**4. Optimization Strategy**

The solution of the mathematical model described from (1) to (8) requires a methodology that works with mixed-integer variables [45]. In the specialized literature, two main approaches have been proposed to deal with MINLP models in power systems. One of them corresponds to the hybridization methods composed by master-slave stages with metaheuristics. Some of them are genetic algorithms [38], the particle swarm optimizer [42], tabu search algorithms [39] and krill-herd algorithms [41]. These methods decouple the problem of PV source location from sizing. The solutions of these problems are reached through iterative procedures; nevertheless, the optimal solution depends on the number of iterations defined, with the main disadvantage that it is not possible to find the same numerical solution each time that the methodology is evaluated [42]. Thus, statistical procedures are needed to determine their efficiencies [46]. The second focus is to solve MINLP models using gradient-based approaches embedded into branch and bound (B&B) methods, where the gradient searches solve the resulting nonlinear programming model, while the B&B guides the discrete search by defining the location of the PVs [43]. These approaches use nonlinear large-scale solvers available in optimization packages such as GAMS [6,44]. Here, due to the contribution of this paper being related to the presentation of the MINLP model to locate and size PV sources in DC grids for isolated areas to reduce greenhouse emissions by diesel generators, we solve the proposed mathematical model using the CONOPT solver available for GAMS.

As mentioned before, this solver works with gradient searches and B&B methods. In the first step, all the binary variables are relaxed to find the best possible solution; then, this relaxation is discretized to recover the nature of the binary variables in order to provide the optimal solution of the problem. It is important to mention that this optimization package has been successfully used in different problems, such as the optimal operation of batteries [6], optimal location of distributed generators in AC and DC grids [44], optimal design of osmotic power plants [55] and economic dispatch analysis [50,56]. Finally, Algorithm 1 resumes the necessary steps to solve the MINLP model defined from (1) to (8) [57].


### **5. Test System and Numerical Validations**

In this section, we present the test system structure and the output behavior of the PV forecasting used as input on the location and sizing process; in addition, all the simulation scenarios are defined and the numerical results are presented using the GAMS optimization package with its large-scale nonlinear solver CONOPT [44].
