Multiple Decoupling Current Control Strategies for LCL Type Grid-Connected Converters Based on Complex Vectors under Low Switching Frequencies

Abstract

In medium-voltage and high-voltage high-power converters, the switching devices need to operate at a low switching frequency to reduce power loss and increase the power capacity. This increases the delay of the signal sampling and PWM. It also makes the cross-couplings of the d-q current components more severe. In addition, the LCL filter has three cross-coupling loops and is prone to resonance. In order to solve these problems, this paper establishes a complex vector model of an LCL type grid-connected converter. Based on this model, two multiple decoupling current control strategies with passive damping / notch damping are proposed for the LCL type grid-connected converter. The proposed strategies can effectively eliminate the cross-couplings of the converter, achieve independent control of the d-q current components, expand the stable region and suppress the resonance of the LCL filter. Simulation and experimental results verify the correctness of the theoretical analysis and the feasibility of the proposed strategies.

I. INTRODUCTION

Grid-connected converters have the advantages of a sinusoidal output current, little harmonic content, a bidirectional power flow, adjustable power factor, and few voltage fluctuations on the DC side. They are widely used in renewable energy power generation systems, electric power transmission systems and AC/DC grid connection interfaces.

As the demand for power capacity is expanding, it is necessary to reduce the switching frequency of converters [1]-[4]. An example of this would be an ABB ACS6000 with 2.3~3.3 kV, 5~36 MVA [5]. Its switching device is an IGCT, and its switching frequency is around 500 Hz. It is generally considered that when the switching frequency is below 1 kHz, it is called low switching frequency control [6], [7]. A lower switching frequency increases the delay of the signal sampling and PWM. Thus, cross-coupling is introduced in the converter in the d-q coordinate system. The cross-coupling affects the dynamic and static control performance of the converter, which makes the design more difficult [8], [9]. On the other hand, due to the cross-coupling, when the active power command of the grid-connected converter changes, the reactive power fluctuates. This restricts the performance of grid-connected systems and can lead to instability of the grid [10]-[12], especially in high-power applications. Therefore, it is hoped that independent control of the active and reactive powers can be realized.

At present, high-performance control strategies for grid-connected converters under a low switching frequency are mainly studied from two aspects: design of the PI parameters and improvement of the control strategy. In [13] and [14], by ignoring the cross-coupling, the system is approximated as a second-order single-input single-output system. A method for parameter design was obtained based on this system. However, the design is not very reasonable. In [15], a parameter design method based on the root locus of a multi-input multi-output system was presented. This method can completely eliminate the static coupling of a system. However, the dynamic coupling still exists. On the other hand, the performance of converters can be improved by some new control strategies. In [16] and [17], a PR controller was used in the stationary coordinate system to avoid the coupling problem. However, the dynamic performance of this PR controller is not as good as a PI controller in the d-q coordinate system [18]. Model predictive control was proposed in [19]. However, this control has a serious problem since the switching frequency has not been fixed. The authors used the relative gain matrix theory to analyze the coupling characteristics of different control methods of L-type grid-connected converters and proposed a novel output feedback decoupling strategy [20].

All of the above strategies use an L type grid-connected converter as a control objective. However, the current trend is to use an LCL type grid-connected converter to improve the capability for suppressing high-frequency harmonics and reducing the size and cost of the filter. A decoupling control strategy for LCL type grid-connected converters is given in [21]. However, it operates at a high switching frequency. At this point, delays of the signal sampling and PWM have little effect on the coupling. As a result, they can be ignored. However, under a low switching frequency, there are four cross-coupling links in LCL type grid-connected converters and there is no corresponding decoupling strategy. In addition, the LCL filter has an inherent resonance point. Therefore, it is liable to cause current resonance. In order to suppress this resonance, the most commonly used methods are passive damping methods [22], such as connecting resistors in series in the capacitor branch. However, resistors increase power loss, and additional cooling equipment is required, especially in high-power applications. The resonance point of an LCL filter changes due to the different impedance of the grid. Moreover, there can be a large capacitive reactance in the grid, resulting in multiple resonance peaks. Damping resistors cannot cope with these situations [23]. Recent research has focused on replacing damping resistors with more flexible correction control algorithms, namely active damping methods. Commonly used methods include active damping strategies based on the phase-lead network method [24], virtual resistors [25], genetic algorithms [26], and so on. These methods require additional sensors in the capacitor branch. An active damping control strategy without additional sensors was presented in [27], where the voltages and currents of the capacitor branch are reconstructed. The stability of these active damping methods is extremely susceptible to delays in signal sampling and PWM, grid impedance, etc. [28]. Under a low switching frequency, the delay of signal sampling and PWM is more serious. There is currently no specific research on active damping methods under a low switching frequency.

In this paper, a model of an LCL type grid-connected converter under a low switching frequency is established based on the complex vector [29]-[32]. Using the complex vector can make the equations more concise and reveal the electromagnetic relationships within the system. Then, a multiple decoupling current control strategy with passive damping is proposed for an LCL type grid-connected converter. A multiple decoupling current control strategy with notch damping is proposed, which can realize decoupling control and resonance suppression at the same time. Finally, an undecoupled control system with passive damping and the two proposed control systems are compared. A comparative analysis of the stability, dynamic performance and parameter sensitivity are carried out. Finally, the correctness of the theoretical analysis and the proposed strategies are verified by simulation and experimental results.

II. LCL TYPE GRID-CONNECTED CONVERTER MODEL BASED ON COMPLEX VECTOR

The main circuit of an LCL type grid-connected converter is shown in Fig. 1. L₁ is the converter-side inductor, R₁ includes the power loss equivalent resistance of the converter and the parasitic resistance of the inductor L₁, C_f is the filter capacitor, R_d is the damping resistor, L₂ is the grid-side inductor, R₂ is the parasitic resistance of the inductor L₂, i_1k (k=a, b, c) is the converter-side current, i_2k (k=a, b, c) is the grid-side current, i_ck (k=a, b, c) is the capacitor current, and u_ck (k=a, b, c) is the capacitor voltage. When using a passive damping strategy, the filter capacitor C_f is often connected in series with the damping resistor R_d to suppress the resonance of the LCL filter. When an active damping strategy is used, the resistor R_d is removed and the resonance of the LCL filter is suppressed by a control algorithm.

Fig. 1. Main circuit of a three-phase LCL type grid-connected converter.

According to Fig. 1, a mathematical model of the LCL type grid-connected converter in the d-q coordinate system can be obtained based on the complex vector.

\(\left\{\begin{array}{l} L_{1} \frac{d \boldsymbol{I}_{1 \mathrm{dq}}}{d t}+\mathrm{j} \omega_{\mathrm{b}} L_{1} \boldsymbol{I}_{1 \mathrm{dq}}+R_{1} \boldsymbol{I}_{1 \mathrm{dq}}=\boldsymbol{U}_{\mathrm{rdq}}-\left(\boldsymbol{U}_{\mathrm{cdq}}+R_{\mathrm{d}} \boldsymbol{I}_{\mathrm{cdq}}\right) \\ L_{2} \frac{d \boldsymbol{I}_{2 \mathrm{dq}}}{d t}+\mathrm{j} \omega_{\mathrm{b}} L_{2} \boldsymbol{I}_{2 \mathrm{dq}}+R_{2} \boldsymbol{I}_{2 \mathrm{dq}}=\left(\boldsymbol{U}_{\mathrm{cdq}}+R_{\mathrm{d}} \boldsymbol{I}_{\mathrm{cdq}}\right)-\boldsymbol{E}_{\mathrm{dq}} \\ \boldsymbol{I}_{\mathrm{cdq}}=\boldsymbol{I}_{1 \mathrm{dq}}-\boldsymbol{I}_{2 \mathrm{dq}} \\ \boldsymbol{I}_{\mathrm{cdq}}=C_{\mathrm{f}} \frac{d \boldsymbol{U}_{\mathrm{cdq}}}{d t}+\mathrm{j} \omega_{\mathrm{b}} C_{\mathrm{f}} \boldsymbol{U}_{\mathrm{cdq}} \end{array}\right.\)       (1)

Where I_1dq, I_2dq, I_cdq and U_cdq indicate complex vectors. For example, I_1dq=i_1d+ji_1q and i_1d, i_1q are the components of i_1k (k=a, b, c) in the d-q coordinate system. Equation (1) is transformed into the transfer function shown in (2), where L_t=L₁+L₂，R_t=R₁+R₂，S_j=s+jω_b. In addition, s is the complex variable, and b is the grid-voltage angular frequency.

\(\boldsymbol{F}(s)=\frac{\boldsymbol{I}_{2 \mathrm{dq}}(s)}{\boldsymbol{U}_{\mathrm{rdq}}(s)}=\frac{R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1}{C_{\mathrm{f}} S_{\mathrm{j}}\left(L_{1} S_{\mathrm{j}}+R_{1}\right)\left(L_{2} S_{\mathrm{j}}+R_{2}\right)+\left(L_{\mathrm{t}} S_{\mathrm{j}}+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1\right)}\)       (2)

Inevitably, there is the delay in the signal sampling and PWM in actual systems, especially under a low switching frequency.

The delay is so long that it cannot be ignored. Therefore, it is necessary to consider the influence of the delay on the control system. Typically, a delay of the signal sampling and PWM includes one and a half switching period [33]-[35]. In other word, τ_d=1.5/f_sw, where f_sw is the switching frequency of the converter. This paper mainly considers the performance of the low-frequency band of a converter system. Therefore, the delay can be regarded as a first-order inertial link. Its transfer function F_d(s) is as follows.

\(\boldsymbol{F}_{\mathrm{d}}(s)=\frac{\boldsymbol{U}_{\mathrm{rdq}}(s)}{\boldsymbol{U}_{\mathrm{rdq}}^{*}(s)}=\frac{1}{\tau_{\mathrm{d}} S_{\mathrm{j}}+1}\)       (3)

Combining (2) and (3), the complex vector transfer function of the controlled object of the current control system is shown in (4). Therefore, a complex vector signal flow diagram of the current control system of an LCL type grid-connected converter can be obtained, as shown in Fig. 2. F_r(s) is the current controller, and \(I_{2dq}^{*}\)  is the complex vector of the grid-side current command in the d-q coordinate system. When the active damping strategy is used, R_d can be removed in Fig. 2. As a result, the terms with R_d should be removed in (2) and (4). According to Fig. 2, there are four complex factors in the system, which indicates that there are four cross-couplings. The four cross-couplings are caused by L₁, L₂, C_f in the LCL filter, and the first-order inertial link.

\(\boldsymbol{F}_{\mathrm{m}}(s)=\frac{\boldsymbol{I}_{2 \mathrm{dq}}(s)}{\boldsymbol{U}_{\mathrm{rdq}}^{*}(s)}=\boldsymbol{F}(s) \boldsymbol{F}_{\mathrm{d}}(s)=\frac{1}{\tau_{\mathrm{d}} S_{\mathrm{j}}+1} \cdot \frac{R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1}{C_{\mathrm{f}} S_{\mathrm{j}}\left(L_{1} S_{\mathrm{j}}+R_{1}\right)\left(L_{2} S_{\mathrm{j}}+R_{2}\right)+\left(L_{\mathrm{t}} S_{\mathrm{j}}+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1\right)}\)       (4)

Fig. 2. Complex vector signal flow diagram of an LCL type grid-connected converter current control system.

It can be seen from the above modeling process that the cross-couplings between variables correspond to the imaginary part. That is, in the mathematical model, the complex factor j determines whether the system has cross-coupling. In order to intuitively analyze the coupling degree of the system, the coupling function is defined as:

\(\boldsymbol{F}_{x y}(\mathrm{j} \omega)=\frac{\operatorname{Im}\{\boldsymbol{F}(\mathrm{j} \omega)\}}{\operatorname{Re}\{\boldsymbol{F}(\mathrm{j} \omega)\}}\)       (5)

If the current control system is undecoupled and a PI controller is used, the current controller F_r(s) is as follows.

\(\boldsymbol{F}_{\mathrm{r}}(s)=\frac{K_{\mathrm{p}}\left(\tau_{\mathrm{r}} s+1\right)}{\tau_{\mathrm{r}} s}\)       (6)

The amplitude of the coupling function |F_xy(jω)| of the current control system can be obtained, as shown in Fig. 3. It can be clearly seen from Fig. 3 that the coupling degree of the system gradually increases with a decrease of the switching frequency. The coupling degree is at its most serious near the fundamental frequency.

Fig. 3. Coupling function amplitude |F_xy(jω)| of a closed-loop current control system.

III. MULTIPLE DECOUPLING CURRENT CONTROL STRATEGY BASED ON COMPLEX VECTOR

It can be seen from the above analysis that the LCL type grid-connected converter is a multivariable, strongly coupled nonlinear system under a low switching frequency. There are four cross-couplings in the system. Due to the existence of the cross-couplings, there are some problems when controlling the system.

1) Since the coupling loops are related to each other, they cannot be considered separately. The parameter design is difficult and there is no universal method for its design.

2) The analysis and design of a decoupled system only requires information on one loop. For an undecoupled system, information from all of the loops needs to be considered. Therefore, the information necessary for analyzing and designing an undecoupled system is far greater than that of a decoupled system.

3) Due to cross-couplings, when the active power command of the grid-connected converter changes, the reactive power fluctuates, especially in high-power applications. This severely restricts the performance of the system and can lead to instability of power grid systems.

A. Multiple Decoupling Strategy with Passive Damping

For the LCL type grid-connected converter with a passive damping strategy, the open-loop complex vector transfer function of the current control system is (7). According to (7), the transfer function of a PI controller does not have a complex zero -jω_b to offset the complex pole. Therefore, the cross-coupling cannot be eliminated. Multiple decoupling units D_m(s) are used to solve the coupling problem of the system. A complex vector signal flow diagram is shown in Fig. 4.

\(\boldsymbol{F}_{\mathrm{oc}_{-} \mathrm{m}}(s)=\boldsymbol{F}_{\mathrm{r}}(s) \boldsymbol{F}_{\mathrm{m}}(s)=K_{\mathrm{p}} \cdot \frac{\tau_{r} s+1}{\tau_{\mathrm{r}} s} \cdot \frac{1}{\tau_{\mathrm{d}} S_{\mathrm{j}}+1} \bullet \frac{R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1}{C_{\mathrm{f}} S_{\mathrm{j}}\left(L_{1} S_{\mathrm{j}}+R_{1}\right)\left(L_{2} S_{\mathrm{j}}+R_{2}\right)+\left(L_{\mathrm{t}} S_{\mathrm{j}}+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1\right)}\)       (7)

Fig. 4. Complex vector signal flow diagram of a multiple decoupling current control system.

Theoretically, the decoupling unit can be designed for an arbitrary target system. In order to preserve the characteristic of the main channel and to eliminate only the coupling parts, the complex vector transfer function of the decoupled current control system is obtained as (8). Therefore, the required multiple decoupling unit is (9). For convenience of implementation, D_m(s) is decomposed into D_m1(s), D_m2(s) and D_m3(s), as shown in (10). D_m1(s) is the decoupling unit for the delay. D_m2(s) is the decoupling unit for the complex zeros of the controlled object of the grid-connected converter. D_m3(s) is the decoupling unit for the complex poles of the controlled object of the grid-connected converter. The decoupling units are shown in (11), and they are separated into real parts and imaginary parts. In order to preserve the characteristic of the main channel, the real part is written in the form of "1+". A block diagram of the decoupling units is shown in Fig. 5.

\(\boldsymbol{G}_{\mathrm{m}}(s)=K_{\mathrm{p}} \bullet \frac{\tau_{\mathrm{r}} s+1}{\tau_{\mathrm{r}} s} \cdot \frac{1}{\tau_{\mathrm{d}} s+1} \bullet \frac{R_{\mathrm{d}} C_{\mathrm{f}} s+1}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+\left(L_{t} s+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)}\)       (8)

\(\boldsymbol{D}_{\mathrm{m}}(s)=\frac{\boldsymbol{G}_{\mathrm{m}}(s)}{\boldsymbol{F}_{\mathrm{oc}_{-} \mathrm{m}}(s)}=\frac{\tau_{\mathrm{d}} S_{\mathrm{j}}+1}{\tau_{\mathrm{d}} s+1} \bullet \frac{R_{\mathrm{d}} C_{\mathrm{f}} s+1}{R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1} \bullet \frac{C_{\mathrm{f}} S_{\mathrm{j}}\left(L_{1} S_{\mathrm{j}}+R_{1}\right)\left(L_{2} S_{\mathrm{j}}+R_{2}\right)+\left(L_{\mathrm{t}} S_{\mathrm{j}}+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} S_{\mathrm{j}}+1\right)}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+\left(L_{\mathrm{t}} s+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)}\)       (9)

\(\boldsymbol{D}_{\mathrm{m}}(s)=\boldsymbol{D}_{\mathrm{m} 1}(s) \boldsymbol{D}_{\mathrm{m} 2}(s) \boldsymbol{D}_{\mathrm{m} 3}(s)\)       (10)

\(\left\{\begin{array}{l} D_{\mathrm{m} 1}(s)=1+\mathrm{j} \frac{\omega_{\mathrm{b}} \tau_{\mathrm{d}}}{\tau_{\mathrm{d}} s+1} \\ D_{\mathrm{m} 2}(s)=1-\frac{\omega_{\mathrm{b}}^{2} R_{\mathrm{d}}^{2} C_{\mathrm{f}}^{2}}{\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)^{2}+\omega_{\mathrm{b}}^{2} R_{\mathrm{d}}^{2} C_{\mathrm{f}}^{2}}-\mathrm{j} \frac{\omega_{\mathrm{b}} R_{\mathrm{d}} C_{\mathrm{f}}\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)}{\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)^{2}+\omega_{\mathrm{b}}^{2} R_{\mathrm{d}}^{2} C_{\mathrm{f}}^{2}} \\ \boldsymbol{D}_{\mathrm{m} 3}(s)=1-\frac{3 \omega_{\mathrm{b}}^{2} L_{1} L_{2} C_{\mathrm{f}} s+\omega_{\mathrm{b}}^{2} C_{\mathrm{f}}\left(L_{1} R_{2}+L_{2} R_{1}\right)}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+\left(L_{\mathrm{t}} s+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)}+\mathrm{j} \omega_{\mathrm{b}} C_{\mathrm{f}} \frac{\left(3 L_{1} L_{2}+2 L_{1} R_{2}+2 L_{2} R_{1}\right) s+R_{1} R_{2}-L_{1} L_{2}}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+\left(L_{\mathrm{t}} s+R_{\mathrm{t}}\right)\left(R_{\mathrm{d}} C_{\mathrm{f}} s+1\right)} \end{array}\right.\)       (11)

Fig. 5. Block diagram of multiple decoupling units.

B. Multiple Decoupling Strategy with Notch Damping

It is well known that a damping resistor increases power loss and reduce the efficiency of a system. Moreover, additional cooling equipment is required, especially in high-power applications. To solve these problems, the damping resistor can be removed so that R_d=0. Then an active damping control algorithm can be used to suppress the resonance of the LCL filter. At present, the capacitive current proportional feedback control method is used to suppress resonance. This method is designed under a high switching frequency without considering the four across-coupling loops. Additional sensors in the capacitor branch are also required. In this paper, a multiple decoupling strategy with notch damping is proposed. This strategy can eliminate the cross-couplings and suppress resonance without additional sensors.

After removing the damping resistor, the controlled object of the current control system of the LCL type grid-connected converter is decoupled by the same method. It can be known from (4) that the complex vector transfer function of this controlled object is (12). Then multiple decoupling units can be obtained by setting R_d=0 in (9), as shown in (13).

\(\boldsymbol{F}_{\mathrm{u}}(s)=\left.\boldsymbol{F}_{\mathrm{m}}(s)\right|_{R_{\mathrm{d}}=0}=\frac{1}{\tau_{\mathrm{d}} S_{\mathrm{j}}+1} \cdot \frac{1}{C_{\mathrm{f}} S_{\mathrm{j}}\left(L_{1} S_{\mathrm{j}}+R_{1}\right)\left(L_{2} S_{\mathrm{j}}+R_{2}\right)+L_{\mathrm{t}} S_{\mathrm{j}}+R_{\mathrm{t}}}\)       (12)

\(\boldsymbol{D}_{\mathrm{u}}(s)=\left.\boldsymbol{D}_{\mathrm{m}}(s)\right|_{R_{\mathrm{d}}=0}=\left.\boldsymbol{D}_{\mathrm{m} 1}(s) \cdot 1 \cdot \boldsymbol{D}_{\mathrm{m} 3}(s)\right|_{R_{\mathrm{d}}=0}\)       (13)

Therefore, the open-loop complex vector transfer function of the current control system after decoupling is shown in (14). This system is called a multiple decoupled system without damping.

\(G_{u}(s)=\boldsymbol{F}_{r}(s) \boldsymbol{D}_{u}(s) \boldsymbol{F}_{u}(s)=K_{\mathrm{p}} \cdot \frac{\tau_{r} s+1}{\tau_{r} s} \cdot \frac{1}{\tau_{d} s+1} \cdot \frac{1}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+L_{t} s+R_{t}}\)       (14)

The multiple decoupled system without damping is unstable since it lacks damping for the resonance peak of the LCL filter. In order to solve the problem, the damping term k_ts² is added to the denominator of (14), as shown in (15). k_t is the coefficient of the notch damping.

\(G_{\mathrm{t}}(s)=\boldsymbol{F}_{\mathrm{r}}(s) \boldsymbol{D}_{\mathrm{t}}(s) \boldsymbol{F}_{\mathrm{u}}(s)=K_{\mathrm{p}} \boldsymbol{\bullet} \frac{\tau_{\mathrm{r}} s+1}{\tau_{\mathrm{r}} s} \bullet \frac{1}{\tau_{\mathrm{d}} s+1}\frac{1}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+L_{\mathrm{t}} s+R_{\mathrm{t}}+\underbrace{k_{\mathrm{t}} s^{2}}_{\text {damping term }}}\)       (15)

Fig. 6 shows the open-loop frequency characteristics of current control systems with different damping strategies when the switching frequency is 1kHz. As can be seen from Fig. 6, the multiple decoupled system without damping has a peak at the resonance frequency of the LCL filter. Thus, the magnitude margin at -180° is negative and the system is unstable. When the multiple decoupling strategy with notch damping is used, the resonance peak is effectively suppressed and the magnitude margin at -180° is positive. The system is stable and the control effect is basically the same as that of the strategy with passive damping.

Fig. 6. Open-loop frequency characteristics of current control systems with different damping strategies.

According to the improved open-loop complex vector transfer function (15), multiple decoupling units can be reversed, as shown in (16). As shown in Fig. 7, D_t2(s) generates a negative resonance peak at the resonance frequency, which can offset the resonance peak of the LCL filter, and realizes the effect of notch damping.

\(\left\{\begin{array}{l} \boldsymbol{D}_{\mathrm{t}}(s)=\boldsymbol{D}_{\mathrm{t} 1}(s) \boldsymbol{D}_{\mathrm{t} 2}(s) \\ \boldsymbol{D}_{\mathrm{t} 1}(s)=\boldsymbol{D}_{\mathrm{m} 1}(s)=1+\mathrm{j} \frac{\omega_{\mathrm{b}} \tau_{\mathrm{d}}}{\tau_{\mathrm{d}} s+1} \\ \boldsymbol{D}_{\mathrm{t} 2}(s)=1-\frac{3 \omega_{\mathrm{b}}^{2} L_{1} L_{2} C_{\mathrm{f}} s+\omega_{\mathrm{b}}^{2} C_{\mathrm{f}}\left(L_{1} R_{2}+L_{2} R_{1}\right)+k_{\mathrm{t}} s^{2}}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+L_{\mathrm{t}} s+R_{\mathrm{t}}+k_{\mathrm{t}} s^{2}}+\mathrm{j} \omega_{\mathrm{b}} C_{\mathrm{f}} \frac{\left(3 L_{1} L_{2}+2 L_{1} R_{2}+2 L_{2} R_{1}\right) s+R_{1} R_{2}-L_{1} L_{2}}{C_{\mathrm{f}} s\left(L_{1} s+R_{1}\right)\left(L_{2} s+R_{2}\right)+L_{\mathrm{t}} s+R_{\mathrm{t}}+k_{\mathrm{t}} s^{2}} \end{array}\right.\)       (16)

Fig. 7. Frequency characteristics of D_t2(s).

In order to unify the design of the coefficient of the notch damping k_t under different system parameters, k_t is written as:

\(k_{\mathrm{t}}=2 \xi_{\mathrm{t}} \omega_{\mathrm{r}} L_{1} L_{2} C_{\mathrm{f}}\)       (17)

where ξ_t is the damping factor, and ω_r is resonant angular frequency of the LCL filter, that is:

\(\omega_{\mathrm{r}}=\sqrt{\frac{L_{1}+L_{2}}{L_{1} L_{2} C_{\mathrm{f}}}}\)       (18)

Fig. 8 shows the open-loop frequency characteristics of a multiple decoupled system with notch damping under different damping factors at f_sw=1kHz. It can be seen from Fig. 8 that when ξ_t increases, the damping effect increases. However, the phase margin and magnitude margin decrease. As a compromise, ξ_t=0.7 is sleeked in this paper. The magnitude margin is K_g=11.7dB and the phase margin is γ=50.8°, which meet the requirements of engineering design.

Fig. 8. Open-loop frequency characteristics of multiple decoupled systems with notch damping under different damping factors.

IV. COMPARISON ANALYSIS OF CURRENT CONTROL SYSTEMS

In this paper, the stability, dynamic performance and parameter sensitivity of an undecoupled system with passive damping and the two proposed multiple decoupled systems are compared under the same main parameters.

A. Analysis of Stability and Dynamic Performance

In Fig. 9(a) - Fig. 9(c), the root loci of the three current control systems are given at f_sw=1kHz. It can be seen that for small values of K_p, some of the poles of the undecoupled current control system are in the right half-plane (RHP). However, the dominant poles of the two multiple decoupled current control systems are always in the left half-plane (LHP). This means that the multiple decoupling strategies increase stability. From Fig. 9(b) and 9(c), two poles move towards the RHP with K_p increasing. That is because the resonance suppression for the LCL filter deteriorates with K_p increasing, instead of coupling.

Fig. 9. Root loci of current control systems. (a) Undecoupling strategy with passive damping. (b) Multiple decoupling strategy with passive damping. (c) Multiple decoupling strategy with notch damping.

In Fig. 10, the closed-loop frequency characteristics of the three current control systems are given at f_sw=1kHz. In this figure, a number of things can be observed. The two multiple decoupled systems have larger bandwidths, faster dynamic responses and stronger ability to track signals. The multiple decoupled system with notch damping has the strongest high-frequency attenuation. The two multiple decoupled systems can eliminate the resonance peak, and their relative stability is greatly improved.

Fig. 10. Closed-loop frequency characteristics of current control systems.

Fig. 11 shows unit step response curves of the three current control systems at f_sw=1kHz. When the d axis current component is fed into the unit step signal, the two multiple decoupled systems have smaller overshoots, shorter rise times, and shorter times to reach the steady state. The d axis and q axis current components are not coupled to each other. However, in the undecoupled system, the d axis current component can affect the q axis current component through cross-coupling, which results in large fluctuations and overshoots in the q axis current component. Although the undecoupled system does not have static coupling, it needs a long time to reach the steady state. This means that when the undecoupled system is operating at the unit power factor, if the active current is given a sudden change, the reactive current is affected and it takes a while before returning to the steady state. This impacts the reactive power and overcurrent of the converter and affect the stability of the power grid.

Fig. 11. Unit step response curves of current control systems.

B. Parameter Sensitivity Analysis

The multiple decoupling strategies proposed in this paper are based on the principle of zero pole cancellation. In practical systems, the parameters of the filter can be affected by temperature, operating condition, etc. Therefore, it is necessary to analyze the performance of the control systems when the parameters are changed. Fig. 12 shows a parameter sensitivity analysis of the two proposed strategies.

Fig. 12. Parameter sensitivity analysis of the current control systems. (a) Multiple decoupling strategy with passive damping. (b) Multiple decoupling strategy with notch damping.

In Fig. 12, "×" represents a pole and "○" represents a zero. Fig. 12 shows the root loci of the dominant poles of the current control systems when L₁, L₂ and C_f change from -30% to +30%. Here, the filter parameters L₁, L₂ and C_f are changed while the other parameters are left unchanged. It can be seen from Fig. 12(a) that when the multiple decoupling strategy with passive damping is used, changes of three parameters have little effect on the position of the dominant pole, and the current control system remains stable. Fig. 12(b) shows the case of the multiple decoupling strategy with notch damping. With a decrease of parameter value, especially the capacitor value, the dominant pole moves towards the imaginary axis. When the capacitor value is reduced by -30%, the dominant pole of the system appears in the RHP and the system is unstable. When compared with the multiple decoupling strategy with passive damping, the parameters of the multiple decoupling strategy with notch damping are more sensitive.

V. EXPERIMENTS AND SIMULATIONS

A. Experimental Results

A laboratory setup of a two-level LCL type grid-connected converter, shown in Fig. 13, has been built to verify the proposed multiple decoupling strategies and to compare their performances with the traditional undecoupled strategy under a low switching frequency. A high-performance NI cRIO-9024 of National Instruments Co. is used as the core controller, IGBTs from INFINEON Company are used, and the driving circuit for the IGBTs is designed on the basis of the dual SCALE driver 2SD315A from CONCEPT Company. The experimental parameters are shown in Table I.

Fig. 13. Images. (a) Laboratory setup of a two-level converter. (b) Power circuit diagram.

TABLE I EXPERIMENTAL PARAMETERS

The DC side of the grid-connected converter is connected to the DC power supply, and the system operates in the inverter state. The reactive current is given as zero when the grid-connected converter is operating at the unit power factor. The d-q components of grid-side current are output to an oscilloscope through a DAC module NI 9263. The whole control system framework of the two-level PWM converter can be seen in Fig. 2.

Experimental waveforms of three different control strategies for the LCL type grid-connected converter are shown in Fig. 14. Waveforms of the undecoupling strategy are in Fig. 14 (a1), (b1) and (c1). Waveforms of the multiple decoupling strategy with passive damping are in Fig. 14 (a2), (b2) and (c2). Waveforms of the multiple decoupling strategy with notch damping are in Figs. 14 (a3), (b3) and (c3).

Fig. 14. Experimental waveforms with different control strategies at f_sw = 1kHz: (a1), (b1), (c1) Undecoupling strategy with passive damping. (a2), (b2), (c2) Multiple decoupling strategy with passive damping. (a3), (b3), (c3) Multiple decoupling strategy with notch damping.

In Fig. 14 (a1), (a1) and (a1), waveforms of the DC voltage U_dc and the d-q components of the grid-side current are given when the d component mutates from 5A to 10A. When compared with the undecoupled system, the two decoupling systems with passive damping and with notch damping have better control effects. The response speed is faster, the overshoot is smaller, and the fluctuation of i_q is smaller. The existence of the fluctuation i_q is mainly because the delay link is regarded as a first-order inertial link and the influence of its high-order terms is ignored.

It can be seen from Fig. 14 (c1), (c2) and (c3) that the THD of the grid-side current of the undecoupled system is 3.7%, and the THDs of the grid-side currents of the two decoupled systems are 3.5% and 3.0%. It is verified that the high-frequency harmonics attenuation of the multiple decoupling strategy with notch damping is better than those of the other two strategies.

The above experimental waveforms verify that the proposed multiple decoupling strategies can effectively eliminate dynamic coupling and improve the control performance of the system. The multiple decoupling strategy with notch damping can effectively suppress the resonance of the LCL filter without adding a damping resistor or additional sensors.

B. Simulation Results

In this paper, the proposed strategies are designed for high-power converters, and the decoupling phenomenon is more obvious in high-power applications. Due to limited laboratory conditions, it is not possible to perform experimental verification on a high-power converter. Therefore, the following simulation was performed and the power level of the converter was 2MW. Simulation waveforms are shown in Fig. 15. It can be seen that the coupling problem of the high-power converter is obvious. In addition, the multiple decoupling strategy with passive damping has fluctuation of the q axis current Δi_q from 1083A down to 286A. The multiple decoupling strategy with notch damping has Δi_q from 1083A down to 430A. They both significantly reduce Δi_q.

Fig. 15. Simulation waveforms of the three current control systems at P=2MW, f_sw = 1 kHz. (a) Undecoupling strategy with passive damping. (b) Multiple decoupling strategy with passive damping. (c) Multiple decoupling strategy with notch damping.

Here, a statistics table is made based on the d axis current step value Δi_d in Fig. 16(a). It is found that the greater Δi_d^*, the greater Δi_q. In addition, the superiority of the two proposed strategies can be reflected. The fluctuation of i_q causes fluctuation of the reactive power Q, as shown in Fig. 16(b). In addition, the two proposed strategies can make it significantly improved.

Fig. 16. Fluctuation comparison of the three current control systems at f_sw = 1 kHz. (a) Statistics of Δi_q based on the difference of Δi_d*. (b) Fluctuation of the reactive power when Δi_d*=1.2kA.

Simulation waveforms of multiple decoupling strategy with notch damping are shown in Fig. 17. When the decoupling strategy is removed at 0.4s, the current control system cannot provide sufficient damping and the currents are resonant. When the strategy is re-added to the system at 0.5s, the current is effectively suppressed. This effectively validates the effectiveness of the strategy for resonance suppression of the LCL filter.

Fig. 17. Simulation waveforms of the multiple decoupling strategy with notch damping.

VI. CONCLUSIONS

In this paper, the complex vector is used to establish a model of an LCL type grid-connected converter. Through analyzing the coupling degree, it was found that the coupling degree of the undecoupled system increases with a decrease of the switching frequency. The coupling is the most serious in the low-frequency band. For this reason, two multiple decoupling strategies with passive damping / notch damping were proposed. Time-domain and frequency-domain analyses showed that the proposed strategies can effectively eliminate the cross-couplings of the d-q current components caused by a low switching frequency. They also showed that the proposed strategies can achieve better dynamic performance and a greater stability margin. The multiple decoupling strategy with notch damping can effectively suppress the resonance of the LCL filter without adding a damping resistor or additional sensors. A parameter sensitivity analysis showed that the proposed strategies have good robustness when the parameters of the LCL filter change. When compared with the multiple decoupling strategy with passive damping, the multiple decoupling strategy with notch damping is more sensitive. Finally, the simulation and experimental results verify the correctness of the theoretical analyses and the superiority of the proposed control strategies.

Under the unit power factor, the proposed decoupling strategies can achieve power decoupling, especially in the dynamic process. It also has a certain practical value for the independent control and flexible adjustment of the active power and reactive power in microgrids.