# Control Strategy of Microgrid Inverter Based onH∞ State Feedback Repeated Deadbeat Control

Authors: Ren Xie; Yougui Guo; Yonghong Lan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Aiming at the voltage distortion at the microgrid public connection point caused by nonlinear loads, aH∞ state feedback deadbeat repetitive control strategy is proposed to rectify the total harmonic distortion of the output voltage. Firstly, through establishing the state space of the repetitive controller, introducing state feedback, combining the H∞ control theory, and reformulating the system stability problem as a convex optimization problem with a set of linear matrix inequality (LMI) constraints to be solved, high stability control accuracy can be guaranteed and antiharmonic interference strengthened. Secondly, by introducing deadbeat control technology to improve the transient response speed of the system, changes in output voltage caused by load changes can be quickly compensated. Compared with the existing methods, the designed control method has the advantages of good stability, low harmonic content, and fast convergence speed, and the results are easier to verify. Finally, the simulation verifies the effectiveness of the proposed control strategy.


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## Body

## 1. Introduction

In recent years, the microgrid has received wide attention for its unique form of maximizing the flexibility and advantages of distributed generation systems (DG) [1–3]. Its internal energy conversion is mainly conducted via power electronics, and the inverter is the core link of the distributed generation system, its operating state is related to the performance of the whole system, and z-source inverter is widely used because of its function of voltage-up and step-down conversion. When the microgrid runs in the island mode, affected by the harmonic current generated by the nonlinear loads, the voltage at the Point of Common Coupling is thus distorted, leading to the degradation of the supply voltage quality and affecting the normal operation of the load and inverter [4–6]. Therefore, it is of great practical significance to study a control strategy for reducing the total harmonic distortion (THD) of the output voltage in a state of satisfying the demand for nonlinear loads.Proportional integral (PI) control, though deficient [7, 8] for harmonic suppression, is widely used in microgrid inverters due to its easily realized structure, and the repetitive control, taking the advantage of its convenient implementation, easily realized structure, and high efficient waveform control, is widely applied to control the inverters, but the response speed is slow [9]. To enhance the robustness of the system, [10] proposes a combined controller for repetitive and sliding mode control and adds feedforward control to improve its dynamic properties, but the design of phase advance units and low-pass filters is complex. Document [11] proposed a method based on mixed sensitivity to determine repetitive control parameters. However, it takes efforts to choose weight function. Document [12, 13] proposed a design method of H∞ control theory to obtain a stable compensator to improve its robustness, but compensators with good performance are often high-end and complex to implement. Most of the repetitive control design techniques mentioned in the literature are based on the transfer function method, which is not sufficient to deal with the time-varying uncertainty caused by the load change. Not only is the solution process complicated, but also the steady-state error and phase angle margin are used to judge the system performance after consulting the benchmark amount, which is often determined through heuristics.Deadbeat control is widely studied in terms of its fast response speed and easy implementation, rather than its poor robustness [13]. Document [14] proposes an improved deadbeat control strategy to compensate for the delay caused by the digital implementation by predicting the system behavior, but it needs to select large orders to ensure the accuracy of the algorithm prediction, which makes the calculation process complicated.Inspired by the above literature, in order to reduce the harmonic content of the PCC point voltage of the z-source inverter with nonlinear loads and achieve power sharing, this paper proposes aH∞ state feedback deadbeat repetitive control based on a parameter-optimized droop controller (H∞SFDBRC) strategy. Aiming at the disadvantages of traditional repetitive control being insufficient to deal with the time-varying uncertainty caused by load changes, the complexity of the solution process, and the difficulty of verifying conditions, this paper designs an improved repetitive control system that optimizes the design of the repetitive controller, constructs a state space expression formula, and introduces H∞ state feedback control, and thus the design of the repetitive controller is transformed into a convex optimization problem with a set of LMI constraints according to the Lyapunov functional; due to the inherent delay characteristics of repetitive control, in order to improve the transient performance of the system to obtain better antidisturbance characteristics, a new conceptual topology is proposed, with the deadbeat control technology, to provide fast dynamic response after the start of the system or during the large load step changes. A simulation model was built on the MATLAB/Simulink simulation platform, and the simulation results verified the effectiveness of the control strategy.


## 2. System Structure

The single-phase off-grid inverter system based on z-source inverter is shown in Figure1. The designed control system has three constituent modules: repeated control, deadbeat control, and droop control modules. Lf and Cf are the filter inductors and capacitors, RLf is an equivalent series resistance of the filter inductance, Ri and Li are the line impedance, ILabc is an inductor current, uC is the capacitor voltage, V0abc is the output voltage, K11, K12, and K2 are competitive controller coefficients calculated by LMI, Q(s)e−sτ is a first-order low-pass filter, K3, K4, K5, and K6 are the coefficients of the deadbeat control system, uk is the SVPWM voltage modulation signal, kp and kq are droop coefficients, f0 is the rated frequency, U0 is the voltage amplitude when output reactive power is zero, and P, Q are measured values of active and reactive power, respectively. The measured inverter output voltage V0abc and filter inductor current ILabc are converted, by Park Formation, into voltage V0abc and current ILabc on the two-phase rotating coordinates. The input of voltage V0dq and current ILabc into the droop control system generates reference voltage r, which is then converted into modulate wave signals under the repetitive and deadbeat control. After controlling the inverter’s on-off switch pipe, the effective tracking of the voltage signal can be realized.Figure 1
Control block diagram of island microgrid inverter.Figure1 shows the simplified topology of the droop control outer loop and the voltage control inner loop (Figure 2).Figure 2
System simplified control topology diagram.The z-source inverter consists of two capacitors and two inductors to form an X-type network, which connects the DC source with the three-phase inverter. The function of the diode is to prevent the current from flowing back to the DC side. In normal operation, there are two states of straight-through and nonstraight-through; when the upper and lower power devices of the same bridge arm are turned on at the same time, it is in the straight-through state, through which the z-source inverter can flexibly boost and step down; the nonstraight-through state refers to the traditional inverter state. Figures3 and 4 are equivalent circuits in both straight-through and nonstraight-through states, respectively.Figure 3
Straight-through commutation state.Figure 4
Nonstraight-through commutation state.Since the three-phase filter circuit parameters are consistent and thedq-axis is independent after the coordinates are transformed, simply analyze the d-axis single-phase LC filter as shown in Figure 5; the DC voltage, z-source network, and inverter are equivalent to a voltage source u(t), where the influence of both the linear and nonlinear loads on the controlled output voltage is modeled by the uncertain load admittance Y0(t) and the external current source id(t).Figure 5
LC filter single-phase equivalent circuit.Select inductor currentILd and capacitor voltage V0d as state variables to establish circuit equations according to Kirchhoff's law(1)ut=LfdILdtdt+RLfILdt+V0dt,ILdt=CfdV0dtdt+V0dtY0t+idt.Organize it into a matrix form(2)dILdtdtdV0dtdt=−RLfLf−1Lf1CfY0tCf+1Lf0ut+01Cfidt,V0dt=01ILdtV0dt.Ifx(t) = [ILd (t) V0d (t)]T, it can be written as a state space expression.(3)x˙t=AY0txt+But+Bdidt,yt=Cxt,where x(t) ∈ R is the state vector of the inverter, u(t) ∈ R is control input, y(t) ∈ R is control output, id(t) ∈ R is periodic interference, and A(Y0(t)) is the matrix function of the uncertain parameter Y0(t).Suppose that the minimum and maximum values of theY0(t) are known(4)Ymin≤Y0t≤Ymax.The parameterY0(t) is usually converted according to its nominal value YN and the deviation YD, as follows:(5)Y0t=YN+δtYd,δt∈−1,1,where YN=Ymin+Ymax/2,Yd=Ymin−Ymax/2.According to (5)(6)AY0t=AYN+HYdδtE,δt∈−1,1,where A(YN), H(YD), and E are the constant matrix of the uncertain structures, as given by the following formula:(7)AYN=−RLfLf−1Lf1Cf−YNCf,HYd=000−YdCf,E=I2.


### 2.1. Voltage Control Strategy

Repetitive control is to reflect the deviation from the previous operation to the present and add it to the controlled object together with the “current deviation” for control to improve tracking accuracy and suppress periodic interference. However, such closed-loop systems have infinite poles on the virtual axis. It is impossible to achieve stability through classical control methods.Figure6 shows a repetitive controller with a low-pass filter; set the low-pass filter Q(s) as(8)Qs=ωcs+ωc,ωc=1T,where ωc and T are the corner frequency and time constants of the first-order low-pass filters, respectively. The method of determining the size of ωc is as follows:(9)qjω≈1,ω≪ω0,qjω<1,ω>ω0,where ω0 is the frequency bandwidth of the modulation reference signal: ω0=2π/τ.Figure 6
Low-pass filter repetitive control topology.From Figure6, the transfer function of the repeated controller is(10)Grcs=11−Qse−sτ.Deduced from formulas (8) and (9), the state space of formula (9) is available(11)x˙rct=−ωcxrct+ωcxrct−τ+ωcet−τ,yrct=xrct+et,where xrc(t) is a low-pass filter state variable.


### 2.2. Design ofH∞ State Feedback Repetitive Controller

Construct the augmentation vector as follows:(12)Zt=xtxrct∈Rn+1.Rules (3) and (11) can be converted to(13)Z˙t=Aa+ΔAatZt+AdZt−τ+Baut+Bqqt,where(14)qt=rtidt′∈R2,Aa=AYN02×101×2−ωc,Ba=B0,Ad=01×20−Cωcωc,ΔAat=HaδtEaHa=HYd01×2,Ea=E′01×2′′,Bq=02×1Bdωc0.As shown from Figure7, the control rate in the closed-loop system (13) is(15)u1t=K1xt+K2yrct,where K1=K11K12.Figure 7
Repetitive control topology.Override theu1(t)(16)u1t=FZt+K2rt,where F∈R1×n+1=K1−K2CK2.Then the augmented system (13) is rewritten as(17)Z˙t=AΔ+BaFZt+AⅆZt−τ+B¯qqt,where ΑΔ=Aa+ΔAat, and B¯q depends on K2(18)B¯q=BK2Bdωc0.To validate the stability of the above system, ignoring the external inputq(t), the closed-loop system can be abbreviated as(19)Z˙t=AΔ+BaFZt+AdZt−τ.Then for a given cutoff frequencyωc, the gain F needs to be determined so that the system is asymptotically stable for any Y0(t).In view of the above augmented system, performance criteria are introduced:(20)Jpt≔pt22=∫0∞pt′ptdt,where p(t) defines the performance output for(21)pt≔Cpzt+Dput,where Cp and Dp are the constant matrix of the appropriate dimensions.Further ensure that the system trajectory has a given exponential decay rate ofα(22)Zt≤βZ0e−αt,t>0,where β is a positive scalar and Z(0) is the initial state.(23)Zϕ=0,∀ϕ∈−τ,0.In order to obtain a sufficient condition for the robust asymptotic stability of system (19), according to [16], the above problem can be solved by the following lemma.Lemma 1 (see [15]).
For the given positive scalarsωc and α, consider formulas (17) and (21). Suppose there is a symmetric positive definite matrix W, S ∈ R3×3, matrix Y ∈ R1×3, and positive scalars λ and ν are satisfied with(24)ΓW,S,υeαtAdWWEaWCp′+Y′Dp′eαtWA′d−S03×303×1Ea′W03×3−υI303×1CpW+DpY01×301×3−λ<0,where ΓW,S,υ=AaW+WAa′+2αW+BaY+Y′Ba′+S+υHaHa′.Then the closed-loop system in equation (17) is gradually stable when the gain F=YW−1=F1F2.Note 1.
Lemma1 gives the stability conditions of H∞ repetitive control when the inverter carries an uncertain load and the design method of the state feedback control rate. It shows that the closed-loop system (19) is progressively stable, so the closed-loop system (17) is internally stable.
In this paper, LMI is used to optimize the design of the repetitive controller, and the MATLAB toolbox is used to solve it. Compared with the traditional repetitive controller design method that determines, by heuristics, reference quantity, like steady-state error and phase angle margin, this paper acquires them by solving the linear matrix inequality and at the meantime simplifies the design process by reducing the number of filters and the cost of the whole system, facilitating the solution of the time-varying uncertainty caused by load change. See Table1 for the inverter parameters involved.
Takeωc = 1000 and α = 155, and the parameters of the repetitive controller obtained by using LMI according to Lemma 1 are(25)K1=−181.2753−153.7798,K2=5.2749×103,


### 2.3.H∞ State Feedback Deadbeat Repetitive Control

Due to the inherent nature of the delay characteristic of the repetitive controller, the dynamic performance of the system is poor. While deadbeat control has the advantages of quick instantaneous reaction and low harmonic distortion rate, in order to improve the dynamic performance of the system, compensate for the distortion caused by the dead time of the switch, introduce deadbeat control technology, and propose a new conceptual topology as shown in Figure8.The sampling principle of deadbeat control is shown in Figure9, the sampling period is expressed as T, the output value of the inverter is expressed as +E and −E, ΔT(K) represents the adjusting width of the K th cycle square wave, and the same ΔT(K) represents the width of the K + 1 th cycle. The voltage value in the sampling period is determined by the sampling value at that time and the reference value at the next time.When the microgrid inverter is working, the introduced deadbeat control technology takes into account the effect of the actual load current, so that the entire system can automatically compensate for load disturbances during the transient performance, so that it can be used during system startup or load step. It provides fast dynamic response during the change period.The system adopts discrete-time simulation. The reason why the digital system can achieve the deadbeat control effect is that the output of the next beat of the system can always be expressed as a linear combination of the current input control quantity and the system state variable. When the system deviates from the reference value, it will respond quickly. The load disturbance is compensated and the pulse width of the next switching period is calculated. According to the state space expression of formula (3), it is to discretize on the equivalent impulse principle; since the deadbeat controller is mainly used to provide rapid dynamic response during system startup or load step responses, to simplify the calculation, taking Y0(t) as a fixed value is available to(26)xk+1=Gxk+M1u2k+M2I0dk,yk=Cxk,where I0d is the d-axis load current.(27)G=eAY0Ts=g11g12g21g22,xk=V0dkILdk,M1=AY0−1eAY0Ts−IB=m11m12T,M2=AY0−1eAY0Ts−IBd=m21m22T.Expanded by formula (26)(28)V0dk+1=g11V0dk+g12ILdk+m11u2k+m21I0dk,ILdk+1=g21V0dk+g22ILdk+m12u2k+m22I0dk.According to formula (28), the capacitor voltage V0d(k + 1) at tk+1 time is determined by V0d(k), ILd(k), and u2(k) at t time. Conversely, if V0d(k + 1), V0d(k), and ILd(k) are known at tk time, then the output voltage u2(k) at tk time can be calculated. The calculation formula can be derived from formula (28):(29)u2k=1m11V0dk+1−g11V0dk−g12ILdk−m21I0dk.In fact, bothV0d(k) and ILd(k) are tk time sampling values, which are known. Now let us determine the capacitor voltage V0d(k + 1) at the time of tk+1. It can be seen from formula (28) that the output of the system is the capacitor voltage and the ideal output voltage of the inverter is the standard sinusoidal reference voltage. Therefore, the reference voltage r(k + 1) of tk+1 can be used instead of the capacitor voltage V0d(k + 1) of tk+1, that is,(30)u2k=1m11rk+1−g11V0dk−g12ILdk−m21I0dk=K3rk+1−K4V0dk−K5ILdk−K6I0dk.System integrated control lawuk = u1 + u2; therefore, it is necessary to obtain the discrete-time model of u1, where K1x(t) is obtained from static feedback and does not require a discretization process. K2yrc(t) is derived from the dynamic compensator and must be discretized to determine the relationship between yrc(t) and e(t). Discretization of equation (10) can obtain the discrete-time transfer function from e(z) to yrc(z)(31)Grcz=2+ωt+ωt−2z−12+ωt+ωt−2z−1−ωtz−γ−ωtz−γ−1,where ωt=ωcTs, in terms of difference equations(32)yrck=ek+−2+ωt2+ωtek−1−yrck−1+ωt2+ωtyrck−γ+yrck−γ−1.From equations (31) and (32),(33)u1k=K11ILdqk+K12V0dqk+K2yrck.The system control law is(34)ukk=K11−K5ILdqk+K12−K4V0dqk+K2yrck+K3rk+1−K6I0dqk.According to the conclusion of the document [17], when the H∞ state feedback repetitive control and deadbeat control act independently, the system is stable, so the composite system is also stable.In summary, the design steps of the control law in this paper are as follows.Table 1
Inverter parameters.
ParameterValueFilter inductanceLf/mH0.6Filter capacitorCf/μF1500Damping resistanceRL/Ω0.01Minimum admittance/S0.0001Maximum admittance/S0.2Switching frequencyf/KHz21.6DC bus voltage/V800Figure 8
H∞ state feedback repeated deadbeat control topology.Figure 9
Structure diagram of deadbeat control sampling.Step 1.
Give suitableωc and α.Step 2.
Solve the following convex optimization problem:(35)minW,Y,S,υ,λλSatisfaction.Step 3.
Compute(36)F=YW−1=F1F2,K1=F1+K2C,K2=F2.Step 4.
Solve the output voltageu2 at t = k, and obtain the state feedback coefficient.


### 2.4. Droop Control

This paper studies the island mode of the inverter; the calculation formula of average active power P and average reactive powerQ is as follows:(37)P=V0dILd+V0qILqwf1S+wf1,Q=V0dILq−V0qILdwf2S+wf2.In the formula,wf1 and wf2 are the cutoff frequency of the low-pass filter.The droop control equation is(38)f=f0+kqQ,U=U0−kpP.In the formula,kq and kp are droop characteristic coefficients, f0 is rated frequency, and U0 is voltage amplitude when output reactive power is 0. The control parameters are shown in Table 2.Table 2
Droop control parameters.
ParameterValueLine impedance/Ω1.284 +j0.0166Filter parameterswf1 = 50; wf2 = 100Rated frequency/Hz50Voltage amplitude/V110Droop coefficientkp = 10−5, kq = 3 × 10−4


### 2.5. Simulation Verification

This paper establishes a microgrid operation simulation model with two DG on the MATLAB/Simulink software platform as shown in Figure10. The DC bus voltage of DG module is maintained by ideal power source, which adopts the same LC filter and line impedance. The standard feeder impedance of low-voltage microgrid is 0.642 + j0.0083 Ω/km. The line impedance seen from DG1 and DG2 is 1.284 + j0.0166 Ω/km. The common load load3 is connected to the common AC bus of the microgrid and through on and off switch K.Figure 10
Microgrid simulation model.In order to verify the effectiveness of the control strategy proposed in this paper, the following simulation experiments are carried out. The load is a rectifier with a 21.8 Ω output in parallel with a set of 10.9 Ω resistors.Figure11(a) shows the phase A output voltage during system startup. It can be seen that, due to the deadbeat control technology, the voltage has achieved good tracking in a very short time. As shown in Figure 11(b), the periodic error signal begins to fluctuate and then quickly converges to a very small level (less than 0.5 V), and the entire system is stable. Then apply a large load step change test to check the system control performance. At the 0.4 s time point, the nonlinear load suddenly increases from one group to two groups. Figure 12(a) shows the corresponding PCC point voltage and current changes. It can be seen that, for large load step changes, the system is robust and withstands variation. Figure 12(b) shows the changes of active power and reactive power when the system load changes step by step. It can be seen that the system can still maintain a stable and equal distribution of active and reactive power when the load step changes, and the power fluctuates in it tending to be stable in a short period of time, indicating that the proposed control strategy has good anti-interference performance and dynamic response performance.Figure 11
Output voltage of phase A during system startup and voltage error signal ofd-axis seen by the repetitive controller. (a) Phase A output voltage. (b) d-axis voltage error obtained.
(a)(b)Figure 12
Three-phase output voltage during steady-state operation and three-phase voltage and current during load step changes. (a) PCC voltage and current. (b) Active and reactive power.
(a)(b)According to the comparison method of [18], the proposed H∞ state feedback deadbeat repetitive control is compared with H∞ repetitive control and PI control (hereinafter referred to as H∞SFDBRC, H∞RC, and PI), in which H∞ repetitive control and PI control are designed by ourselves, and other conditions such as inverter parameters and line impedance are consistent. It is proved that the designed control strategy can improve the performance of inverter PCC point voltage THD under nonlinear load.When using PI control, the PCC point voltage waveform, voltage error, and spectrum analysis results are shown in Figure13(a). It can be seen from the figure that the PCC point voltage waveform has obvious distortion, the steady-state error is the largest, and the tracking performance is poor. The quality of the grid-connected voltage waveform is poor, and the voltage harmonic content is 4.91%.Figure 13
PCC point voltage, d-axis voltage error, and its THD value. (a) PI controller. (b)H∞RC controller. (c) H∞SFDBRC controller.
(a)(b)(c)When usingH∞RC, the PCC point voltage waveform, voltage error, and spectrum analysis results are shown in Figure 13(b). The introduction to the internal model link enables the system to better compensate for the harmonic voltage, and the PCC point voltage waveform quality is better than PI control strategy, the voltage waveform is smoother, the harmonic compensation effect is good, the voltage harmonic content is 2.44%, and the steady-state error is better than PI control, but the dynamic response speed is slow.When usingH∞SFDBRC, the PCC point voltage waveform, voltage error, and spectrum analysis results are shown in Figure 13(c). At this time, the system steady-state error is the smallest, and the excellent tracking performance significantly improves the voltage waveform quality. The deadbeat control technology with the introduction to the system enhances the anti-interference performance of the system, enabling the system to quickly respond to various sudden problems. The quality of the voltage waveform is better than the first two control strategies, and the harmonic compensation effect is very good, and the voltage harmonic content is only 0.67%.In order to further verify the feasibility of the proposed scheme, the PCC point voltage waveform and power changes in the following two cases are simulated. Under the situation, only the inverter DG1 runs when the system starts, and the DG2 switch is closed at 0.4 s, and the simulation time is 1s. The simulated waveforms are shown in Figures14(a)–14(c). When the microgrid is operating normally, it is connected to DG2 for interconnection at 0.4 s. DG1 and DG2 maintain a good coordinated operation. The PCC voltage quickly returns to a stable sinusoidal curve after slight fluctuations, during which active power and reactive power can also move quickly to achieve power sharing and maintain stability. It can also move quickly to achieve power sharing and maintain stability.Figure 14
PCC voltage, DG output active, and reactive power. (a) PCC voltage waveform. (b) DG output active power. (c) DG output reactive power. (d) PCC voltage waveform. (e) DG output active power. (f) DG output reactive power.
(a)(b)(c)(d)(e)(f)In case two, the two inverters run in parallel, the simulation time is 1 s, the switchKn is disconnected at 0.4 s, and DG2 exits operation. From the simulation waveform diagrams 14(d)–14(f), it can be seen that when the microgrid is running in parallel, when one of the DGs quits operation due to a fault, the microgrid can react quickly and reach a new stable state, where the PCC voltage is almost invariant and remains as a sine curve, and the distribution and transformation of active power and reactive power in this process also maintain extremely high accuracy.In summary, the control strategy proposed in this article can still ensure the stability and normal operation of the microgrid when the DG is connected or disconnected or the load changes.


## 2.1. Voltage Control Strategy

Repetitive control is to reflect the deviation from the previous operation to the present and add it to the controlled object together with the “current deviation” for control to improve tracking accuracy and suppress periodic interference. However, such closed-loop systems have infinite poles on the virtual axis. It is impossible to achieve stability through classical control methods.Figure6 shows a repetitive controller with a low-pass filter; set the low-pass filter Q(s) as(8)Qs=ωcs+ωc,ωc=1T,where ωc and T are the corner frequency and time constants of the first-order low-pass filters, respectively. The method of determining the size of ωc is as follows:(9)qjω≈1,ω≪ω0,qjω<1,ω>ω0,where ω0 is the frequency bandwidth of the modulation reference signal: ω0=2π/τ.Figure 6
Low-pass filter repetitive control topology.From Figure6, the transfer function of the repeated controller is(10)Grcs=11−Qse−sτ.Deduced from formulas (8) and (9), the state space of formula (9) is available(11)x˙rct=−ωcxrct+ωcxrct−τ+ωcet−τ,yrct=xrct+et,where xrc(t) is a low-pass filter state variable.


## 2.2. Design ofH∞ State Feedback Repetitive Controller

Construct the augmentation vector as follows:(12)Zt=xtxrct∈Rn+1.Rules (3) and (11) can be converted to(13)Z˙t=Aa+ΔAatZt+AdZt−τ+Baut+Bqqt,where(14)qt=rtidt′∈R2,Aa=AYN02×101×2−ωc,Ba=B0,Ad=01×20−Cωcωc,ΔAat=HaδtEaHa=HYd01×2,Ea=E′01×2′′,Bq=02×1Bdωc0.As shown from Figure7, the control rate in the closed-loop system (13) is(15)u1t=K1xt+K2yrct,where K1=K11K12.Figure 7
Repetitive control topology.Override theu1(t)(16)u1t=FZt+K2rt,where F∈R1×n+1=K1−K2CK2.Then the augmented system (13) is rewritten as(17)Z˙t=AΔ+BaFZt+AⅆZt−τ+B¯qqt,where ΑΔ=Aa+ΔAat, and B¯q depends on K2(18)B¯q=BK2Bdωc0.To validate the stability of the above system, ignoring the external inputq(t), the closed-loop system can be abbreviated as(19)Z˙t=AΔ+BaFZt+AdZt−τ.Then for a given cutoff frequencyωc, the gain F needs to be determined so that the system is asymptotically stable for any Y0(t).In view of the above augmented system, performance criteria are introduced:(20)Jpt≔pt22=∫0∞pt′ptdt,where p(t) defines the performance output for(21)pt≔Cpzt+Dput,where Cp and Dp are the constant matrix of the appropriate dimensions.Further ensure that the system trajectory has a given exponential decay rate ofα(22)Zt≤βZ0e−αt,t>0,where β is a positive scalar and Z(0) is the initial state.(23)Zϕ=0,∀ϕ∈−τ,0.In order to obtain a sufficient condition for the robust asymptotic stability of system (19), according to [16], the above problem can be solved by the following lemma.Lemma 1 (see [15]).
For the given positive scalarsωc and α, consider formulas (17) and (21). Suppose there is a symmetric positive definite matrix W, S ∈ R3×3, matrix Y ∈ R1×3, and positive scalars λ and ν are satisfied with(24)ΓW,S,υeαtAdWWEaWCp′+Y′Dp′eαtWA′d−S03×303×1Ea′W03×3−υI303×1CpW+DpY01×301×3−λ<0,where ΓW,S,υ=AaW+WAa′+2αW+BaY+Y′Ba′+S+υHaHa′.Then the closed-loop system in equation (17) is gradually stable when the gain F=YW−1=F1F2.Note 1.
Lemma1 gives the stability conditions of H∞ repetitive control when the inverter carries an uncertain load and the design method of the state feedback control rate. It shows that the closed-loop system (19) is progressively stable, so the closed-loop system (17) is internally stable.
In this paper, LMI is used to optimize the design of the repetitive controller, and the MATLAB toolbox is used to solve it. Compared with the traditional repetitive controller design method that determines, by heuristics, reference quantity, like steady-state error and phase angle margin, this paper acquires them by solving the linear matrix inequality and at the meantime simplifies the design process by reducing the number of filters and the cost of the whole system, facilitating the solution of the time-varying uncertainty caused by load change. See Table1 for the inverter parameters involved.
Takeωc = 1000 and α = 155, and the parameters of the repetitive controller obtained by using LMI according to Lemma 1 are(25)K1=−181.2753−153.7798,K2=5.2749×103,


## 2.3.H∞ State Feedback Deadbeat Repetitive Control

Due to the inherent nature of the delay characteristic of the repetitive controller, the dynamic performance of the system is poor. While deadbeat control has the advantages of quick instantaneous reaction and low harmonic distortion rate, in order to improve the dynamic performance of the system, compensate for the distortion caused by the dead time of the switch, introduce deadbeat control technology, and propose a new conceptual topology as shown in Figure8.The sampling principle of deadbeat control is shown in Figure9, the sampling period is expressed as T, the output value of the inverter is expressed as +E and −E, ΔT(K) represents the adjusting width of the K th cycle square wave, and the same ΔT(K) represents the width of the K + 1 th cycle. The voltage value in the sampling period is determined by the sampling value at that time and the reference value at the next time.When the microgrid inverter is working, the introduced deadbeat control technology takes into account the effect of the actual load current, so that the entire system can automatically compensate for load disturbances during the transient performance, so that it can be used during system startup or load step. It provides fast dynamic response during the change period.The system adopts discrete-time simulation. The reason why the digital system can achieve the deadbeat control effect is that the output of the next beat of the system can always be expressed as a linear combination of the current input control quantity and the system state variable. When the system deviates from the reference value, it will respond quickly. The load disturbance is compensated and the pulse width of the next switching period is calculated. According to the state space expression of formula (3), it is to discretize on the equivalent impulse principle; since the deadbeat controller is mainly used to provide rapid dynamic response during system startup or load step responses, to simplify the calculation, taking Y0(t) as a fixed value is available to(26)xk+1=Gxk+M1u2k+M2I0dk,yk=Cxk,where I0d is the d-axis load current.(27)G=eAY0Ts=g11g12g21g22,xk=V0dkILdk,M1=AY0−1eAY0Ts−IB=m11m12T,M2=AY0−1eAY0Ts−IBd=m21m22T.Expanded by formula (26)(28)V0dk+1=g11V0dk+g12ILdk+m11u2k+m21I0dk,ILdk+1=g21V0dk+g22ILdk+m12u2k+m22I0dk.According to formula (28), the capacitor voltage V0d(k + 1) at tk+1 time is determined by V0d(k), ILd(k), and u2(k) at t time. Conversely, if V0d(k + 1), V0d(k), and ILd(k) are known at tk time, then the output voltage u2(k) at tk time can be calculated. The calculation formula can be derived from formula (28):(29)u2k=1m11V0dk+1−g11V0dk−g12ILdk−m21I0dk.In fact, bothV0d(k) and ILd(k) are tk time sampling values, which are known. Now let us determine the capacitor voltage V0d(k + 1) at the time of tk+1. It can be seen from formula (28) that the output of the system is the capacitor voltage and the ideal output voltage of the inverter is the standard sinusoidal reference voltage. Therefore, the reference voltage r(k + 1) of tk+1 can be used instead of the capacitor voltage V0d(k + 1) of tk+1, that is,(30)u2k=1m11rk+1−g11V0dk−g12ILdk−m21I0dk=K3rk+1−K4V0dk−K5ILdk−K6I0dk.System integrated control lawuk = u1 + u2; therefore, it is necessary to obtain the discrete-time model of u1, where K1x(t) is obtained from static feedback and does not require a discretization process. K2yrc(t) is derived from the dynamic compensator and must be discretized to determine the relationship between yrc(t) and e(t). Discretization of equation (10) can obtain the discrete-time transfer function from e(z) to yrc(z)(31)Grcz=2+ωt+ωt−2z−12+ωt+ωt−2z−1−ωtz−γ−ωtz−γ−1,where ωt=ωcTs, in terms of difference equations(32)yrck=ek+−2+ωt2+ωtek−1−yrck−1+ωt2+ωtyrck−γ+yrck−γ−1.From equations (31) and (32),(33)u1k=K11ILdqk+K12V0dqk+K2yrck.The system control law is(34)ukk=K11−K5ILdqk+K12−K4V0dqk+K2yrck+K3rk+1−K6I0dqk.According to the conclusion of the document [17], when the H∞ state feedback repetitive control and deadbeat control act independently, the system is stable, so the composite system is also stable.In summary, the design steps of the control law in this paper are as follows.Table 1
Inverter parameters.
ParameterValueFilter inductanceLf/mH0.6Filter capacitorCf/μF1500Damping resistanceRL/Ω0.01Minimum admittance/S0.0001Maximum admittance/S0.2Switching frequencyf/KHz21.6DC bus voltage/V800Figure 8
H∞ state feedback repeated deadbeat control topology.Figure 9
Structure diagram of deadbeat control sampling.Step 1.
Give suitableωc and α.Step 2.
Solve the following convex optimization problem:(35)minW,Y,S,υ,λλSatisfaction.Step 3.
Compute(36)F=YW−1=F1F2,K1=F1+K2C,K2=F2.Step 4.
Solve the output voltageu2 at t = k, and obtain the state feedback coefficient.


## 2.4. Droop Control

This paper studies the island mode of the inverter; the calculation formula of average active power P and average reactive powerQ is as follows:(37)P=V0dILd+V0qILqwf1S+wf1,Q=V0dILq−V0qILdwf2S+wf2.In the formula,wf1 and wf2 are the cutoff frequency of the low-pass filter.The droop control equation is(38)f=f0+kqQ,U=U0−kpP.In the formula,kq and kp are droop characteristic coefficients, f0 is rated frequency, and U0 is voltage amplitude when output reactive power is 0. The control parameters are shown in Table 2.Table 2
Droop control parameters.
ParameterValueLine impedance/Ω1.284 +j0.0166Filter parameterswf1 = 50; wf2 = 100Rated frequency/Hz50Voltage amplitude/V110Droop coefficientkp = 10−5, kq = 3 × 10−4


## 2.5. Simulation Verification

This paper establishes a microgrid operation simulation model with two DG on the MATLAB/Simulink software platform as shown in Figure10. The DC bus voltage of DG module is maintained by ideal power source, which adopts the same LC filter and line impedance. The standard feeder impedance of low-voltage microgrid is 0.642 + j0.0083 Ω/km. The line impedance seen from DG1 and DG2 is 1.284 + j0.0166 Ω/km. The common load load3 is connected to the common AC bus of the microgrid and through on and off switch K.Figure 10
Microgrid simulation model.In order to verify the effectiveness of the control strategy proposed in this paper, the following simulation experiments are carried out. The load is a rectifier with a 21.8 Ω output in parallel with a set of 10.9 Ω resistors.Figure11(a) shows the phase A output voltage during system startup. It can be seen that, due to the deadbeat control technology, the voltage has achieved good tracking in a very short time. As shown in Figure 11(b), the periodic error signal begins to fluctuate and then quickly converges to a very small level (less than 0.5 V), and the entire system is stable. Then apply a large load step change test to check the system control performance. At the 0.4 s time point, the nonlinear load suddenly increases from one group to two groups. Figure 12(a) shows the corresponding PCC point voltage and current changes. It can be seen that, for large load step changes, the system is robust and withstands variation. Figure 12(b) shows the changes of active power and reactive power when the system load changes step by step. It can be seen that the system can still maintain a stable and equal distribution of active and reactive power when the load step changes, and the power fluctuates in it tending to be stable in a short period of time, indicating that the proposed control strategy has good anti-interference performance and dynamic response performance.Figure 11
Output voltage of phase A during system startup and voltage error signal ofd-axis seen by the repetitive controller. (a) Phase A output voltage. (b) d-axis voltage error obtained.
(a)(b)Figure 12
Three-phase output voltage during steady-state operation and three-phase voltage and current during load step changes. (a) PCC voltage and current. (b) Active and reactive power.
(a)(b)According to the comparison method of [18], the proposed H∞ state feedback deadbeat repetitive control is compared with H∞ repetitive control and PI control (hereinafter referred to as H∞SFDBRC, H∞RC, and PI), in which H∞ repetitive control and PI control are designed by ourselves, and other conditions such as inverter parameters and line impedance are consistent. It is proved that the designed control strategy can improve the performance of inverter PCC point voltage THD under nonlinear load.When using PI control, the PCC point voltage waveform, voltage error, and spectrum analysis results are shown in Figure13(a). It can be seen from the figure that the PCC point voltage waveform has obvious distortion, the steady-state error is the largest, and the tracking performance is poor. The quality of the grid-connected voltage waveform is poor, and the voltage harmonic content is 4.91%.Figure 13
PCC point voltage, d-axis voltage error, and its THD value. (a) PI controller. (b)H∞RC controller. (c) H∞SFDBRC controller.
(a)(b)(c)When usingH∞RC, the PCC point voltage waveform, voltage error, and spectrum analysis results are shown in Figure 13(b). The introduction to the internal model link enables the system to better compensate for the harmonic voltage, and the PCC point voltage waveform quality is better than PI control strategy, the voltage waveform is smoother, the harmonic compensation effect is good, the voltage harmonic content is 2.44%, and the steady-state error is better than PI control, but the dynamic response speed is slow.When usingH∞SFDBRC, the PCC point voltage waveform, voltage error, and spectrum analysis results are shown in Figure 13(c). At this time, the system steady-state error is the smallest, and the excellent tracking performance significantly improves the voltage waveform quality. The deadbeat control technology with the introduction to the system enhances the anti-interference performance of the system, enabling the system to quickly respond to various sudden problems. The quality of the voltage waveform is better than the first two control strategies, and the harmonic compensation effect is very good, and the voltage harmonic content is only 0.67%.In order to further verify the feasibility of the proposed scheme, the PCC point voltage waveform and power changes in the following two cases are simulated. Under the situation, only the inverter DG1 runs when the system starts, and the DG2 switch is closed at 0.4 s, and the simulation time is 1s. The simulated waveforms are shown in Figures14(a)–14(c). When the microgrid is operating normally, it is connected to DG2 for interconnection at 0.4 s. DG1 and DG2 maintain a good coordinated operation. The PCC voltage quickly returns to a stable sinusoidal curve after slight fluctuations, during which active power and reactive power can also move quickly to achieve power sharing and maintain stability. It can also move quickly to achieve power sharing and maintain stability.Figure 14
PCC voltage, DG output active, and reactive power. (a) PCC voltage waveform. (b) DG output active power. (c) DG output reactive power. (d) PCC voltage waveform. (e) DG output active power. (f) DG output reactive power.
(a)(b)(c)(d)(e)(f)In case two, the two inverters run in parallel, the simulation time is 1 s, the switchKn is disconnected at 0.4 s, and DG2 exits operation. From the simulation waveform diagrams 14(d)–14(f), it can be seen that when the microgrid is running in parallel, when one of the DGs quits operation due to a fault, the microgrid can react quickly and reach a new stable state, where the PCC voltage is almost invariant and remains as a sine curve, and the distribution and transformation of active power and reactive power in this process also maintain extremely high accuracy.In summary, the control strategy proposed in this article can still ensure the stability and normal operation of the microgrid when the DG is connected or disconnected or the load changes.


## 3. Conclusion

Aiming at the problem of voltage distortion at the public grid connection point of island microgrid caused by nonlinear load, this paper designs aH∞ state feedback deadbeat repetitive control strategy to reduce the total harmonic distortion of the output voltage. Through theoretical analysis and research, the following conclusions can be drawn.(1)
Lyapunov functional is used to optimize the design problem of repetitive controllers, the robustness of the closed-loop system is ensured by introducing state feedback, combined withH∞ control theory, the design problem is transformed into a set of linear matrix inequality constraints’ convex optimization problem, which can simplify the repetitive control design, and compared with the traditional design method, the parameters representing the stability of the system can be accurately obtained and verified. It does not need to be obtained through repeated tests by trial method. It is more suitable for practical engineering applications and has the advantages of good stability, low harmonic content, and fast convergence speed.(2)
The introduction of deadbeat control technology not only effectively improves the response speed of the system, but also compensates for the distortion introduced by the switch dead time. Through simulation, we can find that the microgrid can quickly react and respond regardless of the actual system startup period or the large load step change period. A stable state is reached.(3)
The simulation results show that, compared with PI control andH∞ repetitive control, the proposed control strategy can effectively reduce harmonics, improve the voltage quality at PCC point, and has good control performance under nonlinear load conditions.



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*Source: 8324926-2021-12-21.xml*