## 4 Three-Phase Inverters
### Grid Supporting Inverters  
Grid supporting inverters deliver power to the grid, while contributing to the stability and reliability of the system, similarly to synchronous generators. This is achieved by means of a droop control mechanism, that maintains an inverse relationship between \(\omega\) and \(P\), and between \(|E|\) and \(Q\). A basic design is shown in Figure 33.  
Figure 32: Example: Reactive power sharing between two generators. Grid supporting inverters may be described as grid forming inverters with an additional droop control mechanism:  
\[ \begin{array}{c}\omega^{*}=\omega_{s}+3D(P_{ref}-P),\\ \\ \frac{u_{d}^{*}}{\sqrt{2}}=|E|=E_{ref}+k_{q}(Q_{ref}-Q),\end{array}  \tag{152}\]  
where \(\omega_{s}\) is the nominal grid frequency, \(\omega_{s}=2\pi\)50 rad/s or \(\omega_{s}=2\pi\)60 rad/s. Such inverters may be viewed as a combination of grid forming and grid feeding inverters, and the exact balance between these two modes of operation depends on the droop parameters \(D\) and \(k_{q}\). For instance:  
* If \(D=0\), then the frequency is constant \(\omega^{*}=\omega_{s}\), as in grid forming inverters. * If \(D\rightarrow\infty\), then the active power is constant \(P=P_{ref}\), as in grid feeding inverters. * If \(k_{q}=0\), then the voltage amplitude is constant \(|E|=E_{ref}\), as in grid forming inverters. * If \(k_{q}\rightarrow\infty\), then the reactive power is constant \(Q=Q_{ref}\), as in grid feeding inverters. * Middle values: the inverter supports the grid by regulating the active power, reactive power, frequency and voltage. As an example, consider a grid supporting inverter connected to an infinite bus with a frequency \(\omega_{g}\), where \(\omega_{g}\
eq\omega_{s}\). We will use this example to demonstrate the typical dynamics of grid supporting inverters, and specifically to explain how the inverter synchronizes to the grid. To simplify, it is assumed that \(k_{q}=0\) (no voltage droop), and that the infinite bus voltage is \(|E|=E_{ref}\) (RMS). Intuitively, if \(\omega^{*}>\omega_{g}\) the voltage angle at the inverter output gradually increases, which increases the active power \(P\), and decreases \(\omega^{*}\). On the other hand, if \(\omega^{*}<\omega_{g}\) the voltage angle gradually decreases, which decreases the active power \(P\), and increases \(\omega^{*}\). If this dynamic process is stable, then in steady state \(\omega^{*}=\omega_{g}\), and the inverter is synchronized to the grid. This dynamic process is modeled as follows. The voltages at the output of the inverter stage are given by \(\tilde{u}_{d}=\sqrt{2}|E|\), \(\tilde{u}_{q}=0\), \(\tilde{u}_{0}=0\), where the reference angle of these \(dq0\) signals is \(\theta(t)\). In addition, the voltages of the infinite bus are \(v_{d}=\sqrt{2}|E|\), \(v_{q}=0\), \(v_{0}=0\) with a reference angle \(\omega_{g}t\). To represent  
Figure 33: Grid supporting inverter operating as a voltage source (conceptual control scheme). the signals in the same reference frame, the voltages \(\tilde{u}_{d}\), \(\tilde{u}_{q}\), \(\tilde{u}_{0}\) are transformed to the reference frame of the infinite bus. The result is  
\[ \begin{bmatrix}u_{d}\\ u_{q}\\ u_{0}\end{bmatrix}  = \begin{bmatrix}\cos(\theta-\omega_{g}t)&-\sin(\theta-\omega_{g}t)&0 \\ \sin(\theta-\omega_{g}t)&\cos(\theta-\omega_{g}t)&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}\sqrt{2}|E|\\ 0\\ 0\end{bmatrix}  \tag{153}\] \[= \begin{bmatrix}\sqrt{2}|E|\cos(\theta-\omega_{g}t)\\ \sqrt{2}|E|\sin(\theta-\omega_{g}t)\\ 0\end{bmatrix} = \begin{bmatrix}\sqrt{2}|E|\cos(\delta)\\ \sqrt{2}|E|\sin(\delta)\\ 0\end{bmatrix} ,\]  
where the angle \(\delta\) is defined as \(\delta=\theta-\omega_{g}t\). To simplify the dynamic equations, we represent the system using time-varying phasors, and estimate the active power \(P\) based on the DC power flow approximation. The two phasors are  
\[ \begin{split} U(t)&=\frac{1}{\sqrt{2}}\left(u_{d}+ju_{q} \right)&=|E|(\cos(\delta)+j\sin(\delta))=|E|\angle\delta,\\ V&=\frac{1}{\sqrt{2}}\left(v_{d}+jv_{q}\right)&=|E| \angle 0^{\circ},\end{split}