## 3 Maximum Power Transfer Capability of Grid-Connected Systems at Varying PCC Voltages  
In the previous section, the situation where the PCC voltage equals the rated value was evaluated. A more general situation where the PCC voltage varies within a small range around the rated value will be analyzed in this section. By substituting Equations (2) and (8) into Equation (4), a standard quadratic equation for the q-axis inverter current \(i_{q}\) can be obtained. Then, the positive solution of the quadratic equation is derived as Equation (18). Because the control law in the voltage PI controller is only suitable for the positive solution, the negative solution does not exist in the actual control system. Hence, it is disregarded here. \[\left\{ \begin{array}{c}i_{q(pu)}=\frac{i_{q}}{I_{\text{inc}(max)}}=\frac{1}{ V_{\text{pc}(pu)}\sqrt{R_{g}^{2}/X_{g}^{2}+1}}(\sqrt{\frac{V_{\text{pc}(pu)} ^{2}+1}{2}}-\frac{K_{1}}{4}\cdot\frac{R_{g}}{X_{g}}-\frac{K_{2}}{2})\\ K_{1}=SCR^{2}\cdot\left({V_{\text{pc}(pu)}}^{2}-1\right)^{2}+\frac{1}{SCR^{2} },K_{2}=SCR\cdot\left({V_{\text{pc}(pu)}}^{2}-1\right)+\frac{1}{SCR}\end{array} \right. \tag{18}\]  
where \(V_{\text{pc}(pu)}\) is the per-unit value of the PCC voltage, expressed as \"\(V_{\text{pc}(pu)}=V_{\text{pc}}/V_{g}\)\"; \(K_{1}\) and \(K_{2}\) are two intermediate variables. If the solution listed in Equation (18) exists, the expression in Equation (19) must be satisfied. \[\frac{1}{\left|V_{\text{pc}(pu)}+1\right|}\leq SCR\leq\frac{1}{\left|V_{\text{ pc}(pu)}-1\right|} \tag{19}\]  
Figure 5: Relationship between inverter power \(P_{inv(pu)}\), grid power \(P_{grid(pu)}\), and \(R_{g}/X_{g}\) ratio with different SCRs at the rated PCC voltage. According to Equation (19), when the SCR is equal to 1, the range of \(V_{pcc(pu)}\) is [0, 2]. When the SCR is equal to 10, the range of \(V_{pcc(pu)}\) is [0.9, 1.1]. Hence, it is noted that the PCC voltage is naturally close to its rated value under strong grid conditions, where reactive power compensation may not be necessary. However, the PCC voltage varies easily under weak grid conditions. Thus, reactive power compensation is necessary to keep the PCC voltage close to its rated value. When Equation (18) is substituted into Equation (8), the expression of the d-axis inverter current \(i_{d}\) can be derived as:  
\[i_{d(pu)}=\frac{i_{d}}{I_{inv(max)}}=\frac{1}{V_{pcc(pu)}}\sqrt{\frac{V_{pcc(pu) }^{2}-1}{2}+\frac{K_{1}}{4}+(1-\frac{K_{1}}{2}+\sqrt{\frac{V_{pcc(pu)}^{2}+1}{ 2}-\frac{K_{1}}{4}\cdot K_{2}\cdot\frac{R_{g}}{X_{g}}})\frac{1}{{R_{g}}^{2}/{X _{g}}^{2}+1}} \tag{20}\]  
where the expressions of \(K_{1}\) and \(K_{2}\) are shown in Equation (18). According to Equations (14) and (20), the output active power of the inverter (per unit) can be derived as:  
\[P_{inv(pu)}=\frac{P_{inv}}{S_{N}}=V_{pcc(pu)}i_{d(pu)}=\sqrt{\frac{V_{pcc(pu)} ^{2}-1}{2}+\frac{K_{1}}{4}+(1-\frac{K_{1}}{2}+\sqrt{\frac{V_{pcc(pu)}^{2}+1}{ 2}-\frac{K_{1}}{4}\cdot K_{2}\cdot\frac{R_{g}}{X_{g}}})\frac{1}{{R_{g}}^{2}/{X _{g}}^{2}+1}} \tag{21}\]  
According to Equations (16) and (21), the transferred active power to the grid (per unit) can be derived as:  
\[P_{grid(pu)}=\sqrt{\frac{V_{pcc(pu)}^{2}-1}{2}+\frac{K_{1}}{4}+(1-\frac{K_{1}} {2}+\sqrt{\frac{V_{pcc(pu)}^{2}+1}{2}-\frac{K_{1}}{4}\cdot K_{2}\cdot\frac{R_{ g}}{X_{g}}})\frac{1}{{R_{g}}^{2}/{X_{g}}^{2}+1}}-\frac{\frac{1}{{\rm SCR}} \cdot\frac{R_{g}}{X_{g}}}{\sqrt{{R_{g}}^{2}/{X_{g}}^{2}+1}} \tag{22}\]  
According to Equations (21) and (22), Figure 6 presents the relationship between inverter power \(P_{inv(pu)}\), grid power \(P_{grid(pu)}\), and PCC voltage \(V_{pcc(pu)}\) with different \(R_{g}/X_{g}\) ratios and SCRs. Moreover, the points (\(A_{1}\), \(A_{2}\), \(A_{3}\)), (\(B_{1}\), \(B_{2}\), \(B_{3}\)), (\(C_{1}\), \(C_{2}\), \(C_{3}\)), and (\(D_{1}\), \(D_{2}\), \(D_{3}\)) in Figure 6 are the maximum value points of \(P_{grid(pu)}\). As shown in Figure 6a,b, when the SCR is low, the grid power \(P_{grid(pu)}\) drops significantly as the R/X ratio increases.