## 4 Three-Phase Inverters
### Droop Control
#### 4.5.2 Voltage Droop Control  
A primary objective of the voltage droop controller is to regulate the reactive power and voltage. To see this, consider a generator that is represented in steady state as a voltage source behind a series reactance, as described in Figure 29.  
The voltage droop control law is  
\[|E|=E_{ref}-k_{q}(Q-Q_{ref}), \tag{145}\]  
where \(E_{ref}\) is the reference voltage, \(Q_{ref}\) is the reference reactive power, and \(k_{q}\) is a constant. In a synchronous machine this control law is implemented in practice by adjusting the current in the field winding. The generator reactive power \(Q\) may be computed from the circuit in Figure 29, and is given by  
\[Q=\frac{|V|}{X}\left(|E|\cos(\delta)-|V|\right), \tag{146}\]  
and if \(\delta\to 0\) and \(|E|\approx|V|\), then  
\[Q=\frac{|E|}{X}\left(|E|-|V|\right). \tag{147}\]  
Figure 29: The generator is modeled in steady state as a voltage source behind a series reactance.  
This relationship between the reactive power and voltage is plotted in Figure 30.  
If the voltage \(|E|\) is fixed, variations in \(|V|\) may result in high and unpredictable reactive power flow. This is potentially dangerous, since high reactive power causes loss, and may lead to stability problems. To solve this, the droop mechanism regulates the reactive power by adjusting \(|E|\) in inverse proportion to \(Q\). Substitution of (145) in (147) yields  
\[Q\approx\frac{|E|}{X}\left(|E|-|V|\right)\approx\frac{E_{ref}}{X}\left(E_{ref} -k_{q}(Q-Q_{ref})-|V|\right), \tag{148}\]  
which may be written as  
\[Q=\frac{E_{ref}}{X+E_{ref}k_{q}}\left(E_{ref}+k_{q}Q_{ref}-|V|\right). \tag{149}\]  
This equation is illustrated in Figure 31.  
In steady state, the controller operates as follows:  
In case \(k_{q}=0\) (no droop control):  
* The voltage \(|V|\) varies in a small range. The generator provides reactive power as needed to maintain a certain voltage amplitude. As \(|V|\) decreases, the reactive power increases to compensate.
* The reactive power varies.
* The generator operates as a voltage source, or like a grid forming inverter.  
In case \(k_{q}\to\infty\):  
Figure 31: Droop characteristics in steady state: reactive power as a function of voltage.  
Figure 30: Reactive power as a function of voltage. Variations in \(|V|\) may result in high and unpredictable reactive power flow.  
* The reactive power is regulated, \(Q\approx Q_{ref}\).
* The voltage \(|V|\) varies.
* The generator operates as a power source, or like a grid feeding inverter.  
Middle values of \(k_{q}\):  
* Combine the properties of these two extreme cases.
* The reactive power is regulated, but is not constant.
* The voltage \(|V|\) is somewhat regulated. The generator provides reactive power to maintain a stable voltage.
* As we shall see, the generator operates like a grid supporting inverter.  
An additional objective of the voltage droop controller is to share the reactive power among the generators. As an example, consider the system in Figure 32, which is described by the following set of equations:  
\[Q_{1} \approx\frac{|E_{1}|}{X}\left(|E_{1}|-|V|\right), Q_{2} \approx\frac{|E_{2}|}{X}\left(|E_{2}|-|V|\right),\] \[|E_{1}| =E_{ref,1}-k_{q,1}\left(Q_{1}-Q_{ref,1}\right), |E_{2}|=E_{ref,2}-k_{q,2}\left(Q_{2}-Q_{ref,2}\right), \tag{150}\] \[Q_{L} =Q_{1}+Q_{2}.\]  
Assume first that there is no droop control (\(k_{q,1}=k_{q,2}=0\)), and \(Q_{L}=0\). The resulting reactive powers in this case are  
\[Q_{1}=-Q_{2}=\frac{|E_{1}||E_{2}|}{X}\frac{|E_{1}|-|E_{2}|}{|E_{1}|+|E_{2}|}. \tag{151}\]  
Here, even though \(Q_{L}=0\), there may be significant reactive power flow between the two generators. Now assume that the droop controller is active, and \(k_{q,1}=k_{q,2}=k_{q}\), \(E_{ref,1}=E_{ref,2}=E_{ref}\), \(Q_{ref,1}=Q_{ref,2}=Q_{ref}\). The resulting reactive powers in this case are \(Q_{1}=Q_{2}=\frac{Q_{L}}{2}\). The controller prevents circulation of reactive power, and shares the reactive power evenly between the two generators.