## 4 Dynamic Behavior Under Predictive Control  
### Solving the Predictive Controller of Photovoltaic Grid-Connected Inverter  
The predictive controller expression can be easily solved from a simplified circuit of the photovoltaic grid-connected inverter. Considering the system shown in Figure 1, \(i_{\mathrm{1}}\approx i_{\mathrm{2}}\equiv i_{\mathrm{O}}\); let \(L=L_{1}+L_{2}\), \(\eta_{\mathrm{0}}\in(0,1.0)\), and  
\[\left\{ \begin{array}{c}L_{1}=\eta_{\mathrm{0}}L\\ L_{2}=(1-\eta_{\mathrm{0}})L\end{array} \right.. \tag{7}\]  
The simplified circuit of the photovoltaic grid-connected inverter is shown in Figure 4, and its differential equation is as follows:  
\[L\frac{di_{\mathrm{O}}}{dt}+R_{\mathrm{L}}i_{\mathrm{O}}+u_{\mathrm{g}}-u_{ \mathrm{inv}}=0. \tag{8}\]Its solution is  
\[i_{\text{O}}(t)=i_{\text{O}}(t_{0})e^{-\frac{R_{\text{L}}}{L}(t-t_{0})}+\frac{1}{L }\int_{t_{0}}^{t}e^{-\frac{R_{\text{L}}}{L}(t-\tau)}\Big{[}u_{\text{inv}}(\tau)- u_{\text{g}}(\tau)\Big{]}d\tau. \tag{9}\]  
Generally, the switching frequency of the inverter is \(\geq\) 10 kHz, so \(T_{\text{S}}\leq\) 10\({}^{-4}\) s, \(R_{\text{L}}\) (m\(\Omega\)), and \(L\) (mH) can be regarded as the same order of magnitude. It can be found that \(e^{-\frac{R_{\text{L}}}{L}T_{\text{S}}}\approx\) 1. Let \(t=(k+1)T_{\text{S}}\) and \(t_{0}=kT_{\text{S}}\), make \(k\) equivalent to \(kT_{\text{S}}\), and in the \(k\)th switching cycle denote the average output voltage of the full-bridge circuit and the grid voltage as \(u_{\text{inv}}(k)\) and \(\widetilde{u}_{\text{g}}(k)\), respectively. Then the discrete form of Equation (9) is  
\[i_{\text{O}}(k+1)=i_{\text{O}}(k)+\frac{T_{\text{S}}}{L}\Big{[}u_{\text{inv}}( k)-\widetilde{u}_{\text{g}}(k)\Big{]}. \tag{10}\]  
Considering that the system switching frequency (>10 kHz) is much higher than the grid voltage frequency (50 Hz), it can be considered that the grid voltage varies linearly within one switching cycle, so  
\[\widetilde{u}_{\text{g}}(k)=\frac{u_{\text{g}}(k)+u_{\text{g}}(k+1)}{2}. \tag{11}\]  
Similarly, because of the role of the DC terminal capacitor \(C_{1}\), \(u_{C1}\) is almost constant during the \(k\)th switching cycle. The average bridge circuit output voltage \(u_{\text{inv}}(k)\) is the control signal to be solved following the below equation:  
\[u_{\text{inv}}(k)=\left[2D(k)-1\right]u_{C1}(k). \tag{12}\]  
From Equations (10)-(12), the expression for the controller is as follows:  
\[D(k)=\frac{1}{2u_{C1}(k)}\Bigg{\{}\frac{L}{T_{\text{S}}}[i_{\text{O}}(k+1)-i_{ \text{O}}(k)]+\frac{u_{\text{g}}(k)+u_{\text{g}}(k+1)}{2}\Bigg{\}}+\frac{1}{2}. \tag{13}\]  
Before time \(k\), the system needs to calculate \(D(k)\) so that it can control the system according to \(D(k)\) at time \(k\) and make the output current \(i_{\text{O}}(k+1)\) at time \((k+1)\) almost equal to the expected output current \(i_{\text{ref}}(k+1)\). Therefore, in Equation (13), \(u_{C1}(k)\), \(u_{\text{g}}(k)\), \(u_{\text{g}}(k+1)\), and \(i_{\text{O}}(k)\) are unknown and can be calculated by the predictive method. Relative to the power utility period (0.02 s) of the grid-connected inverter, the switching period is usually very small (on the microsecond scale). According to the principle of linear prediction, the voltage signal changes in two adjacent switching periods are considered to be equal, with the following difference equations:  
\[u_{\text{Cl}}(k+1)-u_{\text{Cl}}(k)=u_{\text{Cl}}(k)-u_{\text{Cl}}(k-1), \tag{14}\]  
\[u_{C1}(k)-u_{\text{Cl}}(k-1)=u_{C1}(k-1)-u_{C1}(k-2), \tag{15}\]  
\[u_{\text{g}}(k+1)-u_{\text{g}}(k)=u_{\text{g}}(k)-u_{\text{g}}(k-1), \tag{16}\]  
\[u_{\text{g}}(k)-u_{\text{g}}(k-1)=u_{\text{g}}(k-1)-u_{\text{g}}(k-2). \tag{17}\]  
\[i_{\text{O}}(t)=i_{\text{O}}(t_{0})e^{-\frac{R_{\text{L}}}{L}(t-t_{0})}+\frac{1}{L }\int_{t_{0}}^{t}e^{-\frac{R_{\text{L}}}{L}(t-\tau)}\Big{[}u_{\text{inv}}(\tau)- u_{\text{g}}(\tau)\Big{]}d\tau. \tag{9}\]0 Simplified circuit of photovoltaic grid-connected inverter.