**5. Case Study Data**

This section describes the data used for the simulations of the problem explained in Section 3. The Benders' decomposition method is applied to a 34-bus three-phase radial feeder [21], which has 1 substation, 29 buses with load, and 5 buses with no load. The topology of the system is shown in Figure 2. The data are based on those provided in [18] and [21], but adding new scenarios. In [18] there are 3 levels of demand, wind production, and solar production per each of the 8 time blocks. Now, there are 4 levels and 12 time blocks.

**Figure 2.** Distribution system under study.

The wind turbines chosen have a capacity of 100 kW and the PV modules have power of 2.5 kW. The candidate buses for each technology are shown in Figure 2 and the maximum power (wind turbine and PV modules) that may be installed at each bus is 250 kW. The maximum number of units is limited to 2 for wind turbines and to 85 for PV modules, and the substation can be expanded adding transformers of sizes ranging from 1 MVA up to a maximum of 5 MVA. Demand increases 2% each year along the planning horizon (20 years). The voltage data of the distribution system is 1 pu in the substation node and the minimum and maximum allowable voltage values are 0.95 pu and 1.05 pu, respectively. The values defined for the interest and discount rates are 8% and 12.5% [6], respectively. Investment data of new devices are shown in Table 2.


**Table 2.** Investment data.

The operation and maintenance costs of the new renewable production units are €0.007/kWh [8]. The annual budget is €150,000 and the maximum portfolio investment for the life time of the devices is €5,500,000. Data on demand, wind speed, and PV factors used per time block are shown in Table 3, where wind and PV production levels are also displayed. The different levels are combined with each other in every time block. There are four levels of demand, wind, and PV production, hence, the total number combinations (scenarios) per time block is 64. Hence, there are 12 time blocks, 6 per season (winter (October–March) and summer (April–September)), accounting for 768 scenarios. Note that energy prices are not used in scenario generation and each price corresponds with a defined demand factor. These prices are shown in Table 3 and increase 1% each year with respect to the base year. All the levels in each time block have a probability of 1/4. Therefore, the weight of each of the scenarios within each time block is 1/64 [22]. Two blocks are used for piecewise linearization, where the cost of energy not supplied is €15,000/MWh. The tolerance ε of the Benders' algorithm is specified as €1.


**Table 3.** Investment data.


**Table 3.** *Cont.*

#### **6. Results Discussion**

Two case studies have been simulated to test the model, where the constraints related to installed power and limits of investment are different. The results of Benders' algorithm for each case study case are compared with the MILP model given in Equations (1)–(56). These results are obtained using CPLEX 11 under GAMS [23] on an Intel Xeon E7-4820 computer with 4 processors at 2 GHz and 128 GB of RAM. Table 4 presents the numerical results of each simulation in each study case for 768 scenarios. The relative gap is set to 0.01 in all simulations for Benders' simulation.



• Case a. Investment limits included: This case represents the most realistic scenario all constraints of the model are taken in account. The first year of the time horizon the renewable technology chosen for investment is photovoltaic (PV), with an installed capacity of 1047.5 kW (see Table 5). The expansion of the substation is carried out in the ninth year, with a new transformer. The new PV devices are installed at the end of the branches because it reduces the costs associated with energies losses. The location of the PV modules within the network is displayed in Table 6. The CPU time decreases by 97.9% when Benders' decomposition is the method chosen for the simulations.




**Table 6.** Nodal allocation of new power installed for Case a.

• Case b. Investment limits not included: in this case, investment constraints (Equations (16) and (17)) are not considered. This new constraint scenario allows the investment in, not only new transformers and PV modules, but also in wind units (see Table 7). Two expansions of the substation are made, in years 9 and 16, PV modules are installed in all candidate nodes, and six wind units, in total, are also installed in the last year. In this case, the new installed capacity is 4725 kW. The location of the PV modules within the network is displayed in Table 8. The CPU time decreases by 94.4% using Benders' algorithm.


**Table 7.** Power installed (kW) in Case b.


**Table 8.** Nodal allocation of new power installed for Case b.

The results are better in Case b than Case a because the problem is less constrained. The introduction of renewable energy in distribution systems reduces the total operation and maintenance (O&M) system costs (see Table 9). The reduction of O&M costs is due a reduction of losses costs and purchase energy by substation. These results highlight the advantages of investing in renewable generation in the long term.


**Table 9.** Total O&M system costs (€) for 768 scenarios.

Finally, Table 10 shows the CPU times required to solve the DGP problem using both the MILP model (1)–(56) and the Benders' algorithm for different number of scenarios. The CPU time required to solve the MILP model (1)–(56) increases drastically with the number of scenarios (Figures 3 and 4). However, its increment is approximately linear if Benders' algorithm is considered. This causes that, up to 64 scenarios, MILP model solves the problem faster than Benders' algorithm but, from 216 scenarios on, Benders' algorithm becomes a much more efficient way to solve it.



**Figure 3.** CPU time for Case a.

**Figure 4.** CPU time for Case b.

## **7. Conclusions**

This paper has considered the DGP problem in a stochastic environment, where both PV and wind technologies as well as demand are subject to random changes. The use of Benders' decomposition to solve the two-stage stochastic investment problem has allowed us to further decompose the problem by scenarios and planning periods, making it a fully decomposable one. The model has been tested for a 34-bus example with excellent results.

In terms of computing time, the increase in the number of scenarios makes the differences between MILP and Benders' models evident. Up to 64 scenarios, the MILP model is much faster. Benders' model needs to perform iterative processes (loops) that increase the computing time. This trend changes as the number of scenarios increases and Benders' method becomes faster than MILP. In general, MILP's computing time behaves in a non-linear way, whereas Benders' model is more linear in terms of computing time. This proves the significant computational advantage of Benders' with respect to a conventional MILP model.

Note that applying Benders' decomposition may also allow extending the investment problem to address other relevant issues in distribution systems, such as switching or network reconfiguration. This would be non-viable using the original mixed-integer linear programming problem due to its high computational burden.

The work developed in this article can help investors to decide the kind and size of renewable technologies and the place where install the new devices in the distribution system. As improvements to the proposed problem, other kinds of producers (biomass, hydraulic) can be introduced, incorporate reliability in generation and electric vehicle. In addition, making a comparative with other uncertainty management methods such as K-means.

**Author Contributions:** Conceptualization, S.M.-B., J.I.M.-H., J.C. and L.B.; methodology, S.M.-B., J.I.M.-H., J.C. and L.B.; software, S.M.-B.; validation, J.I.M.-H., J.C. and L.B.; formal analysis, S.M.-B., J.I.M.-H., J.C. and L.B.; investigation, S.M.; resources, S.M.-B., J.I.M.-H., J.C. and L.B.; data curation, S.M.-B.; writing—original draft preparation, S.M.-B.; writing—review and editing, J.I.M.-H., J.C. and L.B.; visualization, S.M.-B.; supervision, J.I.M.-H., J.C. and L.B.; project administration, J.C.; funding acquisition, J.C.; J.I.M.-H., and L.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministry of Science, Innovation, and Universities of Spain under Projects ENE2015-63879-R, RTI2018-096108-A-I00 and RTI2018-098703-B-I00 (MCIU/AEI/FEDER, UE), and the Junta de Comunidades de Castilla—La Mancha, under Project POII-2014-012-P.

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