Saudi Journal of Biological Sciences (2017) xxx, xxx-xxx

King Saud University Saudi Journal of Biological Sciences

www.ksu.edu.sa www.sciencedirect.com

ORIGINAL ARTICLE

Dynamic population artificial bee colony algorithm for multi-objective optimal power flow

Man Ding a, Hanning Chenb'*, Na Linc, Shikai Jingb, Fang Liub, Xiaodan Liangb, Wei Liud

a School of Architecture and Art Design, Hebei University of Technology, Tianjin 300130, China b School of Computer Science and Software, Tianjin Polytechnic University, Tianjin 300387, China cBeijing Shenzhou Aerospace Software Technology Co. Ltd., Beijing 110000, China d College of Information and Technology, Jilin Normal University, Siping 136000, China

Received 11 October 2016; revised 25 December 2016; accepted 7 January 2017

KEYWORDS

Artificial bee colony algorithm;

Life-cycle evolving model; Optimal power flow; Multi-objective optimization

Abstract This paper proposes a novel artificial bee colony algorithm with dynamic population (ABC-DP), which synergizes the idea of extended life-cycle evolving model to balance the exploration and exploitation tradeoff. The proposed ABC-DP is a more bee-colony-realistic model that the bee can reproduce and die dynamically throughout the foraging process and population size varies as the algorithm runs. ABC-DP is then used for solving the optimal power flow (OPF) problem in power systems that considers the cost, loss, and emission impacts as the objective functions. The 30-bus IEEE test system is presented to illustrate the application of the proposed algorithm. The simulation results, which are also compared to nondominated sorting genetic algorithm II (NSGAII) and multi-objective ABC (MOABC), are presented to illustrate the effectiveness and robustness of the proposed method.

© 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

In many fields of science and engineering, there are always multiple conflicting objectives, which are formulated as multi-objective (MO) optimization problems in order to mini-

Corresponding author. E-mail address: chenhanning@tjpu.edu.cn (H. Chen). Peer review under responsibility of King Saud University.

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mize or maximize these conflicting objective functions simultaneously. In MO optimization domain, the set of Pareto optimal solutions, namely several optimal solutions with different trade-offs in the objective space, is called the Pareto optimal front (Fonseca and Fleming, 1998; Cruz et al., 2014). Optimal power flow (OPF) is one of the most important MO problems in power system. The main goal of OPF is to find the optimal adjustments of the control variables to minimize the selected objective function while satisfying various physical and operational constraints imposed by equipment and network limitations (Kumari and Maheswarapu, 2010). Since the real power generation levels and voltage magnitudes are continuous variables whereas the transformer winding

http://dx.doi.org/10.1016/j.sjbs.2017.01.045

1319-562X © 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

ratios and shunt capacitors are discrete variables, the OPF problem is considered as a non-linear multi-modal optimization problem with a combination of the discrete and continuous variables (Abou El Ela et al., 2010).

Many mathematical models and conventional techniques, such as gradient-based optimization algorithms, linear programming, interior point method, and Newton method, have been applied to solve the OPF problem (Momoh et al., 1999a,b). However, these methods suffer from severe limitations in handling non-linear, discrete and continuous functions, and constraints. In order to overcome the limitations of classical optimization techniques, a wide variety of the heuristic methods have been proposed to solve the OPF problem, such as genetic algorithm (GA) (Lai et al., 1997), tabu search (TS) (Abido, 2002), differential evolution (DE) algorithm (Sayah and Zehar, 2008), and biogeography based optimization (BBO) (Roy et al., 2010). The reported results are promising and encouraging for further research in this field (Abou El Ela et al., 2010). However, all the mentioned heuristic mathematical techniques have some drawbacks such as being trapped in local optima or each of them is only suitable for solving a specific objective function in the OPF problem (AlRashidi and El-Hawary, 2007).

Swarm intelligence (SI) is an innovative artificial intelligence technique for solving complex optimization problems (Xu et al., 2013). Among them, artificial bee colony algorithm (ABC) is a relatively new optimization technique which simulates the intelligent foraging behavior of a honeybee swarm (Ma et al., 2013; Eberchart and Kennedy, 1995). Recently, two multi-objective approaches based on ABC model were proposed in (Passino, 2002; Gao and Liu, 2011). However, compared to the huge in-depth studies of other multi-objective evolutionary and swarm intelligence algorithms, such as nondominated sorting genetic algorithm II (NSGAII) (Deb et al., 2002), strength Pareto evolutionary algorithm (SPEA2) (Zitzler et al., 2001), and multi-objective particle swarm optimization (MOPSO) (Coello Coello and Pulido, 2004), how to improve the diversity of swarm or overcome the local convergence of multi-objective ABC (MOABC) is still a challenging to the researchers in MO optimization domain.

In this paper, a novel artificial bee colony algorithm with dynamic population (ABC-DP) is proposed to synergize the idea of extended life-cycle evolving model, which can balance the exploration and exploitation tradeoff in artificial bee colony foraging process. The proposed ABC-DP is a more bee-colony-realistic model that the bee can reproduce and die dynamically throughout the foraging process and population size varies as the algorithm runs. By incorporating this new degree of complexity, ABC-DP can accommodate a considerable potential for solving complex MO problems. Then we applied ABC-DP to solve two and three objective OPF cases considering the cost, loss, and emission impacts as the objective functions respectively on the 30-bus IEEE test system. The simulation results, on both benchmarks and OPF cases, prove that ABC-DP has better optimization performance than the NSGA-II and MOABC algorithms.

The rest of the paper is organized as follows. Section 2 first gives a review of the original ABC algorithm. Section 3 proposes the novel ABC-DP algorithm with the life-cycle model. In Section 4, the multi-objective OPF problem is formulated,

and then the implementation of the ABC-DP on OPF is presented. Simulation results and comparison with other algorithms are given in Section 5. Finally, Section 6 outlines the conclusions.

2. The original artificial bee colony algorithm

From Fig. 1, we can understand the basic behavior characteristics of bee colony foraging behaviors better. Assume that there are two discovered food sources: A and B. At the very beginning, a potential bee forager will start as unemployed bee. That bee will have no knowledge about the food sources around the nest.

There are two possible options for such a bee:

i. It can be a scout and starts searching around the nest spontaneously for a food due to some internal motivation or possible external clue ('S' in Fig. 1).

ii. It can be a recruit after watching the waggle dances and starts searching for a food source ('R' in Fig. 1).

After finding the food source, the bee utilizes its own capability to memorize the location and then immediately starts exploiting it. Hence, the bee will become an ''employed forager". The foraging bee takes a load of nectar from the source and returns to the hive, unloading the nectar to a food store. After unloading the food, the bee has the following options:

iii. It might become an uncommitted follower after abandoning the food source (UF).

iv. It might dance and then recruit nest mates before returning to the same food source (EF1).

v. It might continue to forage at the food source without recruiting after bees (EF2).

It is important to note that not all bees start foraging simultaneously. The experiments confirmed that new bees begin

Figure 1 Behavior of honeybee foraging for nectar.

foraging at a rate proportional to the difference between the eventual total number of bees and the number presently foraging. In mathematical terms, the original ABC algorithm can be formulated as follows.

In the initialization phase, the ABC algorithm generates a randomly distributed initial food source positions of SN solutions, where SN denotes the size of employed bees or onlooker bees. Each solution xi (i = 1, 2,..., SN) is a D-dimensional vector. Here, D is the number of optimization parameters. And then evaluate each nectar amount fit. In ABC model, nectar amount is the solution value of benchmark function or real-world problem.

In the employed bees' phase, each employed bee finds a new food source i in the neighborhood of its current source xi. The new food source is calculated using the following expression:

Vj = Xij + /¡j(xij - xj

where k 2 (1,2, ..., SN) and j 2 (1,2, ..., D) are randomly chosen indexes, and k has to be different from i. / is a random number between [—1,1]. And then employed bee compares the new one against the current solution and memorizes the better one by means of a greedy selection mechanism.

In the onlooker bees' phase, each onlooker chooses a food source with a probability which related to the nectar amount (fitness) of a food source shared by employed bees. Probability is calculated using the following expression:

pi = fitJ Yfü'

In the scout bee phase, if a food source cannot be improved through a predetermined cycles, called "limit", it is removed from the population and the employed bee of that food source becomes scout. The scout bee finds a new random food source position using the equation below:

x = xL+rand[°, 1i(xm ax- x

where xmin and xmax are lower and upper bounds of parameter j, respectively.

These steps are repeated through a predetermined number of cycles, or until a termination criterion is satisfied. The pseudo code of original ABC algorithm is illustrated in Fig. 2.

3. The dynamic population ABC algorithm with life-cycle model

In biology, the term life-cycle refers to the various phases an individual passes through from birth to maturity, reproduction, and death. This process often leads to drastic transformations of the individuals with stage-specific adaptations to a particular environment. Inspired by this phenomenon, this work assumes that the computational life-cycle model of bee colony has five major stages, namely the born, forage, reproduction, death, and migration. The bee state transition diagram is shown in Fig. 3.

N (t +1) =

Ni(t) + 1 ffW) < fit(X) Ni(t) - 1 else

where fit (X\) is the fitness of the i' bee X at time t for a minimum problem, Ni (t) is the nutrient obtained by the ith bee Xi at time t. In initialization stage, nutrients of all bees are zero.

For each Xt at onlooker bee phase, if the new position is better than the last one, it is regarded that the bee will gain nutrient from the environment and the nutrient is added by one. Otherwise, it loses nutrient in the foraging process and its nutrient is reduced by one. Then the information rate F deciding to reproduce or die for each bee Xt at time t is computed as:

H(t) =

fit(X) -fit\

fitLt- IK

tworst

F = gg^L

g ^ S'

ELffj(')

(1 - ; 'c[<x 11

where fit'worst and fit'best are the current worst and best fitness of the whole bee colony at time t.

In the foraging process, if the bee Xt converts enough information rate Ft as:

Ft > max i^Freproduce; Freproduce

(s' - S)

it will reproduce an offspring by using best-so-far solution information in search equation of employed and onlooker bees steps based on the works of:

xnewij = Xij + u(Xbestj - xh

where xnew is the new offspring, xt is the ith bee, xbest is best individual of current colony, j is a randomly chosen indexes; / is a random number in range [—1, 1].

If the bee enters bad environment, and its information rate drops to a certain threshold as:

F < min 0

(S - S)

The pseudo-code of the proposed ABC-DP is listed in Table 1.

It will die and be eliminated from the population. Here S is the initial population size and St is the current colony size, Fsplit and Fadapt are two control parameters used to adjust the bee reproduction and death criterions.

It should be noticed that the population size will increase by one if a bee reproduces and reduce by one if it dies. As a result, the population size dynamically varies in the foraging process. At the beginning of the foraging process, the bee will reproduce when its information rate is larger than Freproduce. In the course of bee foraging, in order to avoid the population size becoming too large or too small, the reproduction and death criteria, namely Eqs. (7) and (9), are delicately designed: if St is larger than S, for each Fadapt of their differences, the reproduce threshold value will increase by one; if St is smaller than S, for each Fdapt of their differences, the death threshold value will decrease by one. The strategy is also consistent with the natural law: if the population is too crowded, the competition between the individuals will increase and death becomes common; if the population is small, the individuals are easier to survive and reproduce. When the nutrient of a bee is less than zero, but it has not died yet, it could migrate with a probability as a scout bee. A random number is generated and if the number is less than migration probability Pe, it will migrate and move to a randomly produced position. Then nutrient of this bee will be reset to zero.

ARTICLE IN PRESS

4 M. Ding et al.

Figure 2 Flowchart of ABC algorithm.

4. The optimal power flow problem formulation

In this paper, the OPF problem is to minimize three competing objective functions, fuel cost, emission, and real power loss, while satisfying several equality and inequality constraints. Generally the problem is formulated as follows.

min f(x, u)

s.t.g(x, u)—0 (10)

h(x, u)—0

where f is the optimization objective function, g is a set of constrain equations, and h is a set of formulated constrain in equations, u is a set of the control variables such as the generator real power output PG expect at the slack bus PG1, x is the vector of dependent variables such as the slack bus power PG1, the load bus voltage VL, generator reactive power outputs QG, and the apparent power flow Sk. x can be expressed as:

xT — [PG1 ; VL1 , ..., VLsg ; Qg ; ...; Q^ , ^ ...; ] (H)

Figure 3 Bee state transition in life-cycle model of ABC-DP.

Table 1 The pseudo-code of ABC-DP algorithm. Algorithm: The proposed HABC algorithm Step.1: Initialization

Step 1.1: Randomly generate SN food sources in the search space

to form an initial population by Eq. (1)

Step 1.2: Evaluate the fitness of each bee

Step 1.3: Set maximum cycles (LimitC)

Step. 2: Iteration = 0

Step. 3: Reproduction and death operations based on life-cycle model

Step.3.1: Calculate the information rate of each bee in the population by Eq. (5) and Eq. (5)

Step.3.2: If the criterion of reproduction determined by Eq. (7) is met, produce a new solution by Eq. (8), the population size increase by one

Step.3.3: If the criterion of death determined by Eq. (9) is met, the population size reduce by one

Step. 4: Employ bee phase: Loop over each food source Step.4.1: Generate a candidate solution Vi by Eq. (2) and evaluate f (Vi)

Step.4.2: Greedy selection and memorize the best solution Step. 5: Calculate the probability value pi by Eq. (3) Step. 6: Onlooker bee phase:

Step.6.1: Generate a candidate solution Vi by Eq. (2) and evaluate f (Vi)

Step.6.2: Greedy selection and memorize the best solution

Step. 9: Iteration = Iteration + 1;

Step. 10: If the iteration is greater than LimitC, stop the

procedure; otherwise, go to step. 3

Step. 11: Output the best solution achieved

0 = Po, - Pd, - V^VjiG.j cos hj + Bj sin hj) i 2 No (13)

o = QGi - QDi - VEVj(G,j cos hj + B,j sin hj) i e N

And the inequality constraints h(x, u) are limits of control variables and state variables which can be formulated as:

P^11 6 pGi 6 p^ Qr 6 qGi 6 Q; Qr 6 QCi 6 Q; TT11 6 Tk 6 TT* V™ 6 Vi 6 vmax ISkl 6 s;

i 2 No i 2 No i 2 Ne k 2 Nt i 2 Nb k 2 Ne

To solve non-linear constrained optimization problems, the most common method uses penalty function to transform a constrained optimization problem into an unconstrained one. The objective function is generalized as follows:

f=f + E kv, (v, - vm)2 + E ko, (Q, - o'om )2

iEN'im

+ E ks, (|S,|-STf

where lVi, kGi, and kSi are the penalty factors. Vfm and Q^m are

'■vt

defined as:

I Va** if V- > Vm™ if V, <vr

4.1. Minimization of total fuel cost

This objective function is to minimize the total fuel cost fcost of the system. The fuel cost curves of the thermal generators are modeled as a quadratic cost curves and can be represented as follows:

fcos t ^YfMP' + biP'i + ch

where at, bt and ct are the fuel cost coefficients of the ith generator, PGi is real power output of the ith generator.

4.2. Minimization of total power losses

where VG is the generator voltages, T is the transformer tap setting, and QC is the reactive power generations of var source. Therefore, u can be expressed as:

[P'2 ; ••• ; PGng ; VG1 ; •

• ; V GNg ;

C1 ; ... ; QCNC

if qGi > om,

if Qot < Q '

The equality constrains g(x, u) are the nonlinear power flow equations which are formulated as below:

The power flow solution gives all bus voltage magnitudes and angles. Then, the total MW active power loss in a transmission network can be described as follows:

fot = Y.Sk( V2 + V2 - 2 V V cos(d - Sj))

where Nl is the number of transmission lines, Vt and Vj are the voltage magnitudes at the ith bus and jth bus, respectively; Sj and Sj are the voltage angles at the ith bus and the jth bus, respectively.

Table 2 Characteristics of the generation units.

G1 G2 G3 G4 G5 G6

Generator limits

PCmax(MW) 150 150 150 150 150 150

PCmin(MW) 5 5 5 5 5 5

Cost coefficients

a 10 10 20 10 20 10

b 200 150 180 100 180 150

c 100 120 40 60 40 100

Emission coefficients

a 4.091 2.543 4.258 5.326 4.258 6.131

ß -5.554 -6.047 -5.094 -3.550 -5.094 -5.555

c 6.490 5.638 4.586 3.380 4.586 5.151

n 2.0e-4 5.0e-4 1.0e-6 2.0e-3 1.0e-6 1.0e-5

k 2.857 3.333 8.000 2.000 8.000 6.667

Emission(ton/h)

Cost($/h)

w 35 (0

Emission(ton/h)

0.2 610

Cost($/h)

1« 3 >

0.22 0.21

Emission(ton/h)

Cost($/h)

Figure 4 Pareto fronts obtained by ABC-DP, MOABC, and NSGA-II on Fuel cost - Emission-Loss (f1-f2-f3). (a) ABC-DP, (b) MOABC and (c) NSGA-II.

4.3. Total emission cost minimization

In this paper, two important types of emission gasses, namely, sulfur oxides SOx and nitrogen oxides NOx, are taken as the pollutant gasses. The emission gasses generated by each generating unit may be approximated by a combination of a quadratic and an exponential function of the generator active power output. Here, the total emission cost is defined as bellow:

./emission = X](ai + ßiPGi + CiP\

where femission is the total emission cost (ton/h) and at, b and c are the emission coefficients of the ith unit.

objectives. Fig. 4 clearly shows the relationships among all presented objective functions. Between the obtained Pareto-optimal solutions, it is necessary to choose one of them as a best compromise for implementation.

To directly analysis the population distribution of ABC-DP, MOABC and NSGA-II, the diversity metric is employed, which measures the extent of spread achieved among the obtained solutions. This metric is defined as:

df + dl + J=1 di - d df + d, + (N - 1)d

where di is the Euclidean distance between consecutive solutions in the obtained non-dominated set of solutions and N is the number of non-dominated solutions obtained by an

5. Results

In order to verify the proposed approach, the IEEE 30-bus system is used as the test systems with ABC-DP, MOABC, and NSGA-II algorithms. The IEEE 30 bus system data are given in (Alsac and Stott, 1974). The active power generation limits are listed in Table 2. The limits of generator buses and load buses are between 0.95-1.1 p.u, and 0.9-1.05 p.u, respectively. The lower and upper limits of transformer taps are 0.9 p.u. and 1.05 p.u., respectively, and the step size is 0.01 p.u.

Experiments were conducted with HMOABC, MOABC, and the nondominated sorting genetic algorithm II (NSGA-II). The NSGA-II algorithm uses Simulated Binary Crossover (SBX) and Polynomial crossover (Deb et al., 2002). We use a population size of 100. Crossover probability pc = 0.9 and mutation probability is pm = 1/n, where n is the number of decision variables.

For the MOABC, as described in (Zou et al., 2011), a colony size of 50, archive size A = 100 was adopted. The ABC-DP algorithm parameters were set as follows: the number of species K is set at 5, the colony size and archive size is N = 10, CR = 0.1, and A = 40, respectively. In the experiment, in order to compare the different algorithms with a fair time measure, the number of function evaluations (FEs) is used for the termination criterion.

In this simulation, three competing objectives are optimized simultaneously by the proposed algorithm and the obtained Pareto-optimal fronts are shown in Fig. 4. Table 3 shows the minimum values for each objective in the three-dimensional Pareto front (f1-f2-f3) (Table 4).

It is clear that cost, emission and loss cannot be further improved without degrading the other two related optimized

Table 4 Diversity of ABC-DP, NSGA-II for OPF problem.

MOABC,

Algorithm Diversity

ABC-DP 9.7129e-002

MOABC 2.8962e-001

NSGA-II 7.5989e-001

400 350 300 Ü 250 d1 200

150 100 50 0

ABC-DP

NSGA-II

OPF Problem

Figure 5 Computing time of ABC-DP, MOABC and NSGA-II on Multi-objective Optimal Power Flow.

Table 3 The best solutions for three-objective OPF.

Three-objective solution Three-objectives (/1-/2-/3) ABC-DP MOABC NSGA-II

Best f1 Bestf2 Bestf3 Best f1 Bestf2 Bestf3 Best f1 Bestf2 Bestf3

PG1 21.2312 47.0978 3.0986 30.1276 24.7689 7.1234 32.4354 16.4456 42.2234

PG2 174.5563 36.8765 13.1254 48.9845 48.9987 38.4457 25.8976 38.7655 62.4456

PG3 67.1351 61.2782 95.7895 41.5532 65.4532 87.9865 97.0871 71.5576 90.5564

PG4 102.4687 45.1243 60.5543 94.2376 34.4543 39.5543 30.5567 30.6675 48.5676

PG5 34.8767 43.6113 3.5563 21.0978 46.4567 23.8876 16.2344 68.5433 51.9878

PG6 43.4140 34.6787 81.7866 41.4445 45.4344 61.5554 81.4555 50.5543 36.7645

fi Fuel cost ($/h) 608.1678 650.1987 661.0433 610.1766 668.1655 632.9985 614.1323 656.9903 633.1543

f2Emission (ton/h) 0.2013 0.1921 0.2356 0.2433 0.1988 0.2309 0.2376 0.2132 0.2109

f3Loss (MW) 2.5432 3.3451 1.5876 2.5676 3.7988 1.7655 4.1231 4.5098 2.8676

algorithm. d is the average value of these distances. df and dl are the Euclidean distances between the extreme solutions and the boundary solutions of the obtained non-dominated set.

It can be proved that the proposed method is giving well distributed Pareto-optimal front for the three-objective OPF optimization. The results confirm that the ABC-DP algorithm is an impressive tool for solving the complex multi-objective optimization problems where multiple Pareto-optimal solutions can be obtained in a single run.

Here, we give a brief analysis on algorithm complexity of the proposed algorithm and other two compared algorithms. Assuming that the computation cost of one individual in the ABC-DP is Cost_a, St is the current population size, D is the problem dimension, then, the total computation cost of ABC-DP for one generation is St * Cost_a. However, due to its dynamical population, the time complexity of this procedure can be roughly estimated as O(S). Through directly evaluating the algorithmic time response on different combinations of objectives as shown in Fig. 5, we can observe that ABC-DP takes the less computing time in most cases. This can be explained that by the life-cycle strategy, the population size of the ABC-DP can be dynamically adaptive. Hence, the proposed ABC-DP algorithms have the potential to solve complex real-world problems.

6. Conclusion

In this paper, different multi-objectives, which consider the cost, loss, and emission impacts, for OPF problem were formed. These multi-objectives have been solved by the proposed ABC-DP. The proposed ABC-DP model extends original artificial bee colony (ABC) algorithm to synergize the idea of extended life-cycle evolving model, which can balance the exploration and exploitation tradeoff in artificial bee colony foraging process. The simulation studies, which conduct on 30-bus IEEE test system, show that the ABC-DP obtains better distributed Pareto optimal solutions than NSGA-II method in terms of optimization accuracy and convergence robust.

In this paper, we only apply the proposed ABC-DP on the optimal power flow problem of the 30-bus IEEE test system, which is a widely adopted simulation model. In the future work, we will analyze the more complex simulation model (e.g. 118-bus IEEE test system) and some virtual power flow system.

Acknowledgements

This research is partially supported by National Natural Science Foundation of China und Grant 61305082, 51378350, 5157518, 61602343 and 51607122.

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