## 4 Dynamic Behavior Under Predictive Control
### Fast-Scale Bifurcation of Photovoltaic Grid-Connected Inverter  
To verify the effectiveness of the model, Equation (6), and the controller, Equation (19), the system parameters shown in Table 1 are selected, where \(U_{\text{OC}}^{md}\) is the open-circuit voltage of the module, \(U_{\text{OC}}^{ar}\) is the open-circuit voltage of the array, \(I_{\text{SC}}^{md}\) is the short-circuit current of the module, and \(I_{\text{SC}}^{ar}\) is the short-circuit current of the array. Generally, the parameters of the PV model can be obtained directly or indirectly from the manufacturer's data manual. \(\alpha_{0}\)-\(\alpha_{3}\), \(I_{\text{ph}}\), \(I_{s}\), \(A\), \(I_{\text{SC}}^{md}\) and \(U_{\text{OC}}^{md}\) in Table 1 are the parameters at STC of commercial PV module, JAM5(I)-72-155. These parameters are determined using the parameterization approach in [8]. It is worth noting that the actual system usually adds a backflow prevention diode to the output end of the photovoltaic array, so the output voltage range of the photovoltaic array will be limited to 0-\(U_{\text{OC}}^{ar}\), and the output current will be limited to 0-\(U_{\text{SC}}^{ar}\). These constraint conditions should be added during numerical simulation.  
With the control parameters set at \(\eta_{0}=0.5\) and \(\eta_{1}=1.0\), the numerical simulation results are shown in Figure 5. The photovoltaic grid-connected inverter output current \(i_{2}\) has the same frequency and phase as the grid voltage, which enables tracking and control of the grid-connected current.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**PV Module**} & \multicolumn{2}{c}{**Inverter Circuit and PV Array**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\alpha_{0}\) & 0.4898 & \(L_{1}\)/mH & 1.5 \\ \(\alpha_{1}\) & 0.8924 & \(L_{2}\)/mH & 1.5 \\ \(\alpha_{2}\) & 0.0064 & \(R_{\text{L}}\)/\(\Omega\) & \(0.2\times 10^{-4}\) \\ \(\alpha_{3}\) & \(-9.5197\times 10^{-5}\) & C\({}_{1}\)/\(\mu\)F & \(4.7\times 10^{3}\) \\ \(I_{\text{ph}}\)/A & 4.9415 & C\({}_{2}\)/\(\mu\)F & 4.7 \\ \(I_{\text{s}}\)/A & \(9.75\times 10^{-5}\) & \(\omega\)/(rad/s) & \(100\pi\) \\ \(N_{\text{S}}\) & 72 & T\({}_{\text{S}}\)/\(\mu\)s & 20 \\ \(A\) & 1.8 & \(n_{\text{S}}\) & 15 \\ \(R_{\text{s}\text{s}\text{/}}\Omega\) & 333.663 & \(n_{\text{P}}\) & 6 \\ \(I_{\text{SC}}^{md}\)/A & 5.57 & \(I_{\text{SC}}^{ar}\)/A & 33.42 \\ \(U_{\text{OC}}^{md}\)/V & 36.0 & \(U_{\text{OC}}^{ar}\)/V & 540.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the photovoltaic grid-connected inverter.  
Figures 6 and 7 show the states of quasiperiodic and chaotic motions of photovoltaic grid-connected inverters, respectively. Figure 6 shows the quasiperiodic motion with the control parameters of \(\eta_{0}=0.5\) and \(\eta_{1}=0.9\). Figure 6a-c are the folded diagrams of output current \(i_{2}\), output voltage \(u_{\text{C2}}\) from the photovoltaic grid-connected inverter, and \(u_{\text{C1}}\) at the photovoltaic array output terminals. Figure 6d is a phase diagram of \(i_{2}\), \(u_{\text{C2}}\), and \(u_{\text{C1}}\). Similarly, Figure 7 shows chaotic motion with the control parameters \(\eta_{0}=0.5\) and \(\eta_{1}=0.5\).  
Figure 5: Time-domain waveform of system variables under predictive control.  
Figure 6: Quasiperiodic motion at \(\eta_{0}=0.5\) and \(\eta_{1}=0.9\). (**a**) Folded diagram of output current \(i_{2}\); (**b**) Folded diagram of output voltage \(u_{\text{C2}}\); (**c**) Folded diagram of output voltage \(u_{\text{C1}}\) of the photovoltaic array; (**d**) Phase diagram of quasi-periodic motion of \(i_{2}\), \(u_{\text{C2}}\) and \(u_{\text{C1}}\).