# Improved PR Control Strategy for an LCL Three-Phase Grid-Connected Inverter Based on Active Damping
## 4 Case Analysis  
In order to verify the correctness and effectiveness of the improved QPIR algorithm proposed in this paper, a system simulation model is built in MATLAB/Simulink simulation software. PR, QPR, and QPIR current controllers combined with capacitor current feedback active damping are used to simulate and compare the three-phase LCL grid-connected inverter. The system simulation parameters are shown in Table 1.  
It can be seen from reference [35] that when the controller parameters are the same, they also have similar performance. Therefore, in order to facilitate comparison, the QPIR controller and PR and QPR controllers are set to have similar parameters, that is, the QPIR controller parameters satisfy the following relationship:  
\[K_{P}=K_{1}=K_{3} \tag{15}\]  
\[K_{i}=K_{2}=K_{4} \tag{16}\]  
For small photovoltaic power stations, the allowable range of grid frequency fluctuation is 49.5-50.5 Hz [36], i.e., the frequency deviation is \(\Delta f=0.5\) Hz. The system phase margin PM > 45\({}^{\circ}\) is selected to ensure a good dynamic response. The gain margin is GM > 3 dB for sufficient robustness. PR, QPR, and QPIR controllers are designed to meet the above conditions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Project** & **Parameter** \\ \hline DC bus voltage \(U_{\text{dc}}\) & 750 V \\ Power grid line voltage \(u_{\text{g}}\) (RMS) & 380 V, \\ Rated power \(P_{\text{N}}\) & 40 kW \\ Inverter side inductance \(L_{1}\) & 700 \(\mu\)H \\ Network side inductance \(L_{2}\) & 110 \(\mu\)H \\ Filter capacitor \(C\) & 15 \(\mu\)F \\ Fundamental frequency \(f_{0}\) & 50 Hz \\ Switching frequency \(f_{s}\) & 15 kHz \\ Capacitance current feedback coefficient \(H_{\text{i1}}\) & 0.12 \\ Power network current feedback coefficient \(H_{2}\) & 0.14 \\ Power network voltage feedback coefficient \(H_{\text{v}}\) & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the LCL-type three-phase grid-connected inverter system. Figure 7: Implementation block diagram of the QPIR controller. Considering the dynamic response and the attenuation of high-frequency noise, the system crossover frequency \(f_{\rm c}\) usually has strict limits. Generally,\(f_{0}<<f_{\rm c}<<f_{\rm s}\), where \(f_{0}\) is the fundamental frequency and \(f_{\rm s}\) is the switching frequency. It can be seen from Figure 6 that when the total parameters of the controller are the same, the PR, QPR, and QPIR controllers have similar amplitude-frequency characteristic curves in the high-frequency part. In other words, for a given converter, irrespective of the controller used, the system crossover frequency is the same, so according to the system loop design requirements, set \(K_{\rm p}=K_{1}=K_{4}=0.65\), \(K_{\rm i}=K_{2}\omega_{\rm i}=K_{4}\omega_{\rm i}=2001\); for the calculation process, refer to the literature [37]. For the QPR controller, considering the fluctuation of the fundamental frequency, take the resonance bandwidth parameter \(\omega_{\rm i}=2\pi\Delta f=\pi\) and further obtain \(K_{\rm\theta}=K_{\rm i}/\omega_{\rm i}=636.94\). For the PR controller, further calculation can be used to obtain \(K_{2}=636.94\). For the QPIR controller, considering the fluctuation of the fundamental frequency, set three different fundamental angular frequencies, namely \(\omega_{\rm 01}=2\pi(f_{0}\cdot\Delta f)=311.02\), \(\omega_{\rm 02}=2\pi f_{0}=314.16\), \(\omega_{\rm 03}=2\pi(f_{0}+\Delta f)=317.3\), making the system gain as large as possible within the frequency fluctuation range. Considering that the fluctuation of the fundamental frequency obeys a normal distribution, in order to make the controller have certain robustness, it is hoped that the bandwidth around the angular frequency \(\omega_{\rm 02}\) should be larger, and the bandwidth around \(\omega_{\rm 01}\) and \(\omega_{\rm 03}\) should be smaller, so set the resonance bandwidth parameter \(\omega_{\rm 11}=\omega_{\rm 13}=0.4\pi\), \(\omega_{\rm 12}=2\pi\).