Skill optimization algorithm for solving optimal power flow problem

ABSTRACT


INTRODUCTION
Solving the optimal power flow problem (OPF) is a top concern in power system.According to Daqaq et al. [1], OPF problem is classified as a large-scale, non-convex, and complicated optimization problem.Therefore, the determination of an optimal solution to OPF that obtains the best value of the objective function and satisfies all related constraints is a huge challenge.In solving OPF problems, the typical objective functions, including minimizing total fuel cost by thermal units, minimizing total power loss in the transmission process, and minimizing total emissions, are commonly considered.Besides, the constraints in this problem, such as the operational limits of the generator, the power balance between the generating side and the receiving side, and the safety constraints, must be satisfied.
An optimal solution to such an OPF problem often includes control and dependent variables.Normally, the control variables include the active power generated by thermal units excluding the one connected with the slack node, the voltage magnitudes at all thermal units, the reactive power of shunt capacitors, and the transformer tap [2].The dependent variables are the active power generated by the thermal unit at the slack node, the voltage magnitude at load nodes, and the reactive power output of all thermal units.These control variables are fully selected by the optimization methods, while these dependent variables are found by using Mathpower.Many researchers have conducted their research on the topic by fully acknowledging the importance of solving OPF problems.However, choosing the proper computing method to unfold the OPF problems accompanied by a series of complicated constraints on a large scale is also a tough question.Luckily, meta-heuristic algorithms have recently demonstrated themselves to be highly affordable and the most trusted computing method to deal with optimization problems, including OPF problems.For instant, different meta-heuristic algorithms have been applied to solve the OPF problems such as backtracking search algorithm (BSA) [1], genetic algorithm (GA) [2], Harris Hawks optimization (HHO) [3], [4], grey wolf optimizer algorithm (GWOA) [5], AMTPG-Jaya algorithm [6], differential evolution (DE) [7], [8], effective Cuckoo search algorithm (ECSA) [9], particle swarm optimization (PSO) [10], Gorilla troops algorithm (GTA) [11], modified coyote optimization algorithm (MCSA) [12], black hole optimization (BHO) [13], the hybrid method of Cuckoo search algorithm (MCOA) and sunflower optimization (SFO) [14], improved multi-objective multi-verse algorithm (IMOMVA) [15], firefly krill herd algorithm (FHHA) [16], improved moth-flame optimization (IMFO) [17], antlion optimization (ALO) and its improved version [18], [19], marine predator algorithm (MPA) [20], social spider algorithm (SSA) [21], slime mould algorithm (SMA) [22], whale optimization algorithm (WOA) [23], and golden ratio optimization (GRO) [24].The effectiveness of meta-heuristic algorithms has shown a huge leap forward when compared with conventional computing methods, such as the Newton-Raphson technique [25], in terms of time response, robustness, and precision degree of the final results.Technically, the OPF problems can be solved by placing different types of flexible alternating current transmission system devices (FACTS).Through experiments, the proper placement of a particular type of FACTS device, such as unified power flow controllers (UPFC) [26] or static synchronous series compensators (SSSC) [27], has brought many positive effects to power system operation, including improving the utilization of the current transmission system, increasing the reliability and availability of the grid, and enhancing the overall power quality.Recently, renewable energy sources such as wind and solar energy have also been evaluated while solving the optimal power flow problems as implemented in [28]- [32].
Basically, these studies tried to reach a better total cost than previous studies or existing methods, which were also implemented in the same work.However, approximately all the studies ignored the predetermined limits of the operation parameters.The shortcomings seem simple, but they are significant to the power system's operation.In fact, all electric components have a limit of operation parameter that must be satisfied all the time when power systems are working.So, the paper shows the limits in detail, and then all the proposed parameters are compared to the range.Two algorithms are applied to unfold the given problem, including one classical meta-heuristic method (PSO) [9] and one modern meta-heuristic called the skill optimization algorithm (SOA) [33].PSO is a very old metaheuristic algorithm, while SOA was recently proposed by Givi and Hubalovska [33].The main inspiration for designing SOA comes from the process of improving skills in human life.According to the authors, SOA proved its high performance over other up-todate meta-heuristic algorithms through various test functions.The IEEE 30-bus configuration system is employed to test the real performance of two applied algorithms.The important implications of the whole research are as follows: -Successfully apply a new meta-heuristic algorithm, the SOA, to minimize the TFC value of all TUs while solving the OPF problem.-Present and prove the superiority of SOA over PSO by using different criteria such as minimum fuel cost value, average fuel cost value, maximum fuel cost value, and standard deviation.Moreover, the minimum convergences and the fifty cost values are also given to support the claims.-Diversify and present a good reference regarding the application of novel meta-heuristic algorithms to deal with the high-complex optimization engineering problems that specifically needed to be solved in the power system.Other sections are organized by: section 2 presents the problem descriptions.Section 3 shows an overview of the applied methods.Section 4 displays the analysis results.Finally, section 5 reveals the conclusions.

PROBLEM DESCRIPTION 2.1. The objective function
The mathematical expression of the considered objective function is presented as (1): Where  stands for the total fuel cost value of the six thermal units in the system;   is the among active power produced by thermal unit n; NTU is the quantities of thermal units; and   ,   and   are, respectively, fuel utilizing coefficients belonging to thermal unit n.

The related constraints
The operational constraints of generators: the constraints are about the active, reactive power supply and the voltage index generated by generators.The expressions of these constraints are as ( 2)-( 4): In ( 2)-( 4),    and    stand for the lowest and highest limits of active power output supplied by thermal units;    and    stand for the lowest and highest limits of reactive power output supplied by thermal units;    and    are the lowest and highest limits of voltage index belonging to thermal units;   ,   , and   are, respectively, the active, reactive power output and voltage magnitude generated by thermal unit n.
The balance constraints: total power supplied by the generating side in both active and reactive power must equal the total power consumed by other electric components.The equations for the constraints are shown in ( 5) and (6). with Where   and   are active and reactive power loss;   and   are active and reactive consumed by receiving side;    and    are the minimum and maximum reactive power of shunt capacitors; and   is the reactive power injected by shunt capacitor m,   is the num quantities of existing capacitors in the considered power transmission configuration.
The safety constraints: the constraints are the boundaries of voltage indexes at load nodes and the apparent power circulating on the transmission lines.

𝑈 𝐿𝐷
≤  ℎ ≤    with ℎ = 1, … ,   (8) Where    and    are the minimum and maximum limits of voltage index at load nodes;    is the maximum apparent power sent through the transmission line g;  ℎ and   are the voltage index at load node h and the apparent power in the transmission line g; and   and   represent, respectively the number of load nodes and the transmission line.

THE APPLIED METHOD
The newly solution update mechanism of two methods is different.This is one of the most important features that greatly affect the overall performance of a particular meta-heuristic method.The process of updating the new solutions of two applied methods is briefly presented as follows:

Particle swarm optimization
New solutions of PSO are updated as (10) and (11): In (10) and (11),    and    are new velocities and positions belonging to particle i.   and   are the present velocities and positions belonging to particle i. E1 and E2 are accelerating ratios. 1 and  2 are the random value between 0 and 1.  , and   are, respectively, the best so far position belonging to particle i and the best position in population.

Skill optimization algorithm
The whole update process of SOA for new solutions consists of the expert acquisition phase and the self-improving phase.The two phases update mechanism provides SOA a better balance between the exploration and exploitation capability for the searching process.The mathematical models for each phase will be described: The experts acquisition phase: new solutions is updated by: In ( 12)   1 and   are the new position and the current position of the individual m;  is the random value between 0 and 1;   is the position of expert individual; and  is randomly selected beween 1 and 2. The self-improving phase: the update process in this phase is implemented by: Where  is the random value;  is the value of current iteration; and   and   are the minumum and the maximum boundaries of individual m.

RESULTS AND DISCUSSION
In part of the study, PSO [10] and SOA [33] are employed to resolve the OPF problem in the IEEE 30-bus configuration network.The network has 6 thermal units connected with nodes 1, 2, 5, 8, 11, and 13, 41 branches, four transformers, 24 loads, and nine capacitors.The single-line description of this system is shown in Figure 1.For a moderate comparison of the real performance of the two computing methods, each method is run for 50 cost values.Besides, the population and the highest iteration number of such methods are 50 and 100 for PSO, while the values of SOA are 25 and 100, respectively.The entire study is executed on a computer with the following configuration: CPU with 2. Figure 2 shows the fitness values obtained by two applied methods on 50 cost values obtained by both PSO and SOA.The observation of the 50 cost values indicates that SOA reaches more optimal values than PSO.Moreover, SOA shows lower fluctuations of fitness values than PSO among 50 cost values.This means that SOA can provide a stable performance when applied to the 30-bus transmission power system.The observation of the best convergences given by the two applied methods in Figure 3 indicates that SOA can provide a faster response capability than PSO.Specifically, SOA can obtain the best value after 100 iterations, while PSO cannot do the same.Clearly, the ability to seek the global optimum solution offered by SOA is more productive than PSO. Figure 2. Fifty costs resulted by PSO and SOA Figure 3.The best convergences given by PSO and SOA Figure 4 presents the comparison between PSO and SOA in terms of the minimum cost (Min.Cost), the average cost (Aver.Cost), the maximum cost (Max.Cost), and standard deviation (Std).In the Figure 4, the results of SOA are better than those of PSO, in terms of the Min.Cost, the result of PSO is 804.1675($/h), but that of SOA is only 803.1437 ($/h).Clearly, the application of SOA has saved approximately $1.04, or 0.13% of TFC, for each hour.Furthermore, Aver.Cost and Max.Cost of SOA are only 811.7451 ($/h) and 872.0927 ($/h), but those of PSO are 830.3455($/h) and 951.2637 ($/h).By taking simple calculations, the savings costs and the corresponding percentages for each hour resulted by SOA over PSO on the two criteria are, respectively, $18.6 and 2.24%, and $79.17 and 8.32%.Finally, the Std value given by SOA is only 11.340, while that of PSO is 33.9804.The Std values once again confirm that SOA is more stable than PSO when dealing with the considered problem.Clearly, SOA outperforms PSO in all comparison criteria.Figure 5 depicts the voltage values of each node in the whole system given by the two applied methods.Clearly, the two methods can provide optimal solutions with the allowable voltage for all nodes between 0.95 and 1.1 pu.However, the voltage profile of SOA is better once the fluctuation of voltage is around 1.0 pu, which is the rated value of voltage and an expected operation voltage.
The optimal variables given by the two applied methods are presented in Table 1.In the Table 1, not only optimal variables but also their operation limits are also reported for checking the validation of the proposed solutions.The observation of Table 1 indicated that all the optimal variables found by the two applied methods are located within the initial limits.
Figure 5.The voltage profile of all nodes in the system achieved by PSO and SOA

CONCLUSION
In this research, a modern SOA was successfully implemented to unfold the OPF problem for the IEEE 30-bus configuration system.The sole objective function of the whole study is to find the best cost of all thermal units.The results obtained by SOA are compared to PSO in different criteria.The evaluation of these criteria indicates that SOA completely outperforms PSO in all compared criteria.Moreover, SOA not only provides a fast response capability but also proves itself to be far more stable than PSO.Therefore, SOA is demonstrated as the high effective search method to deal with the OPF problem.Besides, there are several drawbacks that is needed to be improved for better quality, as follows: i) the scale of the considered problem is relatively small when compared with other theoretical transmission networks, such as the IEEE 57, IEEE 118-bus systems, or an existing transmission network in practice; ii) the study only considers one objective function, minimizing TFC.Other objective functions, such as minimizing the total emissions, minimizing the voltage deviation, and many others, are not evaluated; and iii) the contribution of wind and solar energy is not considered.
By acknowledging all these drawbacks, a future study should be conducted on the larger scale of the OPF problem by using a larger configuration of transmission networks with the consideration of different objective functions and the presence of clean energy.

3 Figure 1 .
Figure 1.The model of the IEEE 30-bus configuration system

Figure 4 .
Figure 4.The specific comparison between PSO and SOA on different criteria

Table 1 .
The limit of control variables and optimal control variable proposed by SOA