الگوریتم هیوریستیک کارآمد برای قرار دادن خازن بهینه در سیستم های توزیع استفاده می شود

An efficient heuristic algorithm is presented in this work in order to solve the optimal capacitor placement problem in radial distribution systems. The proposal uses the solution from the mathematical model after relaxing the integrality of the discrete variables as a strategy to identify the most attractive bus to add capacitors to each step of the heuristic algorithm. The relaxed mathematical model is a non-linear programming problem and is solved using a specialized interior point method. The algorithm still incorporates an additional strategy of local search that enables the finding of a group of quality solutions after small alterations in the optimization strategy. Proposed solution methodology has been implemented and tested in known electric systems getting a satisfactory outcome compared with metaheuristic methods. The tests carried out in electric systems known in specialized literature reveal the satisfactory outcome of the proposed algorithm compared with metaheuristic methods.

In the radial distribution systems, shunt capacitors must be installed in order to compensate the reactive power of the loads and reduce the reactive power provided by the system, to diminish energy losses, to improve the voltage magnitude profile and, on a smaller scale, to release system capacity [1], [2], [3], [4] and [5]. Thus, the general problem of optimal capacitor placement (OCP) appears in radial distribution systems. This problem consists in determining the optimal number, locations, types, sizes and switching times, for the case of switched capacitors, in order to maximize costs savings while maintaining requested good operation conditions. This problem is interesting because its mathematical model is a mixed integer non-linear programming problem (MINLP) with a non-differentiable objective function due to the fact that the costs of the capacitor vary in a discrete manner. The system load also varies continuously throughout the day. In this way, practically all the research related to the OCP assumes the demand variation in a determined number of discrete load levels in order to solve the problem using existing optimization techniques. Usually the demands have three load levels, that is, peak load, medium load, and light load. Therefore, for each load level, the control scheme of the switched capacitors must be determined. The OCP has been carefully analyzed in the mathematical model as well as in the development of solution techniques. At the early stage, simplified network models that allowed the use of optimization algorithms were used. The mathematical modeling was improved and a greater contribution in this aspect was carried out in Ref. [1] where an optimization mathematical model that is nowadays used by the researchers is shown. The optimization techniques proposed to solve the OCP problem, nowadays, can be grouped into three large groups: (1) analytical methods, (2) numerical programming algorithms, (3) approximate algorithms. Among numerical programming algorithms, methods such as Benders decomposition techniques and the Branch and Bound algorithms, used in the initial phases but limited to small systems and simplified mathematical models [1] and [6], can be included. The approximated algorithms can be grouped into two groups: (i) heuristic algorithms and (ii) metaheuristics. Heuristic algorithms generally use a performance criterion or a sensitivity indicator incorporated into an optimization strategy in order to find quality solutions for complex problems [7], [8] and [9]. These algorithms present the advantage of finding quality solutions with small processing efforts and are generally robust and simple to understand and to implement. These algorithms differ in two fundamental aspects: (1) the performance criterion used and (2) the proposed optimization strategy. The performance criterion can be a very simple decision or it can be obtained by solving a complex sub-problem that, in many cases, implies in solving the real problem itself after relaxing the characteristics of some variables and/or constraints. Metaheuristics have been the most used algorithms in the last few years to solve complex problems in the field of operational research. Recent research using metaheuristics shows the great popularity of these techniques to solve OCP problems [2], [5], [10], [11], [12] and [13]. The mathematical model proposed in [1] was used in this work, which is the most used model for the OCP, and a heuristic algorithm was presented to solve this problem. Our proposal uses an interior point method to solve a non-linear programming problem (NLP), yielded after relaxing integer variables of the original MINLP problem, as an indicator to add or place new capacitors in the most attractive bus of the system. This process is controlled by an constructive heuristic algorithm. Another proposal presented in this work is to leave the substation voltage (S.S. Volt.) as a decision variable, finding better quality solutions when compared to the optimization solutions that set the magnitude voltage of the substation to the nominal value. Our experience shows that in highly stressed systems, that is, with elevated violation in the voltage magnitude, it is very difficult to find a feasible solution only adding capacitors. Additionally, in case these feasible solutions exist, they generally represent operation points where losses are greater than in the case of operation without capacitors. In this context, the capacitors diminish the real energy losses and improve the voltage magnitude profile but do not have the capacity to correct the voltage magnitude in highly charged systems. It must be observed that almost all the metaheuristics presented in specialized literature keep the value of the substation voltage fixed [1], [4], [14] and [15].

A heuristic algorithm was presented in this work, to solve the optimum capacitor placement problem in radial distribution systems. The basic idea of the algorithm is to solve a relaxed version of the exact mathematical model, that is, after relaxing the integrality of the variable related to the capacitor and approximating the objective function by a derivable function. The relaxed problem, that is, a NLP, is solved by using an efficient non-linear programming algorithm. From this solution, a heuristic strategy finds a group of near-optimum points of the original problem. The algorithm presented satisfactory results in terms of quality for the solution found and with a relatively small processing effort as it solves a reduced number of non-linear programming problems. The algorithm works with no difficulty in systems where different cost switched and fixed capacitors can be placed. However, the presented proposal corresponds to a heuristic strategy and, at least from a theoretical point of view, may be not compete in quality with the solutions found with recent metaheuristics, but the solution obtained in the test is better in quality than [11], mainly because it has represented less investment and more savings, and close to the solution reported in [5]. This is true taking into account that main substations “usually” have voltage regulators or load tap changer transformers. In the tests, it was also observed that the capacitor placement is efficient in the reduction of losses of the system but it can not always solve problems of voltage drop, especially, in systems highly loaded. This problem must be overcome by placement of voltage regulators. In this work, the voltage drop problem was adequately overcome considering that the voltage magnitude in the substation can be seen as a variable that was optimized by the non-linear programming algorithm. In the case of highly loaded systems, it is believed that the most adequate strategy is the optimum capacitor placement and optimum operation point setting of voltage regulators, being that the capacitors solve the diminishing of losses more adequately and the voltage regulators solve the voltage drop problems in a better manner.