## 3 Case Study
The energy-saving, \(ES\), in the analyzed period has been calculated with the formula:  
\[ES(MWh)=\sum\limits_{h=1}^{T_{H}}F^{*}{}_{Loss,h}-\sum\limits_{h=1}^{T_{H}}F^ {**}{}_{Loss,h};\quad ES(\%)=\frac{\sum\limits_{h=1}^{T_{H}}F^{*}{}_{Loss,h}- \sum\limits_{h=1}^{T_{H}}F^{**}{}_{Loss,h}}{\sum\limits_{h=1}^{T_{H}}F^{*}{}_{ Loss,h}}\cdot 100 \tag{32}\]  
Three time intervals resulted in the optimization process of the tap positions, namely: (1-7,8-21), and (22-24). Thus, transformer \(T_{(1,2)}\) has the tap changer mainly at position 8 in the first time interval (6 out of 7 h), at position 10 in the second time interval (13 out of 14 h), and at position 9 in the third time interval. The last transformers, \(T_{(5,6)}\) and \(T_{(8,9)}\)  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Hour**} & \multirow{3}{*}{\(F^{*}{}_{Loss,h}\)} & \multirow{3}{*}{\(F^{**}{Loss,h}\)} & \multirow{3}{*}{\(\Delta F_{Loss}\)} & \multirow{3}{*}{\(\Delta F_{Loss}\)} & \multirow{3}{*}{\(\Delta F_{Loss}\)} \\  & & & & & & \(T_{(1,2)}\) & \(T_{(5,6)}\) & \(T_{(8,9)}\) \\  & & & & & & **(Tap * = 15)**  
have the same positions of the tap changer most of the time, namely: position 7 in the first interval (7 out of 8 h), position 9 in the second interval (12 out of 14 h), and position 8 in the third interval, see Figure 8. The number of turns for the tap changers of transformers \(T_{(1,2)}\) and \(T_{(5,6)}\) is 6 and for transformer \(T_{(8,9)}\) is 8 in the analyzed period to keep the voltage at the level of each bus within an optimal range corresponding to minimum power losses in the network. Figure 9 shows the voltage variation at the 110 kV buses in the base scenario 50, without and with coordinated control of the OLTCs. Table 10 from Appendix A presents the hourly values of the voltage at each bus, without and with coordinated control of the OLTCs. A statistical analysis based on a boxplot has been considered to highlight details on the voltage variations in the network. The boxplot includes the most significant statistical features of the voltage, providing a better understanding and comparison for each bus and whole network [43]. The boxplot graphically represents five values of distribution regarding the minimum value, \(Q0\), the first quartile, \(Q1\) (or lower quartile), the median, \(Q2\), the third quartile, \(Q3\) (or upper quartile), and the maximum value, \(Q4\)[44], see Figure 10a. The median voltage at the buses (drawn with a red line inside each box), without coordinated control of the OLTCs, has a significant multi-step decrease identifying four groups: _Group_ 1 (Buses 2), with \(Q2\) = 1.091; _Group_ 2 (Buses 3, 4, and 6) with \(Q2\) between 1.078 and 1.082; _Group_ 3 (Bus 7) with \(Q2\) = 1.063, and _Group_ 4 (Buses 8 and 10) with \(Q2\) = 1.048. The interquartile ranges are reasonably similar for all buses, as seen by the lengths of the boxes represented  
Figure 8: The radar display of the tap positions resulted from the optimization process. Figure 9: The variation of the voltage at the 110 kV buses in the analyzed period (24 h), base scenario 50.  
with the blue color in the figure (lower and upper sides of the box, named quartiles \(Q1\) and \(Q3\)). The voltage distribution is positioned towards the upper side of the box for all buses, observable through longer whiskers and larger areas over the median, quartile \(Q2\) (red line from inside the box), see Figure 10a. Regarding the coordinated control of the OLTCs, the median voltage (Q2) at buses is closer, with a reduced difference of about 0.017 between _Group_ 1 (Bus 2, \(Q2=1.051\)) and _Group_ 4 (Buses 8 and 10, \(Q2=1.034\)). The interquartile ranges are similar for all buses, as seen by the lengths of the boxes, except Bus 2, which is smaller, see Figure 10b. The voltage distribution is positioned towards the lower side of the box for all buses, observable through longer whiskers and larger areas under the median, quartile Q2.