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 13-bus 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 derivative-based and non-derivative-based 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 non-derivative-based 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 population-based 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 123-bus test systems. The node maps of the circuits are shown in Figures
IEEE 13-bus node map.
IEEE 123-bus node map.
The different load conditions are given in Tables
Active and reactive loads on IEEE 13-bus 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 123-bus 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 13-bus test system with no controls.
Two different simulation bus voltage magnitudes of IEEE 123-bus test system with no controls.
The minimum and maximum voltage magnitudes in pu with no control of IEEE 123-bus 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 13-bus compared to uncontrolled case in simulation I condition.
PSO method output of IEEE 13-bus compared to uncontrolled case in simulation I condition.
MFO method output of IEEE 13-bus compared to uncontrolled case in simulation I condition.
ICA method output of IEEE 123-bus compared to uncontrolled case in simulation II condition.
PSO method output of IEEE 123-bus compared to uncontrolled case in simulation II condition.
MFO method output of IEEE 123-bus compared to uncontrolled case in simulation II condition.
The numerical results in Tables
Best results values of IEEE 13-bus 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 123-bus 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 13-bus 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 123-bus 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 13-bus test system in simulation I.
The proposed algorithm iteration versus best fitness value of 123-bus 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.