## 2 Maximum Power Transfer Capability of Grid-Connected Systems at the Rated PCC Voltage
### Realistic Maximum Power with Current Limitation
As seen in Figure 4, because the amplitude of the inverter current is invariable, the larger the d-axis current component, the smaller the q-axis current component, and vice versa. Thus, as the output reactive power increases, the output active power decreases. This characteristic is more obvious in a weaker grid, where the amplitude of \(Z_{g}\) is higher. The output active power of the inverter \(P_{inv}\) shown in Figure 2 can be calculated as:  
\[P_{inv}=\frac{3}{2}V_{pcc}i_{d} \tag{14}\]  
Thus, according to Equations (13) and (14), the output active power of the inverter (per unit) can be derived as:  
\[P_{inv(pu)}=\frac{P_{inv}}{S_{N}}=\frac{3/2\cdot V_{g}\cdot i_{d}}{3/2\cdot V_ {g}\cdot I_{inv(max)}}=i_{d(pu)}=\frac{\sqrt{SCR^{2}-\frac{1}{4}}+\frac{R_{g }}{2X_{g}}}{SCR\cdot\sqrt{1+\frac{R_{g}^{2}}{X_{g}^{2}}}} \tag{15}\]  
Moreover, according to Equations (1) and (2), the power loss introduced by the transmission line (per unit) is derived as:  
\[P_{loss(pu)}=\frac{P_{loss}}{S_{N}}=\frac{3/2\cdot I_{inv(max)}^{2}\cdot R_{g }}{3/2\cdot V_{g}\cdot I_{inv(max)}}=\frac{\frac{R_{g}}{X_{g}}}{SCR\cdot\sqrt {1+\frac{R_{g}^{2}}{X_{g}^{2}}}} \tag{16}\]  
Thus, according to Equations (15) and (16), the transferred active power to the grid (per unit) can be derived as:  
\[\frac{1}{{{R_{g}}^{2}}+{X_{g}}^{2}}\cdot{i_{q}}^{2}+\frac{{{X_{g}}{I_{inv(max) }}^{2}}}{{{({R_{g}}^{2}}+{X_{g}}^{2}){V_{g}}}}\cdot{i_{q}}+[\frac{{{I_{inv(max) }}^{4}}}{{4{V_{g}}^{2}}}-\frac{{{R_{g}}^{2}{I_{inv(max)}}^{2}}}{{{({R_{g}}^{2 }}+{X_{g}}^{2})}^{2}}]=0 \tag{9}\]0  
Based on Equations (15) and (17), Figure 5 presents the relationship between the inverter power \(P_{inv(pu)}\), grid power \(P_{grid(pu)}\), and \(R_{g}/X_{g}\) ratio with different SCRs at the  
\[\frac{1}{{{R_{g}}^{2}}+{X_{g}}^{2}}\cdot{i_{q}}^{2}+\frac{{{X_{g}}{I_{inv(max) }}^{2}}}{{{({R_{g}}^{2}}+{X_{g}}^{2}){V_{g}}}}\cdot{i_{q}}+[\frac{{{I_{inv(max) }}^{4}}}{{4{V_{g}}^{2}}}-\frac{{{R_{g}}^{2}{I_{inv(max)}}^{2}}}{{{({R_{g}}^{2 }}+{X_{g}}^{2})}^{2}}]=0 \tag{9}\]1 Voltage and current vector diagrams of the grid-connected system with various impedance characteristics of the power grid. (**a**) Purely inductive impedance; (**b**) purely resistive impedance; (**c**) hybrid impedance. rated PCC voltage. Since the SCR is usually higher than 1 in practice, the situation where SCR is lower than 1 is omitted. It can be seen that a higher SCR leads to higher grid power, \(P_{grid(pu)}\). However, a higher R/X ratio causes lower grid power, \(P_{grid(pu)}\). Therefore, inductive grids with a high SCR have a high power transfer capability, while resistive grids with a low SCR have a limited power transfer capability. Moreover, in the resistive grid (\(R_{g}/X_{g}>1\)) case, the higher the SCR, the lower the output power of the inverter, \(P_{inv(pu)}\). It seems counterintuitive. This is because more reactive power has to be supplied to maintain the PCC voltage so that the output active power of the inverter is limited. This indicates that it is uneconomical to use reactive power to compensate the PCC voltage in resistive grids because much of the potential active power capacity is sacrificed. Aiming at this problem, tuning the PCC voltage within a small range can reduce reactive power consumption and increase the power transfer capability in resistive grids, which will be analyzed in the next section.