## 4 Three-Phase Inverters
### Droop Control
#### 4.5.1 Frequency Droop Control  
Consider a lossless generator with a single-phase active power \(P\) and a frequency \(\omega\). At steady state, the droop control law is  
\[3P=3P_{ref}-\frac{1}{D}(\omega-\omega_{s}), \tag{136}\]  
where \(P_{ref}\) is the reference power, \(D\) is the damping constant, and \(\omega_{s}\) is the nominal grid frequency (a constant). A primary objective of the frequency droop controller is to stabilize the grid frequency, by regulating the active power. Consider first a synchronous generator, in which the mechanical power is  
\[p_{m}=3P_{ref}-\frac{1}{D}(\omega-\omega_{s}). \tag{137}\]  
As the frequency increases, the mechanical power decreases, and vice versa. This behavior regulates the generated power, and matches it to the actual load, as described in Figure 27.  
As an example, assume that a single synchronous generator is feeding a load with active power \(P_{L}(t)\). The angular acceleration of the rotor is given by  
\[\frac{\mathrm{d}}{\mathrm{d}t}\omega=K(p_{m}(t)-3P_{L}(t)), \tag{138}\]  
where \(K\) is the swing equation constant. With no droop control (\(D=\infty\)),  
\[\frac{\mathrm{d}}{\mathrm{d}t}\omega=3K(P_{ref}-P_{L}(t)), \tag{139}\]  
Figure 27: Conceptual operation of the frequency droop mechanism. and since in general \(P_{ref}\
eq P_{L}(t)\), the frequency \(\omega\) is unstable. On the other hand, when the droop control mechanism is active (finite \(D\)), the resulting dynamics equation is  
\[\frac{\mathrm{d}}{\mathrm{d}t}\omega=K\left(3P_{ref}-\frac{1}{D}(\omega-\omega_ {\mathrm{s}})-3P_{L}(t)\right)=-\frac{K}{D}\omega+K\left(3P_{ref}-3P_{L}(t)+ \frac{\omega_{\mathrm{s}}}{D}\right). \tag{140}\]  
The term \(-(K/D)\omega\) provides negative feedback, and the frequency \(\omega\) is stabilized. The relationship between the active power and frequency at steady state is illustrated in Figure 28.  
The controller operates as follows:  
In case \(D\to 0\):  
* The frequency \(\omega\) is constant such that \(\omega\approx\omega_{\mathrm{s}}\). The generator provides active power as needed to stabilize the frequency. * The active power varies. * The generator operates like a grid forming inverter, or an infinite bus. In case \(D\to\infty\):  
* The active power is constant, \(P\approx P_{ref}\). * The frequency \(\omega\) varies. * The generator operates as a power source, like a grid feeding inverter. Middle values of \(D\):  
* Combine the properties of these two extreme cases. * The active power is regulated, but is not constant. The generator provides variable active power to adjust the frequency. * The frequency \(\omega\) is regulated, but is not constant. * As we shall see, the generator operates like a grid supporting inverter. An additional objective of the controller is to share the active power among the generators. Assume a system with \(N\) generators, such that in steady state, \(3P_{i}=3P_{ref,i}-\frac{1}{D_{i}}(\omega-\omega_{\mathrm{s}})\). Also assume that the total load in the system is \(P_{L}\), such that \(\sum_{i=1}^{N}P_{i}=P_{L}\). These equations may be written as  
\[ \begin{bmatrix}3D_{1}&&0&1\\ &\ddots&&\vdots\\ 0&&3D_{N}&1\\ 1&\cdots&1&0\end{bmatrix}\begin{bmatrix}P_{1}\\ \vdots\\ P_{N}\\ \omega-\omega_{\mathrm{s}}\end{bmatrix} = \begin{bmatrix}3D_{1}P_{ref,1}\\ \vdots\\ 3D_{N}P_{ref,N}\\ P_{L}\end{bmatrix} , \tag{141}\]  
Figure 28: Droop characteristics in steady state: active power as a function of frequency. and the solution is  
\[ \begin{split}\omega-\omega_{s}&=-3\left(\sum_{i=1}^{N} \frac{1}{D_{i}}\right)^{-1}\left(P_{L}-\sum_{i=1}^{N}P_{ref,i}\right),\\ P_{i}&=P_{ref,i}-\frac{1}{3D_{i}}(\omega-\omega_{s}).\end{split}  \tag{142}\]  
As an example, for two generators in steady state:  
\[ \begin{split}\omega&=\omega_{s}-\frac{3D_{1}D_{2}}{ D_{1}+D_{2}}\left(P_{L}-P_{ref,1}-P_{ref,2}\right),\\ P_{1}&=P_{ref,1}-\frac{1}{3D_{1}}(\omega-\omega_{s }),\\ P_{2}&=P_{ref,2}-\frac{1}{3D_{2}}(\omega-\omega_{s }).\end{split}