Application of Modified Artificial Bee Colony Algorithm for Optimization of Reactive Power

Abstract

Reactive power is beneficial to a power system's voltage stability. As a result, optimizing reactive power is required to enhance voltage profile. RPO also aids in the minimizing of active power losses while being compliant with all limitations. Honey bees’ seeking behaviour is influenced by the ABC algorithm. Honeybee behaviour is used to a variety of RPO challenges. In this study, standard ABC is changed by using a weighted technique to improve exploitation capability. This updated ABC approach is used to solve the RPO problem as well as standard benchmark functions. For both regular ABC and MABC, the benchmark functions are tried with different dimensional sizes and the results are studied and compared. MABC is determined to be more successful than normal ABC for the majority of activities. To prove its effectiveness, the modified method is implemented on standard IEEE 30 and 57 test bus systems. Data from the literature is compared to the findings.

1 Introduction

In power systems, reactive power is a crucial feature. Reactive power generation and distribution have a significant impact on the performance, efficiency and reliability of power systems [1]. The voltage in power system sags down when there isn’t enough reactive power. This drop in voltage affects power delivered to load through transmission lines. Hence, reactive power is essential for improving voltage stability and to avoid voltage sag in power systems [2].

Optimization deals with the minimization of cost of production and maximization of efficiency of production. Reactive power optimization is essential for reduction in active power losses [3]. RPO helps in voltage profile improvement and for minimization of real power losses under various operating conditions.

There are different methods that can be used for RPO like artificial neural networks, Evolutionary programming,, Genetic Algorithm, Particle Swarm Optimization Algorithm and Artificial Bee Colony Algorithm [4]. As compared to other algorithms, ABC algorithm is simply effective in solving non-linear equation of reactive power. The ABC algorithm is more effective as compared to other algorithms. Also, ABC algorithm operates well in finding superior solutions of the objective functions. Hence, we have chosen ABC algorithm with modified method to study its exploitation capability for RPO problems. We are going to implement modified ABC with weighted method on benchmark functions. With the outcomes of benchmark functions, we will be able to understand the possibility of use of modified weighted method of ABC algorithm for reactive power optimization problems [4]. Further, this modified ABC approach is tested on IEEE 30 and 57 standard systems for RPO problems. Obtained outcomes of RPO problems with modified ABC are compared with existing results available in literature data.

The rest paper is organized as follows. In Sect. 2, information about ABC algorithm and its literature review is described. Section 3 gives description of modified ABC approach and process with flow chart. Section 4 describes mathematical expression of objective function for reactive power optimization problems with constraints. Section 5 includes experimental setup and results for the modified approach on benchmark functions. It includes simulation results of objective function of RPO problem on IEEE test bus system. The modified approach and its experimental results with comparison are concluded in Sect. 6.

2 Related Work

2.1 Artificial Bee Colony Algorithm

This is an optimization technique that stimulates finding behaviour of honeybees. The technique is implemented to various practical problems.

Employed bees, onlooker bees and scout bees are the three sorts of bees that exist. These three types worked together to find better solutions. The employed bee searches for food sources and informs onlooker bees. The onlooker bee continued to work on these selected food sources in order to locate better ones. Fitness values of sources determine the quality of food sources. The scout bee generates a new solution (food source) if limited iterations cannot give better fitness value of existing food source. This new solution generated by scout bee is used to update old solution.

Assume that $X_{i} = \left\{ {x_{i,1} , x_{i,2 } ,...,x_{i,D } } \right\}$ is the ith solution where i = 1, 2, 3, …, N. A solution $X_{i}$ is selected by each employed bee and search is conducted in the neighbourhood to find new solution $V_{i}$ [4].

$$v_{i,j} = x_{i,j} + \emptyset_{i,j} . \left( {x_{i,j} - x_{k,j} } \right)$$

(1)

where $j \in [1,D]$ is a dimensional index chosen randomly. $X_{k}$ is an arbitrarily selected solution and $k \in \left[ {1, N} \right]$. The weight $\emptyset_{i,j}$ is selected arbitrarily and value is between [−1.0, 1.0].

We know that equation Min f(x) is used by employed bees to find food sources. Further sharing of information of found food sources is done from employed bees to onlooker bees. Based on quality of found food sources, onlooker bees select some superior food sources for further search.

The selection method is as follows:

$$p_{i} = \frac{{f{\text{i}}t_{i} }}{{\mathop \sum \nolimits_{i = 1}^{N} f{\text{i}}t_{i} }}$$

(2)

where $f{\text{i}}t_{i}$ indicates fitness value of ith solution $X_{i} .$ In standard ABC, the same equation Min $f\left( x \right)$ is used by the onlooker bees to find superior solution (food source).

If limit iterations fail to improve the fitness value of an existing food source, the scout bee creates a new solution (food source). Scout bee generates a new solution that is utilized to update an existing one.

$$x_{i,j} = {\text{low}}_{j} + {\text{rand}}_{j} \cdot \left( {{\text{up}}_{j} - {\text{low}}_{j} } \right)$$

(3)

where $\left[ {{\text{low}}_{j} , {\text{up}}_{j} } \right]$ is the frame of constraint and ${\text{rand}}_{j}$ is a randomly generated values between [0, 1].

The standard ABC algorithm process is mentioned in following flowchart.

Flowchart 1. ABC Algorithm Process

2.2 Literature Review

Various benchmark functions and practical problems can be resolved using ABC algorithms. There were different types of algorithms proposed along with standard algorithm. The literature review of this work is summarized as follows.

Ozturk et al. [2] have proposed the ABC technique to obtain multi-objective RPO. IEEE 10 bus system is used for implementation of multi-objective RPO with ABC technique. In this paper, minimization of active power losses is achieved. Along with it, cost optimization and voltage quality are achieved. Ayan and Kılıc [5] have proposed ABC algorithm technique. It is implemented on standard IEEE 30 and 118-bus systems and further results are analyzed. They have discussed about why ABC algorithm is advantageous over other types of algorithm (like genetic algorithm and differential evolution). In this paper, ORPF supported ABC algorithm technique is discussed. It is implemented to reduce active power loss. The effectiveness and computing efficiency of the proposed technique for ORPF problems are verified and investigated. MATLAB software is used for result analysis.

Ettappan et al. [6], have proposed ABC algorithm technique with approach of optimal reactive power dispatch. This method is used for reduction in active power losses and voltage stability improvement. Simulations of standard IEEE 30 and 57 test bus are used for RPO problem study. Results obtained are compared with other methods.

Thorat A. R. [3] have proposed an approach of gbest-Guided ABC Algorithm for RPO problems. This algorithm is applied on IEEE test bus systems and comparative analysis of obtained results is done. Standard IEEE 30, 57 and 118 test bus systems are used for analysis on MATLAB software. The analysis shows that gbest-Guided ABC is superior over standard ABC.

Cao et al. [4] have proposed a modified ABC to increase efficiency of GABC method. In this paper, they proposed a modified approach to increase exploitation capability. The modification is done at onlooker bee phase in search equation. Modified approach of ABC algorithm is implemented on different benchmark functions. The obtained results are compared with other algorithms.

Zhu and Kwong [7] mentioned gbest-Guided ABC algorithm in paper. It is implemented for numerical function optimization. The proposed gbest-guided method increases the exploitation capability. The method is implemented on benchmark functions with suitable parameters. Results show that GABC outperforms ABC algorithm. Li et al. [8] proposed hybrid differential evolution algorithm along with ABC algorithm. It is applied on reactive power optimization problems. Results of simulation show this method finds better solutions and having more convergence speed and same method is suitable for RPO problems.

3 Modified ABC Approach

Standard ABC possesses good exploration but it shows poor exploitation. Poor exploitation results in slow convergence speed. Hence, exploitation capability of ABC can be improved by using some different ABC variants. One of them is GABC—global best guided ABC—$G_{{{\text{best}}}}$ is introduced in search equation and expected that $G_{{{\text{best}}}}$ helps to find best solution.

In GABC, search equation is given by

$$v_{i,j} = x_{i,j} + \emptyset_{i,j} . \left( {x_{i,j} - x_{k,j} } \right) + \varphi_{i,j} . \left( {G_{{{\text{best}}}} - x_{i,j} } \right)$$

(4)

where $G_{{{\text{best}}}}$ stands for the global best solution obtained using GABC algorithm. The weight $\emptyset_{i,j}$ is a selected randomly between [0, C]. We have chosen value of C = 1.5 [7] and it is a predefined value.

In $G_{{{\text{best}}}}$ guided ABC algorithm, we are making use of Eq. (4) to find new solutions. The probability of each solution is derived by onlooker bees. These probabilities are used to make further searches and find better solutions. To find new solutions, same Eq. (4) is used by onlooker bees.

In standard ABC algorithm, the same equation has been used in employed and onlooker bee phase to find new solutions. As a result of it, employed bees possess good exploration capability and onlooker bees show lack of exploitation capability. Hence, we are choosing a modified version to study its exploitation capability for RPO problems. In modified approach, we are using two search phases: exploration and exploitation. This approach will help to improve exploration capability by employed bees. Onlooker bees help to improve exploitation using modified approach. To implement modified approach, we are making use of a new solution-finding equation with onlooker bees [4].

$$v*_{i,j} = a_{1} \cdot x_{i,j} + a_{2} \cdot G_{{{\text{best}}\,j}} + a_{3} \cdot \left( {x_{k,j} - x_{i,j} } \right)$$

(5)

where $X_{i }$ stands for current solution, $G_{{{\text{best}}}}$ denotes global best solution and $X_{k}$ is arbitrarily generated solution for which ($i \ne k$). Weights $w_{1} , w_{2 }$ and $w_{3 }$ are three random values and $a_{1} + a_{2 } + a_{3} = 1.$ Firstly, we need to generate three values $a_{1} , a_{2 } {\text{and}} a_{3}$ in the range [0, 1]. Then we use following Eqs. (6), (7) and (8) to generate new weights.

$$a_{1} = \frac{{w_{1} }}{{w_{1} + w_{2} + w_{3} }}$$

(6)

$$a_{2} = \frac{{w_{2} }}{{w_{1} + w_{2} + w_{3} }}$$

(7)

$$a_{3} = \frac{{w_{3} }}{{w_{1} + w_{2} + w_{3} }}$$

(8)

In standard ABC algorithm, slow convergence speed can be obtained by change in one dimension of parent solutions. In this paper, the MR parameter is used in modified method which is useful for controlling the number of updating dimensions. For each food source solution, all dimensions j, $\left( {j = 1,2,...,D} \right)$ are checked. If ${\text{rand}}_{j}$ < MR, then Eq. (5) is used to revise $v_{i,j}$. Then, new search equation is given by

$$v_{i,j} = \left\{ {\begin{array}{*{20}c} {v*_{i,j} ,} & {{\text{if rand}}_{j} < {\text{MR}}} \\ {v_{i,j} ,} & {{\text{Otherwise}}} \\ \end{array} } \right.$$

(9)

where MR is an already defined parameter of probability, ${\text{rand}}_{j}$ is a randomly generated value between [0,1] for the jth dimension. The value $v*_{i,j}$ is as mentioned in Eq. (5).

Step-wise process for modified ABC method is as follows [4].

• Step 1. Make selection of N solutions randomly. Fitness values of N solutions are calculated. Then, put FEs = N (no. of fitness evaluations and it is equal to no. of solutions generated).

• Step 2. For every $X_{i }$ solution, a latest solution $V_{i }$ is derived by using Eq. (4). Calculate fitness value of $V_{i }$ and further FEs = FEs + 1. Superior value between $X_{i }$ and $V_{i }$ is selected as new $X_{i } .$ Following to this if $f\left( {V_{i } } \right) < f\left( {X_{i } } \right)$, then ${\text{trial}}_{i } = 0$; otherwise ${\text{trial}}_{i } = {\text{trial}}_{i} + 1.$

• Step 3. Calculate the probability $p_{i }$ consistent with Eq. (2). If $X_{i }$ is chosen, then $V_{i }$ is derived by Eq. (9). Fitness value of $V_{i }$ is calculated and FEs = FEs + 1. Superior value between $X_{i }$ and $V_{i }$ is selected as new $X_{i } .$ If $f\left( {V_{i } } \right) < f\left( {X_{i } } \right)$, then ${\text{trial}}_{i } = 0$; otherwise ${\text{trial}}_{i} = {\text{trial}}_{i} + 1.$

• Step 4. Solution $X_{i }$ is replaced by Eq. (4), if $\max \left( {{\text{trial}}_{i } } \right) > {\text{ limit}}$.

• Step 5. Fix $t = t + 1$. If FEs < maxFEs, then go to Step 2; else terminate algorithm.

We have summarized process for modified ABC algorithm in the flowchart as below.

Flowchart II. Modified ABC Algorithm Process

In comparison with standard ABC algorithm, modified ABC approach makes change in the search equation at onlooker bees phase. The weights $a_{1} , a_{2 }$ and $a_{3 }$ introduced in the modified method make helps to improve capability of standard ABC algorithm. So, in terms of computational time complexity, it is same for both MABC and ABC.

4 Mathematical Expression of Objective Function for RPO

4.1 Objective Function

Main aim of RPO is active power loss minimization. It is given by following Eq. (10). The control variables selected are as follows- capacity of reactive power compensation device, tap position of on-load tap changer and terminal voltage of the generator. PV bus reactive power output and PQ bus voltage are chosen as state variables.

$$\min F = \min \mathop \sum \limits_{k = 1}^{n} G_{k} \left( {i,j} \right)\left[ {u_{i}^{2} + u_{j}^{2} - 2u_{i} u_{j} \cos \left( {\theta_{i} - \theta_{j} } \right)} \right]$$

(10)

where n stands for total no. of branches, $G_{k} \left( {i,j} \right)$ denotes conductance of branch k connecting bus i and j, $\theta_{i}$ and $\theta_{j}$ denote phase angles of bus i and j, voltages of bus i and j are denoted by $u_{i}$ and $u_{j}$ [8].

4.2 Equality Constraints

Equations. (11) and (12) show power flow balance equations. Equation (11) is of active power and Eq. (12) is of active power and reactive power. The same equations are used as equality constraints in optimization problems.

$$Pi = u_{i} \mathop \sum \limits_{j \in Ni} u_{j} (G_{ij} \cos \delta_{ij} + B_{ij} \sin \delta_{ij} )$$

(11)

$$Qi = u_{i} \mathop \sum \limits_{j \in Ni} u_{j} (G_{ij} \sin \delta_{ij} - B_{ij} \cos \delta_{ij} )$$

(12)

where $u_{i}$ is voltage of bus $i$ and $u_{j}$ is voltage of bus $j$, $Pi$ denote active power of bus $i$ and $Qi$ denote reactive power of bus $i$. Conductance and susceptance of bus $i$ and $j$ are denoted by $G_{ij}$ and $B_{ij}$, respectively. $\delta_{ij}$ denotes voltage phase angle difference between bus $i$ and $j.$ $i \in N,$ N denotes the set of all buses in distribution system.

4.3 Inequality Constraints

The both state variable constraints (13) and control variable constraints (14) fall into inequality constraints.

$$\left. {\begin{array}{*{20}c} {Q_{Gi\min } \le Q_{Gi } \le Q_{Gi\max } } \\ {U_{i\min } \le U_{i } \le U_{i\max } } \\ \end{array} } \right\}$$

(13)

$$\left. {\begin{array}{*{20}c} {U_{Gi\min } \le U_{Gi } \le U_{Gi\max } } \\ {T_{i\min } \le T_{i } \le T_{i\max } } \\ {Q_{Ci\min } \le Q_{Ci } \le Q_{Ci\max } } \\ \end{array} } \right\}$$

(14)

where $Q_{Gi }$ denotes reactive power output of generator bus $G_{i}$. $U_{i}$ stands for voltage load bus $i$. The voltage of generator bus $G_{i}$ is denoted by $U_{Gi }$ and the ratio of on-load tap changer is denoted by $T_{i }$. $Q_{Ci }$ stands for reactive power compensation capacity of reactive power device $C_{i} .$

5 Experimental Verification

This section includes two parts—(1) Experimental verification of MABC algorithm on standard benchmark functions. The outcomes are compared with standard ABC algorithm. (2) Modified method of ABC is implemented on standard IEEE 30 and 57 test bus systems to check its effectiveness. The outcomes are analyzed with data from existing literature. The comparative analysis of the outcomes reflects the effectiveness of MABC algorithm. The outcomes are tested in MATLAB software.

5.1 Experimental Setup

The performance of different optimization algorithms is checked by using various benchmark functions. We have to check performance of MABC algorithm with the help of four benchmark functions. Here four benchmark functions are selected for testing in experiments. Table 1 indicates the parameters used for benchmark functions such as their global optimum, mathematical definition, search scope [4]. The selected benchmark functions are minimization problems. We have tested all benchmark functions on both MABC and standard ABC algorithms in MATLAB software. The above-mentioned parameters are set to particular values for all benchmark functions and run both algorithms in MATLAB. The outcomes of algorithms are compared in respective sections.

Table 1 Benchmark functions used in experiments

Here, dimension D has been set to 10, 30 and 50, respectively. Algorithms are run with benchmark functions for each dimension and obtained best optimum solution. The obtained results on both algorithms (MABC and ABC) are compared.

To have better clarity in outcomes, we are using same parameters and values in both algorithms. The number of employed bees and onlooker bees is equal in this case, at 50. For every benchmark function, each algorithm is going for 30 trials. For ABC and MABC, the parameters N, limit and maxFEs values are as follows, respectively - 50, 100 and 1.5E + 05. As mentioned earlier, parameter C = 1.5 and MR to be fixed to 0.3 in MABC [4]. For each value of D, outcomes are mentioned in respective table. The superior outcome is mentioned in Bold in each table.

5.2 Outcomes for D = 10

With D = 10, both standard ABC and MABC algorithms are run on benchmark functions. Table 2 represents the outcomes obtained on the benchmark set for D = 10. We get to know that modified ABC method gives better solutions than standard ABC except for function $f_{2}$. For function $f_{2}$, modified ABC drops into local minima and find inferior solutions than standard ABC algorithm. It shows that the modified ABC strategy is not appropriate for this problem. For function $f_{3}$, MABC outperforms on standard ABC and algorithm converge to global optimum. Modified ABC shows much superior functioning than ABC. Also, MABC shows much faster convergence speed than standard ABC.

Table 2 Outcomes for D = 10

5.3 Outcomes for D = 30

With D = 30, both standard ABC and MABC algorithms are run on benchmark functions. Table 3 gives outcomes obtained for D = 30. For function $f_{3}$, MBC finds the overall minimum, but ABC falls into local minimum range. For function $f_{2}$, ABC shows better solutions than MABC. It seems that modified approach may limit the search for this function. This also shows that MABC converges much faster than standard ABC on function $f_{1}$.

Table 3 Outcomes for D = 30

5.4 Outcomes for D = 50

With D = 50, both standard ABC and MABC algorithms are run on benchmark functions. Table 4 shows the outcomes obtained when D = 50. With the increase in dimensions, the performance of the algorithms is severely affected. Related to ABC, MABC achieves superior solutions on benchmark functions. Like D = 30, ABC gives better results as compared to MABC for function $f_{2}$. The convergence speed is more for MABC as compared to standard ABC.

Table 4 Outcomes for D = 50

5.5 Simulation Results in IEEE-30 Bus System

For simulation, we're using data from the IEEE-30 bus system, which we got from MATPOWER[3]. It includes six thermal generators, four transformer taps and nine reactive power compensators. Here, 283.4 MW is an active power load for 30-bus system whereas 126.2 MVAR is reactive power load[5]. The test system has total of 41 transmission lines. 100 MVA is selected as base MVA. Limit variables data for state variables and control variables are given in Table 5.

Table 5 Limits of variables for IEEE-30 bus system

Figure 1 depicts the convergence characteristics of a 30-bus system. Here, it shows that MABC converges at 12th iteration. Obtained outcomes are compared with available literature data in Table 6. Total active power loss using MABC is obtained as 4.280 MW. When compared and analyzed the outcomes, it is observed that MABC outcome is good at tracking global optimum solutions.

Fig. 1

Convergence characteristics of IEEE-30 bus system

Table 6 Comparison table for IEEE-30 bus system

5.6 Simulation Results in IEEE-57 Bus System

For simulation, we're using data from the IEEE-57 bus system, which we got from MATPOWER [3]. It includes six thermal generating units, three reactive power compensators and seventeen transformer taps [6]. 100 MVA is selected as base MVA. Figure 2 shows convergence characteristics of test bus system. Here, MABC algorithm converges at 19th iteration. Total active power loss is reduced to 17.50 MW. Obtained outcomes are compared with available literature data in Table 7 [6].

Fig. 2

Convergence characteristics of IEEE-57 bus system

Table 7 Comparison table for IEEE-57 bus system

6 Conclusion

We worked on a modified approach to the ABC method and its implementation in this study. GABC has been enhanced by MABC. It aids in the enhancement of exploitation capabilities as well as the acceleration of convergence. This strategy can be used to tackle difficulties with reactive power optimization in power systems. Onlooker bees construct a weighted technique solution-finding equation for optimization purposes in a modified manner. ABC and MABC are tested using a set of benchmark functions. On both techniques, functions are examined with varied dimensions (10, 30 and 50). It compares the performance of the modified approach to the standard ABC method. On benchmark functions, MABC beats ordinary ABC, according to the results. Experimental studies support the use of MR in the MABC approach.

On IEEE 30 and 57 test bus systems, the modified ABC is used. We used MATLAB to run the system data from MATPOWER. The results of the test bus system demonstrate that MABC provides the best solution for the problem of reactive power optimization.