## 2 Theoretical Behavior of Distributed versus Concentrated Volt/var Local Control Strategy
### Theoretical Background  
Figure 3 shows an overview of a feeder segment with feedback active power supply. Figure 3a shows its equivalent circuit. The feeder segment is modeled through its series impedance, while the parallel impedance is neglected because it plays only a small role in voltage regulation, since the nominal voltage of LVGs is very small. In a typical LV feeder with large-scale PV penetration, active power \(P\) is supplied back to the medium-voltage grid (MVG), while reactive power \(Q\) is withdrawn from inverters to maintain the voltage below the upper limit. Figure 3b shows the corresponding phasor diagram. The reference is the phasor voltage \(\overline{U}_{1}\), i.e., \(\overline{U}_{1}=U_{1}\).  
The impedance, \(\overline{Z}\), which carries the current of the prosumer, causes the voltage drop as in:  
\[\Delta\overline{U}_{1}=\overline{U}_{1}-\overline{U}_{2}=\overline{Z}. \overline{I}, \tag{1}\]  
where \(\Delta\overline{U}_{1}\)--phasor voltage drop; \(\overline{U}_{1}\)--phasor voltage at the load bus or at the end of the feeder segment; \(\overline{U}_{2}\)--phasor voltage at the head of the feeder segment; \(\overline{Z}\) = R + jX--feeder segment series impedance; \(\overline{I}\)--phasor current.  
The apparent power of the prosumer is calculated as in:  
\[\overline{S_{1}}=\overline{U}_{1}.\overline{I}^{*}=P_{1}-jQ_{1}, \tag{2}\]  
where \(\overline{S_{1}}\)--phasor apparent power of the prosumer; \(P_{1}\)--active power of the prosumer; \(Q_{1}\)--reactive power of the prosumer.  
Since the load current is expressed as in:  
\[\overline{I}=\frac{P_{1}+jQ_{1}}{U_{1}}. \tag{3}\]  
Figure 3: Overview of a feeder segment with back-up active power supply: (**a**) Equivalent circuit; (**b**) Phasor diagram.  
Figure 2: Overview of the concentrated \(L(U)\) local control strategy or rather concentrated var-sink.  
Substituting \(\overline{I}\) into Equation (2) results in:  
\[\Delta\overline{U}_{1}=(\mathrm{R}+j\mathrm{X})\cdot\frac{P_{1}+jQ_{1}}{U_{1}}= \frac{P_{1}\cdot R-Q_{1}\cdot X}{U_{1}}+j\frac{P_{1}\cdot X+Q_{1}\cdot R}{U_{1}}= \Delta U_{R}+j\Delta U_{X}, \tag{4}\]  
where \(\Delta U_{R}\)--real part of the voltage drop; \(\Delta U_{X}\)--imaginary part of the voltage drop.  
As illustrated in Figure 3b, the voltage drop through the feeder segment has two components: one in phase (\(\Delta V_{R}\)) modifies the voltage amplitude, and the other one in phase quadrature (\(\Delta V_{X}\)) rotates the voltage vector. The latter has no significant impact on the amplitude of the voltage drop; therefore, \(\Delta V_{X}\) is neglected. Under these assumptions, the amplitude of voltage drop is expressed as:  
\[\Delta U_{1}=\Delta U_{R}=\frac{P_{1}\cdot R-Q_{1}\cdot X}{U_{1}}. \tag{5}\]  
Although \(P\) has the most significant influence on the voltage drop amplitude on the feeder, the purpose of this chapter is to investigate only the impact of the reactive power on it. Therefore, \(P\) is neglected, and Equation (5) is simplified as follows:  
\[\Delta U_{1}=-\frac{Q_{1}\cdot X}{U_{1}}. \tag{6}\]  
Since the \(X\) of the feeder segment is very small, reactive losses are neglected (\(\Delta Q\approx 0\)), and the feeder segment flows fulfill the equation:  
\[Q_{1}=Q_{2}. \tag{7}\]  
Based on Equations (6) and (7), an equation is derived below to calculate the voltage drop at a feeder with distributed var-sinks.