A new model reduction method based PBC control for grid-connected inverter with LCL-filter

Abstract

Passivity-based control (PBC) exhibits robustness against parameters shift, providing a substantial level of stability. However, for the LCL-filtered grid-connected inverter (GCI), the conventional PBC (called C-PBC) controller has a narrow control bandwidth due to the control time delay, resulting in poor dynamic performance, especially in the weak grid state. This paper proposes a new PBC (called P-PBC) method to increase the closed-loop bandwidth of the system by setting an appropriate feedback proportional coefficient between the grid-side current and the inverter-side current. By extending the closed-loop bandwidth of the system, the proposed P-PBC method offers improved dynamic performance, particularly in challenging grid conditions. In addition, a state observer is also employed to reduce sensors, which saves costs and enhances the system's reliability. A 110 V/50 Hz/3 kW/3-phase experimental setup has been developed using the dSPACE DS1202 platform to validate the effectiveness of the proposed control method.

Introduction

With the advancements in renewable energy generation, a grid-connected inverter (GCI) has been adopted in distributed power systems^1,2. To suppress the harmonics at high switching frequencies, a low-pass output power filter is indispensable for GCI. In comparison with the L filter, the LCL filter demonstrates enhanced performance in harmonic attenuation and has a more economical price^3,4. Thus, the application of the LCL-type filtered GCI has found extensive use in the industry.

To ensure a high quality of the grid-side current, the use of a suitable current controller is essential. The most common linear current controller for LCL-filtered GCI system is based on the PI control in a synchronous rotating coordinate system^5,6, or the proportional-resonance control in a static coordinate system⁷. These two types of linear controllers are not only simple in control structure but also can follow the reference current without steady-state error. Note that resonance is a potential issue in the LCL-filtered GCI when utilizing a linear current controller, which affects the stability of the system. In Ref.⁸, a comprehensive overview of the additional damping measures to suppress the possible resonance of LCL-filtered GCI has been presented, which can effectively ensure the system stability. However, it can be found that as the equivalent grid impedance increases, the bandwidth of the system will inevitably be reduced, resulting in poor dynamic performance under weak grid operating conditions⁹.

Recently, due to increasingly potent digital processors with high computational speeds, a great deal of nonlinear control strategies, including the sliding mode control (SMC)^10–12, model predictive control (MPC)^13,14 and passivity-based control (PBC)^15–17, have found application in GCIs system. Among these control strategies, the chattering problem inherent in SMC may lead to mechanical wear and noise in the system in practical applications. In addition, delays and unmodeled high-frequency modes induced by chattering on the SMC surface can degrade the control accuracy. MPC is computationally complex and relatively complicated to implement. These two control methods do not consider the inverter's energy dissipation characteristics, which are inherent to its physical structure. Energy dissipation characteristics have a crucial part in the control of GCIs with respect to system stability, dynamic response, thermal management and reliability. Utilizing the considered system's energy dissipation characteristics, energy shaping-based PBC has attracted much attention¹⁸.

PBC can be modeled using the Euler Lagrange^19–21 or the port-controlled Hamiltonian (PCH) equations^22,23 to produce a smooth, asymptotically stabilizing controller with a guaranteed attraction domain. The stability of the network is guaranteed by rendering the system to be passive in terms of the desired storage and damping functions. With the advantages of clear physical meaning, straightforward and intuitive modeling process, fast dynamic response and high robustness, PBC has been widely used in systems containing a high percentage of power electronic converters²⁴.

As for the application in LCL-filtered GCI systems, the PBC strategy has also been studied. In Refs.^25,26, a control method employing three control loops is introduced for the regulation of the grid-side current, utilizing a PBC system. This strategy demonstrates robustness in the face of variations in system parameters and external disturbances. However, since the PBC controller is multi-loop, the nested inner loop should be much faster in response speed than the related outer loop²⁷. To meet the controller's design requirements, the dynamic performance of the outer loop is reduced to achieve the required response speed of the inner loop. In Ref.²⁸, the PBC controller using the inverter-side inductor current feedback control is presented. Nevertheless, due to the control delay, the closed-loop bandwidth is not so high, resulting in unsatisfactory dynamic performance. Therefore, enhancing the dynamic performance of the PBC controller and increasing the system bandwidth remains a big challenge for engineers.

Note that in the conventional linear control area, the model reduction method has been applied in the LCL-filtered GCIs to increase the control bandwidth of the entire system. For example, He et al. in Ref.²⁹ presented the weighted average current control (WACC) method for a three-phase LCL-filtered GCI. By appropriately setting the sampling coefficients of the grid-side current and the inverter-side current, the open-loop transfer function is reduced from the third to the first order, enabling an increase in the control loop gain and bandwidth. However, whether and how to use the model reduction method on PBC to enhance the dynamic performance of LCL-filtered GCI needs to be further explored.

The contributions of this paper are summarized as follows.

1. A new model reduction method based on PBC control will be proposed for a three-phase GCI system with an LCL-type filter to achieve high control bandwidth. Based on an improved PBC (called P-PBC) control method, the system has good stability performance and dynamic performance.

2. The state observer and PI controller are also adopted in this paper to reduce the number of sensors and achieve zero steady-state error.

The subsequent sections are arranged as follows. Firstly, in section “Modeling and conventional PBC controller for three-phase GCI with LCL filter”, the PCH model will be built, while the conventional PBC (called C-PBC) controller using inverter-side inductor current feedback control is briefly presented. And analyzed the reason why the dynamic performance of the C-PBC controller is not so good through the system's closed-loop transfer function. Secondly, a P-PBC controller is proposed in section “Proposed PBC controller”, where the design details, dynamic performance and robustness of the controller, are investigated. Moreover, it is demonstrated that the zero steady-state error is achieved, accompanied by the introduction of a state-observer. Thirdly, section “Experimental verification” demonstrates the effectiveness of the P-PBC controller by comparing experimental results. Finally, section “Conclusion” draws some conclusions from the paper.

Modeling and conventional PBC controller for three-phase GCI with LCL filter

PCH model for LCL-filtered GCI

Figure 1 shows the full topology and control block for a three-phase GCI system with an LCL filter. L₁ and L₂ are the inverter-side inductor and the grid-side inductor, respectively, where R₁ and R₂ denote the equivalent resistances of the two inductors. C is the capacitor between the two inductors. L_g represents the equivalent grid inductor. i_1k, i_ck and i_2k (k = a, b, c) are the inverter-side current, the capacitor current and the grid-side current, respectively. ${\widehat{i}}_2$ represents the grid-side current estimated by the state observer. u_k, u_ck and u_gk (k = a, b, c) respectively represent the inverter output voltage, capacitor voltage and ideal grid voltage. v_pcc represents the voltage at the point of common coupling (PCC). Since the relatively slow dynamics of the DC system, which are justifiably disregarded, a constant value for the DC-link voltage (U_dc) is assumed. The SVPWM technique is implemented to derive the driving signals.

Fig. 1.

Full topology and control block for a three-phase GCI system with an LCL filter.

Applying Kirchhoff's law to the circuit topology of the LCL-type filter leads to its circuit equation in the three-phase static coordinate system, as shown in (1).

$$\left\{\begin{array}{c}{L}_1\frac{d{i}_{1\text{k}}}{\mathit{dt}}=-R_1{i}_{1\text{k}}+u_{\text{k}}-u_{\text{ck}}\\ L_2\frac{d{i}_{2\text{k}}}{\mathit{dt}}=-R_2{i}_{2\text{k}}+u_{\text{ck}}-v_{\text{pcck}}\\ C\frac{d{u}_{\text{ck}}}{\mathit{dt}}=i_{1\text{k}}-i_{2\text{k}}\\ \end{array}\right)(\text{k}=\text{a},\text{b},\text{c})$$ 1

The mathematical model of a three-phase LCL-filtered GCI is transformed from the three-phase static coordinates to the two-phase rotating coordinates^25,26, which is written as

$$\left\{\begin{array}{c}{L}_1\frac{d{i}_{1\text{d}}}{\mathit{dt}}=\omega L_1{i}_{1\text{q}}-R_1{i}_{1\text{d}}+u_{\text{d}}-u_{\text{cd}}\\ L_1\frac{d{i}_{1\text{q}}}{\mathit{dt}}=-\omega L_1{i}_{1\text{d}}-R_1{i}_{1\text{q}}+u_{\text{q}}-u_{\text{cq}}\\ L_2\frac{d{i}_{2\text{d}}}{\mathit{dt}}=\omega L_2{i}_{2\text{q}}-R_2{i}_{2\text{d}}+u_{\text{cd}}-v_{\text{pccd}}\\ L_2\frac{d{i}_{2\text{q}}}{\mathit{dt}}=-\omega L_2{i}_{2\text{d}}-R_2{i}_{2\text{q}}+u_{\text{cq}}-v_{\text{pccq}}\\ C\frac{d{u}_{\text{cd}}}{\mathit{dt}}=\omega C{u}_{\text{cq}}+i_{1\text{d}}-i_{2\text{d}}\\ C\frac{d{u}_{\text{cq}}}{\mathit{dt}}=-\omega C{u}_{\text{cd}}+i_{1\text{q}}-i_{2\text{q}}\\ \end{array}\right)$$ 2

To describe the mathematical model for a three-phase GCI with an LCL filter using the PCH model, the variables x are defined as $x={\left(L_1{i}_{1\text{d}}{L}_1{i}_{1\text{q}}{L}_2{i}_{2\text{d}}{L}_2{i}_{2\text{q}}C{u}_{\text{cd}}C{u}_{\text{cq}}\right)}^T$. The Hamiltonian formulation H(x) is set as the system's total energy function, which is the sum of the energy of capacitor C, inductors L₁ and L₂. Thus, the function is expressed as

$$H(x)=\frac{x_1^2}{2L_1}+\frac{x_2^2}{2L_1}+\frac{x_3^2}{2L_2}+\frac{x_4^2}{2L_2}+\frac{x_5^2}{2C}+\frac{x_6^2}{2C}=\frac{1}{2}{x}^T{D}^{-1}x,$$ 3

where $D=diag(L_1,L_1,L_2,L_2,C,C)$.

Finally, the PCH model of a three-phase GCI with LCL filter can be expressed as:

$$\left\{\begin{array}{c}\dot{x}=(J-R)\frac{\partial H(x)}{\partial x}+g(x)u\\ y=g^T(x)\frac{\partial H(x)}{\partial x}\\ \end{array}\right),$$ 4

where

$$\begin{array}{cc}\hfill R& =diag(R_1,R_1,R_2,R_2,0,0);\hfill \\ \hfill u& ={\left(\begin{array}{cccccc}{u}_{\text{d}}& u_{\text{q}}& -v_{\text{pccd}}& -v_{\text{pccq}}& 0& 0\end{array}\right)}^T;\hfill \\ \hfill J& =\left(\begin{array}{cccccc}0& \omega L_1& 0& 0& -1& 0\\ -\omega L_1& 0& 0& 0& 0& -1\\ 0& 0& 0& \omega L_2& 1& 0\\ 0& 0& -\omega L_2& 0& 0& 1\\ 1& 0& -1& 0& 0& \omega C\\ 0& 1& 0& -1& -\omega C& 0\end{array}\right)\hfill \\ \hfill \end{array}$$

g(x) is a unit matrix; J represents the interaction between state variables and it is an antisymmetric matrix satisfying J = − J^T; R denotes the system's dissipation characteristic, which is required as a positive-semidefinite matrix to satisfy x^TRx ≥ 0; u and y are the input and the output of the PCH model system, which represent the exchange of energy between the system and its surroundings.

Conventional PBC controller design

The passivity of the GCI with an LCL filter can be investigated by the passive theory mentioned in Ref.³⁰. According to the PCH modeling approach, the interconnection and damping assignment design methodology can be adopted to deduce the control law, which is briefly described in the following.

Figure 2 depicts the equivalent transformation of the mathematical model for the C-PBC controller, where e^−τs denotes the total delay, including calculation delay and modulation delay, typically selected as τ = 1.5T_s (T_s is the inverter sampling period). Further, Padé approximation is used to approximate the time delay to a third-order transfer function^31,32. Note that only the d-axis component is plotted in Fig. 2, given that the structure of the q-axis component is identical to that of the d-axis.

Fig. 2.

Equivalent transformation of the mathematical model for the C-PBC controller.

Desired equilibrium points are $x_{}^{\ast }=\left(L_{1}i_{1\text{d}}^{\ast }{L}_{1}i_{1\text{q}}^{\ast }{L}_{2}i_{2\text{d}}^{\ast }{L}_{2}i_{2\text{q}}^{\ast }Cu_{\text{cd}}^{\ast }Cu_{\text{cq}}^{\ast }\right)_{}^T$. The goal of the design methodology is to bring the PCH model (4) with feedback loop to a closed-loop system, which has a new form as

$$\dot{x}=[J_{\text{d}}-R_{\text{d}}]\frac{\partial H_{\text{d}}(x)}{\partial x},$$ 5

where $J_{\text{d}}=J+J_{\text{a}},R_{\text{d}}=R+R_{\text{a}}$, and $x_{}^{\ast }=argmin{H}_{\text{d}}(x)$.

The energy function H_d(x) indicates the new system's total energy. Without losing generality, J_a and R_a are also antisymmetric and positive-semidefinite matrices. In order to force the variables x to quickly converge to the desired equilibrium points x^*, the extra positive-semidefinite matrix of R_a is set as diag{r₁ r₁ r₂ r₂ r₃ r₃} (where r₁, r₂, r₃ ≥ 0).

Combining (4) and (5), the control law β(x) satisfies

$$u=\beta (x)=g^{-1}(x)\left[(J_{\text{d}}-R_{\text{d}})\frac{\partial H_{\text{d}}(x)}{\partial x}-(J-R)\frac{\partial H(x)}{\partial x}\right]$$ 6

For design convenience, normally, the H_d (x) and J_a are determined as

$$H_{\text{d}}(x)=\frac{1}{2}{x}_{\text{e}}^T{D}^{-1}{x}_{\text{e}},x_{\text{e}}=x-x_{}^{\ast },J_{\text{a}}=0$$ 7

Substituting (7) into (6), the control law is obtained as

$$\left\{\begin{array}{cc}{u}_{1\text{d}}=(R_1+r_1)i_{1\text{d}}^{\ast }-r_1{i}_{1\text{d}}-\omega L_{1}i_{1\text{q}}^{\ast }+u_{\text{cd}}^{\ast }& 1)\\ u_{1\text{q}}=(R_1+r_1)i_{1\text{q}}^{\ast }-r_1{i}_{1\text{q}}+\omega L_{1}i_{1\text{d}}^{\ast }+u_{\text{cq}}^{\ast }& 2)\\ u_{\text{cd}}^{\ast }=(R_2+r_2)i_{2\text{d}}^{\ast }-r_2{i}_{2\text{d}}-\omega L_{2}i_{2\text{q}}^{\ast }+v_{\text{pccd}}& 3)\\ u_{\text{cq}}^{\ast }=(R_2+r_2)i_{2\text{q}}^{\ast }+\omega L_{2}i_{2\text{d}}^{\ast }-r_2{i}_{2\text{q}}+v_{\text{pccq}}& 4)\\ 0=r_3(u_{\text{cd}}^{\ast }-u_{\text{cd}})-i_{1\text{d}}^{\ast }+i_{2\text{d}}^{\ast }-\omega Cu_{\text{cq}}^{\ast }& 5)\\ 0=r_3(u_{\text{cq}}^{\ast }-u_{\text{cq}})-i_{1\text{q}}^{\ast }+i_{2\text{q}}^{\ast }+\omega Cu_{\text{cd}}^{\ast }& 6)\end{array}\right)$$ 8

To verify the system's stability, the derivative of energy function H_d(x) can be obtained shown below

$${\dot{H}}_{\text{d}}(x)=\frac{\partial H_{\text{d}}(x)}{\partial x}\dot{x}=-{\left(\frac{\partial H_{\text{d}}(x)}{\partial x}\right)}^T{R}_{\text{d}}(x)\frac{\partial H_{\text{d}}(x)}{\partial x}\le 0$$ 9

and

$${\dot{H}}_{\text{d}}(x_{}^{\ast })=0,\frac{{\partial }^2{H}_{\text{d}}}{\partial x^2}(x_{}^{\ast })>0$$ 10

Therefore, the system is asymptotically stable according to the Lyapunov stability conditions^30,33.

As shown in the control law of (8), the sub-equations 1) and 2) are often used as a C-PBC controller to control the inductor current on the inverter-side^16,28. The block diagram of the C-PBC controller in the S-domain is depicted in Fig. 2.

Drawback of C-PBC controller in closed-loop

The closed-loop transfer function of the C-PBC controller from the reference current $i_{1\text{d}}^{\ast }$ to the grid-side current i_2d can be represented as

$$G_{\text{c}}(s)=\frac{i_{2\text{d}}^{}}{i_{1\text{d}}^{\ast }}=\frac{(R_1+r_1)e^{-\tau s}}{s^3{M}_1+s^2{M}_2+s{M}_3+M_4+e^{-\tau s}{r}_1(s^2{N}_1+s{N}_2+1)}$$ 11

where $M_1=L_1{L}_{2}C,M_2=L_1{R}_{2}C+L_2{R}_{1}C,M_3=L_1+L_2+R_1{R}_{2}C,M_4=R_1+R_2,N_1=L_{2}C,N_2=C{R}_2.$

The Bode diagrams for the C-PBC controller's closed-loop transfer function are depicted in Fig. 3. When the time delay is neglected, the closed-loop bandwidth f_2b of G_c(s) can reach about 610 Hz; when e^−τs is selected as τ = 1.5T_s, the closed-loop bandwidth f_1b of G_c(s) is only about 30 Hz. It can be found that the closed-loop bandwidth of the C-PBC controller has been significantly reduced due to the influence of time delay. In a real application, the extra damping methods are usually added to suppress the resonance peak, resulting in an improved control bandwidth. However, the problem of narrow bandwidth still exists for C-PBC, especially in weak grids.

Fig. 3.

Bode diagrams of closed-loop transfer functions of the C-PBC controller with different time delays.

Note that the PBC controller only uses two sub-equations from the control law of (8). The information of the control law has not been fully utilized. In order to use the remaining four sub-equations, the control law will be modified as analyzed in the next section. Moreover, the parameters of the P-PBC controller are also designed to ensure the stability of the overall GCI system.

Proposed PBC controller

In this section, the proper coefficients of the grid-side current and the inverter-side current are set, utilizing the information of the control law (8). Also, the capacitor current feedback is introduced from control law (8). To ensure the stability and good dynamic performance of the proposed controller, the parameters of the damping gains are designed through linear analysis. In addition, the robustness of the LCL filter to parameters shift is analyzed. Furthermore, the proportional term k_p of the PI controller is adopted instead of the damping gain r to attain zero steady-state error, and a state observer is utilized to minimize sensor count. The equivalent transformation of the mathematical model for the P-PBC controller is shown in Fig. 4.

Fig. 4.

Equivalent transformation of the mathematical model for the P-PBC controller.

Design of damping gains k_p

According to Fig. 4, the damping gain r (that is k_p) reflects the ability of the feedback current to track the reference current. In linear analysis, a higher gain k_p means a higher control bandwidth and faster dynamic response, but the gain margin (GM) and phase margin (PM) are smaller, which can easily destroy the system's stability. To balance the stability and dynamic performance of the system, the damping gain k_p is determined by linear analysis in this paper. Note that, since the system's open-loop transfer function does not exhibit a resonance peak according to Fig. 5, the damping gain r₃ only reduces the magnitude around the resonant frequency, which has little effect on GM and PM. Thus, the damping gain r₃ can be ignored when k_p is designed. The open-loop transfer function of the feedback system without r₃ is described as

$$G_{\text{open}}(s)=\frac{e^{-\tau s}(s^2{Q}_1+s{Q}_2+Q_3)}{s^3{M}_1+s^2{M}_2+s{M}_3+M_4}$$ 12

where $Q_1=k_{\text{p}}{L}_1{L}_{2}C,Q_2=k_{\text{p}}{L}_{1}C{R}_2,Q_3=k_{\text{p}}(L_1+L_2),M_1=L_1{L}_{2}C,M_2=L_1{R}_{2}C+L_2{R}_{1}C,M_3=L_1+L_2+R_1{R}_{2}C,M_4=R_1+R_2.$

Fig. 5.

Bode diagrams of the open-loop transfer function of G_open(s) as k_p varies.

The Bode diagrams of the open-loop transfer function are shown in Fig. 5, where the preferred region of the k_p range is between 1000 and 4000. It is evident that the resonance characteristic of the LCL filter is not revealed in the open-loop transfer function. The explanation is that the P-PBC controller has the same characteristics as the weighted average current controller, which reduces the open-loop transfer function from third order to first order²⁹. Therefore, the control loop gain of the P-PBC controller can be set higher than in the C-PBC controller.

From Fig. 5, with the increase of k_p, the crossover frequency f_c increases from 160 Hz to 636 Hz, the GM decreases from 20.4 dB to 8.4 dB, and the PM also decreases from 86.2° to 56.8°. Based on Ref.³⁴ , a small percentage of overshoot can be achieved, when f_c is chosen to be about 4% of the sampling frequency (i.e., 400 Hz in the example under study) with a PM over 60°. Therefore, the damping gain k_p can be determined as 2500, where f_c is about 400 Hz, GM is 12.5 dB and PM is 70.4°.

Increasing the closed-loop bandwidth by weighting proper coefficients

Firstly, by combining the sub-equations 1)–4) of the control law (8), the equation can be obtained as

$$\left\{\begin{array}{c}{u}_{1\text{d}}=r_1(i_{1\text{d}}^{\ast }-i_{1\text{d}})-\omega L_{1}i_{1\text{q}}^{\ast }+R_{1}i_{1\text{d}}^{\ast }+r_2(i_{2\text{d}}^{\ast }-i_{2\text{d}})-\omega L_{2}i_{2\text{q}}^{\ast }+R_{2}i_{2\text{d}}^{\ast }+v_{\text{pccd}}\\ u_{1\text{q}}=r_1(i_{1\text{q}}^{\ast }-i_{1\text{q}})+\omega L_{1}i_{1\text{d}}^{\ast }+R_{1}i_{1\text{q}}^{\ast }+r_2(i_{2\text{q}}^{\ast }-i_{2\text{q}})+\omega L_{2}i_{2\text{d}}^{\ast }+R_{2}i_{2\text{q}}^{\ast }+v_{\text{pccq}}\\ \end{array}\right)$$ 13

Then, assuming that the ratio of resistance R to inductance L is the same for different inductors ($\delta =R_1\left./\phantom{R_1{L}_1}\right)L_1=R_2\left./\phantom{r_2{L}_2}\right)L_2$)^35,36, the new variables are defined as

$$\left\{\begin{array}{c}{i}_{\text{wd}}=L_1{i}_{1\text{d}}\left./\phantom{L_1{i}_{1\text{d}}(L_1+L_2}\right)(L_1+L_2)+L_2{i}_{2\text{d}}\left./\phantom{L_2{i}_{2\text{d}}(L_1+L_2}\right)(L_1+L_2)\\ i_{\text{wq}}=L_1{i}_{1\text{q}}\left./\phantom{L_1{i}_{1\text{q}}(L_1+L_2}\right)(L_1+L_2)+L_2{i}_{2\text{q}}\left./\phantom{L_2{i}_{2\text{q}}(L_1+L_2}\right)(L_1+L_2)\\ \end{array}\right)$$ 14

Thus, Eq. (13) can be rewritten as

$$\left\{\begin{array}{c}\frac{u_{1\text{d}}}{L_1+L_2}=\frac{r_1}{L_1+L_2}(i_{1\text{d}}^{\ast }-i_{1\text{d}})+\frac{r_2}{L_1+L_2}(i_{2\text{d}}^{\ast }-i_{2\text{d}})-\omega i_{\text{wq}}^{\ast }+\delta i_{\text{wd}}^{\ast }+\frac{v_{\text{pccd}}}{L_1+L_2}\\ \frac{u_{1\text{q}}}{L_1+L_2}=\frac{r_1}{L_1+L_2}(i_{1\text{q}}^{\ast }-i_{1\text{q}})+\frac{r_2}{L_1+L_2}(i_{2\text{q}}^{\ast }-i_{2\text{q}})+\omega i_{\text{wd}}^{\ast }+\delta i_{\text{wq}}^{\ast }+\frac{v_{\text{pccq}}}{L_1+L_2}\\ \end{array}\right)$$ 15

Subsequently, Eq. (16) is used to simplify Eq. (15).

$$r=k_{\text{p}}=r_1\left./\phantom{=r_1{L}_1}\right)L_1=r_2\left./\phantom{r_2{L}_2}\right)L_2(r>0)$$ 16

It should be noted that since the inductances L₁ and L₂ are greater than zero, r₁ and r₂ are also greater than zero (as introduced in section “Conventional PBC controller design”), then r (that is k_p) is greater than zero. The extra positive-semidefinite matrix R_a (as introduced in section “Conventional PBC controller design”) still meets the requirement of positive definite to ensure the convergence of the system.

Applying (16) into (15), the sub-equations 1)–4) in (8) are rewritten as

$$\left\{\begin{array}{c}{U}_{1\text{d}}=k_{\text{p}}(L_1+L_2)(i_{\text{wd}}^{\ast }-i_{\text{wd}})-(L_1+L_2)\omega i_{\text{wq}}^{\ast }+(L_1+L_2)\delta i_{\text{wd}}^{\ast }+v_{\text{pccd}}\\ U_{1\text{q}}=k_{\text{p}}(L_1+L_2)(i_{\text{wq}}^{\ast }-i_{\text{wq}})+(L_1+L_2)\omega i_{\text{wd}}^{\ast }+(L_1+L_2)\delta i_{\text{wq}}^{\ast }+v_{\text{pccq}}\\ \end{array}\right)$$ 17

Based on (2) and (17), the closed-loop transfer function from the reference current $i_{\text{wd}}^{\ast }$ to the grid-side current i_2d is shown below.

$$G_{1\text{c}}(s)=\frac{i_{2\text{d}}}{i_{\text{wd}}^{\ast }}=\frac{(\delta +k_{\text{p}})(L_1+L_2)e^{-\tau s}}{s^3{M}_1+s^2{M}_2+s{M}_3+M_4+e^{-\tau s}(s^2{Q}_1+s{Q}_2+Q_3)},$$ 18

where $Q_1=k_{\text{p}}{L}_1{L}_{2}C,Q_2=k_{\text{p}}{L}_{1}C{R}_2,Q_3=k_{\text{p}}(L_1+L_2),M_1=L_1{L}_{2}C,M_2=L_1{R}_{2}C+L_2{R}_{1}C,M_3=L_1+L_2+R_1{R}_{2}C,M_4=R_1+R_2.$

A comparison of the Bode diagrams of closed-loop transfer functions under 1.5 time delay between the C-PBC controller and the P-PBC controller is shown in Fig. 6. It is clear that the P-PBC controller's closed-loop bandwidth f_3b is about 850 Hz, which is significantly higher than that of the C-PBC controller.

Fig. 6.

Comparison of the Bode diagrams of closed-loop transfer functions with 1.5 time delay between the C-PBC controller and the P-PBC controller.

Table 1 demonstrates the closed-loop bandwidths of the C-PBC controller and P-PBC controller at 0 time delay and 1.5 time delay. It can be concluded that the improved method is an effective method to increase the closed-loop bandwidth in the presence of time delay.

Table 1.

Comparison of closed-loop bandwidths.

Controller	0 time delay	1.5 time delay
C-PBC	f2b ≈ 610 Hz	f1b ≈ 30 Hz
P-PBC	–	f3b ≈ 850 Hz

However, the information provided in Fig. 6 indicates that both the C-PBC controller and the P-PBC controller have resonance peaks, which will amplify the noise disturbance at the resonant frequency. Naturally, capacitor current feedback is used to suppress resonance⁷. Note that sub-equations 5) and 6) in the control law (8) show the effect of PBC on the capacitor. The derivation of the sub-equations 5) and 6) in (8) can be used as a capacitor current feedback loop as follows

$$\left\{\begin{array}{c}0=r_{3}C(d{u}_{\text{cd}}\left./\phantom{d{u}_{\text{cd}}\mathit{dt}}\right)\mathit{dt})\\ 0=r_{3}C(d{u}_{\text{cq}}\left./\phantom{d{u}_{\text{cq}}\mathit{dt}}\right)\mathit{dt})\\ \end{array}\right)\to \left\{\begin{array}{c}0=r_3\omega C{u}_{\text{cq}}+r_3{i}_{\text{cd}}\\ 0=-r_3\omega C{u}_{\text{cd}}+r_3{i}_{\text{cq}}\\ \end{array}\right)$$ 19

Substituting (19) into (17) and using Kirchhoff's current law (i_cd = i_1d − i_2d), the P-PBC controller can finally be deduced as

$$\left\{\begin{array}{c}{U}_{1\text{d}}=k_{\text{p}}(L_1+L_2)(i_{\text{wd}}^{\ast }-i_{\text{wd}})-(L_1+L_2)\omega i_{\text{wq}}^{\ast }\\ +(L_1+L_2)\delta i_{\text{wd}}^{\ast }+r_3(i_{1\text{d}}-i_{2\text{d}})+r_3\omega C{u}_{\text{cq}}+v_{\text{pccd}}\\ U_{1\text{q}}=k_{\text{p}}(L_1+L_2)(i_{\text{wq}}^{\ast }-i_{\text{wq}})+(L_1+L_2)\omega i_{\text{wd}}^{\ast }\\ +(L_1+L_2)\delta i_{\text{wq}}^{\ast }+r_3(i_{1\text{q}}-i_{2\text{q}})-r_3\omega C{u}_{\text{cd}}+v_{\text{pccq}}\\ \end{array}\right)$$ 20

By setting the proper coefficients of the grid-side current and inverter-side current, coupled with the incorporation of capacitor current feedback, the information of the control law (8) is fully utilized. This leads to a wider closed-loop bandwidth than using only sub-equations 1) and 2).

Design of damping gains r₃

In order to further discuss the effect of active damping r₃, the closed-loop transfer function from the reference current $i_{\text{wd}}^{\ast }$ to the grid-side current i_2d is written as

$$G_{2\text{c}}(s)=\frac{i_{2\text{d}}}{i_{\text{wd}}^{\ast }}=\frac{(\delta +k_{\text{p}})(L_1+L_2)e^{-\tau s}}{s^3{M}_1+s^2{M}_2+s{M}_3+M_4+e^{-\tau s}(s^2{P}_1+s{P}_2+P_3)},$$ 21

where $P_1=k_{\text{p}}{L}_1{L}_{2}C-r_3{L}_{2}C,P_2=k_{\text{p}}{L}_{1}C{R}_2-r_{3}C{R}_2,P_3=k_{\text{p}}(L_1+L_2),M_1=L_1{L}_{2}C,M_2=L_1{R}_{2}C+L_2{R}_{1}C,$ $M_3=L_1+L_2+R_1{R}_{2}C,M_4=R_1+R_2.$

The Bode diagrams of the closed-loop transfer function of (21) are presented in Fig. 7. Apparently, the resonance peak of the closed-loop transfer function can be effectively suppressed after adding the capacitor current feedback loop. However, as r₃ increases, the resonance frequency has shifted and the peak first decreases and then increases. The reason is that due to the time delay, r₃ is not a pure resistance in the equivalent circuit, which can be represented in the form of a parallel connection of a resistor and a reactance³⁷. As r₃ increases, the reactance also increases, causing the resonance frequency to deviate. In order to derive a suitable r₃ value, in real applications, the trial-and-error way is usually used. Based on minimizing the amplitude of the resonance peak as shown in Fig. 4, it is appropriate that the value of r₃ is set equal to 5 in the example under study.

Fig. 7.

Bode diagrams of the closed-loop transfer function of G_2c(s) with an additional loop for capacitor current feedback.

The unit step response of the C-PBC controller and P-PBC controller is illustrated in Fig. 8, where the setting times are approximately 1.6 ms and 5.4 ms, respectively. It can be seen that the control method based on the P-PBC controller has a shorter time to reach the steady state compared to the C-PBC controller.

Fig. 8.

Unit step response of the C-PBC controller and the P-PBC controller.

The parameter values used for generating the Bode plots are summarized as follows, which can be seen in Table 2.

Table 2.

Parameter values used for Bode plots in Figs. 3, 5, 6 and 7.

	Figure 3	Figure 5	Figure 6	Figure 7
Inverter-side Inductance L1	1.2 mH	1.2 mH	1.2 mH	1.2 mH
Grid-side inductance L2	1.2 mH	1.2 mH	1.2 mH	1.2 mH
Filter capacitance C	6 μF	6 μF	6 μF	6 μF
Resistance R1	0.1 Ω	0.1 Ω	0.1 Ω	0.1 Ω
Resistance R2	0.1 Ω	0.1 Ω	0.1 Ω	0.1 Ω
Damping coefficient r1	1.5 Delay: 0.24 0 Delay: 8	–	–	–
Damping coefficient kp	–	1000, 1500, 2500, 4000	2500	2500
Resistance-to-inductance ratio $\delta$	–	–	R1/L1 or R2/L2	R1/L1 or R2/L2
Active damping coefficient r3	–	–	–	0, 1, 3, 5, 7

Strong robustness of LCL filter to parameters shift

This section aims at analyzing the robustness when using the P-PBC controller. For analysis purposes, the pole map of the closed-loop system is used here. And the pole map shown in Fig. 9 is based on the G_2c closed-loop transfer function. The nominal values of parameters and parameter drift are indicated in red and blue, respectively. The inductance L₁ varies from 0.6 mH to 1.8 mH. In order to consider the robustness against grid impedance variations, the inductance L₂ varies from 0.6 mH to 6 mH. And the capacitance C varies from 3 μF to 9 μF. In every instance, only one parameter changes, while the rest of the parameters remain unchanged. It is clear that even though the values of L₁, L₂ and C change over a wide range in Fig. 9, the system's all closed-loop poles are still within the unit circle. This observation demonstrates the P-PBC controller's strong robustness in the face of parameter deviations.

Fig. 9.

Pole maps when LCL parameters vary (a) L₁ varying from 0.6 mH to 1.8 mH, (b) L₂ varying from 0.6 mH to 6 mH, (c) C varying from 3 μF to 9 μF.

Achievement of zero steady-state error and less sensors

In practical applications, the PBC controller suffers from steady-state error when parameters vary, since it depends on a mathematical model of LCL-filtered GCI. To eradicate the steady-state error, in the previous studies, various modified PBC strategies have been employed, including the utilization of disturbance observer²⁷, PI regulator^25,38 and PR regulator³⁹, etc. To avoid increasing the complexity of the GCI system, the damping gain, that is the r mentioned above, can be replaced by the proportional term k_p of the PI regulator.

In addition, aiming to minimize the sensor count, a state observer is used to predict the grid-side current, which is ${\widehat{i}}_2$. The state observer's design principles can be found in Refs.^40,41. Figure 10 shows the overall equivalent system diagram of GCI with an LCL filter adopting the P-PBC controller.

Fig. 10.

Overall equivalent system diagram of GCI with LCL filter adopting the P-PBC control method.

The state observer adopted in this paper aims to predict the grid-side current and reduce the number of sensors. Figure 11 compares estimated and measured A-phase grid-side current waveforms. As shown in Fig. 11, the values estimated by the state observer are almost the same as the measured ones. Therefore, employing a state observer is a feasible design method for the P-PBC controller.

Fig. 11.

Comparison between the estimated A-phase grid-side current and the measured A-phase grid-side current.

Experimental verification

A 110 V/50 Hz/3 kW/3-phase experimental device has been established. An image illustrating the experimental device, incorporating dSPACE DS1202-to realize control algorithm, Danfoss FC320-inverter, Chroma 61830-to realize three-phase grid simulator, Yokogawa DL1640-to show experiment waveform, Chroma 62150H-600S-to provide DC power and so on, is displayed in Fig. 12. The detailed experimental parameter values are detailed in Table 3.

Fig. 12.

Experimental device with a 110 V/50 Hz/3 kW/3-phase GCI.

Table 3.

Experimental parameter values.

Symbol	Description	Value
Vg	Grid voltage	110 V (RMS)
f, fs	Switching and sampling frequency	10 kHz
fr	LCL resonance frequency	2.7 kHz
Udc	DC bus voltage	350 V
L1	Inverter-side inductance	1.2 mH
C	Filter capacitance	6 μF
L2	Grid-side inductance	1.2 mH
Lg	Equivalent grid impedance	0 mH, 4.8 mH
R1	Equivalent series resistance of L1	0.1 Ω
R2	Equivalent series resistance of L2	0.1 Ω
$i_{\text{1d}}^{\ast }$, $i_{\text{wd}}^{\ast }$	Reference current	12.86 A
r1	Damping coefficient of C-PBC in 0Ts or 1.5Ts delay	8 or 0.24
r(kp)	Damping coefficient of P-PBC in 1.5Ts delay	2500
r3	Active damping coefficient	5

To verify the effectiveness and advantages of the proposed control method in this paper, three sets of experiments have been performed. Experiment #1 shows the steady-state waveforms with L_g = 0 mH and L_g = 4.8 mH of the grid-side current, confirming the efficacy of the control strategy. Experiment #2 is conducted to demonstrate the P-PBC's capability to reject the grid disturbances including experiments on unbalanced grid voltage experiment and background harmonic voltage experiment. Experiment #3 is performed to compare the dynamic performance of the P-PBC controller with C-PBC controller under the conditions of different grid impedances and parameters shift, respectively.

Experiment #1: Steady-state performance with P-PBC control method when L_g = 0 mH and L_g = 4.8 mH.

In this part, experiments are carried out to demonstrate the steady-state performance of grid-side current when L_g = 0 mH and L_g = 4.8 mH, respectively. The experimental waveforms are depicted in Fig. 13. The achievement of a zero steady-state error and unit power factor by the P-PBC control method ensures the high quality of the grid-side currents. And the total harmonics distortion (THD) is approximately 1.87% and 1.74%, respectively.

Fig. 13.

Steady-state waveforms of the grid-side current with (a) L_g = 0 mH, (b) L_g = 4.8 mH.

Experiment #2: Steady-state performance with P-PBC control method under unbalanced grid voltages and background harmonic voltages.

Usually, in weak grids, the presence of unbalanced grid voltages and background harmonic voltages will often distort the grid-side current. To verify the P-PBC's ability to reject the grid disturbances, the experiments of unbalanced grid voltages and background harmonic voltages are shown in Fig. 14. For the experiment of unbalanced grid voltages experiment, L_g is set equal to 4.8 mH, and the voltage in phase B of the grid is decreased by 10% and 20% in phase C. Due to equipment reasons, only the measured grid voltage and grid-side current of phases A and B are shown in Fig. 14a. For the experiment of background harmonic voltages, the grid inductance L_g is also set equal to 4.8 mH, and the 3rd, 5th and 7th harmonics distort the grid voltage. The magnitudes of these harmonics, relative to the fundamental voltage of the grid, are all 3%. Figure 14b illustrates that the grid-side current manifests a satisfactory sinusoidal waveform, which is almost not affected by unbalanced grid voltages and background harmonic voltages.

Fig. 14.

Experimental waveforms of the grid-side current with (a) unbalanced grid voltages, (b) background harmonic voltages.

Experiment #3: Dynamic performance of the P-PBC controller and C-PBC controller under different grid impedances and parameters shift.

Experiments are developed to further investigate the dynamic performance of the P-PBC controller. The q-axis reference current is established at 0, whereas the d-axis current reference is suddenly changed from 6.43  A to 12.86 A at some point. Figures 15 and 16 show the waveforms of the grid-side current with L_g = 0 mH and L_g = 4.8 mH when using the P-PBC controller and C-PBC controller. The grid-side current of the P-PBC controller reaches the steady-state in only approximately 1.5 ms, regardless of whether it is in L_g = 0 mH and L_g = 4.8 mH. However, it takes approximately 5 ms for the grid-side current of the C-PBC controller to reach a steady-state under L_g = 0 mH and approximately 10 ms to reach a steady-state under L_g = 4.8 mH. Thus, the transient response of the P-PBC controller is better than that of the C-PBC controller. Under the C-PBC controller, the transient performance significantly deteriorates because the control bandwidth is substantially reduced owing to the high value of the grid inductance. However, under the P-PBC controller, the transient process of the grid-side current is very smooth and fast. Thus, the dynamic results indicate that the P-PBC controller can achieve improved transient performance with characteristics of high control bandwidth when facing a broad range of grid impedance variations.

Fig. 15.

Dynamic waveforms of grid-side current when L_g = 0 mH using (a) P-PBC controller, (b) C-PBC controller.

Fig. 16.

Dynamic waveforms of grid-side current when L_g = 4.8 mH using (a) P-PBC controller, (b) C-PBC controller.

In evaluating the robustness of the P-PBC control method under a reference step-changed condition, deviations of − 33.3% in L₁, − 33.3% in L₂, and − 5% in C from their nominal values are set. Figure 17a,b display the grid-side current measured for the P-PBC controller and C-PBC controller under conditions of parameter uncertainties. Similarly, the P-PBC controller exhibits a faster settling time compared to the C-PBC controller. The grid-side current for the P-PBC controller achieves a steady-state in approximately 1.5 ms, while for the C-PBC controller, it takes around 12 ms to reach a steady-state. Although that harmonic current ripple exists, the current waveforms are still sinusoidal and attain zero steady-state error. In this part, the experiment demonstrates that the P-PBC controller exhibits stronger robustness against variations in parameters, which completely agrees with the theoretical analysis.

Fig. 17.

Dynamic performances with parameters variation using (a) P-PBC controller, (b) C-PBC controller.

Conclusion

In this paper, an improved PBC controller is proposed to increase the closed-loop bandwidth of the system, which achieves good dynamic and steady-state performance. The major contributions include the following:

1. The design method of the PBC controller based on the PCH model is introduced and the shortcoming of the narrow bandwidth of the C-PBC is analyzed, laying the groundwork for the proposed improvements.

2. The improved PBC controller is constructed, in which the control law of the PBC is modified by setting the appropriate proportional coefficients for the grid-side current and inverter-side current. The system model can be reduced from the third order to the first order, significantly enhancing the closed-loop bandwidth and overall system performance.

3. A state observer is adopted to minimize sensor count, and a PI regulator replaces the damping coefficient in the PBC controller to achieve a zero steady-state error. The experimental results can prove that the P-PBC exhibits outstanding dynamic performance and strong robustness against the disturbances including parameters variation, unbalanced grid voltages and background harmonic voltages.

All analyses, as well as conclusions, have been verified by experiments on a 110 V/50 Hz/3 kW/3-phase lab setup.

Author contributions

C.G. and F.T. designed the research, performed the experiments, and analyzed the data. C.G. wrote the manuscript. F.T. provided study oversight and assistance in the revision of the paper. All authors reviewed the manuscript.

Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.