# Modified Passivity-Based Control for LCL-Filtered Grid-Tied Inverter with Output Admittance Reshaping

Authors: Han Wang; Gang Shi; Jianwen Zhang; Miao Zhu; Xu Cai
Journal: Journal of Control Science and Engineering
 (2023)

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## Abstract

As a nonlinear control method, the Euler–Lagrange- (EL-) based passivity-based control (PBC) has been studied for grid-tied converters based on energy function to achieve better performance. However, the EL-based PBC method is dependent on an accurate mathematical model. In the traditional EL-based PBC research for LCL-filtered grid-tied inverter, the effect of the digital control delay is rarely considered and the stability under the grid impedance uncertainties is not discussed, especially in the capacitive grid or complex weak grid. To address these concerns, this study proposes a modified EL-based PBC method based on the output admittance reshaping for LCL-filtered grid-tied inverters. The system’ passive region is expanded by adding capacitor current feedback loop up to the Nyquist frequency. The potential resonance is thus eliminated irrespective of the grid impedance. Additionally, the stable region and control parameters design methods of the modified EL-based PBC method with inverter-side control are also carried out. To verify the correctness of the theoretical analysis, both simulation and experimental results are investigated from a 3 kW grid-tied inverter prototype.


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## Body

## 1. Introduction

Under the influence of energy shortage and global warming, the distributed renewable energy generation system has become more and more important. Generally, renewable energy resources are interfaced to the power grid by grid converters [1, 2]. To get a better dynamic performance and attenuate the injected current harmonics effectively, the LCL filter is frequently selected to filter out switching harmonics caused by pulse width modulation (PWM) [3]. Even though the LCL filter can save total inductance compared to the L filter, the inherent resonance problems caused by the high-order filter may destabilize the entire system and pose challenges to the design of the controller, especially when the variations of grid impedance are considered.Proportional-integral (PI) and proportional-resonant (PR), as conventional linear control methods, are usually implemented with passive damping [4–6] or active damping [7–12] techniques to eliminate the possible resonance problems. The passive damping method involves directly adding physical resistors in the filter circuit, which will certainly lead to extra losses and complexity to the resistor design. The active damping method is usually achieved by using state feedback variables to construct a virtual resistor, but additional sensors or complex algorithms will be necessary in the realization. What is more is that filter-based damping and inherent damping methods are also proposed with the digital implementation [13, 14]. Given the time delay of 1.5T, where T is the sampling period, it was found that if the LCL resonance frequencies are placed below one-sixth of the sampling frequency, the extra damping is not required for inverter-side current control. Instead, for the grid-side current control, the LCL resonance frequencies should be placed above one-sixth of the sampling frequency. But control effects of these damping methods are strongly dependent on grid impedance, LCL filter parameters, and sampling frequency [15].Apart from the linear control methods, some nonlinear control methods have been increasingly studied for DC-DC converters and DC-AC converters considering the nonlinear characteristics of converters, [16] including passivity-based control (PBC), [17–20] predictive control (PC), [21, 22] sliding mode control (SMC), and so on [23, 24]. Among these nonlinear control strategies, the passivity-based control (PBC) approach has attached wide attention owing to its capability to realize better performances and strong robustness based on the energy dissipation and modification methods. The PBC method was first proposed in the electrical system by Ortega et al. [25] which described the physical system by the Euler–Lagrange (EL) model or port-controlled Hamiltonian systems with dissipation (PCHD) equation. The error energy function based on Lyapunov’s theory is normally constructed and the controller is designed with damping injection. The authors in [26] adopted the PBC-based EL model to suppress the low-frequency oscillations with L filter in the railway power system. The authors in [17] implemented the passivity-based decoupling control for the neutral point clamped photovoltaic inverter to gain parameters perturbations, but the grid impedance is not considered. In order to decrease the steady-state error and improve the robustness of the system, a design procedure of the damping gains for the LCL-filtered grid-tied inverter based on PBC is proposed in reference [18], but the delay is approximated. In some cases, grid-tied inverters are normally interfaced together to the grid at the point-of-common-coupling (PCC) in the distributed generation system, and the equivalent grid impedance could be capacitive in long cable or more complex grid [27]. The conventional EL-based PBC methods of the LCL-type filter-based grid-tied voltage source inverter (VSI) system rarely take the capacitive grid condition and digital delay into consideration [28, 29]. The probably varying resonance frequencies of the system will easily lead to interactive resonances in the digital implementation. Therefore, the controller design of PBC becomes very important facing the complex grid impedance.Although EL-based PBC is defined as the nonlinear control based on energy dissipation, the control law can be implemented with linear control block, which combine feedforward control, negative feedback control, and decoupling control. To oppose grid impedance uncertainties, the impedance-based method has been applied to assess stability and dynamic interactions of paralleled VSIs systems with linear control methods [21, 30–33]. The authors in [33] analyzed passive characteristics of the single-loop grid-side current control and converter-side current control with traditional PI or PR, but it is hard to ensure all regions passively. Inspired by the previous work, the modified EL-based PBC strategy has the following contributions:(i)
For conventional EL-based PBC, the passivity is only ensured for inverter itself when designing the control law. Nonpassive regions caused by the control delay of the system and grid impedance are not considered. In this study, the impedance model of EL-based PBC-based LCL-filtered VSI with control delay is derived and it reveals that the inverter output admittance is not passive with the EL-based PBC method.(ii)
It still has lack of the specific design method for the parameters of EL-based PBC. In this study, a frequency-domain design method is implemented to ensure the stability of the inverter under the complex grid impedance.(iii)
As compared to conventional linear PI control, this study proposed a simple admittance reshaping strategy for LCL-filtered VSI with EL-based PBC, which easily ensure the passivity of the inverter output admittance up to the Nyquist frequency.This study is organized as follows: First, the Euler–Lagrange model is built for LCL-filtered three-phase VSI, and EL-based PBC control law is deduced with inverter-side current control. Then, the impedance of LCL-filtered three-phase grid-tied VSI with EL-based PBC is analyzed and control output admittance is deduced. Based on the frequency-domain passivity theory, the influence of time delay and the nonpassive regions of the inverter output admittance are identified. Then, the phase reshaping is applied to expand the passive region with the capacitor current feedback method. The proposed strategy can achieve passive output admittance up to the Nyquist frequency with inverter-side current control. Finally, simulated and experimental results are presented to confirm the performance and correctness of the proposed control method.


## 2. EL-Based PBC Control for LCL-Filtered Three-Phase Grid-Tied Inverter

### 2.1. System Modelling

The structure of the three-phase LCL-filtered VSI system with the PBC controller is illustrated in Figure1. The LCL filter is expressed by L1, C, and L2, where R1 and R2 denote the equivalent parasitic resistances of L1 and L2, respectively. The output current and voltage of inverter created by PWM are notated as i1 and ui. uC and ic represent the voltage and current of capacitor, respectively. In this study, the inverter-side current is regulated and synchronized with capacitor voltage. Udc is noted as the DC link voltage and i2 is the injected grid current. vpcc is the PCC joint voltage. Zg is the impedance of the grid, which is composed of inductor Lg and capacitor Cg. The mathematical model of the LCL-filtered three-phase VSI can be described by applying Kirchhoff voltage law (KVL). The three-phase variables are usually formulated in the synchronously rotating dq-coordinate frame, and then, the system voltage equations are expressed in function (1), where ω represents the grid angular frequency. The three-order LCL system can be decomposed into three subsystems to decouple the cascaded three-order LCL system in d-channel and q-channel, respectively.(1)L1di1ddt+R1i1d=uid+ωL1i1q−uCdL1di1qdt+R1i1q=uiq−ωL1i1d−uCq,L2di2ddt+R2i2d=uCd+ωL2i2q−vpccdL2di2qdt+R2i2q=uCq−ωL2i2d−vpccq,CduCddt+i2d=i1d+ωCuCqCduCqdt+i2q=i1q−ωCuCd.Figure 1
Configuration of LCL-filtered three-phase grid-tied VSI based on PBC with inverter-side current control.As an important part of the nonlinear PBC theory, the EL method is used to describe the system model. Since inverter-side current is controlled, the filter capacitor and filter inductor can be seen as together with the grid impedance. So, only the model of the inverter-side current variables should be designed for the simplification. The first two equations of function (1) can be written in the form with the EL model as follows:(2)Mx˙+Jx+Rx=u,where(3)M=L100L1,J=0−ωL1ωL10,R=R100R1,x=i1di1qT,u=uid−uCduiq−uCq.Equation (2) is seen as the state equation, in which i1d and i1q represent the corresponding d-component and q-component of state variable i1, respectively. M and R are the positive definite diagonal matrices, where R represents the internal energy dissipation characteristics of the system; J is the skew-symmetric matrix which describes the internal coupled structure between d-axis and q-axis current component, and J = −JT; u is the control input matrix, which represents the system and external energy input.


### 2.2. Traditional EL-Based PBC Control Law and Damping Injection

The system is considered strictly passive if there exists a positive definite functionQ (x), and a nonnegative storage function V (x) can satisfythe equation [17].(4)Vxt−Vx0≤∫0tuTydτ−∫0tQxdτ,t>0,or(5)V˙x≤uTy−Qx,where u and y are noted as system input and output variables, respectively. uTy is the injected rate of external energy. Function (4) indicates that the sum of the increased energy of the system is always less than the sum of the external injected energy. For the LCL-filtered grid-tied inverter, the system energy storage function is defined as follows:(6)Vx=12xTMx,where positive definite function V (x) represents the energy storage of the internal system. Combining with (2), the time derivative of system energy function (6) is taken. Then, it deduces the following:(7)V˙x=xTMx˙=xTu−Jx−Rx=xTu−xTRx.Lettingy = x, Q (x) = xTRx, and combining with function (5), it can be seen that the system is passive. The integration term ∫0txTudt is the external supply energy of the system. ∫0txTRxdt is the dissipative term caused by the parasitic resistance of the system.According to the passivity situation, the object described in (2) is strictly passive where the EL-based PBC can be adopted to design the controller. For inverter-side current control, the current reference variables can be defined as follows:(8)x∗=i1d∗,i1q∗T.The error vector of the current is denoted asxe=x∗−x; then, an error EL equation is obtained by (2) as(9)Mx˙∗−x˙e+Jx∗−xe+Rx∗−xe=u.(10)Mx˙∗+Jx∗+Rx∗−u=Mx˙e+Jxe+Rxe.If the errorxe is equal to zero, the desired control objective and stable equilibrium point will happen. A positive damping matrix Rd can be added to the error EL model (10) to accelerate the error to reach zero. The injected damping and new dissipation term are described as(11)Rd=diagr1r1,Rnew=R+Rd,where r1 > 0. Taking function (11) into the error equation, the new error equation with injected damping is rewritten as(12)Mx˙e+Jxe+Rnewxe=Mx˙∗+Jx˙∗+Rx∗+Rdxe−u=ζ.In this study,ζ is set as Jxe to eliminate the coupling terms of system [26]. Thus, the control law from (12) is derived as follows:(13)ud=L1di1d∗dt−ωL1i1q+R1i1d∗+r1i1d∗−i1d+uCd,uq=L1di1q∗dt+ωL1i1d+R1i1q∗+r1i1q∗−i1q+uCq.According to the analysis mentioned before, the controller diagram in Laplace domain of the grid-tied VSI with the EL-based PBC method is depicted in Figure2, where L1 and R1 present the actual values of the LCL filter physical model, and L1e and R1e present the designed LCL filter parameters in the controller. In the implementation, r1 can be substituted by PI controller to reduce the steady-state error. e−1.5Ts indicates the computational and modulation delays, and T is the sampling period [9]. Since the delays are not considered in the design of the conventional EL-based PBC controller, dissipative properties of its output admittance should be investigated in the following discussion.Figure 2
Control structure for LCL-filtered three-phase grid-tied VSI with EL-based PBC with inverter-side current control.


## 2.1. System Modelling

The structure of the three-phase LCL-filtered VSI system with the PBC controller is illustrated in Figure1. The LCL filter is expressed by L1, C, and L2, where R1 and R2 denote the equivalent parasitic resistances of L1 and L2, respectively. The output current and voltage of inverter created by PWM are notated as i1 and ui. uC and ic represent the voltage and current of capacitor, respectively. In this study, the inverter-side current is regulated and synchronized with capacitor voltage. Udc is noted as the DC link voltage and i2 is the injected grid current. vpcc is the PCC joint voltage. Zg is the impedance of the grid, which is composed of inductor Lg and capacitor Cg. The mathematical model of the LCL-filtered three-phase VSI can be described by applying Kirchhoff voltage law (KVL). The three-phase variables are usually formulated in the synchronously rotating dq-coordinate frame, and then, the system voltage equations are expressed in function (1), where ω represents the grid angular frequency. The three-order LCL system can be decomposed into three subsystems to decouple the cascaded three-order LCL system in d-channel and q-channel, respectively.(1)L1di1ddt+R1i1d=uid+ωL1i1q−uCdL1di1qdt+R1i1q=uiq−ωL1i1d−uCq,L2di2ddt+R2i2d=uCd+ωL2i2q−vpccdL2di2qdt+R2i2q=uCq−ωL2i2d−vpccq,CduCddt+i2d=i1d+ωCuCqCduCqdt+i2q=i1q−ωCuCd.Figure 1
Configuration of LCL-filtered three-phase grid-tied VSI based on PBC with inverter-side current control.As an important part of the nonlinear PBC theory, the EL method is used to describe the system model. Since inverter-side current is controlled, the filter capacitor and filter inductor can be seen as together with the grid impedance. So, only the model of the inverter-side current variables should be designed for the simplification. The first two equations of function (1) can be written in the form with the EL model as follows:(2)Mx˙+Jx+Rx=u,where(3)M=L100L1,J=0−ωL1ωL10,R=R100R1,x=i1di1qT,u=uid−uCduiq−uCq.Equation (2) is seen as the state equation, in which i1d and i1q represent the corresponding d-component and q-component of state variable i1, respectively. M and R are the positive definite diagonal matrices, where R represents the internal energy dissipation characteristics of the system; J is the skew-symmetric matrix which describes the internal coupled structure between d-axis and q-axis current component, and J = −JT; u is the control input matrix, which represents the system and external energy input.


## 2.2. Traditional EL-Based PBC Control Law and Damping Injection

The system is considered strictly passive if there exists a positive definite functionQ (x), and a nonnegative storage function V (x) can satisfythe equation [17].(4)Vxt−Vx0≤∫0tuTydτ−∫0tQxdτ,t>0,or(5)V˙x≤uTy−Qx,where u and y are noted as system input and output variables, respectively. uTy is the injected rate of external energy. Function (4) indicates that the sum of the increased energy of the system is always less than the sum of the external injected energy. For the LCL-filtered grid-tied inverter, the system energy storage function is defined as follows:(6)Vx=12xTMx,where positive definite function V (x) represents the energy storage of the internal system. Combining with (2), the time derivative of system energy function (6) is taken. Then, it deduces the following:(7)V˙x=xTMx˙=xTu−Jx−Rx=xTu−xTRx.Lettingy = x, Q (x) = xTRx, and combining with function (5), it can be seen that the system is passive. The integration term ∫0txTudt is the external supply energy of the system. ∫0txTRxdt is the dissipative term caused by the parasitic resistance of the system.According to the passivity situation, the object described in (2) is strictly passive where the EL-based PBC can be adopted to design the controller. For inverter-side current control, the current reference variables can be defined as follows:(8)x∗=i1d∗,i1q∗T.The error vector of the current is denoted asxe=x∗−x; then, an error EL equation is obtained by (2) as(9)Mx˙∗−x˙e+Jx∗−xe+Rx∗−xe=u.(10)Mx˙∗+Jx∗+Rx∗−u=Mx˙e+Jxe+Rxe.If the errorxe is equal to zero, the desired control objective and stable equilibrium point will happen. A positive damping matrix Rd can be added to the error EL model (10) to accelerate the error to reach zero. The injected damping and new dissipation term are described as(11)Rd=diagr1r1,Rnew=R+Rd,where r1 > 0. Taking function (11) into the error equation, the new error equation with injected damping is rewritten as(12)Mx˙e+Jxe+Rnewxe=Mx˙∗+Jx˙∗+Rx∗+Rdxe−u=ζ.In this study,ζ is set as Jxe to eliminate the coupling terms of system [26]. Thus, the control law from (12) is derived as follows:(13)ud=L1di1d∗dt−ωL1i1q+R1i1d∗+r1i1d∗−i1d+uCd,uq=L1di1q∗dt+ωL1i1d+R1i1q∗+r1i1q∗−i1q+uCq.According to the analysis mentioned before, the controller diagram in Laplace domain of the grid-tied VSI with the EL-based PBC method is depicted in Figure2, where L1 and R1 present the actual values of the LCL filter physical model, and L1e and R1e present the designed LCL filter parameters in the controller. In the implementation, r1 can be substituted by PI controller to reduce the steady-state error. e−1.5Ts indicates the computational and modulation delays, and T is the sampling period [9]. Since the delays are not considered in the design of the conventional EL-based PBC controller, dissipative properties of its output admittance should be investigated in the following discussion.Figure 2
Control structure for LCL-filtered three-phase grid-tied VSI with EL-based PBC with inverter-side current control.


## 3. Impedance Model and Stability Analysis of LCL-Filtered Grid-Tied VSI with EL-Based PBC

### 3.1. Impedance Model of LCL-Filtered Grid-Tied VSI with EL-Based PBC

The common Norton equivalent model of the LCL-filtered grid-tied inverter is shown in Figure3. Yo(s) is the control output admittance for the inverter-side current control and Zg is the grid impedance mentioned in Figure 1.Figure 3
Norton equivalent model.The voltage across the filter capacitor can be regarded as the disturbance input. Then, overall closed-loop response of the inverter current loop can be described with two parts as follows:(14)i1=Gsi1∗−YosuC,where G (s) is the closed-loop transfer function from output current to reference current, which determines the dynamic response and steady-state error of LCL-filtered grid-tied VSI with inverter-side current control. Filter capacitor and filter inductor are regarded as external grid impedance, together with grid admittance Zg. Ygt is the total external grid admittance, as shown in the following function.(15)Ygts=s2LgCg+1s3CgLgL2+sL2+Lg+sC.Based on the closed-loop response of the inverter current in (14), the closed-loop inverter current response seen from the grid voltage can be deduced in the following equation.(16)i1=Gs1+Yo/Ygtii∗−Yos1+Yo/Ygtvg.Function (16) can also be divided in two parts. The dynamic response and external disturbance stability of the whole system are also related to the transfer function Yos/Ygts. Yo (s) and Ygts should be both passive to guarantee the stability of the interconnected inverters and the grid system. Since Ygts is composed of resistor, inductor, and capacitor, it is obviously a passive network. Thus, the impedance model of inverter-side control with EL-based PBC should be derived to make sure that Yo (s) is passive.The current and voltage equations of the LCL filter can be obtained from Kirchhoff law, which are given by following matrices:(17)i1di1q=0ωL1−ωL10D00Di1di1q+D00Duduq,where ω=2πf, D=1/sL1+R1. Based on Figure 2, the inverter side current control law u∗ based on EL-based PBC in d-q axis is written as follows:(18)ud∗uq∗=E00Ei1d∗i1q∗−r1ωL1e−ωL1er1i1di1q+uCduCq,where E=sL1e+R1e+r1. Considering the computing delay and sampling delay, the control law u after delay is written in function (19) and T = 1/fs.(19)uduq=e−1.5Tsud∗uq∗.Combined with functions (17)–(19), the closed-loop transfer function matrix of the system is deduced in (20). Then, the performance of the LCL-filtered three-phase grid-tied inverter with EL-based PBC with inverter-side current control can be evaluated.(20)i1di1q=DEe−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts21+r1De−1.5TsωL1D−ωL1eDe−1.5Ts−ωL1D+ωL1eDe−1.5Ts1+r1De−1.5Tsi1d∗i1q∗+De−1.5Ts−11+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts21+r1De−1.5TsωL1D−ωL1eDe−1.5Ts−ωL1D+ωL1eDe−1.5Ts1+r1De−1.5TsuCduCq.Due to the steady-statei1d/i1d∗=1, d-q axis coupling terms are neglected. Then, Gs and Yo (s) of the proposed method can be expressed in the following equations.(21)Gs=DEe−1.5ts1+r1De−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts,(22)Yos=−De−1.5ts−11+r1De−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts.


### 3.2. Passivity-Based Stability Analysis

Passivity theory can also be described in frequency domain based on the impedance model, which provided a sufficient but not necessary condition. To define a passive system, the closed-loop transfer functionGs should have no RHP poles, and it is internally stable. Then, the output admittance of the control loop, Y (jω), should have a nonnegative real part or the phase of Y (jω) within [−90, 90], i.e., Re Yjω≥0⇔argYjω∈−90°,90°,∀ω>0.Hence, to make the LCL-filtered three-phase grid-tied VSI to be passive considering the control delay, the internal stability of the system should be analyzed. The system parameters used for analysis are described in Table1.Table 1
System parameters.
SymbolsComponentsValueUdcDC link voltage350 VL1Inverter-side inductor1.2 mHL2Grid-side inductor1.2 mHCCapacitor of LCL filter6μFfsSampling frequency10 kHzfswSwitching frequency10 kHzR1,R2Equivalent resistance of filter inductorL1, L20.1 Ω, 0.1 ΩVgGrid voltage110 V (RMS)foGrid frequency50 HzCgGrid capacitor3μFLgGrid inductor3.6 mHFigure4 indicates the z-domain zeros-poles based on the damped close-loop plant to illustrate the boundary for internal stability. As long as the gain r1 is less than 12, all the poles are placed inside the stable circle. It can be noticed that the poles move inside and a better internal stability can be obtained with less damping coefficient r1. But the dynamic response and the bandwidth will be better with larger r1. Figure 5 shows the step responses of Gs with different r1 to help with the selecting of r1. The suitable range of r1 can be selected between 3 and 6 based on the performance of overshoot and setting time.Figure 4
Closed-loop poles, whenr1 is increased from 0 to 12.Figure 5
Step responses of closed-loopGs when r1 = 1, 3, 4, 6, 8, 10.The passive characteristic of the system has been analyzed when applying the EL-based PBC in Section2. However, the delay and control will bring the nonpassive region in the digital realization. It is worth noting that the grid impedance values can change in a wide range depending on the situation. Thus, the external interaction stability with grid impedance should be evaluated. Based on the frequency-domain passivity theory, the EL-based PBC output admittance can be deduced to analyze the passivity of the overall system. The influences of coefficient r1 on the output admittance passivity region of the EL-based PBC method and system stability are shown in Figure 6, and Ygt (s) is the equivalent network admittance.Figure 6
The output admittanceYo (s) with different r1 and the equivalent network admittance Ygt (s).The magnitude ofYo (s) is sufficiently small in low frequency range, which means the inverter rejection capability for grid voltage harmonics is sufficient with the adoption of the EL-based PBC method. It can be seen that the equivalent negative resistor also exists which makes the system unstable with the traditional EL-based PBC control. Compared to traditional PI control, the 1/6fs is no longer the critical frequency for the LCL-filtered grid-tied VSI with the EL-based PBC method. When r1 is greater than 14, the phase curve of Yo crosses 90 degree no matter what the Ygt value is. Thus, r1 should be chosen between the region 4 and 9 to achieve a better stability margin, also considering internal stability and system dynamic response. Here, r1 is selected as 6.


## 3.1. Impedance Model of LCL-Filtered Grid-Tied VSI with EL-Based PBC

The common Norton equivalent model of the LCL-filtered grid-tied inverter is shown in Figure3. Yo(s) is the control output admittance for the inverter-side current control and Zg is the grid impedance mentioned in Figure 1.Figure 3
Norton equivalent model.The voltage across the filter capacitor can be regarded as the disturbance input. Then, overall closed-loop response of the inverter current loop can be described with two parts as follows:(14)i1=Gsi1∗−YosuC,where G (s) is the closed-loop transfer function from output current to reference current, which determines the dynamic response and steady-state error of LCL-filtered grid-tied VSI with inverter-side current control. Filter capacitor and filter inductor are regarded as external grid impedance, together with grid admittance Zg. Ygt is the total external grid admittance, as shown in the following function.(15)Ygts=s2LgCg+1s3CgLgL2+sL2+Lg+sC.Based on the closed-loop response of the inverter current in (14), the closed-loop inverter current response seen from the grid voltage can be deduced in the following equation.(16)i1=Gs1+Yo/Ygtii∗−Yos1+Yo/Ygtvg.Function (16) can also be divided in two parts. The dynamic response and external disturbance stability of the whole system are also related to the transfer function Yos/Ygts. Yo (s) and Ygts should be both passive to guarantee the stability of the interconnected inverters and the grid system. Since Ygts is composed of resistor, inductor, and capacitor, it is obviously a passive network. Thus, the impedance model of inverter-side control with EL-based PBC should be derived to make sure that Yo (s) is passive.The current and voltage equations of the LCL filter can be obtained from Kirchhoff law, which are given by following matrices:(17)i1di1q=0ωL1−ωL10D00Di1di1q+D00Duduq,where ω=2πf, D=1/sL1+R1. Based on Figure 2, the inverter side current control law u∗ based on EL-based PBC in d-q axis is written as follows:(18)ud∗uq∗=E00Ei1d∗i1q∗−r1ωL1e−ωL1er1i1di1q+uCduCq,where E=sL1e+R1e+r1. Considering the computing delay and sampling delay, the control law u after delay is written in function (19) and T = 1/fs.(19)uduq=e−1.5Tsud∗uq∗.Combined with functions (17)–(19), the closed-loop transfer function matrix of the system is deduced in (20). Then, the performance of the LCL-filtered three-phase grid-tied inverter with EL-based PBC with inverter-side current control can be evaluated.(20)i1di1q=DEe−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts21+r1De−1.5TsωL1D−ωL1eDe−1.5Ts−ωL1D+ωL1eDe−1.5Ts1+r1De−1.5Tsi1d∗i1q∗+De−1.5Ts−11+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts21+r1De−1.5TsωL1D−ωL1eDe−1.5Ts−ωL1D+ωL1eDe−1.5Ts1+r1De−1.5TsuCduCq.Due to the steady-statei1d/i1d∗=1, d-q axis coupling terms are neglected. Then, Gs and Yo (s) of the proposed method can be expressed in the following equations.(21)Gs=DEe−1.5ts1+r1De−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts,(22)Yos=−De−1.5ts−11+r1De−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts.


## 3.2. Passivity-Based Stability Analysis

Passivity theory can also be described in frequency domain based on the impedance model, which provided a sufficient but not necessary condition. To define a passive system, the closed-loop transfer functionGs should have no RHP poles, and it is internally stable. Then, the output admittance of the control loop, Y (jω), should have a nonnegative real part or the phase of Y (jω) within [−90, 90], i.e., Re Yjω≥0⇔argYjω∈−90°,90°,∀ω>0.Hence, to make the LCL-filtered three-phase grid-tied VSI to be passive considering the control delay, the internal stability of the system should be analyzed. The system parameters used for analysis are described in Table1.Table 1
System parameters.
SymbolsComponentsValueUdcDC link voltage350 VL1Inverter-side inductor1.2 mHL2Grid-side inductor1.2 mHCCapacitor of LCL filter6μFfsSampling frequency10 kHzfswSwitching frequency10 kHzR1,R2Equivalent resistance of filter inductorL1, L20.1 Ω, 0.1 ΩVgGrid voltage110 V (RMS)foGrid frequency50 HzCgGrid capacitor3μFLgGrid inductor3.6 mHFigure4 indicates the z-domain zeros-poles based on the damped close-loop plant to illustrate the boundary for internal stability. As long as the gain r1 is less than 12, all the poles are placed inside the stable circle. It can be noticed that the poles move inside and a better internal stability can be obtained with less damping coefficient r1. But the dynamic response and the bandwidth will be better with larger r1. Figure 5 shows the step responses of Gs with different r1 to help with the selecting of r1. The suitable range of r1 can be selected between 3 and 6 based on the performance of overshoot and setting time.Figure 4
Closed-loop poles, whenr1 is increased from 0 to 12.Figure 5
Step responses of closed-loopGs when r1 = 1, 3, 4, 6, 8, 10.The passive characteristic of the system has been analyzed when applying the EL-based PBC in Section2. However, the delay and control will bring the nonpassive region in the digital realization. It is worth noting that the grid impedance values can change in a wide range depending on the situation. Thus, the external interaction stability with grid impedance should be evaluated. Based on the frequency-domain passivity theory, the EL-based PBC output admittance can be deduced to analyze the passivity of the overall system. The influences of coefficient r1 on the output admittance passivity region of the EL-based PBC method and system stability are shown in Figure 6, and Ygt (s) is the equivalent network admittance.Figure 6
The output admittanceYo (s) with different r1 and the equivalent network admittance Ygt (s).The magnitude ofYo (s) is sufficiently small in low frequency range, which means the inverter rejection capability for grid voltage harmonics is sufficient with the adoption of the EL-based PBC method. It can be seen that the equivalent negative resistor also exists which makes the system unstable with the traditional EL-based PBC control. Compared to traditional PI control, the 1/6fs is no longer the critical frequency for the LCL-filtered grid-tied VSI with the EL-based PBC method. When r1 is greater than 14, the phase curve of Yo crosses 90 degree no matter what the Ygt value is. Thus, r1 should be chosen between the region 4 and 9 to achieve a better stability margin, also considering internal stability and system dynamic response. Here, r1 is selected as 6.


## 4. Passivity-Based Damping Enhancement Method

To narrow the nonpassive region, the damping enhancement method should be investigated. To reshape the control output admittance, the capacitor current feedback is adopted for EL-based PBC in this study, as shown in Figure2 with dotted box, and is given as follows:(23)Ic=sCuC.After adding capacitor current feedback, the function (18) is replaced by the following function.(24)uduq=E00Ei1d∗i1q∗−r1ωL1e−ωL1er1i1di1q+uCduCq−KsCuCduCq.Thus, system admittance equation is replaced by function (25), where K is the feedback coefficient.(25)Yos=−D1−KsCe−1.5Ts−11+r1De−1.5Ts1+r1De−1.5Ts2+ωL1D−ωL1eDe−1.5Ts2.The Bode plot of the control output admittance with capacitor current proportional feedback is shown in Figure7. The proportional coefficient k should be negative to ensure the stability of the system. It can be seen that the nonpassive regions of high frequency are narrowed after adding proportional feedback, and a better mitigation effect is achieved with higher feedback coefficient, but the nonpassive region cannot be totally removed. Even worse, it will cause another nonpassive in the low and middle frequencies if the feedback value is high.Figure 7
The output admittanceYo (s) with different proportional feedback coefficients and the equivalent network admittance Ygt (s).In order to compensate the negative effect of delays on the system phase in the high frequency, there are many filter-based methods, such as the lead-lag compensator, high pass filter (HPF), and the infinite impulse response (IIR) filter with the phase lead characteristics [34]. But it is hard to achieve all passive within the Nyquist frequency for the traditional PI controlled grid-connected inverter [9].Here, to extend the passivity range of the output phase, a simple high pass filter is added to the capacitor current feedback loop based on EL-based PBC. TheK (s) is expressed as follows:(26)Ks=kss+a.The Bode plots in Figure8 show that the high pass has a negligible effect at low frequency, but it can compensate the nonpassive region near to the Nyquist frequency or even higher. The corner frequency a is selected as 10000 to reach the compensation effect. The coefficient k cannot be too high or too small, and it is selected as −7 to keep the consistency with the proportional feedback method. Here, Lg = 3.6 mH, Cg = 2 μF, and the potential unstable point is around 3000–4000 Hz. If the Lg or Cg varies, the interaction points could move forward or backward. Within Nyquist frequency, the modified EL-based PBC control can ensure the robustness of the LCL-filtered grid-tied inverter in the weak grid.Figure 8
The output admittanceYo (s) with different HPF feedback coefficients and the equivalent network admittance Ygt (s).


## 5. Simulations and Experimental Studies

### 5.1. Simulation Results

The overall control structure is depicted in Figure2. Based on the theoretical analysis mentioned above, a three-phase LCL-filtered grid-tied inverter system with EL-based PBC is simulated to verify the effectiveness of the proposed control method using Matlab/Simulink platform. The detailed parameters of the system and controller are shown in Tables 1 and 2, respectively.Table 2
Controller parameters.
SymbolsComponentsValuekGain of capacitor current feedback−7r1Damping gain6Figure9 depicts the simulated grid currents of the LCL-filtered three-phase grid-tied inverter system with EL-based PBC control when different feedback strategies of capacitor current are enabled in the strong grid. At the beginning, the system is stable with proposed HPF feedback and the system is stable when k feedback is enabled at 0.12 s. However, the grid currents start to oscillate when damping controllers are deactivated at 0.16 s. Figure 10 illustrates Bode plot of the control output admittance Yo (s) and the equivalent network admittance Ygt (s) without capacitor current feedback when Lg = 0 and Cg = 0.Figure 9
Simulated grid currents with different capacitor current feedback methods whenLg = 0 and Cg = 0.Figure 10
The output admittanceYo (s) without feedback and the equivalent network admittance Ygt (s) when Lg = 0 and Cg = 0.It can be seen that there is one interaction point ofYo (s), and Ygt (s) falls into the nonpassive region, where the phase of Y (jω) is out of the region [−90, 90]. Hence, three-phase LCL-filtered grid-tied VSI with EL-based PBC could be unstable due to the influence of the delay and the damping enhancement method should be performed.Figure11 shows the simulated grid currents of the LCL-filtered three-phase grid-tied inverter system with different capacitor current feedback methods when Lg = 3.6 mH and Cg = 0. The system keeps stable with two different capacitor current feedback methods in the inductive grid impedance.Figure 11
Simulated grid currents of the three-phase LCL-filtered grid-tied inverter system with different capacitor current feedback methods whenLg = 3.6 mH and Cg = 0.The simulated grid currents of the three-phase LCL-filtered grid-tied inverter system with different capacitor current feedback methods are shown in Figure12, when Lg = 3.6 mH and Cg = 2 μF. As mentioned before, as shown in Figure 7, the system nonpassive region can be narrowed with higher k, but cannot be fully compensated before Nyquist frequency with k feedback. Here, the k is selected as −7, where the smaller Cg would bring higher interaction resonance frequency and easier stimulate oscillations. In Figure 12, the system is unstable in capacitive grid with Lg = 3.6 mH and Cg = 2 μF and then turns to stability at the time instant of 0.12 s with the proposed HPF feedback.Figure 12
Simulated grid currents with different capacitor current feedback methods whenLg = 3.6 mH and Cg = 2 μF.


### 5.2. Experimental Results

Apart from the simulated results, the experimental results are also investigated on a 3 kW three-phase LCL-filtered grid-tied inverter system to validate the performance of the proposed control method. The hardware prototype is with Danfoss FC302 inverter, as shown in Figure13. The proposed control scheme and sampling data are implemented in a dSPACE DS1202 controller. The DC voltage is supplied with a Chroma 62150H-600S, and the power grid is generated by a Chroma 61830 three-phase grid simulator.Figure 13
Experimental system prototype.System parameters adopted in the experiments are same as the simulations, as shown in Table1. HPF feedback coefficient of the proposed method is −7 and the injected damping r1 is 6. In the steady state test, Figure 14 shows the experimental results of grid currents with the proposed controller, when Lg = 3.6 mH and Cg = 2 μF. The grid currents are sinusoidal and synchronized with the PCC voltages, which gives a consistent current injection performance with analysis in the capacitive grid. It verifies the analysis that the interaction between the grid and inverter is avoided before the Nyquist frequency with the proposed control method shown in Figure 8. The dynamic responses of grid-currents are observed in Figure 15, when the reference current steps from 6.4 A to 12.8 A. The experimental results prove again that the system can achieve fast performance recovery and stable operation with proposed controller in the capacitive grid.Figure 14
Experimental waveforms with proposed control whenLg = 3.6 mH and Cg = 2 μF.Figure 15
Measured waveforms of grid currents with reference step whenLg = 3.6 mH and Cg = 2 μF.Figure16 shows the measured results of PCC voltages and grid currents with the proposed control and a zoom-in view of the test results is provided when the HPF feedback loop is deactivated instantaneously. The system turns unstable because the interaction points between the grid and the inverter fall into the nonpassive region without the damping enhancement method, as shown in Figure 6. Experimental results confirm the theoretical analysis and the controller design again.Figure 16
Measured waveforms of PCC voltages and grid currents if the HPF feedback is disabled, whenLg = 3.6 mH and Cg = 2 μF.


## 5.1. Simulation Results

The overall control structure is depicted in Figure2. Based on the theoretical analysis mentioned above, a three-phase LCL-filtered grid-tied inverter system with EL-based PBC is simulated to verify the effectiveness of the proposed control method using Matlab/Simulink platform. The detailed parameters of the system and controller are shown in Tables 1 and 2, respectively.Table 2
Controller parameters.
SymbolsComponentsValuekGain of capacitor current feedback−7r1Damping gain6Figure9 depicts the simulated grid currents of the LCL-filtered three-phase grid-tied inverter system with EL-based PBC control when different feedback strategies of capacitor current are enabled in the strong grid. At the beginning, the system is stable with proposed HPF feedback and the system is stable when k feedback is enabled at 0.12 s. However, the grid currents start to oscillate when damping controllers are deactivated at 0.16 s. Figure 10 illustrates Bode plot of the control output admittance Yo (s) and the equivalent network admittance Ygt (s) without capacitor current feedback when Lg = 0 and Cg = 0.Figure 9
Simulated grid currents with different capacitor current feedback methods whenLg = 0 and Cg = 0.Figure 10
The output admittanceYo (s) without feedback and the equivalent network admittance Ygt (s) when Lg = 0 and Cg = 0.It can be seen that there is one interaction point ofYo (s), and Ygt (s) falls into the nonpassive region, where the phase of Y (jω) is out of the region [−90, 90]. Hence, three-phase LCL-filtered grid-tied VSI with EL-based PBC could be unstable due to the influence of the delay and the damping enhancement method should be performed.Figure11 shows the simulated grid currents of the LCL-filtered three-phase grid-tied inverter system with different capacitor current feedback methods when Lg = 3.6 mH and Cg = 0. The system keeps stable with two different capacitor current feedback methods in the inductive grid impedance.Figure 11
Simulated grid currents of the three-phase LCL-filtered grid-tied inverter system with different capacitor current feedback methods whenLg = 3.6 mH and Cg = 0.The simulated grid currents of the three-phase LCL-filtered grid-tied inverter system with different capacitor current feedback methods are shown in Figure12, when Lg = 3.6 mH and Cg = 2 μF. As mentioned before, as shown in Figure 7, the system nonpassive region can be narrowed with higher k, but cannot be fully compensated before Nyquist frequency with k feedback. Here, the k is selected as −7, where the smaller Cg would bring higher interaction resonance frequency and easier stimulate oscillations. In Figure 12, the system is unstable in capacitive grid with Lg = 3.6 mH and Cg = 2 μF and then turns to stability at the time instant of 0.12 s with the proposed HPF feedback.Figure 12
Simulated grid currents with different capacitor current feedback methods whenLg = 3.6 mH and Cg = 2 μF.


## 5.2. Experimental Results

Apart from the simulated results, the experimental results are also investigated on a 3 kW three-phase LCL-filtered grid-tied inverter system to validate the performance of the proposed control method. The hardware prototype is with Danfoss FC302 inverter, as shown in Figure13. The proposed control scheme and sampling data are implemented in a dSPACE DS1202 controller. The DC voltage is supplied with a Chroma 62150H-600S, and the power grid is generated by a Chroma 61830 three-phase grid simulator.Figure 13
Experimental system prototype.System parameters adopted in the experiments are same as the simulations, as shown in Table1. HPF feedback coefficient of the proposed method is −7 and the injected damping r1 is 6. In the steady state test, Figure 14 shows the experimental results of grid currents with the proposed controller, when Lg = 3.6 mH and Cg = 2 μF. The grid currents are sinusoidal and synchronized with the PCC voltages, which gives a consistent current injection performance with analysis in the capacitive grid. It verifies the analysis that the interaction between the grid and inverter is avoided before the Nyquist frequency with the proposed control method shown in Figure 8. The dynamic responses of grid-currents are observed in Figure 15, when the reference current steps from 6.4 A to 12.8 A. The experimental results prove again that the system can achieve fast performance recovery and stable operation with proposed controller in the capacitive grid.Figure 14
Experimental waveforms with proposed control whenLg = 3.6 mH and Cg = 2 μF.Figure 15
Measured waveforms of grid currents with reference step whenLg = 3.6 mH and Cg = 2 μF.Figure16 shows the measured results of PCC voltages and grid currents with the proposed control and a zoom-in view of the test results is provided when the HPF feedback loop is deactivated instantaneously. The system turns unstable because the interaction points between the grid and the inverter fall into the nonpassive region without the damping enhancement method, as shown in Figure 6. Experimental results confirm the theoretical analysis and the controller design again.Figure 16
Measured waveforms of PCC voltages and grid currents if the HPF feedback is disabled, whenLg = 3.6 mH and Cg = 2 μF.


## 6. Conclusion

This study proposes a robust control strategy based on passivity-based control and reshaping of control output admittance within Nyquist frequency for LCL-filtered VSI. The overall conclusion can be summarized in the following points:(1)
LCL-filtered three-phase grid-tied VSI based on EL-based PBC with single inverter-side current control guarantees the robustness of the system but cannott damp the interactive resonance effectively due to the delay, especially in the capacitive grid(2)
In order to ensure the passivity of PBC-based LCL-filtered grid-tied inverter, the output admittance analysis methodology is utilized to enhance the stability and expand the passive region(3)
Even though the potential resonances can be damped effectively with capacitor current proportional feedback, there still have some nonpassive regions within Nyquist frequency. The conventional EL-based PBC method is then modified by adding a HPF feedback of the capacitor current to ensure the output control admittance which is passive within Nyquist frequency.Finally, the simulated and experimental results are both provided from a 3 kW system to verify the effectiveness of the proposed modified controller.



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*Source: 3373863-2023-01-06.xml*