## 2 Problem Description and Model Analysis
### State Space Model  
Compared with the application in practical control engineering, the uncertainty of system inductance and capacitance inductance (LCL) were studied. It was proposed that adding components with impedance or parameters to the filter instead of uncertain factors would change the characteristics of the nominal system.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline Converter side inductance of filter & \(L_{1}\) & 1.83 mH \\ Converter side resistance of filter & \(R_{1}\) & 0.52 m\(\Omega\) \\ Grid side inductance of filter & \(L_{2}\) & 1.75 mH \\ Grid side resistance of filter & \(R_{2}\) & 12 m\(\Omega\) \\ Damping resistance of filter & \(R_{f}\) & 0.6 \(\Omega\) \\ Capacitance of filter & \(C_{f}\) & 270 μF \\ \hline \hline \end{tabular}
\end{table}
Table 1: System parameters.  
Figure 3: A separate inverter unit: (**a**) Frequency; (**b**) Voltage.  
Figure 2: Configuration of the MAF-PLL scheme.  
For the accurate modeling and performance analysis of an uncertain intelligent microgrid system. Combined with Figure 6 and the specific filter uncertain parameters were given to establish the state-space model of the inverter in the synchronous coordinate system, as shown below:  
\[\frac{di_{1,d}}{dt}=\frac{1}{L_{1}}\Big{(}v_{d}-\Big{(}R_{1}+R_{f}\Big{)}i_{1,d} ++\omega L_{1}i_{1,d}-v_{f,d}\Big{)} \tag{3}\]  
\[\frac{di_{1,q}}{dt}=\frac{1}{L_{1}}\Big{(}v_{q}-\Big{(}R_{1}+R_{f}\Big{)}i_{1,q }+\omega L_{1}i_{1,d}-v_{f,d}\Big{)} \tag{4}\]  
\[\frac{di_{2,d}}{dt}=\frac{1}{L_{2}}\Big{(}v_{f}+R_{f}i_{1,d}-\Big{(}R_{2}+R_{f} \Big{)}i_{2,q}+\omega L_{tot}i_{2,d}\Big{)} \tag{5}\]  
\[\frac{di_{2,d}}{dt}=\frac{1}{L_{2}}\Big{(}v_{f}+R_{f}i_{1,d}-\Big{(}R_{2}+R_{f }\Big{)}i_{2,q}-\omega L_{tot}i_{2,d}\Big{)} \tag{6}\]  
\[\frac{dv_{f,d}}{dt}=\frac{1}{C_{f}}\Big{(}i_{1,d}-i_{2,d}+\omega C_{f}v_{f,d} \Big{)} \tag{7}\]  
\[\frac{dv_{f,d}}{dt}=\frac{1}{C_{f}}\Big{(}i_{1,d}-i_{2,q}-\omega C_{f}v_{f,d} \Big{)} \tag{8}\]  
where \(L_{1}\), \(R_{1}\), \(L_{2}\), \(R_{2}\) are the impedance, \(C_{f}\) and \(R_{f}\) are the capacitance and impedance, \(v_{f,d}\), \(v_{f,d}\) are the capacitance \(v_{c}\) shunt voltage at both ends, \(v_{g,d}\), \(v_{g,d}\) are the grid voltage \(v_{g}\) shunt voltage at both ends, between the microgrid side and the grid side branch current \(i_{1,d}\), \(i_{1,d}\), \(i_{2,d}\), \(i_{2,d}\), \(\omega\) is the power system frequency.  
Figure 4: Voltage changes after the parallel connection of inverter \(\mathrm{V_{DGi}}=\mathrm{V1}\;(\mathrm{i}=1,2)\).  
Figure 5: Active and reactive power under nominal model: (a) DG1; (b) DG2.  
At this time, the continuous time representation of the system can be re represented by the state-space model of the following formula:  
\[\left\{\dot{x}(t)=A^{*}(t)x(t)+{B_{1}}^{*}(t)U_{dq}(t)+{B_{2}}^{*}(t)e_{gdq}(t)\right. \tag{9}\]  
where \(x_{i}=\begin{bmatrix}v_{fd}\ v_{fq}\ i_{1d}\ i_{1q}i_{2d}\ i_{2q}\end{bmatrix}^ {\mathsf{T}}\), \(A^{*}=A_{dq}+\Delta A\), \({B_{1}}^{*}=B_{1 dq}+\Delta B_{1}\),  
\[{B_{2}}^{*}=B_{2 dq}+\Delta B_{2}\]  
\[{B_{1 dq}}=\begin{bmatrix}0&0&1/L_{1}&0&0&0\\ 0&0&0&1/L_{1}&0&0\end{bmatrix}^{\mathsf{T}}\text{, }{B_{2 dq}}=\begin{bmatrix}0&0&0&0&1/L_{2}&0\\ 0&0&0&0&0&1/L_{2}\end{bmatrix}^{\mathsf{T}}\text{, }\]  
\[A_{dq}=\begin{bmatrix}0&\omega_{c}&1/C_{f}&0&-1/C_{f}&0\\ -\omega_{c}&0&0&1/C_{f}&0&-1/C_{f}\\ -1/L_{1}&0&-\Big{(}R_{1}+R_{f}\Big{)}/L_{1}&\omega_{c}&R_{f}/L_{1}&0\\ 0&-1/L_{1}&-\omega_{c}&-\Big{(}R_{1}+R_{f}\Big{)}/L_{1}&0&R_{f}/L_{1}\\ 1/L_{2}&0&R_{f}/L_{2}&0&-\Big{(}R_{2}+R_{f}\Big{)}/L_{2}&\omega_{c}\\ 0&1/L_{2}&0&R_{f}/L_{2}&-\omega_{c}&-\Big{(}R_{2}+R_{f}\Big{)}/L_{2}\end{bmatrix}\]  
The grid side current was the measured state, and the inverter side current and capacitor voltage were the unmeasured parts of the state vector. To facilitate the use of data acquisition equipment and PWM signal acquisition in the simulation experiment, the bounded range analysis of parameter uncertainty was conducted.