This study presents a novel hybrid multiobjective particle swarm optimization (HMOPSO) algorithm to solve the optimal reactive power dispatch (ORPD) problem. This problem is formulated as a challenging nonlinear constrained multiobjective optimization problem considering three objectives, that is, power losses minimization, voltage profile improvement, and voltage stability enhancement simultaneously. In order to attain better convergence and diversity, this work presents the use of combing the classical MOPSO with Gaussian probability distribution, chaotic sequences, dynamic crowding distance, and selfadaptive mutation operator. Moreover, multiple effective strategies, such as mixedvariable handling approach, constraint handling technique, and stopping criteria, are employed. The effectiveness of the proposed algorithm for solving the ORPD problem is validated on the standard IEEE 30bus and IEEE 118bus systems under nominal and contingency states. The obtained results are compared with classical MOPSO, nondominated sorting genetic algorithm (NSGAII), multiobjective evolutionary algorithm based on decomposition (MOEA/D), and other methods recently reported in the literature from the point of view of Pareto fronts, extreme, solutions and multiobjective performance metrics. The numerical results demonstrate the superiority of the proposed HMOPSO in solving the ORPD problem while strictly satisfying all the constraints.
The optimal reactive power problem (ORPD) has attracted great attention in the past decades because it can greatly improve economy and security of power system. Generally, the aim of the ORPD is to minimize the network real power loss and improve voltage profile by regulating generator bus voltages, switching on/off static VAR compensators, and changing transformer tapsettings, while satisfying various constraints.
Voltage stability is a major aspect of power system security analysis. It is well known that voltage instability and collapse have led to major system failures. Due to the continuous growth in the demand of electricity with unmatched generation and transmission capacity expansion, power systems are operated nearer against the voltage stability limit than before. Voltage instability is emerging as a new challenge to power system planning and operation. Hence, there is a need to consider voltage stability enhancement as one of the objectives in the ORPD. Voltage stability analysis often requires examination of a wide range of system conditions and a large number of contingency scenarios. For such application, many analysis methods of system voltage stability determination have been developed on static analysis techniques based on the power flow model, since these techniques are simple, fast, and convenient to use. In this work, a fast indicator of voltage stability,
In the literature, a large number of optimization techniques have been proposed to solve the ORPD problem. Many conventional methods such as nonlinear programming (NLP) [
In the last few years, the use of evolutionary algorithms for multiobjective optimization has significantly grown. These multiobjective evolutionary algorithms (MOEAs) can find multiple Paretooptimal solutions in one single run. Nondominated sorting genetic algorithm II (NSGAII) [
Particle Swarm optimization (PSO) [
Recently, some of these multiobjective optimization algorithms have been proposed to the MORPD problem. Strength Pareto evolutionary algorithm (SPEA) [
In this work, a new hybrid MOPSO algorithm (HMOPSO) is proposed for solving the multiobjective ORPD problem, in which three competing objectives such as real power loss minimization, voltage profile improvement, and voltage stability enhancement are optimized simultaneously in a single run. MOPSO is one of the modern heuristic algorithms that has already been applied successfully to solve complicated problems. With few parameters, MOPSO is simple to implement and efficient and robust compared to other algorithms. However, the classical MOPSO often converges to local optimal solutions so as to be premature convergence. Moreover, it suffers the problem of maintaining good diversity and uniform spacing among the solutions in the Pareto front. Therefore, some modifications have been incorporated into the classical MOPSO to improve its performance. In order to enhance overall stochastic search capability of algorithm and prevent premature convergence, the proposed approach employs Gaussian probability distribution to generate random numbers, chaotic sequences to tune the inertia weight, and a time variant structure for adjusting acceleration coefficients. In addition, a selfadaptive mutation operator is adopted to enhance the diversity of the population of MOPSO, which can avoid getting trapped in local optima. In order to obtain Pareto front with good diversity, dynamic crowding distance is adopted. The performance of the proposed approach is tested and evaluated on the standard IEEE 30bus and IEEE 118bus systems in both nominal and contingency situations. To validate the performance of the proposed algorithm, three performance metrics, namely, generational distance (GD) [
The objective is to minimize the real power loss in transmission lines expressed as
The objective is to minimize the deviations in voltage magnitude at load buses that can be expressed as
In this work, enhancing voltage stability can be achieved through minimizing the voltage stability indicator
For multinode system,
By segregating the load buses (PQ) from generator buses (PV), (
Rearranging the above equation, we get
The
Thus, a global system indicator describing the stability of the complete system is
These constraints represent load flow equation as
The inequality constraints represent the system operating constraints:
Up to now, there have been several proposals to extend PSO to handle multiobjective optimization problems. Here, we choose one of popular MOPSO algorithms developed by Coello Coello et al. [
In PSO, the group is a community composed of all the particles, which fly around in a multidimensional search space. During each iteration, each particle adjusts its position according to its past experiences (the best local position
Though MOPSO has been applied to the MOO problem with impressive success, it suffers from premature convergence and bad diversity of the Pareto front. In this paper, MOPSO is improved by incorporating Gaussian probability distribution, chaotic descending inertia weight, time variant acceleration coefficients, selfadaptive mutation operator, and dynamic crowding distance, which are explained below in detail.
In this paper, we employ Gaussian distribution sequences to generate the random numbers
In MOPSO, the parameter
This strategy can enhance the convergence velocity of MOPSO, but it pays out cost: MOPSO usually gets into the local optimum when it solves the question of more apices’ function. In order to overcome these draws, this paper implements chaotic inertial weight approach (CIWA) defined as follows:
In addition to
In order to diversify the population of MOPSO and avoid being trapped in local optimal solutions, a selfadaptive mutation strategy is newly introduced to the algorithm. The mechanism mixes several different mutation functions; each mutation function has different searching capability. This feature causes searching of the entire search space in different directions to be improved. Consequently, the diversity of the swarm increases intensely and the probability of being trapped in local minima decreases drastically; that will insure the global optimal value being found by the proposed HMOPSO. In this mutation technique, four different mutation functions are put forward to generate different mutant vectors described as follows:
In order to improve the diversity of the external archive, this paper employs a diversity maintenance strategy (DMS) based on dynamic crowding distance (DCD) [
The application of DCD to maintain the diversity of the external archive is explained below. At each iteration, the archive gets updated by incorporating nondominated solutions from the current population into archive. If the archive size
Calculate DCDs of particles in the archive
Sort the nondominated solutions in
Remove the individual with the lowest DCD value from
If
In the decision variables of the ORPD problem, generator voltages are continuous variables, which can be running at any real number within the limit boundary; transformer taps and reactive compensations are discrete variables, which can only be given a value from a fixed discrete values set. Here, we use real numbers to represent continuous variables and the integer to other variables. The vector of new control variables can be written as
Because the basic forms of MOPSO can only handle continuous variables, we employ a mixedvariable handling method [
From Section
The feasible solution is better than the infeasible solution.
For two feasible solutions, the one who has the better objective function value is better.
For two infeasible solutions, the one who has the smaller constraint violations is better.
According to the above criteria, objective and constraint violation information are considered separately. The update process of every iteration generation can be completed by comparing using the objective function value and the sum of all the constraints violations. In this paper, the sum of constraint violations is calculated as follows:
The most commonly used stopping criterion in MOEAs is a priori fixed number of generations. Unfortunately, it may involve a waste of computational resources, as the optimal algorithm goes on, being applied after a point where iterations get no improvement over the current solution. Thus, this paper adopts another efficient stopping criterion [
In addition, the follow chart of the proposed approach for solving ORPD problem is shown in Figure
Computational flow of the proposed HMOPSO for ORPD problem.
In this section, the effectiveness and efficiency of the proposed HMOPSO algorithm for solving the ORPD problem are tested on the standard IEEE 30bus and 118bus power systems and the results are compared with existing popular algorithms, MOPSO, NSGAII, MOEA/D, and other previous methods. All the techniques and simulations developed in this study are implemented on 1.83 GHz PC using MATLAB language. The load flow is run using MATPOWER 4.1 software [
After conducting a series of experiments with different parameter settings, the optimal parameters are selected as follows. For the IEEE 30bus system, the population size
HMOPSO: inertia weight
MOPSO:
NSGAII: crossover probability
MOEA/D: the neighborhood size
The standard IEEE 30bus test system consists of 6 generators, 4 transformers, and 2 VAR compensators. Thus the number of optimized variables is 12 in this problem. Four branches (6, 9), (6, 10), (4, 12), and (27, 28) are under load tap setting transformer branches within the range [
To demonstrate the effectiveness of the proposed algorithm, three different cases have been considered as follows:
Case 1: minimization of power loss and voltage deviation.
Case 2: minimization of power loss and voltage stability index.
Case 3: minimization of power loss, voltage deviation, and voltage stability index.
Owing to the randomness of the proposed HMOPSO, MOPSO, NSGAII, and MOEA/D, each case has 30 independent runs for different algorithms.
In this case, two objectives are considered, that is, power loss and voltage deviation. The best Pareto fronts obtained out of 30 runs using HMOPSO, MOPSO, NSGAII, and MOEA/D are shown in Figure
Optimal settings of control variables for
Variable  Initial  MOPSO  NSGAII  MOEA/D  HMOPSO  

Best 
Best VD  Best 
Best VD  Best 
Best VD  Best 
Best VD  

1.05  1.0708  1.0183  1.0658  1.0086  1.0800  1.0112  1.0715  1.0120 

1.04  1.0630  1.0084  1.0561  1.0008  1.0703  1.0041  1.0642  1.0022 

1.01  1.0415  1.0030  1.0346  1.0000  1.0482  1.0025  1.0401  0.9998 

1.01  1.0405  0.9995  1.0351  1.0013  1.0494  1.0060  1.0409  1.0000 

1.05  1.0744  1.0300  1.0417  0.9933  1.1000  1.0261  1.0773  1.0982 

1.05  1.0620  1.0418  1.0694  1.0650  1.0995  1.0500  1.0598  1.0376 

1.078  1.075  1  1.025  1  1.0125  0.9875  1.0375  1.05 

1.069  0.9  0.9125  0.9  0.9125  0.9125  0.9875  0.95  0.9875 

1.032  1  1  0.9875  1.025  1.0125  1  0.975  0.9875 

1.068  0.9625  0.925  0.95  0.925  0.95  0.925  0.9625  0.925 

0.0  0.05  0.01  0.01  0.04  0.1  0.02  0.02  0.01 

0.0  0  0.05  0.02  0  0.1  0.03  0.01  0.05 

5.8327  4.9849  5.5968  5.0465  5.6749  4.9595  5.6553  4.8536  5.8650 
VD (pu)  1.1582  0.7196  0.1222  0.7248  0.1274  0.7452  0.1237  0.8842  0.0913 
Best Pareto fronts obtained using HMOPSO, MOPSO, NSGAII, and MOEA/D for
For a fair and complete comparison, the extreme solutions obtained by the proposed approach are compared with SPEA [
Comparison of best solutions of different algorithms for IEEE 30bus system (Case 1).
Objective  Initial  SPEA [ 
GPAC [ 
CA [ 
BBO [ 
GSA [ 
OSAMGSA [ 
DE [ 
MDE [ 
HMOPSO 


5.8327  5.1167  5.0923  5.0921  4.9650  5.0924  5.0713  4.8750  4.8736  4.8536 
VD (pu)  1.1582  0.1442  0.1274  0.1225  0.15499  0.1133  0.1126  0.0927  0.0910  0.0913 
To validate the effectiveness of the proposed HMOPSO algorithm for enhancing the voltage security under stress condition and contingency states, two different conditions are considered in this case. Case 2(a) is heavy load condition. The load on each bus is uniformly increased to 150% of the base load condition to analyze the voltage stability level of the system under severe conditions; Case 2(b) is contingency state and the single line outrage (27–30) is simulated. Tables
Optimal settings of control variables for
Variable  Initial  MOPSO  NSGAII  MOEA/D  HMOPSO  

Best 
Best 
Best 
Best 
Best 
Best 
Best 
Best 


1.05  1.1000  1.0965  1.0978  1.0947  1.1000  1.0959  1.0923  1.0714 

1.04  1.0807  1.0844  1.0865  1.0882  1.0871  1.0771  1.0871  1.0545 

1.01  1.0496  1.0393  1.0547  1.0522  1.0400  1.0241  1.0400  1.0142 

1.01  1.0522  1.0547  1.0346  1.0313  1.0502  1.0525  1.0502  1.0103 

1.05  1.0962  1.1000  1.0864  1.0897  1.0831  1.0626  1.0831  1.0881 

1.05  1.0918  1.0784  1.0323  1.0344  1.0560  1.0815  1.0560  1.0678 

1.078  1.0125  1  0.9875  0.9875  1.0125  1  1.0125  0.9875 

1.069  0.95  0.9625  0.9  0.9  0.925  0.9  0.925  0.925 

1.032  1.025  0.9375  0.9375  0.9  0.9875  1  0.9875  0.9375 

1.068  0.95  0.9  0.925  0.925  0.925  0.9  0.925  0.925 

0.0  0.03  0.03  0  0.01  0.02  0.02  0  0.02 

0.0  0.03  0.03  0.04  0.03  0.01  0.03  0.01  0.01 

13.9971  12.1380  12.5101  12.8269  13.0480  12.1136  12.7399  12.0156  12.9451 

0.2410  0.1837  0.1715  0.1856  0.1824  0.1858  0.1739  0.1883  0.1636 
Optimal settings of control variables for
Variable  Initial  MOPSO  NSGAII  MOEA/D  HMOPSO  

Best 
Best 
Best 
Best 
Best 
Best 
Best 
Best 


1.05  1.0844  1.0837  1.0848  1.0803  1.0888  1.0754  1.0772  1.0512 

1.04  1.0735  1.0729  1.0723  1.0689  1.0734  1.0668  1.0733  1.0451 

1.01  1.0537  1.0533  1.0525  1.0456  1.0504  1.0297  1.0624  1.0175 

1.01  1.0558  1.0566  1.0525  1.0475  1.0506  1.0413  1.0555  1.0136 

1.05  1.0963  1.0962  1.0688  1.0674  1.0762  1.0771  1.0862  1.0747 

1.05  1.0793  1.0788  1.0805  1.0758  1.0649  1.0656  1.0646  1.0531 

1.078  1.0125  1.0125  0.9875  0.9875  0.9875  1.0375  1.0125  0.9875 

1.069  0.9  0.9  0.9125  0.925  0.9  0.9375  0.9875  0.9125 

1.032  0.9875  0.9875  1  1  1  1.05  1.025  1.05 

1.068  0.95  0.95  0.95  0.9375  0.95  0.925  0.95  0.925 

0.0  0.01  0  0.01  0.02  0.02  0.01  0.02  0.02 

0.0  0.04  0.3  0.02  0.02  0.01  0.05  0.01  0.03 

6.5179  5.4436  5.5493  5.5094  5.5548  5.4504  5.6932  5.3076  5.9774 

0.2956  0.2162  0.2061  0.2177  0.2168  0.2197  0.1980  0.2386  0. 1842 
In this case, three objectives, that is, power loss, voltage deviation, and voltage stability index are optimized simultaneously. Table
Optimal solutions of control variables for
Variable  Initial  NSGAII  MOEA/D  HMOPSO  

Best 
Best VD  Best 
Best 
Best VD  Best 
Best 
Best VD  Best 


1.05  1.0688  1.0256  1.0594  1.0828  1.0176  1.0706  1.0886  1.0425  1.0748 

1.04  1.0570  1.0246  1.0053  1.0727  1.0044  1.0642  1.0813  1.0316  1.0655 

1.01  1.0338  1.0248  1.0780  1.0478  0.9998  1.0349  1.0547  1.0143  1.0452 

1.01  1.0340  1.0057  1.0518  1.0480  0.9999  1.0362  1.0591  1.0132  1.0461 

1.05  1.0877  1.0543  1.0831  1.0938  1.0938  1.0627  1.0986  1.0664  1.0456 

1.05  1.0538  1.0287  1.0503  1.0951  1.0359  1.0597  1.0714  1.0238  1.0542 

1.078  0.9875  1  1.025  1.025  1.025  0.9875  1.025  1.05  1 

1.069  0.9375  0.9  0.9625  0.9  1  0.9625  0.9375  0.9125  0.9 

1.032  0.9625  0.95  0.9375  1.0125  0.9875  1.0125  1.0125  0.975  0.9375 

1.068  0.95  0.925  0.95  0.975  0.925  0.9  0.9625  0.925  0.925 

0.0  0.03  0.02  0  0  0  0.02  0.02  0.01  0.04 

0.0  0.01  0.03  0.02  0.08  0.06  0.01  0.03  0.05  0.04 

5.8327  5.2064  5.8076  5.8909  4.9803  5.6340  5.3730  4.8638  5.8846  5.4867 
VD (pu)  1.1582  0.6114  0.1268  0.6963  0.7145  0.1137  0.876  0.9625  0.1098  0.9886 

0.1728  0.1394  0.1437  0.1315  0.1385  0.1472  0.1301  0.1367  0.1567  0.1203 
To validate the possibility of the proposed HMOPSO to the largescale power system, the IEEE 118bus system is adopted. This system consists of 54 generators, 9 transformers, and 12 capacitor banks. Thus, the dimension of control variables in this case is 75. For more information about the system, one can refer to [
The best extreme
Optimal solutions for
Object  Initial  NSGAII  MOEA/D  HMOPSO  

Best 
Best VD  Best 
Best 
Best VD  Best 
Best 
Best VD  Best 


132.8629  124.1573  131.781  128.5268  121.458  134.647  131.332  120.0564  138.8756  132.4865 
VD (pu)  1.4393  1.3478  0.5841  0.9013  1.255  0.5231  0.968  1.5536  0.4237  1.2524 

0.0694  0.0569  0.0571  0.0552  0.0571  0.0586  0.0534  0.0582  0.0598  0.0512 
Comparison of the best solutions of different algorithms for IEEE 118bus system.
Object  Initial  GPAC [ 
CA [ 
APSODV [ 
BBO [ 
GSA [ 
OGSA [ 
HMOPSO 


132.8629  131.9083  131.8638  —  128.97  122.6762  126.99  120.0564 
VD (pu)  1.4393  1.2849  1.2756  0.4666  0.5020  0.5091  0.3666  0.4237 

0.0694  —  —  0.0538  0.0523  0.0600  0.0512 
In multiobjective optimization processes, two goals are normally taken into account:
In order to evaluate the performance metrics of these algorithms, each algorithm is executed 100 times for two test systems. The statistic comparison results of performance metrics for two test systems are shown in Table
Statistical results of performance metric for IEEE 30bus and IEEE 118bus systems.
Performance measures  IEEE 30bus system  IEEE 118bus system  

NSGAII  MOAE/D  MOPSO  HMOPSO  NSGAII  MOAE/D  MOPSO  HMOPSO  
GD  
Mean  0.113627  0.015378  0.015862 

0.330574  0.022916  0.024811 

Std. dev.  0.037141  0.00414  0.003135  0.002283  0.064578  0.003867  0.005095  0.003456 
MSP  
Mean  0.263546  0.32752  0.289632 

0.287658  0.41543  0.325624 

Std. dev.  0.040234  0.01564  0.02466  0.011338  0.058462  0.003133  0.038236  0.031247 
HV  
Mean  0.185746  0.22676  0.211651 

0.132385  0.153652  0.165727 

Std. dev.  0.035541  0.031572  0.041701  0.025813  0.042356  0.055814  0.063217  0.040472 
Table
Computational time of different algorithm for IEEE 30bus and 118bus systems.
Time  IEEE 30bus system  IEEE 118bus system  

NSGAII  MOAE/D  MOPSO  HMOPSO  NSGAII  MOAE/D  MOPSO  HMOPSO  
Best (s)  168.42  54.34  40.16  23.52  405.78  167.59  142.65  110.17 
Mean (s)  195.76  65.29  48.35 

451.69  191.58  174.34 

Worst (s)  213.84  78.67  55.37  41.78  510.36  215.28  191.28  165.54 
Std. dev.  7.48  4.13  3.18  3.46  23.57  11.38  16.55  13.76 
In this paper, a multiobjective ORPD problem with three conflicting objectives, such as real power loss minimization, bus voltage profile improvement, and voltage stability enhancement, is considered. To improve convergence and maintain good diversity, a new HMOPSO algorithm is proposed by incorporating Gaussian probability distribution, chaotic descending inertia weight, time variant acceleration coefficients, selfadaptive mutation operator, and dynamic crowding distance into the classical MOPSO. The proposed approach has been successfully applied to solve the ORPD problem on IEEE 30bus and IEEE 118bus systems under both normal and contingency states. The performance of the proposed approach is compared with existing popular MOEAs including NSGAII and MOEA/D and other methods recently reported in the literature with respect to various multiobjective performance measures. The comparison results demonstrate the superiority of HMOPSO in terms of solution quality and computational efficiency and confirm its potential to solve the ORPD problem in the practical power system operation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National High Technology Research and Development Program (2012AA050215) of the Ministry of Science and Technology of China.