Current PI Control of the Single-Phase Grid Inverter

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 (PI 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. .e quality of the inverter grid-connected current is analyzed by using the fast Fourier transform (FFT). .e 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.


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. erefore, 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 gridconnected current is as small as possible.
e 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. e 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 currenttracking 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. e 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. is 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 PI λ controller flexible control structure; when it acts on the controlled object, the system has better dynamic, steadystate performance, and robustness [6,7]; the PI λ 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, PI λ controller is designed, and the MATLAB/ Simulink simulation model is set up; the performance of the control system is simulated and compared with the PI λ controller and the integer-order PI controller, and the effectiveness of the control strategy is verified.

Single-Phase PV Grid-Connected Inverter
Control Strategy e 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. e output voltage of the inverter is automatically clamped to the grid voltage.
erefore, 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. is paper adopts the control strategy of current instantaneous value feedback and triangular wave comparison, and the controller uses the fractional-order proportional-integral (FO-PI λ ) control method.

Grid-Connected Inverter Control Structure.
In the singlephase PV grid-connected inverter control structure as shown in Figure 1, 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. e latter stage is composed of four IGBT switches to form a DC/AC fullbridge inverter circuit, which can realize the grid-connected current and grid voltage with the same frequency and phase control and DC bus voltage stability. C pv for the PV output side of the storage capacitor is used to stabilize the PV module output DC voltage and C dc is the storage capacitor of the inverter DC side.
e specific control process is as follows: compared with the difference between grid-connected current reference value i ref and the actual value of the grid-connected current i s adjusted by the PI λ 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 i s .

e 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.
In Figure 2, G c (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, K inv /(sT inv + 1), where K inv is the gain of the inverter and T inv 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

PI λ D μ Controller. PI λ D μ controller is proposed by
Professor I. Podlubny [12], and the transfer function is PI λ D μ 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 K p , K i , K d , 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. e value plane of λ and μ is shown in Figure 3. As can be seen from the figure, the conventional PID controller is a special case of the PI λ D μ controller. Since λ and μ can vary continuously, the PI λ D μ 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.

PI λ Controller Design.
e PI λ D μ controller transfer function is expressed by formula (2), which makes μ � 0, and then the transfer function of the PI λ controller can be obtained: In order to facilitate the analysis, formula (3) can be rewritten as 2 Mathematical Problems in Engineering For the specific control object G s (s), the parameters of the PI λ controller are optimized by using the frequency characteristic of the control system and the given cut-off frequency ω c and the phase margin φ m . e open-loop transfer function G k (s) can satisfy the following performance index by designing the PI λ controller G c (s) [15]: (1) Phase margin: (2) Gain robustness: (3) Amplitude criterion: e frequency response of the PI λ controller is obtained by formula (4) combined with the Euler's formula: Its phase and amplitude are For the controlled object G s (s), the inverter gains K inv and 1/R can be transferred to the proportional gain K p of the controller without affecting the gain of the whole inverter control link [16]. erefore, for the sake of generality, the gain and 1/R in formula (1) are normalized to 1. At this point, there is e frequency response of the controlled object is Its phase and amplitude are e open-loop transfer function is From formulas (9), (13), and (15)  Mathematical Problems in Engineering 3 From formulas (10), (14), and (15), the amplitude of the open-loop transfer function is From formula (16) and the phase angle margin criterion, the relationship between K′ I and λ can be obtained as From formula (16) and the robustness criterion, another relation between K'i and λ can also be established: where From Formula (17) and the amplitude criterion, we obtain the equation for K p : On the basis of satisfying the phase angle margin criterion, the robustness criterion, and the amplitude criterion, the relations of λ and K p and λ and K i are obtained. It is shown that, in the PI λ controller, the change of λ will affect the response speed, overshoot, adjustment time, and steadystate 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]. e given phase angle margin and the expected value of the cut-off frequency are shown in Table 1.
By using the plotting method, three parameters of the PI λ controller are obtained: K p � 7.89, K i � 73.25, and λ � 0.535. According to formulas (18)-(21) and K i � K p ·K' i , we can write the PI λ controller transfer function as

e Oustaloup Approximation of Integral Operator s λ .
e integral operator sλ is approximated by the Oustaloup algorithm in the transfer function expression of the FO-PI controller [17]. e 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 ω b and ω h , the algorithm is described as In this formula, 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. e 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. e high-order term 1/s λ of the FO-PI controller is approximated in the frequency band of interest (ω b � 0.001 Hz and ω h � 1000 Hz) with n � 3, i.e., N � 1. e approximate result is

e Dynamic and Steady-State Performance Analysis.
In the integral PI controller, the integral K i s −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 K i s −λ can be adjusted arbitrarily, and the adjustment accuracy is high. e 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. From Figure 4, 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. e value of λ continues to increase, and the adjustment time becomes larger. e steady-state accuracy is reduced, and oscillation occurs. erefore, 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.
Let λ � 1 and μ � 0 in formula (2), then the integral-order PI controller transfer function can be expressed as e parameters of the PI controller are obtained by the method of engineering design, that is, K p1 � 0.13 and K i1 � 10.79, and the transfer function of the integer-order PI controller is 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 Figure 5.
e main performance indexes of the FO-PI controller and the IO-PI controller are obtained from the step response curve: rise time t r , overshoot δ, and adjustment time t s (±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.
To 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 Figure 6.
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 Table 3.
As the results shown in Figure 6 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) e 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 Figure 6 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.

System Antidisturbance Analysis.
Considering the influence of grid voltage, the transfer function of the output current of the grid-connected inverter is produced in Figure 2.
Substituting the fundamental frequency ω 0 into equation (26), the amplitude of IO-PI controller is obtained: which is a finite value. Let the first term in equation (28) be ε·i ref , where ε � (KG(s)/(Ls + R + KG(S))) � 1/(1 + (Ls + R L /KG(s))); 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 frequency ω0, the amplitude is A(ω 0 ) 2 � ������������ K 2 P + (K I /ω λ 0 ) 2 ; the integral order λ can take any real number greater than 0, and when λ gets the appropriate value on (0, 1), A(ω 0 ) 2 ≫ A(ω 0 ) 1 ; the first term of equation (28) is closer to i ref and the second term is closer to 0; thus, ≈i ref . So, compared to the IO-PI controller, the FO-PI controller largely reduces the system steadystate 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 Figure 7.
From Figure 7, 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. e result shows that the FO-PI controller has better anti-interference ability, and the system possesses strong robustness.

Simulation Result Analyses
e 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. e simulation model of the single-phase PV grid-connected current control is built using MATLAB/Simulink, as shown in Figure 8. e simulation parameters are shown in Table 4.
On 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 Figure 9. e 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.
e 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. e 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 gridphase current and grid voltage with the same frequency and phase, and the current distortion rate is small.

Conclusion
In this paper, the PI λ 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. e PI λ controller is designed by using the amplitude phasefrequency method, and the dynamic performance, steadystate performance, and robustness of the control system are simulated and compared with those of the integer-order PI controller. e results show that the PI λ 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 antiinterference of the system, and makes the system realize grid operation better.
Data Availability e inverter system parameter data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.