Abstract
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.
1. Introduction
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 [1]. High demand used to be managed by voltage reduction [2]. Another way of helping system operation is using capacitors to improve the power factor and the voltage profile and reduce power losses [3]. Furthermore, tap operations of voltage regulators are helpful in enhancing voltage profiles. Capacitors and voltage regulators are integrated, and improved voltage profiles are obtained. However, the life span of these devices is shortened by frequent operation because they are based on mechanical switch operations. New technological developments have made electronics-based voltage regulators and capacitors available [4], thereby bringing additional flexibility into the operation of smart grids.
On the planning side, optimal capacitor locations are sought [4]. For instance, in an algorithm that depends on dynamic programming, fuzzy logic and genetic algorithm (GA) approaches are used for capacitor distribution in distribution feeders. Gravitational search algorithm was used for optimal capacitor placement in [5], whereas a teaching-learning-based optimization was used for the same aim in [6]. Capacitors can also be used to reduce the effects of harmonics in distribution systems; the harmony search algorithm was applied for this goal in [7]. Capacitor location and sizing problem have been solved by other heuristics, such as clonal selection algorithm [8], ant colony optimization algorithm [8], and PSO [9].
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 [10]. Subsequently, methods inspired by the behaviors of birds and fish (particle swarm optimization [PSO]) [11], improvisation process of musicians (harmony search) [12], and the navigation approach of moths in nature, which is named transverse orientation (moth flame optimization [MFO]) [13], were developed.
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 1 and 2 are applicable to operation, and Case 3 is applicable to planning in distribution systems.(i)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.(ii)The second model uses only the capacitors and optimizes the sizes of these devices for given load conditions.(iii)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 [14, 15] are used in the simulations.
The rest of the paper is organized as follows. Section 2 proposes the voltage optimization models. Section 3 briefly explains ICA, PSO, MFO, and modified algorithm-based voltage deviation. Section 4 presents the experiments and the simulation results. Section 5 presents the conclusions.
2. Model
We model three different cases.
Case 1. This case considers tap changers for the voltage regulators to minimize voltage deviations. The optimization model is as follows: where denotes the fitness values (cost), is the number of buses, is the voltage magnitude of bus , Tapi is the tap position of the regulator, and and represent the minimum and maximum positions that a tap in a regulator can take, respectively. These values are in the range of .
Case 2. This case considers changing the size of the capacitors, and the model is as follows: where represents the fitness values (cost), is the number of buses, is the voltage magnitude of bus , is the size of the bank capacitor, and is the maximum size of the bank capacitor.
Case 3. This case integrates Cases 1 and 2 and changes the locations of the capacitors. The mathematical model is as follows: where represents the fitness values (cost), is the number of buses, is the voltage magnitude of bus , is the tap position of the regulator, and represent the minimum and maximum positions that a tap in a regulator can take, respectively (these values are in the range of ), is the size of the bank capacitor, is the maximum size of the bank capacitor, represents the location of the capacitors, and represents the maximum bus location.
3. Heuristic Algorithms
3.1. Imperialist Competitive Algorithm (ICA)
3.1.1. General Approach
ICA was recently developed in 2007 by Esmaeil Gargari and Caro Lucas for continuous optimization problems [16]. The working philosophy corresponds to other evolutionary algorithms and initially creates random solution candidates called countries. The cost function of each solution candidate shows the power of each country. Hence, populations are composed of either colonized or imperialist countries. According to random rules, a part of a population is selected as the imperialists or the powerful countries, and the remaining part of the population comprises the colonized. Figure 1 presents a flowchart of ICA [16].
The method is conducted as follows:(i)Form countries: the th country is formed as follows: where DN denotes the problem variables or dimensions. Initial random values for should be within the lower and upper ranges for each variable.(ii)Find the powers of each country by evaluating the objective function of the optimization problem as follows:(iii)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 is the cost of th imperialist and is the normalized value.(iv)Then, the colony countries move to the imperialist ones to start the optimization process. The DN country population is generated, and represents the most powerful population, whose members are selected as imperialists (the sets of controller coefficients with smaller cost function in this problem). The remaining countries are the colonies (the sets of controller coefficients with a high cost function in this problem), each of which is a part of one of the above-mentioned empires. In the attraction policy, the colonies move toward the imperialists along Mx units and are situated in a new position. Mx is a random variable with regular distribution and can be expressed as follows: where β is a constant number greater than 1 and ds is the space between imperialist and colony. To search different points around the imperialist, we add a random amount of deviation to the direction of movement as follows: , where ץ is a parameter to adjust the deviation value .(v)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. = cost (imperialist ) + · mean (cost (colonies of empire)), where T.C. is the total cost of th empire and is a positive number that is considered to be less than 1 (). The weakest colony of the weakest empire is picked out.
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, α is a number in the interval of and denotes the importance of mean minimum compares to the global minimum, and revolution rate is a positive number (0.3) representing the process in which the sociopolitical characteristics of a country change suddenly.
3.1.2. ICA-Based Voltage Deviation Algorithm
The flowchart of the modified ICA algorithms is shown in Figure 2, and the steps are as follows.
Step 1. Initialize the ICA parameters, namely, population size , maximum iteration number , number of imperialist countries , and number of colony countries . Set the voltage magnitude limits, and set the possible capacitor locations, capacitor size limits, and minimum and maximum tap positions depending on the case being solved.
Step 2. Randomly create the size and location of the capacitors and tap positions of the regulators, and form the initial country as follows: where and represent the numbers of bank capacitors and voltage regulators, respectively.
Step 3. 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 (1), (4), and (6).
Step 4. Determine the imperialist and colonized countries depending on the fitness value as in (9) and (10).
Step 5. Update the size and location of the capacitors and the tap position of the regulators for all empires as in (11).
Step 6. Repeat Steps 3–5 until the stopping condition is met.
3.2. Particle Swarm Optimization (PSO)
3.2.1. General Approach
PSO was originally developed in 1995 by Kennedy and Eberhart and inspired by the social behavior of schooling fish and flocking birds [17]. The birds in a group are considered an individual in the PSO method. These particles can be flown through a search space. The location of a particle in the search problem represents one solution for the problem. A new and different solution is created when the individual moves to a new location in the search space. Each solution can be evaluated using an objective function that supplies a cost of the benefit of the solution. The direction and velocity of each individual can move along all dimensions of the search space and thus can change with all generation of movement. PSO is generally considered an evolutionary computation (EC) sample. Other EC examples include evolutionary strategies, genetic programming, evolutionary programming, and GA [18]. Each individual maintains the following information [19]:(i) is the individual current position.(ii) is the individual current velocity.(iii) is the local better position of the individual (pbest), the better position visited yet by the individual.(iv) is the global better position of the swarm (gbest), the better position visited yet by the entire swarm.
Figure 3 shows a flowchart of the PSO algorithm.
By using the above notation, the method is implemented as follows:(1)Initialize the set constants, such as swarm size, dimension of the problem, maximum number of iterations, and upper and lower bounds.(2)Randomly initialize the individual positions.(3)Randomly initialize the individual velocities.(4)Repeat until the stopping condition is met.(5)Evaluate the fitness values using the objective function.(6)Determine pbest and gbest.(7)Determine the alteration particle velocity vector as follows: where, represents current iteration, and represent uniform random numbers between 0 and 1, acceleration coefficients are and , usually between 0 and 4, and represents the inertia weight; a damping factor, usually decreasing from around 0.9 to around 0.4 during the computation, is calculated as follows: where maximum number of iterations is .(8)Determine the alteration particle position vector as follows:
3.2.2. PSO-Based Voltage Deviation Algorithm
A flowchart of the modified PSO algorithms is shown in Figure 4, and the steps are as follows.
Step 7. Initialize the PSO parameters, namely, swarm size , dimension of the problem , maximum number of iterations , cognitive parameter , social parameter , upper bound value ub, lower bound value lb, and maximum velocity value . Set the voltage magnitude limits, and set the possible capacitor locations, capacitor size limits, and minimum and maximum tap positions depending on the case being solved.
Step 8. 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:where and represent the numbers of bank capacitors and voltage regulators, respectively.
Step 9. 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 (1), (4), and (6).
Step 10. Select local best (lb) and global best (gb), and then determine alteration particle velocity vector and particle positions as in (13)–(15).
Step 11. Repeat Steps 3 and 4 until the stopping condition is met.
3.3. Moth Flame Optimization (MFO)
3.3.1. General Approach
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 [13]. Approximately 160,000 different groups of insects, including moths, are present in nature. Moths have two life phases: larvae and adult phases. These insects are considerably similar to the family of butterflies but possess a special feature when moving at night [20]. Moths fly straight lines over long distances by preserving a fixed angle with the moon. This mechanism is effective for traveling, especially when the light source is far. When the light source is close, moths fly around it in a spiral path and ultimately converge with it. These insects represent the candidate solutions, and their position in the search space represents the problem variables. Therefore, moths can fly in one or more dimensions by updating the position vectors. Figure 5 presents a flowchart of the MFO algorithm.
This model is implemented as follows:(1)Initialize the set constants, such as number of moths, number of variables (dimension), and upper and lower bounds.(2)Randomly initialize the population of moths depending on the number of moths, number of variables, and upper and lower bounds as follows: where and represent the numbers of moths and variables, respectively.(3)Calculate and store the corresponding fitness values for all the moths as follows: where represents the number of moths.(4)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 and represent the numbers of moths and variables, respectively.(5)Repeat until the stopping condition is met.(6)Calculate the distance between the th flame and the th moth as follows: (7)Update the position of moths using a spiral function as follows: where represents the distance, is a random value in , and is a constant number.(8)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 represents the current iteration.
3.3.2. MFO-Based Voltage Deviation Algorithm
A flowchart of the modified MFO algorithms is shown in Figure 6, and the steps are as follows.
Step 12. Initialize the MFO parameters, namely, the number of moths , variable number , maximum number of repetitions , upper bound value ub, and lower bound value lb. Set the voltage magnitude limits, and set the possible capacitor locations, capacitor size limits, and minimum and maximum tap positions depending on the case being solved.
Step 13. Randomly create the size and location of the capacitors and tap positions of the regulators, and form the initial moth position as in (17).
Step 14. 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 (18). Use (1), (4), and (6) depending on the case number.
Step 15. Select the best moth position as a flame position and the best moth fitness value as the flame fitness value using (23) and (24), as shown in (19) and (20), respectively.
Step 16. Calculate the distance between moths and flames, and then calculate new moth position using (21) and (22).
Step 17. Repeat Steps 3–5 until the stopping condition is met.
4. Experiments and Simulation Results
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 7 and 8, respectively.
The different load conditions are given in Tables 1 and 2 and are denoted as simulation I (minimum load) and simulation II (maximum load) on the IEEE 13- and IEEE 123-bus test systems, respectively.
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 9 and 10 for the IEEE 13- and IEEE 123-bus test systems, respectively. The minimum and maximum voltage magnitudes in pu with no control of IEEE 13-bus test system in simulation I are 0.9081 and 0.99995, respectively, and in simulation II they are 0.89236 and 0.99993, respectively.
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 11–16 for the IEEE 13- and IEEE 123-bus test systems, respectively.
The numerical results in Tables 3 and 4 support graphically results in Figures 11–16, respectively.
The smooth curves in Figures 11–16 represent the performance of Cases 1–3. Good voltage profile is observed in Case 3, in which tap regulator position and capacitor size and location are controlled. Through the curves in Figures 11–16, the voltage magnitudes can be obtained within the admissible range in any of the cases. The comparison of the simulation results that are based on ICA, PSO, and MFO is given in Figures 17–19. The numerical results in Tables 3–6 support graphically results in Figures 17–19.
The performance curves in Figures 17–19 demonstrate that improved voltage is achieved in Cases 2 and 3 using ICA as shown in Tables 5 and 6. Meanwhile, as shown in Tables 5 and 6, Case 1 has the same values under all algorithms. The proposed algorithm iteration versus best fitness value in Case 3 of simulation I and simulation II of 13- and 123-bus system is shown in Figures 20 and 21, respectively. Tables 5 and 6 present comparison results, namely, best fitness values, mean voltage magnitudes for best fitness values, and standard deviation voltage magnitudes for best fitness values, in three phases under the different cases and modified heuristic approaches for the IEEE 13- and IEEE 123-bus test systems, respectively.
5. Conclusion
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 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. To prove the implementation of the proposed approach, it is applied and demonstrated on the IEEE 13- and IEEE 123-bus test systems. The numerical simulation results show that the voltage deviation problem is solved and the best solution is obtained in Case 3, which considers tap changers for the voltage regulators and the sizes and locations of the capacitors. Moreover, the ICA method provides improved results.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.