## 4 Three-Phase Inverters
### Grid Forming Inverters  
A basic control scheme for grid forming inverters is shown Figure 19. \begin{table}
\begin{tabular}{c c l} \hline Grid Forming & \(\bullet\) & Also called Voltage Source Inverters. \\  & \(\bullet\) & The inverter operates as a voltage source. The voltage amplitude \(|E|\) and the frequency \(\omega\) are directly controlled. \\  & \(\bullet\) & The active power \(P\) and reactive power \(Q\) are determined by the interaction of the inverter with the grid. \\  & \(\bullet\) & The inverter usually cannot operate in parallel to other grid forming inverters, since the frequency \(\omega\) is constant. \\  & \(\bullet\) & Typical applications are standby UPS systems and isolated small networks. \\  & \(\bullet\) & In power flow studies: represented as a reference bus (slack bus), with a constant voltage amplitude \(|E|\) and angle \(\delta=0\). \\ \hline Grid Feeding & \(\bullet\) & Also called Grid Following Inverters, or inverters with P-Q control. \\  & \(\bullet\) & The inverter operates as a power source. The active power \(P\) and reactive power \(Q\) are directly controlled. \\  & \(\bullet\) & The frequency \(\omega\) and voltage amplitude \(|E|\) are determined by the interaction of the inverter with the grid. \\  & \(\bullet\) & Suitable for parallel operation. \\  & \(\bullet\) & Cannot operate in isolation. The system must include other generators (e-g., other inverters, synchronous machines) that control the voltage amplitude and frequency. \\  & \(\bullet\) & Typical applications are renewable energy systems, and distributed generation systems. \\  & \(\bullet\) & In power flow studies: represented as a P-Q bus, with constant active power \(P\) and reactive power \(Q\). \\ \hline Grid Supporting & \(\bullet\) & Delivers power to the grid, while promoting stability and reliability. Regulates the frequency and voltage, and balances the active and reactive power generation. \\  & \(\bullet\) & Implements a linear relationship between \(P\) and \(\omega\), and between \(Q\) and \(|E|\). \\  & \(\bullet\) & Suitable for parallel operation. \\  & \(\bullet\) & Suitable for isolated operation. \\  & \(\bullet\) & Combines well with energy storage systems, and online UPS systems. \\  & \(\bullet\) & In power flow studies: in general, cannot be represented as a standard bus. However, if the frequency \(\omega\) is known, and the voltage droop mechanism can be ignored, then the inverter is represented as a P-V bus with constant active power and voltage amplitude \(|E|\), similarly to a synchronous machine. \\ \hline \end{tabular}
\end{table} 
Table 2: Typical operation modes of grid-connected inverters. Figure 19: A basic control scheme for grid forming inverters. The main inverter stage often consists of three Buck converters connected in parallel to a DC source, as shown in Figure 20. Assuming these converters are lossless and store no energy, the average voltages \(u_{a}(t)\), \(u_{b}(t)\), \(u_{c}(t)\) are given by  
\[u_{a}(t) =V_{dc}d_{a}(t), \tag{125}\] \[u_{b}(t) =V_{dc}d_{b}(t),\] \[u_{c}(t) =V_{dc}d_{c}(t).\]  
Note that the duty cycles are defined in the range \([-1,1]\) such that \(-V_{dc}\leq u_{x}(t)\leq+V_{dc}\). The LC filter at the output removes the switching harmonics, and delivers sinusoidal voltages and currents to the grid. Grid forming inverters are controlled as voltage sources with a constant frequency. A possible control law is  
\[d_{d}=V_{dc}^{-1}u_{d}^{*},\;d_{q}=0,\;d_{0}=0, \tag{126}\]  
where \(u_{d}^{*}\) is the desired voltage amplitude. In addition, the reference angle for the \(dq0\) transformation is \(\theta(t)=\omega^{*}t\), where \(\omega^{*}\) is the desired frequency. The resulting output voltage is  
\[u_{d}=u_{d}^{*},\;u_{q}=0,\;u_{0}=0 \tag{127}\]  
with a reference angle \(\omega^{*}t\). Therefore, the grid forming inverter implements a voltage source with constant frequency and amplitude. In practical designs, the inverter currents should be controlled to improve the dynamic performance and efficiency, and to avoid over currents.