# Improved PR Control Strategy for an LCL Three-Phase Grid-Connected Inverter Based on Active Damping
## 3 Analysis of the Control Strategy
### Improved Controller QPIR  
As can be seen from the previous analysis, the LCL-type three-phase grid-connected inverter has no coupling in the two-phase stationary coordinate system, which creates favorable conditions for independent control of active power and reactive power. However, the control quantity in the two-phase stationary coordinate system is AC quantity, and the traditional PR and QPR controllers have their own defects, which decreases the control effect of the inverter. In order to solve this problem, this paper proposes an improved controller with proportional, integral, and quasi-resonant hybrid structures. Its transfer function is shown in Equation (12):  
\[G_{QPIR}=K_{P}+\frac{K_{i0}}{s}+\frac{K_{r1}\omega_{i1}s}{s^{2}+2\omega_{i1}s+ \omega_{o1}^{2}}+\frac{K_{r2}\omega_{i2}s}{s^{2}+2\omega_{i4}s+\omega_{o2}^{2 }}+\frac{K_{r3}\omega_{i3}s}{s^{2}+2\omega_{i3}s+\omega_{o3}^{2}} \tag{12}\]  
\[K_{i}=K_{i0}+K_{r1}\omega_{i1}+K_{r2}\omega_{i2}+K_{r3}\omega_{i3}=\alpha K_{i} +\beta K_{i}+\gamma K_{i}+\eta K_{i} \tag{13}\]  
In Equations (12) and (13), \(K_{P}\) is the proportional coefficient, \(K_{i0}\) is the integral coefficient, \(K_{r1}\), \(K_{r2}\), \(K_{r3}\) are quasi-resonant coefficients, \(K_{i}\) is the total integral resonant coefficient, \(\alpha\), \(\beta\), \(\gamma\), \(\eta\) are the weight coefficients of \(K_{i0}\), \(K_{r1}\), \(K_{r2}\), \(K_{r3}\), respectively, \(\omega_{o1}=2\pi(f_{0}-\Delta f)\), \(\omega_{o2}=2\pi f_{0}\), \(\omega_{o3}=2\pi(f_{0}+\Delta f)\) are the fundamental angular frequencies considering the frequency fluctuation of the power grid, \(\omega_{i1}\), \(\omega_{i2}\), \(\omega_{i3}\), \(\omega_{i4}\) represent the resonant term bandwidth, and \(\Delta f\) is the frequency fluctuation range.  
By inserting Equation (13) into Equation (12), another form can be obtained:  
\[G_{QPIR}=K_{P}+K_{i}(\frac{\alpha}{s}+\frac{\beta s}{s^{2}+2\omega_{i1}s+ \omega_{o1}^{2}}+\frac{\gamma s}{s^{2}+2\omega_{i4}s+\omega_{o2}^{2}}+\frac{ \eta s}{s^{2}+2\omega_{i3}s+\omega_{o3}^{2}}) \tag{14}\]  
Figure 5: Bode diagram of PR and QPR controllers.  
The Bode diagram of the improved controller \(G_{\text{QPRR}}\)(s) is shown in Figure 6. In Figure 6, the total integrated resonance coefficients of the three controllers \(G_{\text{PRR}}\)(s), \(G_{\text{QPRR}}\)(s), and \(G_{\text{QPRR}}\)(s) are all the same, but the improved controller \(G_{\text{QPRR}}\)(s) has different weight coefficients and resonance bandwidths. As can be seen from the Bode diagram of Figure 6, the improved QPIR controller is basically consistent with the Bode diagram of the QPR controller in other frequency bands except near the fundamental frequency. It can be seen from the local amplified Bode diagram that the Bode diagram (red) of the improved QPIR controller has a wider frequency range than the traditional PR controller (blue), and at the same time has a higher gain near the fundamental frequency compared with the traditional QPR controller (green), which can not only improve the steady-state accuracy of the grid-connected current but also has certain robustness to the fluctuation of the grid frequency.