###### Abstract

Table 1: _Cont._The TFWO algorithm imitates the physical behavior of the turbulent flow of water, in which water follows a circular path with a changing magnitude and speed. In TFWO, a whirlpool represents water's behavior seen in the ocean, sea, and river. A hole in the center of the whirlpool attracts the particles and elements around it by applying a centripetal force. Such a force pulls the moving object toward the center of the whirlpool while the object's speed remains unchanged. This algorithm has been adopted in many applications, several of which can be seem in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Reference** & **Year** & **Contribution** & **Area of the Application** \\ \hline
[34] & 2021 &
\begin{tabular}{c} \(\Theta\)-turbulent flow of \\ water-based optimization \\ (\(\theta\)-TFWO) \\ \end{tabular} & Reactive power control of a \\ \hline
[35] & 2022 & TFWO & Slope reliability evaluation, \\  & & & estimate the correlation \\  & & & parameter of Kriging method \\ \hline
[36] & 2022 & TFWO & Optimal sizing of different \\  & & & energy sources in an isolated \\  & & & hybrid micro-grid \\ \hline
[37] & 2020 & TFWO & Optimal placement of \\  & & & parallel compensators at the \\  & & & distribution level \\ \hline
[38] & 2022 & Chaotic TFWO & Optimal reactive power \\  & & & dispatch (ORPD) problems in \\  & & & the power systems \\ \hline
[39; 40; 41] & 2021 & TFWO & Estimating parameters of \\  & & & photovoltaic models \\ \hline
[42] & 2021 & TFWO &
\begin{tabular}{c} Finding optimal parameters \\ of the back-to-back voltage \\ source converters (BTB-VSC) \\ \end{tabular} \\ \hline
[43] & 2021 & TFWO & Color aerial image multilevel \\  & & thresholding \\ \hline
[44] & 2022 & \
\begin{tabular}{c} Quasi-oppositional \\ TFWO \\ \end{tabular} &
\begin{tabular}{c} Short-term hydrothermal \\ scheduling (SHTS) \\ \end{tabular} \\ \hline
[45] & 2021 &
\begin{tabular}{c} A hybrid of TFWO and \\ battle royale optimization \\ (BRO), called TFW-BRO \\ \end{tabular} & Power flow in smart grids \\  & & & using renewables \\ \hline
[46] & 2021 & TFWO &
\begin{tabular}{c} Economic load dispatch \\ (ELD) problems in the power \\ systems \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of some applications of the TFWO algorithm in the recent literature. The optimal power flow problem is very complex, nonlinear, and non-convex. Thus, the present study combines the power of TFWO and TLBO algorithms to propose a novel robust algorithm for various OPF problems in integrated systems. Here are the main contributions of this paper:  
1. Hybridizing teaching and learning algorithms with turbulent flow optimizations developed a novel, efficient, and robust optimization algorithm named TLTFWO. This method is used to optimize optimal power flow (OPF) problems involving conventional thermal power plants, solar photovoltaics, and distributed wind power. 2. This work addresses the uncertainties of renewable generation by using the Weibull probability density function to model wind distribution and the lognormal probability density function to model solar radiation. 3. In addition to fuel costs, emissions, power losses, and voltage deviations, OPF also considers fuel costs, emissions, power losses, and voltage deviations. Factors such as economics, technology, and safety limit these functions. Furthermore, this study examined reserve, direct, and penalty costs in addition to thermal power unit production costs. 4. An optimal scheduling of thermal power plants based on renewable energy is determined by the amount of carbon tax associated with the goal function. In order to demonstrate the validity and effectiveness of the proposed TLTFWO algorithm, it is compared to other recently published algorithms on the IEEE 30-bus test system. The rest of the study is organized as follows. Section 2 formulates the OPF problem. Section 3 states the optimization steps of the proposed algorithm.