Optimal power flow solution using HFSS Algorithm

The optimal power flow is a pragmatic problem in a power system with complex behavior that includes many control parameters. Many metaheuristic algorithms with different search methodologies have been proposed for solving the OPF problem. However, existing algorithm faces challenges such as stagnation, premature convergence, and local optima trapping during the optimization process, which provide low-quality and misleading results for real-world problems. In this paper, a novel algorithm which is inspired by the natural foraging phenomenon of the flying squirrel named as Squirrel Search Algorithm is used and it is hybridized with arithmetic crossover operation to enhance its effectiveness and be used for solving the OPF problem. So, the proposed algorithm is named the Hybrid Flying Squirrel Search Algorithm (HFSSA). The capability and performance of the proposed algorithm are observed on benchmark test functions and on the IEEE-30 bus system. Generation fuel cost, emission, and transmission losses are considered objectives of an optimal power flow problem. We got optimal values by handling the control parameters; Generation fuel cost as 799.86 $/h, Emission as 0.20374 ton/h, and transmission loss as 3.1687 MW. The obtained results corroborate that the proposed algorithm outperforms the existing algorithms for solving the OPF problem.


INTRODUCTION
In recent years, development is taking place at a good pace which has led to an increase in the demand for electricity at a rigorous rate.Establishing new plants to fulfill the demand is not a good alternative, thus there is a need to utilize the existing system to its level best.To identify the optimal control variables in the power system the proposed optimal control technique is used and it helps to automatically adjust to minimize instantaneous generation fuel cost, emission, transmission losses, or any other objectives, and at those control variables power system can be planned and operated (Hermann and Williams, 1968).
In this paper, a nature-inspired algorithm is considered which is the squirrel search algorithm (SSA) (Mohit et al., 2019) inspired by the foraging process of flying squirrels (Thomas and Weigl, 1998), is hybridized with arithmetic crossover operation (Yalcinoz and Halis, 2005;Qun et al., 2014) and proposed a new algorithm named as hybrid flying squirrel search algorithm (HFSSA).This proposed algorithm is tested on benchmark test function and used for solving the single objective problems of OPF that is, generation fuel cost, emission, and transmission loss.

Objective functions
The considered objective functions such as cost minimization, emission minimization and power loss minimizations are described below: where,  )) min( 3(3)

Constraints
To solve power system problem needs to consider following constraints: a. Equality constraints:

Proposed Hybrid Flying Squirrel Search Algorithm (HFSSA)
This paper developed the proposed algorithm with the hybridization of existing squirrel search algorithm and arithmetic crossover operation to enhance the accuracy in dealing with power systems.

Existing Squirrel Search Algorithm (SSA)
SSA basically works on the flying squirrels locomotive mechanism for optimize the problem.According (20), there are two available food sources for flying squirrels, one is hickory nut tree which are in limited numbers.The behavioral routine of flying squirrel is to adapt the changes according to seasons as they become very active and searches for food during summer season while during winter, they become dormant but they do not sleep.Thus, during summer when they are active, they will scrupulously search for food.Flying squirrel's gliding locomotion with hickory nut is shown in Figure 1.In this algorithm n numbers of flying squirrels and n number of total trees are considered with the assumption that one flying squirrel will be on one tree.

Random initialization
Initially all the flying squirrels are randomly allocated with any of the tree.Mathematically it can be represented as: and max.values and U(0,1) is a random number(0,1).

Generation of new location in order to search food
It can be mathematically modelled according to the three situations which are as follows: Case A: Flying squirrels which are on acron trees and target to moves towards nut tree.Calculate new location as below equations: = Random position will be allocated if  Case B: Flying squirrels and target to moves towards acron trees.So, the new location can be determined as follows: = Random position will be allocated if Case C: Flying squirrels and target to moves towards hickory nut trees.So, the new location can be determined as follows: = Random position will be allocated if

Seasonal monitoring of flying squirrel
As discussed, the behaviour change of flying squirrel with season changes is modelled as:

At the end of winter season flying squirrel are randomly re-allocated-
As the levy distribution makes exploration more effective so it is calculated by following equation: where  1 and  2 are random number between (01,1) and where, α is a constant 1.5(1).
The re-allocation is mathematically modelled as:

Arithmetic crossover operation
As in existing method Levy distribution control the exploration and to equalize the exploitation and exploration, this arithmetic crossover operation is implemented so that final result can be achieved efficiently in less iteration.As arithmetic crossover operation (38, 39) updates the generated locations and thus convergence occurs faster.Thus, the hybridization helps to get the optimum values can be expressed as (18): where, ξ is a random number between 0 and 1.After

Pseudo code of proposed Hybrid Squirrel Search Algorithm (HSSA)
Step 1: Input-Read bus data, line data, generator data and cost data for the given electric system.
Step 3: Initially generate the random location for the flying squirrels using Equation (4).
Step 4 Map the algorithm variables with the load flow data and then evaluate them for obtaining the solution of the single objective problems.
Step 5: Calculate each flying squirrels fitness and sort ascending order of their fitness values.
Step 6: The 3 fit values will be of flying squirrel on acron trees and then remaining will be on normal trees.
Step 7: Randomly select one of the flying squirrel which are on normal trees and target them to move toward hickory nut tree and remaining to the acron trees.While (the condition not satisfied) Step 8: for i=1 to no.of flying squirrels if R1 ≥ 0.1

Random position will be allocated end
Step 9: for i=1 to fs2 if R2 ≥ 0.1

Random position will be allocated end
Step 10: for i=1 to fs3 (number of flying squirrels which are on normal trees and target to moves towards hickory nut tree)

Random position will be allocated end
Step 11: Find seasonal constant using Equation (8).
Step 12: Verify the seasonal monitoring condition, if it found valid then find levy flight operator using Eq. ( 10) and randomly relocate the flying squirrel using Equation (10).
Step 13: Update the location of flying squirrels with arithmetic crossover operation using Equation ( 12).
Step 14: Determine the minimum value of seasonal constant using Equation ( 9).Go to step 8. end Step 15: Output-The flying squirrel which is on hickory  nut tree will be the optimal value of considered objective function.

Validation of proposed algorithm on benchmark test functions
Two standard test functions were consider to justify the effectiveness of the proposed algorithm, which is mentioned below: 1. Himmelblau's function It can be seen from Tables 1 and 2, that the proposed algorithm optimizes the function more efficiently as compared to the other methods available.Booth's function achieves its minimum value that is, 0.0, which is the desired output with best minima and for Himmelblau's function least value i.e. 2.8075e-10 is obtained which is less as compare to the existing methods.The convergence curves are shown in Figures 2 and 3 for both function from which, it can be seen that proposed algorithm converges at lesser number of iterations and initially it starts will less value of function as compared to the existing algorithm.Thus, this validates the robustness and efficiency of the proposed HFSS algorithm.

Electric IEEE-30 bus system
The IEEE 30 bus system is considered to test the proposed HFSA algorithm.In this system total 6 generators and 4 tap changing transformers and 2 shunt capacitors are available.To optimize considered objective functions proposed method runs 100 iterations.

Minimization of generation fuel cost
The considered control variables are optimized for IEEE-30 bus system is shown in Table 3.It could be seen from Table .3 and Figure 4, that the generation cost obtained by proposed algorithm is 799.86 $/h which is lower than the generation cost obtained by existing algorithms.It could be analyzed from Figure 4 that proposed algorithm converges at 22nd iteration by which it is proven that its convergence is very fast as compared to other algorithms and least value starts early.

Minimization of emission
Minimization of emission is second objective function.The considered control variables are optimized for IEEE-30 bus system is shown in Table 4.It could be observed from Table 4 and Figure 5, that emissions got minimized to 0.20374 ton/h which is less in comparison to the emissions obtained by existing algorithms.It can also be observe from Figure 5 that convergence curve starts will less value of emission and converge at 18th iteration which is the best result in comparison to the existing observations.

Minimization of transmission losses
Minimization of transmission loss is third objective function.The considered control variables are optimized for IEEE-30 bus system is shown in Table 5.It could be observed from Table 5 and Figure 6, that the transmission loss minimized to 3.167MW which is the best result obtained with the implementation of proposed algorithm.This also converges fast as compared to other algorithms.
Observation can be made from Tables 3-5 that when generation fuel cost is to minimize then power generated from slack bus is maximum and power generated for second PV bus is minimum while the minimization of transmission loss and emission power generated from slack bus is minimum and power generated at second PV

Figure 1 .
Figure 1.Gliding locomotion of flying squirrel with hickory nut in mouth.Source:Thomas and Weigl (1998) random gliding distance of flying squirrel.

S
is the previous position of flying squirrel at acron tree and t h S is the current position of flying squirrel at hickory nut tree at th t iteration.

Table 2 .
Optimal result of Booth's function.

Table 3 .
OPF result for minimizing generation fuel cost $/h.

Table 4 .
OPF result for minimizing emission ton/h.

Table 5 .
OPF result for minimizing transmission loss