## 3 Nonlinear Dynamic Model of Photovoltaic Grid-Connected Inverter  
PV arrays have the characteristics of nonlinear output, wide range of voltage and current fluctuations, and are susceptible to environmental influences. Therefore, when modeling photovoltaic grid-connected inverters, PV arrays affect the accuracy of photovoltaic grid-connected inverter models. Using the mathematical model of the PV module [8] and denoting the number of PV modules in the PV array in series and in parallel as \(n_{\text{S}}\) and \(n_{\text{P}}\), respectively, the mathematical model of the PV array is composed of the two following equations:  
\[i_{\text{PV}}=G(u_{\text{C1}}), \tag{2}\]  
\[G(u_{\text{C1}})=n_{\text{P}}I_{ph}-n_{\text{P}}I_{s}\Bigg{[}e^{ \frac{\sigma_{(0}+\frac{\alpha_{1}}{n_{\text{S}}}\text{C1}+\frac{\alpha_{2}}{n_ {\text{S}}^{2}}\text{C1}+\frac{\alpha_{3}}{n_{\text{S}}^{2}}\text{C1})}{N_{ \text{S}}kTA}}}-1\Bigg{]} \tag{3}\]  
where \(I_{ph}\) is the photocurrent, \(I_{s}\) is the saturation current, \(N_{s}\) is the number of series units inside the module, \(T\) is the module temperature, \(A\) is the diode ideality factor, \(R_{sh}\) is the shunt resistance, and \(\alpha_{0}\), \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are polynomial fitting coefficients. These parameters can be calculated from the specification parameters provided by the module manufacturer. Also in Equation (3), \(q\) the quantity of elementary charge is \((1.60217657\times 10^{-19})\), and \(K\) is the Boltzmann constant \((1.3806488\times 10^{-23})\). The PV array model is shown in Figure 3.  
Figure 3: PV array model. (**a**) Equivalent circuit model; (**b**) I-U curve. Figure 2: Sinusoidal Pulse Width Modulation (SPWM) principle. (**a**) Unipolar SPWM; (**b**) Bipolar SPWM. Now we consider the two switching states, \(\mathrm{S}=0\) or \(1\), of the circuit in Figure 1 and establish the state equation of the photovoltaic grid-connected inverter, using \(u_{\mathrm{C1}}\), \(u_{\mathrm{C2}}\), \(i_{\mathrm{1}}\), and \(i_{\mathrm{2}}\) as the state variables. In the switch state \(\mathrm{S}=1\), the switching combination of the power switches \(\mathrm{M_{1}}\)-\(\mathrm{M_{4}}\) yields \(i_{\mathrm{S}}=i_{\mathrm{1}}\). The KCL (Kirchoff's Current Law) equations on the nodes a and \(b\) are derived. The KVL (Kirchhoff's Voltage Law) equations are derived on the \(C_{1}\)\(\rightarrow\)\(L_{1}\)\(\rightarrow\)\(C_{2}\)\(\rightarrow\)\(C_{1}\) loop and on the \(C_{2}\)\(\rightarrow\)\(R_{\mathrm{L}}\)\(\rightarrow\)\(L_{2}\)\(\rightarrow\)\(u_{\mathrm{g}}\)\(\rightarrow\)\(C_{2}\) loop. Similarly, when \(\mathrm{S}=0\), the switching combination of the power switches \(\mathrm{M_{1}}\)-\(\mathrm{M_{4}}\) yields \(i_{\mathrm{S}}=-i_{\mathrm{1}}\), and the same KCL and KVL equations are derived. The nonlinear mathematical model of the photovoltaic grid-connected inverter is finally obtained:  
\[\left\{ \begin{array}{c}\frac{di_{\mathrm{1}}}{dt}=-\frac{1}{L_{1}}u_{\mathrm{ C2}}+\frac{1}{L_{1}}(2S-1)u_{\mathrm{C1}}\\ \frac{di_{\mathrm{2}}}{dt}=-\frac{R_{\mathrm{L}}}{L_{2}}i_{\mathrm{2}}+\frac{1 }{L_{2}}u_{\mathrm{C2}}-\frac{1}{L_{2}}u_{g}\\ \frac{du_{\mathrm{C1}}}{dt}=\frac{1}{C_{1}}G(u_{\mathrm{C1}})-\frac{1}{C_{1}}( 2S-1)i_{\mathrm{1}}\\ \frac{du_{\mathrm{C2}}}{dt}=\frac{1}{C_{2}}i_{\mathrm{1}}-\frac{1}{C_{2}}i_{ \mathrm{2}}\end{array} \right., \tag{4}\]  
where and \(u_{g}=sin(\omega t)\), where \(\omega\) represents the angular frequency (rad/s) of the power grid signal. To reflect the influence of PLL on the model, it is necessary to introduce the power grid frequency \(\omega\) into the model as a parameter of the model. The power grid voltage signal \(u_{\mathrm{g}}\) cannot be a fixed sine wave signal, but a state variable. Therefore, we need to build an intermediate variable \(\delta\), which has the following relationship with the grid signal \(u_{\mathrm{g}}\):  
\[\left\{ \begin{array}{c}\frac{du_{\mathrm{g}}}{dt}=\omega\delta\\ \frac{d\delta}{dt}=\omega u_{\mathrm{g}}\end{array}