# Research on Optimal Voltage Control of Distribution Network with the Participation of EVs and PVs
## 2 Materials and Methods
### Grid Voltage Support Capability of PV Systems
#### 2.2.2 The Volt/Var Control Scheme of PV Systems  
PV systems are connected to distribution networks via power electronic converters, which can real-time monitor and control the current injected into the grid. By adequately adjusting the control circuits, PV inverters can provide a bidirectional reactive power exchange between the device and the distribution grid. This makes them ideal reactive power support resources for the voltage regulation of distribution networks. As PV operating power fluctuates with the varying solar irradiations absorbed by the PV panels and the operating power factor of PV systems has to be above a specific threshold value (e.g., 0.9),  
Figure 3: The typical configuration of a distribution network feeder with EVs and PVs connected.  
the reactive power control capability of a PV inverter at time instance \(t\) is determined by its operating power \(P_{t}\), its minimum and maximum operating power \(P_{min}\), and \(P_{max}\) and its minimum power factor \(pf_{min}\) (corresponds to a maximum phase angle difference between grid voltage and grid current, \(\theta_{max}\)). As illustrated in Figure 5, the maximum reactive power can be achieved when \(P_{t}\) equals \(S_{t}\)cos(\(\theta_{max}\)) (denoted as \(P_{s}\) and \(S_{r}\) is the rated power of the PV system). When \(P_{t}\) is greater than \(P_{s}\), the maximum achievable reactive power, \(Q_{max,t}\), equals \(\sqrt{S_{r}^{2}-P_{t}^{2}}\) (illustrated by the yellow triangle in Figure 5). When \(P_{t}\) is less than \(P_{s}\), the maximum reactive power equals \(P_{t}\)tan(\(\theta_{max}\)) to ensure the power factor, \(pf_{t}\), is no less than \(pf_{min}\). To sum up, the maximum achievable reactive power at time instance \(t\), \(Q_{max,t}\), equals \(\min\biggl{\{}\sqrt{S_{r}^{2}-P_{t}^{2}},P_{t}tan(\theta_{max})\biggr{\}}\) when \(P_{t}\) is within the range of [\(P_{min}\), \(P_{max}\)]. Equivalently, the minimum power factor at time instance \(t\), \(pf_{min,t}\), equals \(\max\{P_{t}/S_{r},cos(\theta_{max})\}\).  
A voltage droop control is typically applied to establish the connection between the reactive power and the voltage of the grid integration point, as illustrated in Figure 6. To avoid the frequent voltage adjustment near the reference voltage, \(V_{0}\), a deadband for the voltage control (i.e., the region marked by D) is utilized. The overall voltage control can be divided into three regions depending on the grid-side voltage level of the PV system, \(V_{t}\). When \(V_{t}\) is within the range of [\(V_{0}-D/2\), \(V_{0}+D/2\)], the PV system operates at the unity power factor. When \(V_{t}\) is within the scope of [\(V_{t}\), \(V_{0}-D/2\)] \(\cup\) [\(V_{u}\), \(V_{0}+D/2\)] (\(V_{1}\) and \(V_{u}\) are the lower and upper voltage limits declared by distribution network operators), the reactive power \(Q_{t}\) varies linearly with the change of \(V_{t}\). When \(V_{t}\) is below \(V_{t}\) or above \(V_{u}\), the PV system operates at its maximum reactive power (and hence the lowest power factor).  
Figure 4: Typical PV generation files under a sunny day and a cloudy day.  
Figure 5: The reactive power regulation range of PV inverters.  
Accordingly, the mathematical representations of the voltage control schemes under the voltage droop control can be derived, as in (7).  
\[Q_{t}=\left\{\begin{array}{c}Q_{max,t},V_{t}\leq V_{t}\\ \frac{V_{0}-V_{t}-\frac{D}{2}}{V_{0}-\frac{D}{2}-0.9}Q_{max,t},V_{l}<V_{t}\leq V _{0}-\frac{D}{2}\\ 0,V_{0}-\frac{D}{2}<V_{t}\leq V_{0}+\frac{D}{2}\\ -\frac{V_{t}-V_{0}-\frac{D}{2}}{1.1-V_{0}-\frac{D}{2}}Q_{max,t},V_{0}+\frac{D} {2}<V_{t}\leq V_{u}\\ \hskip 142.26378 pt-Q_{max,t},V_{t}>V_{u}\end{array}\right. \tag{7}\]  
where \(Q_{t}\) refers to the reactive power of the PV system at time instance \(t\), \(Q_{max,t}\) is the maximum reactive power that the PV system can provide, D is the control deadband, \(V_{0}\) and \(V_{t}\) are the per-unit nominal voltage and the grid-side supply voltage of a PV system at time instance \(t\), and eps refers to the precision of floating numbers.