL/R+ϕm−π.From formula (16) and the robustness criterion, another relation between K’i and λ can also be established:(19)Ki′=−B±B2−4A2ωc−2λ2Aωc−2λ,where(20)A=Tinv1+ωcTinv2+L/R1+ωcL/R2,B=2Aωc−λcosλπ2−λωc−λ−1sinλπ2.From Formula (17) and the amplitude criterion, we obtain the equation for Kp:(21)Kp=1+ωcL/R21+ωcTinv21+Ki′ωc−λcosλπ/22+Ki′ωc−λsinλπ/22.On the basis of satisfying the phase angle margin criterion, the robustness criterion, and the amplitude criterion, the relations ofλ and Kp and λ and Ki are obtained. It is shown that, in the PIλ controller, the change of λ will affect the response speed, overshoot, adjustment time, and steady-state error of the system through the proportional and integral links.In order to obtain a relatively satisfactory transient response, the photovoltaic grid-connected control system phase angle margin is generally between 30°∼60°; the cut-off frequency is 200 rad/s∼300 rad/s [17]. The given phase angle margin and the expected value of the cut-off frequency are shown in Table 1.Table 1
System parameter.
System parametersParameter valueCutoff frequency desired value,ωc200 rad/sPhase angle margin expected value,φm60°Switching time period,Tinv100μsEquivalent resistance, R0.5 ΩFilter inductance, L6 mHBy using the plotting method, three parameters of the PIλ controller are obtained: Kp = 7.89, Ki = 73.25, and λ = 0.535. According to formulas (18)–(21) and Ki = Kp·K’i, we can write the PIλ controller transfer function as(22)Gcs=7.89+73.25s0.535.


## 3.1. PIλDμ Controller

PIλDμ controller is proposed by Professor I. Podlubny [12], and the transfer function is(2)Gcs=Kp+Kisλ+Kdsμ.PIλDμ controller includes an integral order λ and a differential order μ, where λ and μ can be arbitrary real numbers. In the design of a fractional-order control system, the controller parameters Kp, Ki, Kd, λ, and μ must be optimized to meet the requirements according to the performance index of the system.From formula (2), when λ and μ take different combinations, respectively, (λ, μ) = {(0, 0), (0, 1), (1, 0), (1, 1)}, it can be obtained as a conventional proportional (P) controller, a PD controller, a PI controller, and a PID controller. The value plane of λ and μ is shown in Figure 3.Figure 3
P-I-D order value plane.As can be seen from the figure, the conventional PID controller is a special case of the PIλDμ controller. Since λ and μ can vary continuously, the PIλDμ controller is more flexible than the integer-order PID controller [13, 14]. By selecting the values of λ and μ reasonably, the dynamic performance of the fractional-order control system can be better adjusted.


## 3.2. PIλ Controller Design

The PIλDμ controller transfer function is expressed by formula (2), which makes μ = 0, and then the transfer function of the PIλ controller can be obtained:(3)Gcs=Kp+Kisλ.In order to facilitate the analysis, formula (3) can be rewritten as(4)Gcs=Kp1+Ki′sλKi=KpKi′.For the specific control objectGs(s), the parameters of the PIλ controller are optimized by using the frequency characteristic of the control system and the given cut-off frequency ωc and the phase margin φm. The open-loop transfer function Gk(s) can satisfy the following performance index by designing the PIλ controller Gc(s) [15]:(1) Phase margin:(5)ArgGkjωc=ArgGcjωcGsjωc=−π+ϕm.(2) Gain robustness:(6)dArgGcjωGsjωdωω=ωc=0.(3) Amplitude criterion:(7)Gkjωc=GcjωcGsjωc=1.The frequency response of the PIλ controller is obtained by formula (4) combined with the Euler’s formula:(8)Gcjω=Kp1+Ki′ω−λcosλπ2−jKi′ω−λsinλπ2.Its phase and amplitude are(9)ArgGcjω=−arctanKi′ω−λsinλπ/21+Ki′ω−λcosλπ/2,(10)Gcjω=KpKi′ω−λsinλπ22+1+Ki′ω−λcosλπ22.For the controlled objectGs(s), the inverter gains Kinv and 1/R can be transferred to the proportional gain Kp of the controller without affecting the gain of the whole inverter control link [16]. Therefore, for the sake of generality, the gain and 1/R in formula (1) are normalized to 1. At this point, there is(11)Kp′=1RKpKinv.The frequency response of the controlled object is(12)Gsjω=1jωTinv+1⋅1jωL/R+1.Its phase and amplitude are(13)ArgGsjω=−arctanωTinv−arctanωLR,(14)Gsjω=11+ωTinv2⋅1+ωL/R2.The open-loop transfer function is(15)GkS=GcS⋅GsS.From formulas (9), (13), and (15), the phase angle of the open-loop transfer function is(16)ArgGkjω=ArgGcjω+ArgGsjω=−arctanKiω−λsinλπ/21+Kiω−λcosλπ/2−arctanωTinv−arctanωLR.From formulas (10), (14), and (15), the amplitude of the open-loop transfer function is(17)Gkjω=GcjωGsjω=Kp1+Kiω−λcosλπ/22+Kiω−λsinλπ/221+ωTinv21+ωL/R2.From formula (16) and the phase angle margin criterion, the relationship between K′I and λ can be obtained as(18)Ki′=−tanarctanωcTinv+arctanωcL/R+ϕm−πωc−λsinλπ/2+ωc−λcosλπ/2tanarctanωcTinv+arctanωcL/R+ϕm−π.From formula (16) and the robustness criterion, another relation between K’i and λ can also be established:(19)Ki′=−B±B2−4A2ωc−2λ2Aωc−2λ,where(20)A=Tinv1+ωcTinv2+L/R1+ωcL/R2,B=2Aωc−λcosλπ2−λωc−λ−1sinλπ2.From Formula (17) and the amplitude criterion, we obtain the equation for Kp:(21)Kp=1+ωcL/R21+ωcTinv21+Ki′ωc−λcosλπ/22+Ki′ωc−λsinλπ/22.On the basis of satisfying the phase angle margin criterion, the robustness criterion, and the amplitude criterion, the relations ofλ and Kp and λ and Ki are obtained. It is shown that, in the PIλ controller, the change of λ will affect the response speed, overshoot, adjustment time, and steady-state error of the system through the proportional and integral links.In order to obtain a relatively satisfactory transient response, the photovoltaic grid-connected control system phase angle margin is generally between 30°∼60°; the cut-off frequency is 200 rad/s∼300 rad/s [17]. The given phase angle margin and the expected value of the cut-off frequency are shown in Table 1.Table 1
System parameter.
System parametersParameter valueCutoff frequency desired value,ωc200 rad/sPhase angle margin expected value,φm60°Switching time period,Tinv100μsEquivalent resistance, R0.5 ΩFilter inductance, L6 mHBy using the plotting method, three parameters of the PIλ controller are obtained: Kp = 7.89, Ki = 73.25, and λ = 0.535. According to formulas (18)–(21) and Ki = Kp·K’i, we can write the PIλ controller transfer function as(22)Gcs=7.89+73.25s0.535.


## 4. PIλ Controller Performance Analysis Diagnosis

### 4.1. The Oustaloup Approximation of Integral Operatorsλ

The integral operator sλ is approximated by the Oustaloup algorithm in the transfer function expression of the FO-PI controller [17]. The algorithm can guarantee the approximation of the fractional transfer function on the basis of stability so that the transfer function is as little as possible to the zero and the pole. Assuming that the selected approximation bands are ωb and ωh, the algorithm is described as(23)sλ≈K∏k=−NNs+ωk′s+ωk.In this formula,(24)ωk′=ωbωhωbk+N+0.51−λ/2N+1,K=ωhλ,ωk=ωbωhωbk+N+0.51+λ/2N+1.Formula (24) can be considered as a recursion filter (IIR) with the order n = 2N + 1. When the approximation order n = 1, the approximation error is large. When the approximation order n = 5, both the amplitude-frequency characteristic and the phase-frequency characteristic can well approximate the fractional order. The higher the approximation order is, the higher the approximation accuracy will be. But when n is more than 5, the approximation accuracy and the approximation order are no longer proportional [18]. So, the third-order inner approximation is considered to use. The high-order term 1/sλ of the FO-PI controller is approximated in the frequency band of interest (ωb = 0.001 Hz and ωh = 1000 Hz) with n = 3, i.e., N = 1. The approximate result is(25)1sλ=0.024831s+342.8s+3.428s+0.03428s+29.17s+0.2917s+0.002917.


### 4.2. The Dynamic and Steady-State Performance Analysis

In the integral PI controller, the integralKis−1 is mainly used to improve the steady-state performance of the system. But because of the 90° lag phase angle of the integral, the overshoot and the adjustment time of the system are increased, and the dynamic performance of the system is poor. λ of Kis−λ can be adjusted arbitrarily, and the adjustment accuracy is high. The stability and dynamic performance of the system can be taken into account under the premise of improving the steady-state characteristics of the system. Figure 4 shows the unit step response curve for different values of λ.Figure 4
Comparison of step response asλ varies.From Figure4, it can be seen that, with the increment of λ, the steady-state accuracy of the system is obviously improved, the adjustment time is gradually reduced, and the overshoot quantity decreases gradually. The value of λ continues to increase, and the adjustment time becomes larger. The steady-state accuracy is reduced, and oscillation occurs. Therefore, the integral order λ mainly affects the steady-state error of the system. Selecting the proper integration order can reduce the steady-state error of the system and obtain a good dynamic performance.Letλ = 1 and μ = 0 in formula (2), then the integral-order PI controller transfer function can be expressed as(26)Gc1s=Kp1+Ki1s.The parameters of the PI controller are obtained by the method of engineering design, that is,Kp1 = 0.13 and Ki1 = 10.79, and the transfer function of the integer-order PI controller is(27)Gc1s=0.13+10.79s.In the MATLAB/Simulink environment, a step-by-step simulation model is set up. Taking the step signal as the input, the control object is controlled by the PI controller and the integer-order PI controller; the simulation results are shown in Figure5.Figure 5
System step response curve. (a) Step response curve (b) Local magnification.
(a)(b)The main performance indexes of the FO-PI controller and the IO-PI controller are obtained from the step response curve: rise timetr, overshoot δ, and adjustment time ts (±2% steady-state value), as shown in Table 2. It can be seen that the dynamic follow-up performance of the system using the FO-PI controller is superior to the system using the IO-PI controller.Table 2
Dynamic performance index.
Controllertr (ms)δ (%)ts (ms)IO-PI0.29121.08FO-PI0.266.91.02To further verify the validity of the designed FO-PI controller parameters and the superiority of the control performance, the Bode diagram of the open-loop system under FO-PI and IO-PI controller is obtained, as shown in Figure6.Figure 6
Open-loop transfer function: Bode diagram.In order to control the two control effects of comparative analysis more intuitively, the performance index values in the chart are tabulated, as shown in Table3.Table 3
Open-loop system performance.
When the system is stablePerformance indexExpected valueFO-PIIO-PIPhase margin (Xo)6058.456.3Amplitude variation (dB)00.00610.0296Cut-off frequency (rad/s)2002016.67e + 003As the results shown in Figure6 and Table 3, the following conclusions can be drawn:(1)
When the system is stable, the FO-PI control system is closer to the expected phase margin and cutoff frequency than the IO-PI control system that the phase margin criterion is met.(2)
The robustness criterion (at the cutoff frequency): the first derivative of the phase is 0, the derivative at point A and point B on the curve of the FO-PI control system in Figure6 is 0, which satisfies the criterion of robustness.(3)
When the FO-PI control system is in the cutoff frequency, the amplitude variation is 0, which meets the criterion of amplitude.


### 4.3. System Antidisturbance Analysis

Considering the influence of grid voltage, the transfer function of the output current of the grid-connected inverter is produced in Figure2.(28)is=KGsLs+R+KGsiref−1Ls+R+KGsugrid.Substituting the fundamental frequencyω0 into equation (26), the amplitude of IO-PI controller is obtained: Aω01=Kp12+Ki1/ω02, which is a finite value. Let the first term in equation (28) be ε·iref, where ε=KGs/Ls+R+KGS=1/1+Ls+RL/KGs; it can be seen that 0<ε<1, the output current is less than the reference current, and the system has steady-state errors. Similarly, the second term is also a limited value, that the inverter output current is affected by the grid voltage; the system’s anti-interference ability is poor.For the FO-PI controller, at the fundamental frequencyω0, the amplitude is Aω02=KP2+KI/ω0λ2; the integral order λ can take any real number greater than 0, and when λ gets the appropriate value on (0, 1), Aω02≫Aω01; the first term of equation (28) is closer to iref and the second term is closer to 0; thus, ≈iref. So, compared to the IO-PI controller, the FO-PI controller largely reduces the system steady-state error and enhances the system’s antidisturbance capability.At the time of 0.05 s, the grid voltage disturbance, the grid voltage dips, and the change in the output current of the inverter is shown in Figure7.Figure 7
output current waveform of the inverter in the grid voltage change. Simulation waveforms under (a) IO-PI control and (b) FO-PI control.
(a)(b)From Figure7, we can see that the output current of the inverter under the control of IO-PI appears small fluctuations, while the output current of the inverter under FO-PI control is almost unchanged, which is not affected by the voltage disturbance of the grid. The result shows that the FO-PI controller has better anti-interference ability, and the system possesses strong robustness.


## 4.1. The Oustaloup Approximation of Integral Operatorsλ

The integral operator sλ is approximated by the Oustaloup algorithm in the transfer function expression of the FO-PI controller [17]. The algorithm can guarantee the approximation of the fractional transfer function on the basis of stability so that the transfer function is as little as possible to the zero and the pole. Assuming that the selected approximation bands are ωb and ωh, the algorithm is described as(23)sλ≈K∏k=−NNs+ωk′s+ωk.In this formula,(24)ωk′=ωbωhωbk+N+0.51−λ/2N+1,K=ωhλ,ωk=ωbωhωbk+N+0.51+λ/2N+1.Formula (24) can be considered as a recursion filter (IIR) with the order n = 2N + 1. When the approximation order n = 1, the approximation error is large. When the approximation order n = 5, both the amplitude-frequency characteristic and the phase-frequency characteristic can well approximate the fractional order. The higher the approximation order is, the higher the approximation accuracy will be. But when n is more than 5, the approximation accuracy and the approximation order are no longer proportional [18]. So, the third-order inner approximation is considered to use. The high-order term 1/sλ of the FO-PI controller is approximated in the frequency band of interest (ωb = 0.001 Hz and ωh = 1000 Hz) with n = 3, i.e., N = 1. The approximate result is(25)1sλ=0.024831s+342.8s+3.428s+0.03428s+29.17s+0.2917s+0.002917.


## 4.2. The Dynamic and Steady-State Performance Analysis

In the integral PI controller, the integralKis−1 is mainly used to improve the steady-state performance of the system. But because of the 90° lag phase angle of the integral, the overshoot and the adjustment time of the system are increased, and the dynamic performance of the system is poor. λ of Kis−λ can be adjusted arbitrarily, and the adjustment accuracy is high. The stability and dynamic performance of the system can be taken into account under the premise of improving the steady-state characteristics of the system. Figure 4 shows the unit step response curve for different values of λ.Figure 4
Comparison of step response asλ varies.From Figure4, it can be seen that, with the increment of λ, the steady-state accuracy of the system is obviously improved, the adjustment time is gradually reduced, and the overshoot quantity decreases gradually. The value of λ continues to increase, and the adjustment time becomes larger. The steady-state accuracy is reduced, and oscillation occurs. Therefore, the integral order λ mainly affects the steady-state error of the system. Selecting the proper integration order can reduce the steady-state error of the system and obtain a good dynamic performance.Letλ = 1 and μ = 0 in formula (2), then the integral-order PI controller transfer function can be expressed as(26)Gc1s=Kp1+Ki1s.The parameters of the PI controller are obtained by the method of engineering design, that is,Kp1 = 0.13 and Ki1 = 10.79, and the transfer function of the integer-order PI controller is(27)Gc1s=0.13+10.79s.In the MATLAB/Simulink environment, a step-by-step simulation model is set up. Taking the step signal as the input, the control object is controlled by the PI controller and the integer-order PI controller; the simulation results are shown in Figure5.Figure 5
System step response curve. (a) Step response curve (b) Local magnification.
(a)(b)The main performance indexes of the FO-PI controller and the IO-PI controller are obtained from the step response curve: rise timetr, overshoot δ, and adjustment time ts (±2% steady-state value), as shown in Table 2. It can be seen that the dynamic follow-up performance of the system using the FO-PI controller is superior to the system using the IO-PI controller.Table 2
Dynamic performance index.
Controllertr (ms)δ (%)ts (ms)IO-PI0.29121.08FO-PI0.266.91.02To further verify the validity of the designed FO-PI controller parameters and the superiority of the control performance, the Bode diagram of the open-loop system under FO-PI and IO-PI controller is obtained, as shown in Figure6.Figure 6
Open-loop transfer function: Bode diagram.In order to control the two control effects of comparative analysis more intuitively, the performance index values in the chart are tabulated, as shown in Table3.Table 3
Open-loop system performance.
When the system is stablePerformance indexExpected valueFO-PIIO-PIPhase margin (Xo)6058.456.3Amplitude variation (dB)00.00610.0296Cut-off frequency (rad/s)2002016.67e + 003As the results shown in Figure6 and Table 3, the following conclusions can be drawn:(1)
When the system is stable, the FO-PI control system is closer to the expected phase margin and cutoff frequency than the IO-PI control system that the phase margin criterion is met.(2)
The robustness criterion (at the cutoff frequency): the first derivative of the phase is 0, the derivative at point A and point B on the curve of the FO-PI control system in Figure6 is 0, which satisfies the criterion of robustness.(3)
When the FO-PI control system is in the cutoff frequency, the amplitude variation is 0, which meets the criterion of amplitude.


## 4.3. System Antidisturbance Analysis

Considering the influence of grid voltage, the transfer function of the output current of the grid-connected inverter is produced in Figure2.(28)is=KGsLs+R+KGsiref−1Ls+R+KGsugrid.Substituting the fundamental frequencyω0 into equation (26), the amplitude of IO-PI controller is obtained: Aω01=Kp12+Ki1/ω02, which is a finite value. Let the first term in equation (28) be ε·iref, where ε=KGs/Ls+R+KGS=1/1+Ls+RL/KGs; it can be seen that 0<ε<1, the output current is less than the reference current, and the system has steady-state errors. Similarly, the second term is also a limited value, that the inverter output current is affected by the grid voltage; the system’s anti-interference ability is poor.For the FO-PI controller, at the fundamental frequencyω0, the amplitude is Aω02=KP2+KI/ω0λ2; the integral order λ can take any real number greater than 0, and when λ gets the appropriate value on (0, 1), Aω02≫Aω01; the first term of equation (28) is closer to iref and the second term is closer to 0; thus, ≈iref. So, compared to the IO-PI controller, the FO-PI controller largely reduces the system steady-state error and enhances the system’s antidisturbance capability.At the time of 0.05 s, the grid voltage disturbance, the grid voltage dips, and the change in the output current of the inverter is shown in Figure7.Figure 7
output current waveform of the inverter in the grid voltage change. Simulation waveforms under (a) IO-PI control and (b) FO-PI control.
(a)(b)From Figure7, we can see that the output current of the inverter under the control of IO-PI appears small fluctuations, while the output current of the inverter under FO-PI control is almost unchanged, which is not affected by the voltage disturbance of the grid. The result shows that the FO-PI controller has better anti-interference ability, and the system possesses strong robustness.


## 5. Simulation Result Analyses

The inverter link is the key to the whole PV grid-connected system, and it must be connected to the grid through an effective control strategy. According to GB/T 19935-2005 technical requirements for the grid connection of the PV system, the grid-connected current and grid voltage with the same frequency and phase and the total harmonic distortion(THD) of the grid-connected current should be less than 5% of the inverter-rated output. The simulation model of the single-phase PV grid-connected current control is built using MATLAB/Simulink, as shown in Figure 8. The simulation parameters are shown in Table 4.Figure 8
Single-phase photovoltaic grid current control strategy using MATLAB/Simulink simulation model.Table 4
Simulation parameters.
Input voltage400 VFilter inductance6 mHEffective value of power network voltage220 VEquivalent resistance of filter inductance0.5 ΩPower grid voltage frequency50 HzSwitching cycle100μsFilter capacitor0.1μFPWM frequency5 kHzMaximum voltage of PV68 VMaximum current of PV59 AChopping frequency5 kHzPower of load2 kWThe inductor of DC/DC3 mHCapacitor of DC/DC220μFOn the same control object, the waveform comparison of the grid voltage and the grid current by using IO-PI control and FO-PI control is shown in Figure9. The corresponding grid current is analyzed by using the fast Fourier transform (FFT) algorithm, and the total harmonic distortion rate (THD) of the grid-connected current is illustrated in Figure 10.Figure 9
Grid current and grid voltage waveform. (a) IO-PI control grid current and grid voltage wave. (b) FO-PI control grid current and grid voltage wave.
(a)(b)Figure 10
FFT analysis of grid-connected current of the (a) IO-PI controller and (b) FO-PI controller.
(a)(b)The simulation results show that compared with the IO-PI controller, the grid-connected current and the grid voltage have smaller phase differences by using the FO-PI controller, which can effectively reduce the tracking error of the system. The THD of the grid-connected current of the IO-PI control and the FO-PI control was 0.87% and 0.35%, respectively, and the FO-PI controller is able to achieve grid-phase current and grid voltage with the same frequency and phase, and the current distortion rate is small.


## 6. Conclusion

In this paper, the PIλ controller is applied to the single-phase PV grid-connected power generation system and tracking control of the output current of the grid-connected inverter. The PIλ controller is designed by using the amplitude phase-frequency method, and the dynamic performance, steady-state performance, and robustness of the control system are simulated and compared with those of the integer-order PI controller. The results show that the PIλ controller has faster response speed and better following performance than the fractional-order PI controller, which reduces the harmonic content of the grid-connected current, enhances the anti-interference of the system, and makes the system realize grid operation better.



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*Source: 7074771-2021-11-08.xml*