## 2 Maximum Power Transfer Capability of Grid-Connected Systems at the Rated PCC Voltage
### Realistic Maximum Power with Current Limitation  
In real applications, the maximum inverter current \(I_{inv(max)}\) is determined by the endurable current of the power semiconductor. Thus, considering the current limitation of the inverter, the d-axis inverter current should be limited by the expression in Equation (8). To analyze the maximum transferred power of a grid-connected inverter, the d-axis inverter current should be equal to the limiting value in Equation (8). It is worth mentioning that since the active power is transferred from the wind or PV power plant to the grid under normal circumstances, the active power \(P_{inv}\) should be positive. Thus, the current \(i_{d}\) should also be positive; hence, a negative \(i_{d}\) is not considered in this paper. \[i_{d}\leq\sqrt{I_{inv(max)}^{2}-{i_{q}}^{2}} \tag{8}\]  
It is worth noting that this paper investigates the system without reactive power compensation devices; thus, all the reactive power is provided by the inverter itself. Due to the flexible control ability of inverters, the suitable reactive power from the inverter can be controlled to make sure the PCC voltage operates near its rated value [23,24]. To be easily understandable, the analysis starts with the case where the PCC voltage equals its rated value. By substituting \"\(V_{pcc}=V_{g}\)\" and Equation (8) into Equation (4), a standard quadratic equation for the q-axis inverter current \(i_{q}\) can be derived as:  
\[\frac{1}{{{R_{g}}^{2}}+{X_{g}}^{2}}\cdot{i_{q}}^{2}+\frac{{{X_{g}}{I_{inv(max) }}^{2}}}{{{({R_{g}}^{2}}+{X_{g}}^{2}){V_{g}}}}\cdot{i_{q}}+[\frac{{{I_{inv(max) }}^{4}}}{{4{V_{g}}^{2}}}-\frac{{{R_{g}}^{2}{I_{inv(max)}}^{2}}}{{{({R_{g}}^{2 }}+{X_{g}}^{2})}^{2}}]=0 \tag{9}\]  
Based on Equation (9), positive and negative solutions of the q-axis inverter current can be obtained. As the control law in the voltage PI controller is only suitable for the positive solution, the negative solution is unattainable. Hence, it is disregarded. The positive solution is given by Equation (10). \[i_{q}=R_{g}{I_{inv(max)}}\cdot\sqrt{\frac{1}{{{R_{g}}^{2}}+{X_{g}}^{2}}-\frac {{{I_{inv(max)}}^{2}}}{{4{V_{g}}^{2}}}}-\frac{{{X_{g}}{I_{inv(max)}}^{2}}}{{2{V _{g}}}} \tag{10}\]  
If the solution listed in Equation (10) exists, Equation (11) must be satisfied. \[\frac{1}{{{R_{g}}^{2}}+{X_{g}}^{2}}-\frac{{{I_{inv(max)}}^{2}}}{{4{V_{g}}^{2} }}\geq 0 \tag{11}\]  
Substituting Equation (2) into Equation (11), the range of the SCR is derived as:  
\[SCR\geq\frac{1}{2} \tag{12}\]  
The expression in Equation (12) shows that the PCC voltage cannot be maintained at the rated value if the SCR is lower than 0.5. Moreover, when Equations (2) and (10) are substituted into Equation (8), the expression of the d-axis inverter current \(i_{d}\) is derived as:  
\[i_{d(pu)}=\frac{i_{d}}{{{I_{inv(max)}}}}=\frac{\sqrt{SCR^{2}-\frac{1}{4}}+ \frac{{R_{g}}}{{2X_{g}}}}{SCR\cdot\sqrt{1+\frac{{R_{g}}^{2}}{{X_{g}}^{2}}}} \tag{13}\]  
In order to visualize the aforementioned analytical analysis, the voltage and current vector diagrams of the grid-connected system at the d-q plane are shown in Figure 4. The purely inductive, purely resistive, and hybrid impedance cases are shown in Figure 4a-c, respectively. The vector amplitudes \(I_{inv}\) and \(V_{g}\) are invariable, while the phase anglesand \(\varphi_{p}\) are adjusted to match different values of \(Z_{g}\). It can be seen from Equation (13) that the d-axis current \(i_{d}\) is always positive because only the inverter mode is considered in this paper. Moreover, according to Equation (10), the q-axis current \(i_{q}\) is negative in the purely inductive case, which is shown in Figure 4a. However, the q-axis current \(i_{q}\) is positive in the purely resistive case, which is shown in Figure 4b. Then, in the hybrid case, the current vector should be in the middle area between these two extreme cases, which is shown in Figure 4c.