Active Disturbance Rejection Control Scheme for Reducing Mutual Current and Harmonics in Multi-Parallel Grid-Connected Inverters

Abstract

With the increasing penetration of renewable energy sources into modern power systems, parallel inverters with LCL filters are commonly employed in the grid interface, giving rise to potential resonance problems. Among the different resonances, interactive resonance is triggered by interaction among inverters when different current references are applied to parallel inverters. It may also feature a mutual current that circulates among inverters instead of flowing into the grid and introduces harmonics or instability in the control system. In this paper, active disturbance rejection control based on a reduced-order extended state observer (RESO) was proposed for parallel inverters. With the proposed scheme, the interaction between inverters is considered as an exogenous disturbance caused by other parallel inverters, estimated by the RESO, and rejected by the controller. In the results, the mutual current and interactive harmonics, calculated via fast Fourier transform, were reduced with the proposed control scheme. Thus, the lower total harmonic distortion of each current was achieved. Additionally, the robust stability and less model-dependent control design are the other additive advantages over the derivative filtered capacitor voltage feedforward-based active damping using PI control. The simulation and real-time experimental results of the conventional and proposed scheme, obtained using the hardware-in-loop, were presented to verify the theoretical analysis under the similar and different current reference cases.

1. Introduction

1.1. Motivation and Incitement

Interest in distributed power-generation systems based on renewable energy sources (RESs) for sustainable development of the environment is growing. In this context, the pulse-width-modulated grid-connected inverter (GCI) has become the most widespread topology for delivering high-quality power from RESs to the grid. To attenuate the high-frequency switching harmonics generated by pulse-width modulation (PWM), the LCL filter has been increasingly adopted between the inverter and power grid [1].

The inherent resonance in the LCL filter can make the controller very complex and require a proper design with consideration of the grid impedance to avoid instability [2,3]. In renewable energy systems, it is common to connect multiple inverters in parallel with the grid to enhance the total generation capacity, leading to the resonance issue becoming more complex [4]. In such systems, internal, series, and interactive resonances have been observed [5]. Among them, interactive resonance, which is also known as parallel resonance, arises, owing to the mutual interaction between parallel GCIs operating under different current references [5,6,7].

1.2. Literature Review

Several studies have been conducted to mitigate the interactive resonance by introducing a virtual harmonic resistance [5], an active damper based on a high-bandwidth power converter [8], or capacitor current-based active damping [9,10,11]. The interactive resonance causes a mutual current—also known as an interactive current—which circulates among inverters instead of flowing into the grid. The relationship between the mutual current and the interactive resonance has been derived, and the adverse effects of the mutual current on the stability of parallel GCIs have been investigated [6,7,12]. However, in these studies, the harmonics introduced by the mutual current were not considered. In [13], the mutual current harmonics were investigated. The harmonics were divided into the low- and high-frequency components. However, the current-reference uncertainty in the RESs was not considered. In previous studies [14,15,16], the instability caused by the mutual current has been investigated under asynchronous PWM carriers or grid-impedance variation. However, the mutual current harmonics may increase the total harmonic distortion (THD) of inverter currents under operation with different references. The general LCL filter design guideline for preventing mutual current instability is provided in [17]. However, the dynamics of currents under different current references were not considered. Thus far, no effective solution for reducing the mutual current and harmonics under different current references in multi-parallel GCIs has been reported.

Active disturbance rejection control (ADRC), which is a disturbance observer-based control method, provides an active and effective way to handle complex/uncertain systems with minimal information about the system [18]. The ADRC estimates all the exogenous and endogenous disturbances, as a value called the lumped disturbance, in the system with the model-independent extended state observer (ESO), which is the main part of ADRC, and compensates them in the control law [19]. The reduction of the resonance harmonics for the LCL filter type GCI was investigated with ADRC in [20,21,22].

1.3. Contribution and Paper Organization

In this paper, a mutual current and resonance harmonics reduction scheme for multi-parallel GCIs with the reduced-order ESO (RESO)-based ADRC, irrespective of the current reference, was proposed. In addition, the common and mutual current expressions were derived with the proposed scheme. The interactive resonance among inverters was treated as an exogenous disturbance, estimated by RESO, and compensated by the proposed scheme. The damping of interactive and common resonances with the proposed method is explained by the frequency response method in the z-domain under different numbers of parallel inverters. The effective resonance damping resulted in greatly reduced mutual current and harmonics in the parallel grid-connected inverters. Furthermore, the stability of the system and disturbance rejection capabilities of the system were analyzed and compared with the conventional PI control using the derivative filter capacitor voltage feedforward-based active damping. To verify the performance of the proposed scheme, several experiments and harmonic analyses were performed for a system composed of two 3kVA inverters in parallel, to consider the worst case, using the real-time hardware-in-loop (HIL) simulation under similar and different reference cases.

The rest of the paper is organized as follows. The mathematical model of the multi-parallel GCI system with LCL filters is derived in Section 2. In Section 3, each element of the overall system with the proposed control scheme is explained, and the design guidelines for the RESO are provided. The common and mutual current expressions in relation to the current reference with the proposed method are derived in Section 4. In Section 5, the resonance damping, stability analysis, and disturbance rejection capabilities of the proposed scheme are presented and compared with those of the conventional derivative filtered capacitor voltage feedforward-based active damping. A simulation and real-time experimental verification for validating the analysis performed in the preceding sections under different current-reference conditions are presented in Section 6. Conclusions are drawn in Section 7.

2. Mathematical Modeling of Multi-Parallel GCIs

The operation of multiple (n) parallel three-phase GCIs with LCL filters is shown in Figure 1. Each inverter is provided with a constant DC voltage V_dci (i = 1,…, n). Z_1i and Z_2i represent the inverter- and grid-side inductor impedances, respectively, Z_3i represents the filter capacitance impedances, and Z_g represents the inductive grid impedance. The inverter-side currents and voltages are represented by i_i and v_i, respectively. e_g represents the ideal and balanced grid voltage, and v_pcc represents the voltage at the point of common coupling (PCC). The inverter parameters are assumed to be identical, i.e.,

$$Z_{1i}=s{L}_1,Z_{2i}=s{L}_2,Z_{3i}=\frac{1}{s{C}_3},V_{dci}=V_{dc}$$

(1)

Each current i_i makes (n − 1) minor contributions to the other j inverters and one major contribution to the grid. The minor contribution is defined as mutual currents i_mj (j = 1,…, n and j ≠ i), which circulate among other inverters. The major contribution is defined as a common current i_c. The multi-parallel GCIs is a multi-input-multi-output system [23]. The relationship between the output currents i_i and input voltages v_i for multi-parallel GCIs can be derived as

$$\left(\begin{array}{c}{i}_1(s)\\ i_2(s)\\ \cdots \\ i_n(s)\end{array}\right)=Y\left(s\right)\cdot \left(\begin{array}{c}{v}_1(s)\\ v_2(s)\\ \cdots \\ v_n(s)\end{array}\right)=\left(\begin{array}{cccc}{Y}_{11}(s)& Y_{12}(s)& \cdots & Y_{1n}(s)\\ Y_{21}(s)& Y_{22}(s)& \cdots & Y_{2n}(s)\\ \cdots & \cdots & \cdots & \cdots \\ Y_{n1}(s)& Y_{n2}(s)& \cdots & Y_{nn}(s)\end{array}\right)\cdot \left(\begin{array}{c}{v}_1(s)\\ v_2(s)\\ \cdots \\ v_n(s)\end{array}\right)$$

(2)

where Y(s) is the n × n admittance matrix of the system. In Equation (2), the impact of each inverter’s own voltage v_i(s) on its own current i_i(s) is represented by diagonal elements Y_ii(s), and the impact of other inverter voltages v_j(s) on the current i_i(s) is represented by non-diagonal elements Y_ij(s). For an n parallel inverter system, an index i corresponds to the inverter under consideration, and an index j corresponds to each of the remaining (n − 1) inverter modules. All the diagonal elements are the same when we assume identical parameters for each inverter. Similarly, all non-diagonal elements are the same. By using the superposition principle, the diagonal and non-diagonal elements can be derived as [6,7,12]

$$\begin{array}{cc}{Y}_{ii}(s)=\frac{n-1}{n}\cdot G_{plant}(s)+\frac{1}{n}\cdot G_{coupling}(s)& \\ Y_{ij}(s)=-\frac{1}{n}\cdot G_{plant}\left(s\right)+\frac{1}{n}\cdot G_{coupling}(s)& \end{array}$$

(3)

where G_plant(s) and G_coupling(s) represent the LCL filter transfer function and grid impedance coupling transfer function, respectively.

$$\begin{array}{l}{G}_{plant}\left(s\right)=\frac{1}{L_1{L}_2{C}_3}\cdot \frac{L_2{C}_3\cdot s^2+1}{s\left(s^2+{\omega }_{res}^2\right)}\\ G_{coupling}\left(s\right)=\frac{1}{L_1\left(L_2+n{L}_g\right)C_3}\cdot \frac{\left(L_2+n{L}_g\right)C_3\cdot s^2+1}{s\left(s^2+{\omega }_{res1}^2\right)}\end{array}$$

(4)

Here, ω_res and ω_res1 are defined as the interactive and common resonance frequencies, respectively.

$$\begin{array}{l}{\omega }_{res}=\sqrt{\left(L_1+L_2\right)/L_1{L}_2{C}_3}\\ {\omega }_{res1}=\sqrt{\left(L_1+L_2+n{L}_g\right)/L_1\left(L_2+n{L}_g\right)C_3}\end{array}$$

(5)

From Equation (4), it can be easily seen that G_plant (s) and G_coupling (s) become identical when L_g is zero, and accordingly, all the non-diagonal elements Y_ij(s) become zero. Consequently, no interaction among inverters is observed [7]. However, when L_g is not zero, each inverter interacts with not only a grid but also the other inverters. In this case, the current i_i(s) can be expressed as

$$i_i(s)={\displaystyle \sum _{j=1,j\ne i}^n{i}_{mj}(s)}+i_c(s)=i_{ms}(s)+i_c(s)$$

(6)

where i_ms(s) represents the sum of (n − 1) contributions of current i_i(s) as a mutual current.

From Equation (2), i_i(s) can be expressed as

$$i_i(s)=Y_{ii}(s)\cdot v_i(s)+{\displaystyle \sum _{j=1,j\ne i}^n{Y}_{ij}(s)\cdot v_j}(s)$$

(7)

From Equations (3), (6) and (7), the currents i_ms(s) and i_c(s) are derived as

$$\begin{array}{l}{i}_{ms}(s)=\frac{1}{n}{G}_{plant}(s)\left(\left(n-1\right)v_i(s)-{\displaystyle \sum _{j=1,j\ne i}^n{v}_j(s)}\right)\\ i_c(s)=\frac{1}{n}{G}_{coupling}(s)\left(v_i(s)+{\displaystyle \sum _{j=1,j\ne i}^n{v}_j(s)}\right)\end{array}$$

(8)

Note that i_ms(s) depends on G_plant(s) and becomes zero when all the inverter voltages are the same. After applying a abc to dq transformation to Equation (8), we obtain

$$\begin{array}{l}{i}_{ms}^{dq}(s)=\frac{1}{n}{G}_{plant}^{dq}(s)\left(\left(n-1\right)v_i^{dq}(s)-{\displaystyle \sum _{j=1,j\ne i}^n{v}_j^{dq}(s)}\right)\pm {\omega }_g(L_1+L_2)i_{ms}^{qd}(s)\\ i_c^{dq}(s)=\frac{1}{n}{G}_{coupliong}^{dq}(s)\left(v_i^{dq}(s)+{\displaystyle \sum _{j=1,j\ne i}^n{v}_j^{dq}(s)}\right)\pm {\omega }_g(L_1+L_2)i_c^{qd}(s)\end{array}$$

(9)

where $i_{ms}^{dq}\left(s\right)$ and $i_c^{dq}$(s) are the dq components of currents i_ms(s) and i_c(s), respectively. Similarly, $v_i^{dq}\left(s\right)$ and $v_j^{dq}\left(s\right)$ are the dq components of voltages v_i(s) and v_j(s), respectively. The term ω_g(L₁+L₂)$i_{ms}^{qd}\left(s\right)$ in the mutual current and ω_g(L₁+L₂)$i_c^{qd}\left(s\right)$ in the common current represent the coupling effect between the respective dq variables. The transfer functions $G_{plant}^{dq}\left(s\right)$ and $G_{coupling}^{dq}\left(s\right)$ can be derived as

$$\begin{array}{l}{G}_{plant}^{dq}(s)=\frac{\left(L_2{C}_3{s}^2-{\omega }_g{}^2{L}_2{C}_3+1\right)}{L_1{L}_2{C}_3\left(s^3+{\omega }_{res}^{2}s-{\omega }_g{}^2\right)}\\ G_{coupling}^{dq}(s)=\frac{\left((L_2+n{L}_g)C_3{s}^2-{\omega }_g{}^2(L_2+n{L}_g)C_3+1\right)}{L_1\left(L_2+n{L}_g\right)C_3\left(s^3+{\omega }_{res1}^{2}s-{\omega }_g{}^2\right)}\end{array}$$

(10)

where ω_g represents the grid frequency. All the parameters of the multi-parallel GCIs are presented in

Table 1. System parameters of multi-parallel GCIs.

Parameter	Symbol	Value
Grid Voltage	eg	120Vrms
Grid Frequency	ɷg	60 Hz
Power Rating	Piref	3 kVA
Reactive Power Rating	Qiref	0
DC Voltage	Vdc	400 V
Switching Frequency	fsw	20 kHz
Sampling Frequency	fs	20 kHz
Inverter-side Impedance	L1	2.5 mH
Grid-side Impedance	L2	1 mH
Filter Capacitance	C3	4 µF
Grid Impedance	Zg(Lg)	1 mH

.

3. Proposed Mutual Current Reduction Scheme

The overall system with the proposed RESO-based ADRC current control scheme for the multi-parallel three-phase GCIs is shown in Figure 2, where $i_i^d$ and $i_i^q$ represent the d and q components of i_i, respectively, and $v_{gi}^d$ and $v_{gi}^q$ represent the d and q components of v_pcc, respectively. The current i_i, which is influenced by all the other j inverters voltages v_j, is chosen as a feedback to the current control. The v_pcc is sensed only for the current and voltage synchronization around ω_g with the phase-locked loop (PLL). The current references $i_{iref}^d$ and $i_{iref}^q$ are generated independently by the active power P_iref and reactive power Q_iref references applied to each inverter, respectively. The six-pulse IGBT gate signals s_i1 to s_i6 are also generated separately for each inverter by applying space-vector modulation (SVM) to the control signals $u_i^{abc}$ provided by each RESO-based ADRC control block after performing inverse dq transformation on control signals $u_i^d$ and $u_i^q$. For $i_i^d$ and $i_i^q$ regulations, the inverters have decentralized control blocks with identical structures and no communication.

The detailed diagram with the proposed control scheme from Figure 2 for any inverter i in the z-domain is shown in Figure 3. In this paper, only $i_i^d$(z) regulation is explained, for simplicity, and a similar configuration for $i_i^q$(z) regulation is adopted. The $i_i^d$(z) is compared with $i_{iref}^d$, followed by proportional control with the gain K_p selected according to the required performance of the control. The estimated lumped disturbance $z_{2i}^d$ calculated by the RESO is subtracted to compensate for the exogenous disturbance, i.e., the effect of the sum of the other j inverters voltages. In the result, the control signal $u_i^d$(z) is generated after dividing by the turning parameter b, which is equal to 1/L₁. The z⁻¹ represents an inherent delay caused by computation and the PWM update, and the inverter is simplified as a linear amplifier with gain K_PWM. Finally, the $v_i^d$(z) is generated. Similarly, $v_j^d$(z) is generated by $i_j^d$(z) current regulation of any other j inverters. The control plant, which comprises multi-parallel GCIs in the discrete time domain is represented as a block-diagram according to Equation (9). With the consideration of the zero-order-hold (ZOH) effect in Figure 3, the discrete representations of $G_{plant}^d\left(s\right)$ and $G_{coupling}^d\left(s\right)$ are given as [24]

$$\begin{array}{l}{G}_{plant}^d\left(z\right)=\left(1-z^{-1}\right)\mathcal{Z}\left[{ℒ}^{-1}\left[\frac{G_{plant}^d\left(s\right)}{s}\right]\right]\\ G_{coupling}^d\left(z\right)=\left(1-z^{-1}\right)\mathcal{Z}\left[{ℒ}^{-1}\left[\frac{G_{coupling}^d\left(s\right)}{s}\right]\right]\end{array}$$

(11)

With the proposed scheme, the interaction influence of other inverters on each current can be treated as an exogenous disturbance, whereas the coupling effect between the dq variables is treated as an endogenous disturbance.

In general, the state-space model of a first-order time-delayed system with an extended state can be represented as

$$\begin{array}{l}\left[\begin{array}{c}{\dot{x}}_1\left(t\right)\\ {\dot{x}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]+\left[\begin{array}{c}b\\ 0\end{array}\right]u_c\left(t-T_s\right)+\left[\begin{array}{c}0\\ 1\end{array}\right]f\left(t\right)\\ y\left(t\right)=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]\end{array}$$

(12)

where x₁(t) and x₂(t) represent the system states and y(t) is the output of the system. b is the control gain parameter, u_c(t − T_s) is the control input with a time delay of sampling time T_s, and f(t) represents the lumped disturbance in the system.

Generally, the second-order ESO, which is the core part of first-order ADRC, is constructed as

$$\left[\begin{array}{c}{\dot{z}}_1\left(t\right)\\ {\dot{z}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ z_2\left(t\right)\end{array}\right]+\left[\begin{array}{c}b\\ 0\end{array}\right]u_c\left(t-T_s\right)+\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]\left(x_1\left(t\right)-z_1\left(t\right)\right)$$

(13)

where z₁(t) and z₂(t) are the estimated values of x₁(t) and x₂(t), respectively, and $\left[\begin{array}{c}{\beta }_1\\ {\beta }_2\end{array}\right]$ is the gain matrix of the ESO.

Assuming that x₁(t) is known through measurement, Equation (13) can be reduced to an estimation of z₂(t):

$${\dot{z}}_2\left(t\right)=\left[0\right]z_2\left(t\right)+\left[0\right]u_c\left(t-T_s\right)+{\beta }_2\left(x_2\left(t\right)-z_2\left(t\right)\right)$$

(14)

Using Equation (12), we can rewrite Equation (14) as

$${\dot{z}}_2\left(t\right)={\beta }_2\left({\dot{y}}_1\left(t\right)-b{u}_c\left(t-T_s\right)-z_2\left(t\right)\right)$$

(15)

where the observer gain β₂ is selected as an observer bandwidth ω_o.

After applying the Laplace transform to Equation (15) and separating the two terms u_c(s) and y(s), the output-to-estimation transfer function G_zy(s) and the control-to-estimation transfer function G_zu(s) are derived as

$$\begin{array}{l}{G}_{zy}\left(s\right)=\frac{z_2\left(s\right)}{y\left(s\right)}=\frac{{\beta }_{2}s}{\left(s+{\beta }_2\right)}\\ G_{zu}\left(s\right)=\frac{z_2\left(s\right)}{u_c\left(s\right)}=\frac{{\beta }_{2}b}{\left(s+{\beta }_2\right)}{e}^{-s{T}_s}\end{array}$$

(16)

Transforming Equation (16) into the discrete time domain with consideration of the zero-order-hold (ZOH) effect and then replacing y(z) with $i_i^d$(z), u_c(z) with $u_i^d$(z), and z₂(z) with $z_{2i}^d$(z) yields the following G_zi(z) and G_zu(z) in Figure 3:

$$\begin{array}{l}{G}_{zi}\left(z\right)=\frac{z_{2i}^d(z)}{i_i^d(z)}={\left(1-z^{-1}\right)\mathcal{Z}\left[{ℒ}^{-1}\left[\frac{G_{zy}\left(s\right)}{s}\right]\right]|}_{y=i_i{}^d,z_2=z_{2i}^d}=\frac{{\omega }_0\left(z-1\right)}{\left(z-e^{-{\omega }_0{T}_s}\right)}\\ G_{zu}\left(z\right)=\frac{z_{2i}^d(z)}{u_c^d(z)}={\left(1-z^{-1}\right)\mathcal{Z}\left[{ℒ}^{-1}\left[\frac{G_{zu}\left(s\right)}{s}\right]\right]|}_{u_c=u_i^d,z_2=z_{2i}^d}=\frac{b\left(1-e^{-{\omega }_0{T}_s}\right)}{z\left(z-e^{-{\omega }_0{T}_s}\right)}\end{array}$$

(17)

4. Modeling of the Proposed Scheme

In this section, the overall dynamics of the system with the proposed control scheme are modeled. From Figure 3, $z_{2i}^d$(z) and $u_i^d$(z) can be expressed as

$$z_{2i}^d(z)=G_{zi}(z)i_i^d(z)-G_{zu}(z)u_i^d(z)$$

(18)

$$u_i^d(z)=\frac{K_P\left(i_{iref}^d-i_i^d(z)\right)-z_{2i}^d(z)}{b}$$

(19)

By substituting Equations (17) and (18) into Equation (19), we obtain

$$u_i^d(z)=G_1(z)\left(i_{iref}^d-i_i^d(z)\right)-G_2(z)i_i^d(z)$$

(20)

where G₁(z) and G₂(z) are defined as follows:

$$\begin{array}{l}{G}_1(z)=\frac{K_P}{b}\frac{z\left(z-e^{-{\omega }_0{T}_s}\right)}{\left(z^2-e^{-{\omega }_0{T}_s}z-1+e^{-{\omega }_0{T}_s}\right)}\\ G_2(z)=\frac{{\omega }_0}{b}\frac{z\left(z-1\right)}{\left(z^2-e^{-{\omega }_0{T}_s}z-1+e^{-{\omega }_0{T}_s}\right)}\end{array}$$

(21)

With a well-tuned RESO, the observer output $z_{2i}^d$(z) closely tracks the lumped disturbances in the multi-parallel GCIs. The proposed scheme can actively compensate for the effect of lumped disturbances in real time. As indicated by Equation (21), the proposed scheme does not rely on detailed knowledge of the system. The expression for $v_i^d$(z) from Figure 3 based on Equation (20) are derived as follows:

$$v_i^d(z)=z^{-1}{K}_{PWM}\left(G_1(z)\left(i_{iref}^d-i_i^d(z)\right)-G_2(z)i_i^d(z)\right)$$

(22)

Similarly, the expression for $v_j^d$(z) can be derived as

$$v_j^d(z)=z^{-1}{K}_{PWM}\left(G_1(z)\left(i_{jref}^d-i_j^d(z)\right)-G_2(z)i_j^d(z)\right)$$

(23)

From Figure 3, the discrete time-domain representation of $i_c^d$(z) can be obtained as

$$i_c^d(z)=\frac{1}{n}{G}_{coupling}^d(z)\left(v_i^d(z)+{\displaystyle \sum _{j=1,j\ne i}^n{v}_j^d(z)}\right)$$

(24)

By substituting Equations (22) and (23) into Equation (24), $i_c^d$(z) can be expressed as

$$i_c^d\left(z\right)=\frac{1}{n}{z}^{-1}{K}_{PWM}{G}_{coupling}^d\left(z\right)\left[G_1\left(z\right){\displaystyle \sum _{i=1}^n{i}_{iref}^d}-\left(G_1(z)+G_2(z)\right){\displaystyle \sum _{i=1}^n{i}_i^d\left(z\right)}\right]$$

(25)

By applying the current-separation scheme to Figure 3 for considering $i_c^d$(z) only, we obtain

$${\displaystyle \sum _{i=1}^n{i}_i^d(z)}=n{i}_c^d(z)$$

(26)

By substituting Equation (26) into Equation (25), the relationship between $i_c^d$(z) and all the current references can be obtained:

$$i_c^d\left(z\right)=\frac{1}{n}\cdot \frac{z^{-1}{K}_{PWM}{G}_{coupling}^d\left(z\right)G_1\left(z\right)}{1+z^{-1}{K}_{PWM}{G}_{coupling}^d\left(z\right)\left[G_1\left(z\right)+G_2\left(z\right)\right]}{\displaystyle \sum _{i=1}^n{i}_{iref}^d}$$

(27)

Similarly, the discrete time-domain representation of $i_{ms}^d$(z) from Figure 3 can be obtained as

$$i_{ms}^d(z)=\frac{1}{n}{G}_{plant}^d(z)\left(\left(n-1\right)v_i^d(z)-{\displaystyle \sum _{j=1,j\ne i}^n{v}_j^d(z)}\right)$$

(28)

By substituting Equations (22) and (23) into Equation (28), $i_{ms}^d$(z) can be expressed as

$$i_{ms}^d(z)=\frac{1}{n}{z}^{-1}{K}_{PWM}{G}_{plant}^d(z)\left[G_1(z){\displaystyle \sum _{j=1,j\ne i}^n\left(i_{iref}^d-i_{jref}^d\right)}-\left(G_1(z)+G_2(z)\right){\displaystyle \sum _{j=1,j\ne i}^n\left(i_i^d(z)-i_j^d(z)\right)}\right]$$

(29)

The current $i_{ms}^d$(z) arises only when dissimilar currents exist among inverters; therefore, we obtain

$${\displaystyle \sum _{j=1,j\ne i}^n\left(i_i^d(z)-i_j^d(z)\right)}=n\times i_{ms}^d(z)$$

(30)

By substituting Equation (30) into Equation (29), the following expression for $i_{ms}^d$(z) is obtained:

$$i_{ms}^d(z)=\frac{1}{n}\cdot \frac{z^{-1}{K}_{PWM}{G}_{plant}^d(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{plant}^d(z)\left[G_1(z)+G_2(z)\right]}{\displaystyle \sum _{j=1,j\ne i}^n\left(i_{iref}^d-i_{jref}^d\right)}$$

(31)

It can be observed from Equation (31) that when $i_{iref}^d$ = $i_{jref}^d$, the mutual current dynamics can be ignored.

By applying the dq transformation and discretizing Equation (6), the expression for the current $i_i^d$(z) can be obtained:

$$i_i^d(z)=i_{ms}^d(z)+i_c^d(z)$$

(32)

Substituting Equations (27) and (31) into Equation (32) gives the complete expression of $i_i^d$(z):

$$\begin{array}{l}{i}_i^d(z)=\left[\frac{n-1}{n}\frac{z^{-1}{K}_{PWM}{G}_{plant}^d(z)G_1(z)}{1+G_{plant}^d(z)\left[G_1(z)+G_2(z)\right]}+\frac{1}{n}\frac{z^{-1}{K}_{PWM}{G}_{coupling}^d(z)G_1(z)}{1+G_{coupling}^d(z)\left[G_1(z)+G_2(z)\right]}\right]i_{iref}^d\\ +\left[-\frac{1}{n}\frac{z^{-1}{K}_{PWM}{G}_{plant}^d(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{plant}^d(z)(z)\left[G_1(z)+G_2(z)\right]}+\frac{1}{n}\frac{z^{-1}{K}_{PWM}{G}_{coupling}^d(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{coupling}^d(z)\left[G_1(z)+G_2(z)\right]}\right]{\displaystyle \sum _{j=1,j\ne i}^n{i}_{jref}^d}\end{array}$$

(33)

The expression in Equation (33) can be simplified by separating the $i_i^d$(z)/$i_{iref}^d$(z) and $i_i^d$(z)/$i_{jref}^d$(z) characteristics and introducing T(z) and R(z):

$$\begin{array}{l}T\left(z\right)=\frac{i_i^d(z)}{i_{iref}^d}=\frac{n-1}{n}\cdot G_{mut}(z)+\frac{1}{n}\cdot G_{com}(z)\\ R\left(z\right)=\frac{i_i^d(z)}{i_{jref}^d}=-\frac{1}{n}\cdot G_{mut}(z)+\frac{1}{n}\cdot G_{com}(z)\end{array}$$

(34)

where G_mut(z) and G_com(z) are the two closed-loop transfer functions representing the $i_{ms}^d$(z) and $i_c^d$(z), respectively, from Equation (34) and are expressed as

$$\begin{array}{l}{G}_{mut}(z)=\frac{z^{-1}{K}_{PWM}{G}_{plant}^d(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{plant}^d(z)\left[G_1(z)+G_2(z)\right]}\\ G_{com}(z)=\frac{z^{-1}{K}_{PWM}{G}_{coupling}^d(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{coupling}^d(z)\left[G_1(z)+G_2(z)\right]}\end{array}$$

(35)

The performance of the multi-parallel GCIs can be evaluated using Equation (34), which includes i_ms(z) and i_c(z) dynamics simultaneously.

5. Resonance Damping and Stability Analysis

The multi-parallel GCIs system is stable if and only if the currents i_ms(z) and i_c(z) are stable. From Equation (35), the loop-gain expressions T_c(z) and T_m(z) for the common and mutual current stability, respectively, are obtained as

$$T_c(z)=\frac{z^{-1}{K}_{PWM}{G}_{coupling}(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{coupling}(z)G_2(z)}$$

(36)

and

$$T_m(z)=\frac{z^{-1}{K}_{PWM}{G}_{plant}(z)G_1(z)}{1+z^{-1}{K}_{PWM}{G}_{plant}(z)G_2(z)}$$

(37)

The resonance damping was analyzed via the frequency-response method using Bode plots with different values of n. The stability analysis was performed on the Bode plot by using the Nyquist stability criterion in the frequency range above 0 dB to find the phase margin (PM) of the system above the −180° phase line. Additionally, the gain margin (GM) of the system is measured to determine how much increase in the control bandwidth is possible. The parameters of the proposed control scheme are given in

Table 2. Parameters of RESO-based ADRC current control.

Parameter	Symbol	Value
Gain Parameter	b	2/L1
Observer Bandwidth	ωo	7 kHz
Proportional Gain	Kp	12,566
PWM Gain	KPWM	~1

. The parameter b in the proposed scheme plays an important role in achieving a larger observer bandwidth to improve the disturbance estimation performance of the RESO. A higher value of ω_o can be obtained when a higher value of b is adopted. Generally, ω_o is limited by the sampling frequency f_s [20]. The Bode plot of the loop gain of the mutual current with the proposed scheme is compared with that of conventional derivative filtered capacitor voltage-based active damping using PI control [7] in Figure 4. The corresponding loop-gain expressions for the conventional scheme are provided in the Appendix A (Equations (A1) and (A2)). The figure indicates that the interactive resonance in the mutual current is independent of the number of parallel inverters. Hence, the frequency response of the mutual current remains the same under different values of n. A small PM of 4° is observed with the conventional scheme owing to the ineffective resonance damping. Additionally, the resonance peak at f_res amplifies the resonance, which introduces the mutual current in the parallel GCIs. Moreover, a further increase in the bandwidth of the conventional scheme may violate the system stability. However, the proposed scheme effectively damps the interactive resonance and achieves a larger PM of 69° and a similar crossover frequency as the conventional scheme. With the proposed scheme, no resonance peak is observed above 0 dB at f_res, which results in a significantly reduced mutual current in the parallel GCIs.

The frequency response of the loop gain of the common current with the conventional scheme under different values of n is shown in Figure 5. The ω_res1 is a variable frequency depending on the number of parallel inverters at the PCC; hence, the frequency-response characteristics of the common current are varied according to n. As indicated by the figure, the conventional scheme cannot achieve enough resonance damping, and its disturbance rejection capabilities are highly affected with an increase in the number of inverters. Additionally, only a small PM in the range of 11°–23° is achieved with the conventional scheme, which limits the bandwidth of the control. In contrast, the proposed scheme effectively damps the common resonance regardless of n and maintains a higher stability of the system, disturbance-rejection capabilities, and the crossover frequency of the control, as shown in Figure 6.

Figure 7a,b show the pole-zero maps of the closed-loop transfer functions G_{com_PI}(z) and G_{mut_PI}(z) with the conventional scheme for two parallel GCIs under k_p variation. As shown in Figure 7, the conventional scheme introduces a pole near the unit circle in the common and mutual current, which causes resonance in the system and reduces the stability margin of the system. However, in the pole-zero map of the proposed scheme shown in Figure 8a,b, we observe that the common and interactive resonances are compensated via pole-zero cancelation. Additionally, the common- and mutual-current poles are well inside the unit circle. Hence, the entire system is commonly and mutually stable.

6. Performance Verification

The proposed scheme was verified by simulation and real-time experimental implementation via the HIL technique with a 1 µs step size for the two-parallel GCIs, where the control algorithm was implemented using real-time digital control circuitry.

The performance of the proposed scheme was investigated under two cases: Case I represents the different current references, i.e., $i_{1ref}^d$ = 5 A and $i_{2ref}^d$ = 0, and Case II represents the same current references, i.e., $i_{1ref}^d$ = 5 A and $i_{2ref}^d$ = 5 A to each inverter. The current references $i_{1ref}^q$ and $i_{2ref}^q$ to the q-axis current control of inverters #1 and #2 remain zero in both cases.

Figure 9 shows the simulation response of the conventional scheme illustrated in [7]. During Case I, the mutual current is generated in the inverter #2 current, as shown in Figure 9b, owing to the ineffective interactive resonance damping. Consequently, more resonance harmonics are observed in i₁ and i_c of Figure 9a,c compared with Case II. Additionally, some oscillations during a step change at 20 ms are observed in the common current shown in Figure 9c.

In Figure 10, the simulation responses of the proposed scheme are shown. The proposed scheme effectively damps the common and interactive resonance in Cases I and II. Consequently, no mutual current is observed in the inverter #2 current of Figure 10b. Additionally, no interactive resonance harmonics are observed in i₁ and i_c of Figure 10a,c for both Cases I and II. The smooth tracking response of the common current shown in Figure 11c during a step change at 20 ms explains the larger stability margin, as described in Section 5.

The real-time experimental results of the conventional scheme for Cases I and II are shown in Figure 11. With the adoption of Case I, the mutual current is activated and circulates in inverter #2, as indicated by i₂. The mutual current introduces harmonics in i₁, i₂ and i_c during Case I. During Case II, the mutual current is zero, and interactive resonance harmonics do not exist. Additionally, during the step change in inverter #2 from Case I to II, oscillations in the common current are observed, as shown in Figure 11b, which depicts the simulation performance shown in Figure 10.

The real-time experimental performance of the proposed scheme during Cases I and II is shown in Figure 12. No mutual current is flowing in i₂ when Case I is adopted. In the results, no harmonic oscillation in the steady state is observed in the currents i₁, i₂ and i_c, regardless of the current reference. Additionally, the oscillations in i_c during the step change (from Case I to Case II) are removed, which explains the larger stability margin obtained with the proposed scheme, as shown in Figure 12b. Furthermore, the proposed control scheme achieves robust performance with minimum information about the system required without adopting grid voltage feedforward. This theory can also be applied to a system with an uncertain gird impedance or imbalance v_pcc.

The harmonic analysis using the fast Fourier transform (FFT) for the conventional scheme on i₁, i₂ and i_c is shown in Figure 13a,b for Cases I and II, respectively. The resonance is excited in the common current around f_res and f_res1, which may result in higher THD of i₁ and i_c during Case I, as shown in Figure 13a. The THD of i₁, i₂ and i_c is slightly improved during Case II, as shown in Figure 13b.

The harmonic analysis using the FFT for the proposed scheme on currents i₁, i₂ and i_c is shown in Figure 14a,b for each case. The harmonics around f_res and f_res1 are effectively damped for both Case I and Case II. Additionally, the THD of i₁, i₂ and i_c is greatly improved. During Case I, the THD values for i₁ and i_c measured at the frequency ω_g are calculated as 5.02% and 0.65%, respectively. Similarly, the THD values of i₁, i₂ and i_c during Case II are calculated as 4.84%, 4.84% and 0.20%, respectively.

7. Conclusions

An RESO-based ADRC control scheme is proposed for damping the interactive and common resonances present in LCL filter-type three-phase multi-parallel GCIs regardless of the current reference. The proposed scheme treats the interaction among inverters as an exogenous disturbance caused by other inverters in parallel, estimates it with the RESO, and compensates disturbances in the control law of ADRC. The larger stability margins for the common and mutual current, the preserved disturbance-rejection capabilities under different values of n, and the less model-dependent control design compared to conventional control scheme are the potential benefits of the proposed scheme. The performance of the proposed control scheme was verified and compared with that of the conventional control scheme presented in [7]. A harmonic analysis using the FFT indicates the low THD performance of the proposed method regardless of the current-reference conditions in the system with two parallel GCIs. Thus, the proposed scheme can be extended to a system with an uncertain grid impedance or grid voltage imbalance.