The efficient planning and operation of power distribution systems are becoming increasingly significant with the integration of renewable energy options into power distribution networks. Keeping voltage magnitudes within permissible ranges is vital; hence, control devices, such as tap changers, voltage regulators, and capacitors, are used in power distribution systems. This study presents an optimization model that is based on three heuristic approaches, namely, particle swarm optimization, imperialist competitive algorithm, and moth flame optimization, for solving the voltage deviation problem. Two different load profiles are used to test the three modified algorithms on IEEE 123 and IEEE 13bus test systems. The proposed optimization model uses three different cases: Case 1, changing the tap positions of the regulators; Case 2, changing the capacitor sizes; and Case 3, integrating Cases 1 and 2 and changing the locations of the capacitors. The numerical results of the optimization model using the three heuristic algorithms are given for the two specified load profiles.
Power systems have been evolving in the last two decades, exhibiting such changes as deregulation and the integration of renewables into the philosophical and operational mentalities. From the operational point of view, control means that involving the coordinated operation of tap changing transformers, such as capacitors, is required because loads are not constant over time and the outputs of renewable energy sources are intermittent. Voltage optimization (VO) is an effective technology that has been saving the industry millions of dollars in wasted electrical energy since the beginning of the new millennium [
On the planning side, optimal capacitor locations are sought [
Producing the best possible result with the available resources is always an objective in engineering problems. Optimization problems are generally solved using two approaches. The first is based on mathematical analysis, and the second is based on numerical calculations. Numerical optimization methods can be divided into derivativebased and nonderivativebased methods. If the derivatives of the encountered model are not easy to find or a mathematical function related to the model does not exist, then nonderivativebased methods are applied. These methods are generally inspired by nature. The most popular model is GA, which reflects the evolution process in nature [
This work models the voltage optimization problem using three different heuristic algorithms, namely, imperialist competitive algorithm (ICA), particle swarm optimization (PSO), and moth flame optimization (MFO). Cases
The first model changes and uses the tap positions of the voltage regulators and obtains the optimal voltage value for given load conditions of the distribution system.
The second model uses only the capacitors and optimizes the sizes of these devices for given load conditions.
The third model uses the voltage regulators and the capacitors and finds the optimal tap positions, capacitor sizes, and locations.
MATLAB and a free power distribution system simulation tool OpenDSS [
The rest of the paper is organized as follows. Section
We model three different cases.
This case considers tap changers for the voltage regulators to minimize voltage deviations. The optimization model is as follows:
This case considers changing the size of the capacitors, and the model is as follows:
This case integrates Cases
ICA was recently developed in 2007 by Esmaeil Gargari and Caro Lucas for continuous optimization problems [
Flowchart of ICA [
The method is conducted as follows:
Form countries: the
where DN denotes the problem variables or dimensions. Initial random values for
Find the powers of each country by evaluating the objective function of the optimization problem as follows:
Select the imperialist and colonized countries. The power of a country is inversely symmetrical to its cost. The division of colonies among imperialists and the normalized value of each imperialist is defined as follows:
where
Then, the colony countries move to the imperialist ones to start the optimization process. The DN country population is generated, and
where
Calculate total power of an empire. It can be determined by the power of imperialist country plus percentage of power of its colonies as follows:
T.C.
There are some other hyperparameters used in the internal calculations of this algorithm; for example, the percent of search space size is a positive number (0.02), which enables the uniting process of two empires,
The flowchart of the modified ICA algorithms is shown in Figure
Flowchart of modified ICA.
Initialize the ICA parameters, namely, population size
Randomly create the size and location of the capacitors and tap positions of the regulators, and form the initial country as follows:
Run a load flow using the specified load profile and the solution candidates, perform a power flow, and calculate the fitness value of the test system depending on the case number as in (
Determine the imperialist and colonized countries depending on the fitness value as in (
Update the size and location of the capacitors and the tap position of the regulators for all empires as in (
Repeat Steps
PSO was originally developed in 1995 by Kennedy and Eberhart and inspired by the social behavior of schooling fish and flocking birds [
Figure
Flowchart of PSO algorithm.
By using the above notation, the method is implemented as follows:
Initialize the set constants, such as swarm size, dimension of the problem, maximum number of iterations, and upper and lower bounds.
Randomly initialize the individual positions.
Randomly initialize the individual velocities.
Repeat until the stopping condition is met.
Evaluate the fitness values using the objective function.
Determine pbest and gbest.
Determine the alteration particle velocity vector as follows:
where,
where maximum number of iterations is
Determine the alteration particle position vector as follows:
A flowchart of the modified PSO algorithms is shown in Figure
Flowchart of modified PSO algorithm.
Initialize the PSO parameters, namely, swarm size
Randomly create the initialized particle velocities, determine the size and location of the capacitors and tap positions of the regulators, and form particle positions as follows:
Run a load flow using the specified load profile, perform power flow using the solution candidates, and compute the corresponding fitness value of the test system depending on the case number as in (
Select local best (lb) and global best (gb), and then determine alteration particle velocity vector and particle positions as in (
Repeat Steps
MFO is a new populationbased algorithm refined in 2015 by Mirjalili; the optimization of this algorithm reflects transverse orientation, which is the method of transmission of moths in nature at night [
Flowchart of MFO algorithm.
This model is implemented as follows:
Initialize the set constants, such as number of moths, number of variables (dimension), and upper and lower bounds.
Randomly initialize the population of moths depending on the number of moths, number of variables, and upper and lower bounds as follows:
where
Calculate and store the corresponding fitness values for all the moths as follows:
where
Initialize the population of flames, which is equal sort population of moths, and flame fitness values, which are the equal sort moth fitness values.
where
Repeat until the stopping condition is met.
Calculate the distance between the
Update the position of moths using a spiral function as follows:
where
Update the flame position, which is equal to the best previous moth position and the current moth position (same as flame fitness values) as follows:
where
A flowchart of the modified MFO algorithms is shown in Figure
Flowchart of modified MFO algorithm.
Initialize the MFO parameters, namely, the number of moths
Randomly create the size and location of the capacitors and tap positions of the regulators, and form the initial moth position as in (
Use the specified load profile to run a load flow, perform power flow using the solution candidates, and calculate the moth fitness value of the test system as in (
Select the best moth position as a flame position and the best moth fitness value as the flame fitness value using (
Calculate the distance between moths and flames, and then calculate new moth position using (
Repeat Steps
The proposed optimization models are experimented on IEEE 13 and IEEE 123bus test systems. The node maps of the circuits are shown in Figures
IEEE 13bus node map.
IEEE 123bus node map.
The different load conditions are given in Tables
Active and reactive loads on IEEE 13bus for simulation I (minimum load) and simulation II (maximum load).
Bus number  Phases  Active load of simulation I (kW)  Active load of simulation II (kW)  Reactive load (kVar)  Load type 

671  a, b, c  854  1153  660  Delta 
634  a  98  160  110  Wye 
634  b  79  120  90  Wye 
634  c  80  120  90  Wye 
645  b  106  170  125  Wye 
646  b, c  160  230  132  Delta 
692  a, b, c  102  170  151  Delta 
675  a  320  485  190  Wye 
675  b  44  68  60  Wye 
675  c  202  290  212  Wye 
611  c  111  169  80  Wye 
652  a  80  128  86  Wye 
670  a  11  17  10  Wye 
670  b  42  66  38  Wye 
670  c  75  117  68  Wye 
Active and reactive loads on IEEE 123bus for simulation I (minimum load) and simulation II (maximum load).
Bus number  Phases  Active load of simulation I (kw)  Active load of simulation II (kw)  Reactive 
Load type  Bus number  Phases  Active load of simulation I (kw)  Active load of simulation II (kw)  Reactive 
Load type 

1  a  30  37  20  Wye  62  c  25  39  20  Wye 
2  b  12  19  10  Wye  63  a  27  39  20  Wye 
4  c  26  38  20  Wye  64  b  50  75  35  Wye 
5  c  13  19  10  Wye  65  a  23  35  25  Delta 
6  c  25  38  20  Wye  65  b  24  35  25  Delta 
7  a  14  19  10  Wye  65  c  52  69  50  Delta 
9  a  24  38  20  Wye  66  c  52  72  35  Wye 
10  a  13  20  10  Wye  68  a  12  20  10  Wye 
11  a  26  40  20  Wye  69  a  25  40  20  Wye 
12  b  14  20  10  Wye  70  a  13  20  10  Wye 
16  c  26  39  20  Wye  71  a  26  40  20  Wye 
17  c  12  20  10  Wye  73  c  27  40  20  Wye 
19  a  26  38  20  Wye  74  c  28  40  20  Wye 
20  a  26  39  20  Wye  75  c  28  38  20  Wye 
22  b  25  39  20  Wye  76  a  62  104  80  Delta 
24  c  26  40  20  Wye  76  b  46  70  50  Delta 
28  a  28  40  20  Wye  76  c  45  70  50  Delta 
29  a  28  39  20  Wye  77  b  26  40  20  Wye 
30  c  24  39  20  Wye  79  a  27  40  20  Wye 
31  c  13  20  10  Wye  80  b  30  40  20  Wye 
32  c  14  20  10  Wye  82  a  29  38  20  Wye 
33  a  26  40  20  Wye  83  c  12  20  10  Wye 
34  c  25  40  20  Wye  84  c  13  19  10  Wye 
35  a  28  40  20  Delta  85  c  25  40  20  Wye 
37  a  28  39  20  Wye  86  b  13  20  10  Wye 
38  b  12  19  10  Wye  87  b  27  40  20  Wye 
39  b  13  20  10  Wye  88  a  29  40  20  Wye 
41  c  12  19  10  Wye  90  b  29  39  20  Wye 
42  a  13  20  10  Wye  92  c  24  40  20  Wye 
43  b  25  40  20  Wye  94  a  26  40  20  Wye 
45  a  15  19  10  Wye  95  b  14  20  10  Wye 
46  a  14  19  10  Wye  96  b  13  20  10  Wye 
47  a, b, c  64  103  75  Wye  98  a  26  40  20  Wye 
48  a, b, c  137  202  150  Wye  99  b  30  40  20  Wye 
49  a  23  35  25  Wye  100  c  28  39  20  Wye 
49  b  45  69  50  Wye  102  c  12  20  10  Wye 
49  c  23  35  20  Wye  103  c  27  39  20  Wye 
50  c  29  39  20  Wye  104  c  26  39  20  Wye 
51  a  15  19  10  Wye  106  b  25  40  20  Wye 
52  a  25  40  20  Wye  107  b  25  40  20  Wye 
53  a  26  40  20  Wye  109  a  29  40  20  Wye 
55  a  13  20  10  Wye  111  a  15  20  10  Wye 
56  b  13  20  10  Wye  112  a  11  20  10  Wye 
58  b  13  20  10  Wye  113  a  25  40  20  Wye 
59  b  15  19  10  Wye  114  a  13  20  10  Wye 
60  a  14  19  10  Wye 
Graphical representations of the bus voltage magnitudes in pu of simulations I and II with no control (test systems do not contain tap regulators and capacitor banks) are shown in Figures
Two different simulation bus voltage magnitudes of IEEE 13bus test system with no controls.
Two different simulation bus voltage magnitudes of IEEE 123bus test system with no controls.
The minimum and maximum voltage magnitudes in pu with no control of IEEE 123bus test system in simulation I are 0.93317 and 0.99999, respectively, and in simulation II they are 0.91934 and 0.99999, respectively. The optimization model results for all cases, which are based on modified heuristic approaches ICA, PSO, and MFO, are graphically compared to uncontrolled results, as shown in Figures
ICA method output of IEEE 13bus compared to uncontrolled case in simulation I condition.
PSO method output of IEEE 13bus compared to uncontrolled case in simulation I condition.
MFO method output of IEEE 13bus compared to uncontrolled case in simulation I condition.
ICA method output of IEEE 123bus compared to uncontrolled case in simulation II condition.
PSO method output of IEEE 123bus compared to uncontrolled case in simulation II condition.
MFO method output of IEEE 123bus compared to uncontrolled case in simulation II condition.
The numerical results in Tables
Best results values of IEEE 13bus test system in simulation I condition.
Load profile condition  Voltage magnitude in pu  Algorithm  No control  Case 
Case 
Case 

Simulation I (minimum load)  Minimum  No control  0.9081  0.9081  0.9081  0.9081 
ICA  0.9081  0.98544  0.98462  0.98684  
PSO  0.9081  0.98544  0.98477  0.97644  
MFO  0.9081  0.98544  0.9846  0.9838  
Maximum  No control  0.99995  0.99995  0.99995  0.99995  
ICA  0.99995  1.0347  1.0209  1.0234  
PSO  0.99995  1.0347  1.022  1.0275  
MFO  0.99995  1.0347  1.0269  1.0227 
Best results values of IEEE 123bus test system in simulation II condition.
Load profile condition  Voltage magnitude in pu  Algorithm  No control  Case 
Case 
Case 

Simulation II (maximum load)  Minimum  No control  0.91934  0.91934  0.91934  0.91934 
ICA  0.91934  0.97758  0.98768  0.9685  
PSO  0.91934  0.97758  0.96908  0.9658  
MFO  0.91934  0.97758  0.98338  0.96869  
Maximum  No control  0.99999  0.99999  0.99999  0.99999  
ICA  0.99999  1.035  1.0408  1.0317  
PSO  0.99999  1.035  1.0404  1.0369  
MFO  0.99999  1.035  1.038  1.0334 
The smooth curves in Figures
Case
Case
Case
The performance curves in Figures
Best results values of IEEE 13bus for 3 phases.
Results type  Load profile condition  Algorithm  No control  Case 
Case 
Case 

Best fitness values; ( 
Simulation I (minimum load)  No control  0.10569  0.10569  0.10569  0.10569 
ICA  0.10569 

0.006764 


PSO  0.10569  0.010363  0.0060784  0.0068729  
MFO  0.10569  0.010363 

0.0066113  
Simulation II (maximum load)  No control  0.14316  0.14316  0.14316  0.14316  
ICA  0.14316 




PSO  0.14316  0.013653  0.0086557  0.014014  
MFO  0.14316  0.013653  0.0085749  0.012991  


Mean voltage magnitudes for best fitness values  Simulation I (minimum load)  No control  0.95284  0.95284  0.95284  0.95284 
ICA  0.95284 

1.0101  1.0065  
PSO  0.95284  1.0073  1.0063  1.0072  
MFO  0.95284  1.0073 



Simulation II (maximum load)  No control  0.94605  0.94605  0.94605  0.94605  
ICA  0.94605 

1.0075  1.0067  
PSO  0.94605  1.0102 

1.0079  
MFO  0.94605  1.0102  1.0072 




Standard deviation voltage magnitudes for best fitness values  Simulation I (minimum load)  No control  0.028612  0.028612  0.028612  0.028612 
ICA  0.028612 




PSO  0.028612  0.015797  0.011767  0.012222  
MFO  0.028612  0.015797  0.011881  0.013154  
Simulation II (maximum load)  No control  0.034842  0.034842  0.034842  0.034842  
ICA  0.034842 




PSO  0.034842  0.017181  0.014732  0.018642  
MFO  0.034842  0.017181  0.014085  0.018644 
Best results values of IEEE 123bus for 3 phases.
Results type  Load profile condition  Algorithm  No control  Case 
Case 
Case 

Best fitness values; ( 
Simulation I (minimum load)  No control  0.39548  0.39548  0.39548  0.39548 
ICA  0.39548 




PSO  0.39548  0.10046  0.092346  0.088997  
MFO  0.39548  0.10046  0.10902  0.079843  
Simulation II (maximum load)  No control  0.57663  0.57663  0.57663  0.57663  
ICA  0.57663 




PSO  0.57663  0.1038  0.13166  0.10153  
MFO  0.57663  0.1038  0.10887  0.085254  


Mean voltage magnitudes for best fitness values  Simulation I (minimum load)  No control  0.96441  0.96441  0.96441  0.96441 
ICA  0.96441 

1.0114 


PSO  0.96441  1.0129 

1.0113  
MFO  0.96441  1.0129  1.0141  1.0124  
Simulation II (maximum load)  No control  0.95697  0.95697  0.95697  0.95697  
ICA  0.95697 




PSO  0.95697  1.0123  1.016  1.0117  
MFO  0.95697  1.0123  1.0122  1.0094  


Standard deviation voltage magnitudes for best fitness values  Simulation I (minimum load)  No control  0.015601  0.015601  0.015601  0.015601 
ICA  0.015601 


0.014012  
PSO  0.015601  0.014796  0.015417  0.014598  
MFO  0.015601  0.014796  0.014815 


Simulation II (maximum load)  No control  0.018723  0.018723  0.018723  0.018723  
ICA  0.018723 




PSO  0.018723  0.015693  0.015685  0.015844  
MFO  0.018723  0.015693  0.01638  0.015424 
: The proposed algorithm iteration versus best fitness value of 13bus test system in simulation I.
The proposed algorithm iteration versus best fitness value of 123bus test system in simulation II.
The proposed optimization model is based on three metaheuristics approaches, namely, particle swarm optimization, imperialist competitive algorithm, and moth flame optimization, for solving the voltage deviation problem. That model uses three different cases: Case
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.