## 4 Optimal Parameters for Maximum Power Operation in Inductive and Resistive Grids  
Based on the aforementioned analysis, the maximum power transfer capability of the grid-connected system is jointly determined by the SCR and the R/X ratio as well as the PCC voltage. Therefore, the optimal values of these parameters for effective power transmission need to be investigated in a mathematical manner. According to Equation (17), when the PCC voltage is at the rated value, the partial derivatives of \(P_{grid(pu)}\) can be derived as:  
\[\frac{\partial P_{grid(pu)}}{\partial\frac{R_{g}}{X_{g}}}=-\frac{\frac{1}{2}+ \frac{R_{g}}{X_{g}}\sqrt{SCR^{2}-\frac{1}{4}}}{SCR\cdot\sqrt{(1+\frac{R_{g} ^{2}}{X_{g}^{2}})}^{3}}<0 \tag{23}\]  
\[\frac{\partial P_{grid(pu)}}{\partial SCR}=\frac{\frac{1}{2}+\frac{R_{g}}{X_{g}} \sqrt{SCR^{2}-\frac{1}{4}}}{2\cdot SCR^{2}\cdot\sqrt{1+\frac{R_{g}^{2}}{X_{g}^ {2}}}\cdot\sqrt{SCR^{2}-\frac{1}{4}}}>0 \tag{24}\]  
It can be seen from Equation (23), the partial derivative from \(P_{grid(pu)}\) to the R/X ratio is smaller than 0, which means that as the R/X ratio decreases, \(P_{grid(pu)}\) increases. Namely, R/X = 0 is the theoretically optimal value for power transmission. Moreover, it can be seen from Equation (24), the partial derivative from \(P_{grid(pu)}\) to the SCR is larger than 0, which means that as the SCR increases, \(P_{grid(pu)}\) increases. Namely, SCR = \(\infty\) is the theoretically optimal value for power transmission. Obviously, these two optimal values, in theory,  
Figure 6: Relationship between inverter power \(P_{int(pu)}\), grid power \(P_{grid(pu)}\), and PCC voltage \(V_{pcx(pu)}\) with different \(R_{g}/X_{g}\) ratios and SCRs. (**a**) SCR = 1; (**b**) SCR = 2; (**c**) SCR = 5; (**d**) SCR = 10.  
cannot be obtained in real applications. Nevertheless, a higher SCR or a lower R/X ratio is beneficial to increase the power transfer capability of the system. Aside from the SCR and the R/X ratio, the PCC voltage is another key parameter affecting the maximum power transfer capability of the system. As aforementioned, the rated PCC voltage in an inductive grid is appropriate for power transmission, while a higher PCC voltage than the rated value in a resistive grid is beneficial for power transmission. Thus, according to Equation (22), the curves for maximum \(P_{grid(pu)}\) with different \(R_{g}/X_{g}\) ratios, SCRs, and PCC voltages are shown in Figure 7. Figure 7a presents the inductive grid impedance case, where the rated PCC voltage is preferable to be used. Figure 7b shows the resistive grid impedance case, where the power curves at two PCC voltages are compared. Obviously, a higher PCC voltage is better than the rated PCC voltage. Based on the above analysis, \(P_{grid(pu)}\) in Equation (22) can be considered a key performance index to quantify the performance of the system at different \(V_{pcc}\), SCRs, and R/X values. Namely, if \(P_{grid(pu)}\) is within [0.8, 1], the performance of the system is good. If \(P_{grid(pu)}\) is within [0.5, 0.8], the performance is medium. If \(P_{grid(pu)}\) is lower than 0.5, the performance of the system is bad. Hence, the suitable \(V_{pcc}\), SCRs, and R/X values can be found according to this definition of performance. It was reported in [21] that the R/X ratio of the high voltage line is around 0.3, while the R/X ratio of the low voltage line is around 7 in real applications. Thus, as shown in Figure 7a, when the R/X ratio is 0.3, an SCR higher than 2 is required in order to make \(P_{grid}\) higher than 0.8 pu (good performance). Moreover, as shown in Figure 7b, when the R/X ratio is 7, an SCR higher than 10 and a PCC voltage higher than 1.05 pu are required in order to make \(P_{grid}\) higher than 0.5 pu (medium performance). In order to apply the above research to practical projects, a case study on wind generation is discussed, as follows. Question: How to increase the penetration level from 20% to 40% for wind generation connected to an inductive grid with \(R_{g}/X_{g}=0.3\) and SCR = 3?