###### Abstract  
The problem of the optimal siting and placement of static compensates (STATCOMs) in power systems is addressed in this paper from an exact mathematical optimization point of view. A mixed-integer nonlinear programming model to present the problem was developed with the aim of minimizing the annual operating costs of the power system, which is the sum of the costs of the energy losses and of the installation of the STATCOMs. The optimization model has constraints regarding the active and reactive power balance equations and those associated with the devices' capabilities, among others. To characterize the electrical behavior of the power system, different load profiles such as residential, industrial, and commercial are considered for a period of 24 h of operation. The solution of the proposed model is reached with the general algebraic modeling system optimization package. The numerical results indicate the positive effect of the dynamic reactive power injections in the power systems on annual operating cost reduction. A Pareto front was built to present the multi-objective behavior of the studied problem when compared to investment and operative costs. The complete numerical validations are made in the IEEE 24-, IEEE 33-, and IEEE 69-bus systems, respectively. annual operative costs minimization; electric power systems; mathematical optimization; mixed-integer nonlinear programming; optimal power flow; static compensators +
Footnote †: journal: Applied Sciences  
of the power system elements (generators, lines, transformers, etc), which produce energy losses, i.e., electrical energy that is dissipated in the form of heat by the resistive effects of mainly the transmission and transformers [7]. In the Colombian context, the amount of power losses in the power system (lines with voltages \(\geq\) 220 kV) is between 1.5% and 2.0% of the energy generated, which implies that, for a peak load scenario, the total power loss oscillates between 141 and 188 MW [8]. To minimize power losses in the electrical power systems, different methodologies are used, such as optimal reactive power dispatch [9; 10], optimal integration of dynamic reactive power compensators [11; 12], and optimal integration of battery energy storage systems [13; 14; 15]. Of the aforementioned methodologies, the literature considers the first two as being more effective in minimizing the losses of the power systems [16] and batteries as being more suitable to help with the integration of renewable energy resources to mitigate their dependence on weather conditions [17; 18]. The dynamic reactive power compensation is an excellent alternative for helping with the energy losses problem in transmission and distribution networks [12; 19], as the devices used for this purpose, i.e., STATCOMs, have long useful-life, low investments and maintenance costs, and high reliability [20]. In the literature, multiple approaches have been proposed for integrating STATCOMs in power systems. Some of them are as follows: [PERSON] and [PERSON] [21] proposed the cuckoo search algorithm to locate and size STATCOMs in multi-machine power systems to increase the system loadability. Numerical results demonstrate the superiority of the proposed cuckoo search algorithm when compared to the classical genetic algorithm and the heuristic open loop method. [PERSON] et al. [22] studied the problem of the optimal siting and sizing of STATCOMs for solving the optimal reactive power flow problem in power systems by implementing the chemical reaction optimization method. Numerical simulations in the IEEE 30- and IEEE 57-bus systems demonstrate the effectiveness of the proposed method regarding total power losses value when compared with classical approaches reported in the literature such as the particle swarm optimizer and differential evolution method.