# Hybrid Modulation Strategy to Eliminate Current Distortion for PV Grid-Tied H6 Inverter
## 4 System Analysis and Control
### Adaptive Current Control under the Full Power Factor  
The fluctuation of both voltage frequency and amplitude in a power system is inevitable. The common rule of \"active power regulates frequency, reactive power regulates voltage\" is often used in power systems to stabilize the frequency and voltage. Therefore, modern grid-tied standards always require that the grid-tied inverter should have the capability of zero error tracking of both reactive and active current set points. As indicated in [18], proportional integral (PI) controllers introduce a steady-state error in the tracking of a sinusoidal reference signal, although the proportional resonant (PR) controller can theoretically achieve zero steady-state error. However, the PR controller's performance can be very sensitive to any mismatch between its resonant frequency and the always slightly varying grid frequency. This can lead to a serious deviation from the expected control effect and can possibly lead to system instability. In order to improve the H6 grid-tied inverter performance in the regulation of frequency and voltage, we integrate the PI and PR controller in this paper and construct a new frequency self-adaptive proportional integral resonant (FSAPIR) controller. Figure 8 displays the self-adaptive current control loop, where the broken arrow indicates the frequency adaptive adjustment. It must be noted that a control delay of one switch period \(T_{s}\) has been considered in this control loop.  
The transfer function \(G_{r}(s)\) of FSAPIR regulator operates according to  
\[Gr(s)=K_{P}+\frac{K_{I}}{s}+\frac{K_{R}s}{s^{2}+\omega^{2}} \tag{10}\]  
where, \(K_{P}\) is the proportional gain, \(K_{I}\) is the integration coefficient, \(K_{R}\) is the resonant coefficient, and \(\omega\) is the fundamental frequency of the grid. In order to realize the FSAPIR, see Figure 9, the improved Nikai Hiroyoshi integral frequency lock loop is used to obtain the grid frequency \(\omega\)[19], and then its adaptive control adjusts the resonant frequency of the circuit.  
Figure 8: The self-adaptive current loop.  
Figure 9: Self-adaptive frequency control.  
Firstly, the resonance section of the circuit is set to zero, and then the tuned the PI controller parameters are obtained from \(G_{r}(s)=K_{P}+K_{I}/s\). The open-loop transfer function is obtained from  
\[G_{O}(s)=\frac{K_{I}(\frac{K_{P}}{K_{I}}s+1)}{Rs(T_{S}s+1)(\frac{I}{R}s+1)} \tag{11}\]  
Using the zero offset and ignoring the poles which are not conducive to the stability of the system results in \(K_{P}/K_{I}=L/R\). From this we obtain the close loop transfer function of the PI controller with the expression  
\[G_{c}(s)=\frac{\frac{K_{P}}{\Gamma I_{s}}}{s^{2}+\frac{1}{I_{s}}s+\frac{K_{P}} {\Gamma I_{s}}} \tag{12}\]  
The damping ratio is set as \(\xi=\frac{\sqrt{\Gamma I_{s}}}{2T_{s}\sqrt{K_{ip}}}=0.707\), the PI parameters are obtained from: \(K_{P}=\frac{L}{2T_{s}},\ K_{I}=\frac{R}{2T_{s}}\). The block diagram for the entire control scheme for the H6 inverter using the HMM and the FSAPIR approach is illustrated in Figure 10.