*5.4. Numerical Results*

To solve the general MINLP model, which represents the problem of optimal location and sizing of DGs in DC systems, we employed the GAMS optimization package with different nonlinear solvers in a desktop computer with an INTEL(R) Core(TM) i5-3550 3.5-GHz processor and 8 GB of RAM running a 64-bit version of Windows 10 Home Single Language.

Table 4 reports the numerical results for all the simulation scenarios previously proposed. Note that in the first scenario, the daily greenhouse emissions of *CO*2 are 13,428.912 pounds, and the minimum emissions occur in the fourth scenario with 10,878.190 pounds per day, which implies en equivalent reduction of 2550.722 pounds of emissions of *CO*2 per day (18.99%). It is important to mention that after using the CONOPT solver for each simulation scenario, the total processing time to reach the optimal solution is lower than 20 s for all the cases. Still, it starts to increase from 6.224 s to 19.063 s (67.35%) depending on the number of PV sources that are considered in the scenario. It is important to mention that the processing times of the ANN training process is not considered in the last column in Table 4, since this is an offline procedure that takes between 5 and 10 min depending on the size of the training set.


**Table 4.** Gas emissions for each simulation case.

Figure 4 presents the total daily reduction per day when a different number of PV sources are located and sized in the 21 nodes DC test feeder. This plot confirms that the best scenario corresponds to the case with three PV sources located inside the network; nevertheless, these bars confirm that the reduction of greenhouse emissions has a nonlinear behavior regarding the number of PV sources; i.e., the reduction tends to have a saturation of 19% approximately. This behavior obeys the fact that the PV sources only work during sunny hours, making it necessary to use diesel sources during the night.

**Figure 4.** Total reductions in *CO*2 emissions per day for each simulation scenario.

Table 5 reports the optimal location of the PV sources in each simulation scenario. These simulations show that the most attractive node to locate a PV source is the node 17, followed by nodes 19 and 12, respectively. In addition, for the third and fourth scenarios, it is important to observe that all the allowed penetration, i.e., 60% of the peak consumption, is used (divided) by the PV generators, while the second scenario only uses about 96.38% of this maximum capability. These results confirm the nonlinear relation between the number of sources available for location and their sizes regarding the minimization of the objective function, which implies that multiple simulations and scenarios need to be taken into account for the grid planner (i.e., utility) to determine its inversions.


**Table 5.** Optimal locations and sizes of the PV sources.

Note that all the simulations were guaranteed through (5) to have all the currents flowing inside of the system be lower than 400 A; i.e., all the possible locations and sizes of the PV sources are feasible to be implemented since conductors of the grid can operate safely. In addition, when we considered the real daily curve reported in Table 3 with the locations and sizes of the PV sources reported in Table 5, it was found that the errors in the estimation of the objective function were lower than 1%, which confirms that ANN are powerful tools for short-term forecasting of renewable energy resources, as reported in [50].

Figure 5 presents the behavior of the diesel generators during the day for all the simulated scenarios. Note that in the first scenario, they support all the power consumption in the DC grid (solid line in Figure 5a,b). Nevertheless, when PV generators are installed, the total generation in the diesel

resources decreases significantly between hours 7 and 18; which clearly corresponds to the periods of time where PV sources can deliver their power to the grid (see dotted and dashed lines in Figure 5a,b).

**Figure 5.** Generation profiles of the diesel sources during the day.

It is also important to mention that the differences between both profiles in the diesel generators are caused by the voltage profiles required at their terminals. In the case of the diesel generator located at node 21, it needs to inject more active power, since it is required to maintain the voltage at 1.05 p.u all the time, while the diesel generator located at node 1 was fixed to 1.0 p.u. This implies that with lower power injections this profile can be sustained.

### **6. Conclusions and Future Works**

A mixed-integer nonlinear programming model for optimal location and sizing of PV sources in DC isolated networks was presented in this paper. The objective of this formulation is to reduce the total greenhouse emissions (i.e., pounds of *CO*2 emissions per day) by diesel generators with the introduction of PV sources considering typical Colombian power profiles and consumption behaviors. Numerical results demonstrate that these gas emissions can be reduced between 17% and 19% depending on the number of PV sources installed and their sizes.

A nonlinear relation between the number of generators and their location was evidenced with the minimization of the objective function, as differences lower than 1.50% were found for all the scenarios that included PV sources (i.e., from the second to the fourth scenario). These situations imply that additional studies are needed in regard to operational costs, useful life and ground lot availability, among other things, to determine the best solution in terms of PV sizes and locations. The objective function, in all scenarios provided in this studio, showed attractive alternatives to be implemented in the case of the 21-node test feeder.

As future work, it would be possible to extend this MINLP model to wind generators by predicting their average power availability with artificial neural networks with high efficiency, as reported in this study of PV sources. In addition, this model can be modified to include battery energy storage systems, to increase the introduction of renewable energy during periods of time with high demand and lower generation availability.

**Author Contributions:** Conceptualization, O.D.M., L.F.G.-N., W.G.-G., G.A. and Q.H.-E.; methodology, O.D.M., L.F.G.-N., W.G.-G., G.A. and Q.H.-E.; formal analysis, O.D.M., L.F.G.-N., W.G.-G., G.A. and Q.H.-E.; investigation, O.D.M., L.F.G.-N., W.G.-G., G.A. and Q.H.-E.; resources, O.D.M., L.F.G.-N., W.G.-G., G.A. and Q.H.-E.; writing—original draft preparation, O.D.M., L.F.G.-N., W.G.-G., G.A. and Q.H.-E. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Universidad Tecnológica de Bolívar under project CP2019P011, and Instituto Tecnológico Metropolitano, Universidad Nacional de Colombia, and Colciencias under the project "Estrategia de transformación del sector energético Colombiano en el horizonte de 2030 - Energética 2030"—"Generación distribuida de energía eléctrica en Colombia a partir de energía solar y eólica" (Code: 58838, Hermes: 38945).

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