A Newly Hybrid Method Based on Cuckoo Search and Sunﬂower Optimization for Optimal Power Flow Problem

: The paper proposes a new hybrid method based on cuckoo search (CSA) and sunﬂower optimization (SFO) approach (called HCSA-SFO) for improving the performance of solutions in the optimization power system operation problem. In the power system, the optimal power ﬂow (OPF) problem is one of the important factors which usually minimizes total cost and total active power losses while satisfying all constraints of the output power of generators, the voltage at buses, power ﬂow on branches, the capacity of capacitor banks and steps of transformer taps. HCSA-SFO utilizes the mutation and selection mechanism in the SFO algorithm to replace the L é vy ﬂights function in CSA. Hence, this makes HCSA-SFO avoid the ﬁxed step size in the CSA from that can reduce run time and improve the quality of solution for the HCSA-SFO algorithm in the OPF problem. The proposed hybrid technique is simulated on the 30-buses and 118-buses systems. The obtained simulation results from the suggested technique are compared to many other approaches. The result comparisons in di ﬀ erent cases showed that the suggested HCSA-SFO can achieve a better result than many other optimization approaches. Therefore, the suggested HCSA-SFO is also an e ﬀ ective approach for dealing with the OPF problem.


Introduction
Electric companies are constantly striving to find ways for improving effectiveness in the operation of power systems to decrease the production cost while still satisfying all security constraints. The optimal power flow (OPF) problem still plays a major role in power system operation, and it has been continuously studied for enhancing effectiveness in solving the above problems. The OPF is a nonlinear optimization issue with several parameters and numerous equations and also inequality constraints. The parameters of the OPF problem are that power generation outputs, switchable capacitor banks, voltages at buses and tap changers of transformers, while the equations and inequality constraints are real and reactive power balance constrains, the maximum and minimum limits of reactive and real power outputs, the voltage at buses, the capacity of capacitor banks and steps of transformer taps. Therefore, the OPF in power systems is one of a more difficult topic which needs an effective method for solving. Several traditional methods and optimization algorithms have been used to find an OPF solution.
A lot of traditional methods in dealing with the problem of OPF were proposed with aims of minimizing fuel cost, including method of interior point [1], a technique of nonlinear programming [2], a linear programming technique [3], quadratic programming [4] and a newton-based approach [5].

•
Dealing with OPF frameworks with several objective functions conditions using a hybrid HCSA-SFO algorithm; • The HCSA-SFO utilizes the mutation and selection mechanism to follow the best orientation to the sun of sunflowers from the SFO algorithm to replace the Lévy flights function in CSA. This technique helps HCSA-SFO to avoid the fixed step size in the CSA, hence the run time is reduced and the quality of solution for the HCSA-SFO algorithm in the OPF problem is improved; • The simulation result is validated on the standard 30-buses and 118-buses systems; • The result is compared to many previous methods, which shows the effectiveness of the suggested HCSA-SFO method in dealing with the OPF problem.
The structure of manuscript are given as follows: Section 2 of manuscript presents the OPF problem formulation, while the original CSA and SFO algorithm is presented in Section 3; Furthermore, Section 3.3 also introduces the HCSA-SFO technique and implementing HCSA-SFO for dealing with the OPF is applied in detail in Sections 3.1 and 3.2. The calculated results and comparisons to other techniques are shown in Section 4. Conclusions are described in Section 5.

Problem Formulation
The OPF problem is one of the optimization problems related to the operation of power systems. It is usually used to minimize the objective functions with many controlled variables while satisfying the security constraints of power systems [25]. The OPF problem can be described as follows: Min ff (x, u) (1) Subject to: -The constraints of equality and inequality.
g(x, u) = 0 (2) h(x, u) ≤ 0 where, ff is the goal function which is optimized; g(x,u) and h(x,u) are the constraints of equality and inequality; x is the state variable vector which includes variables of slack bus's active power P G1 , the voltage of load bus V L , reactive generation power Q G and apparent power at branch S l as shown in Equation (4); u is the control variable vector which includes variables of active generation power P G , generator voltages V G , tap ratio of transformer T and shunt compensation capacitor Q c as shown in Equation (5).
where, N L , N G , N TL , N T and N C are the number of load nodes, generator nodes, transmission lines, tap transformers and the number of VAR compensators, respectively.

OPF Objective Functions
The objectives functions are minimized in the study and include fuel cost, power loss and deviation of voltage.
-Fuel cost: -Total real power losses -Voltage deviation where, a i , b i and c i are cost factors of the generator i; g k is the conductance at k th line; V i , V j is voltages amplitude of bus i and j; θ ij is voltage angle difference between bus i and j.

Constraints
-Constraints of power balance -The limits of power generation: -The limits of generator voltage bus and load voltage bus: -The limits of switchable capacitor capacity: -The limits of transformer tap: -The limits of transmission line: where, N B is the total number of nodes; P Di , Q Di are active and reactive power of load at bus i; G ij , B ij are the real and imaginary parts of the admittance between bus i and j; δ i , δ j are the voltage angles at bus i and j; P Gi,max , P Gi,min, and Q Gi,max, Q Gi,min are the limits of active and reactive capacity outputs of generator i; V Gi,max , V Gi,min and V Li,max , V Li,min are the limits of the voltage magnitude of generator i and load i, respectively; Q ci,max , Q ci,min and T k,max , T k,min are the limits of the capacity of switchable capacitor bank and tap changer of transformer i; S l,max is the maximum capacity of transmission line i.

SFO Method
The SFO approach is inspired by nature and was proposed by G. F. Gomes, et al. in 2019 [38]. The SFO algorithm simulates the movement of the sunflower toward the sun. Sunflowers' activity is repeated every morning based on their behavior. These sunflowers search for the best orientation to the sun and move themselves to best catch the sun's radiation. In the morning, the sunflowers move toward the sun and the opposite orientation at the end of the day. The sunflowers' growing rule is repeated for the next morning. The sunflowers which are close to the sun's direction will collect more heat than those far from the sun's direction; hence they remain still in this region. On the contrary, those which are located in the region far from the sun's direction will take larger steps for moving as close to the sun as possible to the global optimum.
The steps of the SFO algorithm are: 1.

2.
The fitness function f (X t i ) of sunflowers is evaluated.

3.
Retain the best solutions in the sunflower population X*.

4.
Modify all sunflowers headed for the best one (called sun) as Equation (18).

5.
Determine the direction for each sunflower by Equation (19).
In which, λ: Inertial displacement of the sunflower plants. p i : Pollination probability. X i , X i-1 : Current position and nearest neighbor position 6.
Examine the highest step of individual as Equation (20).
where, X min , X max : The lower and upper limits. N pop : the number of populations.
The position of new generated individual (sunflower) is updated using the as Equation (21).

CSA Method
The CSA method was developed based on the behavior of some cuckoo breeds. The cuckoo leaves her eggs in the bird nests selected at random from other host birds. The cuckoo's egg will be brooded with a host birds' eggs by the host birds. The processing of laying and moving of cuckoos is performed according to the Lévy flight function. There are two crucial search capabilities in the CSA algorithm, global and local search, which are evaluated by a discovery rate. The Lévy flight function with infinite mean and variance is used for global search rather than the random walk technique.
There are three principle rules that are used in CSA: -A cuckoo lays its one egg into a bird nest which is selected at random from other host birds. - The best nests will bear to the next generation.
-A host bird may detect a strange egg by a probability pa є[0, 1]. For this situation, the host bird can throw out the cuckoo's egg or leave the nest and find another place for building a new one (with new random solutions).
The CSA maintains a balance between global and the local search random which is controlled by the parameter Pa∈ [0, 1]. Equations (22) and (23) present the local and global random walks, respectively [36,37]: where: X i , X j and X k : Current positions selected randomly α > 0: Scaling coefficient The steps of are in Table 1:

Implementation of the Hybrid CSA and SFO Method
The effective solution of the optimization approaches will be improved with a balance between exploitation and exploration. Exploration is used to ensure finding the global solution, while exploitation is performed to search the best optimal values around current good solutions. So, finding a suitable balance between exploitation and exploration from the combination of the CSA and SFO promises to be an effective technique for dealing with the optimization problem. With that viewpoint, this paper suggests a hybrid CSA and SFO (HCSA-SFO) technique for the OPF problem with several objective functions. The main objective of the suggested technique is to replace Lévy flight function in CSA by using mutation and selection mechanism in the SFO algorithm to avoid the fixed step size in CSA, in order to increase the effective global search and improving the quality of candidate solution.
The steps of the suggested HCSA-SFO technique for dealing with the OPF problem are given as below: Step 1: Set HCSA-SFO parameters Before performing the procedure, it is necessary to set the control parameters of HCSA-SFO, such as the population size Np, mortality rate m, pollination rate p, maximum number of iterations Nmax, probability Pa∈ [0, 1].
Step 2: Generate a population of solutions Each solution in the population is initialized by where Sol i is the ith solution in population; Sol ijmax and Sol ijmin are upper and lower limits of the jth element in candidate solution; d is the problem's dimension and rand 1 is the random numbers in [0, 1].
Step 3: Evaluate the initial solutions in the population: The quality of initialized solutions is evaluated by the fitness function Equation (25) via solving the power flow problem. Find the best solution (Sol best ) with the corresponding best fitness value FF best.
where, F is the objective function of each case (F C , F TL , F V ) that is defined by Equations (6)- (8).
Set the iteration counter n = 1.
Step 4: Generate the first new solutions: Create new solutions by using the mechanism of SFO. The step of each solution towards the best solution is calculated by Equations (18)- (20). The new solution of the population is updated using Equation (21). . Update the best solution (Sol best ) with the corresponding best fitness function value FF dbest Step Step 8: Check the sopping condition: If n < Nmax, n = n + 1, the searching process will return to Step 4 for finding the optimal solution. Otherwise, the searching process will stop.

The IEEE 30-Bus Test System
The system includes six generators, 24 load buses and 41 lines, as in Figure Table 2 and in [25,39]. shunt capacitors is in the range of [0-5 MVAR]. The system, generator data and operating conditions for the IEEE 30-bus test system are given in Table 2 and in [25,39].   Table 3 presents obtained optimal values using CSA and HCSA-SFO for cases 1-3, consisting of fuel cost, power loss and voltage deviations. In addition, these control parameters are also presented in this table. From this table, the total generator cost obtained is 799.118 ($/h) using the HCSA-SFO technique, while the total generator cost using the CSA approach is 799.129 ($/h) for case 1. For case 1, the total generator cost of the CSA approach approximates that of the HCSA-SFO approach; however, the run time of the suggested HCSA-SFO technique is shorter than that of the CSA approach for all of simulation cases. Wherein, the run time of HCSA-SFO is 9.0261, 7.0549 and 7.3082 s, which are less than those of CSA for solving the problem in case 1, case 2 and case 3, respectively. The convergence curve of the total fuel cost objective function is demonstrated in Figure 2. From this figure, convergence ability to the optimal value of the HCSA-SFO algorithm is better than CSA in terms of optimal value.   Table 3 presents obtained optimal values using CSA and HCSA-SFO for cases 1-3, consisting of fuel cost, power loss and voltage deviations. In addition, these control parameters are also presented in this table. From this table, the total generator cost obtained is 799.118 ($/h) using the HCSA-SFO technique, while the total generator cost using the CSA approach is 799.129 ($/h) for case 1. For case 1, the total generator cost of the CSA approach approximates that of the HCSA-SFO approach; however, the run time of the suggested HCSA-SFO technique is shorter than that of the CSA approach for all of simulation cases. Wherein, the run time of HCSA-SFO is 9.0261, 7.0549 and 7.3082 s, which are less than those of CSA for solving the problem in case 1, case 2 and case 3, respectively. The convergence curve of the total fuel cost objective function is demonstrated in Figure 2. From this figure, convergence ability to the optimal value of the HCSA-SFO algorithm is better than CSA in terms of optimal value.  In order to evaluate the effectiveness of the suggested HCSA-SFO technique in dealing with the OPF problem, simulations results of the HCSA-SFO technique is compared with many other approaches as demonstrated in Table 4. For case 1, it can be seen from the Table 4, total fuel cost obtained by HCSA-SFO is 799.11 ($/h), which is better than many other methods in the literature. This is the demonstration of the robustness of the hybrid HCSA-SFO technique in dealing with OPF. Table 4. Compared results of HCSA-SFO and other methods for cases 1, 2 and 3.

Method Case 1 Case 2 Case 3 Total Generator Cost
Voltage Profile Total Active Power Loss Gradient method [17] 804.853 NR 10.486 DE-OPF [35] 802.394 NR 9.466 MDE-OPF [35] 802.375 NR 9.459 MSFLA [34] 802.287 NR 9.6991 IGA [16] 800.805 NR NR ABC [15] 800.66 0.1381 3.1078 GSA [9] 798.675143 NR NR SCA [25] 800.1018 0.1082 2.9425 Hybrid PSO-GSA [19] 800.49859 0.12674 9.0339 Jaya [18] 800  In order to evaluate the effectiveness of the suggested HCSA-SFO technique in dealing with the OPF problem, simulations results of the HCSA-SFO technique is compared with many other approaches as demonstrated in Table 4. For case 1, it can be seen from the Table 4, total fuel cost obtained by HCSA-SFO is 799.11 ($/h), which is better than many other methods in the literature. This is the demonstration of the robustness of the hybrid HCSA-SFO technique in dealing with OPF. The simulation results of the HCSA-SFO technique compared with many other approaches for case 2 is also shown in Table 4. As observed from Table 4, the voltage deviation of the suggested HCSA-SFO technique is better than those of many other methods. Moreover, the voltage deviation obtained by HCSA-SFO is 0.0945 pu, which is also better than that of CSA as shown in Table 4.
For case 3, the total active power loss achieved using HCSA-SFO is 2.8748 (MW), while the total active power loss reduces to 2.8752 (MW) using CSA as shown in Table 4. From Table 4, it can be observed that the total power loss of the suggested HCSA-SFO technique obtains a better minimum value compared with other approaches. Besides, the results of fuel cost, voltage deviation and active power loss using CSA and HCSA-SFO also are given in Figures 3-5. From the figures, the results of HCSA-SFO are better than those of CSA for all three cases. In addition, the statistical results of HCSA-SFO and CSA in Table 5 show that HCSA-SFO outperforms CSA in terms of the best, average and the worst fitness values as well as the standard deviation. The analytical results show that HCSA-SFO is also an effective method to find an optimized solution with fast convergence ability. The simulation results of the HCSA-SFO technique compared with many other approaches for case 2 is also shown in Table 4. As observed from Table 4, the voltage deviation of the suggested HCSA-SFO technique is better than those of many other methods. Moreover, the voltage deviation obtained by HCSA-SFO is 0.0945 pu, which is also better than that of CSA as shown in Table 4.
For case 3, the total active power loss achieved using HCSA-SFO is 2.8748 (MW), while the total active power loss reduces to 2.8752 (MW) using CSA as shown in Table 4. From Table 4, it can be observed that the total power loss of the suggested HCSA-SFO technique obtains a better minimum value compared with other approaches. Besides, the results of fuel cost, voltage deviation and active power loss using CSA and HCSA-SFO also are given in Figures 3-5. From the figures, the results of HCSA-SFO are better than those of CSA for all three cases. In addition, the statistical results of HCSA-SFO and CSA in Table 5 show that HCSA-SFO outperforms CSA in terms of the best, average and the worst fitness values as well as the standard deviation. The analytical results show that HCSA-SFO is also an effective method to find an optimized solution with fast convergence ability.    The simulation results of the HCSA-SFO technique compared with many other approaches for case 2 is also shown in Table 4. As observed from Table 4, the voltage deviation of the suggested HCSA-SFO technique is better than those of many other methods. Moreover, the voltage deviation obtained by HCSA-SFO is 0.0945 pu, which is also better than that of CSA as shown in Table 4.
For case 3, the total active power loss achieved using HCSA-SFO is 2.8748 (MW), while the total active power loss reduces to 2.8752 (MW) using CSA as shown in Table 4. From Table 4, it can be observed that the total power loss of the suggested HCSA-SFO technique obtains a better minimum value compared with other approaches. Besides, the results of fuel cost, voltage deviation and active power loss using CSA and HCSA-SFO also are given in Figures 3-5. From the figures, the results of HCSA-SFO are better than those of CSA for all three cases. In addition, the statistical results of HCSA-SFO and CSA in Table 5 show that HCSA-SFO outperforms CSA in terms of the best, average and the worst fitness values as well as the standard deviation. The analytical results show that HCSA-SFO is also an effective method to find an optimized solution with fast convergence ability.

The IEEE 118-Bus Test System
The larger power system with the standard IEEE 118-bus is used to test the robustness of HCSA-SFO for dealing with the OPF problem. Parameters of the 118-bus system are given in [25,39]. The 118-bus system includes 118 buses, which are 99 load buses, 54 thermal units, 186 branches, 9 transformers and 12 reactive compensations with size within (0-30) MVAr each. The system is considered as a large-scale OPF problem which is usually used to test the robustness of many other algorithms.
The optimal value of objective function and control optimal parameters for the IEEE 118-bus system using CSA and suggested HCSA-SFO is presented in Table 6. From Table 6, the total generator cost achieved is 129619.848 ($/h) using HCSA-SFO, while the total generator cost achieved by CSA is 129847.86 ($/h). Moreover, Table 6 also shows that the run time to the obtained optimal value of the suggested HCSA-SFO method is 234.2190 s, which is 44.2609 s less than of CSA. Besides, the variation of the total generator cost is also presented in Figure 6. From this figure, the convergence ability of the HCSA-SFO technique for the OPF problem with large scale systems can be demonetized. For additional effective confirmation, the results of HCSA-SFO are also compared with many other approaches, as shown in Table 7. From Table 7, HCSA-SFO achieved the solution better than many other methods. Table 8 presents the values of optimal objective functions for cases 2 and 3 obtained by HCSA-SFO compared to CSA. As observed, the suggested HCSA-SFO achieved better optimal results than the CSA algorithm. The voltage deviation decreases to 0.3836 in case 2 and the power loss of 11.2784 MW in case 3 using HCSA-SFO, while the voltage deviation is 0.6117 in case 2 and the power loss 21.3664 MW in case 3 using CSA.

The IEEE 118-Bus Test System
The larger power system with the standard IEEE 118-bus is used to test the robustness of HCSA-SFO for dealing with the OPF problem. Parameters of the 118-bus system are given in [25,39]. The 118-bus system includes 118 buses, which are 99 load buses, 54 thermal units, 186 branches, 9 transformers and 12 reactive compensations with size within (0-30) MVAr each. The system is considered as a large-scale OPF problem which is usually used to test the robustness of many other algorithms.
The optimal value of objective function and control optimal parameters for the IEEE 118-bus system using CSA and suggested HCSA-SFO is presented in Table 6. From Table 6, the total generator cost achieved is 129,619.848 ($/h) using HCSA-SFO, while the total generator cost achieved by CSA is 129,847.86 ($/h). Moreover, Table 6 also shows that the run time to the obtained optimal value of the suggested HCSA-SFO method is 234.2190 s, which is 44.2609 s less than of CSA. Besides, the variation of the total generator cost is also presented in Figure 6. From this figure, the convergence ability of the HCSA-SFO technique for the OPF problem with large scale systems can be demonetized. For additional effective confirmation, the results of HCSA-SFO are also compared with many other approaches, as shown in Table 7. From Table 7, HCSA-SFO achieved the solution better than many other methods. Table 8 presents the values of optimal objective functions for cases 2 and 3 obtained by HCSA-SFO compared to CSA. As observed, the suggested HCSA-SFO achieved better optimal results than the CSA algorithm. The voltage deviation decreases to 0.3836 in case 2 and the power loss of 11.2784 MW in case 3 using HCSA-SFO, while the voltage deviation is 0.6117 in case 2 and the power loss 21.3664 MW in case 3 using CSA.

Conclusions
In the next years, OPF problems will still be one of the important issues of power system operation, especially in the electricity market. Many research teams still continue developing other methods to enhance the performance solution of the OPF problem. This is a nonlinear issue with many control parameters that requires an effective technique in dealing with it. A newly robust hybrid technique, which is successfully applied for dealing with OPF for large-scale systems, is

Conclusions
In the next years, OPF problems will still be one of the important issues of power system operation, especially in the electricity market. Many research teams still continue developing other methods to enhance the performance solution of the OPF problem. This is a nonlinear issue with many control parameters that requires an effective technique in dealing with it. A newly robust hybrid technique, which is successfully applied for dealing with OPF for large-scale systems, is presented in this paper. In order to evaluate the ability to find an optimal solution of the suggested technique, the hybrid HCSA-SFO technique is compared with CSA and many other approaches. The simulation results are tested and evaluated for the IEEE 30-and IEEE 118-bus system with the objective function of minimizing generator costs, power loss and voltage deviation. For both of the testing systems, HCSA-SFO reaches better solutions and faster convergence than CSA for all of cases of the objective functions in each independent run as well as in 50 runs. In comparison with other methods, the proposed HCSA-SFO method outperforms other methods. The simulation results achieved show that HCSA-SFO can be a potential approach for dealing with large-scale OPF problems or the OPF problems considering to distributed generation sources and FACTS devices.