Optimal Power Flow Incorporating Thyristor-Controlled Series Capacitors Using the Gorilla Troops Algorithm

Department of Electrical Engineering, Faculty of Engineering, Suez University, Suez 43533, Egypt Department of Electrical Engineering, College of Engineering, Taif University, Taif 21944, Saudi Arabia Department of Electrical Engineering, Faculty of Engineering, Damietta University, Damietta 34517, Egypt Department of Electrical Engineering, Faculty of Engineering, Kafrelsheikh University, Kafrelsheikh 33516, Egypt Department of Electrical Engineering, Cairo University, Giza 12613, Egypt


Introduction
e optimal power flow issue (OPFI) is considered a primary tool for managing electrical power networks, where it provides electrical power at the lowest possible cost while maintaining high quality [1,2]. Its major goal is to optimize objective functions such as system losses, cost of fuel, and emissions with meeting a set of equality and inequality constraints such as generator bus voltage magnitude, valvepoint constraints, generator real power, transformer taps, and reactive power of shunts while optimizing a given objective function [3].
A numerous optimization approaches were developed to solve the OPFI over the last few decades. Classic optimization methods and evolutionary algorithms are the two categories of optimization approaches [4,5]. A plethora of mathematical approaches was addressed to tackle the OPFI such as programming based on semidefinite [6], linear [7,8], fuzzy linear [9], nonlinear [10], and quadratic [11,12] frameworks, sequential unconstrained minimization technique [13], Newton-based method [14,15], and interior point approach [16][17][18]. Although these approaches have produced encouraging outcomes, they do have certain drawbacks.
Numerous researchers have turned for utilizing several nature-inspired algorithms to solve the shortcomings of traditional optimization methods [19,20]. ese algorithms do not require derivative information, and instead, they employ random probabilities for optimization rather than deterministic probabilities. It can be noticed that these nature-inspired algorithms are capable of solving large-scale nonlinear problems and jumping out of the local optimum. As a result, a variety of evolutionary algorithms have been used to solve OPFI in recent years [21,22].
Various algorithm strategies have been demonstrated in the literature to identify the optimal OPFI solution as illustrated in Table 1.

Research Gap and Contributions.
Although these algorithms have been widely employed in the literature to deal with OPFI, there is a lack of comparative analysis of these algorithms in terms of solution quality exists. is research investigates the computing efficiency of GTA on the standard IEEE 57-bus EPN and WD-EPN with and without the inclusion of the TCSC devices. e active power losses throughout the system, fuel cost, and emissions are all factors in the objective function. ere are two goals in this paper. e first is to use the gorilla troops algorithm (GTA) [51,52], a recently developed metaheuristic optimization algorithm, to solve the OPFI in EPNs with and without TCSC devices. e second goal is to solve the OPFI with four different objectives which are fossil fuel cost, transmission losses, voltage stability, and emission. Simulations were run on normal and modified IEEE 57-bus EPN and WD-EPN, as well as comparative examinations of other approaches in the literature are conducted in this study. e simulation results demonstrated the GTA's effectiveness in solving the OPFI using TCSC devices. e key contributions of this work are summarized as follows: (i) e GTA is adopted to handle the OPF including TCSC devices.
(ii) e GTA is utilized to reduce numerous functions such as minimizing the fuel rates, pollutant emissions, voltage deviation, and power losses related to EPNs. (iii) e proposed GTA is applied on the standard IEEE 57-bus EPN and a WD-EPN with and without the inclusion of the TCSC devices. In addition, the proposed GTA is applied on a large-scale IEEE 118 bus system with higher outperformance than the particle swarm optimization (PSO). (iv) e OPFI, including TCSC devices, is solved with diverse objective functions which are minimizing thermal generation cost, voltage stability, transmission power loss, and emission. (v) A comparative study is conducted between the proposed GTA and recently developed algorithms such as ISSA, QRJFS, SSA, IHBA, SNSA, EFO, and MICA.
e remainder of the work is divided into the following sections: the OPFI construction is elaborated in Section 2, while Section 3 characterizes the intended GTA for OPFI. In addition, Section 4 contains the simulated findings and comments, while Section 5 has the concluding notes.

OPFI Formulation considering TCSC Devices Incorporation
In OPFI, the reactive power injections of switching capacitors and reactors and the generators' real power output are denoted by (Qcr 1 , Qcr 2 , . . ., Qcr Nqr ) and (Pgr 1 , Pgr 2 , . . ., Pgr Ngr ), respectively. e voltages of the generators and the tap changer settings are designated by (Vgr 1 , Vgr 2 , . . ., Vgr Ngr ) and (Tp 1 , Tp 2 , . . .. . ., Tp Ntr ), respectively. Here, N qr , N tr , and N gr represent the number of reactive power sources, on-load tap changers, and generators, respectively. e dependent variables are load bus voltage magnitudes, generator reactive power outputs, and transmission flow limits, as shown by (VL 1 , . . ., VL NPQ ), (Qgr 1 , Qgr 2 , . . ., Qgr Ngr ), and (SF 1 , . . ., SF NF ), where N F and N PQ represent the number of transmission lines and load buses, respectively. is problem can be expressed numerically as follows: Subject to where OJF characterizes the modeled objective function of numerous u aims and the symbols (l and x) denote the control and state variables, whereas C (l, x) and D (l, x) illustrate the equality and inequality constraints of the OPFI, respectively.

Modelling of TCSC Devices.
e TCSC is one of the most prominent series of FACTS devices, with several advantages such as high performance, quick reaction, and inexpensive cost. It offers several benefits over series capacitors. TCSC 2 International Transactions on Electrical Energy Systems devices have two reactive operational modes which are inductive and capacitive. In both modes, the reactance of the corresponding transmission line can be, accordingly, lowered or raised. Figure 1 depicts TCSC modeling in power systems that are connected in series with a line. erefore, the reactance of the TCSC is represented as a function of the transmissionline reactance (X Line ). e needed value of the TCSC device (X TCSC ) to avoid transmission line overcompensation may be computed using the following equation [53,54]:

Aim 1: Fuel
Cost. e goal of (Aim 1) is to keep costs down while meeting electricity demands. A quadratic relationship exists between fuel cost and generated power (OJF 1 ) which can be expressed in dollars per hour and formulated as follows: where A p , B p, and C p elaborate the cost coefficients of generator p.

Aim 2:
Emissions. e goal of (Aim 2) is to reduce emissions produced by the power plants.
e emissions (OJF 2 ) can be expressed in ton/h and formulated as follows: where c p , β p , α p , ξ p , and λ p designate the emission coefficients of generator p. e goal of (Aim 3) is to reduce the overall transmission system power loss. e emissions (OJF 3 ) can be formulated as follows [55]: where G pq illustrates the transfer conductance between buses p and q, V is the voltage; N bq is the number of buses, and θ refers to the phase angle.

Aim 4: Voltage
Stability. e goal of (Aim 4) is to improve voltage stability by minimizing the maximum voltage stability index (L-index) that is illustrated in Reference [56]. To illustrate, the L-index for each bus j (L j ) can be mathematically formulated as follows: e voltage stability of the system can be maximized using the maximum L-index, and this index can be mathematically formulated as follows:

System
Constraints. e equality constraints are as follows: Vgr p G jp cos θ jp + B jp sin θ jp 0, j 1, . . . , N pq , where QL and PL illustrate the consumed power consumption of its reactive and active components, respectively. Furthermore, G jk and B jk characterize the mutual conductance and susceptibility of a transmission line connected between buses j and k. e inequality constraints are as follows: Pgr min p ≤ Pgr p ≤ Pgr max p , p 1, 2, . . . , Ngr, Tp min Tr ≤ Tp Tr ≤ Tp max Tr , Tr 1, 2, . . . , Nt, Qgr min p ≤ Qgr p ≤ Qgr max p , p 1, 2, . . . , Ngr, Vgr min p ≤ Vgr p ≤ Vgr max p , p 1, 2, . . . , Ngr, VL min where S illustrates the power ow via line and VL j signi es the load voltage at bus j.

Gorilla Troops Optimization Technique
e gorilla troops algorithm (GTA) relies on gorillas' group dynamics throughout ve strategic options. Such strategic options involve movement with other gorilla, migrating to an unknown place, migrating toward an established site, competing for female adults, and accompanying the silverback. ese strategic options can be categorized into two stages which are the exploration and exploitation stages as explained in the next paragraphs.

Exploration Stage.
In the exploratory stage, three distinct tendencies are characterized: the rst tendency is to advance GTA exploration which is called (movement to an unknown destination), while the second tendency is to augment the consistency between exploratory and exploitation which is called (movement of other gorillas). Moreover, the third tendency is to promote GTA capabilities to determine myriads of computation spaces which are called (gorilla's movement in the path of a familiar destination).
ese three tendencies can be mathematically represented as depicted in equation (17). In this equation, the movement to an unknown destination tendency is chosen once a factor (Pr) is greater than a random value. Furthermore, the tendency of (movement in the path of an identi able place) is chosen once a random value is more than or equal to 50%, whilst the tendency of (a movement in the path of a recognized site) is selected once a random value is less than 50%.
where X (Itr) and GoX (Itr+1) describe the entire and prospective vectors of gorilla location in the next iterations, respectively, whereas rv, rv 1 , rv 2 , and rv 3 signify random values between [0, 1]. e factor (Pr) represents the likelihood of selecting a migrating method to an undetermined location and can be given in the range [0 : 1]. e variables X r and GoXr depict a gorilla among the current group and a candidate position that can be arbitrarily assigned, accordingly. e variables' minimum and maximum bounds are denoted by LB and UB, correspondingly. Equations (2)

Exploitation Stage.
According to the factor D × (1 − Itr/MxItr) and by comparing it to the variable (Y), two tendencies could be determined. To manifest, if the value of D × (1 − Itr/MxItr) is greater than or equal to the value of Y, the approach of the silverback that can direct the others to sources of food could be chosen. is can be illustrated in equation (22) that can be used to exemplify this tendency as follows: X (Itr) is the vector of gorilla location, and X silverback indicates the best solution which is the silverback. Furthermore, GoX i (Itr) represents the location of each potential gorilla vector in iteration Itr, while Nog represents the population of gorillas.
If the term D × (1 − Itr/MxItr) is less than Y, the approach of competing for female adults could be chosen [57].
is can be illustrated in equation (22) which can be used to exemplify this tendency as follows: L represents the force of impact, which could be expressed as in equation (23), whereas rv 5 represents random value within bound [0 : 1]. e factor (A) indicates a vector that provides the level of violence in a fight and may be calculated by equation (26). From that equation, β refers to a preoptimization value, and E is utilized to mimic the violence efficacy.
If the cost of GoX (Itr), at the end of the exploitation stage, is less than X(Itr), the GoX(Itr) solution will replace X (Itr). Figure 2 illustrates the key procedures of the GTA [52].

Proposed Solution-Based GTA for OPFI in EPNs.
e Newton-Raphson Approach (NRA) could be utilized to meet the equality principles that describe power flow balancing models. It meets the balance restrictions and illustrates the operation of electric grids in a steady state. As a result, MATPOWER uses the NRA as a crucial framework for demonstrating three-phase systems [58]. Also, the dependent/independent variable constraints must be maintained.
e following are the operational limits of the independent variables:

International Transactions on Electrical Energy Systems
As can be seen, the variables dramatically continue to reach their limits, and if one of them exceeds the ratings, it is regenerated at random within the necessary bounds. Furthermore, the target cost objective widens and penalizes the restrictions of the second category. Consequently, in the next round, if the gorilla's location surpasses any of the proper limitations, it will be discarded.

Simulation Results
First of all, the performance analysis of GTA for standard well-known mathematical functions is performed and compared with well-known established optimization algorithms such as the particle swarm optimization (PSO) [59] and grey wolf optimization (GWO) [60] as illustrated in Table 2 in the Appendix. It is manifested from this table that the proposed GTA has a higher performance and e ciency than PSO and GWO in the tested mathematical functions which verify the robustness of GTA in obtaining the optimal solution of these mathematical functions. In the next subsections, the proposed GTA is applied to three test EPNs with di erent scenarios.
Firstly, the standard IEEE 57-bus EPN is considered as described in Figure 3, whereas a practical Egyptian WD-EPN is manifested in Figure 4. In addition to that, the proposed GTA is applied to the large-scale IEEE 118-bus EPN.
irty simulation runs are conducted using the proposed GTA with gorillas' group of 50 members and peak iterations of 500. e rst EPN involves 17 on-load tap changing transformers; 57 buses; three capacitive sources on buses 18, 25, and 53; and seven generators on buses 1, 2, 3, 6, 8, 9, and 12, 80 lines. e data of this test system are extracted from [63]. For this EPN, two TCSC devices are considered to be installed on lines 8 and 15 [54]. Secondly, the WD-EPN involves 52 buses with voltages in the ranges between 1.06 and 0.94 p.u. For this EPN, two TCSC devices are considered to be installed on lines 4 and 9 [53]. Lastly, the standard IEEE 118-bus EPN is considered which has 118 buses, 9 on-load tap transformers, 186 routes, 14 capacitor devices, and 54 generators [64]. erefore, the proposed GTA is applied for 130 decision variables where each gorilla position comprises a vector of 130 elements.  (18) Update the gorilla position using (22) Update the gorilla position using (24)          International Transactions on Electrical Energy Systems e simulation runs were conducted using MAT-LABR2017b with 8 GB of RAM and CPU of Core (TM) i7-7200U -(2.5 GHz) Intel(R).

Outcomes of the GTA Applications for the First EPN.
For this EPN, four scenarios are examined with and without considering the TCSC devices: (i) Scenario 1: OJ 1 minimization described in equation (4) (ii) Scenario 2: OJ 2 minimization described in equation (5) (iii) Scenario 3: OJ 3 minimization described in equation (6) (iv) Scenario 4: OJ 4 minimization described in equation (7) 4.1.1. Minimization of the Fuel Costs (Scenario 1). Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices, and the results are shown in Table 3. Furthermore, Figure 5 depicts the convergent properties of the suggested GTA for this scenario considering both cases. As illustrated, the suggested GTA reduces, in the first case without considering the TCSC devices, the fuel costs from 51345 $/h in the initial scenario to 41668.02 $/h with a reduction percentage of 18.847%. In the second case by considering the TCSC devices, the suggested GTA reduces the fuel costs from 51345 $/h in the initial scenario to 41682.64 $/h with a reduction percentage of 18.818%. For these reasons, the suggested GTA is operating the TCSC devices on lines 8 and 15 at compensation levels of − 32.768% and 49.9686%, respectively.

Minimization of the Emissions (Scenario 2).
Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices in order to minimize the produced emissions, and the results are shown in Table 4. Furthermore, Figure 7 illustrates the convergent properties of the suggested GTA for this scenario considering both cases. As illustrated, the suggested GTA reduces, in the first case without considering the TCSC devices, the produced emissions from 2.528 ton/h in the initial scenario to 1.036393 ton/h with a reduction percentage of 59%. In the second case by considering the TCSC devices, the suggested GTA reduces the produced emissions from 2.528 ton/h in the initial scenario to 1.037115 ton/h with a reduction percentage of 58.97%. For this case, the suggested GTA is operating the TCSC devices at lines 8 and 15 at compensation levels of − 24.486% and 19.6084%, respectively.

Minimization of the Power Losses (Scenario 3).
Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices in order to minimize the power losses, and the results are shown in Table 5. Furthermore, Figure 9 depicts the convergent properties of the suggested GTA for this scenario considering both cases. As illustrated, the suggested GTA reduces, in the first case without considering the TCSC devices, the power losses from 27.835 MW in the initial scenario to 9.92662 MW with a reduction percentage of 64.337%. In the second case by considering the TCSC devices, the suggested GTA reduces the power losses from 27.835 MW in the initial scenario to 9.692474 MW with a reduction percentage of 65.178%. For this case, the suggested GTA is operating the TCSC devices at lines 8 and 15 at compensation levels of 49.9995% and 34.7962%, respectively.

Minimization of the L max (Scenario 4). (1) L max Minimizing (Scenario 4).
Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices in order to improve the voltage stability by minimizing the index (L max ), and the results are shown in Table 6.
International Transactions on Electrical Energy Systems Furthermore, Figure 11 illustrates the convergent properties of the suggested GTA for this scenario considering both cases. As illustrated, the suggested GTA reduces, in the first case without considering the TCSC devices, the stability index from 0.3000 in the initial scenario to 0.259785 with a reduction percentage of 13.405%. In the second case by considering TCSC devices, the suggested GTA reduces the stability index from 0.3000 in the initial scenario to 0.265479 with a reduction percentage of 11.507%. For this case, the suggested GTA is operating the TCSC devices at lines 8 and 15 at compensation levels of 24.1205% and -50%, respectively.

Outcomes of the GTA Applications for the Second EPN.
For this EPN, three scenarios are examined with and without considering the TCSC devices: Scenario 5: OJ 1 minimization described in equation (4) Scenario 6: OJ 3 minimization described in equation (6) Scenario 7: OJ 4 minimization described in equation (7)

Minimization of the Fuel Costs (Scenario 5).
Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices, and the results are shown in Table 7. Furthermore, Figure 12 depicts the convergent properties of the suggested GTA for this scenario considering both cases. As illustrated, the suggested GTA reduces, in the first case without considering the TCSC devices, the fuel costs from 25098.70 $/h in the initial scenario to 22953.42 $/h with a reduction percentage of 8.547%. In the second case by considering the TCSC devices, the suggested GTA reduces the fuel costs from 25098.70 $/h in the initial scenario to 22948.95 $/h with a reduction percentage of 8.565%. For these reasons, the suggested GTA is operating the TCSC devices at lines 8 and 15 at compensation levels of − 45.836% and − 35.71%, respectively. In addition, Figure 13 shows the comparisons of the outcomes of Scenario 5 with numerous alternative techniques without considering the TCSC devices. e described results are reported to be related to the grey wolf algorithm (GWA) [76], SSA [77], novel bat algorithm (NBA) [78], improved spotted-hyena algorithm (ISHA) [79], crow search algorithm (CSA) [80], and modi ed CSA [80]. As shown, the proposed GTA provides the superior performance in minimizing the fuel costs with the least value of 22953.42 $/h. On the other side, the minimum costs related to SSA [77], GWA [76], NBA [78], ISHA [79], CSA [80], and modi ed CSA [80]

Minimization of the Power Losses (Scenario 6).
Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices in order to minimize the power losses, and the results are shown in Table 8. Furthermore, Figure 14

Minimization of the L max (Scenario 7)
. Using such a situation, the suggested GTA is used both with and without the inclusion of the TCSC devices in order to improve the voltage stability by minimizing the index (L max ), and the results are shown in Table 9. As illustrated, the suggested GTA reduces, in the rst case without considering the TCSC devices, the stability index from 0.173 in the initial scenario to 0.149417 with a reduction percentage of 13.641%. In the second case by considering the TCSC devices, the suggested   International Transactions on Electrical Energy Systems GTA reduces the stability index from 0.173 in the initial scenario to 0.149417. For this case, the suggested GTA is operating the TCSC devices at lines 8 and 15 at compensation levels of 6.0923% and -28.321%, respectively.

Outcomes of the GTA Applications for the ird EPN.
For the IEEE 118-bus large-scale network, the proposed GTA is applied in comparison to the PSO technique, which is one of the well-established algorithms to deal with such OPF problems, for minimizing the fuel costs. Based on those circumstances, the obtained results using the proposed GTA and PSO techniques are tabulated in Table 10, while the regarding convergence properties are described in Figure 15. As illustrated, the proposed GTA achieves lower fuel cost value of 129752.2 $/hr than the implemented PSO that achieves fuel costs of 129990.8$/hr. Also, Table 11 describes a comparison with reported results of well-established techniques of the differential algorithm [74] and PSO [81]. As shown, the proposed GTA provides better performance than the differential algorithm [74] and PSO [81], which obtain fuel costs of 130518.5 and 130288.21 $/h, respectively.   International Transactions on Electrical Energy Systems 13   37  55  73  91  109  127  147  163  181  199  217  235  253  271  289  307  325  343  361  379  397  415  433  469  451  487   49  73  97  121  145  169  193  217  241  265  289  313  337  361  385  409  457  443  481 Without TCSC With TCSC Figure 9: Convergent characteristics of the proposed GTA for Scenario 3 with and without considering the TCSC devices.   Figure 11: Convergent characteristics of the proposed GTA for Scenario 4.       37  55  73  91  109  127  145  163  181  199  217  235  253  271  289  307  325  343  361  379  397  415  433  469  451  487   49  73  97  121  145  169  193  217  241  265  289  313  337  361  385  409  457  443  481 Without TCSC With TCSC Figure 14: Convergent characteristics of the proposed GTA for scenario 6 with and without considering the TCSC devices.

Conclusion
is research investigates a new nature-inspired algorithm called the gorilla troops algorithm (GTA) to solve the OPFI with and without the inclusion of the yristor-Controlled Series Capacitor (TCSC) devices in the system. Numerous objective functions are minimized throughout regulating the generator bus voltage magnitude, generator real power, reactive power of shunts, and transformer taps. Seven scenarios with and without the inclusion of the TCSC devices in the system are evaluated, and each scenario involves the goal function of transmission losses, fuel costs, voltage stability, and emissions. e e ciency of the GTA has been veri ed using the standard IEEE 57-bus EPN and a practical Egyptian West Delta-EPN (WD-EPN) with and without the inclusion of the TCSC devices to appraise the GTA algorithm's performance in the OPFI. In addition to that, the proposed GTA is applied on the large-scale IEEE 118-bus EPN. e simulation ndings demonstrate that integrating TCSC devices in an OPFI framework can signi cantly improve voltage stability and reduce the operation cost, active power losses, and emissions. In addition to this, the ndings obtained by the GTA have outperformed the recently developed powerful and well-known algorithms in the literature. With the incredible e cacy of the provided GTA in this paper, it is suggested that the proposed GTA can be tested on the OPFI for compensated power systems by considering static var converters (SVCs) or a combination of TCSC and SVC. Moreover, it is recommended that the proposed GTA can be investigated in the future for resolving the OPFI in power grids with high renewable energy penetration. e future extension can be added for the GTA improvement and the possible network limitations under competitive systems.
Data Availability e complete data of this IEEE 30-bus EPS are extracted from [1,40,61], while the data for Real WD-EPS are extracted from [1,9,62]

Conflicts of Interest
e authors declare that they have no con icts of interest.