These mathematical methods have been developed for over 40 years, starting with classical mathematical optimization techniques [9] - [11] , and more recently reaching genetic and artificial intelligence algorithms [4] [6] [9] [12] [13] . OPF mathematical methods are shown in Figure 1 and Figure 2.

This section presents the most important contributions in OPF methods.

In the year of 1968 H. W. Dommel and W. F. Tinney introduced the concept of OPF in power systems. The Authors present a solution method for OPF using the Newton method with an optimal adjustment of the control parameters through the gradient method; this is done with the purpose of reducing losses of a three-node system [10] .

In 1973 A. M. H. Rashed and D. H. Kelly presented a mathematical algorithm to minimize losses implementing a combination of Lagrange multipliers and Newton’s method, replacing the Jacobian for one Hessian matrix. The proposed method has a faster convergence, requiring fewer steps compared to other methods [14] .

In 1993 Yu-Chi Wu, et al. introduced a method to resolve OPF in order to optimize an economic dispatch. The study combines an algorithm based on the interior point method, justifying that the methods proposed years ago did not meet the requirements of large-scale power systems. The authors concluded that the method provided robust and accurate results [15] .

Subsequently, in 1995 J. Kennedy and R. Eberhart conducted a research in optimization of nonlinear functions using the “Particle Swarm Optimization” (PSO) method. Thus, the authors state that this method could be used to solve a myriad of problems, including the OPF in power systems [12] .

In the year of 2001 C.A. Roa-Sepúlveda and B.J. Pavez-Lazo introduced an OPF method to resolve the economic dispatch problem based on linear Programming. This method is applied to tow IEEE systems of six and thirty nodes without considering the losses; finally, they concluded that OPFs could be solved without a Jacobian matrix, contributing to determine any objective function [7] .

In 2004 A. EL-Dib, et al. focused their research in an application (PSO) to figure out OPF as optimization problems. The authors argue that these studies are the basis to determine the voltage stability and the maximum system power [13] .

In 2006 R. Mageshvaran, et al. presented three different techniques to solve OPF (PSO, CPSO, HDE), proposing useful and encouraging solutions, that were verified in Matlab. The objective of the research was the development of new techniques that allow a better operation; this was done with the purpose of ensuring an economic generation and also meeting the demand of active and reactive power through simple and efficient algorithms [9] .

In the year of 2008 J. M. LópezLezema and L. A. GallegoPareja developed an algorithm to solve OPF, employing the gradient method for achieving reduction of active power losses in power systems. In their research the authors reduced the quantity of iterations by means of the adjustment parable method [11] .

In 2012 C. H. Fujisawa, et al. concentrated their research on direct current OPF with losses and into alternating current; they further presented the mathematical characteristics of each one and their respective advantages and disadvantages. The authors determined that the alternating current method is the most interesting model, since it solves problems through an iterative process and applying various methods of linear programming [16] .

In Section 2.2 are exposed OPF applications in power systems, specifically in electrical microgrids.

The implementation of OPF in electrical microgrids started years ago at first, to control the optimal power dispatch [8] [11] and subsequently to achieve the loss minimization in power systems [4] [5] .

In 2009 E. Sortomme and M. A. El-Sharkawi conducted studies in a microgrid through the Particle Swarm Optimization (PSO); this was done in order to achieve an optimal dispatch of controllable loads and effectively use the storage energy devices that microgrids contain. The authors emphasized that the research was a preliminary study where they concluded that peak loads were reduced in 10 MV, generating an economic improvement of 14% [8] .

In the year 2010 P. Wan and M. D. Lemmon considered an optimization algorithm for distributed event to solve OPF in microgrids. The optimization was performed by simulations in Sim Power System (Matlab) implementing the microgrid model: CERTS. In this study the authors reduced the error by the algorithm and conclude that through the method the optimization problem in microgrids could be solved [6] .

In 2013 E. Dall’Anese, et al. applied several OPF to electrical microgrids in order to minimize losses and the cost of the energy supplied by distributed generation (small energy sources capable of supplying user demand), this was done by SDR techniques (Semidefinite Programming), which the possibility of obtaining the optimal solution for the microgrid [5] .

In 2013 Yoash Levron, Josep M. Guerrero and Yuval Beck introduced a solution for OPF that considered a microgrid with energy storage. The authors claim that classical solutions of OPF were not adapted for the microgrid analysis, in particular due to the presence of distributed generation and of the storage devices. The Optimal Power Flow developed by authors considered the limitations of the storage devices and those of the microgrid regarding to the voltages, currents and powers [4] .

According to [10] , the OPF is based on Equation (1).

where (x) are state variables and (u) control variables [17] :

The control variables can be adjusted without altering the equality and inequality. Therefore, the problem of OPF is to adjust the control variables u in order to optimize a specific parameter of the system represented by the objective function f(x, u) [10] .

The equality restrictions g(x, u) of Equation (1), correspond to the active and reactive power of the system nodes equations:

Finally, the inequality restrictions are the minimum and maximum allowed values for the control variables.

Improving the voltage profiles and reducing losses in the microgrid helps to improve performance, to enhance the useful life of the conductors, to improve energy quality and reduce system costs [18] .

Control variables are defined as generation nodes voltages and Slack as it look for improving the system voltages profiles.

In a power system, slack power node is not specified, because the node assumes the losses of the system [6] . Consequently, the microgrid losses are minimized when the powers of the nodes are kept constant and when the power generation of the Slack is minimized.

According to the previous idea, function f(x, u) is going to be the Slack node power of the microgrid. Equality restrictions g(x, u) of Equation (1) will be the typical power equations which define each node in a power system [19] :

The inequality restrictions (5) are the minimum and maximumvoltages allowed on themicrogrid. Considering that the proposed OPF is particularly for low voltage microgrids and also that for the Colombian case, this levels are 120/208 with limits of +5% (126) y −10% (108) [20] :

Finally, Equation (6) represents the formulation of an OPF to improve the voltage profiles and reduce the active power losses in a low voltage electrical microgrid.

where

In [10] , they are based on an equation named Lagrangian (7) to solve OPF, which has a Lagrange multiplier λ.

If minimun necessary conditions or Karush Kunh-Tucker conditions are applied to Equation (7) [11] , the systems of Equations (8), (9) and (10) are obtained. Optimal microgrid solution is determined by solving the equations system.

The gradient in (9) measures the sensitivity of the objective function with regard to the changes in control variables. Originating at an opposite direction to the gradient, a minimum feasible point with a lower function value is reached [11] . Repeating this process leads to the optimal solution of a system.

The new control values are updated by (11) to continue the iterations until the error will reduce to a minimum.

In Equation (11), the variable is the value that determines the step size that should result from the algorithmin the opposite directionto the gradient. If a low value for c is chosen, a convergence is ensured, however it makes that a lot of iterations be necessary. Instead, if a high value for c is chosen, it will produce oscillations around the optimum [10] .

Figure 4 shows the proposed microgrid for the case of study. This system is proposed for its simplicity and the presence of an unconventional generator. The microgrid has a power of 800 w, which is fed by the main grid and a photovoltaic generator, which to a specific temperature and irradiance generates 450 w (interconnected microgrid).

In order to illustrate the improvements in the system with the OPF, Figure 5 shows the losses of active power before (13.1589) and after (11.7640) the application of the algorithm.

The mathematical algorithm of the proposed OPF was developed in Matlab, selecting different values for the

step size (c = 0.1, c = 0.3 and c = 1.38) with an error tolerance of 0.0001 (ε = 0.0001).

The mathematical algorithm minimizes the difference between the power generated by the grid and the photovoltaic panel, so that losses are minimized. The results show that there is a range of values that minimizes microgrid losses, which is shown in (12). This occurs because the rate of change of the voltage decreases each iteration, until the point where a change in the control variables exerts a change in the power of the Slack node.

The results with different step sizes were the following (Table 1):

As shown in Table 1, the number of iterations required to reach the optimum of the system depends on the size of step (c) chosen. Selecting a value c = 0.1 the algorithm meets the limit of error in 22 iterations; however, if this value increases to 0.3, iterations will be reduced to 8. If the step size increases highly (c = 1.38), the algorithm will present a premature convergence with only 4 iterations generating oscillations.

In order to verify the obtained results in the microgrid, an analysis of profile voltages and system losses was realized by means of power flows through the Newton-Raphson method in Power System Analysis Toolbox (PSAT) by Matlab. The results are represented by Figure 6 and Figure 7.

To examine the voltage profiles in the microgrid, power flows for different cases which can be present in the system were realized, considering the limits in Equation (8), subsequently comparing graphically with the optimal values found with the OPF.

The cases mentioned above are shown in Table 2. The voltage profiles of the microgrid are illustrated in

Figure 7. In this figure, the flattening of the curve (black straight) with the optimal voltage values determined by the OPF can be observed (Case 25).

The analysis of losses was performed in the same way that the voltage profiles, executing various power flows for cases described on Table 2 and comparing with the resulting values of the OPF. As Figure 7 shows, the losses decrease when the optimal voltages are applied (V1 = 121.1844 v and V2 = 118.5517) compared to normal operating conditions (V1 = 120 v and V2 = 120 v). The black dot indicates the maximum reduction of system losses.