Congestion cost estimation using adaptive red fox algorithm in restructured electricity markets

Abstract

Congestion of the power system is the most common challenge an Independent System Operator (ISO) faces in restructured electricity markets. It affects the efficiency of the market when transmission lines are congested causing transmission costs to rise. To prevent transmission line congestion, ISO needs to take the necessary steps. To solve these issues, this paper introduces a new method namely the Adaptive Red Fox Optimization algorithm (ARFOA) to compute the congestion cost considering the power losses in the transmission line system. Initially, all the generators in the system are selected to reschedule real power outputs. Second, by establishing a proposed optimization issue, ARFOA is employed to control transmission line congestion. The implementation of the proposed method is evaluated on the IEEE 30 bus system. The algorithm’s adaptability is tested using several case studies involving the base case and line outages, also compared with the other existing techniques such as PSO, ASO, and GSO approaches. The simulation outcomes indicate that the proposed strategy outperforms existing techniques in terms of congestion cost, power loss, generation rescheduled power, and computational time.

Keywords

Restructured power systems congestion management generator rescheduling Adaptive Red Fox Optimization algorithm optimal power flow

1 Introduction

Due to the increasing demand for electric power around the globe, electric utilities have increased their power generation capacity to meet the need [1]. Indian states are trading electricity with each other when they have a surplus electricity supply. The available capacity on the line cannot handle all of the load demanded by many traders [2]. Electric power can be transmitted between two locations in a transmission network only under certain conditions, including thermal, voltage, and stability limits, whichever is most restrictive at the time [3]. Overloading the line beyond its transfer capability limit causes congestion, which may override thermal limits, violate constraints, and result in violation of constraints [4]. Congestion is largely caused by sudden changes in load demand, line outages or generator outages, and a lack of coordination between generation (GENCOs) and transmission (TRANSCOs) companies. A congested network can result in higher electricity prices, heightened security limits, and tripping of overloaded lines which then result in tripping of healthy lines, causing the system to black out. Power loss, an unwanted voltage profile, and temperature violations are caused by congestion, all of which can cause the system to fail. In addition, congestion becoming more challenging for independent system operators (ISOs) to maintain system synchronism [5 –8].

A constraint is based on either the requirement to minimize transmission costs or the requirement that the power flow be violated. However, reducing congestion costs is the major objective [9]. Optimal power flow (OPF), improved power quality, preventing equipment failure, and avoiding more blackouts require reducing or avoiding congestion. Some approaches have been proposed for estimating the congestion costs of power systems. To alleviate transmission line congestion, real power generation can be adjusted, load shedding can be implemented, and Flexible AC Transmission Systems (FACTS) can be installed [10]. A few ways are given below.

Salkuti, S.R et. al proposed restructured power markets, an optimal power flow-based congestion solver using the multiobjective-GSO (Glowworm Swarm Optimization) algorithm [11]. Deb, S. et. al proposed an Atom Search Algorithm (ASO) for generator real power rescheduling for Congestion Management in Transmission Line systems [12]. Bashir, M.U. et. al presented a hybrid of Teaching learning-based optimization –Particle swarm optimization (TLBO-PSO) approach of congestion estimation using the optimum rescheduling of a generator in the pool electricity market [13]. Multi-objective-PSO [14], Hybrid Grey Wolf Optimizer and Cuckoo Search Algorithm [15], a hybrid of gravitational search algorithm (GSA) –fuzzy adaptive particle swarm optimization (FAPSO) [16], and Slime Mould Algorithm [17] techniques are presented for generator rescheduling, which is used to estimate congestion while taking generator fuel costs into account. Due to the transmission lines’ restricted power flow limitation, the generators in this approach are unable to provide the appropriate quantity of electricity to the load or customers. Sarwar, M. et. al created a congestion management system based on exploiting distributed generators’ optimum capabilities via hybrid swarm optimization (HSO) [18]. Chabok, H. et. al proposed best allocation of a cost wind turbine and storage system for energy units is achieved by utilizing a bi-level model with Karush-Kuhn-Tucker (KKT) criteria. The key disadvantage of this work is that the time required for the convergence of the presented approach is longer [19]. Namilakonda, S. et. al presented a Chaotic Darwinian Particle Swarm Optimization (CDPSO) is a method for establishing the best schedules for load response loads and rescheduling traditional generators to reduce overloading which is integrated with renewable energy sources [20]. Behera, S.K. et. al designed an improved GWO approach for dealing with congestion concerns in a deregulated electricity system by utilizing tap changer settings, reactive power compensation settings, and Thyristor controlled series compensator (TCSC) reactance setup. Here the optimal location of TCSC is calculated by using 25% and 50% overloading and line outage [21].

Srivastava, J. et. al designed a hybrid Lion Algorithm with Moth-based Mutation (LA-MM) algorithm-based Congestion management of minimized rescheduling cost and congestion cost [28]. Sometimes Generator Sensitivity Factors (GSF) are considered in CM power systems using Improved Crow Search Algorithm (ICSA) [29]. The location of TCSC can change with changes in generation and load levels in CM in power systems using Multi-Objective Genetic Algorithm (MOGA) [30]. The generation rescheduling is analyzed using hybrid Deep Neural Networks and modified backpropagation algorithms based on ANN techniques for congestion management [31]. Project management problem is solved using multi-objective optimization (MOO), the MOO not only satisfies the demands of taking into account the complexity and conflict between the desired multi-objectives but can also be capable of offering optimal solutions to realize a better project performance improvement [32]. A Whale Optimisation Algorithm (WOA) approach is utilized to sensorless control the speed of a permanent magnet brushless DC (PMBLDC) motor fed by solar photovoltaic (PV) energy under a variety of operating conditions. This allows for the best tuning of fractional-order proportional integral and integral-order controllers [33].

From the literate review, many approaches are proposed for relieving congestion, such as rescheduling generators, using regulating transformers or tap setting transformers, and using FACTS devices. The system architecture is kept unchanged by rescheduling generators instead of using other strategies. Consequently, congestion management costs will increase because bus power generation will change significantly. Many nonlinearity and multimodality difficulties cannot be addressed by most classic optimization strategies. Thus, to manage congestion effectively, it is necessary to utilize a fast convergence technique. Therefore, the above-mentioned research studies have a long convergence criterion (computation time or the number of iterations) which is one of their main drawbacks [34]. Due to the massive amount of transmission data and overwhelming of data transfer into the buses and network, there would be the congestion in the transmission lines. In order to reduce the congestion and reduces losses in the transmission network, the novel method should be developed. Also, it is important to obtain locally optimal solutions instead of globally optimal ones.

Considering the limitations outlined above, this study proposes transmission congestion costs in restructured energy markets using the Adaptive Red Fox Optimization algorithm (ARFOA) algorithm. ARFOA has been developed for generation rescheduling to minimize congestion costs and power loss. The problem of data congestion in the transmission line should be considered seriously. This can affect the electricity which causes the blackouts. Due to the huge amount of data transfer in the transmission grid causes the fault data transfer in the network. It affects the transmission efficiency in the bus system. So the total power grid(transmission line)gets affected.The existing search approach involves updating the solution in every iteration, and most deterministic transition rules are used to reach the optimal solution. ARFOA has the fastest convergence mobility and enhances the flexibility of the system by resolving the multiobjective problems associated with large-scale power systems. The primary goal of this work is to support ISO in removing line congestion optimally. This method of congestion cost calculation has been tested on IEEE 30 bus system.

The contributions of the present work are described below,

An Adaptive Red Fox Optimization algorithm (ARFOA) has been studied for the optimization challenge of resolving transmission line congestion.

By rescheduling each generating station’s generation, congestion can be minimized.

Two objective functions are taken into account; total congestion cost and total transmission losses minimization is considered

ARFOA is an effective optimization tool for minimizing rescheduling costs for the IEEE 30-bus system under various conditions.

The proposed ARFOA algorithm proves superior to existing algorithms in the congestion cost estimation problem.

The rest of the paper is arranged as follows; section 2 defines the problem formulation and the objective function; Section 3 explains the ARFO algorithm-based problem-solved issues and Section 4 describes the optimal power flow implementation steps of the proposed algorithm. Section 5 discusses the simulation results and comparison performance finally section 6 provides the conclusion.

2 Problem formulation

Power system economic evaluation often involves the analysis of optimal power flow, which is defined as an optimization problem containing an objective function and a set of constraints.

The standard for OPF problems can be written as follows: $Min {f (x)}, h (x) = 0; g (x) ⩾ 0$ (1)

Where f (x) signifies the objective function of the proposed work, g (x) signifies the constraints of inequality, h (x) denotes the constraints of equality, and x denotes the control variable vector that can be changed by the control center operator (i.e., the voltage level on the generation bus, active and reactive power, etc.).

2.1 Objective function

A key multi-objective function (MOF) in the formulation of the optimal power flow is to reduce the total cost/congestion cost by rescheduling (decreasing or increasing) generator active power output, as well as minimizing power losses throughout the network. The MOF of the proposed work is expressed below;

2.1.1 Total Congestion cost (Obj 1)

In this multi-objective congestion management procedure, cost minimization is one of the objectives. Also, every generation unit’s cost function is determined by its active power generated [22]. The congestion cost is calculated by increments or decreases in price bids from generators and demands. $C_{c} = [\sum_{i = 1}^{TG} C_{i}^{+} (Δ P_{i}^{+}) - \sum_{i = 1}^{TG} C_{i}^{-} (Δ P_{i}^{-})]$ (2)

Where, C_c –Congestion cost, P_{G
_i} –ith bus generation active power, TG –number of generation units, $Δ P_{i}^{+}$ and $C_{i}^{+} (Δ P_{i}^{+})$ –incremental change in generator power outputs and cost function respectively, $Δ P_{i}^{-} {andC}_{i}^{-} (Δ P_{i}^{-})$ –decremental change in generator power outputs and cost function respectively.

2.1.2 Minimization of transmission loss (Obj 2)

The second objective of this work is to optimize the power losses in the system as much as possible. From the load function solution, the loss is calculated for a congested line [23]. The below equation represents the total power loss as the sum of the power losses on a congested line.

$\begin{matrix} T_{loss} \\ = [\frac{1}{2} \sum_{i, j} [R_{ij} (V_{i}^{2} + V_{j}^{2} - 2 V_{i} V_{j} cos (δ_{i} - δ_{j}))]] \end{matrix}$ (3)

Where, R_ij –actual portions of the bus admittance matrix element, V_i and V_j- ith and jth bus voltage magnitude, δ_i and δ_j –ith and jth bus voltage angle.

2.2 Constraints

2.2.1 Equal constraints

It is essential to ensure that output equals electricity demand and power loss while minimizing the cost function [22]. Power flow equations are regarded as equal limitations in this regard. $[\begin{matrix} Δ P_{i} \\ Δ Q_{i} \end{matrix}] = [\begin{matrix} P_{i} (V, ϑ) - (P_{Gi} - P_{Di}) \\ Q_{i} (V, ϑ) - (Q_{Gi} - Q_{Di}) \end{matrix}] = 0$ (4)

Assuming the bus i is injected with active and reactive power, it follows equations (6): $P_{i} (V, ϑ) = \sum_{j = 1}^{NS} V_{i} V_{j} (R_{ij} cos ϑ_{ij} + S_{ij} sin ϑ_{ij})$ (5) $Q_{i} (V, ϑ) = \sum_{j = 1}^{NS} V_{i} V_{j} (R_{ij} sin ϑ_{ij} - S_{ij} cos ϑ_{ij})$ (6)

Where, R_ij –conductance, S_ij –susceptance, ϑ_ij –bus voltage angle, and V_i- ith bus magnitude of voltage.

2.2.2 inequality constraints

A generator’s reactive and active power outputs are dependent on its upper and lower limits, and are expressed as follows: $P_{Gi}^{\min} ⩽ P_{Gi} ⩽ P_{Gi}^{\max} i = 1, 2, \dots, TG$ (7) $Q_{Gi}^{\min} ⩽ Q_{Gi} ⩽ Q_{Gi}^{\max} i = 1, 2, \dots, TG$ (8) $V_{Gi}^{\min} ⩽ V_{Gi} ⩽ V_{Gi}^{\max} i = 1, 2, \dots, TG$ (9)

Limitations apply to transformer taps [24], both at minimum and maximum settings, $T_{i}^{\min} ⩽ T_{i} ⩽ T_{i}^{\max} i = 1, 2, \dots, N_{TF}$ (10)

VAR switchable sources have the following restrictions: $Q_{ci}^{\min} ⩽ Q_{ci} ⩽ Q_{ci}^{\max} i = 1, 2, \dots, N_{sv}$ (11)

Boundaries on load bus voltage magnitude (V_Di) and line flow (S_Li) are included in the security constraints, which are expressed below; $V_{Di}^{\min} ⩽ V_{Di} ⩽ V_{Di}^{\max} i = 1, 2, \dots, N_{D}$ (12) $S_{Li} ⩽ S_{Li}^{\max} i = 1, 2, \dots, N_{line}$ (13)

3 Proposed Adaptive Red Fox Optimization Algorithm (ARFOA)

The presented ARFO algorithm is used to resolve the given multi-objective problem with objective functions and constraints. Here, the optimal power flow problem with congestion cost and transmission loss minimizes objective functions. The outline of the proposed method is depicted in Fig. 1.

Fig. 1

Outline of the proposed methodology.

3.1 Analysis of Red Fox Optimization Algorithm

The red fox is a good predator of tiny animals, including both wild and domestic species. Red foxes are classified into two types: those that leave the well-defined territory and those that spend nomadic lives [25]. Under the alpha couple’s arrangement, each herd shears a specified region. If the young have a good chance of gaining control of another region when they reach adulthood, they may decide to leave the herd and create their herd. They either stay in the family or obtain fox-hunting land from a blood relative.

Fig. 2

Proposed ARFO algorithm for OPF process.

Each population is characterized by an n-coordinate point $\bar{X} = (X_{0}, X_{1}, \dots, X_{n - 1})$ . Initially, some randomly distributed individuals (in this case, red foxes) start it up. Thus, the best solution will be obtained when the global optimum is provided by f((X) ⁱ) ^t. Among the crew members, each is assigned a specific task to complete so that the family can be helped. Whenever there is not enough bait in one territory, the individuals move to another territory. A family member shares the location of a better territory with another member of the family if it is found. Exploration is modeled as global searching. After moving over the territory, it will attack the prey once it has been closed enough to it. Local searching is modeled in this phase as exploitation. Cost value determines the sort order of the population. Euclidean distance square can be used to accomplish this. To obtain this term, follow these steps: $({({(X)}^{i})}^{t}, {(X_{best})}^{t}) = \sqrt{{({(X)}^{i})}^{t} - {(X_{best})}^{t}}$ (14)

The candidates then move through the following steps to determine the best solution: ${({(X)}^{i})}^{t} = {({(X)}^{i})}^{t} + γ \times sgn ({(X_{best})}^{t} - {({(X)}^{i})}^{t})$ (15)

Where, γ –random value.

In this case, if the new location provides a better solution, it will replace the previous location, otherwise, the former location will remain. To catch its prey, the red fox gets close to it after observing it. This is achieved by exploiting the Red Fox Optimization algorithm, which assumes an initial value of r in the range [0, 1]: ${\begin{matrix} move closer if r > 3 / 4 \\ stay and hide if r ⩽ 3 / 4 \end{matrix}$ (16)

A first parameter is a random number between [0, 0.2], while the second variable, φ₀, describes the p-value in the range of [0, 2π] that sets the angle of the fox observation [25]. As a result, the visual radius of the hunting fox is as follows: $m = {\begin{matrix} p \times sin (φ_{0}) / φ_{0} φ_{0} not equal to 0 \\ γ φ_{0} equal to 0 \end{matrix}$ (17)

Where, γ –random amount between 0 and 1. The following are the individual moves of the candidates:

$\begin{matrix} {\begin{matrix} x_{0}^{new} = m \times p \times cos (φ_{1}) + x_{0}^{act} \\ x_{1}^{new} = m \times p \times sin (φ_{1}) + m \times p \times cos (φ_{2}) + x_{1}^{act} \\ \begin{matrix} x_{1}^{new} = m \times p \times sin (φ_{1}) + m \times p \times sin (φ_{2}) + m \times p \times \cos (φ_{3}) + x_{2}^{act} \\ ⋮ \\ x_{n - 1}^{new} = m \times p \times \sum_{k = 1}^{n - 2} sin (φ_{1}) + m \times p \times \cos (φ_{n - 1}) + x_{n - 2}^{act} \\ x_{n - 1}^{new} = m \times p \times sin (φ_{1}) + m \times p \times sin (φ_{2}) + \dots \\ + m \times p \times \sin (φ_{n - 1}) + m \times p \times \sin (φ_{n - 1}) + x_{n - a}^{act} \end{matrix} \end{matrix} \end{matrix}$ (18)

This equation represents the fox’s behavior as it approaches the bait and moves quietly to attack it. At the end of the procedure, 5 percent of the worst-case scenarios are discarded, and instead, some new candidates are introduced to the population by regenerating the alpha pair. In addition, the two top candidates are chosen as (X (1)) ^t and (X (2)) ^t to be the alpha couple in iteration t. The habitat center was then generated by doing the following: ${({Hab}_{cen})}^{t} = \frac{{(X (1))}^{t} - {(X (2))}^{t}}{2}$ (19)

Based on Euclidean distance, the habitat diameter is as follows:

${({Hab}_{dia})}^{t} = \sqrt{{(X (1))}^{t} - {(X (2))}^{t}}$ (20)

During the iteration, then choose a random variable, s, between 0 and 1. ${\begin{matrix} New nomadic candidate & if s > 0.45 \\ Reproduction of the alpha couple & if s ⩽ 0.45 \end{matrix}$ (21)

Random positions are chosen in the solution space and outside the habitat. The alpha pair then generates additional candidates, i.e. ${(X_{rep})}^{t} = s \frac{{(X (1))}^{t} - {(X (2))}^{t}}{2}$ (22)

3.2 Adaptive Red Fox Optimization Algorithm

The RFO algorithm incorporates various tuning parameters that have a significant impact on the algorithm’s performance. The randomization parameter (γ) is used to regulate the method for a random search when the provided fox does not observe the nearby foxes. Control of the random movement of each prey, which is picked at random from the range [0, 1]. To improve the algorithm’s capabilities both locally and globally, an adaptive control procedure is presented in this paper. The Adaptive RFO algorithm’s main idea is to automatically adjust three parameters, the fox size, the coefficient, and each particle, depending on the fitness values of the particles during the process. To execute an effective leap, the optimization is based on approaches for estimating the distance between the prey and the fox. Also, the exploration and exploitation tendencies are achieved in a moderate balance.

4 Optimization implementation process

In this proposed method, an optimization problem is solved by the ARFO algorithm. Here, the optimal power flow process and congestion cost estimation of a transmission line is solved by the proposed algorithm [26]. The implementation process steps are as follows:

4.1 Optimal power flow (OPF) implementation process

This process using the proposed Adaptive Red Fox Optimization algorithm is shown in Fig. 1 and the process steps are given below.

Step 1: The input data should comprise the bus voltage, the generator’s fuel cost coefficients, the generator’s real power, and the switchable VAR capacitor’s reactive power.

Step 2: Initialize the number of foxes and the number of iterations.

Step 3: Run load flows

Step 4: Randomly generate the population of foxes (iteration, k = 1)

Step 5: Set constraint which is mentioned in Eq. (4-13).

Step 6: Choose the random sample of γ red foxes by finding the values of the objective functions using Eq. (1–3).

Step 7: Update the distance of each fox using Eq. (19–22).

Step 8: Check the convergence criteria (stopping criteria). If yes, calculate the optimal objective function value and stop. Else go to step 5.

5 Results and discussions

The proposed work is implemented in MATLAB R2020b (64-bit) software on a system powered by an Intel Pentium Gold Processor with a clock speed of 4.01 GHz and 4 GB RAM and evaluated on IEEE 30-bus test systems shown in Fig. 3. This test system’s total active and reactive load power is 283.4 MW and 126.2 MVAR, respectively. The IEEE 30 bus consists of 6 generator systems [27]. To demonstrate the performance of the proposed ARFO algorithm in solving the congestion cost problem, the proposed method is compared with existing approaches such as Glowworm Swarm Optimization (GSO) [11], Atom Search Optimization (ASO) [12], and Multi-Objective Particle Swarm Optimization (PSO) [14].

Fig. 3

Single line diagram of IEEE 30 bus system.

Here, the objective problem is solved by concerning the congestion cost minimization, and power loss, which is solved by using the proposed ARFOA approach. The population of 50 foxes is used in this paper for solving the congestion cost problem., and 100 iterations are set to three test cases. For simulation purposes, congestion is generated in the lines by overloading them. The present study results can be summarized as follows.

Case 1: The system is not congested, and generation scheduling is used (i.e., base case).

Case 2: The line between buses 1-2 is unavailable with normal loading.

Case 3: The line connecting buses 1-7 is unavailable, and all buses have a 50% load increase.

5.1 Analysis of case 1

As previously stated, the bidding costs are not taken into account in the market clearing procedure in this scenario. Table 1 illustrates the optimal power flow data before and after the optimization of generator power for Case 1 (base case). Table 1 shows the best generating schedules with no congestion in the system by concurrently optimizing total generation cost and transmission losses.

Table 1
Optimal power flow results for case 1(base case)

SI. NO Parameters Before ARFOA Optimization After ARFOA optimization

1. PG1 (in MW) 177.0644 172.9633

2. PG2 (in MW) 48.7004 46.1677

3. PG5 (in MW) 23.3499 21.3041

4. PG8 (in MW) 21.0812 14.7344

5. PG11 (in MW) 14.3322 11.8834

6. PG13 (in MW) 22.0621 12.0000

7. Optimum generation cost ($/h) 799.8456 798.8466

8. Power losses (MW) 3.3014 2.8803

9. Computational time (s) 18.7383 16.5624

SI. NO	Parameters	Before ARFOA Optimization	After ARFOA optimization
1.	PG1 (in MW)	177.0644	172.9633
2.	PG2 (in MW)	48.7004	46.1677
3.	PG5 (in MW)	23.3499	21.3041
4.	PG8 (in MW)	21.0812	14.7344
5.	PG11 (in MW)	14.3322	11.8834
6.	PG13 (in MW)	22.0621	12.0000
7.	Optimum generation cost ($/h)	799.8456	798.8466
8.	Power losses (MW)	3.3014	2.8803
9.	Computational time (s)	18.7383	16.5624

The optimum generating cost derived by ARFOA may be shown in this table, which has a power loss of 2.8803MW and 798.8466 $/h. This demonstrates that meta-heuristic approaches outperform traditional procedures in terms of objective function values, although they are computationally demanding. Table 2 also includes the computational times for GSO, ASO, and PSO. By using the classical optimization techniques, GSO, ASO, and PSO are 799.069$/h, 799.688$/h, and 799.703$/h, respectively. The proposed techniques required less computation time when compared with the existing methods.

Table 2

Comparative analysis of optimal power flow results for case 1(base case)

Generator number and parameters	Techniques
	PSO	ASO	GSO	Proposed ARFOA
PG1 (in MW)	174.261	173.848	173.260	172.963
PG2 (in MW)	45.303	45.564	44.145	46.167
PG5 (in MW)	20.291	20.341	21.767	21.304
PG8 (in MW)	14.447	15.066	15.733	14.734
PG11 (in MW)	11.593	11.381	11.022	11.883
PG13(in MW)	13.321	13.029	13.312	12
Total generation (MW)	279.216	279.229	279.339	279.49
Total Generation cost ($/h)	799.703	799.688	799.069	798.8466
Transmission losses (MW)	3.75	3.209	3.0012	2.8803
Computational time (s)	25.62	22.37	20.82	16.5624

Figure 4(a) shows the case 1 optimum results of generation cost before and after ARFOA optimization.

Fig. 4

Optimum results of before and after ARFOA optimization (a) Generation cost (b) power loss for case 1.

The overall generation cost of the optimal method derived using the ARFOA technique is 798.8466$/h and the loss is 2.8803MW. It can be seen from these data that the computed objective values from the ARFO algorithm are extremely near the optimal value. This demonstrates the ARFO algorithm’s capacity to get closer to the optimum value. These findings demonstrate the ARFO algorithm’s resilience when used for the suggested congestion cost estimate problem.

Figure 4(b) shows the case 1 optimum results of power loss before and after ARFOA optimization. From the simulation results, the power is slightly reduced after applying the proposed ARFOA algorithm.

5.2 Analysis of case 2

This situation is developed by the interruption of line 1 between bus-1 and bus-2, which results in congestion. Congestion occurs on bus lines 1-6 and 7-8 when the line between bus 1 and bus 2 is outed. A proposed ARFO algorithm of power flow is used to estimate the amount of overloading on congested lines. There is power passing along such lines of 147.46 MW and 136.29 MW, for congested lines 1-6, and 7-8 correspondingly and a net power flow limit of 130 MW for both.

A transmission network’s power can’t be transmitted beyond its permitted limits for the power system to operate efficiently and reliably. For this reason, mandatory steps must be taken to reduce the congestion on the lines. Utilizing the generation rescheduling technique, the study aims to ease the overloading of the transmission line. Hence, the congestion cost can be improved by optimizing the generator rescheduling output. Table 3 and Fig. 5 show the Optimal generation schedule results before and after optimization for case 2. After applying the proposed ARFOA algorithm, the generation cost is reduced from 812.1002 $/h to 802.214 $/h.

Table 3
Optimal generation schedule results before and after optimization for case 2

Sl. NO Parameters Before ARFOA Optimization After ARFOA optimization

1. PG1 (in MW) 58.3761 160.8138

2. PG2 (in MW) 80 51.1187

3. PG5 (in MW) 49.3120 21.1658

4. PG8 (in MW) 34.8313 28.9262

5. PG11 (in MW) 30 10.9080

6. PG13(in MW) 34.2508 19.3936

7. Total Generation (in MWs) 286.7703 292.326

8. Optimum Generation Cost (in $/h) 812.1002 802.214

9. Transmission losses (in MW) 3.2623 3.37213

Sl. NO	Parameters	Before ARFOA Optimization	After ARFOA optimization
1.	PG1 (in MW)	58.3761	160.8138
2.	PG2 (in MW)	80	51.1187
3.	PG5 (in MW)	49.3120	21.1658
4.	PG8 (in MW)	34.8313	28.9262
5.	PG11 (in MW)	30	10.9080
6.	PG13(in MW)	34.2508	19.3936
7.	Total Generation (in MWs)	286.7703	292.326
8.	Optimum Generation Cost (in $/h)	812.1002	802.214
9.	Transmission losses (in MW)	3.2623	3.37213

Fig. 5

Optimal generation cost analysis before and after Optimization for case 2.

Table 4 summarizes the results obtained by applying the proposed ARFO to case 2 for the estimation of congestion costs. In Table 4, results obtained using GSO, PSO, and ASO techniques are also included for comparison. Table 4 shows that the proposed ARFOA provides the best results, with minimum congestion costs without overloading other lines, as compared to other methods. Based on the proposed ARFOA, the best solution is 435.013 $/h. The convergence of congestion cost using ARFOA for case 2 is represented in Fig. 6. From the results, the proposed ARFOA method produces a better solution in the 34th iteration. As a result, the ARFOA algorithm-based generator rescheduling solution mitigates this congestion scenario. A comparison of active power rescheduling and congestion costs obtainable by different methods such as GSO, PSO, and ASO is presented in Figs. 7 8. As provided by the proposed ARFOA method, the active power generated by generators can be adjusted up or down.

Table 4

Comparative analysis of optimal power flow results for case 2

Generator number and Parameters	Techniques
	PSO	ASO	GSO	Proposed ARFOA
ΔPG1 (MW)	–9.6112	–10.589	–8.863	–9.1494
ΔPG 2 (MW)	10.2356	10.5669	11.635	14.1918
ΔPG 3 (MW)	3.4897	3.5365	4.6342	0.53264
ΔPG 4 (MW	1.6558	1.5258	0.9421	0.04575
ΔPG5 (MW)	1.05015	1.2581	0.9445	0.34678
ΔPG6 (MW)	0.2821	0.9125	1.0251	0.950943
Power flow on earlier congested lines 1–7 (MW)	129.751	129.99	129.459	129.3053
Power flow earlier congested line 7–8 (MW)	109.2544	109.3304	109.5524	109.6952
Total generation rescheduled (MW)	24.0235	24.9554	24.5962	25.21727
Total congestion cost ($/h)	446.354	448.568	449.658	435.013

Fig. 6

Convergence of congestion cost using ARFOA for case 2.

Fig. 7

Comparative active power rescheduling of generators for case 2.

Fig. 8

Comparative analysis of Congestion cost for case 2.

All of the generators in the system are rescheduled, lowering both the cost of rescheduling and the line power flow. A generation rescheduling alone completely relieves line congestion without affecting load curtailment. Comparing the proposed method to existing GSO, ASO, and PSO-based techniques, the proposed approach not only mitigates congestion but also yields an optimum solution.

5.3 Analysis of case 3

Congestion is induced in this scenario by considering the interruption of line 2, which connects buses 1 and 7, as well as a 50% increase in load on all buses. This calculated scenario causes congestion of lines linked between buses 1-2, bus 2-8, and bus 2-9 with power flows of 310.9 MW, 97.3 MW, and 103.5 MW, correspondingly, which exceed the extreme power flow limitations of their respective buses. This circumstance occurs when one of lines 1-7 is forced out of service, resulting in a 50% increase in the total bus load. The line limit of congested lines 1-2, 2-8, and 2-9 is 130, 65, and 65 MW, respectively. To improve this overloading, the optimal rescheduling of generators is performed using ARFOA, and the results are shown in Table 5. According to the literature assessment, the rescheduling costs reported by several of the methods are significantly greater than the suggested approach.

Table 5
Comparative analysis of optimal power flow results for case 3

Parameters Techniques

PSO ASO GSO ARFOA

Power flow on earlier congested lines 1-2 (MW) 129.06 128.59 129.42 126.0307

Power flow on previously congested lines 2-8 (MW) 61.77 61.91 62.05 68.72213

Buses 2-9 Power flow on earlier congested line (MW) 65.14 64.80 65.29 69.78821

ΔPG1 (MW) NR –9.9971 NR –9.119

ΔPG 2 (MW) NR 77.9425 NR 79.52688

ΔPG 3 (MW) NR 0.0018 NR 0.663739

ΔPG 4 (MW) NR 44.046 NR 47.23817

ΔPG5 (MW) NR 23.6766 NR 26.00169

ΔPG6 (MW) NR 17.3866 NR 19.09192

Total generation rescheduled (MW) 164.55 173.0506 167.974 172.6414

Total congestion cost ($/h) 5788.05 5535.5 5369.58 5239.16

NR –Not reported in Literature

Parameters	Techniques
Power flow on earlier congested lines 1-2 (MW)	129.06	128.59	129.42	126.0307
Power flow on previously congested lines 2-8 (MW)	61.77	61.91	62.05	68.72213
Buses 2-9 Power flow on earlier congested line (MW)	65.14	64.80	65.29	69.78821
ΔPG1 (MW)	NR	–9.9971	NR	–9.119
ΔPG 2 (MW)	NR	77.9425	NR	79.52688
ΔPG 3 (MW)	NR	0.0018	NR	0.663739
ΔPG 4 (MW)	NR	44.046	NR	47.23817
ΔPG5 (MW)	NR	23.6766	NR	26.00169
ΔPG6 (MW)	NR	17.3866	NR	19.09192
Total generation rescheduled (MW)	164.55	173.0506	167.974	172.6414
Total congestion cost ($/h)	5788.05	5535.5	5369.58	5239.16
NR –Not reported in Literature

Table 5 compares several techniques for congestion control in Case 3. All of the published findings have the same test data. The total number of postponed generators is 172.6414 MW. From the table, the proposed ARFOA algorithm increases or decreases power on each generator bus to maintain the line power flow within its limit while satisfying load demands while taking power loss into account. Figure 9 depicts the comparative analysis of active power rescheduling of generators for case 3. From the result, the total rescheduling cost is 5239.16 $/hr for the proposed ARFOA techniques and the existing GSO, ASO, and PSO have a cost of 5369.58$/hr, 5535.5$/hr, and 5788.05$/hr, respectively shown in Fig. 10. The proposed ARFOA total rescheduling cost is quite less than GSO, ASO, and PSO-based techniques. An ARFOA-derived fitness function is shown in Fig. 11. After 100 iterations, the proposed ARFOA model reaches the least cost function of the existing algorithm, where the cost function decreases as iterations decrease. ARFOA has some additional advantages, such as a random reduction, a shorter time required to produce an optimum value, and an automatic subdivision of the foxes.

Fig. 9

Comparative active power rescheduling of generators for case 3.

Fig. 10

Comparative analysis of Congestion cost for case 3.

Fig. 11

ARFOA-based convergence for case 3.

In Fig. 12, convergence characteristics of PSO, ASO, GSO, and Proposed ARFO are shown. The maximum iteration limit is set to 1000. The proposed ARFO could converge to a lower redispatch cost compared to existing PSO, ASO, and GSO.

Fig. 12

Convergence characteristics on the IEEE 30-bus system.

6 Conclusion

This research presents a novel Adaptive Red Fox optimization algorithm (ARFOA) for estimating congestion costs in a restructured energy market. ARFOA is used to reduce the cost of rescheduling to eliminate congestion. This work takes into account contingencies like line outages and load variations. The suggested approach is evaluated on IEEE 30 bus systems, and the outcomes are compared with Glow worm Swarm Optimization (GSO), Atom Search Optimization (ASO), and Multi-objective Particle Swarm Optimization (PSO). According to the simulation findings, the suggested ARFOA efficiently dismisses congestion, and the rescheduling cost achieved is significantly lesser than the costs indicated by the existing approaches. Furthermore, the overall rescheduling and loss amounts are determined to be smaller. As a result, it is possible to infer that ARFOA is a powerful solution to optimizing issues, giving the most inexpensive, dependable, and secure working conditions. ARFOA is effectively used to manage system congestion and minimize total congestion costs and transmission losses. Performing sensitivity analysis on the selection of contributing generators, as well as rescheduling, may be of interest in a future study.

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Parameters	Techniques
	PSO	ASO	GSO	ARFOA
Power flow on earlier congested lines 1-2 (MW)	129.06	128.59	129.42	126.0307
Power flow on previously congested lines 2-8 (MW)	61.77	61.91	62.05	68.72213
Buses 2-9 Power flow on earlier congested line (MW)	65.14	64.80	65.29	69.78821
ΔPG1 (MW)	NR	–9.9971	NR	–9.119
ΔPG 2 (MW)	NR	77.9425	NR	79.52688
ΔPG 3 (MW)	NR	0.0018	NR	0.663739
ΔPG 4 (MW)	NR	44.046	NR	47.23817
ΔPG5 (MW)	NR	23.6766	NR	26.00169
ΔPG6 (MW)	NR	17.3866	NR	19.09192
Total generation rescheduled (MW)	164.55	173.0506	167.974	172.6414
Total congestion cost ($/h)	5788.05	5535.5	5369.58	5239.16
NR –Not reported in Literature