## 4 Case Study  
To verify the feasibility and the effectiveness of the proposed VOP, numerical studies are carried out on the IEEE 123-bus system [31], shown in Figure 2. A 115/4.16 kV substation supplies a main feeder, branching off into several laterals and sublaterals, with unbalanced loads, voltage regulators, shunt capacitor banks, and switches. To apply the proposed approach, the test distribution system is changed into a three-phase balanced network. In particular, at each node, the unbalanced loads have been replaced by balanced loads with the same rated three-phase power, modeled as constant shunt impedances; for each line, the single-phase equivalent pi-model has been adopted, assuming, for its parameters, the average values of the three-phase model parameters. The total balanced rated load connected to the network is equal to about 3.49 MW and 1.17 MVAr. Furthermore, the action of the voltage regulators has been disabled, that is, constant ratios have been fixed, and the status of the switches has been assumed in the basic configuration.  
In addition, 15 DERs are connected to the grid at nodes 3, 9, 17, 2, 160, 105, 63, 112, 18, 25, 40, 51, 38, 77, and 87, as shown in Figure 2. In particular, seven DERs are DGs connected to the grid by inverters with a rectangular capability chart. DGs inject a fixed active power equal to 200 kW and a reactive power that vary in the range (\(-100\), \(100\)) kVAr, independently from the value of the active powers; in the absence of any optimization, DGs inject active power with a power factor equal to 1 that is with null reactive power. The remaining six DERs are DGs with battery energy storage systems: DGs generate a  
Figure 2: The IEEE 123-bus system including 15 DERs.  
fixed active power equal to 200 kW; the state-of-charge of the storage allows for varying the active power injected by DER in the range (100, 300) kW. In addition, in this case, the inverter is equipped with a rectangular capability chart that varies the reactive power in the range (\(-\)150, 150) kVAr; in the absence of any optimization, the injected active power is equal to 200 kW and no reactive power is injected or absorbed. The admissible range of variation of the active and reactive powers injected/absorbed by DERs are summarized in Table 1.  
Three different operational cases are considered by varying the amplitude of the nodal voltage at the HV busbar (slack bus) and the total load connected to the grid:  
* _Case 1_: the voltage amplitude of the HV busbar is fixed at 1.025 p.u.; the loads are equal to 100% of their rated values;
* _Case 2_: the voltage amplitude of the HV busbar is fixed at 1.03 p.u.; the loads are equal to 50% of their rated values;
* _Case 3_: the voltage amplitude of the HV busbar is fixed at 1.02 p.u.; the loads are equal to 120% of their rated values.  
The linearized DistFlow Equation (2) has been obtained by applying the method in [21] and assuming the same operating conditions as the ones in _Case 1_, as far as loads and slack bus are concerned, but disconnecting all the DERs. The obtained matrix \(\mathbf{\Gamma}\) is the only sensitivity matrix that has been used in all the VOP problems and in all the three considered cases.  
To test the proposed voltage optimization methods, numerical analysis was performed in MatLab environment through the use of the MatPower package, by a CPU Intel Core i7-2670 QM with 8 GB of RAM.