## 5 Discussion  
The output of the photovoltaic array has strongly nonlinear characteristics. The inverter circuit contains power switching devices, which is also a typical strongly nonlinear circuit. Therefore, the photovoltaic grid-connected inverter model, Equation (6), is a nonlinear model. The accuracy of the model is very important for the optimal design of the controller and the stability analysis of the system. Figures 6 and 7 provide analyses of the fast-scale bifurcations of photovoltaic grid-connected inverters with folded diagrams. They show that at certain parameter values, the system is in quasiperiodic and chaotic motion states. Fast-scale bifurcation analysis can be used to understand the dynamic behavior of each phase of the system state variable in the entire power frequency cycle with specific parameters for photovoltaic grid-connected inverters. Figures 8 and 9 show the bifurcation and chaos phenomena of the system using slow-scale bifurcation diagrams. It shows that as the control parameters change, the system switches between periodic, bifurcation, and chaotic motion states. The analysis of slow-scale bifurcation can be used to understand the influence of the changes in control parameters on the nonlinear dynamic behavior of the system, and it can also be used as a design guide for control parameters. Figures 8 and 9 indicate that when using \(\eta_{0}(\eta_{1})\) as the bifurcation parameter, the value of \(\eta_{1}(\eta_{0})\) affects the size of the chaotic motion window of the system. Therefore, in order to fully understand the influence of control parameters on the nonlinear dynamic behavior of the system, a three-dimensional bifurcation diagram, as shown in Figure 10, is constructed using \(\eta_{0}\) and \(\eta_{1}\) as the bifurcation parameters. The top view of the three-dimensional bifurcation diagram as shown in Figure 10b clearly shows the parameter area corresponding to the chaotic motion and periodic motion of the system. This servers as the guidance for selection of control parameters. When designing a controller, the control parameters \(\eta_{0}\) and \(\eta_{1}\) should be selected from the area of the periodic orbits shown in Figure 10b to ensure that the system is in a periodic motion state. In the design, the parameter values can be further optimized according to the objectives. Taking the reduction of the total harmonic distortion (THD) of the output current as an example, just take the parameter corresponding to the minimum value of THD from the output current THDs of the inverter calculated in the area of the periodic orbits shown in Figure 10b. A specific case is as follows. Figure 11a shows the method of parameter selection through the bifurcation diagram and the total harmonic distortion. Let \(\eta_{0}\)=0.8 and \(\eta_{1}\) be the bifurcation parameters; for example, when \(\eta_{1}\) is in the range of 0.8-1.0, the system is in the periodic motion state. In the region of this periodic motion, the total harmonic distortion of the output current \(i_{2}\) of the photovoltaic grid-connected inverter is calculated for each parameter value. The purple curve in Figure 11a shows that when the control parameter \(\eta_{1}\) is 0.93, the total harmonic distortion of \(i_{2}\), at 2.87%, is at its lowest. Accordingly, the control parameters are selected as \(\eta_{0}=0.8\) and \(\eta_{1}=0.93\), and the time-domain waveform of the output current \(i_{2}\) from the photovoltaic  
Figure 10: (a) Three-dimensional bifurcation diagram and (b) its top view with both \(\eta_{0}\) and \(\eta_{1}\) being bifurcation parameters. grid-connected inverter is shown in Figure 11b. Based on the control parameters selected in this example, a 1 kW experimental prototype is used to verify the effectiveness of the controller, as shown in Figure 12. The parameters of the experimental prototype are shown in Table 2. The experimental results show that the output current of the inverter can track the voltage signal of the power grid well without obvious distortion.