In [26], equality constraints for congestion management are given as:

P G k − P D k = | V j | | V k | | Y K j | cos ( δ k − δ j − θ k j ) ;     j = 1 , 2 , ⋯ , N b (2)

Q G k − Q D k = | V j | | V k | | Y K j | sin ( δ k − δ j − θ k j ) ;     j = 1 , 2 , ⋯ , N b (3)

P G k = P G k c + Δ P G k + − Δ P G k − ;       k = 1 , 2 , ⋯ , N g (4)

P D j = P D j c ;       j = 1 , 2 , ⋯ , N d (5)

where;

P G k = real power produced at bus k.

Q G k = reactive power produced at bus k.

P D k = real power available at bus k.

Q D k = reactive power available at bus k.

V j , V k = voltages at busj and k.

δ j , δ k = voltage angle of busesj and k.

θ k j = admittance angle of the line between bus j and k.

N b , N g , N d = number of buses, generators and loads.

P G k c = real power provided by generator k.

P D j c = real power employed by load bus j.

The Equation (2) represents active power while the Equation (3) represents the reactive power of respective nodes similarly the Equations (4) and (5) are related to power market prices.

The Equations (6)-(9) give the inequality constraints which describe the maximum permissible limits under which the power system components like transformers, transmission lines, and generators must be operated for efficient operation. If these limits are violated, it is consequences are very serious causing serious damage to the power system element.

P G k min ≤ P G k ≤ P G k max ,       ∀ k ∈ N g (6)

Q G k min ≤ Q G k ≤ Q G k max ,       ∀ k ∈ N g (7)

( P G k − P G k min ) = Δ P G k min ≤ Δ P G k ≤ Δ P G k max = ( P G k max − P G k ) (8)

V n min ≤ V n ≤ V n max ,       ∀ n ∈ N l (9)

P i j ≤ P i j max (10)

where max and min represent maximum and minimum values and N₁ denote the number of transmission lines.

Developed in [19], TLBO algorithm is influenced by the method of knowledge sharing in the classroom. The learning process in the classroom is an exchange of information between a teacher and a learner. TLBO brief overview is given in subsequent sections. The flowchart of the TLBO is given in the Figure 1. Novel

TLBO algorithm is motivated by the process of knowledge sharing in the classroom. The process of teaching-learning is very important to bring fruitful changes in the students. The grades or marks of the learners i.e., the students are considered as the product of the TLBO algorithm. In this approach, the teacher is presumed to be a highly knowledgeable person who imparts his knowledge to his students. Students are considered as learners whose aim is to try to acquire knowledge in order to ameliorate their grades. Thus, if grades got ameliorated, the process of learning gets enhanced. Besides learning from the teacher, the students also improve their learning by interacting with other fellow students in the classroom to enhance their grades.

TLBO is a metaheuristic family member algorithm based on population. In this approach, a batch of students in the class is treated as a population while design variables are taken as the subjects learned by the students in the classroom. The objective of our work i.e., the fitness function is comparable to the grades obtained by students. TLBO algorithm operates in two phases which are elaborated in the below sections.

PSO algorithm, developed in 1995 was introduced by Dr. Kennedy and Dr. Eberhart [22] [27]. Simulation of the social behavior of organisms was carried out and it was found to be very effective for obtaining the solutions of optimization problems of continuous non-linear nature. PSO is a searching algorithm based on population, in which individual particles update their position with time. Each particle in a PSO can serve as a likely solution to the complication in an N-dimensional search space. Amid searching, each particle modifies its state corresponding to its own knowledge and the knowledge of the neighboring particle. The sum of particles in the search algorithm is randomly generated (usually between 10 to 100 is considered ample). Each particle is associated with the parameter of position X i = ( x i 1 , x i 2 , x i 3 , ⋯ , x i N ) and also the parameter of velocity V i = ( v i 1 , v i 2 , v i 3 , ⋯ , v i N ) . Corresponding to the objective or fitness function, the position of particles is appraised at the end of every iteration. During the course of searching, the particles are presumed to hold on to the thought of best positions. It is then desired to share these thoughts containing the best positions with the rest of the particles. The cumulative of all the best particle thoughts are referred to as the global best position (GB), where G B = ( g b 1 , g b 2 , g b 3 , ⋯ , g b N ) while the best position attained by the individualistic particle is referred as local best. The local best for an ith particle in search space is given as P i = ( p i 1 , p i 2 , p i 3 , ⋯ , p i N ) . Information obtained from these two parameters is used by particles to update their values of position and velocity as given in [28]

V i j + 1 = W V i j + c 1 ∗ r 1 ∗ ( P i j − X i j ) + c 2 ∗ r 2 ∗ ( G B j − X i j ) (14)

X i j + 1 = X i j + V i j + 1 (15)

where the coefficient w represents inertia weight, c₁ and c₂ are cognitive and social parameters and value of both c₁ and c₂ are equal to 2, r₁ and r₂ indicate random numbers lying in the range of 0 and 1, and the coefficient j denotes the iteration number.

Acceleration constant c₁ helps the particles to attain the local best position quickly while the acceleration constant c₂ aids the particles to attain the global best position quickly. The appropriate choice of the inertia weight factor “w” helps in quick convergence. Sensibly, the maximum velocity Vmax of each particle should be selected otherwise the particle may not be able to find the best solution [29]. Similarly, the choice of inertia weight parameter is also of great importance since its selection directly controls the influence of past values of velocities on the prevailing values of particle velocities and hence improves the local and global expedition capacities of the particles. Larger the inertia weight, better is the exploration tendency of the particle. Generally, inertia weights are considered in the range from 0.4 to 0.9. PSO is a meta heuristic technique as either few or much less hypotheses are made by it regarding the optimized problems and tries to search for further options. Figure 2 shows the flow chart of the PSO technique algorithm and its application to physical systems.

To validate our work, we have firstly chosen modified IEEE 30 bus test system. This bus system combines generator buses six in number, load buses twenty-four in number, and transmission lines forty-one in number. The total active power of the load is 283.4 MW, and the total reactive power is 126.2 MVAR. PG and PD are taken as the values for generation and load are also given in Appendix A as the initial market clearing values. In a power system, contingencies are mainly due to the line outages and hence for the purpose of simulation, we have taken line outages along with load variations. Two cases are studied for this purpose:

Case 1.1: Considering the unavailability of the line between bus 1&2 with normal loading.

When the line between bus 1&2 is being outed, the lines between bus 1&7, and 7&8 are overloaded. For obtaining the information about the amount of overloading in the congested lines, Newton-Raphson algorithm of power flow [24] is performed and the results are mentioned in Table 1. The real power flowing in these transmission lines are 150.46 MW and 138.78 MW, respectively, while 130 MW is the net power flow limit in both these lines. Thus, the total power violation encountered is 29.24 MW. For the efficient and reliable working of the power system, transmission network should not transmit power beyond their permissible limits. Hence to alleviate the congestion in the lines, mandatory steps should be taken. The main motive of the research is to ease the transmission line from overloading using the technique of generation rescheduling.

The results obtained by implementing the TLBO-PSO hybrid optimization technique for managing the CM problem for case 1.1 are mentioned in Table 2. The acquired results are also compared with TLBO and PSO techniques as given in [1] and [18] in the same table.

It is worthy to note that the solution obtained from the suggested method as given in Table 2 successfully alleviates the overload of 29.24 MW without affecting other lines in the system. By the observation of Table 2, it is clearly understood that the proposed hybrid algorithm technique gives the best solution as 100.69 $/h which is less as compared to the other techniques. Prior to CM, the total system loses was 8.8094 MW which effectively decreased to 7.5168 MW. The objective or fitness function as obtained by the hybrid technique for the present case is shown in Figure 3. Real power rescheduled from the participating generators is shown in Figure 4.

Case 1.2: Considering the unavailability of the transmission line between bus 1&7 and load increment of 50% at all buses.

When the line between bus 1&7 is being outed and 50% increment in load at each bus is done, the effect is that the lines between bus 1&2, bus 2&8 and bus 2&9 are overloaded. The actual power flowing in these lines is 202.57 MW, 66.11 MW, and 69.46 MW, respectively, while 130 MW is the net power flow limit in line between bus 1&2 and for lines between buses 2&8 and 2&9, it is 65 MW for both. Thus, the total power violation encountered is 92.745 MW (Table 1). Hence to alleviate the congestion in the overloaded lines, optimal real power generation rescheduling is done by using the suggested hybrid algorithm as

mentioned in Table 3. The rescheduling of the real power of the participating generators as cited by the suggested hybrid algorithm is shown in Figure 5.

Thus, it is noticeable that the cost of managing congestion is lower for the proposed hybrid technique when compared with the other methods mentioned. The total losses in the system have also decreased to 9.8123 MW from the initial value of 15.2915 MW during congestion. The proposed hybrid algorithm-based convergence of fitness function with the iteration number is shown in Figure 6.

Further to validate our work, we have also chosen a modified IEEE 57 bus test

system. This bus system combines generator buses seven in number, load buses fifty in number, and transmission lines eighty in number. The total active power of the load is 1250.8 MW, and the total reactive power is 336 MVAR. Two cases are also studied for this purpose. For obtaining the information about the measure of overloading in the congested lines, Newton-Raphson power flow [19] is performed for the mentioned test case and the analysis of overloaded lines are given in Table 4.

Case 2.1: Simulating overload of lines between buses 5&6 and 6&12 by reducing their capacity.

In the present case, the real power flowing in the line between bus 5&6 is 184.62 MW and in line between bus 6&12 is 46.985 MW. The baseload power limit of the line between bus 5&6 is 200 MW while that of the line between bus 6&12 is 50 MW. To perform overload simulation, the limit of the line between bus 5&6 is taken as 175 MW while that of the line between bus 6&12 as 35 MW.

The effect of reducing the capacity of lines is that the line between bus 5&6 gets overloaded by 5.49% while the line between bus 6&12 gets overloaded by 34.24% and net power violation encountered is 21.605 MW. To manage this overload of 21.605 MW, optimal rescheduling of generation is accomplished using the proposed hybrid algorithm. The results obtained are tabulated in Table 5. The amount of rescheduled power is presented in Figure 7.

From Table 5, the CM cost achieved from the suggested hybrid algorithm is 2787.70 $/h, which is lowest as compared to mentioned optimization algorithms. The total losses in the system have also decreased to 34.1438 MW from the initial value of 37.1555 MW during congestion. The proposed hybrid algorithm-based convergence of fitness function with the count of iterations is presented in Figure 8.

Case 2.2: Simulating overload of the line between bus 2&3 by reducing its capacity.

To simulate the overload in this case, the capacity of the line between bus 2&3

is reduced from an outset value of 85 MW to an ending value of 20 MW. The baseload power flow in the line is 38.6 MW and for that reason, the line is overloaded by 93%. The net power violation in the line is 18.6 MW. To manage this congestion of 18.6 MW, optimal rescheduling of generation is done according to the suggested method. The results obtained along with other comparative methods are presented in Table 6.

From Table 6, the CM cost obtained from the proposed hybrid algorithm is 939.3927 $/h, which is lowest as compared to mentioned optimization algorithms. The total losses in the system have also decreased to 36.4281 MW from the initial value of 36.9342 MW during congestion. The amount of rescheduled power is presented in Figure 9. The proposed hybrid algorithm-based convergence of fitness function with the count of iterations is presented in Figure 10.