*3.1. Multiperiod Simulation and Scenario Selection*

In most of the approaches, the maximum grid hosting capacity is determined while analyzing the worst-case scenario or by selecting a few representative scenarios of DG production/load consumption. This approach usually does not cover all possible operation scenarios, nor does it account for the probability or duration of scenarios.

In the proposed approach, the objective is to maintain a normal system operating state for the entire set of operating states considered in the model. Therefore, the input data used model is generalized considering the possibility of obtaining multiple operating conditions based on demand/production variations. Furthermore, this analysis does not take into account dynamic reconfiguration but rather gives a unique static topology solution, which is applied for all considered operation scenario set.

The input data for this case study is the time-series data of network load and DG production, which accounts for the correlation between these relevant parameters. Using this raw data in the optimization problem would introduce a large number of variables, which would result in high

computational time. On the other hand, limiting the number of scenarios to only a few extreme cases does not account for the probability of a system operating state. The approach proposed here constructs blocks of scenarios using available time-series data based on a load/DG generation duration curves, which are constructed from available time-series data. Simulation is conducted based on real measured data related to wind power plant production, sun intensity, and network demand.

In the case study, we considered a few optimization models based on the level of the operational flexibilities, which are considered in the models. These submodels that do not consider all previously mentioned flexibilities could be easily derived from the proposed full optimization model by fixing the values of certain variables. Models implemented for comparison are defined as follows:


The initial parameters of the simulation are given in Table 1.



One of the most used network models for distribution grid analysis and simulation is the IEEE 33 bus test network. The network consists of 37 branches and 32 demand buses with one supply point. It contains 5 elementary loops and is presented in Table 2 [26]. The offline branches indicated in Table 2 are related to the initial network radial topology. In this case study, we assumed that all branches are available for network reconfiguration if they are part of the network elementary cycles.


The full meshed modified IEEE 33 bus test network, with all possible DG connection points, is shown in Figure 1. The maximum install capacity of all DG units (wind and solar) is 10 MW and for every DG unit, only one connection point can be realized.

**Figure 1.** IEEE 33 bus test network with potential distributed generation (DG) connection buses.

Figure 2 shows the wind power plant (WPP) and PV plant relative production for one year with an hourly resolution. This real time-series data together with network consumption data was used to construct a representative scenario set of operating scenarios that were used in the proposed model for maximization of network hosting capacity.

**Figure 2.** Wind power plant (WPP) and photovoltaic (PV) plant relative production.

The available time-series data related to network consumption and normalized DG production is represented by its load/production duration curves [27]. The load duration curve was used as a key for sorting DG production duration curves. In order to reduce the model computational burden, the load duration curve, solar power curve and wind power curve were divided into 4 demand/production blocks as shown in Figure 3. The demand variations within each demand block were approximated by a set of scenarios, namely, high, average, and low. The DG production duration curve associated with each demand block was also approximated with a set of 3 scenarios, namely, high, average, and low. This way, originally measured time-series data was approximated by jointly considering the demand and DG levels, which in turn results with a representative set of 36 operating scenarios: three demand levels by three DG levels by four demand blocks (Table 3). Using this clustering technique it is possible to maintain a correlation between network consumption and DG production data while significantly reducing the model computational burden. The number of levels, and blocks, can be differently defined to achieve a better approximation of original time-series data at the expense of computational time.


**Table 3.** Parameters for 36 operating scenarios considered in the analysis.

**Figure 3.** Load consumption—DG production duration curves divided into 4 blocks.

#### *3.2. Implementation of the Mathematical Model and Results Discussion*

To assess grid hosting capacity based on upper data settings, and provide adequate numerical results, we implemented a mixed-integer second-order cone programming (MISOCP) mathematical model described in the previous chapter in the general algebraic modeling system (GAMS) [28] and solved underlying problems using CPLEX software (IBM, Armonk, NY, USA) [29]. Tests were performed on a Windows machine equipped with an Intel Core i3 (2.27 GHz) processor and 4 GB of RAM. By implementing a mathematical model we obtained results that are presented in Table 4.


**Table 4.** Grid hosting capacity and optimal DG allocation and capacity results.

WPP—wind power plant, PVP—photovoltaic power plant.

We can see from the results shown in Table 4 that considered submodels produce different results related to optimal DG allocation and capacity.

The base model, which did not include any of the considered flexibilities and was simulated for 36 different operating scenarios, reached a total hosting level of 10.444 MW. According to the results of the base model, the total hosting capacity was distributed as follows: WPP1 optimal install capacity was 1.54 MW with connection to bus 15, WPP2 optimal install capacity was 4.019 MW with connection to bus 28, and PVP optimal install capacity was 4.884 MW with connection to bus 21. Further increment of DG power, in this case, would lead to violation of DN operational constraints (bus voltage constraints would be violated in certain operating scenarios). This total hosting capacity represents the referent value for comparison with other submodels.

Model "a" considers the possibility of a DG power factor control in the range cosϕ = 0.95 (leading/lagging), which leads to an increment of network hosting capacity to 12.935 MW with the highest increment of install power for WPP2 (optimal installed power increased by 1.274 MW). Model "b", in addition to DG power factor control, considers the possibility of OLTC voltage control. Based on this submodel the limits for DG penetration were additionally increased to 13.75 MW, with the highest increase of install capacity for WPP2. Furthermore, in the submodel "c" we included the possibility of network reconfiguration in addition to DG power factor and OLTC control. By redistributing power from DGs through optimal power network topology modification while maintaining radial network operation, further increase of network hosting capacity can be achieved. To increase network hosting capacity the model suggests topology modification by switching offline branches 9, 16, 21, 25, and 33. In this case, we reached the maximal level of DG integration equal to 14.272 MW, making this approach the most convenient for the maximization of network hosting capacity. This level of DG penetration represents an increment of 37% compared to the base model.

Total network active power losses for different operating scenarios are shown in Figure 4. The analysis shows an increase in power losses due to DG power penetration. It is interesting to note that network power losses were lower for submodel "c" in comparison with submodels "a" and "b" although total install DG capacity was higher. The reason for this is network topology optimization included in submodel "c", which not only did it increase network hosting capacity but it also reduced network active power losses.

**Figure 4.** Total power losses for different operating scenarios and submodels.

Figure 5 shows minimum/maximum bus voltage ranges for four different submodels and all considered operating scenarios. It is evident that the voltage level stayed within the limits in all operating scenarios. Moreover, the limiting factor for a further increment of network hosting capacity was visible from the figures and was directly related to the increase of the voltage in DG connection buses and adjacent grid at least in one scenario included in the simulation.

**Figure 5.** Voltage ranges for different operating scenarios and submodels.

Figure 6 shows a comparison of maximal line loadings for power lines '1–17 (maximal power rating 10 MVA) and in Figure 7 for lines '18–37 (maximal power rating 5 MVA) for all considered operating scenarios and four submodels. It is evident that upper power ratings of the lines were not surpassed but rather within the element power rating limits. Looking at the bus voltage results in parallel with the results of line loading it is obvious that in the base model main limitation factor for DG penetration was not connected with line overloading but rather to the voltage rise problem. Models "a", "b", and "c" on the other hand show limitations related both to voltage rise problems and line loading problems. In these models, upper loading limits of the power lines were reached making additional barriers along with the voltage level for further DG power penetration.

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**Figure 6.** Maximal power line loading (branches '1–17') for different operating scenarios and submodels.

**Figure 7.** Maximal power line loadings (branches '18–37') for different operating scenarios and submodels.

#### **4. Conclusions**

This paper proposed a MISOCP mathematical model for the multiperiod maximization of DGs penetration into the existing DN. The model includes different network flexibilities such as the DG power factor control, OLTC transformer control, and network topology reconfiguration. The proposed approach includes the discretization of demand and DG production duration curves and the construction of a representative operating scenario set. This approach accounts for the correlation between relevant system variables and at the same time considers fewer scenarios thus reducing the level of operating conditions to a level appropriate for mathematical modeling optimization. The proposed method was tested on a modified IEEE 33 bus test network. Modifications refer to three DGs added to the grid with a presumed maximum rated power and defined set of potential grid

connection points. Simulation analysis covered four different submodels and compared results related to maximum grid hosting capacity and optimal DG allocation and installed capacity. The model that includes network topological flexibility gave a maximal increment of grid hosting capacity in relation to the base submodel. The other submodels that did not include topological flexibility were affected with line overloading and hence allowed lower grid hosting capacity. Results show a significant increment in network hosting capacity in all three submodels compared to the base model, with the largest increase of grid hosting capacity in a model that considered network topology reconfiguration, DG power factor control, and OLTC voltage control with a 37% increase of grid hosting capacity in relation to the base case model.

**Author Contributions:** D.J. and R.C. defined the main concepts and implemented the proposed algorithms. ˇ All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
