Transmission Congestion Management Using Generator Sensitivity Factors for Active and Reactive Power Rescheduling Using Particle Swarm Optimization Algorithm

Independent System Operators have difficulty in fulfilling all contractual power transactions in a competitive energy market due to transmission network congestion. As a result, applications of generator rescheduling become one of the antidotes in alleviating this difficulty in the consequence of ever-increasing numerous power transactions. The goal of this research is to lower the cost of active and reactive power of the generators by reducing the deviation of rescheduled active and reactive power from scheduled values. The inclusion of reactive power rescheduling and voltage stability in this paper is innovative, as compare to other existing methodologies solely examine active power rescheduling. This paper made the following contributions: formulated a multi-objective function for congestion control in an electric transmission network. Furthermore, formulated the generator sensitivity factors to identify overloaded lines and which generators will be involved in congestion management. Developed a particle swarm optimization (PSO) algorithm to solve the multi-objective function of the transmission congestion management system. In addition, the developed PSO method for CM approach was validated on three IEEE standard test system networks (14, 30, and 118). The simulation results prove that reduces active and reactive power, lowering the cost of generator rescheduling, and demonstrating the usefulness of developed PSO method for transmission network congestion. Furthermore, voltage stability and voltage profile improvements demonstrate the performance effectiveness of the PSO algorithm used in this work.


C Pg
Cost of rescheduling active power by the partaking generator in congestion management C Qg Cost of rescheduling reactive power by the partaking generator in congestion management P g Generator's active power adjustments Q g Generator's reactive power adjustments L max Maximum voltage stability indicator The associate editor coordinating the review of this manuscript and approving it for publication was Sotirios Goudos .

PF
Penalty factor S Gmax Generator's maximum nominal apparent power C P g Cost of generating reactive power by the generator ϕ Profit rate of reactive power generation NB Number of buses P Gi Active power produced at bus i Q Gi Reactive power produced at bus i P Di Active power demand at bus i Q Di Reactive power demand at bus i Bus j complex voltage Y ij < δ ij Bus i and j mutual admittance θ ij Bus i and j impedance angle Overall number of buses x min Minimum variable limit x max Minimum variable limit k n Penalty function constant (i.e n = 1, 2, . . . n) PG gn Generator's active power QG gn Generator's reactive power V i Voltage at bus i θ i Phase angle at bus i G ij Line k conductance B ij Line k susceptance V [] Particles velocity ω Inertia weight N p Overall particles number in the swarm n Overall, members number in one particle V g,i Initial velocity of the particles P g,i Initial [] Global best solution of all solutions Iter max Overall number of iterations P present [] Position of the current particle c1andc2 Are acceleration coefficients a n , b n , c n Are generators predetermining cost coefficients h 1 , h 2 , h 3 , h 4  The electrical power structure has traditionally been divided into three parts: generation (GENCOs), transmission (TRAN-COs), and distribution (DISCOs) [1]. Initially, all three divisions of the power system were monitored and controlled by a single authority known as a vertically integrated utility. However, with rapid population growth, rapid industrialization, and technological advancement, there is a tremendous and exponential demand for more clean and reliable energy at the consumer end. As a result, the global electric power industries are experiencing restructuring and deregulation in several countries [2]. The reforming and deregulating of the electric power industry have ensued in various consumers' high open-access penetration of the network. Each customer competes to access a very affordable and consistent supply from the cheapest available generator, regardless of distance. Aside from the primary goal of restructuring the electric power system to meet the high demand for electricity supply, some other disadvantages contribute significantly to the system's unhealthy state, such as ancillary service, inefficient market, congestion, and market power. Congestion is the most serious of the disadvantages associated with the deregulation of the electric power industry, and more attention is being paid to it.
Congestion occurs when a transmission line's thermal, voltage, and stability limits are violated or exceeded due to overloading. Other unexpected scenarios that contributed to the electric transmission network congestion included a sudden power outage, equipment failure, and an unexpected increase/decrease in demand. Congestion management (CM) techniques are used to mitigate these scenarios. However, any traces of congestion must be dealt with immediately to maintain a good and well-functioning system and avoid total system collapse, which can lead to total blackout. Therefore, a proper congestion management strategy must also be implemented.
Congestion management is critical for system balancing, system security, and reliability, as well as providing solutions to any financial problems caused by congestion. Therefore, several CM methods have been discussed in the literature, and a significant amount of research efforts have been devoted to identifying the appropriate congestion management techniques that can alleviate congestion in transmission networks while causing no or little disruption in consumer demand for electricity. According to the literature, popular CM techniques include Generators Rescheduling (GR), Flexible Alternating Current Transmission System (FACTS) devices, Optimization techniques, Re-dispatch, and CM-based Available Transfer Capability (ATC) [3], [4], [5]. Fig. 1 depict various types of congestion management methods.
Ref [6] presented a technique for verifying the best position and size of FACTS devices (TCSC) with GA for CM. The method was successfully validated by examining the effects of TCSC on the IEEE 30-bus network, and it was confirmed that the compensated network with TCSC reduced congestion. It was also determined that the device was adequate for long-term congestion control. [7] investigated active power Performance Index (PI)-based optimum allocation of FACTS devices to address CM technical issues in power system deregulation. The approach was validated using MATLAB Simulink on an IEEE 14 bus system.
Ref [8] presented a cost-free approach to optimal global allotment of FACTS devices using GA to mitigate overloading in a liberalized power system network. The objective function in congestion management was solved with GA because it was nonlinear. This strategy's applicability to a real-world practical system was demonstrated using the IEEE 30-bus system network. In [9], a novel approach to alleviating power system congestion was validated by combining a Power Oscillation Damper (POD) with FACTS devices (SSSC and UPFC). The proposed solution decongests the congested lines while also increasing the capability power of the lines. Ref [10] applied phase shift transformer method to relieve congestion on congested lines. The method was validated on a modified IEEE 24-bus test system network. [11] proposed a thyristor-controlled phase shifting transformer technique using GAMS solver to alleviate congestion. The viability of the technique was examined on the IEEE 24-bus network.
Authors in Ref [12] developed a cost-free approach to CM by investigating TCSC potentials. The FACTS results, as mentioned earlier, were compared and validated using the IEEE 14-bus system network and MATLAB software. [13] proposed a congestion management solution based on simulated annealing (SA) algorithm-based optimal placement of UPFC. The purported method was utilized to unravel the multi-objective function problem of UPFC placement. MATLAB software was employed to validate the proposed method. Ref [14] presented a unique technique for optimum allocation of FACTS controllers-based GA and the Strength Pareto Evolutionary Algorithm (SPEA). Both approaches were used concurrently for single-objective and three-objective optimization on power systems. The method was validated using MATLAB software on an IEEE 30 bus test setup. Ref [15] created a modified UPFC control circuit to alleviate congestion. A sensitivity-based technique was used to pinpoint the UPFC's location. In PSCAD/EMTDC, a 5-buses, 7-line transmission network was utilized to simulate the model. Authors in [16] describe numerous index techniques for optimal FACTS device placement. An IEEE 30-bus test system network was utilized to validate the method.
Ref [2] proposed a survey on methodologies and approaches to reducing congestion on power transmission lines and reviewed several major CM strategies used by researchers. In [17], the author makes use of a novel demand response programs to alleviate congestion. The optimal time of execution of DRPs using wind power and the proposed model was validated on IEEE 39-bus system network. Ref [18] suggested a generator rescheduling method to avoid congestion. The authors make use of firefly optimization algorithm to reschedule the output power of the participating generator to the congested lines. The model's capability was tested on the IEEE 39-bus system network. Authors [19] created a novel method for CM in transmission networks by creating a control algorithm that manages active power flow in the network and validating it on an IEEE 5-bus test system network. Ref [20] proposed a probability occurrence method for CM, which analyzed the most critical lines and validated the technique's performance using the IEEE 14-bus system network. Many of these techniques/methods relieve transmission network congestion by rescheduling only active output power of participating generators to congested lines.
A method of optimal placement of energy storage systems (ESSs) for the mitigation of congestion in electric power transmission network was proposed in [21]. The authors solved these multi-objective function by utilizing generalized algebraic modelling system (GAMS) based security constraint unit commitment (SCUC) and MATLAB. The effectiveness of this method was validated on the IEEE 24-bus RTS. The system operational cost was reported to be minimized in GAMS by SCUC, while the investment and storage costs are minimized in MATLAB by the NSGA-II algorithm which gives a set of Pareto optimal solutions. Authors in [22] proposed a novel real-time hierarchical congestion management (RHCM) method. The proposed method mitigate congestion by reschedules generators in two stages based on Available Congestion Clearing Time (ACCT) of the transmission lines in presence of renewable energy sources (solar and wind). The proposed two-stage RHCM method provides feasible solution to ISO to mitigate congestion in terms of minimum cost of relieving congestion.
Ref [23] proposed a potential probabilistic method based on wind power outputs to manage the congestion problems of power grids caused by variability of the loads. For the implementation and validation of the proposed approach, the author's utilized the historical data from the wind farms located on Jeju Island in South Korea to fit the Weibull distribution and implement Monte Carlo simulations. Authors in [24] proposed an hourly method of congestion management in deregulated power market. The authors utilized transmission congestion rent (TCR) to determined optimal location of the placement of DGs (solar PV and energy storage system) whereas optimal size of the DGs were determined by a hybrid deferential evolution and particle swarm optimization technique. The proposed method was carried out on IEEE 30-bu test system.
In [25], the authors proposed a novel transmission switching based cost-effective technique to alleviate congestion (overloading). The model was design to provides minimum voltage security index while easing transmission lines congestion. The proposed model was implemented on a 6-bus IEEE test system and 93-bus real test network (Transmission network of Fars province in Iran) to show the validity and authenticity of the work. Ref [26], developed a new congestion management model based on power system partitioning technique. The proposed model used congestion index to identify the congested lines and congestion management was performed by the identification of the candidate zones, in order to alleviate congestion on the critical lines. The model was implemented on IEEE 39-bus test system. Ref [27], proposed a probabilistic security-constrained optimal power flow (SCOPF) model for congestion management which was based on the non-linear ac formulation. For proper controlling power system devices, the proposed method used a second-order cone (soc) relaxation. The method was validated on modified IEEE-118 bus test system, and the results of the soc and the traditional ac power flow were compared. The state-of-the-art of the literature review for transmission congestion management systems based on technical and non-technical methods using the state-of-the-art devices and novel algorithm to relive the transmission congestion proposed by authors from the literature is summarized in Table 1 below. Therefore, this paper presents an optimal power flow (OPF) analysis-based PSO algorithm method to identify participating generators to congestion and optimally reschedule their output (active and reactive power) while managing congestion at the lowest possible rescheduling cost. Furthermore, because the conventional method of OPF is premised on the exploration path, which is obtained from the function derivative, the output of the participating generators was optimally rescheduled to mitigate congestion using the PSO algorithm. The following are the study's contributions: i) Formulated multi-objective function mathematical model for congestion control in an electric transmission network. ii) Formulated the Generator Sensitivity Factor (GSF) for both active and reactive power in order to detect the congested lines. iii) Developed particle swarm optimization (PSO) algorithm for the multi-objective function to solve the transmission congestion management system. iv) The developed PSO method for CM approach was validated on three IEEE standard test system networks (14, 30, and 118) and its simulation results are presented.

II. PROBLEM FORMULATION FOR CONGESTION MANAGEMENT A. OBJECTIVE FUNCTION
This work aimed at alleviating the electric transmission network congestion by reducing the rescheduling cost of the output power of the generators involved in congestion. The PSO algorithm is employed to unravel this nonlinear OPF problem. The sum of the overall amount of rescheduling needed by the designated generator can be written as (1) [28]: C P g PG gn = a n PG 2 gn + b n PG gn + c n Normalization for multi-objective functions can be made by utilizing weighting strategy (weighted fitness function) to convert both economic and technical parameters into a single objective function [29]. Any multi-objective function solutions without weighting strategy have a higher tendency to divert towards conflicting solutions. In this proposed work, the authors make use of normalized weights to form final fitness function for (1) to be optimized. The weighted multiobjective fitness function is expressed as: These are system power balance constraints, and they can be written as (4) and (5):

2) INEQUALITY CONSTRAINTS
These are constraints for control variables, and they can be written as (6) to (10): N g g GS k Pgn × P g + P ij The penalty function PF is expressed in (11) and (12) to control the limits of all the inequality constraint variables.

B. FORMATION OF THE GENERATOR SENSITIVITY FACTORS
With different sensitivity of generators to power flow on the overloaded lines, a change in power flow on transmission line k joining buses i and j subjected to unit variation in active and reactive power injection by generator-g at bus-n can be term as the generator sensitivity to the congested line (GSF).

1) GENERATOR SENSITIVITY FACTORS FOR ACTIVE POWER
Mathematically, GSF for active power at line k can be stated as (13) [28], [30]: By disregarding the P-V coupling, (13) can be further uttered as (14): The congested line power flow equation can be stated as (15): Differentiating (15) gives the first and the third term of (14) and can be written as (16) and (17).
The injected real power at bus i can be stated as (18): P i can be conveyed as (19): Differentiating equation (19) w.r.t θ i and θ j gives (20) and (21), and disregarding the P-V coupling, the expression that governs the dependency of the incremental variation in active power at the system buses on the phase angles of voltages is given in matrix form as (22) to (24): 2) REACTIVE POWER GENERATOR SENSITIVITY FACTORS Mathematically, GSF for reactive power at line k can be conveyed as (25) [22]: By neglecting the Q − δ coupling, (25) can be further expressed as (26): The congested line reactive power flow equation can be penned as (27): By differentiating (27), gives first and the third term of (26) and can be given as (28) and (29).
Therefore, injected reactive power at bus i can be written as (30): Q i can be expressed as (31): Differentiating equation (31) w.r.t θ i and θ j gives (32) and (33). The matrices of the partial derivatives for (32) and (33) w.r.t magnitude voltages at buses i and j can be stated as (34) and (35), respectively.

III. OVERVIEW OF PSO AND ITS CONGESTION MANAGEMENT SOLUTION
Eberhart and Kennedy proposed PSO as a high-speed, uncomplicated, and effective population-based optimization technique [31], [32]. It was stirred by organism actions such as fish schooling. A 'SWARM' in PSO is a collection of particles representing various solutions. Each particle's coordinates are linked to two vectors: position and velocity vectors. Each position and velocity possess equal capacity with the capacity of the problem space. Swarm particles all  fly through the search space in search of optimal solutions by updating the generation. The two best values update each particle in every iteration. The first value is called the personal best P best solution of the particle at each iteration, and the second value is called the global best G best solution of all the best particle solutions. For each particle, the velocity and positions are updated using (36) and (37), respectively [33]:   The inertia weight can be expressed as (38): Without a limit enacted on the particles' maximum velocity (V max ), the particles may break away from the search space. Therefore, each particle velocity is coordinated between (−V max ) to (V max ). Also, a correct range of inertia weight in (38) gives good stability between global and local explorations.     The GSF for active and reactive power rescheduling is stated in (13) and (25), respectively. Inequality and equality constraints are given in sub-sections 1 and 2 of section II(A), and the penalty function (11) and (12) are utilized to formulate the objective function (1) for the congestion management problem. By incorporating the specific CM problem, the problem mentioned above is mitigated using the PSO algorithm. To map the CM problem to suit the PSO formation of velocity and position in (36) and (37). The following assumption was made: 1. Member's numbers in different particles in the swarm were assumed to equal the number of the generators. 2. The active and reactive power was made to represent the velocities variables utilized to explore the constraint's domain. 3. Finally, the particle number in the swarm was denoted by N p .
Step 1: Input systems data for all the three IEEE networks (14, 30, and 118) considered.  Step 2: Run the Power Flow Analysis method by Newton-Raphson to determine the congested lines.
Step 3: Calculate Generator Sensitivities Factors (GSF) for all generators to the overloaded line using (13) and (25), respectively. This is done by checking out for both active and reactive power GSF of all generators matching the overloaded lines.
Step 5: Based on active and reactive power limits constraints, minimum and maximum initial velocities values were computed using (6) and (7) and are further expressed as follows: The particle's velocity and position are calculated using (n-1) generators because one of the generators is selected as the slack generator. Step 6: Except slack bus generator, the initial particle velocity is calculated using (41).
Traditionally, the electric power system buses are categorized into three, which are slack bus, voltage control (PV) bus, and load (PQ) bus. In addition, the nearer bus to the generator with higher generating capacity is called the slack bus. The function of a slack bus in implementing the Particle Swarm VOLUME 10, 2022  Optimization algorithm is to comply with the power balance constraint stated in (4) and (5).
Step 8: Compute the objective function for the initial positions using (1).  Step 9: Compute the personal best and the global best as follows: i) The personal best of the particles is computed using (43) ii) The global best is calculated using (44) Step 10: New velocity is computed using (45): Step 11: New position in the particles is computed using (46):  Step 12: Repeat step 2 to compute new line flows, new rescheduling active and reactive power, line losses, and new voltage magnitude in all buses.
Step 13: Compute penalty function for each particle using (11). This is done by finding constraint violations.
Step 14: Compute fitness function for each particle using (1) Step 15: Find out the ''global best'' (G best ) particle and ''personal best'' (P best ) of all particles.
Step 17: Repeat steps 3, 10 to 18 until the convergence criterion is met.
Step 18: Stop simulation.    rescheduling of active and reactive power of the generators, and cost of rescheduling were the performance metrics considered. The simulation was carried out using MATLAB 2022a.

A. CASE 1: IEEE 14-BUS SYSTEM NETWORK
The network data were acquired from [34]. The network comprises 14 buses, 20 interconnected lines, and 5 generators. Fig. 3 depicts its single-line diagram.  According to the power flow results, line number 6 (between buses 2 and 5) was identified as the congested line. Table 2 shows the detailed result for the power flow of the congested line. Fig. 4 also shows the detailed results of  generator sensitivity factors, which were utilized to identify any generators that are or are not participating in congestion. Any generator with a negative sensitivity factor for both active and reactive power indicates that increasing the generation of such a generator reduces the power flow in congested lines. Also, positive values of sensitivity factor for both active and reactive power of the generator indicate an increase in the power flow in a such generator. As can be seen in Fig. 4, generators 1, 2, 6, and 8 are the generators that would help to alleviate congestion on the congested line. Therefore, to alleviate congestion, the output power of the generators was optimally rescheduled using the PSO Algorithm. The detailed results of PSO optimally rescheduling the output power of the partaking generators to alleviate congestion are shown in Table 3.
Generator rescheduling for congestion mitigation can sometimes result in significant or low load bus voltage deviation. To address the issue of voltage deviation on the load buses, generator voltages were rescheduled to maintain voltages at all load bus within allowable boundaries. In addition, reactive power rescheduling significantly improves the voltage profile of all load buses and protects the system from voltage collapse. The Pre-CM and Post-CM voltage profile improvement is shown in Fig. 5. Fig. 6 and 7 also depict the convergence characteristics of the PSO-based active and reactive power rescheduling cost for the test system network. As shown in Figures 6 and 7, the cost of rescheduling both active and reactive powers for IEEE 14 bus system decreases as the converge characteristics (iteration number) increases.

B. CASE 2: IEEE 30 BUS SYSTEM NETWORK
The network data were obtained from [36]. The network comprises 30 buses, 41 interconnected lines, and 6 generators. Fig. 8 depicts its single-line diagram.
According to the power flow results, lines 1 and 5 are the most congested. The detailed result for the power flow of the congested line is shown in Table 4 below. In addition, Fig. 9 and 10 show the detailed results of generator sensitivity factors (GSF), which were used to identify any generators contributing to congestion on lines 1 and 5.
Based on the GSF principle explained in sub-section IV (A) of case 1 above, generators 1, 2, 5, 8, and 13 are the generators that would participate in alleviating congestion from the congested line. In addition, the generator output powers have been optimally rescheduled using the PSO Algorithm to reduce congestion. The detailed results of PSO optimally rescheduling the output power of the partaking generators to alleviate congestion are shown in Table 5.
Also, to conquer the hassle of voltage deviation at the load buses, generator voltages were rescheduled to hold load bus voltages within acceptable boundaries. Reactive power rescheduling helps enhance the voltage stability in all load buses and ensures the system out of voltage collapse point. Fig. 11 shows the before and after voltage profile improvement. Also, Fig. 12 and 13 describe the convergence characteristics of PSO-based active and reactive power rescheduling costs for the test network.
As shown in Figures 12 and 13, the cost of rescheduling both active and reactive powers of the IEEE 30 bus system decrease as the converge characteristics (iteration number) increases.

C. CASE 3: IEEE 118 BUS SYSTEM NETWORK
Ref [39] describes the system in detail. The system has 118 buses, 179 interconnected lines, and 54 generators. Its single-line diagram is shown in Fig. 14 below. The detailed power flow result of the congested lines is shown in Table 6 below. Fig. 15 to 20 show the details of the generator sensitivity factors (GSF) for each congested line. Table 7 and 8 show the details of PSO optimally rescheduling the output active and reactive power of the participating generators to reduce congestion. According to the tables, only generators 6, 24, 34, 54, 66, 85, and 105 are not involved in congestion. Table 9 also provides a detailed summary of both active and reactive power loss before and after congestion management. The diagrammatic representation of voltage profile improvement before (Pre) and after (Post) congestion management is shown in Fig. 21. Fig. 22 and 23 also depict the convergence characteristics of PSO-based active and reactive power rescheduling costs for the test network.
As shown in Figures 22 and 23, the cost of rescheduling both active and reactive powers of the IEEE 118 bus system decrease as the converge characteristics (iteration number) increases.

V. CONCLUSION
This research presented a novel generator rescheduling approach for transmission system network congestion control. The rescheduled generators were identified based on their sensitivity to the congested line, as shown by their active and reactive power characteristics. Then, to save money, a PSO-based algorithm was employed to restrict the divergence of the rescheduled generation's active and reactive power from the scheduled generator. This approach's applicability was examined utilizing IEEE 14, 30, and 118 standard network buses. The simulation results prove that after rescheduling the cost of both active and reactive powers is less expensive. The active power losses for each of the considered IEEE 14, 30, and 118 cases are 4.7%, 11.03%, and 10.87% respectively, while the reactive power losses are 3.67%, 15.39%, and 12.31% respectively. The results suggest that decreasing the divergence of active and reactive power of rescheduled generators from planned generators can minimize the total cost of congestion management. Furthermore, attaining enhanced voltage stability and voltage profile while reducing the transmission system operation cost. The future researchers work to develop a classical method and compare the exiting heuristics method TCM solutions, secondly apply parallel computing approaches for the solution of transmission congestion control as part of the future study.