## 2 Maximum Power Transfer Capability of Grid-Connected Systems at the Rated PCC Voltage  
### System Configuration and Mathematical Model  
Figure 1 shows the typical configuration of grid-connected wind and PV generation systems. Generally, a wind generation unit contains an AC/DC converter for maximum power point tracking (MPPT) control and a DC/AC inverter for grid connection. Similarly, a PV generation unit mainly contains a DC/DC converter for MPPT control and a DC/AC inverter for grid connection. In the entire wind or PV power plant, all these generation units are connected at the PCC. The total output power of inverters at the PCC is injected into the high-voltage transmission network connected to the main grid. To simplify the analysis, the transmission network is represented by a transmission line, as shown in Figure 1. In the following sections, this paper will mainly focus on the impact of transmission-line impedance on the maximum power transfer capability of the system.  
Grid-connected inverters usually operate in current-controlled mode, so the whole generation power plant can be treated as an ideal current source. Moreover, the grid can be represented by its Thevenin's equivalent circuit; the equivalent circuit of the grid-connected generation system is shown in Figure 2. Assuming the PCC voltage vector is aligned to the d-axis, \(V_{pcc}\angle 0\) is the vector of the PCC voltage, \(I_{inv}\angle\varphi_{i}\) is the vector of the inverter current, \(V_{g}\angle\varphi_{v}\) is the vector of the grid voltage, and \(Z_{g}\) is the grid impedance. It is noted that series-connected resistor \(R_{g}\) and inductor \(X_{g}\) are used to represent grid impedance \(Z_{g}\).  
The stiffness of the grid at the PCC can be described by the SCR, which is expressed as:  
\[SCR=\frac{S_{SC}}{S_{N}}=\frac{3/2\cdot{V_{g}}^{2}/\left|Z_{g}\right|}{3/2\cdot V _{pcc(rated)}\cdot I_{inv(rated)}} \tag{1}\]  
where \(S_{SC}\) is the short-circuit apparent power at the PCC, \(S_{N}\) is the total rated apparent power of inverters, \(V_{g}\) is the amplitude of grid voltage, \(V_{pcc(rated)}\) and \(I_{inv(rated)}\) are the amplitude of the rated PCC voltage and rated inverter current.  
Generally, the rated PCC voltage should be equal to the grid voltage, namely, \"\(V_{pcc(rated)}\) = \(V_{g}\)\", and the rated current \(I_{inv(rated)}\) is the same as the maximum inverter current \(I_{inv(max)}\), which is determined by the endurable current of the power semiconductor. Thus, the grid impedance can be represented by the SCR according to Equation (1), which is expressed as:  
\[\left|Z_{g}\right|=\sqrt{{R_{g}}^{2}+{X_{g}}^{2}}=\frac{V_{g}}{I_{inv(max)} \cdot SCR} \tag{2}\]  
According to the equivalent circuit in Figure 2, the relationship between the grid voltage and the inverter current can be derived as:  
\[V_{pcc}\angle 0=V_{g}\angle\varphi_{v}+(R_{g}+jX_{g})\cdot I_{inv}\angle\varphi_{i} \tag{3}\]  
To simplify the vector expression in Equation (3), the current vector \(I_{inv}\angle\varphi_{i}\) can be expressed by the d-axis component \"\(i_{d}=I_{inv}\cdot\cos(\varphi_{i})\)\" and the q-axis component \"\(i_{q}=I_{inv}\cdot\cos(\varphi_{i})\)\".  
Figure 1: Typical configuration of grid-connected wind and PV generation systems.  
Figure 2: Equivalent circuit of the grid-connected generation system.  
\(I_{inv}\cdot\sin(\varphi_{i})\)\". Thus, the relationship between the two current components and the PCC voltage can be derived as:  
\[({R_{g}}^{2}+{X_{g}}^{2})({i_{d}}^{2}+{i_{q}}^{2})+2V_{pcc}(X_{g}{i_{q}}-R_{g}{i_ {d}})+{V_{pcc}}^{2}-{V_{g}}^{2}=0 \tag{4}\]