## 6 Conclusions and Future Works  
The problem of the optimal siting and sizing of STATCOMs in electrical power systems with radial and mesh structures was addressed in this research from the exact mathematical optimization point of view with the help of the BONMIN solver in the GAMS optimization environment. Numerical results for the IEEE 24-, IEEE 33-, and IEEE 69-bus test systems  
Figure 7: Reactive power behavior of the STATCOMs in nodes 14, 30, and 32 for the IEEE 30-bus system.  
demonstrate a reduction in the annual operating cost of 30.56%, 21.84%, and 23.72%, respectively, when the STATCOMs are optimally sized and located.  
In the IEEE 24-bus system, the multi-objective nature of the problem of the optimal placement and sizing of STATCOMs in power systems was verified, as the cost of annual energy loss is an objective function in conflict with the investment costs in STATCOMs. This Pareto front conformed with the multi-objective optimization approach using weighting factors, where the difference between both extreme solutions are MUS $9.5 regarding the cost of energy losses and MUS $3.8 regarding the inversions in STATCOMs. The single-objective function optimization model and the minimum annual operating costs reported by the multi-objective optimization model (see Solution 9 in Table 9) are practically the same solution, as the difference between these was 0.02%. In both cases, the nodes 6 and 10 were identified as the most promising nodes for installing STATCOM devices with capacities of about 140 and 30 MVAr, respectively.  
The processing time of the single-objective model in the IEEE 24-bus system was about 11 s, while the multi-objective model was solved in approximately 242 s. In the case of the single-objective model for the IEEE 33-bus system, this processing time was about 23 s, whereas, for the IEEE 69-bus system, this processing time was about 71 s. These results confirm the effectiveness and robustness of the GAMS optimization package and the BONMIN solver in solving the MINLP models with more than 1000 variables between the integer and continuous ones in a few seconds.  
In the future, the following studies can be conducted: (i) the development of a mixed-integer convex reformulation of the studied problem to ensure the global optimum finding via conic and/or semidefinite programming; (ii) the development of a hybrid master-slave optimization algorithm based on metaheuristics and power flow to solve the optimization problem sequentially; and (iii) the extension of the studied problem to three-phase unbalanced networks.  
Conceptualization, O.D.M., J.E.F., F.D.M., J.A.B., and H.R.C.; Methodology, O.D.M., J.E.F., F.D.M., J.A.B., and H.R.C.; Investigation, O.D.M., J.E.F., F.D.M., J.A.B., and H.R.C.; and Writing--review and editing, O.D.M., J.E.F., F.D.M., J.A.B., and H.R.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by the Centro de Investigacion y Desarrollo Cientifico de la Universidad Distrial Francisco Jose de Caldas under grant 1643-12-2020 associated with the project \"Desarrollo de una metodologia de optimizacion para la gestion optima de recursos energeticos distribuidos en redes de distribucion de energia electrica.\" and in part by the Direccion de Investigaciones de la Universidad Tecnologica de Bolivar under grant PS2020002 associated with the project \"Ubicacion optima de bancos de capacitores de paso fijo en redes electricas de distribucion para reducion de costos y perdidas de energia: Aplicacion de metodos exactos y metaheuristics\".