# Coordinated Mitigation Control for Wideband Harmonic of the Photovoltaic Grid-Connected Inverter
## 3 Resonance Suppression Method of LCL Filter
### Active Damping Feedforward Control Method  
Due to the existence of the LCL filter, a traditional inner loop is not adequate to maintain the control system's stability. Thus, an additional loop with the capacitor current feedforward is used to improve the ability to suppress resonance. It is difficult to quantify the influence on the subsequent control loop after integration through the PI controller. Therefore, the feedforward signal is directly added to the output signal of the modulation voltage. Figure 7 shows the active damping feedforward control method, and \(K_{\text{c}}\) is the capacitive current feedforward gain factor.  
Figure 6: Error voltage in phase A.  
Figure 7: Active damping feedforward strategy.  
Where \(G_{\text{PWM}}(s)\) is equivalent to the production of PWM gain with control and calculation delay, and its transfer function can be shown as Equation (3). Approximation of the pure delay link can be carried out by using the three-order Pade formula.  
\[\left\{\begin{array}{l}G_{\text{PWM}}(s)=K_{\text{PWM}}G_{\text{delay}}(s) \\ G_{\text{delay}}(s)=e^{-1.5T_{\text{z}}}=\frac{-T_{\text{z}}^{3}s^{2}+12T_{ \text{z}}^{2}s^{2}-60T_{\text{z}}s+120}{T_{\text{z}}^{3}s^{3}+12T_{\text{z}}^ {2}s^{2}+60T_{\text{z}}s+120}\end{array}\right. \tag{3}\]  
where \(K_{\text{PWM}}=1\) represents PWM modulation gain.  
The LCL filter's equivalent transfer function after capacitive current feedforward is shown in Figure 8.  
The transfer function of the LCL filter can be rewritten as Equation (4); compared with Equation (2), the constructed loop of capacitive current feedforward adds a second-order pole to the closed-loop characteristic equation, which changes the LCL filter's closed-loop poles distribution.  
\[G_{\text{ LCL}}^{\prime}(s)=\frac{1}{s^{3}L_{1}L_{2}C+s^{2}L_{2}CK_{\text{c}}G_ {\text{PWM}}(s)+s(L_{1}+L_{2})} \tag{4}\]  
Assuming that the constructed new pole can be equivalent to the shunt resistance with the filter capacitor, the topological equivalent diagram is shown in Figure 9.  
The equation for the grid-connected current and the inverter's output voltage can be constructed with Kirchhoff's voltage law as shown in Equation (5).  
\[G_{\text{LCL}}^{\prime}(s)=\frac{1}{s^{3}L_{1}L_{2}CR_{\text{active}}+s^{2}L_{1 }L_{2}+sR_{\text{active}}(L_{1}+L_{2})} \tag{5}\]  
Combining Equation (4) with Equation (5), active damping can establish the expression relationship with \(K_{\text{c}}\) and \(G_{\text{PWM}}(s)\) in Equation (6).  
\[R_{\text{active}}=\frac{L_{1}}{CK_{\text{c}}G_{\text{PWM}}(s)} \tag{6}\]  
The harmonics of a specific resonant frequency passes through the active shunt resistance, so the grid current quality can be improved significantly. The response characteristics of other frequencies are not affected due to virtual resistance only affecting the frequency  
Figure 8: Active damping feedforward control.  
Figure 9: Active damping equivalent model.  
segment near the resonance point. This conclusion is verified in the next section through the Bode diagram. Meanwhile, the proposed active resistance reduces the input of passive resistance, making many contributions to the economical operation of the power distribution network.