Comparison of the evolutionary algorithm's performances on power flow analysis

Öz Power flow in energy systems is one of the major problems. Several classical analysis methods are utilized for solving this problem. However, power generation limits, valve loading effects of units also makes the power flow problem become much harder to solve in the system. In this case, it is possible to achieve the most appropriate solutions with evolutionary algorithms. In this study, optimal power flow problems are solved under same beginning conditions, comprehensively performed with evolutionary algorithms which are recently used and associated algorithm performance is analyzed in IEEE 30-bus test system for two cases. Energy gains of algorithms are obtained; the best, worst and mean values found from optimization are evaluated; convergence analyses are performed comparatively. Thus the effectiveness and efficiency of evolutionary algorithms are clearly demonstrated on solution of optimal power flow problems. Enerji sistemlerinde güç akışı, önemli problemlerden biridir. Bu problemin çözümü için farklı klasik çözümleme yöntemlerinden faydalanılmaktadır. Ancak sistemdeki jeneratörlerin güç üretme limitleri, valf yükleme etkileri gibi parametreler güç akışı probleminin ilgili yollarla çözümünü zorlaştırmaktadır. Bu durumda evrimsel algoritmalarla en uygun çözümleri gerçekleştirmek mümkün olabilmektedir. Gerçekleştirilen çalışmada optimal güç akışı problemlerinin çözümü, iki farklı durumda 30 baralı IEEE test sisteminde güncel evrimsel algoritmalar ile eşit başlangıç şartlarıyla karşılaştırmalı olarak gerçekleştirilmiş ve ilgili algoritmaların performans değerlendirmeleri yapılmıştır. Algoritmaların enerji kazanımları elde edilmiş; optimizasyon sonucunda elde edilen en iyi, en kötü ve ortalama değerleri hesaplanmış; yakınsama analizleri karşılaştırmalı olarak gerçekleştirilmiştir. Böylece optimal güç akışı probleminin çözümünde evrimsel algoritmaların etkinlik ve verimlilikleri açıkça ortaya konulmuştur.


Introduction
The optimal operation of the electricity grid in Carpentier's work since 1962 has become a topic of continuous improvement [1]. As technology developing, the customers need more energy and the optimal power flow (OPF) problem becomes more important. Therefore, many researchers have interested growingly in optimizing OPF problem [2]- [9]. In general, objective of the OPF problem is to minimize total fuel cost by adjusting optimal power system control variables while satisfying given set of power system constraints.
Several conventional optimization methods have been used to reduce the overall production cost for power flow problems in the literature [2]- [4]. Some of these methods have disadvantages because of the OPF problem which is nonlinear, non-convex and multimodal. In most cases, these methods have slow convergence, may suffer from local minima and they have unpredictable run time if number of variables increases. To overcome these deficiencies and limitations in conventional methods, evolutionary algorithms are developed. The major advantage of the evolutionary algorithms is that they can be applicable to various problems without getting stuck in local minima as well as have the ability to self-adapt a solution space when improving a solution as compared with conventional methods. Genetic algorithm (GA), differential evolution algorithm (DE), particle swarm optimization algorithm (PSO), simulated annealing algorithm (SA), and harmony search algorithm (HS) are often used in solution of OPF problem in several studies [5]- [9]. However in the studies above related to solving OPF problems, the algorithms may not show own efficiency and robustness. When these studies are analyzed in terms of population size, number of execution and initial point, it is seen that different population size is preferred in some of studies and different number of execution is used as the stopping criterion. On the other hand, it is not possible to use same beginning point in different computers due to nature of algorithms which initial randomly. Using same initial seed gives support to algorithms for making comparison uniform. The aim of this study is to detailed comparative analysis in between recently-used eight optimization algorithms to solve power flow problems. All the algorithms were performed on the same platform to prevent misjudgment of the algorithms employed. By using same initial values, population size and number of execution, eight different algorithm analyses were carried out on IEEE-30 test bus system for two different cases of OPF problem. This paper is organized as follows: In Section 2, formulation of OPF problem is declared. In Section 3, general structure of evolutionary algorithms is defined. In Section 4, implementation of OPF problem is described. Finally, In Section 5, simulation results are shown and discussed.

Optimal power flow
Optimal power flow optimization is a nonlinear constrained optimization problem and defined as follows: Subject to ( , ) = 0 and ℎ( , ) ≤ 0 In these functions, represents the state variables that can be seen in Eq. 3 and represents the control variables as shown in Eq. 4 In Eq. 3, consists of slack bus , load bus voltage , generator reactive power and transmission line loading (line flow) .
Where is generator active power, is generation bus voltage, is shunt VAR compensation and is transformers tap settings.
( , ) is the function of equality constraints and relies on typical load flow equations which are formulated in Eq. 5 and Eq. 6, where and are the active and reactive load demand of ℎ line and is the number of buses in power system. and are real (conductance) and imaginary (susceptance) part of admittance matrix of power system. Ɵ is the difference in voltage angle between bus and bus [7]. ℎ( , ) is inequality constraints of the OPF problem. System operating constraints reflect the limits on physical devices in the power system. These constraints are given as follows: Where , and are the total number of generators, transformers and compensator devices, respectively. , are lower and upper limits of active power for each generator.
, are lower and upper limits of reactive power for each generator.
, are lower and upper limits of voltage for each bus.
, are lower and upper limits of the transformer tap setting for each transformer [7].

Objective function
( , ) refers to the desired objective function to be minimized and represents total fuel cost in Eq. 12: Where, , and are the cost coefficients of ℎ generator. Objective function is considered as the total fuel cost. Total fuel cost depends on the power of each generator and the cost coefficients.

Valve point effect
Considering the valve point effects of generators, a recurring rectifying sinusoidal term is joined to the principal quadratic cost function, as follows in Figure 1. With the valve point effects, cost function has some ripples because of these ripples; the number of local optima is increased in solution space [10]. Figure 1 represents the cost function of generators with/without the valve point effect [11]. Total fuel cost with valve point loading is given by Eq. 13.
Where, and are coefficients from valve point effect of ℎ generator [11].

Evolutionary algorithms
In Table 1, the structure of algorithms used in OPF problems are summarized:

Implementation of the algorithms to OPF problem
This part makes a general statement of the all algorithms which is about how to solve OPF problem using objective function as gathered from Eq. 12. To solve this problem, algorithms generally follow these steps: Step 1 : Define algorithm parameters and load input data from IEEE 30 bus test system.
Step 2 : Initialize algorithm randomly with initial population , as follows: Where, is the population size and is the dimension of the problem. At the beginning, population is taken by algorithm randomly. It should be noted that whole algorithms use same initial population for equally comparison in this paper.
Step 3: Check the all constraints.
If any limit of constraint is violated, produce new individual with respect to Eq. 14 and transform the constrained OPF problem into unconstrained one by adding penalty term to ( , ) for each violated constraints.
Otherwise, go to Step 4.
Step 4: a) Calculate the objective function in Eq. 12 using , b) Process the algorithm according to the obtained results from objective function and own searching criteria. The structures of all algorithms can be shown in Section 3.
c) Choose best individual in the population that is found so far. This individual represents the variables of problem which are obtained from objective function.
Step 5: If the current number of execution reaches the predetermined maximum execution number, algorithm process is stopped, otherwise go to Step 4.

Simulation results
In this study, eight different algorithms are applied to OPF problem on IEEE 30-bus test system. Data of the 30-bus system can be achieved from [20]. As shown in Figure 2, IEEE 30-bus test system consists of four transformers and six generators, which has 283.4 MW total active demands, 126.2 MVAr total reactive demands. Related data of generators for IEEE 30 bus system which is online available in [20] are given in

Case 1: OPF problem solution without valve point effect of generators
In the first case, OPF problem is solved without valve point effects by minimizing the objective function using Eq. 12. Active power outputs for generators found by each algorithm are given in Table 3. The obtained best, worst, mean values over 10 runs are shown in Table 4. It can be seen that in Table  3, there is no big difference (less than %1) in terms of fuel cost values of ABC, CS, FF, PSO, and DE. However, DE and PSO find lesser costs compared to other three algorithms.
As shown from Figure 3 which shows the convergence performance of different algorithms, CS is not more successful in solving case 1 of OPF problem compared with PSO and DE. Despite that, CS can converge to own optimum point (local solution) in lesser executions. On the other hand, ABC is more robust algorithm dealing with stability and accuracy. It can be also observed that SA and FF are not stable with big fluctuation and they make more number of executions to reach optimum point than other algorithms. Table 4 show that if accuracy of all algorithms are analyzed just based on best values, PSO is better choice to solve case 1 of OPF problem. It can be easily seen in Table 4, SA and RCGA find significantly worst solution than other algorithms.   As given from Figure 4, energy saving ratios are alike but when the convergence of the different algorithms are compared, each algorithm converges to different points in different number of executions as shown in Figure 3.

Case 2: OPF problem solution by considering valve point effect of generators
In this case, valve point effect is considered and the data is listed in Table 2. Active power outputs for generators are given in Table 5 and convergence performance of different algorithms related to case 2 is shown in Figure 5. Best, mean, worst values for the results of case 2 obtained by associated algorithms are given in Table 6. From the results in Table 6, best, mean, worst solutions found by each algorithm are different from the solution of case 1 due to valve point effect which increases computational costs of algorithms. As an example, without considering valve point effect, mean value of costs found by all algorithms is almost 803.22 $/h but considering this effect, the cost rises to almost 831.88 $/h.
As shown from Figure 5, HS, SA and FF have more number of executions (more than 80) to settle at the minimum point. Moreover, significant differences exist among these three algorithms for best cost values and HS gives better results compared with two of them.
As seen from Table 6 unlike to case 1, RCGA finds worst result and CS gives best result with small difference compared with ABC. On the other hand, similar to case 1 as shown in Figure 5, FF and SA don't only make more numbers of executions but also give worse result compared to rest of algorithms. Of course, FF, SA and RCGA are not suitable for solving case 2 as compared to other algorithms. In general, if the balance between best and worst values of algorithms are analyzed which are shown in Table 6, the results of ABC, PSO, DE and HS are very identical and if small differences are not taken into account (less than %1), this problem can be solved with these four algorithms for case 2.  As shown in Figure 6, energy saving ratios according to initial cost values are very similar except RCGA but the number of executions taken to convergence solution is different when compared with each other.

Conclusions
The OPF problems become difficult to solve because of large number of constraints and nonlinearity of OPF problems for mathematicians as well as for engineers. In this paper, eight different algorithms are successfully performed on two different cases of OPF problem. Discussing valve point effect and complex constraints create a challenge for algorithms to reach minimum cost. We have pointed out equal competition between algorithms in this challenge and also compared to all algorithms comprehensively on the OPF problem under equal conditions for both cases. The equally analysis showed that each algorithm has own strength and weakness when dealing with OPF problem. The outcome of this research helps not only provide an equal comparison between whole algorithms but also helps to show us which algorithm works more efficiently on which cases.