The power loss in electrical power systems is an important issue. Many techniques are used to reduce active power losses in a power system where the controlling of reactive power is one of the methods for decreasing the losses in any power system. In this paper, an improved particle swarm optimization algorithm using eagle strategy (ESPSO) is proposed for solving reactive power optimization problem to minimize the power losses. All simulations and numerical analysis have been performed on IEEE 30-bus power system, IEEE 118-bus power system, and a real power distribution subsystem. Moreover, the proposed method is tested on some benchmark functions. Results obtained in this study are compared with commonly used algorithms: particle swarm optimization (PSO) algorithm, genetic algorithm (GA), artificial bee colony (ABC) algorithm, firefly algorithm (FA), differential evolution (DE), and hybrid genetic algorithm with particle swarm optimization (hGAPSO). Results obtained in all simulations and analysis show that the proposed method is superior and more effective compared to the other methods.
The reactive power optimization approach is important for power quality, system stability, and optimal operation of electrical power systems. Reactive power control can be set with adjusting the voltage levels, tap positions of transformers, shunt capacitors, and other control variables. The reactive power optimization approach can minimize the power losses and improve the voltage profiles. Many conventional methods such as dynamic programming, linear and nonlinear programing, interior point method, genetic algorithm, and quadratic programming have been employed for solving reactive power optimization problem [
Moreover, in the last years, various intelligence computation methods have proposed for the reactive power optimization such as particle swarm optimization, differential evolution, ant colony, and BC [
In this study, a particle swarm optimization algorithm using eagle strategy (ESPSO) has been developed to implement reactive power optimization for reducing power losses. Eagle strategy (ES) has been originated by the foraging behavior of eagles such as golden eagles. This strategy has two important parameters: random search and intensive chase. At first it explores the search space globally, and then in the second case the strategy makes an intensive local search with using an effective local optimizer method [
Moreover, simulations and analysis of reactive power optimization problem have been performed on IEEE 30-bus test system, IEEE 118-bus test system, and a real power subsystem, and the proposed approach has been compared with various algorithms to show performance. It can be seen that, in case studies, the proposed approach has been outperformed compared to other methods mentioned.
This paper defines the objective function as reducing power losses of power system. The objective function given in [
Note that superscripts “min” and “max” in (
Eagle strategy (ES) is a two-stage process, developed by Yang et al. [
Note that ES is not an algorithm; it is a method. In fact, various algorithms can be used at the different stages. This provides that it combines the advantages of these different algorithms so as to obtain better results.
Lévy distribution [
Pseudocode of ES can be given as in Pseudocode
Load objective function Initial population
Random global search (Levy walks) Local search by using PSO Update the current best
This technique processing is so easy that PSO utilizes some parameters and definitions of the optimization process and then it starts the process with an initial random population, named particles. Each of these particles has a possible solution for the main problem and is processed as a part in
Pseudocode of PSO see Pseudocode
Load objective function Generate the initial population and velocity of Find global best (at
Calculate new velocity and position of each particle via ( Update weight
We know that for local search we can use an algorithm such as PSO, DE, and ABC; therefore, PSO is applied to the local search stage of ES method. On the other side, randomization with Lévy walks can be used in the global search. The proposed method is a population-based algorithm.
We used the parameters of PSO used in most applications [
There are several stopping criteria given in the literature: a fixed number of generations, the number of iterations since the last change of the best solution being greater than a specified number, the number of iterations reaching maximum number, a located string with a certain value, and no change in the average fitness after some generations. In this paper, the stopping criteria are chosen as the maximum number of iterations and the tolerance value for fitness where
Steps of proposed method have been explained in Algorithm
performing random global search using Levy Flight ( Then, find a promising solution
between global search and local search. (We set
Calculate new velocity and position of each particle via ( Then evaluate new fitness (Use the objective function based on Newton–Raphson power flow for reactive power optimization problem)
Maximum number of iterations or a given tolerance (tolerance set as for reactive power optimization problem)
For testing proposed method, seven well-known benchmark functions are handled with comparison of widely used six algorithms (GA, PSO, ABC, FA, and hGAPSO). These benchmark functions are listed in Table
Benchmark functions.
Equation | Name |
|
Feasible bounds |
---|---|---|---|
|
Sphere/parabola | 30 |
|
|
Ackley | 30 |
|
|
Rosenbrock | 30 |
|
|
Generalized Griewank | 30 |
|
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Hartman 3 | 3 |
|
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Six-hump camel-back | 2 |
|
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Goldstein-Price | 2 |
|
All the methods run in 500 iterations and over 50 times for each function. The results obtained by proposed ESPSO algorithm on some benchmark functions are statistically different from the other algorithms and ESPSO has a good performance on the test functions. Results point out that ESPSO algorithm is appropriate for optimizations of unimodal and multimodal functions. Table
Comparison of ESPSO with other methods on test functions.
Function | Index | GA | PSO | ABC | FA | hGAPSO | ESPSO |
---|---|---|---|---|---|---|---|
|
Best | 1.1091 | 0.9206 | 0.8991 | 0.0037 | 0.8874 | 0.0026 |
Mean | 1.7623 | 1.0038 | 1.2978 | 0.0126 | 1.1172 | 0.9032 | |
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Best | 0.9398 | 0.7646 | 0.8991 | 0.0317 | 0.0918 | 0.0125 |
Mean | 1.6693 | 1.6237 | 2.6842 | 0.1004 | 1.9044 | 1.0005 | |
|
|||||||
|
Best | 101.9834 | 24.0076 | 38.5673 | 17.2783 | 22.9814 | 16.5622 |
Mean | 217.0324 | 65.3482 | 94.0345 | 19.2953 | 28.0912 | 54.8731 | |
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Best | 0.1983 | 0.0078 | 0.0659 |
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|
|
Mean | 0.9219 | 0.0396 | 0.15415 | 0.0117 | 0.0335 | 0.1931 | |
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Best | −3.8664 | −3.8642 | −3.8652 | −3.8628 | −3.8629 | −3.8628 |
Mean | −3.8824 | −3.8659 | −3.8739 | −3.8656 | −3.8728 | −3.8638 | |
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|
Best | −1.0525 | −1.0393 | −1.0353 | −1.0317 | −1.0318 | −1.0316 |
Mean | −1.1316 | −1.0626 | −1.0445 | −1.0319 | −1.0336 | −1.0330 | |
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Best | 3.1870 | 3.0782 | 3.0390 | 3.0256 | 3.0280 | 3.0130 |
Mean | 3.7683 | 3.5900 | 3.4802 | 3.3380 | 3.4517 | 3.2300 |
Unlike other algorithms, ESPSO performs global search and local search by using parameter
In reactive power optimization problem, transformer tap positions and shunt capacitor banks are discrete or integer variables. However, the proposed method only uses continuous variable. To use these variables, each particle of ESPSO explores in the space as searching for continuous variable, and objective function is evaluated by cutting the corresponding dimensions of particles into integers. This means that the real position values of particles consist of values of capacitor banks and transformers tap positions represented as a vector to be used for calculating objective function.
Here, inequality constraints are handled in iterations as follows: if variables violate their limit, they are clamped to their upper limit and the remaining mismatch is taken by another one not on limit based on their inertia, and during computation if any PV bus reactive power is violated then PV bus is assumed as PQ bus fixing at the brink value.
Furthermore, the procedure of solving reactive power problem can be explained as follows.
The proposed approach of ESPSO loads parameters of power system and initial condition and specifies upper and lower limits. Initial particles are determined randomly via uniform distribution to locate their initial positions, and initial velocities of particles are constituted. The algorithm evaluates the fitness using objective function and adjusts local bests and global best from locals.
While stopping criteria are provided, ESPSO performs that searching global best by Lévy flight and then determining a random number. If this determined random number is less then
To verify the ability, capability, and performance of the proposed ESPSO on reactive power optimization problem, it is implemented to IEEE 30-bus test system, IEEE 118-bus test system, and a real distribution subsystem. For testing and proving the performance of proposed method, it is compared with various algorithms such as GA, PSO [
The population-based algorithms GA, PSO, DE, and their improved version have a great interest in engineering optimization problems and they all have been successfully implemented to reactive power optimization problems [
In [
The results of studies in [
The proposed method is coded in MATLAB software for 30-bus and 118-bus test systems. Maximum number of iterations is set as 100 and ESPSO is run over 50 times. The best solutions for both systems are tabulated after 50 times running.
Power system includes 41 branches, 6 generators, 21 load buses, and 9 shunt compensators. Branches of 4–12, 6–9, 6–10, and 28-27 are adjustable tab under load and symbolized as
Limits of variables (pu).
Bus | 1 | 2 | 5 | 8 | 11 | 13 |
---|---|---|---|---|---|---|
|
0.500 | 0.200 | 0.150 | 0.100 | 0.100 | 0.120 |
|
2.000 | 0.800 | 0.500 | 0.350 | 0.300 | 0.400 |
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−0.200 | −0.200 | −0.150 | −0.150 | −0.100 | −0.150 |
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2.000 | 1.000 | 0.800 | 0.600 | 0.500 | 0.600 |
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Variable |
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Limit | 0.90 | 1.10 | 0.90 | 1.10 | 0.00 | 0.50 |
For comparing the proposed approach with other algorithms mentioned previously, index related to performance consists of the minimization of power loss given in Table
Control variables values and power loss (control variables given pu).
Variables | GA | PSO | ABC | FA | DE | hGAPSO | ESPSO |
---|---|---|---|---|---|---|---|
|
1.0345 | 1.0606 | 1.0927 | 1.1000 | 1.0500 | 1.0300 | 1.0770 |
|
1.0463 | 1.0524 | 1.0880 | 1.0967 | 1.0446 | 1.0400 | 1.0775 |
|
1.0294 | 1.0284 | 1.0695 | 1.0850 | 1.0247 | 1.0400 | 1.0700 |
|
1.0283 | 1.0289 | 1.0722 | 1.0895 | 1.0265 | 1.0200 | 1.0700 |
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1.0517 | 0.9833 | 1.0860 | 1.0930 | 1.1000 | 1.0300 | 1.0800 |
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1.0214 | 0.9924 | 1.0926 | 1.0969 | 1.1000 | 0.9500 | 1.0810 |
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1.0314 | 1.0530 | 0.9983 | 1.0478 | 1.0000 | 1.0300 | 1.0050 |
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1.0994 | 1.1000 | 0.9994 | 0.9439 | 1.1000 | 1.0800 | 1.0050 |
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1.1000 | 1.0745 | 0.9984 | 1.0318 | 1.0800 | 1.7000 | 0.9800 |
|
0.9071 | 0.9247 | 1.0034 | 1.0044 | 0.9200 | 1.0400 | 0.9720 |
|
0.0153 | 0.1174 | 0.0155 | 0.0534 | 0.2600 | — | 0.2050 |
|
0.0063 | 0.0056 | 0.0371 | 0.0663 | 0.1000 | — | 0.1170 |
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4.78 | 4.39 | 3.09 | 4.71 | 5.01 | 3.69 | 3.079 |
Generators powers (MW).
Variables | ESPSO |
---|---|
|
51.50 |
|
80.00 |
|
50.00 |
|
35.00 |
|
30.00 |
|
39.98 |
The convergence curve of power loss (MW) for 30-bus power system.
The variation of generator voltages for IEEE 30-bus power system.
The variation of tap ratios for IEEE 30-bus power system.
The variation of shunt compensators for IEEE 30-bus power system.
The IEEE 118-bus system data are given in [
The results obtained by proposed method are given in Table
Comparison of the power loss for IEEE 118-bus test system.
Variables |
|
---|---|
GA | 139.16 |
PSO | 135.64 |
ABC | 119.6923 |
FA | — |
DE | 128.318 |
hGAPSO | — |
ESPSO | 119.5500 |
Tap ratio.
Variables | ESPSO |
---|---|
|
0.9850 |
|
0.9900 |
|
0.9910 |
|
0.9850 |
|
0.9500 |
|
1.0000 |
|
0.9950 |
|
0.9450 |
|
1.0050 |
The convergence curve of power loss (MW) for 118-bus power system.
Variables of IEEE 118-bus power system: (a) value of shunt compensator (pu), (b) voltages of generators (pu), and (c) power of generators (MW).
The proposed method has been used to minimize the power loss by using reactive power optimization on Eregli Distribution Subsystem in Turkey. All data and parameter have been supplied from Eregli Branch Office of Meram Electricity Distribution Corporation.
The power subsystem considered in this paper includes 1311 buses at 0.4 kV, 9 buses at 31.5 kV, and 3 buses at 15.8 kV and 12 branches with adjustable tab under load. First bus is selected as the slack bus, 2–9 buses are PV buses, and others are selected as PQ buses (switching capacitor banks are located on all 0.4 kV buses). In this respect, vector dimensions become 1333 dimensions composed of nine voltage magnitude values, 12 tab settings, and 1311 switchable capacitor banks. The discrete control variables given in Table
The limits of control variables.
Parameter | Min. val. (pu) | Max. val. (pu) |
---|---|---|
Voltage | 0.9 | 1.1 |
Capacitor | 0.1 | 0.61 |
Tap position | 0.9 | 1.1 |
For comparing the proposed method with GA, PSO, ABC, FA, and hGAPSO, the performance index including minimum active power losses is illustrated in Figure
Control variables values and power loss (control variables given pu).
Variables | GA | PSO | ABC | FA | DE | hGAPSO | ESPSO |
---|---|---|---|---|---|---|---|
|
1.0100 | 1.0200 | 1.0400 | 1.0600 | 1.0600 | 1.0600 | 1.0600 |
|
1.0300 | 1.0400 | 1.0200 | 1.0100 | 0.9900 | 1.0300 | 1.0500 |
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0.9800 | 1.0200 | 1.0400 | 1.0100 | 1.0600 | 1.0600 | 1.0300 |
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1.0400 | 1.0300 | 1.0200 | 1.0400 | 0.9800 | 1.0300 | 1.0500 |
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0.9900 | 1.0600 | 1.0400 | 0.9900 | 0.9500 | 1.0600 | 1.0200 |
|
0.9600 | 1.0600 | 1.0200 | 1.0500 | 1.0100 | 1.0300 | 1.0100 |
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1.0600 | 1.0600 | 1.0400 | 1.0600 | 1.0100 | 1.0600 | 1.0100 |
|
1.0600 | 1.0500 | 1.0200 | 1.0500 | 0.9700 | 1.0300 | 1.0400 |
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1.0200 | 1.0400 | 1.0000 | 1.0000 | 0.9900 | 1.0000 | 1.0500 |
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1.07 | 1.03 | 1 | 0.98 | 1.05 | 1 | 1 |
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1.05 | 1 | 1 | 0.98 | 1.04 | 0.97 | 0.98 |
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1 | 1.06 | 1 | 1.04 | 1.05 | 0.97 | 1.04 |
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1.01 | 1 | 1 | 1.01 | 1 | 0.98 | 1.01 |
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1.03 | 1 | 1.05 | 0.97 | 0.98 | 1.01 | 1 |
|
1 | 1 | 1.03 | 1.01 | 0.98 | 1.05 | 1 |
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0.98 | 0.97 | 1.03 | 0.99 | 0.97 | 1.04 | 0.98 |
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0.96 | 0.98 | 0.97 | 1.03 | 1 | 0.98 | 1.03 |
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1 | 1 | 0.98 | 0.98 | 0.98 | 1 | 0.99 |
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0.98 | 0.98 | 0.97 | 1.04 | 1.04 | 1 | 1 |
|
0.97 | 0.96 | 1.06 | 1 | 1.05 | 1.01 | 1 |
|
0.99 | 0.07 | 1.05 | 1 | 1.01 | 1.01 | 1 |
|
0.91576 | 0.91189 | 0.91758 | 0.90622 | 0.91760 | 0.90922 | 0.89730 |
Convergence of active power loss (MW).
In order to clarify the ability of ESPSO that is statistically more robust and better than other methods, it can be shown that the minimum power loss obtained by ESPSO is smaller than all others listed and it conducts global searches accurately. After 50 times running it can be seen that the power losses are decreased from 1.0961 MW to 0.8973 MW by the proposed ESPSO which is the highest power loss reduction. The results obtained by all algorithms handled in this paper are given below, which indicates minimum power loss, voltage magnitude of buses, and tap ratios.
Eagle strategy is a method of combination of global search and intensive local search for optimization. In this paper, an approach, ES with PSO based reactive power optimization method, has been implemented to a reactive power optimization problem for minimizing the power losses of 30-bus test system, 118-bus test system, and a real distribution subsystem. It can be seen that two case studies in this paper which are performance test on benchmark functions and the minimization of power losses of various power systems may help to clarify the capability of proposed approach for optimization problem. Furthermore, ESPSO is effectively solving optimization problems and finding the optimum more successfully than the other algorithms. On the other hand, about benchmark functions performance, the proposed approach has a better performance than all other algorithms. So, it can be seen that the proposed ESPSO algorithm is influential and able to solve power loss minimization problem and may become a good candidate for other optimization problems such as reactive power dispatch, cost minimization, or multiobjective optimization.
Total power losses
Conductance between nodes
Admittance between nodes
Angle difference between nodes
Number of power system buses
Number of slack buses
Number of PQ buses
Number of transformer branches
Number of capacitor installed buses
Number of generator buses
Tap position of transformers at branches
Shunt capacitor value
Voltage magnitude of bus
Voltage magnitude of bus
Voltage magnitude of each generator
Voltage magnitude of each bus
The injected reactive power
The injected active power
The reactive power demanded by load
The active power demanded by load
Transmission line current flow capacity at line between nodes
Lévy distribution function
Standard gamma function
Gamma function parameter
The initial step length value
Step length
Number of iterations
Velocity of a particle in PSO
Position of a particle in PSO
Local best position
Global best
The weight function
Selected randomly in the range
Acceleration coefficients
Control parameter of switching
Objective function.
The authors declare that they have no conflicts of interest.