## 4 Problem Formulation
### Test Network Model  
A tested network model of a real radial LV distribution network, shown in Figure 3, has been modeled in DigSILENT Power Factory. The modeled test network is used to simulate an actual rural distribution feeder with a total length of 870 m, consisting of a 170 m section of PPOOA 4 \(\times\) 120 mm\({}^{2}\) cable and a 700 m section of overhead line FR-N1 XD9-AR 3 \(\times\) 70 + 70 + 2 \(\times\) 16 mm\({}^{2}\). A total of 74 different consumers are connected to the tested network. The consumer models represented as loads in this study are generally divided into uncontrollable and controllable loads. Uncontrollable loads are modeled as households with constant and uninterrupted electricity supply whose consumption cannot be actively managed or modified. Real measurement data from four different types of households, considering their characteristics and connection methods, are implemented in uncontrollable loads models. The controllable load model allows for shifting consumption to a desired time period, enabling load management. Figure 3: Model of test network. An EV home charging station has been taken as a controllable load model in the test network. This approach enhances grid flexibility and allows for determining the maximum charging power available for EVs. Two controllable load models have been used in test networks, depending on their connection type (three-phase or single-phase loads). The test network uses three-phase and single-phase PV power plants as the DGs. The DG in this paper does not use parameters such as solar irradiance, ambient temperature, orientation, or PV panel tilt angle as a direct input variable. Instead, the parameters mentioned are included within real measured data conducted across various types of PV plants. Two PV system models have been developed, each with corresponding inverters: a single-phase model with an installed power of 3.2 kWp and a three-phase model with an installed power of 6 kWp. Uniform continuous distribution determines the layout of loads (uncontrollable and controllable) and sources across the test network. This approach ensures that load locations are assigned uniformly, enabling a balanced representation of demand points across the feeder. A total of 74 uncontrollable loads (households), 8 controllable loads (home EV charging stations), and 14 DGs (PV power plants) are modeled within the test network. The results of the uniform continuous distribution, representing the connection points (buses) of these consumers and DGs, are presented in Table 2.  
The distance between consumers is defined by the mutual distance calculated as the total feeder length (cable or overhead line) ratio to the number of consumers connected along each section. Consequently, consumers connected to the substation by cable are 14 m apart, while those connected by overhead line are 11 m apart. The distance between the substation and the first bus (i.e., the first consumer) is 16 m. The radial feeder's branching is designed to increase the feeder's complexity and broaden the search space for the optimization method. A double-tariff electricity pricing system is also considered while modeling the test network. The higher tariff is active from 8:00 a.m. to 10:00 p.m., while the lower tariff is active from 10:00 p.m. to 8:00 a.m.  
Three specific terminals have been observed. The first characteristic terminal, marked as KTS 16 (0.4 kV), stands for the connection point of the radial network to the transformer. This terminal is considered a PCC where voltage conditions are monitored according to the EN 50160 standard, which defines voltage characteristics and limitations for the PCC. The second characteristic terminal is terminal 12, marked as Terminal_12 in Figure 3. It represents the point where the cable transitions into an overhead line, where the conductor cross-section is reduced. The point marked as Terminal_31 is the last characteristic terminal, and it is considered a point where multiple overhead lines are branching.