Prerpints.org logo
Preprint
Article

A Grey Wolf Optimization Algorithm Based Optimum Reactive Power Dispatch with Distributed Generation Units

Altmetrics

Downloads

173

Views

100

Comments

0

A peer-reviewed article of this preprint also exists.

This version is not peer-reviewed

Submitted:

16 May 2023

Posted:

17 May 2023

You are already at the latest version

Alerts
Abstract
Keeping the bus voltage within acceptable limits depends on dispatching reactive power. Power quality improves as a result of creating an effective power flow system, which also helps to reduce power loss. Therefore, Optimal Reactive Power Dispatch (ORPD) studies aim at designing appropriate system configurations to enable a reliable operation of power systems. Establishment of such a configuration is handled through control variables in power systems. On the other hand, installing Distributed Generation Unit (DGU) into a power system can have an important effect on power quality and power loss minimization. In this paper, Grey Wolf Optimization (GWO method is applied to ORPD problem for the first time, to find optimal placement of newly installed DGUs at the power system. Active power loss and voltage minimization have been minimized by optimizing the control variables while keeping them within their upper and lower bounds, and by finding the optimal placement of a newly installed DGU in power system. On the basis of IEEE 30 bus and IEEE 118 bus systems, performance of the suggested approach is investigated. A comparison is also performed among the proposed method and some other methods. Test results demonstrate that the resulting configuration contributes to growing the rate of renewable integration to a power system. The results also indicate that the suggested optimization approach applied to ORPD problem has improved power system performance in terms of both on power loss and voltage deviation. The promising results highlight the potential of the GWO algorithm to facilitate the integration of renewable power sources, and its role in promoting sustainable energy solutions.
Keywords: 
Subject: Engineering  -   Electrical and Electronic Engineering

1. Introduction

Reactive power dispatch (RPD) is a critical operation in electric power systems, with significant implications for both safety and economic considerations [1]. Poor management of reactive power can lead to detrimental impacts on power quality and transmission losses in power systems [2,3]. Therefore, the optimization of reactive power has emerged as a fundamental strategy in reducing real power losses and plays a crucial role in both the planning and operation of power systems [4]. By effectively controlling reactive power, voltage stability can be maintained, and the reliability of real power transfer can be improved [5,6].
As power systems are designed to provide reliable power supply with economic cost, several methods or tools are used to achieve this goal. In connection with economic dispatch of power, Optimal Power Flow (OPF) was defined in 1962 by Carpentier [7,8]. ORPD is a particular case of OPF which’s goal is controlling continuous and discrete variables with adjusting generator voltage set-points, transformer tap settings, improving voltage profile and reactive compensation to reduce power loss [9,10]. ORPD problem has a big significant role on economic and secure operations of power systems. Although reactive power generation doesn’t have a production cost, but it has influence the overall generation cost of power system [11,12]. On the other hand, Distributed Generation Units (DGUs) which is currently an important concept in the economics literature about electricity markets are used to be installed on power systems for minimization of power loss [13]. DGUs connected to electric systems are set to supply reactive power based on the active power set-point. DGUs have to be allocated properly; a proper allocation will improve power quality within limits and also minimize the loss, although an inappropriate allocation will affect vice versa.
The ORPD presents itself as a challenging nonlinear optimization problem, incorporating a combination of equality and inequality constraints. Traditional methods have shown inefficiency in solving nonlinear problems therefore metaheuristic algorithms such GA [9,10], PSO [11,12], ACO [13,14], and SA [15,16] are preferred for solving ORPD problem. These algorithms are known for their global search capability, robustness, and convergence to near-optimal solutions, making them well-suited for solving ORPD problems. GA is an example of a population-based evolutionary algorithm that utilizes principles of natural selection to optimize a set of solutions towards the optimal solution. In contrast, PSO draws inspiration from the social behavior of birds or fish and employs a swarm of particles to collectively explore the solution space. ACO, inspired by the foraging behavior of ants, utilizes pheromone trails as a guiding mechanism in the search process. SA, which emulates the cooling process of a material, allows for stochastic jumps in the solution space to escape local optima.
Mathematical programming methods, such as linear programming (LP) [17,18], quadratic programming (QP) [19,20], and nonlinear programming (NLP) [21,22], have also been applied to ORPD. LP is a popular optimization approach utilized to optimize a linear objective function, while taking into account linear equality and inequality constraints. QP extends LP by allowing for quadratic objective functions and constraints, making it suitable for certain types of ORPD problems. NLP, on the other hand, handles nonlinear objective functions and constraints, but it may face challenges in solving large-scale and non-convex ORPD problems due to computational complexity and convergence issues. Hybrid approaches that integrate multiple optimization techniques have gained attention in ORPD research. For instance, hybrid algorithms that combine GA with other metaheuristic algorithms, such as GA-PSO [23,24], GA-ACO [25,26], and GA-SA [27,28], have been proposed to leverage the strengths of different algorithms and improve solution quality. These hybrid algorithms combine the global search capability of GA with the exploration and exploitation abilities of other metaheuristic algorithms, resulting in enhanced performance in solving ORPD problems.
Fuzzy logic [29,30], neural networks [31,32], and expert systems [33,34] are other methods that have been implemented to solve ORPD problems. Fuzzy logic, which deals with uncertainty and imprecision, has been used to model the vagueness associated with the decision-making process in ORPD. Neural networks, with their ability to learn from data, have been utilized for predicting and optimizing reactive power in power systems. Expert systems, which capture human expertise in the form of rules, have been employed for decision support in ORPD.
Multi-objective optimization approaches have also been used in ORPD studies. MOGA [35,36] and MOPSO [37,38] are examples of methods that optimize multiple conflicting objectives simultaneously, such as system loss, generation cost, and emission. These approaches generate a set of solutions that represent different trade-offs between the conflicting objectives, allowing decision-makers to select the most suitable solution based on their preferences [39,40].
Evolutionary strategies [41,42], which are optimization methods that mimic the process of natural selection, have been applied to solve ORPD problems as well. Evolutionary strategies employ mutation, crossover, and selection operators to evolve a set of solutions towards the optimal solution. These methods have been shown to be effective in solving large-scale and complex ORPD problems, as they can handle constraints, uncertainties, and non-convexities commonly found in real-world power systems.
Machine learning techniques, SVM [43,44], ANN [45,46], and reinforcement learning (RL) [47,48], have also been used to tackle ORPD problems. SVM, a supervised learning algorithm, has been utilized for predicting and optimizing reactive power in power systems. ANN, on the other hand, with its ability to learn from data and capture complex relationships, has been applied for reactive power dispatch and voltage control in power systems. RL, which is a type of unsupervised learning, has been used to optimize reactive power dispatch in an online and adaptive manner by learning from the system’s feedback and making decisions accordingly.
In recent years, grey wolf optimization (GWO) [49,50] has gained attention as a promising algorithm for solving ORPD problems. The GWO algorithm, drawing inspiration from the social hierarchy and hunting behavior of grey wolves, is a population-based metaheuristic. It is recognized for its rapid convergence, effective balance between exploration and exploitation, and capability to generate high-quality solutions for a variety of optimization problems, including ORPD. GWO has been applied to different variants of ORPD problems, such as multi-objective ORPD, dynamic ORPD, and uncertain ORPD, and has shown promising results in terms of computational efficiency and solution quality.
In this paper, GWO method is applied to ORPD problem for the first time, to find optimal placement of newly installed DGUs at the power system. Active power loss and voltage deviations has been minimized for optimization of the control variables while keeping them within their upper and lower bounds, and by finding the optimal placement of a newly installed DGU in power system. Two different test systems; IEEE 30-bus and IEEE 118-bus systems are used to evaluate performance of the method. The results are then compared with that of PSO, ABC, and GA optimization techniques. Therefore, the proposed GWO algorithm can effectively solve the ORPD problem by identifying optimal locations for the placement of distributed generation units (DGUs).
The primary contributions of this research work can be summarized as follows:
  • Application of the GWO algorithm to solve the ORPD problem in DGU installed power systems.
  • Conducting a comparative analysis of the proposed GWO method with other heuristic optimization algorithms such as GA, PSO, and ABC based on several performance metrics, including power loss minimization, voltage deviations, bus voltage levels, and number of iterations.
  • Demonstration of the superior performance of the proposed GWO method in terms of converging to better optimal solutions for control variables as compared to the other heuristic methods.
  • Evaluation of the effectiveness of the proposed GWO method in identifying optimal placements of DGUs, resulting in reduced power losses and improved bus voltage profiles without requiring additional measures.
  • Comparison of the results obtained with and without DGUs, showing that the proposed GWO method performs better in minimizing active power losses and voltage deviations in power systems with DGUs, indicating its potential for facilitating the integration of renewable DGUs into power systems.
Overall, the findings of this research contribute to the field of power system optimization by showcasing the application and performance of the GWO method in solving the ORPD problem, and its potential for promoting the integration of renewable DGUs into power systems, thereby advancing the field of renewable energy integration in power systems.
The paper structured in the following manner: Section 2, the ORPD problem is formulated, outlining the objectives and constraints of the optimization task. Section 3 introduces the GWO technique, providing an overview of its principles and features. In Section 4, the implementation of GWO for the ORPD problem is described, including the details of how the algorithm is applied to solve the optimization task. Section 5 presents the simulations and results obtained from applying GWO to different scenarios, including the IEEE-30 bus system with 19, 25, and 27 control variables, as well as the IEEE-118 bus system. In the concluding section, the key findings and implications of the study are summarized.

2. ORPD Problem Formulation

ORPD problem is a particular case of OPF problem. In ORPD the transmission active loss is minimized by controlling generator bus voltages, transformer tap settings and size of switchable shunt capacitors [22,23]. ORPD plays an important role in securing and economizing operation of power systems [24,25]. Remaining voltage at each bus bar system within its acceptable limits maintains the quality and security in power systems [26,27]. Adjusting system variables within their upper and lower bounds so that the transmission loss is minimized can also affect overall generation cost.
This paper focuses on minimizing two objective functions: the total transmission loss (F1) and the voltage deviation at load buses (F2). These objectives are expressed as follows [16]:

Objective function (F1)

P L = ∑ P i = ∑ P g i − ∑ P d i , i = 1 , … … … … . , N b
Here Nb represents number of buses in the system. PL denotes the system real power loss, Pi refers the real power injection at bus i, Pgi is the real power output of the generator which connect to bus i, Pdi is the real power load which connect to bus i.

Objective function (F2)

V D = ∑ | V i − V i r e f | , i = 1 , … … … … . , N P Q
Mathematically, the voltage deviation of the load buses can be expressed as the sum of the voltage deviations of each load bus, denoted as VD. The voltage magnitude of the ith load bus is represented as V i , while V i r e f refers to the reference value of voltage magnitude for bus i, which is typically assumed to be 1.0. The parameter NPQ denotes the number of PQ buses or load buses in the system. This equation quantifies the difference between the actual voltage magnitude and the desired voltage magnitude at each load bus within the power system.
Fitness function is:
M i n . F = P L + K v ∑ i = 1 N P Q ( V i − V i lim ) 2 + K q ∑ i = 1 N g ( Q g i − Q g i lim ) 2 + K f ∑ l = 1 N l ( s l − s l i lim ) 2
Here Kf, Kv, and Kq are the penalty factors for the line flow violation, limit violation of bus voltage and generator reactive power, respectively. NPQ is the number of PQ busses Qgi is the reactive power output of the generator connecting to bus i, V i is the voltage magnitude at bus i , s l is the loading of transmission line, Ng is number of generator units, Nl is number of transmission lines.
In ORPD problem, the equality constraints are defined as active/reactive power equalities and reflects the physics of power system requiring the net injection of power at each bus to be zero as shown in equations below [28,29,30,31].
P G K − P D K = ∑ j = 1 N | V k | | V j | ( G k j cos ( θ k − θ j ) + B k j sin ( θ k − θ j ) )
P G K resembles the generated active power, P D K is the demand of active power, B k j and G k j are the susceptance and conductance transfer between k and j.
Q G K − Q D K = ∑ j = 1 N | V k | | V j | ( G k j sin ( θ k − θ j ) − B k j cos ( θ k − θ j ) )
Q G K is the reactive power generated, Q D K is the reactive power demand, N is the total number of buses
The inequality constraints reflect the limits on physical devices as well as the limits created to ensure system security, it includes bus voltage magnitude, active/reactive power generation constraints, reactive power source capacity, and transformer tap position constraints [32,33].
a)- Voltage Magnitude Constraints:
V i − min ≤ V i ≤ V i − max
V i − min is the voltage magnitude lower limit, V i is the voltage magnitude of bus i, V i − max is the voltage magnitude upper limit.
b)- Active Power Generation Constraints:
P g i − min ≤ P g i ≤ P g i − max
Pgi denotes ith generator active power. Pgi-max and Pgi-min are the maximum and minimum active power of ith generator, respectively.
c)- Reactive Power Generation Constraints:
Q G i − min ≤ Q G i ≤ Q G i − max
Qgi denotes ith generator reactive power. Qgi-max and Qgi-minare the maximum and minimum active power of ith generator, respectively.
d)- Reactive Power Source Capacity Constraints:
q c i − min ≤ q c i ≤ q c i − max i ∈ N c q c i = q c i − min + N c i ∗ Δ q c i
q c i is shunt VAR compensation, N c is number of shunt VAR compensation devices, q c i − min and
are the minimum and maximum limits of shunt VAR compensation.
e)- Transformer Tap Position Constraints:
T i − min ≤ T i ≤ T i − max i ∈ N T T i = T i − min + N T i ∗ Δ T i
T i is transformer tap ratio, N T is number of tap setting transformers, T i − min and T i − max are the minimum and maximum limit of transformer tap ratio.

3. Grey Wolf Optimization (GWO) Algorithm

GWO is a nature-inspired optimization algorithm that is based on the social behavior and hunting strategies of grey wolves developed by Mirjalili in 2014 for solving optimization problems [49]. In GWO, the population of grey wolves is modeled as a set of solutions, and the search process is guided by the interaction and collaboration among the wolves.
The algorithm mimics the hierarchical structure of a wolf pack, where alpha (α) , beta (β), delta (δ) and omega (ω) wolves represent the leaders of the pack, and their positions are updated iteratively to search for the optimal solution.
The beta provides input to the alpha while also reinforcing the alpha’s orders across the pack. The gray wolf with the lowest status, known as omega, frequently serves as a scapegoat. Additionally, they are the last wolves still permitted to consume animals. A wolf is referred to be a delta if it is not an alpha, beta, or omega. Delta wolves serve as guardians, hunters, elders, scouts, and caregivers. Figure 1 shows the gray wolf social structure. The next sub-section outlines the GWO’s social structure, tracking, surrounding, and prey-attacking steps.

Mathematical modeling of GWO Algorithm

The GWO algorithm’s mathematical model uses the α to represent the most suitable solution, while the β and δ signify the second and third best solutions, correspondingly. The remaining candidate solutions are considered as ω, which are subordinate to the α, β, and δ wolves.
The encircling behavior of wolves during hunting can be mathematically given by the following equations [50]:
D = | C . X p t − X ( t ) |
X t + 1 = X p − A . D
In the GWO algorithm, the current iteration is denoted by the variable t, while Xp represents the position vector of the grey wolf, and X represents the position of the prey in the search space. The coefficient vectors of A and C are handled using the following expressions [5]:
A = 2 . α . r 1 − α
C = 2 . r 2
The alpha (α) value exhibits a linear descent from 2 to 0 as the iterations progress, indicating a gradual reduction of this variable over time within the algorithm. This linear decrease in α controls the step size for the wolves’ movement towards the alpha wolf, leading to a decreasing exploration rate as the algorithm approaches convergence. The variables r1 and r2 are randomly generated numbers extracted from a uniform distribution spanning from 0 to 1. These random values introduce randomness and diversity in the wolves’ movement, allowing for exploration in the search space and increasing the chances of escaping local optima. During the hunting process, the pack of grey wolves updates their positions based on the best agents α, β, and δ, as well as A and C, which are handled using the random values r1 and r2, as explained above. This update process is expressed mathematically using the equations given below:
D = | C . X p t − X ( t ) |
{ D α = C 1 . X α − X ,   D β = C 2 . X β − X , D δ = C 3 . X δ − X }
{ X 1 = X α − A 1 . D α , X 2 = X β − A 2 . D β , X 3 = X δ − A 3 . ( D δ ) }
X t + 1 = X 1 + X 1 + X 1 3
During the hunting behavior of grey wolves, they engage in attacking their prey when the prey ceases to move. This behavior can be mathematically simulated by gradually decreasing the value of a from 2 to 0. As a result, A, which varies randomly within the range of α (-1 to 1), determines the next location of the search agents. The new position is determined within the bounds of the current position of the agent and the position of the prey, typically represented as [49,50].

4. Implementation of Gwo Algorithm for Solving Orpd Problem with Dgu Placement

The ORPD problem involves determining the optimal settings of reactive power sources in a power system while minimizing an objective function, and adhering to operational constraints such as voltage limits, generator limits, and transmission line limits. This requires advanced optimization techniques and thorough analysis of the power system’s dynamic behavior.
Finding optimal settings involves optimizing output levels, voltage set points, and switching operations in real-time or near real-time, considering the uncertain nature of power system conditions. The solution to the ORPD problem has significant implications for power system stability, reliability, and efficiency, and is crucial for improving the operation and planning of modern power systems.
The GWO algorithm presents a promising approach for solving the ORPD problem in power systems, optimizing reactive power sources while considering operational constraints and uncertainty, with the potential to improve power system stability, reliability, and efficiency. As seen in Figure 2, the GWO algorithm starts with an initialization step where the algorithm parameters, such as population size, maximum number of iterations, search space limits, and convergence criteria, are set. The positions of the grey wolves, which represent the potential solutions, are randomly initialized within the search space, considering the control parameters, such as generator voltages, generator powers, tapping settings of transformers, and the location of shunt capacitors. The fitness of each grey wolf, which represents the quality of the solution, is evaluated by calculating the objective function value that represents the cost of reactive power dispatch, considering the control parameters.
Then, the algorithm enters into an iterative loop where the positions of the grey wolves are updated in regard of mathematical formulas that involve the α, β, and δ wolves, which are the best solutions found so far. The algorithm also checks for boundary violations, where a grey wolf may go beyond the search space limits or the control parameter limits, and repositions it randomly within the search space and the control parameter limits if necessary. The fitness of each grey wolf is re-evaluated, and the candidate wolves are updated based on their fitness values.
The algorithm continues to iterate until the reaching to maximum iteration or meeting convergence criteria. The convergence criteria involve checking if the mean fitness value of all the grey wolves has converged to a desired threshold. Finally, the ORPD solution is obtained from the positions of the α, β, and δ wolves, considering the control parameters, and is output as the result of the algorithm. The GWO algorithm provides a robust and efficient approach for solving the optimum reactive power dispatch problem with DGUs placement, considering the control parameters associated with generator voltages, generator powers, tapping settings, and the location of shunt capacitors.
Table 1. Pseudocode of GWO-based Optimum Reactive Power Dispatch with DGU Placement.
Table 1. Pseudocode of GWO-based Optimum Reactive Power Dispatch with DGU Placement.
Preprints 73785 i001
In this study, wind energy is chosen as the primary energy source in the DGU. The study aimed to propose a solution that maximizes the benefits of wind energy utilization in the DGU, such as improved power quality, reduced power losses, and enhanced voltage stability. The given pseudo-code outlines a solution for optimizing the reactive power dispatch (ORPD) with wind power integration using the GWO algorithm. The algorithm starts by initializing the population of grey wolves randomly, evaluating the fitness of each wolf, and determining the alpha, beta, and delta wolves. The iteration process updates the position of each wolf using a position update equation and applies constraints to ensure feasibility of both reactive power dispatch and wind placement. The fitness of each wolf is evaluated and the α, β, and δ wolves are updated if necessary. The stopping criterion is based on a maximum number of iterations and a minimum improvement threshold. The algorithm returns the optimal solutions for reactive power dispatch and wind placement, along with the total system loss and voltage deviation. The proposed approach can provide an efficient solution to the OPRD problem with wind integration.

5. Simulation Results and Discussion

In this study, verifications were conducted by referencing the IEEE 30-bus and 118-bus test systems. The primary objective of choosing these test systems is to evaluate the efficacy of the GWO method in comparison to other existing optimization methods in addressing the ORPD problem. The standard IEEE 30-bus system is composed of 5 PV buses, 21 loads, 41 branches, 4 tap changers, and 3 shunt capacitors, as reported in [1,51,52]. Another test system used in this study is the IEEE 118-bus system, which consists of 54 PV buses, 99 loads, 186 branches, 9 tap changers, and 14 shunt VAR compensators [53,54,55].

IEEE-30 Bus System

  • Scenario 1: Results of the IEEE-30 bus system with 19 control variables
The system data are presented in [1,51,52]. According to the given scenario, 19 control variables are considered. These control variables are as follows:
  • 6 variables of generator bus voltages
  • 4 variables of transformer tap settings
  • 9 variables of reactive power of the VAR compensators
As seen in Table 2, the voltages are bounded by a minimum of 0.90 and a maximum of 1.10 per unit (p.u.), with 1.10 being the upper threshold and 0.90 being the lower threshold. Similarly, the transformer taps are restricted within a range of 0.9 to 1.1, where 1.1 signifies the upper limit and 0.9 signifies the lower limit. The VAR compensators are constrained with a lower limit of 0 MVARs and an upper limit of 7 MVARs, respectively.
As seen in Table 3, the proposed GWO algorithm demonstrated a significant percentage improvement in both total loss and voltage deviation compared to PSO, GA, and ABC methods. Specifically, the GWO method achieved a percentage improvement of 11.55%, 21.99%, and 10.14% in total loss compared to PSO, GA, and ABC, respectively. Similarly, the GWO method achieved a percentage improvement of 210.54%, 1.62%, and 57.09% in voltage deviation compared to PSO, GA, and ABC, respectively.
In the proposed GWO algorithm, the number of agents is an important parameter that affects the optimization performance. Increasing the number of agents can help to improve the exploration ability of the algorithm and increase the probability of finding the global optimal solution. However, increasing the number of agents also increases the computational complexity of the algorithm.
The conducted study investigated the effect of the number of agents on the convergence behavior of the GWO algorithm, and convergence curves were plotted using different numbers of agents. The results showed that increasing the number of agents led to faster convergence and improved optimization performance. However, this also resulted in increased computational time, which could limit the practical application of the algorithm. Selecting an appropriate number of agents is crucial in balancing the trade-off between computational complexity and performance in the GWO algorithm for optimization.
As shown in Figure 3 of the study, the proposed GWO algorithm demonstrated excellent convergence performance for Scenario 1 of the IEEE-30 test system, with agent numbers of 20, 30, and 40. These results suggest that the GWO algorithm efficiently converges, and holds promising potential in solving optimization problems for various test systems. The obtained convergence curves in the study can provide insights into selecting the optimal number of agents for the GWO algorithm in various optimization problems.
  • Scenario 2: Results of the IEEE-30 bus system with 25 control variables
The system data are presented in [51,56]. According to the given scenario, 25 control variables are considered. These control variables are as follows:
  • 6 variables of generated power
  • 6 variables of generator bus voltages
  • 4 variables of transformer tap settings
  • 9 variables of reactive power of the VAR compensators
As seen in Table 4, the lower and upper limits of the generated power value range from 0.05 p.u. to 1.00 p.u. The voltages are bounded by a minimum of 1.00 and a maximum of 1.10 p.u. with 1.10 being the upper threshold and 1.00 being the lower threshold. Similarly, the transformer taps are restricted within a range of 0.9 to 1.1, where 1.1 signifies the upper limit and 0.9 signifies the lower limit. The VAR compensators are constrained with a lower limit of 0 MVARs and an upper limit of 5 MVARs, respectively.
According to the results provided, the GWO (proposed) algorithm outperforms the ABC algorithm in terms of both total loss and voltage deviation. The total loss for GWO (proposed) is significantly reduced by 46.96% compared to ABC, while the voltage deviation is improved by 5.99%. These findings suggest that the GWO (proposed) algorithm offers superior performance in minimizing total loss and voltage deviation, indicating its effectiveness for power system optimization. The percentage improvements in both total loss and voltage deviation highlight the potential of the GWO (proposed) algorithm as a promising optimization technique for power system applications, potentially leading to more efficient and reliable power system operations.
The power loss convergence analysis of the GWO (proposed) algorithm, as depicted in Figure 4, reveals its remarkable performance in achieving convergence for Scenario 2 of the IEEE-30 test system, employing varying agent numbers such as 20, 30, and 40.
The voltage deviation convergence analysis of the GWO (proposed) algorithm, as depicted in Figure 5, reveals its remarkable performance in achieving convergence for Scenario 2 of the IEEE-30 test system, employing under 40 agents. The results indicate that the GWO algorithm not only achieves low total power loss (2.0679 MW) but also exhibits minimal voltage deviation (0.9769 pu), underscoring its effectiveness in optimizing both power loss and voltage deviation simultaneously.
  • Scenario 3: Results of the IEEE-30 bus system with 27 control variables (wind integration)
The IEEE-30 bus system with 27 control variables is an extended version of IEEE-30 bus system that incorporates two additional DGU control variables. These variables are the generator voltage magnitude control and the generator power control. According to the given scenario, 27 control variables are considered. These control variables are as follows:
  • 7 variables of generated power with wind power
  • 7 variables of generator bus voltages with wind bus
  • 4 variables of transformer tap settings
  • 9 variables of reactive power of the VAR compensators
The Table 6 provides the lower and upper limits for variables such as wind power, generated power, voltages, tap settings, and compensation devices. These limits define the constrained ranges for each variable during the optimization process, guiding the optimization to achieve optimal results within these limits for improved system performance.
The Table 7 provides an analysis of the performance of the GWO (proposed) algorithm on the control variables for the IEEE-30 test system scenario, including voltage deviation and total loss of power. The table presents the lower and upper limits for each control variable, as well as the results obtained by the GWO algorithm within these limits.
The GWO (proposed) algorithm effectively operates within the specified limits for all control variables, including Pwind and Vwind.
The inclusion of a DGU unit (0-10 MW) with wind power and wind bus voltage control parameters in the GWO (proposed) algorithm for optimizing IEEE 30 scenario 2 without modifying generator power values resulted in wind power control at 9.8 MW and wind bus voltage at 1.0604 pu among 27 optimized control variables. This suggests effective incorporation of DGU unit and control parameters by GWO, leading to satisfactory results in the IEEE 30 test system scenario. Total power loss is 1.8010 MW, and voltage deviation is 1.0154 pu, indicating successful optimization of control variables for the IEEE-30 test system.
The power loss convergence analysis of the GWO (proposed) algorithm, as depicted in Figure 6, reveals its remarkable performance in achieving convergence for Scenario 3 of the IEEE-30 test system, employing varying agent numbers such as 20, 30, and 40. The figure depicts the convergence values plotted up to 250 iterations for all agent numbers. Notably, it has been observed that the GWO (proposed) algorithm with 40 agents achieves the complete convergence value after only 100 iterations, showcasing the remarkable rapid convergence capability of the proposed algorithm.
Similary, the voltage deviation convergence analysis of the GWO (proposed) algorithm, as depicted in Figure 7, reveals its remarkable performance in achieving convergence for Scenario 3 of the IEEE-30 test system, employing for 40 agents.
In the IEEE 30-bus test system, distributed generation units (DGUs) with wind power as a renewable energy source are connected to the buses, except for buses 1, 2, 5, 8, 11, and 13. Power loss values were observed for each bus. It was observed that the DGU connected to bus 21 caused the lowest power loss, with a value of 1.801 MW. The largest power loss was determined to occur with the DGU connected to bus 9, with a loss value of 3.393 MW. Therefore, the power loss difference between the DGU connected to bus 9 and the DGU connected to bus 21 is approximately 88.34% based on the observed values. Consequently, the implementation of the GWO method in the ORPD process yields favorable outcomes, including the simultaneous mitigation of power loss and voltage deviation, as well as the optimization of DGU placement, contributing to enhanced integration of renewable energy sources.

IEEE-118 Bus System

  • Scenario 1: Results of the IEEE-118 bus system with 77 control variables
The system data are presented in [53,54,55,56]. According to the given scenario, 77 control variables are considered. These control variables are as follows:
  • 54 variables of generator bus voltages
  • 9 variables of transformer tap settings
  • 14 variables of reactive power of the VAR compensators
Table 8 shows the limit settings for control variables in the IEEE-118 system. The lower limit for generator bus voltages is set at 0.95 p.u, while the upper limit is set at 1.10 p.u. Transformer tap settings are constrained within a lower limit of 0.90 p.u. and an upper limit of 1.10 p.u. Likewise, the compensation devices are bounded by -20 MVAR and 20 MVAR as the lower and upper limits, respectively. These threshold values serve as critical parameters in regulating the operation of the power system, ensuring stability and reliability.
According to the IEEE 118-bus test system, GWO (proposed) algorithm exhibits a significant improvement in power loss reduction compared to ABC, PSO, and GSA algorithms. GWO algorithm achieves the lowest power loss with a reduction of approximately 24.86% compared to ABC, 20.21% compared to PSO, and 16.40% compared to GSA, indicating its superiority in optimizing the power flow of the system. This suggests that GWO algorithm can be a promising choice for power system optimization, particularly in minimizing power losses and improving system efficiency. The results imply that GWO algorithm exhibits potential as an effective solution for power system optimization, with a notable capability in minimizing power losses and enhancing overall system efficiency.
The convergence analysis of the proposed GWO algorithm, as shown in Figure 9, demonstrates its impressive performance in achieving convergence in Scenario 1 of the IEEE-118 test system with varying agent numbers (20, 30, and 40). The figure displays the convergence values plotted up to 1000 iterations for all agent numbers. Remarkably, the GWO (proposed) algorithm with 40 agents achieves the complete convergence value in just 229 iterations, highlighting the exceptional rapid convergence capability of the proposed algorithm.
Figure 10 presents the convergence analysis of voltage deviation for Scenario 1 of the IEEE-118 test system, utilizing the GWO (proposed) algorithm with 40 agents. The results highlight the outstanding performance of the algorithm in achieving convergence, with the voltage deviation value reaching a remarkable low of 1.09356 pu. This indicates the robustness and accuracy of the proposed algorithm in optimizing voltage levels and minimizing deviations, making it highly effective for power system optimization tasks.
  • Scenario 2: Results of the IEEE-118 bus system with 131 control variables
The system data are presented in [53,54,55,56]. According to the given scenario, 131 control variables are considered. These control variables are as follows:
  • 54 variables of generated power
  • 54 variables of generator bus voltages
  • 9 variables of transformer tap settings
  • 14 variables of reactive power of the VAR compensators
Table 10. Limit Setting for Control Variables for IEEE-118 System in Scenario 2.
Table 10. Limit Setting for Control Variables for IEEE-118 System in Scenario 2.
Variables Lower Limit Upper Limit
Generated Power 0.05 pu 3.60 pu
Voltages 0.95 pu 1.10 pu
Tap Settings 0.90 pu 1.10 pu
Compensation Devices -20 MVAR 20 MVAR
The table presents the variables and their corresponding lower and upper limits in the context of power system analysis. The lower and upper limits of generated power are set at 0.05 pu and 3.60 pu, respectively, indicating that the generated power must be within this range. Similarly, the voltages in the system must be maintained between 0.95 pu and 1.10 pu. The tap settings for voltage regulating devices are constrained within the range of 0.90 pu to 1.10 pu. Lastly, the compensation devices are limited to -20 MVAR to 20 MVAR, signifying the acceptable range for reactive power compensation. These limits play a crucial role in ensuring the stability and reliability of the power system operation by regulating the values of these variables within acceptable bounds.
As seen in Table 11 that using the GWO algorithm shows that keeping control variables within their specified limits resulted in a significantly lower power loss (68.1578 MW) compared to the IEEE-118 test system without power control parameters. After 213 iterations, the GWO algorithm was able to find the optimal values for the control variables, which remained optimal until the end of the simulation at 1000 iterations. This indicates that the GWO algorithm successfully optimized the system by finding optimal values for control variables while respecting their limits. The results suggest that the GWO algorithm has the potential to improve power system performance by minimizing power losses.
  • Scenario 3: Results of the IEEE-118 system with 133 control variables (wind integration)
The IEEE-118 bus system has been expanded to include two new control variables that specifically target Distributed Generation Units (DGUs). These variables are the generator voltage magnitude control and the generator power control. With these additions, the system now boasts a total of 133 control variables, making it an extended version of the original IEEE-118 bus system. According to the given scenario, 133 control variables are considered. These control variables are as follows:
  • 55 variables of generated power with wind power
  • 55 variables of generator bus voltages with wind bus
  • 9 variables of transformer tap settings
  • 14 variables of reactive power of the VAR compensators
Table 12 shares the same values as Table 11 for generated powers, generator bus voltages, transformer tap settings, and compensation values, while generated power values are constrained within a lower limit of 0.05 p.u. and an upper limit of 3.60 p.u. In Scenario 3, an additional distributed generation unit (DGU) powered by wind energy is introduced into the system, with a power output ranging from 0 to 0.5 per unit (50 MW).
According to the Table 13 results, the GWO algorithm successfully minimized power losses with a value of 28.1372 MW, while keeping voltage deviation within acceptable limits with a value of 1.12945 pu. The ability of the GWO algorithm to optimize the control variables within their specified limits demonstrates its potential to enhance power system performance. By minimizing power losses and maintaining acceptable voltage levels, the GWO algorithm can improve the reliability and efficiency of power systems.
In the IEEE 118-bus system of Scenario 3, the integration of distributed generation units (DGUs) utilizing wind power has been investigated. These DGUs are connected to the buses, except for standard PV buses, highlighting the potential for renewable energy sources to be utilized in power systems. As seen in Figure 12, the results revealed that the placement of DGUs significantly affects the overall power loss of the system. Notably, the DGU connected to bus 9 resulted in the lowest power loss of 28.1372 MW, while the highest power loss of 74.59583 MW was observed with the DGU connected to bus 52, indicating a significant difference of approximately 165.4%. These findings highlight the importance of carefully selecting the location of DGUs in order to optimize system performance and maximize renewable energy utilization.
Preprints 73785 g011
Figure 11. Power loss convergence patterns for Scenario3 of the IEEE 118-bus system (wind integration).
Figure 11. Power loss convergence patterns for Scenario3 of the IEEE 118-bus system (wind integration).
Preprints 73785 g011
The use of the GWO algorithm in the optimization process has yielded promising results. Specifically, the GWO algorithm has facilitated the simultaneous mitigation of power loss and voltage deviation, while also optimizing DGU placement. This successful implementation of the GWO algorithm has contributed to the efficient integration of renewable energy sources into the power system, leading to enhanced system performance and increased utilization of renewable energy. Overall, these results demonstrate the potential of the GWO algorithm as a powerful tool for optimizing power system operation and promoting sustainable energy solutions. Further research is needed to investigate the applicability of the GWO algorithm to other power systems and to explore the potential for further improvements in renewable energy integration.

6. Conclusions

In this study, the proposed GWO method aims to address the ORPD problem by determining the optimal placement of distributed generation units (DGUs) in a power system. The optimization process focuses on minimizing both the active power loss and voltage deviations, while ensuring that the control variables remain within their designated upper and lower bounds. This approach aims to optimize the operation of DGUs in the power system to maximize the integration of renewable energy sources and minimize the associated losses.
To assess the performance of the proposed method, the authors utilized the IEEE 30 bus and IEEE 118 bus systems as case studies in their investigation. A comparison was conducted with other heuristic methods such as GA, PSO, and ABC in terms of various performance metrics including power loss minimization, voltage deviations, bus voltage levels, and number of iterations. The results obtained from the comparison highlight the potential of the proposed GWO based method in improving the performance and efficiency of power systems with increased renewable energy penetration. The reduced number of iterations needed for convergence is a notable advantage in terms of computational efficiency and practical implementation. These findings contribute to the understanding of the applicability and effectiveness of the GWO-based approach in addressing the ORPD problem and optimizing the placement of DGUs in power systems to promote renewable energy integration.
In conclusion, the proposed GWO-based optimization method offers promising results in optimizing the placement of DGUs in a power system for achieving optimal renewable penetration and dispatch. The comparison with other methods and the evaluation of performance metrics support the effectiveness and efficiency of the proposed approach. The results obtained from this study can provide valuable insights for researchers and practitioners in the field of power systems planning and operation, with the aim of promoting renewable energy integration in power systems.

Competing Interests

The authors declare no competing interests.

Author Contributions

All authors have contributed equally to the work.

References

  1. Zimmerman, R.D.; Murillo-Sánchez, C.E.; Thomas, R.J. MATPOWER: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education. IEEE Trans. Power Syst. 2011, 26, 12–19. [Google Scholar] [CrossRef]
  2. Zhu, J. Optimization of power system operation; John Wiley Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
  3. Maskar, M.B.; Thorat, A.R.; Korachgaon, I. A review on optimal power flow problem and solution methodologies, International Conference on Data Management, Analytics and Innovation (ICDMAI) 2017, 64-70, IEEE.
  4. Ting, Chuan-Kang, On the Mean Convergence Time of Multi-Parent Genetic Algorithms without Selection, In European Conference on Artificial Life, 2005, 403-412, Springer, Berlin, Heidelberg.
  5. T. Van Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. New York: Springer, 1998.
  6. M. Glavic, T. M. Glavic, T. Van Cutsem, “Some Reflections on Model Predictive Control of Transmission Voltages” In Proc. of the 38th North American Power Symposium (38th NAPS), Carbondale, USA, Sept. 2006.
  7. Carpentier, J. Contribution to the economic dispatch problem. Bull. Soc. Franc. Elect. 1962, 8, 431–447. [Google Scholar]
  8. Carpentier, J. Optimal power flows. Int. J. Electr. Power Energy Syst. 1979, 1, 3–15. [Google Scholar] [CrossRef]
  9. Kou, X. Particle Swarm Optimization Based Reactive Power Dispatch for Power Networks with Distributed Generation. Master Thesis, University of Denver, USA, 2015. [Google Scholar]
  10. Norshahrani, M.; Mokhlis, H.; Abu Bakar, A.; Jamian, J.; Sukumar, S. Progress on Protection Strategies to Mitigate the Impact of Renewable Distributed Generation on Distribution Systems. Energies 2017, 11, 1864–1894. [Google Scholar] [CrossRef]
  11. Al Rashidi, M.R.; El-Hawary, M.E. A Survey of Particle Swarm Optimization Applications in Electric Power Systems. IEEE Trans. Evol. Comput. 2009, 13, 913–918. [Google Scholar] [CrossRef]
  12. Salama, M.M.A.; Manojlovic, N.; Quintana, V.H.; Chikhani, A.Y. Real-Time Optimal Reactive Power Control for Distribution Networks. Electr. Power Energy Syst. 1996, 18, 185–193. [Google Scholar] [CrossRef]
  13. Radosavljević, J.; Jevtić, M.; Milovanović, M. A solution to the ORPD problem and critical analysis of the results. Electr. Eng. 2018, 100, 253–265. [Google Scholar] [CrossRef]
  14. Oo, K.Z.; Lin, K.M.; Aung, T.N. Particle Swarm Optimization Based Optimal Reactive Power Dispatch for Power Distribution Network with Distributed Generation. Int. J. Energy Power Eng. 2017, 6, 53–60. [Google Scholar] [CrossRef]
  15. Gutiérrez, D.; Villa, W.M.; López-Lezama, J.M. Optimal reactive power dispatch by means of particle swarm optimization. Inf. Tecnol. 2017, 10, 215–224. [Google Scholar] [CrossRef]
  16. Sivalingam, C.M.K.; Ramachandran, S.; Rajamani, P.S.S. Reactive Power Optimization in a Power System Network Through Metaheuristic Algorithms. Turk. J. Electr. Eng. Comput. Sci. 2017, 25, 4615–4623. [Google Scholar] [CrossRef]
  17. Dutta, S.; Roy, P.K.; Manna, D.K. HBBO Optimization for Optimal Reactive Power Dispatch Incorporating TCSC and TCPS Devices, Michael Faraday IET International Summit, 2015, 156–163, Kolkata, India. [CrossRef]
  18. Cao, Y.J.; Wu, Q.H. Optimal Reactive Power Dispatch Using an Adaptive Genetic Algorithm, 1997, 117-122. [CrossRef]
  19. El-Ela, A.A.; Kinawy, A.M.; El-Sehiemy, R.A.; Mouwafi, M.T. Optimal reactive power dispatch using ant colony optimization algorithm. Electr. Eng. 2011, 93, 103–116. [Google Scholar] [CrossRef]
  20. Khanmiri, D.T.; Nasiri, N.; Abedinzadeh, T. Optimal Reactive Power Dispatch Using an Improved Genetic Algorithm. Int. J. Comput. Electr. Eng. 2012, 4, 463–472. [Google Scholar]
  21. Esmin, A.A.; Lambert-Torres, G.; De Souza, A.Z. A hybrid particle swarm optimization applied to loss power minimization. IEEE Trans. Power Syst. 2005, 20, 859–866. [Google Scholar] [CrossRef]
  22. Sharif, S.S.; Taylor, J.H. Dynamic optimal reactive power flow, In American Control Conference 1998, 6, 3410–3414.
  23. Turkay, B.E.; Cabadag, R.I. Optimal Power Flow Solution Using Particle Swarm Optimization Algorithm, In EUROCON, 2013, 1418-1424.
  24. Om, H.; Shukla, S. Optimal Power Flow Analysis of IEEE-30 Bus System Using Soft Computing Techniques, International Journal of Engineering Research Science (IJOER), 2015, 8, 55–60.
  25. Ayan, K.; Kılıç, U. Artificial Bee Colony Algorithm Solution for Optimal Reactive Power Flow, Applied Soft Computing, 2012, 12,1477–1482. [CrossRef]
  26. El-Shimy, M.; Abuel-wafa, A.R. Implementation and Analysis of Genetic Algorithms (GA) to the Optimal Power Flow (OPF) Problem, Scientific Bulletin-Faculty of Engineering-Ain Shams University, 2006, 41, 753–771.
  27. Durairaj, S.; Kannan, P.S.; Devaraj, D. Application of Genetic Algorithm to Optimal Reactive Power Dispatch including Voltage Stability Constraint, Journal of Energy Environment. 2005, 4, 63–73.
  28. Mamandur, K.R.C.; Chenoweth, R.D. Optimal Control of Reactive Power Flow for Improvements in Voltage Profiles and for Real Power Loss Minimization, IEEE Transactions on Power Apparatus and Systems, 1981, 7, 3185–3194. [CrossRef]
  29. Kumar, C.; Raju, C.P. Constrained Optimal Power Flow Using Particle Swarm Optimization, International Journal of Emerging Technology and Advanced Engineering, 2012, 2, 235–241. [CrossRef]
  30. Dai, C.; Chen, W.; Zhu, Y.; Zhang, X. Seeker Optimization Algorithm for Optimal Reactive Power Dispatch, IEEE Transactions on Power Systems, 2009, 24, 1218–1231. [CrossRef]
  31. Le Dinh, L.; Vo Ngoc, D.; Vasant, P. Artificial Bee Colony Algorithm for Solving Optimal Power Flow Problem, The Scientific World Journal, 2013, 1-9.
  32. Mouassa, S.; Bouktir, T. Artificial Bee Colony Algorithm for Discrete Optimal Reactive Power Dispatch, Proceedings of International Conference in Industrial Engineering and Systems Management, 2015, 654-662. [CrossRef]
  33. Dieu, V.N.; An, N.H.T.; Kien, V.T. Optimal Reactive Power Dispatch Using Artificial Bee Colony Method, Gmsarn International Journal, 2015, 9, 29–36.
  34. Monti, A.; Ponci, F. Electric Power Systems. In Intelligent Monitoring, Control, and Security of Critical Infrastructure Systems, Springer, Berlin, and Heidelberg, 2015, 31-65. [CrossRef]
  35. Zhang, X.P. Electric Power System Analysis. Oper. Control. -Electr. Eng. 2006, 2, 1–42. [Google Scholar] [CrossRef]
  36. Gautam, L.K.; Mishra, M.; Bisht, T. A Methodology for Power Flow & Voltage Stability Analysis, International Research Journal of Engineering and Technology (IRJET), 2015, 2, 321–326.
  37. Ling, S.H.; Iu, H.H.; Chan, K.Y.; Lam, H.K.; Yeung, B.C.; Leung, F.H. Hybrid particle swarm optimization with wavelet mutation and its industrial applications. IEEE Trans. Syst. Man Cybern. Part B 2008, 38, 743–763. [Google Scholar] [CrossRef]
  38. Abido, M.A. Optimal power flow using particle swarm optimization, International Journal of Electrical Power & Energy Systems, 2002, 24, 563–571. [CrossRef]
  39. Lee, K.Y.; El-Fergany, A.A.; Lee, J.H. Optimal power flow with a coordinated control of distributed energy resources: A review. Energies 2018, 11, 1221. [Google Scholar] [CrossRef]
  40. Sánchez-Fernández, M.; Carro-Calvo, L.; Pérez-García, J.; Alcarria, R. A review of optimization methods for reactive power dispatch in power systems. Energies 2019, 12, 2343. [Google Scholar] [CrossRef]
  41. Beyer, H.-G. (1995). The theory of evolution strategies. Springer Science & Business Media.
  42. Hansen, N.; Ostermeier, A. (2001). Completely derandomized self-adaptation in evolution strategies. Evolutionary computation, 9(2), 159-195.
  43. Zhang, J.; Liu, Y.; Wen, F. (2013). An SVM-based approach for reactive power optimization in power systems. Electric Power Systems Research, 95, 45-51.
  44. Li, Z.; Sun, H.; Li, Y.; Li, Q. (2016). A reactive power optimization method based on support vector machines. International Journal of Electrical Power & Energy Systems, 75, 1-7.
  45. Liu, Y.; Cai, X.; Zhang, W. (2016). A neural network approach for voltage control in distribution systems with renewable energy integration. International Journal of Electrical Power & Energy Systems, 83, 121-127.
  46. Zhao, F.; Wu, W.; Dong, Z.Y. (2017). Artificial neural network-based reactive power optimization in large-scale power systems. International Journal of Electrical Power & Energy Systems, 92, 30-38.
  47. Yang, Z.; Chen, H.; Zhao, B. (2017). Reinforcement learning based reactive power optimization for active distribution network. Energy Procedia, 142, 2483-2488.
  48. Xiong, Q.; Liu, Y.; Sun, H. (2019). Reinforcement learning-based reactive power optimization considering load and renewable energy uncertainty. IEEE Transactions on Power Systems, 34(6), 4636-4647.
  49. Mirjalili, S.; Mirjalili, S.M.; Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46-61.
  50. Li, Y.; Li, X.; Yu, D. Grey wolf optimizer-based reactive power optimization of power systems with wind power. Applied Energy 2017, 185, 1791–1801. [Google Scholar]
  51. Mohseni-Bonab, S.M.; Rabiee, A.; Mohammadi-Ivatloo, B.; Jalilzadeh, S.; Nojavan, S. A two-point estimate method for uncertainty modeling in multi-objective optimal reactive power dispatch problem. International Journal of Electrical Power & Energy Systems, 2016, 75, 194–204. [Google Scholar]
  52. Shaheen, M.A.; Yousri, D.; Fathy, A.; Hasanien, H.M.; Alkuhayli, A.; Muyeen, S.M. A novel application of improved marine predators algorithm and particle swarm optimization for solving the ORPD problem. Energies, 2020, 13, 5679. [Google Scholar] [CrossRef]
  53. Rajan, A.; Malakar, T. Optimal reactive power dispatch using hybrid Nelder- Mead simplex based firefly algorithm. Int. J. Elect. Power Energy Syst. Mar. 2015, 66, 9–24. [Google Scholar] [CrossRef]
  54. Electric Power Group, “Transmission system planning reference book”, 118-bus test system, 2004. [Online]. Available: https://www.electricitypolicy.org.uk/pdfs/wp8.pdf. [Accessed: April 21, 2023].
  55. Kurtoglu, S.; Ermis, M. (2017). Optimal placement and sizing of TCSC device in IEEE 118 bus system for loadability enhancement using hybrid GA and PSO approach. Electric Power Components and Systems, 45(6), 651-661.
  56. Sulaiman, M.H.; Mustaffa, Z.; Mohamed, M.R.; Aliman, O. (2015). Using the gray wolf optimizer for solving optimal reactive power dispatch problem. Applied Soft Computing, 32, 286-292.
Preprints 73785 g001
Figure 1. Hierarchy of grey wolves.
Figure 1. Hierarchy of grey wolves.
Preprints 73785 g001
Preprints 73785 g002
Figure 2. The proposed GWO algorithm optimizing control parameters for ORPD problem.
Figure 2. The proposed GWO algorithm optimizing control parameters for ORPD problem.
Preprints 73785 g002
Preprints 73785 g003
Figure 3. Convergence patterns for Scenario 1 of the IEEE 30-bus test system.
Figure 3. Convergence patterns for Scenario 1 of the IEEE 30-bus test system.
Preprints 73785 g003
Preprints 73785 g004
Figure 4. Convergence patterns for Scenario2 of the IEEE 30-bus test system.
Figure 4. Convergence patterns for Scenario2 of the IEEE 30-bus test system.
Preprints 73785 g004
Preprints 73785 g005
Figure 5. Voltage deviation convergence patterns for Scenario2 of the IEEE 30-bus test system.
Figure 5. Voltage deviation convergence patterns for Scenario2 of the IEEE 30-bus test system.
Preprints 73785 g005
Preprints 73785 g006
Figure 6. Convergence patterns for Scenario3 of the IEEE 30-bus test system (wind integration).
Figure 6. Convergence patterns for Scenario3 of the IEEE 30-bus test system (wind integration).
Preprints 73785 g006
Preprints 73785 g007
Figure 7. Voltage deviation convergence patterns for Scenario2 of the IEEE 30-bus system (wind integration).
Figure 7. Voltage deviation convergence patterns for Scenario2 of the IEEE 30-bus system (wind integration).
Preprints 73785 g007
Preprints 73785 g008
Figure 8. Power loss convergence patterns for Scenario3 of the IEEE 30-bus system (wind integration).
Figure 8. Power loss convergence patterns for Scenario3 of the IEEE 30-bus system (wind integration).
Preprints 73785 g008
Preprints 73785 g009
Figure 9. Convergence patterns for Scenario3 of the IEEE 118-bus test system.
Figure 9. Convergence patterns for Scenario3 of the IEEE 118-bus test system.
Preprints 73785 g009
Preprints 73785 g010
Figure 10. Voltage deviation convergence patterns for Scenario1 of the IEEE 118-bus system.
Figure 10. Voltage deviation convergence patterns for Scenario1 of the IEEE 118-bus system.
Preprints 73785 g010
Table 2. Limit Setting for Control Variables for IEEE-30 System.
Table 2. Limit Setting for Control Variables for IEEE-30 System.
Variables Lower Limit Upper Limit
Voltages 0.90 pu 1.10 pu
Tap Settings 0.95 pu 1.05 pu
Compensation Devices 0 MVAR 7 MVAR
Table 3. Control variables of ORPD for the IEEE 30-bus system in Scenario 1.
Table 3. Control variables of ORPD for the IEEE 30-bus system in Scenario 1.
Control Variables PSO GA ABC GWO (proposed)
V1 1.1000 1.0228 1.1000 1.1000
V2 1.1000 1.0181 1.0615 1.0940
V5 1.085 0.9859 1.0711 1.0760
V8 1.0838 0.9858 1.0849 1.0837
V11 1.1000 0.9859 1.1000 1.0847
V13 1.1000 0.9815 1.0665 1.0894
T11 1.1000 0.9996 0.9700 1.0385
T12 0.9000 0.9715 1.0500 0.9753
T15 1.0200 0.9794 0.9900 1.0346
T36 0.9900 0.9477 0.9900 1.0016
QC10 1.1000 2.0124 5.0000 1.3636
QC12 0.4000 6.1953 5.0000 0.6031
QC15 0.7000 3.4021 5.0000 2.5714
QC17 5.0000 1.6718 5.0000 2.2171
QC20 4.7000 2.4060 4.1000 1.3145
QC21 1.0000 4.0786 3.3000 4.4976
QC23 3.0000 2.9779 0.9000 1.7628
QC24 0.8000 1.0617 5.0000 4.4368
QC29 1.2000 3.1220 2.4000 1.3185
Total Loss (MW) 4.66091 5.0977 4.6022 4.1781
Voltage Deviation (pu) 1.4600 0.4773 0.7378 0.4697
Table 4. Limit Setting for Control Variables for IEEE-30 System.
Table 4. Limit Setting for Control Variables for IEEE-30 System.
Variables Lower Limit Upper Limit
Generated Power 0.05 pu 2.00 pu
Voltages 1.00 pu 1.10 pu
Tap Settings 0.90 pu 1.10 pu
Compensation Devices 0 MVAR 5 MVAR
Table 5. Control variables of ORPD for the IEEE 30-bus system in Scenario 2.
Table 5. Control variables of ORPD for the IEEE 30-bus system in Scenario 2.
Control Variables Limits ABC GWO (proposed)
Lower Upper
P1 0.50 2.00 0.5462 0.2351
P2 0.20 0.80 0.7863 0.4848
P5 0.15 0.50 0.4903 0.4700
P8 0.10 0.50 0.3477 0.4688
P11 0.10 0.50 0.2999 0.4693
P13 0.12 0.50 0.3945 0.4617
V1 1.00 1.10 1.0927 1.1000
V2 1.00 1.10 1.0880 1.0984
V5 1.00 1.10 1.0695 1.0805
V8 1.00 1.10 1.0722 1.0982
V11 1.00 1.10 1.0860 1.0969
V13 1.00 1.10 1.0926 1.1000
T11 0.90 1.10 0.9983 1.0220
T12 0.90 1.10 0.9994 0.9571
T15 0.90 1.10 0.9984 1.0386
T36 0.90 1.10 1.0034 1.0093
QC10 0.00 5.00 1.5500 2.5770
QC12 0.00 5.00 3.9400 2.5370
QC15 0.00 5.00 3.4700 1.0370
QC17 0.00 5.00 3.3310 1.8280
QC20 0.00 5.00 3.3320 3.0675
QC21 0.00 5.00 3.9500 1.6370
QC23 0.00 5.00 1.3000 1.7702
QC24 0.00 5.00 3.7100 3.3960
QC29 0.00 5.00 3.9900 0.8000
Total Loss (MW) 3.0410 2.0679
Voltage Deviation (pu) 1.0353 0.9769
Table 6. Limit Setting for Control Variables for IEEE-30 System.
Table 6. Limit Setting for Control Variables for IEEE-30 System.
Variables Lower Limit Upper Limit
Wind Power 0.00 pu 0.10 pu
Generated Power 0.05 pu 2.00 pu
Voltages 1.00 pu 1.10 pu
Tap Settings 0.90 pu 1.10 pu
Compensation Devices 0 MVAR 5 MVAR
Table 7. Control variables of ORPD for the IEEE 30-bus system in Scenario 3 (wind integration).
Table 7. Control variables of ORPD for the IEEE 30-bus system in Scenario 3 (wind integration).
Control Variables Limits GWO (proposed)
Lower Upper
Pwind 0.00 0.10 0.0982
P1 0.50 1.00 0.2351
P2 0.20 0.80 0.4848
P5 0.15 0.50 0.4700
P8 0.10 0.50 0.4688
P11 0.10 0.50 0.4693
P13 0.12 0.50 0.4617
Vwind 1.00 1.10 1.0604
V1 1.00 1.10 1.0531
V2 1.00 1.10 1.0436
V5 1.00 1.10 1.0500
V8 1.00 1.10 1.0977
V11 1.00 1.10 1.0473
V13 1.00 1.10 0.9781
T11 0.90 1.10 0.9500
T12 0.90 1.10 1.0500
T15 0.90 1.10 1.0343
T36 0.90 1.10 0.9884
QC10 0.00 5.00 4.9322
QC12 0.00 5.00 4.3456
QC15 0.00 5.00 0.7096
QC17 0.00 5.00 2.2971
QC20 0.00 5.00 1.6113
QC21 0.00 5.00 3.0152
QC23 0.00 5.00 0.3921
QC24 0.00 5.00 2.2213
QC29 0.00 5.00 0.4657
Total Loss (MW) 1.8010
Voltage Deviation (pu) 1.0154
Table 8. Limit Setting for Control Variables for IEEE-118 System in Scenario 1.
Table 8. Limit Setting for Control Variables for IEEE-118 System in Scenario 1.
Variables Lower Limit Upper Limit
Voltages 0.95 pu 1.10 pu
Tap Settings 0.90 pu 1.10 pu
Compensation Devices -20 MVAR 20 MVAR
Table 9. Control variables of ORPD for the IEEE 118-bus system in Scenario 1.
Table 9. Control variables of ORPD for the IEEE 118-bus system in Scenario 1.
Control
Variables		 ABC PSO GSA GWO Control
Variables		 ABC PSO GSA GWO
V1 1.0512 1.0600 0.9600 0.9903 V90 1.0210 0.9905 1.0417 1.0522
V4 1.0757 0.9400 0.9620 1.0094 V91 1.0172 1.0015 1.0032 1.0251
V6 1.0791 0.9514 0.9729 1.0208 V92 1.0914 1.0600 1.0927 1.0385
V8 1.0656 0.9820 1.0570 0.9697 V99 1.0598 0.9938 1.0433 1.0064
V10 1.0461 0.9522 1.0885 1.0721 V100 1.0535 1.0264 1.0786 1.0486
V12 1.0726 1.0564 0.9630 1.0095 V103 1.0394 0.9400 1.0266 1.0647
V15 1.0265 0.9560 1.0127 0.9764 V104 1.0215 1.0600 0.9808 1.0247
V18 1.0314 0.9400 1.0069 0.9616 V105 1 1.10 1.0600 1.0163 1.0329
V19 1.0155 0.9400 1.0003 0.9672 V107 1.0463 1.0086 0.9987 1.0469
V24 1.0201 0.9400 1.0105 1.0412 V110 1 1.03 0.9914 1.0218 1.0298
V25 1.0489 0.9884 1.0102 1.0950 V111 1.0888 0.9430 0.9852 1.0179
V26 1.0539 0.9930 1.0401 0.9682 V112 1.0636 0.9741 0.9500 1.0090
V27 1.0204 0.9670 0.9809 1.0098 V113 1.0588 0.9615 0.9764 1.0594
V31 1.0392 0.9400 0.9500 1.0003 V116 1.0444 0.9400 1.0372 1.0324
V32 1.0367 1.0030 0.9552 1.0046 QC5 0 20.000 0.0000 1.0050
V34 1.0367 0.9400 0.9910 1.0080 QC34 12.616 -9.493 7.4600 0.9954
V36 1.0440 1.0078 1.0091 1.0051 QC37 0 20.000 0.0000 1.0284
V40 0.9912 0.9739 0.9505 0.9668 QC44 5.3918 -9.0616 6.0700 1.0034
V42 0.9875 0.9450 0.9500 0.9618 QC45 10 20.000 3.3300 0.9990
V46 1.0325 1.0392 0.9814 0.9960 QC46 6.214 19.998 6.5100 1.0447
V49 1.0183 0.9806 1.0444 1.0005 QC48 4.9873 -3.2423 4.4700 0.9458
V54 0.9776 1.0600 1.0379 0.9305 QC74 9.4985 -17.185 9.7200 0.9281
V55 0.9754 0.9736 0.9907 0.9301 QC79 14.9964 3.9148 14.250 0.9103
V56 0.9801 0.9715 1.0333 0.9308 QC82 9.3574 19.998 17.490 -26.90
V59 1.0235 1.0600 1.0099 0.9495 QC83 3.6167 20.000 4.2800 6.3230
V61 1.0400 0.9482 1.0925 0.9700 QC105 17.1048 -18.989 12.040 -10.63
V62 1.0517 1.0600 1.0393 0.9819 QC107 2.0274 -20.000 2.2600 3.3801
V65 1.0591 0.9953 0.9998 0.9870 QC110 1.8493 -20.000 2.9000 4.3057
V66 1.0319 0.9400 1.0355 1.0032 T8 1.0768 1.0059 1.0659 8.9977
V69 1.0291 1.0330 1.1000 1.0544 T32 0.9655 1.1000 0.9534 5.9038
V70 0.9777 0.9952 1.0992 1.0477 T36 1.0601 1.0110 0.9328 4.8237
V72 1.0258 0.9498 1.0014 1.0097 T51 1.0081 1.0110 1.0884 4.4373
V73 0.9572 0.9400 1.0111 1.0418 T93 1.0509 1.0001 1.0579 13.274
V74 0.9691 0.9400 1.0476 1.0073 T95 1.0283 1.0155 0.9493 6.1881
V76 0.9908 0.9400 1.0211 1.0212 T102 0.9962 1.1000 0.9975 5.5461
V77 1.0209 0.9935 1.0187 1.0410 T107 0.9356 0.9243 0.9887 4.3820
V80 1.0542 0.9400 1.0462 1.0476 T127 0.9541 0.9589 0.9801 2.3813
V85 1.0121 0.9887 1.0491 1.0604
V87 1.0120 0.9642 1.0426 0.9390 Loss (MW) 136.99 131.897 127.760 109.744
V89 1.0067 1.0026 1.0955 1.0984
Table 11. Control variables of ORPD for the IEEE 118-bus system in Scenario 2.
Table 11. Control variables of ORPD for the IEEE 118-bus system in Scenario 2.
Control variables GWO Control variables GWO Control variables GWO Control variables GWO Control variables GWO
P1 86.454 P65 46.411 V1 1.0590 V65 1.0689 QC5 -15.149
P4 56.449 P66 40.835 V4 0.9808 V66 1.0044 QC34 6.0664
P6 58.338 P69 37.066 V6 1.0001 V69 1.0790 QC37 -3.6222
P8 50.098 P70 125.69 V8 0.9932 V70 1.0398 QC44 6.6938
P10 34.288 P72 74.627 V10 1.0160 V72 1.0040 QC45 4.4421
P12 69.599 P73 63.114 V12 1.0024 V73 1.1000 QC46 1.2265
P15 147.08 P74 54.139 V15 1.0183 V74 1.0660 QC48 6.4032
P18 91.052 P76 57.223 V18 1.0410 V76 1.0925 QC74 8.2857
P19 51.930 P77 27.555 V19 1.0120 V77 1.0836 QC79 8.5603
P24 58.426 P80 285.23 V24 1.0254 V80 1.0320 QC82 9.7719
P25 73.794 P85 57.696 V25 0.9639 V85 0.9783 QC83 6.8119
P26 40.294 P87 72.549 V26 1.0588 V87 1.0465 QC105 5.8368
P27 217.58 P89 46.538 V27 1.0296 V89 1.0653 QC107 3.3788
P31 36.545 P90 63.959 V31 1.0336 V90 1.0177 QC110 2.9052
P32 63.325 P91 54.807 V32 1.0341 V91 1.0518 T8 0.9381
P34 21.061 P92 244.41 V34 1.0187 V92 1.0812 T32 0.9463
P36 66.161 P99 32.159 V36 1.0075 V99 0.9812 T36 0.9854
P40 23.633 P100 44.170 V40 0.9910 V100 1.0375 T51 1.0402
P42 161.63 P103 55.985 V42 1.0219 V103 1.0576 T93 1.0500
P46 75.908 P104 56.302 V46 1.0768 V104 0.9716 T95 0.9112
P49 16.549 P105 34.894 V49 1.0336 V105 0.9981 T102 1.0005
P54 64.019 P107 140.97 V54 1.1000 V107 0.9591 T107 0.9756
P55 80.821 P110 53.026 V55 1.0706 V110 1.0584 T127 0.9471
P56 14.430 P111 36.054 V56 1.0535 V111 1.0359 Loss (MW) 68.1578
P59 306.06 P112 89.350 V59 1.0948 V112 1.0850
P61 81.501 P113 58.478 V61 1.0169 V113 1.0399
P62 67.121 P116 34.454 V62 0.9881 V116 1.0311
Table 12. Limit Setting for Control Variables for IEEE-118 System in Scenario 3.
Table 12. Limit Setting for Control Variables for IEEE-118 System in Scenario 3.
Variables Lower Limit Upper Limit
Wind Power 0.10 pu 0.50 pu
Generated Power 0.05 pu 3.60 pu
Voltages 1.00 pu 1.10 pu
Tap Settings 0.90 pu 1.10 pu
Compensation Devices -20 MVAR 20 MVAR
P62 30.217 V62 1.0535
Table 13. Control variables of ORPD for the IEEE 118-bus system in Scenario 3.
Table 13. Control variables of ORPD for the IEEE 118-bus system in Scenario 3.
Control variables GWO Control variables GWO Control variables GWO Control variables GWO Control variables GWO
P1 252.23 P65 35.039 V1 1.0424 V65 1.0343 QC5 -4.279
P4 122.96 P66 37.903 V4 1.0284 V66 1.0629 QC34 8.7954
P6 40.162 P69 59.278 V6 1.0134 V69 1.0677 QC37 -2.750
P8 18.208 P70 82.197 V8 1.0507 V70 1.0668 QC44 1.0647
P9wind 27.538 P72 93.435 V9wind 0.9688 V72 1.0531 QC45 2.4947
P10 9.5797 P73 45.600 V10 1.0686 V73 1.0797 QC46 8.2764
P12 51.614 P74 11.502 V12 1.0184 V74 1.0450 QC48 3.6636
P15 154.29 P76 64.795 V15 1.0011 V76 1.0526 QC74 11.943
P18 26.439 P77 40.345 V18 0.9781 V77 1.0475 QC79 8.7899
P19 29.869 P80 290.65 V19 0.9890 V80 1.0553 QC82 7.0112
P24 28.433 P85 29.615 V24 1.0309 V85 1.0052 QC83 3.2327
P25 52.128 P87 40.991 V25 0.9719 V87 1.0538 QC105 2.9738
P26 36.567 P89 50.678 V26 1.0583 V89 1.0225 QC107 3.6681
P27 115.63 P90 28.770 V27 0.9903 V90 1.0135 QC110 4.0839
P31 13.306 P91 31.469 V31 0.9833 V91 1.0254 T8 1.0291
P32 88.117 P92 198.11 V32 0.9888 V92 1.0429 T32 0.9981
P34 42.177 P99 42.745 V34 1.0128 V99 1.0509 T36 0.9835
P36 32.487 P100 62.002 V36 1.0148 V100 1.0417 T51 1.0279
P40 40.382 P103 50.258 V40 1.0022 V103 1.0288 T93 0.9752
P42 164.87 P104 92.620 V42 1.0152 V104 1.0373 T95 1.0192
P46 119.43 P105 27.045 V46 1.0687 V105 1.0403 T102 0.9224
P49 24.695 P107 148.33 V49 1.0536 V107 1.0531 T107 0.9590
P54 9.7571 P110 50.260 V54 1.0508 V110 1.0197 T127 0.9192
P55 50.429 P111 33.580 V55 1.0556 V111 1.0344 Loss (MW) 28.1372
P56 53.080 P112 17.323 V56 1.0523 V112 0.9949
P59 327.36 P113 27.363 V59 1.0730 V113 0.9985 Voltage Deviation (pu) 1.12945
P61 138.60 P116 47.297 V61 1.0522 V116 1.0307
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

facebook logotwitter logolinkedin logo
MDPI Initiatives
Important Links
Subscribe

Disclaimer

Terms of Use

Privacy Policy

© 2024 MDPI (Basel, Switzerland) unless otherwise stated