An enhanced turbulent flow of water-based optimization for optimal power flow of power system integrated wind turbine and solar photovoltaic generators

This paper uses enhanced turbulent flow in water-based optimization (TFWO), specifically ETFWO, to achieve optimal power flow (OPF) in electrical networks that use both solar photovoltaic (PV) units and wind turbines (WTs). ETFWO is an enhanced TFWO that alters the TFWO structure through the promotion of communication and collaboration. Individuals in the population now interact with each other more often, which makes it possible to search more accurately in the search area while ignoring local optimal solutions. Probabilistic models and real-time data on wind speed and solar irradiance are used to predict the power output of WT and PV producers. The OPF and solution methods are evaluated using the IEEE 30-bus network. By comparing ETFWO to analogical other optimization techniques applied to the same groups of constraints, control variables, and system data, we can gauge the algorithm’s robustness and efficiency in solving OPF. It is shown in this paper that the proposed ETFWO algorithm can provide suitable solutions to OPF problems in electrical networks with integrated PV units and WTs in terms of energy generation costs, improved voltage profiles, emissions, and losses, compared to the traditional TFWO and other proposed algorithms in recent studies.

system network is called optimal power flow (OPF).The primary goal of this problem is to minimize a specific objective function under the conditions of feasibility and security.OPF has been widely used in previous works due to its characteristics, including a highly constrained and large-scale nonlinear convex optimization test problem.Over the past few decades, many OPF formulations have been created to improve the performance of an electric power system that is subject to physical limitations 1 .Various names and multiple objective functions are used to describe the newly developed optimization problem.There are many different OPF solution methods, each with its math properties and processing needs 2 .
OPF optimization issues have been focused on recently due to the network's quick deployment of distributed energy resources 2 .Traditional optimization techniques, such as Newton's method, quadratic programming (QP) and nonlinear programming (NLP), show excellent convergence rates for solving OPF problems.However, these methods rely on theoretical assumptions unsuitable for real-world systems with non-smooth, non-differentiable, and non-convex cost functions.The above limitations can be overcome by using metaheuristics, which are based on a shared set of principles that allow the construction of solution algorithms.However, most metaheuristics take inspiration from nature, including stochastic components, and usually have many parameters that must be adapted to the situation.point for multi-objective OPF (MOOPF) functions.The fuzzy-based Pareto front technique with the chaotic invasive weed was tested by, Ghasemi et al. 24 on the standard IEEE 30-bus test system (optimizers CIWOs, based on chaos, were researched and assessed using various objective functions).Chen et al. 25 showed, by the hybrid firefly-bat algorithm (HFBA), that ten MOOPF cases minimizing active power loss, total emission, and fuel cost are simulated on the IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems to address the strictly limited MOOPF challenges.Islam et al. 26 used the Harris hawks optimization (HHO) for resolving single-objective and multi-objective OPF problems to manage the emissions from thermal-producing sources.Weighted sums and the no-preference method have been used to resolve the various conflicting framed multi-objective functions.Avvari et al. 27 used an evolutionary algorithm (EA) for OPF with four conflicting objectives, including minimization of total cost (TC), total emission (TE), active power loss (APL), and voltage magnitude deviation (VMD), using a novel constraint handling strategy for OPF incorporating Wind, PV, and PEVs uncertainty.Kyomugisha et al. 28 applied the mayfly algorithm (MA) to show how well the voltage stability indices performed in the MOOPFs of the power systems.A new and effective improved turbulent flow of water-based optimization (ITFWO) to solve OPF in systems with PV and WT units was introduced in 29 .Mouassa 30 used the slime mould algorithm (SMA) to control the power flow between various generating resources in order to reduce the main grid's total operating costs.Khorsandi et al. 31 have applied a modified artificial bee colony (MABC) to solve discrete OPF problems on the IEEE 30-bus and IEEE 118-bus test systems that contain both discrete and continuous variables and take valve point effects (VPE) into account.Elattar et al. 32 , by a modified JAYA (MJAYA) examined two separate test systems, i.e., the IEEE 30-bus system and the IEEE 118-bus system.Saha et al. 33 used the two-point estimate method (2PEM) to deal with the unpredictability of renewable generation in OPF.As stated in Table 1, various algorithms have been presented and explored in the literature in relation to the solution to OPFs.
Contribution and paper organization.OPF minimizes objective function values by finding control variables and equation system values.The technical and economic characteristics of the electrical power system must be considered while choosing this function.Due to the network's development and efficiency needs, system operators always look for fast and efficient ways to operate and design the electric power system.Classical techniques require convex and derivable goal functions and limits to find optimal solutions quickly.Deterministic algorithms get stuck in locally optimal solutions.The OPF problem is a large-scale optimization problem with multiple local and one global optimum in the search space, and also, this is multimodal, non-linear, non-convex, and non-smooth.OPF's nonlinearity traps it in a locally optimal solution.Some methods struggle with continuous and discontinuous data.Classical methods with MOOPF considerations are difficult to design.Heuristicbased optimization algorithms have been developed to overcome this issue.The MOOPF technique may help build power grid operations that best handle optimization concerns.The MOOPF solution seeks the best values for several control variables concerning several objective functions while conforming to equality and inequality constraints.Hence, the goals begin to conflict.Thus, the specified aims become competing in nature.
The concept of abnormal oscillations in water turbulent flow serves as the basis for the turbulent flow of water-based optimization (TFWO) presented in 2021 by Ghasemi et al. 38 .The velocity and direction of this type of turbulent flow constantly fluctuate in a circular pattern.From there, the water spirals downward.In this technique, a whirlpool represents an occurrence in the ocean, river, or sea.Particles are believed to be drawn from the whirlpool's periphery into the centre.The whirlpool employs the centripetal force of a stream of moving water caused by the ocean tide to exert pressure on them.When applied to a moving object, the centripetal force acts in a circle in a direction perpendicular to and in the same plane as the centre of motion.Because of the centripetal force, the object's path is changed, but its velocity is unaffected.Many applications have been proposed for TFWO, such as estimating parameters of photovoltaic models 35,39,40 , optimal sizing of the energy sources in an isolated hybrid microgrid 41 , and economic load dispatch (ELD) problems 34,42 .
This work presents a powerful novel algorithm that applies to various problems of optimal load dispatch in hybrid systems by improving the TFWO algorithm.The rest of this paper is organized as follows.In "Formulation of the problem" section, the formulation of the optimal load dispatch problem is discussed."The proposed algorithm: enhanced TFWO (ETFWO)" section describes the optimization steps of the proposed algorithm.In "Discussions" section, the proposed method has been implemented on the IEEE standard 30-bus network with various load distribution functions, and the results have been analysed.Finally, in "Conclusions" section, the general conclusion of the proposed method is stated.

Formulation of the problem
Objective functions.The goal of OPF studies is to determine the constant operational point that minimizes efficiency and minimizes waste.This optimization issue is nonconvex, nonlinear, and massive in scale.OPF can ascertain the current and voltage distributions throughout an electrical grid.Many different OPF formulations and approaches to solving them have been developed in this setting.In addition, modern electricity markets and incorporating of renewable energy sources have contributed to an increase in OPF research.
Many objective functions, which are listed below, are minimized by our proposed algorithm.

Energy of wind (EoW) cost function.
Wind energy is gradually included in the power system because of its continuously falling cost and emission-free properties.The EoW's direct cost of electricity can be expressed as in 11 by: (1) C d,w,i = d w,i P w,i .The operators of wind power facilities are fined if they cannot deliver the volume of energy they had anticipated generating from the wind.P w,i is the scheduled power from the same plant, and d w,i is the direct cost coef- ficient associated with the i th wind power plant.There are two components to penalty fees: 1. the underestimation cost that must be included when available wind farm power is not utilized, and 2. the over-estimated cost that is computed for purchasing power from alternative sources (reserves) or loadshedding.It is possible to model these expenditures in the following way 11 : where i is from the set {1, 2, . . ., n w } , the n w is the number of the wind power plants.The PDF, i.e., the prob- ability density function of EoW output power can be represented by f (P i ) for i th wind power plant.And, K oe,w,i is the reserve cost coefficient pertaining to the i th wind power plant.Also, K ue,w,i is the penalty cost coefficient for the i th wind power plant, P w,r,i is rated output power from the same windfarm.
The following cost function describes the overall expenditures for EoW: where C ue,w,i is the reserve cost for the i th wind plant, C oe,w,i is the penalty cost of wind power surplus.
We employ the Weibull distribution to describe the randomness of wind speed.Hence the probability density function (PDF) f (V w ) and the cumulative distribution function (CDF) F(V w ) can be defined as in 6 by for the K th solar EoW plant: Calculating the energy that EoW produces entails: where V w , V w,r , V w,in , and V w,out represent the speed, the rated speed of EoW generators, the cut-in speed, and the cut-out speed, respectively.Moreover, the contour and scaling variables of the Weibull distribution are shown by K and C , respectively.
PV cost function.Because of their low price and easy implementation, photovoltaic (PV) systems are becoming increasingly popular as a sustainable energy option.Predicting PV's power output is challenging because its features rely highly on a wide range of variables.Following is a breakdown of how to figure out how much PV power will cost to generate C ue,pv,i and how much that will cost in penalties C oe,pv,i 11 : where d pv,i is the direct cost coefficient associated with the i th solar PV plant, P pv,i is the scheduled power from the same plant.i = 1, ..., n v and f (P) represent the PDF of the PV unit's output power.nv shows the number of the PV plants.
Using this method, we may calculate the full price of photovoltaics.
(2) C ue,w,i = K ue,w,i P w,r,i P w,i To calculate the PDF of the i th PVs' output power, one can use the following: photovoltaic (PV) cells, also known as solar cells, are susceptible to sunlight.When modeling the probability density function (PDF) of solar radiation f (R) , a beta distribution is a good fit 11 : where Ŵ , is the gamma function and α, β are parameters of the beta distribution.Moreover, the solar radiation is denoted by R.
The following equation can be used to determine the relationship between the power output of PV and the power output of the solar cell generator, which is connected to solar radiation 11 : where R C is solar radiation in W/m 2 and R STD is the rate of solar radiation transfer to the earth.A standard solar radiation point is typically established at 150 W/m 2 and normal conditions call for 100 W/m 2 .
Basic fuel cost function.Primarily, OPF is used to reduce the cost of essential gasoline.The fuel cost of a power plant is typically described as a quadratic function 42 : where i and n G represent the i th power plant and the number of power plants, respectively.Furthermore, the coef- ficients a i , b i , and c i represents the cost coefficients for the i th power plant and P Gi is power of the i th power plant.
Piecewise quadratic fuel cost function.Typically, power plants will utilize the lowest fuel available for a given operating range.
Piecewise quadratic is the form of the fuel cost function for such a setup.
The following function can be used to figure out the cost of each quadratic component of fossil fuels 42 : where n f is the number of fossil fuel options for the i th power plant.Moreover, the coefficients a i,k , b i,k , c i,k are costs of the i th power plant for the k th fuel option.
Piecewise quadratic fuel cost with valve point loading.Generator cost is a cumulative heat rate graph that is convex but is interrupted by the entrance level valves in large turbines.For each generator's true cost to be determined, the valve point impact must be included in 42 : where coefficients e i and f i represent valve point cost of the i th power plant.

Emission cost function.
Power plants that rely on fossil fuels, such as petroleum, coal, and natural gas, burn these materials to generate electricity.When something is burned, a lot of pollution is released.The sum of all forms of emission taken into account, such as SOX and NOX, with appropriate pricing or weighting on each pollutant emitted, can be used to represent the minimization of emission function for OPF problems.In this study, we use the following function to represent NOx and SOx emissions, two significant air pollutants 42 : The emission coefficients of the i th power plant are shown by α i (ton/h), β i (ton/h MW), γ i (ton/h MW2), ξ i (ton/h) and θ i (1/MW).
Power loss cost function.The generation of electricity and the global demand for it are inversely correlated.Therefore, limiting energy loss is crucial for OPF issues.Power systems inevitably have transmission losses because of the resistance of the transmission cables.Transmission systems will inevitably experience power (12) Gi + e i sin f i P min Gi − P Gi , www.nature.com/scientificreports/losses.The majority of active power loss happens when the power system is in use.The cost of energy production directly relates to the power loss.The next power loss function must be satisfied in order to decrease transmission lines' active power loss 43 : All parameters mentioned in Eq. ( 19) are defined in 44 .
Voltage deviation (VD) cost function.One of the most significant security and service indicators is bus voltage.
In an OPF problem, considering simply a cost-based objective will lead to a workable solution with an unacceptable voltage profile.Consequently, a dual goal function is required to reduce fuel costs and enhance voltage profiles by reducing load bus voltage deviations from 1.0 per unit to provide a desired voltage profile 42 : where n PQ and VL i are the numbers of load bus bars and the i th voltage of load buses, respectively.
Constraints.The following are some requirements that the OPF optimization issue must meet 42 : 1. Active and reactive power balances where P Gi , Q Gi , P Di , and Q Di are the active and reactive power generations and load demands at the i th bus, respectively.31) is the general objective function of optimal load dispatch, in which all kinds of problem constraints are applied In this equation, part n G i=1 f i (P Gi ) is the general objective function that P Gi can be different types of power generation sources, which in this article can include thermal, wind, or solar generators.The rest of Eq. ( 31) is the part added to consider the constraints of the load dispatch problem, i.e.: In this paper, the penalty factors symbolized by P , V , Q , and S , which have been chosen to have large positive values, have been set to 10,000,000.Furthermore, 'limit values' P lim G1 , VL lim i , Q lim Gi , and S lim i in Eq. ( 31) are variables that are defined by the following equation: The proposed algorithm: enhanced TFWO (ETFWO) Turbulent flow of water-based optimization (TFWO).Following the above mentioned introductory paragraphs, we present and simulate the various stages of the original TFWO algorithm's implementation and performance.
Formation of whirlpools.The algorithm begins by random partitioning its initial population X 0 , that consist from N pop members, into N Wh whirlpool sets S Wh j , j = 1, ..., N Wh , of the same cardinality (thus, some disjunct subsets of the set X 0 of the same cardinality ⌈N pop /N Wh ⌉ ).Next, a member, denoted Wh j , with the best fitness value f (Wh j ) , is selected in each whirlpool set S Wh j and it is called as the whirlpool of the whirlpool set S Wh j or the center of the set S Wh j (or hole of S Wh j ).This whirlpool Wh j pulls the other objects in the j th whirlpool set S Wh j , j = 1, ..., N Wh .
The whirlpools effects on objects.Every whirlpool Wh j , j = 1, ..., N Wh , serves as a drinking hole and tends to apply a centripetal force on the other surrounding objects in the set S Wh j to bring them into alignment with Wh j .Because of this, the j th whirlpool Wh j , j = 1, ..., N Wh , with its local position in S Wh j , behaves in a way that its location is merged with that of the i th object X i from S Wh j , i.e., X i = Wh j .Furthermore, we will also assume that individual whirlpools will also influence each other.Thus, whirlpools based on at mutual distances Wh − Wh j between them and cost merits f (•) create various differences X i .Therefore, the new location of the i th particle is X new i = Wh j − X i .Figure 1 depicts the impacts of a whirlpool on the objects in its whirlpool set.In Fig. 1 we can see that the object X i is moving at an angle δ i around and toward its whirlpool Wh j .Therefore, this angle fluctuates at each iteration of the algorithm, which can be indicated as By determining the strongest and weakest whirlpools (thus, whirlpools with the lesser and greater weighted distance to all objects) by Eq. (33).Hence, we can compute using Eqs.( 34) and ( 35) the new position of the i th object X new i .
where Wh with minimum and maximum values of t are denoted by Wh f and Wh w , respectively, and δ new i is the new angle of the i th object.Pseudocodes 1 and 2 show the proposed mathematical model for updating the position of the i th object: Where the function sum(A) returns the sum of all elements of the vector A.
Centrifugal force.Sometimes the centripetal or traction force of the whirlpool is defeated by the centrifugal force FE i , and the object is arbitrarily transferred to a new site.To account for the unpredictable nature of the centrifugal force experienced by each object, there is a random variable along a single dimension of the objectives' space (or the solution).To do this, we first determine the centrifugal force according to the angle it makes with the center of the hole by Eq. (36) and if this force is greater than a random value r from the interval [0, 1] , we conduct the centrifugal action for the chosen the p th dimension of the i th object using Eq. ( 37).Pseudo-code 3 is described in Fig. 2.

Pseudo-code 3. end for;
Wh f = Wh with the minimum value of t ; ; Wh w = Wh with the maximum value of t ; Figure 1.An optimization model by the whirlpool Wh j (Ghasemi 38 ).
Interactions between the whirlpools.Whirlpools interact and move one another, much like how they affect the objects nearby.We somehow approximated this phenomenon to be similar to the effects of whirlpools on objects and particles, where each whirlpool has a tendency to draw other whirlpools, exert centripetal force on them, and cause them to fall into their holes.Using the objective function of the nearest whirlpool and the minimum quantity from Eq. ( 38), we can model and compute Wh j .Then, the variation of the position of the j th whirl- pool to decrease in its objective function is stated by Eqs. ( 39) and (40), where δ j is the j th whirlpool's angle.
Pseudo-code 5. ( end for; Wh f = Wh with the minimum value of t ;  Enhanced TFWO (ETFWO).TFWO has poor solution accuracy, slow convergence speed, and immature behavior when solving complex optimization problems.In this research, a new ETFWO algorithm is developed to improve TFWO's weak points, thereby facilitating information exchange between individuals.Because each participant in the population interacts more frequently, the search can advance more precisely in the search area while ignoring local optimal solutions.This method significantly improves the TFWO algorithm's performance by allowing it to explore the search space more effectively and increasing its ability to exploit.The proposed ETFWO algorithm's new search is described in Eq. (41).In this equation, the local search, X r − X i , is added in an effective way to the search equation of members in the main TFWO algorithm.This new average has a big effect on how well the proposed algorithm searches locally and, as a result, solves problems in general. where where X r is a random member selected haphazardly from the entire population.The flowchart of the proposed ETFWO is provided in Fig. 3.
ETFWO for different OPF problems in the IEEE 30 bus test system.In eight cases of OPF, the TFWO and ETFWO have been implemented on the IEEE 30 bus test system.For both the TFWO and ETFWO was used N pop = 30 and the maximum number of iterations set as 400.In 42 , we can find information about the power systems' parameters.

OPF solutions IEEE 30-bus network.
As shown in Fig. 4, see 42 , four transformers with non-nominal tap ratios are situated on lines 6-9, 6-10, 4-12, and 28-27, whereas six generators are placed on buses 1, 2, 5, 8, 11, and 13.The overall network demand at a base capacity of 100 MVA is 2.834 p.u.The highest and lowest voltages of each load bus were adjusted to 1.05 and 0.95 p.u., respectively.All of the final ETFWO results for the six OPF's scenarios without WT and PV producers of the 30-bus power system are presented in Table 2.
Case 1: Minimizing fuel costs.This objective function takes into account reducing the overall fuel cost of producing electricity, which is represented by the quadratic cost curve shown below.
The fuel cost using ETFWO is 800.4792($/h), according to simulation results shown in Table 2.In comparison to solutions from cutting-edge existing optimization approaches in Table 2, such as MSA 42 , MICA-TLA 45 , MHBMO, HFAJAYA 46 , IEP 47 , EP 43 , JAYA, GWO, DE 48 , AO 49 , MGBICA 44 , ARCBBO 50 , PSOGSA 51 , MRFO 52 , SKH 53 , TS 54 , ABC 55 , PPSOGSA 56 , SFLA-SA 57 , FA 58 , FPA 59 , MFO 60 , MPSO-SFLA 16 , and TFWO, the proposed ETFWO has meaningfully reduced the total cost.Table 3 shows that the minimum fuel cost, as determined by the ETFWO algorithm, is 800.4792dollars per hour.Compared to the best result in the literature, the results in Table 3 show that the best fuel cost determined by the ETFWO algorithm is fairly cheap.Figure 5 displays the convergence rate for this scenario.As can be observed, the acquired data supports the capacity of the proposed ETFWO algorithm to identify precise OPF solutions in this case study.
Case 2: Minimization of piecewise quadratic fuel cost.By burning fossil fuels like coal, natural gas or petroleum, thermal generators are able to make electricity.Equation ( 44) provides a model for the fuel cost curve.www.nature.com/scientificreports/(44) The fuel cost in this case employing the proposed approach is 646.4860($/h), according to simulation results shown in Table 2.In comparison to existing optimization methods reported in recent literature shown in Table 4, such as GABC 61 , LTLBO, IEP, MFO, MICA-TLA, SSA 62 , MDE 63 , MSA, SSO, FPA, and TFWO, it is obvious that ETFWO has considerably decreased the total cost.The proposed ETFWO algorithm outperforms stochastic strategies in terms of satisfactory solutions for the OPF issues, as shown by a comparison of the results from LTLBO and the other methods listed.Table 4 shows that the minimal fuel cost is 646.4860$/h, which is lower than the results that have been reported in the literature.The convergence trend by TFWO and ETFWO algorithms for Case 2 is shown in Fig. 6.
Case 3: Fuel cost with VPEs.The cost function of Eq. ( 45) includes the valve point loading effects (VPEs).
The fuel cost using ETFWO is 832.1625($/h), according to optimal results given in Table 2.For this case, ETFWO has effectively diminished the fuel cost when compared to SP-DE 64 , HFAJAYA, PSO, FA, and TFWO optimization methods in Table 5.As can be observed, the acquired data supports the capacity of the proposed (45) ETFWO algorithm to identify precise OPF solutions in this case study.Table 5 demonstrates that the ETFWO algorithm is the best method to minimize the objective function for Case 3's OPF problem. Figure 7 depicts the convergence graph of the total cost ($/h) by the TFWO and ETFWO algorithms for Case 3.
We applied the proposed technique to Cases 4-6 in order to improve our ability to solve multi-objective OPF problems.The top ETFWO-based simulation solutions for Cases 4-6 are also summarized in Table 2.
What determines the value of these objective functions is the value of their control parameters.These control parameters are optimized and selected here by the optimization algorithm by observing all the limitations of the load dispatch problem and various constraints of the power system.As you can see in Eqs. ( 18), (19), and ( 20), the value of these functions depends on different parameters.And it is possible in the optimization that the best control parameters for one function, e.g., VD , give a weak value to the function n G i=1 f Ei (P Gi ) .In fact, these two objective functions are not in the same direction and have different properties.Therefore, researchers usually use multi-objective optimization, and all the objective functions are considered, making each of them give common control parameters in the algorithm's output based on their importance, which is not the best optimal state for any objective functions.
Case 4: fuel cost and real power loss.Using the multi-objective function provided in Eq. ( 46), where Eqs. ( 14) and ( 19) have been integrated to minimize both.
where, the factor φ p is set to a value of 40 (see 42 ).( 46) The results represent that the fuel cost and power loss using ETFWO are 859.0080($/h) and 4.5295 (MW), and the recommended ETFWO provided the smallest value of the objective function, 1040.1880, which is 1.4615 less than the original TFWO's minimum value of 1041.6495.ETFWO has drastically decreased fuel cost and power loss and additionally achieved the optimum objective function compared with MJaya 36 , QOMJaya 36 , MSA, EMSA 65 , and MOALA approaches in Table 6. Figure 8 depicts the overall cost ($/h) convergence graph through the TFWO and ETFWO methodologies for Case 4.     where the factor φ v is set to a value of 100 (see 43 ).
Using the proposed method, we obtained results that reveal that the fuel cost and voltage discrepancies are 803.7868dollars per hour and 0.0945 per unit, respectively.The multi-objective fitness function that ETFWO discovered to have a minimum value of 813.2368 is the smallest and best one available compared to SpDEA 66 , SSO, PSO 67 , PSO-SSO 67 , MPSO 68 , BB-MOPSO 69 , MOMICA 70 , MNSGA-II 71 , EMSA, DA-APSO 72 , TFWO, and TFWO approaches, which is shown in Table 7. Figure 9 depicts the convergence graph of the overall cost ($/h) through the TFWO and ETFWO methodologies for Case 5.
Case 6: Minimizing the fuel cost, emissions, voltage deviation and losses.This problem, provided by Eq. ( 48), integrates instances 1, 5, and 6 to simultaneously reduce fuel expenditure, voltage variation, emission, and power loss.
(47)   The weight variables are used to manage the problem's several objectives as in 42 with φ v = 21, φ p = 22, and φ e = 19.
The simulation findings depicted in Table 8 demonstrate that ETFWO has greatly optimized all the objects compared to MNSGA-II, MOALA, J-PPS1 73 , J-PPS2 73 , J-PPS3 73 , SSO, MSA, BB-MOPSO, PSO, MODA 74 , I-NSGA-III 75 , MFO, and TFWO approaches.This table demonstrates that, in this example, the proposed ETFWO optimizer outperformed the other optimization techniques.Table 8 demonstrates that the suggested ETFWO's objective function J 6 , which is 964.2683, is the lowest of all the approaches.Figure 10 illustrates the total cost convergence graph for Case 6 for both TFWO and ETFWO.
(48) www.nature.com/scientificreports/ In Cases 1-6, the presented approach did a better job of exploring than newly reported methods that seem to be trapped at a local solution.Overall, the performance of TFWO and ETFWO is quite competitive and consistently outperforms most algorithms.

The solution of OPF involving WT and PV producers in the IEEE 30 bus test system. Case 7:
Cost-cutting through the use of stochastic wind and solar power.In this scenario, the goal is to employ the ET-FWO to get the fuel, wind, and PV costs from Eq. ( 49) to be as low as possible for a system that uses renewable energy sources like solar and wind.
The cost coefficients for this type of OPF have been chosen the same as in Case 1 with the PDF parameters in 11 .The best optimal solutions obtained from 30 independent runs by the TFWO and ETFWO algorithms in this article are given in Table 9.It is worth mentioning that P ws1 shows the wind generator planned power W G1 , and so on.
As it is clear from this table, the proposed algorithm of ETFWO was able to achieve optimal solutions with more quality than the TFWO algorithm.The best total cost was achieved by the proposed ETFWO, which came in at 781.9250 ($/h), which is 0.6251 less than the original TFWO's minimum total cost of 782.5501 ($/h).Figure 11 depicts the total cost convergence graph for Case 7.
Case 8: OPF with carbon tax.This case study aims to lower total generation costs by placing a carbon price on traditional thermal power companies' emissions.It is well known that conventional energy sources create hazardous gases when used to generate electricity.In recent years, a great deal of pressure has been placed on the energy industry, in general, to cut carbon emissions by numerous nations.The reason for this is global warming. (49) Table 9.The optimal values of the variables were achieved for Case 7. Carbon tax ( C tax ) is imposed on each generator of greenhouse gas emissions to encourage investment in solar and wind.The value of emissions 11 is calculated by: where C tax is expected to cost $20 per tonne per hour (t/h).

Variables
Table 10 displays the optimal generation schedule, reactive generator power, total generation cost, and additional calculated factors.In Case 8, where a carbon tax has been imposed, wind and solar energy penetration are greater than in Case 7, in which no carbon tax is regarded.The amount of growth in the optimal schedule for producing renewable energy is contingent on the number of emissions and the speed with which a carbon price is implemented.As it is clear from this table, the best objective function, 810.7341, was produced by the proposed ETFWO, which is 0.3177 less than the 811.0518 minimum value of the objective function obtained by the original TFWO. Figure 12 also displays the TFWO and ETFWO convergence rate for Case 8 of this study.
When addressing the OPF problem while considering the integration of stochastic wind and solar power, the proposed ETFWO demonstrates excellent resilience and superior efficiency compared to the original TFWO and other recent techniques based on the analysis of the optimal results in Cases 7 and 8.
ETFWO for different OPF problems in the IEEE 118 test system.The effectiveness of the proposed algorithm was assessed utilizing an extensive power system.Its outcomes were compared to those of the other approved algorithms to ensure the suggested strategy's viability.
Case 9: Classical OPF.Comparing the IEEE 30-bus system, the number of control variables rises from 24 to 130. Adding more generators makes the OPF issue more challenging.The system needs 4242 MW of active power and 1439 MVAR of reactive power, respectively.The aim function in instances 1 and 9 is to minimize the fuel cost in order to be comparable to other cutting-edge heuristic algorithms presented in the literature.The top results obtained with the suggested ETFWO method are listed in Table 11.Table 11 compares this finding to the outcomes of other algorithms being studied and a few other strategies that have been published in the literature, including MSA 52 , FPA 52 , MFO 52 , PSOGSA 52 , IABC 76 , MCSA 77 , MRao-2 and Rao algorithms 78 , SSO 37 , FHSA 64 , ICBO 3 , GWO 23 , EWOA 79 , and CS-GWO 80 .
Table 11 shows that the ETFWO performs better than several optimization methods for solving large-scale OPF.Working with such a massive system, the CS-GWO delivers remarkable results.

Discussions
All cases that were looked into by both TFWO and ETFWO were put through a comparison study with other well-known algorithms, including WSO (war strategy optimization) 81 , HHO (Harris Hawks optimizer) 82 , and FDA (flow direction algorithm) 83 in this section.Reliability evaluation is a non-empirical way to confirm something and is essential for studying complex things.For Cases 1 and 8, a sensitivity analysis was performed to see how stable the metaheuristics under consideration were.Table 12 shows the results of this analysis according to the best value of the solution (Min), the average value of the solutions (Mean), the worst solution obtained (Max), the standard value of the standard deviation (Std.), and the median value of the time of 30 data runs.This table demonstrates that the proposed ETFWO algorithm is more stable and reliable than the TFWO, WSO, HHO, and FDA algorithms.The ETFWO algorithm surpassed the examined algorithms in terms of Min, Mean, Max, and Std.values for all the objective functions of OPF, proving its superior efficiency.Furthermore, the ( 50)   proposed ETFWO algorithm came out on top in every OPF task, demonstrating its better performance than the alternatives.The TFWO and FDA algorithms showed strong efficacy, placing second and third for the majority of cases, respectively.

Conclusions
ETFWO is a new enhanced turbulent flow of water-based optimization (TFWO) method that has been proposed, developed, and used effectively to rectify eight various testing cases of OPF problems in the IEEE 30-bus system with a mix of photovoltaic units and energy of wind.The findings demonstrate that the current metaheuristic works well for large-scale applications because it quickly converges and doesn't get stuck at local minima very often.Solutions analysis and a comparative study with recently published OPF methods showed that ETFWO is a valid, effective, and robust method for calculating a set of steady optimal solutions for a hybrid electrical network under real-world conditions.This is a very important part of running modern power systems, which use a growing number of different types of energy.The proposed metaheuristic did better than current wellknown optimizers, which proves that it is better and has the potential to yield reliable and precise remedies for multi-objective optimization.In fact, ETFWO could be used as a tool to respond to many particular questions about large, complex systems in general, which would lead to more research.The outcomes demonstrated the applicability and potential of the proposed algorithms for resolving various MOOPF problems.Simulation results validated the ability of the proposed algorithms to produce accurate and well-distributed results for all multi-objective optimization problems considered.Based on the findings of this study, the proposed ETFWO optimization tools are efficient, trustworthy, and quick at solving various MOOPF problems.Environmentally speaking, the proposed optimization technique results in the lowest emission levels.In particular, the proposed ETFWO algorithm is an excellent candidate for solving MOOPF problems in real-world power systems due to its high-quality solutions and excellent convergence properties.In addition, the comparative analysis with the proposed algorithms and published OPF solution methods validates the superiority of the proposed paradigm and its ability to locate valid and accurate solutions, particularly for multi-objective optimization problems.
The ETFWO optimization technique can be used to find a solution to the OPF problem for the future, considering the power system's unpredictability and its components (such as load demand and alternative renewable energy sources).To improve the quality of solutions in the optimization domain, researchers have proposed creating a new hybrid version of the ETFWO algorithm by combining it with other optimization methods, such as PSO and DE approaches.However, the anticipated research activity will extend into the future to examine the ranking of contingencies and the placement of scattered generations.In conclusion, AI algorithms present a formidable new tool for solving complex engineering optimization and MOOPF challenges.

Figure 3 .
Figure 3.The flowchart of the suggested ETFWO algorithm.

Table 1 .
Summary of the proposed methods for solution of the OPF problems in recent literature.
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Table 2 .
The optimal variables for OPF without stochastic renewable energy obtained by ETFWO.

Table 3 .
The results for Case 1: Minimizing fuel costs.

Table 4 .
The results for Case 2.

Table 5 .
The results for Case 3.

Table 6 .
The results for Case 4.

Table 7 .
The results for Case 5.

Table 8 .
The results for Case 6.

Table 10 .
The optimal values of the variables were achieved for Case 8.

Table 11 .
Optimal results for Case 9.

Table 12 .
Optimal results to represent the performance of algorithms.