Optimal design of wind,solar,and hydel units for solving probabilistic optimal power flow using artificial hummingbird algorithm

Abstract

The need to transition to renewable energy sources (RESs) including wind, solar photovoltaic (PV), and hydro is highlighted by the accelerated depletion of fossil fuel reserves and the increasing need for sustainable power generation. In this regard, the present study explores the use of a probabilistic optimal power flow (POPF) framework that addresses the inherent uncertainties related to the power outputs of various RES technologies. The artificial hummingbird algorithm (AHA), inspired by hummingbirds adaptive foraging behavior, is used to solve the ensuing highly nonlinear, multi-modal problem. Several IEEE 57-bus test systems are used to thoroughly validate the efficacy of the suggested AHA-based POPF model. Comparative studies indicate that the AHA consistently provides notable cost and emission reductions, outperforming advanced metaheuristic approaches in all cases. When compared to coral reef optimization and gray wolf optimizer, the AHA reduces emissions by 1.74%–3.08% and costs by 0.27%–0.33% in the traditional IEEE 57-bus system. Additional gains are shown with cost reductions of up to 0.148% and emission reductions of up to 2.40% when RES units are integrated. AHA achieves the greatest results in the most complicated Test System 3, with an ideal generating cost of 4949 $\mathrm{\$}/\mathrm{h}$ and emissions of 1.0299 t/h. The development of a comprehensive POPF system that concurrently incorporates wind, solar, and hydro power with precise uncertainty modeling and related cost penalties is what makes the study new. The superiority of the suggested solution over sophisticated modern approaches is further shown by extensive statistical assessment of 24-hour dynamic load tests across both IEEE 57-bus system as well as IEEE 118-bus system.

Introduction

Energy is delivered from generators to loads via power systems, which are networks. Through the electric power transmission network, generated energy is supplied to loads. The current power system network is growing increasingly complicated for planning and operation due to massive power transfers over longer distances, intricate coordination, the challenge of integrating numerous system controllers, and reduced power reserves. A power system is considered secure when it can tolerate unanticipated disruptions with minimal loss in service quality. Following a disruption, the system goes through the following transient phase before stabilizing into a condition where all operational constraints are maintained within the allowed bounds. In an interconnected power system, the generators’ real and reactive powers must be adjustable within realistic constraints to consume the least amount of fuel while still satisfying the load requirements.

However, optimal power flow, or OPF, is used to alleviate these problems arising from the industry’s liberalization. One of the core problems with power system planning and operation is OPF. In the beginning, Dommel and Tinney (1968) presented it. Thereafter it spreads like a wildfire in the researcher community. The OPF method is now a widely used power system analysis technique that has been demonstrated to be sufficiently reliable for practical applications.Therefore, the primary objective of the OPF formulation is to determine a power generation system that reduces fuel costs while meeting all equality-and inequality constraints. The OPF problem is recognized as a multi-objective, highly nonlinear, non-convex optimization problem. To minimize a desired objective function while still satisfying the power balancing equations and specific inequality constraints in the system, a power system’s optimal steady-state operation must be determined.

Technological developments and the expanding global population have led to a dramatic rise in power consumption in modern electrical power networks. This has resulted in environmental degradation and climate change caused by the excessive utilization of fossil fuels. Hydropower (Sulaiman and Mustaffa, 2021), solar (Li et al., 2022b), and wind farms (Rabiee and Soroudi, 2013) are examples of environmentally friendly renewable energy sources (RESs) that have been incorporated into the system. However, the inclusion of RESs have invited a new challenge in the power system (Hassan et al., 2022). RESs introduce uncertainties in the practical power system resulting in hindrances in achieving reliable and economic operation (Shaheen et al., 2022). These uncertainties have led to a renewed interest in the probabilistic approach to solving OPF. In this approach, the worst-case event scenario and the RES uncertainties are considered via the probabilistic optimum power flow technique. These approaches differ from deterministic methods in that they represent uncertainty using probability density functions. The objective is to reduce the system’s expected generation expenditure in probabilistic OPF (POPF), where the expectation is based on the probability distribution of the RER outcome (Sumair et al., 2022). The primary difference between POPF and normal OPF by Li et al. (2022a) is that the former represents the uncertainty of the RES result using a probabilistic strategy, while the latter employs a fundamental deterministic approach. Additionally, POPF can provide a more accurate assessment of the probability of operational limitations being violated. Because of its deep-well structure, closed-loop U-shaped geothermal wells have great promise. To better understand their thermal behavior, this study constructs an analytical model that integrates quasi-steady borehole heat transfer with transient heat conduction in soil. The paper of Xiao et al. (2022 ) provides a theoretical basis for creating effective operating parameters for U-shaped geothermal wells by using exergy-based optimization to determine that optimal flow rates change over the heating period. In order to increase the frequency stability in power systems, work by Khadanga et al. (2025) presents a modified gorilla troops optimizer (mGTO). By providing improved stability, greater solution quality, and shorter calculation times, the algorithm performs better than conventional optimization techniques (Feng et al., 2025; Su et al., 2019; Wang et al., 2024; Xiao et al., 2025; Yu et al., 2023; Zhu et al., 2025). A Phillips-Heffron model featuring an interline power flow controller and an auxiliary damping controller was created, and mGTO was used to adjust the controller’s parameters. The suggested mGTO-based method outperforms traditional controllers in stabilizing a single machine infinite bus power system, according to simulation results.

Numerous methods based on contemporary control theory have been employed to address POPF issues (Singh et al., 2022). These include the cumulant method (Schellenberg et al., 2005), Monte Carlo simulation (Hashish et al., 2023), and heuristic approaches (Nikmehr and Najafi Ravadanegh, 2015). The POPF problem is solved in the paper of Zou and Xiao (2013) using the cumulative distribution function (CDF) of the output variables. For the mean, standard deviation, and CDF to be as precise as possible, this paper’s statistical formulation makes use of the outcomes of the intricate Monte Carlo simulation. The study of Schellenberg et al. (2005) provides two enhancements to the cumulant method (CM) for POPF studies to better address constraints in POPF applications: the first is an improvement to allow for the inclusion of correlated variables.

Researchers’ focus has shifted in recent years towards optimization methods for the POPF problem. Many optimization algorithms namely flow direction algorithm (FDA) (Maheshwari et al., 2023), cuckoo search algorithm (CSA) (Sarda et al., 2023), teaching–learning-based optimization (TLBO) (Sulaiman et al., 2023), particle swarm optimization (PSO) (Papazoglou and Biskas, 2023), quasi-reflection jellyfish optimization algorithm (QRJFO) (Shaheen et al., 2023), modified artificial rabbit optimizer (MARO) by Khan et al. (2024b) is the most effective method for solving nonlinear OPF problems. They have been effectively used to solve a variety of issues where it is preferable to find global solutions rather than local ones or if there are non-differentiable areas in the problem. The authors of Shaheen et al. (2023) has suggested that the potential of QRJFO may be utilized and investigated to lessen the multi-objective function’s impact of petroleum cost, transmission loss, and emissions. The large-scale IEEE 118-bus and the practical West Delta Region system have all been used to validate the stability. In order to avoid stagnation and boost global optimization efficiency, this paper by Khan et al. (2024a) presents an improved liver cancer algorithm (ILCA) with improved exploration and exploitation techniques. ILCA is very effective in reducing power losses and improving voltage stability in an IEEE 57-bus system that is integrated with renewable energy. The authors of Ebeed et al. (2023) proposed a modified RUNge kutta optimizer (MRUN) to handle the uncertainties of wind and solar integration in OPF. By incorporating cauchy mutation and quasi-oppositional learning, MRUN improves accuracy and robustness. Tests on the IEEE 57-bus system show significant reductions in losses and voltage deviations, along with improved stability, outperforming the conventional RUN method. An adaptive lightning attachment procedure optimizer (ALAPO) (Adhikari et al., 2023) to address the uncertainties from RESs in OPF. Tests on the IEEE 57-bus system show that ALAPO achieves better performance and stability than conventional optimization methods. This study of Jamal et al. (2024) introduced a modified gorilla troops optimizer (MGTO) to solve the issues of stagnation and local optima in the traditional GTO. When evaluated on benchmark functions and IEEE test systems, MGTO yields better and more dependable optimum power flow solutions than existing optimization methods. PSO was utilized by the authors of the paper (Mohamed et al., 2025) to solve the OPF issue while keeping to a strict objective function of decreasing the cost of fuel provision for utility and industrial businesses. To show the superiority of the suggested PSO, the calculated results were compared with several well-known optimization techniques, including the backtracking search algorithm (BSA), hybrid SFLA-SA, differential evolution (DE), improved GA (EGA), and monarch butterfly optimization (MBO). The whale optimization algorithm (WOA) was devised by the author Naidu et al. (2023) to solve the OPF in addition to RES. To determine the optimal capacity of RES in conjunction with thermal generators, the total production cost has been taken into account as an objective function. The IEEE-30 system is used to test the proposed method in order to confirm its suitability. The results collected show that WOA is more effective than other methods, including non-dominated sorting genetic algorithm (NSGA-II), gray wolf optimization (GWO), and PSO. The ideal capacity of RES in combination with thermal generators has been determined by taking the whole generating cost into account as an objective function. The IEEE-30 system is used to test the proposed method in order to confirm its suitability. The study’s findings show how effective WOA is when compared to other algorithms like GWO and PSO-GWO (PSOGWO). Three RESs—wind, solar, and a combination of solar and small hydro—have been paired with an additional thermal power plant to deal with the OPF problem (Farhat et al., 2023). Both the IEEE 30-bus and IEEE 57-bus standard systems use the weighted mean of vector (INFO) optimization technique, which was introduced recently. The outcomes so obtained demonstrate that the suggested by INFO performs better than the other algorithms when compared to other well-known published work. Using the IEEE 57-bus as the test system, the authors of the paper (Adhikari et al., 2023) used the ALAPO approach to solve the stochastic OPF problem. The ALAPO reacts better to the processes of exploration and exploitation since it is the most advanced stage of the conventional LAPO. Comparing the corresponding results from other well-known algorithms, such as the sand cat swarm optimizer (SCSO), GOW, WOA, black widow optimization algorithm (BWO), and capuchin search algorithm (CAPSA), has demonstrated the superiority of the proposed ALAPO. For nonlinear environments, many evolutionary algorithms are unable to produce the best results.

The artificial hummingbird algorithm (AHA) was employed to solve the highly nonlinear, multi-dimensional POPF issue in this work, which incorporates many uncertain renewable energy sources (wind, solar, and hydel). In these situations, classical deterministic approaches and certain conventional metaheuristics often either converge slowly or get trapped in local optima. The authors proposed AHA for optimal tuning parameters of POPF problem incorporating with solar, wind, and hydel power. The reason for selecting the AHA has been stated below:

Strong balance between exploration and exploitation.

Ability to avoid local optima.

Fewer control parameters.

Superior convergence characteristics.

High robustness under uncertainty.

The greater the number of RESs that are included in the proposed study system, the more difficult it becomes. The AHA is an excellent match for multi-modal scenarios including both thermal and RESs because it better balances exploration and uexploitation. For multi-dimensional optimization problems, its adaptive search method improves convergence and produces superior solutions. Additionally, AHA performs better in terms of parameter adjustment than other population-based techniques and provides an easy-to-use implementation. The resilience of AHA is enhanced by its capacity to manage intricate, high-dimensional situations; it can steer clear of local optima. Due to the POPF problem’s highly nonlinear, multi-dimensional character and the uncertainties surrounding the generation of wind, solar, and hydropower, the optimization method selection is critical to its solution. AHA has been chosen as the optimization tool in this work because to its capacity to successfully prevent premature convergence, balance exploration and exploitation, and have fewer control parameters. The directed, territorial, and migratory flight foraging behaviors depicted in AHA facilitate effective search operations that aid in navigating multi-modal solution domains. Additionally, 30 independent runs along with statistical performance evaluations using box-plot analysis and one-way ANOVA are conducted to confirm the robustness and consistency of AHA. As a result, AHA is a highly efficient and reliable optimization framework for solving the POPF problem with multiple renewable sources.

The growing use of RESs in contemporary power systems adds a great deal of variability and uncertainty to OPF calculations. This makes the POPF problem more complex, as both the operational cost and environmental emissions must be minimized while satisfying system constraints under uncertain wind, solar, and hydropower outputs. Therefore, the core problem addressed in this work is to develop an efficient and robust optimization framework capable of managing these uncertainties and achieving cost-effective, reliable, and environmentally sustainable system. The study formulates the POPF problem considering renewable generation uncertainties and solves it using the AHA, with performance evaluated across multiple test conditions to demonstrate its effectiveness. The following are the key contributions as per work carried out in the past:

Because to renewable uncertainties, OPF changed from deterministic to probabilistic.

Used models based on PDFs to depict solar and wind power.

Enhanced cost, loss, and emission optimization using a variety of metaheuristic methods.

For improved economic and environmental performance, a hybrid mix of renewable sources is taken into consideration in standard IEEE systems.

Based on wide literature survey, it has been found that many gaps need to be explored further, a few has been listed below:

Under POPF, integrated wind, solar, and hydro modeling is required.

More accurate probabilistic power output representations are required.

Inadequate methods to manage large levels of uncertainty and nonlinearity.

Insufficient multi-objective cost-emission analysis.

Limited statistical validation and repeatability studies.

The following are the paper’s primary contributions:

The POPF problem is addressed in this work using the AHA.

Because of its effectiveness in managing intricate, nonlinear, and uncertain optimization issues, the AHA, a bio-inspired metaheuristic, is utilized.

In order to lower generating costs and emissions, it incorporates RESs—wind, solar, and hydro—into conventional power networks.

The method is tested under different combinations of thermal and renewable units using three test systems based on the IEEE 57-bus model.

For both renewable (wind, sun, and water) and non-renewable-based IEEE 57-systems, the intended study project accounts for dynamic load, which fluctuates throughout the day.

This experiment aims to reduce emissions and generating costs concurrently for both single- and multi-objective functions in a range of combinations (also known as varied cases).

According to the computed outcomes, AHA outperforms other algorithms in terms of emissions and generating cost.

The computed results from 30 independent trials were evaluated using a box plot, one-way ANOVA, and additional statistical measures. This demonstrates the resilience and effectiveness of AHA in addressing the POPF problem in over-tested settings when several instances are generated based on various objectives.

The following sections of the research article are arranged as follows: the “Problem formulation” section outlines the problem formulation procedure for the present project. The uncertainty modeling, along with a general overview of wind and solar has been covered in the “Problem formulation” section. The “AHA optimization” section demonstrates the proposed “AHA.” The “Results and discussion” section presents the simulation results of several test systems using the proposed algorithm. Finally, the conclusions of the study are provided.

Modeling of uncertainties

This study examines the uncertainty related to energy produced by wind farms, solar photovoltaic (PV) arrays, and hydroelectric power sources. Due to its reliance on a variety of variables, including wind speed and solar irradiation, the electricity produced by the aforementioned RESs is subject to considerable unpredictability. This results in the power output from these RES units being uncertain. These uncertainties make it challenging for the system operators to ensure that the connected grid can consistently provide the required demand. The output power generated by wind turbines (WTs) and PV arrays must thus be simulated. This may be done effectively using a probabilistic model, which is discussed in the following sections.

Overview of WP (WP) generation

Wind energy is widely accessible, produces no emissions, and has very low operational costs. Conventional energy sources, on the other hand, have substantial operating costs and environmental challenges. This advantage has prompted researchers to explore alternative methods of using RESs to generate electricity.

WP model

When describing wind speed, the Weibull PDF (Hazra et al., 2025) is commonly used. It is given by the following equation:

$${PDF}_{rand}(U_{wind})={\displaystyle \frac{f}{s}{\left({\displaystyle \frac{U_{wind}}{s}}\right)}^{f-1}\times e^{-{\left({\displaystyle \frac{U_{wind}}{s}}\right)}^f}}$$

(1)

If the form factor is denoted by $F>0$ and $S>0$ is the corresponding scale factor; the likelihood of obtaining wind velocity of $U_{Wind}$ is $P_{rand}$ . Wind speed PDF is shown in Figure 1. The following represents the corresponding CDF:

$${CDF}_{rand}\left(U_{Wind}\right)=1-e^{-{\left({\displaystyle \frac{U_{Wind}}{s}}\right)}^f}$$

(2)

The wind unit’s WP output is determined by the following equation:

$$\begin{array}{rl}{OP}_{Wind}& =\{\begin{array}{cc}0& U_{Wind}<U_{in}or{U}_{Wind}>U_{out}\\ {\displaystyle \frac{{OP}_{Wirated}({OU}_{wind}-{OU}_{in})}{{OU}_{rated}-U_{in}}}& U_{in}\le U_{Wind}<U_{rated}\\ {OP}_{Wirated}& U_{rated}\le U_{wind}<U_{out}\\ \end{array}\end{array}$$

(3)

where $U_{Rated}$ represents the rated wind velocity, $U_{in}$ and $U_{out}$ represent the cut-in and cut-out wind velocities, respectively, and $O{P}_{Wind}$ and ${OP}_{Wirated}$ represent the indicated output power and rated power, respectively. The WP PDF is as follows:

$$F_{O{P}_{Wind}}(O{P}_{Wind})={\displaystyle \frac{FU}{S{P}_{Wirated}}{\left({\displaystyle \frac{U_{in}+U{\displaystyle \frac{O{P}_{Wind}}{{OP}_{Wirated}}}}{S}}\right)}^{F-1}\times e^{-{\left({\displaystyle \frac{U_{in}+U{\displaystyle \frac{O{P}_{Wind}}{{OP}_{Wirated}}}}{S}}\right)}^F}}$$

(4)

where $U=U_{Rated}-U_{in}$ .

Figure 1.

Weibull-based wind velocity probabilistic density function (PDF).

When $O{P}_{Wind}$ = 0 or $O{P}_{Wind}$ = ${OP}_{Wirated}$ , the probability distributions are presented as follows:

$$\begin{array}{rl}{I}_{Rated}(O{P}_{Wind}=0)=& I_{Rated}(U<U_{in})\\ & +I_{Rated}(U>U_{out})=1-e^{-{\left({\displaystyle \frac{U_{in}}{S}}\right)}^F}+e^{-{\left({\displaystyle \frac{U_{out}}{S}}\right)}^F}\end{array}$$

(5)

$$\begin{array}{rl}{I}_{Rated}(O{P}_{Wind}={OP}_{Wirated})& =I_{Rated}(U_{Rated}\le U<U_{out})\\ & =e^{-{\left({\displaystyle \frac{U_{Rated}}{S}}\right)}^F}-e^{-{\left({\displaystyle \frac{U_{out}}{S}}\right)}^F}\end{array}$$

(6)

The corresponding $O{P}_{Winvd}$ CDF is expressed as follows:

$$f_{O{P}_{Wind}}(O{P}_{Wind})=\{\begin{array}{cc}0& OPWIND<0\\ {\displaystyle \frac{FU}{S{OP}_{Wirated}}{\left({\displaystyle \frac{U_{in}+U{\displaystyle \frac{O{P}_{Wind}}{{OP}_{Wirated}}}}{S}}\right)}^{F-1}\times e^{-{\left({\displaystyle \frac{U_{in}+U{\displaystyle \frac{O{P}_{Wind}}{{OP}_{Wirated}}}}{S}}\right)}^F}}& 0\le O{P}_{Wind}<{OP}_{Wirated}\\ 1& O{P}_{Wind}\ge {OP}_{Wirated}\\ \end{array}$$

(7)

WP cost computation

During times of high demand, WP must be integrated into the existing electrical grid. Two cost categories—overestimation and underestimation—are considered when WP is implemented with the current power grid. This is because WP generation is inherently unpredictable. Here, WP generation is predicted using the Weibull distribution. The total cost of producing power from wind is stated as follows:

$$\begin{array}{rl}{Tota{l}_{c}ost}_{Wind}& =\sum _{m=1}^{N_{Wind}}{cost}_{Windm}({OP}_{Windm})\\ & =\sum _{m=1}^{N_{Wind}}({cost}_{Windm}^S+{cost}_{Windm}^O+{cost}_{Windm}^U)\end{array}$$

(8)

where ${Tota{l}_{c}ost}_{Wind}$ is the total cost of wind; $N_{Wind}$ is the quantity of wind units; ${cost}_{Windm}^S$ is the direct cost; ${cost}_{Windm}^O$ is the overestimation cost; ${cost}_{Windm}^U$ is the underestimation cost; and Suffix $m$ is the wind unit indix.

Direct cost

The direct cost for the $m\mathrm{th}$ WP unit is provided by the following equation:

$${cost}_{Windm}^S=S_m^{Wind}{P}_{Windm},wherem=1,2,3,...,N_w$$

(9)

Here, $S_m^{Wind}$ is the coefficients of direct cost and ${OP}_{Windm}$ is the power scheduled from $m\mathrm{th}$ unit.

Overestimation cost

When actual WP exceeds expectations, underestimation costs arise. Because they would otherwise lose power, any excess electrical energy generated by WTs will be stored in batteries. The following formula is used to calculate the underestimation cost, as shown by Yin et al. (2020):

$$\begin{array}{rl}{cost}_{Windm}^O=& {Pf}_{Windm}^O\times {OP}_{Windm}[1-e^{-{\left({\displaystyle \frac{U_{in}}{I}}\right)}^j}+e^{-{\left({\displaystyle \frac{U_{out}}{I}}\right)}^j}]\\ & +\left({\displaystyle \frac{{OP}_{Wiratedm}{U}_{in}}{U_{rated}-U_{in}}+{OP}_{Windm}}\right)[e^{-{\left({\displaystyle \frac{U_{in}}{c}}\right)}^j}-e^{-{\left({\displaystyle \frac{U_{in}+{OP}_{Windm}{\displaystyle \frac{U_{rated}-U_{in}}{{OP}_{Wirated}}}}{I}}\right)}^j}]\\ & +\left({\displaystyle \frac{{OP}_{Wirated}s}{U_{rated}-U_{in}}}\right)[\zeta \{1+{\displaystyle \frac{1}{j},{\left({\displaystyle \frac{U_{in}+{OP}_{Windm}{\displaystyle \frac{U_{Rated}-U_{in}}{{OP}_{Wirated}}}}{I}}\right)}^j}\}-\zeta \{1+{\displaystyle \frac{1}{j},{\left({\displaystyle \frac{U_{in}}{I}}\right)}^j}\}]\end{array}$$

(10)

Underestimation cost

When the actual WP exceeds expectations, underestimation costs arise. To avoid loss of energy, any excess electrical energy generated by WTs will be stored in batteries. The following formula used to determine the underestimation cost is given by Roy and Hazra (2015):

$$\begin{array}{rl}{cost}_{Windm}^U=& {Pf}_{Windm}^U\times (P_{Wirated}-P_{Windm})[e^{-{\left({\displaystyle \frac{V_{Rated}}{I}}\right)}^j}-e^{-{\left({\displaystyle \frac{V_{out}}{I}}\right)}^j}]\\ & +\left({\displaystyle \frac{P_{Wirated}{V}_{in}}{V_{Rated}-V_{in}}+P_{Windm}}\right)[e^{-{\left({\displaystyle \frac{V_{Rated}}{I}}\right)}^j}-e^{-{\left({\displaystyle \frac{V_{in}+P_{Windm}{\displaystyle \frac{V_{Rated}-V_{in}}{P_{Wirated}}}}{I}}\right)}^j}]\\ & +{\displaystyle \frac{P_{Wirated}I}{V_{RATED}-V_{in}}[\zeta \{1+{\displaystyle \frac{1}{j},{\left({\displaystyle \frac{V_{in}+P_{Windm}{\displaystyle \frac{V_{Rated}-V_{in}}{P_{Wirated}}}}{I}}\right)}^j}\}-\zeta \{1+{\displaystyle \frac{1}{j},{\left({\displaystyle \frac{V_{Rated}}{I}}\right)}^j}\}]}\end{array}$$

(11)

Here, specifically, ${cost}_{Windm}^O$ and ${cost}_{Windm}^U$ are the overestimation and underestimation costs of the $m\mathrm{th}$ wind unit; ${OP}_{Wirated}$ and $U_{Rated}$ are the rated output power and rated velocity, respectively; $U_{in}$ and $U_{out}$ are the WT cut-in and cut-out velocities, respectively; ${Pf}_{Windm}^U$ and ${Pf}_{Windm}^O$ are the underestimation and overestimation cost coefficients.

Overview of solar PV generation

The following is the mathematical formula for the electrical power generated by a solar panel array, which is dependent on its solar irradiance:

$$P_{PV}\left(\omega \right)=\{\begin{array}{cc}{P}_{PVn}\left(\frac{B_s^2}{B_{sd}{R}_c}\right)& \text{,for}{B}_s<R_c\\ P_{PVn}\left(\frac{B_s}{B_{sd}}\right)& \text{,for}{B}_s>R_c\end{array}$$

(12)

Here $P_{PVn}$ denotes the rated power; $P_{PV}(\omega )$ is the solar power generation; $B_{sd}$ represents the solar irradiation under standerd environment of 1000 ${\mathrm{W}/\mathrm{m}}^2$ . $R_c$ signifies a certain irradiance point at a set value of 120 ${\mathrm{W}/\mathrm{m}}^2$ .

Now, the use of beta probabilistic density function (PDF) to calculate solar irradiance has been formulated as follows:

$$F_s(S)=\{\begin{array}{cc}{\displaystyle \frac{\mathrm{\Gamma }(\beta +\gamma )}{\mathrm{\Gamma }(\beta )\mathrm{\Gamma }(\gamma )}\times S^{(\beta +1)}\times (1-S{)}^{\gamma -1},for}& 0\le S\le 1;\beta \ge 0;\gamma \ge 0\\ 0,\mathrm{otherwise}\end{array}$$

(13)

Finally, the power generation using the beta function as depicted by the stated equation.

$$P_{PV}\left(\omega \right)=\frac{{\displaystyle \sum _{i=1}^{N_S}{P}_{PV}\times F_S\left(S_g^t\right)}}{{\displaystyle \sum _{i=1}^{N_S}{F}_S\left(S_g^t\right)}}$$

(14)

where $S_g^t$ denotes the solar irradiance, and $F_S(S_g^t)$ is the corresponding probability.

Cost function: Solar energy

The cost of electricity generation for a solar unit is estimated as follows by adding three different cost functions (Paul et al., 2021):

$$C_{sl}\left(P_{sl}\right)=C_{sl}^d+C_{sl}^O+C_{sl}^U$$

(15)

where $C_{sl}^U$ , $C_{sl}^O$ , and $C_{sl}^d$ , are underestimation cost, overestimation cost, and direct cost of the respective $l\mathrm{th}$ solar unit.

Determination of solar unit’s direct cost

The cost of producing solar energy is known as the ‘‘direct cost.” The following formula determines the direct cost of solar energy:

$$C_{sl}^d=d_l^s{P}_{sshl},\mathrm{where}l=1,2,3,...,n_s$$

(16)

Here the direct cost coefficients for the $l\mathrm{th}$ solar unit is denoted by $d_l^s$ ; schedule power of the $l\mathrm{th}$ unit is denoted by $P_{sshl}$ .

Calculation of the overestimated cost of the solar unit

If the estimated power exceeds the available solar power, the overestimation cost is calculated using the formula below:

$$\begin{array}{rl}{C}_{sl}^O& ={PF}_{sl}^O(P_{sshl}-P_{savl})\\ & ={PF}_{sl}^O{\int }_0^{P_{sshl}}(P_{sshl}-P_s)f_{P_s}(P_s)d{P}_s\end{array}$$

(17)

where the PDF of the solar unit’s power output is represented by $f_{p_s}(P_s)$ ; ${PF}_{sl}^O$ is the overestimation penalty cost coefficient of the $l\mathrm{th}$ solar unit; $P_{sshl}$ is the scheduled power of the $l\mathrm{th}$ solar unit; the $l\mathrm{th}$ solar unit’s available power is $P_{savl}$ .

Calculation of underestimation cost of solar unit

If the available solar power exceeds the scheduled power, the underestimating cost of the $l\mathrm{th}$ solar unit is calculated as follows:

$$\begin{array}{rl}{C}_{sl}^U& ={PF}_{sl}^U(P_{savl}-P_{sshl})\\ & ={PF}_{sl}^U{\int }_{P_{sshl}}^{P_{srl}}(P_s-P_{sshl})f_{p_s}(P_s)d{P}_S\end{array}$$

(18)

where ${PF}_{sl}^U$ is the underestimation panalty cost coefficient of the $l\mathrm{th}$ solar unit; $P_{srl}$ is the rated power of the $l\mathrm{th}$ solar unit.

Problem formulation

The authors of this study proposed the POPF to reduce the total cost and emission of power generation. In order to address the POPF problem using the marine predator algorithm (MPA), current research has been organized into three approaches, each further divided into three categories: two single objective functions for minimizing generation costs and emissions, and, finally, an appropriate multi-objective function for simultaneously minimizing overall generation costs and emissions. First, POPF has traditionally been studied as three distinct case studies to minimize the given objective functions. The problem is modified to account for the creation of hydropower, wind speed, and solar PV, all of which directly affect output power. Here, the planned number of generators in the typical IEEE 57-bus systems has been replaced with a few number of solar, wind, and hydroelectric power generating units. Finally, in addition to the expected variety of generators (1, 2, 3, 6, 8, 9, and 12) in the IEEE 57-bus systems, the autonomous inclusion of solar PV panels, WTs, and hybrid power generation has further reduced the overall cost and emissions of power generation.

Conventional OPF

The traditional OPF, which was initially presented by Dommel and Tinney (1968), aims to minimize the objective function as it is defined below while adhering to all security and physical restrictions. The traditional OPF issue may be expressed as follows:

$$\mathrm{minimize}\mathrm{obj}(u,v)$$

(19)

subjects to:

$$\begin{array}{rl}{h}_m(u,v)& =0\\ g_{n_l}(u,v)& \le g_n(u,v)\le g_{n_u}(u,v)\end{array}$$

(20)

where $h_m$ is the corresponding equality constraint and $g_n(u,v)$ denotes the inequality constraint. A vector of state variables is denoted by $u$ , and a vector of control variables by $v$ . State variables consists of:

$$u=[P_{G1},V_{l1}\cdots V_{lNL},Q_{G1}\cdots Q_{GNG},S_{L1}\cdots S_{LNTL}]$$

(21)

$$v=[P_{G2}\cdots P_{GNG},V_{G1}\cdots V_{GNG},T_1\cdots T_{NT},Q_{C1}\cdots Q_{CNC}]$$

(22)

where the total generation cost is represented by $obj$ , and the accompanying equality and inequality constraints are represented by $h_m$ and $g_n$ . $v$ stands for vectors of control variables, while $u$ is a vector of state variables. The entire number of generator and load buses is indicated by $NG$ and $NL$ ; the number of transmission lines is represented by $NTL$ ; the number of transformers and reactive shunt restitution is indicated by $NT$ and $NC$ . State variables $u$ consists of slack bus active power, load voltages, generators’ reactive powers and transmission lines’ loadings; the vector of independent variables, $v$ , includes the voltages of the generators, the tap settings of the transformers, the reactive power injections, and the active powers of the generators.

Probabilistic OPF

Power systems are open systems rather than closed ones. This implies that any external parameter might affect its operational variables, which makes the system quite unpredictable. The notion of probability theory, or probabilistic techniques, is a commonly used approach to assess the uncertainty of a given system. The main objective of these methods is to determine the state of the system with respect to the uncertainty of the input variables, which are described as follows:

$$Y=f(X)$$

(23)

Because the input variables are unclear, the output variables are also uncertain. It suggests that even if there was just one uncertain input variable, all of the output variables would be uncertain.

Objective functions of POPF problem

Two distinct objective functions are taken into account while evaluating the efficacy of the suggested AHA approach in a single-objective POPF situation. The objective functions are as given below:

Minimization of fuel cost

In a further phase of the study, the POPF issue involves an independent re-evaluation of the cost function every hour. The main objective of these methods is to determine the state of the system with respect to the uncertainty of the input variables, which are described as follows:

$$MinimumCost=\sum _{H=1}^{24}\sum _{i=1}^{NG}{F}_G(P_G(i,H))$$

(24)

$P_{Gi,H}$ symbolizes the actual power produced during the time $H$ at point $i$ . With a quadratic cost function, the total fuel cost of producing units is determined by the following equation:

$$F_G(P_G(i,H))=\sum _{H=1}^{24}\sum _{i=1}^{NG}{a}_i{P_G(i,H)}^2+b_i{P}_G(i,H)+C_i$$

(25)

For flexible operating facilities, multiple valve steam turbines are used to produce a more precise and useful cost function model. The following is the total generation cost for units with valve point discontinuities:

$$\begin{array}{rl}{F}_G(P_G(i,H))=& {\displaystyle \sum _{H=1}^{24}\sum _{i=1}^{NG}{a}_i{P_G(i,H)}^2+b_i{P}_G(i,H)+C_i}\\ & +|d_i\times \sin (e_i\times (P_{Gi}^{min}-P_{Gi}))|+{\displaystyle \sum _{k=1}^{N_W}\sum _{t=1}^{T_L}(E_{u,k}^t+E_{o,k}^t)}\\ & +{\displaystyle \sum _{m=1}^{N_S}\sum _{t=1}^{T_L}F(P_{S,k}^t)}\end{array}$$

(26)

where $a_i$ , $b_i$ , $c_i$ , $d_i$ , and $e_i$ are the cost coefficient of $i\mathrm{th}$ generators; the $i\mathrm{th}$ generator’s actual active power is denoted by $P_{Gi}$ and the $i\mathrm{th}$ generator’s minimum active power generation limit is $P_{Gi}^{min}$ . $N_S$ represents the total number of solar units; $P_{S,k}^t$ is the solar power generation at time $t$ ; $N_W$ represents the total number of wind units; $E_{o,k}^t$ and $(E_{u,k}^t$ underestimation cost and overestimation cost for wind power plant.

Minimization of emission

The emission problem is modeled mathematically as follows:

$$Minimumemission=E\left(P_G(i,H)\right)$$

(27)

The emission generated by each generating unit may be approximated as a quadratic function of the power output of the generator. The same may be formulated as follows:

$$\begin{array}{rl}E(P_G(i,H))=& {\displaystyle \sum _{i=1}^T\sum _{i=1}^H{\alpha }_i(P_G(i,H){)}^2+{\beta }_i(P_G(i,H))+{\gamma }_i}\\ & +\zeta \exp (\xi P_G(i,H))\end{array}$$

(28)

where $\alpha ,\beta ,\gamma ,\zeta ,\mathrm{and}\xi$ are the emission coefficients of the $i\mathrm{th}$ generating unit.

Multi-objective function of POPF problem

Previously single-objective functions are lowered on an individual basis. However, in order to evaluate the efficacy of the proposed strategy in a multi-objective context, active generation cost with valve point loading and emission are simultaneously reduced after the first simultaneous decrease of producing cost and emission. The importance of emissions and generation costs has been balanced using the penalty component ( $ϵ$ ) of the multi-objective function. The mathematical representation of the multi-objective function ( $F$ ) is formulated in the following equation:

$$F=(MinimumCost)+ϵ(Minimumemission)$$

(29)

Constraint

The constraints related to the issue are displayed as follows:

Equality constraints

The power system’s power balance equation, the hydro unit’s water dynamic balance equation, the hydro unit’s water discharge continuity equation, and other equations are among the equality constraints of the issue.

Power balance constraint: The power balancing equation states that the system’s total load demand and transmission losses are supplied by the total power generated by thermal, hydro, wind, and solar sources.

$$\begin{array}{rl}{P}_{Gi}+& \sum _{j=1}^{N_H}{P}_{Hj}^t+\sum _{n=1}^{N_W}{P}_{Wn}+\sum _{n=1}^{N_W}F(P_{S,k}^t)=\left|V_i\right|\\ & \sum _{j=1}^N\left|V_j\right|(G_{ij}\cos {\delta }_{ij}+B_{ij}\sin {\delta }_{ij})+P_{Li}\end{array}$$

(30)

where $V_i$ and $V_j$ are the $i\mathrm{th}$ bus and $j\mathrm{th}$ bus voltages, respectively. There is a set total number of thermal, hydro, wind, and solar generating units given by $N_G$ , $N_H$ , $N_W$ , and $N_S$ , respectively; $P_{Gi}$ , $P_{Wn}$ , and $P_{S,k}$ signify the generation of power in thermal, wind, and solar units for scheduling time $t$ , respectively; and $P_{Hj}$ represents in hydro generation which is formulated by the following equation:

$$\begin{array}{rl}{P}_{Hj}=& C_{1j}{\left(V_{Hj}^t\right)}^2+C_{2j}{\left(Q_{Hj}^t\right)}^2+C_{2j}{\left(V_{Hj}^t\right)}^2\\ & +C_{2j}\left(V_{Hj}^t{Q}_{Hj}^t\right)+C_{4j}\left(V_{Hj}^t\right)+C_{5j}\left(Q_{Hj}^t\right)+C_{6j}\end{array}$$

(31)

where $(V_{Hj}^t)$ and $Q_{Hj}^t$ are volume and discharge of the $j\mathrm{th}$ hydro generator, respectively, and generation coefficients of the $j\mathrm{th}$ hydro generator are $C_{1j}$ , $C_{2j}$ , $C_{3j}$ , $C_{4j}$ , $C_{5j}$ , and $C_{6j}$ .

Water dynamic balance of the hydro reservoir unit: The hydro reservoir unit’s dynamic water balance may be shown in the following equation:

$$V_{Hj}^t=V_{Hj}^{t-1}+I_{Hj}^t-Q_{Hj}^t-S_{Hj}^t+\sum _{p=1}^{N_u}(Q_{Hm}^{t-t_d}+S_{Hm}^{t-t_d})$$

(32)

where $I_{Hj}^t$ , $Q_{hj}^t$ , and $S_{Hj}^t$ represent inflow, discharge, and spillage at $t\mathrm{th}$ hour; $V_{Hj}^t$ and $V_{Hj}^{t-1}$ represent $j\mathrm{th}$ reservoir volume at $t$ and $(t-1)$ time intervals.

Continuous water flow illustration of a hydro generator: Water discharge continuity of hydro generator is represented by the following equation:

$$\begin{array}{rl}{Q}_{Hj}^t& =V_{HJ}^{ini}-V_{Hj}^{fin}+{\displaystyle \sum _{t=1}^{T_l}{I}_t,Hj-\sum _{t=1}^{T_l-1}{Q}_t,Hj}\\ & +{\displaystyle \sum _{t=1}^{T_l}\sum _{m=1}^{N_u}{Q}_{Hm}^{t-t_d}}\end{array}$$

(33)

where $Q_{Hj}^t$ denotes the ultimate hydro it discharge; $V_{HJ}^{ini}$ and $V_{Hj}^{fin}$ first and last volumes of reservoir, respectively.

Inequality constraints

Generator constraints: The generator voltage, active power, and reactive power of the $i\mathrm{th}$ bus fall between their corresponding upper and lower limitations, as shown by the following equation:

$$\begin{array}{rl}{P}_{Gi}^{min}& \le P_{Gi}\le P_{Gi}^{max}i=1,2,\dots ,N_G\\ V_{Gi}^{min}& \le V_{Gi}\le V_{Gi}^{max}i=1,2,\dots ,N_G\\ Q_{Gi}^{min}& \le Q_{Gi}\le Q_{Gi}^{max}i=1,2,\dots ,N_G\end{array}$$

(34)

where $P_G$ and $Q_G$ are the active and reactive power; $V_G$ represents the voltage at the generating units.

Load bus constraints

$$V_{li}^{min}\le V_{li}\le V_{li}^{max}i=1,2,\dots ,N_l$$

(35)

where $V_l$ is the voltage at load bus.

Transmission line constraints

$$S_{Li}^{flow}\le S_{Li}^{max}i=1,2,\dots ,NTL$$

(36)

where $S_{Li}^{flow}$ is the apparent flow of power of the $ith$ branch and $S_{Li}^{max}$ is the apparent power at transmission line.

Transformer tap constraints Transformer tap settings fall between the following upper and lower bounds:

$$T_i^{min}\le T_i\le T_i^{max}i=1,2,\dots ,N_C$$

(37)

where $T_i$ denotes the position of the tap changing transformer.

Shunt compensator constraints Shunt compensation has the following limitations:

$$Q_{ci}^{min}\le Q_{Ci}\le Q_{ci}^{max}$$

(38)

where $Q_C$ is the reactive power of capacitor bank.

AHA optimization

Zhao et al. (2022) created the AHA, a bio-based swarm algorithm that takes inspiration from hummingbird foraging behavior to address single and multi-objective optimization problems. Previous research indicates that it is more competitive than other optimization strategies and has fewer control parameters (Zhao et al., 2022). Additionally, the benefit of this approach in terms of computation load and accuracy of the outcome is seen below. AHA is chosen for this investigation due to its proven effectiveness in resolving challenging, multi-modal optimization issues. Because the POPF scenario under consideration incorporates uncertainty modeling, several dependent variables, and competing objectives (emissions vs. generation cost), various metaheuristics and standard deterministic approaches are vulnerable to local entrapment or slow convergence. Through its biologically inspired flight strategies—directed, territorial, and migratory foraging—AHA naturally strikes a balance between exploration and exploitation, enabling it to search extensively while honing in on favorable areas of the solution space. Additionally, compared to other algorithms like PSO, GA, and so on, AHA has fewer control parameters, which lowers computational tuning effort and increases stability. Three foraging behaviors are simulated by the artificial hummingbird program during the optimization process: migratory, territorial, and directed foraging. These actions also mimic omnidirectional, axial, and diagonal flight patterns. Additionally, the hummingbird memory is replicated using a visit table to choose the required food sources (Wang et al., 2022). Food sources: In reality, a hummingbird selects a suitable food source from a range of possibilities by taking into account the nectar volume and quality of certain flowers, the rate at which nectar replenishes, and the duration since the last visit. The quantity and kind of flowers in each food source are considered to be the same in AHA; a food source is a vector of solutions, and its nectar-refilling rate is represented by a function fitness value.

Hummingbird: Every hummingbird has a specific food source that it may eat from, and the hummingbird and the food source are in the same location. Hummingbirds may share their knowledge of a particular food source’s location and nectar replenishment rate with other members of their community. Additionally, each hummingbird has the ability to remember the duration of time that it has been without visiting each food source on its own.

Visit table: Each hummingbird’s visit level to each food source is recorded by the visit table which is one of the most important components of the AHA. The higher the value in the table, the more nectar volume collected by this source, and hence the higher the visit priority. The visit table is updated after each foraging period.

Initialization

A colony of $l$ hummingbirds is randomly assigned $l\mathrm{th}$ food sources.

$$X_k=low+R.(up-low)k=1,\dots ,l$$

(39)

where $R$ is a random vector in [0, 1], $up$ and $low$ represent the upper and lower boundaries of an s-dimensional problem, respectively, and $X_k$ indicates the position of the $k\mathrm{th}$ food source that symbolizes the solution to a specific problem. The food source visit table is set up as follows:

$${VisitTab}_{k,m}=\{\begin{array}{cc}0& ifk\ne m\\ l& m=1,\dots ,l\\ null& k=m\end{array}$$

(40)

where for $k=m$ , ${VisitTab}_{k,m}=null$ indicates that a hummingbird is consuming food at its individual food source; for $k\ne m$ , ${VisitTab}_{k,m}=0$ indicates that the $k\mathrm{th}$ hummingbird has just visited the $m\mathrm{th}$ food source in the current iteration.

By adding a direction switch vector, the AHA adequately models and exploits three flight capabilities while foraging, namely omnidirectional, diagonal, and axial flights. This vector determines whether or not one or more directions in the s-dimension space are available.

Axial flight:Axial flight demonstrates a hummingbird’s capacity to fly along any coordinate axis. To extend these flight patterns to a $S$ space, the axial flight may be described as follows:

$${\mathrm{AF}}^k=\{\begin{array}{ccccc}1& if& k=randk\left([1,s]\right)& & k=1,\dots ,s\\ 0& & else\end{array}$$

(41)

where $randk([1,s])$ generates a random integer between 1 and $s$ .

Diagonal flight: A hummingbird may go diagonally from one corner of a rectangle to the other using any two of the three coordinate axes. The definition of diagonal flight is as follows:

$${\mathrm{DF}}^k=\{\begin{array}{cc}1& ifk=p(m),m\in [1,n]\\ p=randperm(n),& n\in [2,[R_1.(s-2)]+1]\\ 0& else\end{array}$$

(42)

where $k=1,\dots ,s$ , $randperm(n)$ generates a random permutation of integers ranging from 1 to n, where $R1$ is a random number in the range [0, 1]. Any 2 to $s-1$ coordinate axis can generate a hyper-rectangle in which diagonal flying takes place in an s-dimensional space.

Omnidirectional flight: The omnidirectional flight illustrates how any flying direction may be projected to any of the three coordinate axes. The text colorblack. In other words, all birds can fly in all directions, but only hummingbirds can fly axially and diagonally. The following is the definition of omnidirectional flying:

$${OD}^{(k)}=1k=1,\dots ,s$$

(43)

Guided foraging

The AHA states that a food supply with a greater fitness rate has a quicker rate of nectar replenishing. During the guided foraging phase, hummingbirds will visit the feeders with the greatest nectar in order of the highest visit level. The following mathematical model is introduced to quantify directed foraging:

$$U_k(T+1)=X_{k,target}(T)+g.d.(X_k(T)-X_{k,target}(T))$$

(44)

$$g\sim N(0,1)$$

(45)

where $g$ is a guided factor that is subject to the normal distribution, $X_k(T)$ is the position of the $k\mathrm{th}$ food source at time $T$ , and $X_{k,target}(T)$ is the position of the target food source that the $i\mathrm{th}$ hummingbird intends to visit. $N(0,1)$ , where the mean is zero and the standard deviation is one. Equation (19) allows each current food source to update its location in relation to the target food source, simulating hummingbird directed foraging using a range of flight patterns.

Territorial foraging

If a hummingbird has already eaten the nectar from the flowers, it is more likely to look for another food source after visiting its favorite. As a result, a hummingbird could just go to a neighboring location inside its own territory, where it might discover a new food source that is better than the current one. The following mathematical formula depicts a potential food source and the territorial foraging pattern of hummingbirds:

$$U_k(T+1)=X_k(T)+c.d.X_k(T)$$

(46)

$$c\sim N(0,1)$$

(47)

where $c$ is a territorial factor with a typical normal distribution ranging from 0 to 1.

Migration foraging

A hummingbird will often go to a more distant food source when the supply at its regular feeding site becomes scarce. The migration coefficient is defined by the AHA. If the number of iterations exceeds the specified value of the migration coefficient, the hummingbird who lands at the food source with the lowest nectar-refilling rate will migrate to a new food source that is randomly generated within the entire search space. The visit table will be switched at this point, and the hummingbird will stop feeding at the old source and start eating at the new one. The migratory foraging of a hummingbird from the source with the lowest nectar-refilling rate to a new one formed at random is described below:

$$X_{wor}(T+1)=low+R.(up-low)$$

(48)

where the food source with the lowest population nectar-refilling rate is $X_{wor}$ .

Solutions update

The hummingbird will switch if the candidate solution developed during the territorial or directed foraging stages performs better than the existing food source. The following model exemplifies this behavior. The $k\mathrm{th}$ food source’s position has been updated as follows:

$$X_k(T+1)=\{\begin{array}{cc}{X}_k(T)& \text{if}f(X_k(T))\le f(U_k(T+1))\\ U_K(T+1)& \text{if}f(X_k(T))>f(U_k(T+1))\end{array}$$

(49)

Results and discussion

The IEEE 57-bus transmission system has also included the recommended method, as shown in Figure 2. System information is available by Chary and Rosalina (2023). The authors in the present study have taken into consideration IEEE 57-bus system. Also to show the superiority in more complex system, the authors have aslo extended the work in IEEE 118-bus system.

Figure 2.

Single line diagram of IEEE 57-bus.

Observation in IEEE 57-bus system

The consequent section by considering IEEE 57-bus delves broadly into three test systems, namely (a) Test System 1 incorporating conventional POPF having bus 1 (swing), 2, 3, 6, 8, 9, and 12 as PV-bus (generator bus) employing AHA; (b) Test System 2, which includes limiting the number of thermal generators at 1 (swing), 3, 8, and 12 and replacing the rest with RES namely wind power at bus-2 and bus-6, SPV at bus-9 and hydro unit at bus-6 and solving the same using the proposed AHA; (c) Test System 3 in which conventional positions of the thermal generators (1 (swing), 2, 3, 6, 8, 9, and 12) have been kept intact in addition to addition to connecting wind power generation at bus-5, solar PV at bus-16 and combined implementation of hydro and wind at bus-49. Finally, the considered problem have been solved by the proposed AHA to yield global optimal solution. A lot of nine cases have been considered in this research work as depicted in Table 1 which includes two different objective function namely single-objective function each minimizing emission cost minimization (Cases 2, 5, and 8) and total fuel cost minimizing with valve point effects (Cases 1, 4, and 7) followed by a multi-objective function containing simultaneous minimization of generation cost and emission (Cases 3, 6, and 9). The study reveals the complete understanding of the efficiency of the proposed AHA in different scenario. The emerging arguments are subject to analysis under different time-varying framework.

Table 1.

Numerous inspected studies of IEEE 57-bus.

Case	Single-objective	Multi-objective	Considered objectives	Constraints	Test System
1	$✓$		Total generation cost minimization incorporating valve point effects	Equality and inequality	IEEE 57-bus system 1
2	$✓$		Emission minimization
3		$✓$	Simultaneous minimization of cost and emission
4	$✓$		Total generation cost minimization incorporating valve point effects for thermal, hydro, and wind–PV energy		IEEE 57-bus system 2
5	$✓$		Emission minimization incorporating wind, PV and hydro energy
6		$✓$	Simultaneous minimization of cost and emission
7	$✓$		Total generation cost minimization incorporating valve point effects for thermal, hydro, and wind–PV energy		IEEE 57-bus system 3
8	$✓$		Emission minimization incorporating wind, PV, and hydro energy
9		$✓$	Concurrent reduction of costs and emissions

The preceding segment focuses on the analysis of the computations results of the POPF problems. The inquiry takes into account the system’s varying loads concurrently. The inoculation of the RES notably improves the generation cost and emission cost considerably. Table 2 displays the overall probability of load demand, solar radiation, wind speed, and river flow rate on a normal day. The probability density function for RESs is displayed in Table 3. The data has been taken for a complete whole day of 24 h. The graphical representation of the same can be furnished in Figures 3 and 4. The load demand and corresponding output powers of solar, both windfarms, hydropower unit in a day is depicted in Table 4 and the graphical representation of the same has been in Figure 5.

Figure 3.

Solar irradiance.

Figure 4.

Wind speed.

Figure 5.

Load curve.

Table 2.

The total probability of solar radiation, wind speed, load demand, and river flow rate during the course of an average day.

Hour	Loadpercentage	Wind farm1 at bus-2 wind speed ( $\mathrm{m}/\mathrm{s}$ )	Solar at bus-9irradiance ( ${\mathrm{W}/\mathrm{m}}^2$ )	Wind farm2 andhydro power at bus-6 wind speed ( $\mathrm{m}/\mathrm{s}$ )	River flow rate ( ${\mathrm{m}}^3/\mathrm{s}$ )
1	70	15.5643	0	16.7654	14.8877
2	70	14.9876	0	15.6512	13.5432
3	70	14.9876	0	14.7809	13.6644
4	70	13.6543	0	14.8765	11.5432
5	75	10.6543	0	15.9876	16.0987
6	80	9.8765	0	12.7654	15.4321
7	80	9.5412	87.7676	13.8765	13.3421
8	90	8.2098	208.7655	12.9876	11.5432
9	90	7.9861	409.6754	11.003	15.3342
10	90	6.8702	710.8765	10.7865	12.6981
11	100	6.731	911.3321	10.1122	11.6543
12	110	6.0654	1012.8908	11.9832	10.7687
13	110	5.8965	1001.5634	11.345	14.7765
14	100	4.8743	932.4454	10.4321	14.5434
15	100	3.8667	713.7609	9.865	16.4409
16	90	3.2751	554.5645	8.9876	15.8976
17	90	3.8764	395.8981	7.872	13.6543
18	110	4.5436	106.9891	8.3241	10.3342
19	110	4.8043	37.8976	6.3421	13.1213
20	110	3.765	0	5.4531	13.3423
21	110	2.6754	0	4.8711	12.098
22	90	2.2342	0	3.9861	14.6548
23	80	3.8877	0	4.2324	15.3377
24	70	4.541	0	6.321	10.9899

Table 3.

Probability density function for renewable energy sources.

Wind power generators plants (bus: 2)			Solar power system (bus: 9)			Combined wind and hydro system (bus: 6)
No. of turbines	Rated power Pwr (MW)	Weibull PDF parameters	Cost coefficient ($/MWh) Reserve, KRw	Cost coefficient ($/MWh) Penalty, KPw	Rated power PPVr (MW)	Lognormal parameters	Wind rated power Pwr (MW)	Weibull PDF parameters	Hydrorated power PHDLr (MW)	Gumbel parameters
25	75	$\alpha =9$ , $\beta =2$	3	1.5	50	$\zeta$ =6, $\xi$ =0.6	45	$\alpha$ =10, $\beta$ =2	5	$\gamma$ =220 $\tau$ =24.52

Table 4.

The hydroelectric unit, wind farm, and solar power’s daily load requirement and matching output powers.

Hour	Loadpercentage	Wind farm1 power at bus-2 $P_{Wind1}(\mathrm{MW}$ )	Solar power at bus-9 $P_{Solar}(\mathrm{MW}$ )	Wind farm2 at bus-6 $P_{Wind2}(\mathrm{MW}$ )	Hydro power at bus-6 $P_{Hydro}(\mathrm{MW}$ )	Wind farm2+ Hydro at bus-6 $P_{Hydro+Wind2}$ (MW)
1	70	72.4863	0	45	3.1035	48.1035
2	70	69.1592	0	43.7926	2.8232	46.6159
3	70	69.1592	0	40.78	2.8485	43.6286
4	70	61.4671	0	41.111	2.4063	43.5173
5	75	44.1594	0	44.9571	3.356	48.3131
6	80	39.6721	0	33.8033	3.217	37.0203
7	80	37.7377	3.2096	37.6494	2.7813	40.4308
8	90	30.0565	10.4383	34.5725	2.4063	36.9788
9	90	28.766	20.4838	27.7027	3.1966	30.8993
10	90	22.3281	35.5438	26.9533	2.6471	29.6003
11	100	21.525	45.5666	24.6192	2.4295	27.0486
12	110	17.685	50	31.0957	2.2449	33.3406
13	110	16.7106	50	28.8865	3.0803	31.9669
14	100	10.8133	46.6223	25.7265	3.0318	28.7583
15	100	5.0002	35.688	23.7635	3.4273	27.1908
16	90	1.5871	27.7282	20.7263	3.3141	24.0404
17	90	5.0562	19.7949	16.8646	2.8464	19.711
18	110	8.9054	4.7694	18.4296	2.1543	20.5839
19	110	10.4094	0.5984	11.5688	2.7353	14.3041
20	110	4.4135	0	8.4915	2.7814	11.2729
21	110	0	0	6.4769	2.522	8.9989
22	90	0	0	3.4134	3.055	6.4684
23	80	5.1213	0	4.266	3.1973	7.4633
24	70	8.8904	0	11.4958	2.291	13.7868

Observation of test system-I

The various IEEE 57-bus system statistics for the POPF issue under investigation are already shown in Table 5. The system consists of 57 buses and 80 branches. In the IEEE 57-bus system under consideration, three compensating devices at buses 18, 25, and 53 have been commissioned, and a total of 17 tap changing transformers located at branches 19, 20, 31, 35, 36, 37, 41, 46, 54, 58, 59, 65, 66, 71, 73, 76, and 80. The fixed load demand is 1250.8 MW and 336.4 MVAr. The cost and emission coefficients of thermal generators for the IEEE 57-bus test system under study are displayed in Table 6. The thermal generators are placed at positions 1, 2, 3, 6, 8, 9, and 12. A standard 500 iterations is the stopping criterion applied to all optimization techniques. Table 7 demonstrates the solution of the POPF for variation in load throughout a standard day for Case 1. Here total thermal cost have been considered as the objective function. After critical examination, it has been found the average thermal generation cost came out to be 5112.500 ( $\mathrm{\$}/\mathrm{h}$ ).

Table 5.

An overview of the IEEE 57-bus system 1 under investigation.

Element	Size	Description
Bus	57	Taher et al. (2019)
Transmission line	80	Taher et al. (2019)
Steam generator	7	$G_1$ (swing), $G_2$ , $G_3$ , $G_6$ , $G_8$ , $G_9$ , and $G_{12}$
On load tap changer	17	$T_{B19}$ , $T_{B20}$ , $T_{B31}$ , $T_{B35}$ , $T_{B36}$ , $T_{B36}$ , $T_{B41}$ , $T_{B46}$ , $T_{B54}$ , $T_{B58}$ , $T_{B59}$ , $T_{B65}$ , $T_{B66}$ , $T_{B71}$ , $T_{B73}$ , $T_{B76}$ , and $T_{B80}$
Optimization parameters	34	$PG$ : 7; $V$ : 7; $TCT$ : 17; $SC$ :3
Load capacity		1250.8 MW, 336.4 MVAr
Load voltage limits	50	[0.94–1.06] $\mathrm{p}.\mathrm{u}.$
Shunt devices	3	$S{C}_{18}$ , $S{C}_{25}$ , and $S{C}_{53}$

Table 6.

Steam generators’ costs and emission coefficients of IEEE 57-bus system 1.

		$a$	$b$	$c$	$d$	$e$	$\alpha$	$\beta$	$\gamma$	$\omega$	$\mu$
Generator	Quantity	( $\times {10}^{-2}$ )
TG1	1	0	200	0.375	1800	3.70	409.1	$-555.4$	649	0.0200	28.6
TG2	2	0	175	1.75	1600	3.80	254.3	$-604.7$	563.8	0.05	33.3
TG3	3	0	300	2.5	1350	4.10	613.1	$-555.5$	515.1	0.001	66.7
TG6	6	0	200	0.375	1800	3.70	349.1	$-575.4$	639	0.02	26.6
TG8	8	0	100	6.25	1400	4.0	425.8	$-509.4$	458.6	0.0001	80.0
TG9	9	0	175	1.95	1500	3.90	275.4	$-584.7$	523.8	0.04	28.8
TG12	12	0	325	0.834	1200	4.5	532.6	$-355.5$	338.0	0.20	20.0

Table 7.

POPF solution for load variation throughout the course of a typical day for Case 1 (IEEE 57-bus test system).

Hour	Loadpercentage	Bus1 $P_{TG1}$   (MW)	Bus2 $P_{TG2}$ (MW)	Bus3 $P_{TG3}$ (MW)	Bus6 $P_{TG6}$ (MW)	Bus8 $P_{TG8}$ (MW)	Bus9 $P_{TG9}$ (MW)	Bus12 $P_{TG12}$ (MW)	Totalgeneration (MW)	Generation cost ( $\mathrm{\$}/\mathrm{h}$ )	Emission $(\mathrm{t}/\mathrm{h})$
1	70	368.19	98.07	40	96.16	100	94.64	100	897.06	3489.7043	0.9354
2	70	339.22	99.33	40	98.44	100	99.16	117.72	893.87	3471.4472	0.8293
3	70	339.38	98.52	40	98.09	100	90.36	94.86	861.21	3473.6615	0.8298
4	70	340.15	100	40	99.15	100	97.51	117.38	894.19	3470.2188	0.8329
5	75	423.39	89.17	40	96.79	100	95.77	116.98	962.1	3809.4761	1.1917
6	80	422.98	94.99	40	99.95	100	96.13	171.19	1025.2	4141.7948	1.2283
7	80	431.14	94.2	40	98.96	100	99.1	162.58	1026	4147.8942	1.2626
8	90	512.21	99.95	65.58	99.24	100	98.88	186.05	1161.9	4925.2265	1.7362
9	90	505.2	99.15	58.59	98.12	100	97.49	201.25	1159.8	4929.4004	1.706
10	90	505.52	99.71	40	98.12	100	98.33	217.37	1159.1	4933.9458	1.7266
11	100	571.92	99.83	71.14	95.62	100	96.85	257.8	1293.2	5816.3987	2.2033
12	110	572.09	99.22	87.81	99.82	100	99.61	361.36	1419.9	6824.7797	2.3933
13	110	571.46	99.68	77.55	99.88	139.79	95.68	334.27	1418.3	7138.4413	2.3537
14	100	568.71	99.34	75.32	98.45	100	99.58	251.96	1293.4	5797.4519	2.177
15	100	569.93	99.95	67.74	99.95	100	99.82	254.99	1292.4	5784.9294	2.1895
16	90	506.14	97.77	46.41	99.04	100	99.82	210.08	1159.3	4926.097	1.722
17	90	512.21	99.95	65.58	99.24	100	98.88	186.05	1161.9	4925.2265	1.7362
18	110	572.09	99.22	87.81	99.82	100	99.61	361.36	1419.9	6824.7797	2.3933
19	110	571.46	99.68	77.55	99.88	139.79	95.68	334.27	1418.3	7138.4413	2.3537
20	110	573.59	99.66	62.33	97.63	152.71	98.23	333.19	1417.3	7294.558	2.3743
21	110	572.43	99.65	70.42	97.55	100	96.88	383.39	1420.3	6891.3811	2.4364
22	90	505.2	99.15	58.59	98.12	100	97.49	201.25	1159.8	4929.4004	1.706
23	80	422.98	94.99	40	99.95	100	96.13	171.19	1025.2	4141.7948	1.2283
24	70	339.38	98.52	40	98.09	100	90.36	94.86	861.21	3473.6615	0.8298
Average										5112.5000	1.6823

The study is further extended in Case 2 as shown in Table 8 in which the present author have considered the emission cost minimization objective function to yield the optimal value as 0.7871 ( $\mathrm{t}/\mathrm{h}$ ).

Table 8.

POPF solution for load variation over the course of a typical day for Case 2 (IEEE 57-bus test system).

Hour	Loadpercentage	Bus1 $P_{TG1}$ (MW)	Bus2 $P_{TG2}$ (MW)	Bus3 $P_{TG3}$ (MW)	Bus6 $P_{TG6}$ (MW)	Bus8 $P_{TG8}$ (MW)	Bus9 $P_{TG9}$ (MW)	Bus12 $P_{TG12}$ (MW)	Totalgeneration (MW)	Generation cost ( $\mathrm{\$}/\mathrm{h})$	Emission( $\mathrm{t}/\mathrm{h})$
1	70	101.35	99.98	125.83	99.44	168.89	95.98	192.81	884.28	4892.4619	0.4087
2	70	116.01	99.57	129.36	98.59	143.58	96.6	200.98	884.69	4517.7165	0.408
3	70	102.62	99.45	131.37	98.44	154.26	98.63	198.96	883.73	4697.7078	0.4059
4	70	120.12	99.3	127.76	96.83	151.86	97.95	190.59	884.41	4606.1208	0.407
5	75	120.09	98.44	138.51	99.96	175.16	97.27	217.83	947.26	5384.5992	0.4683
6	80	148.64	99.9	138.02	99.11	171.8	98.77	255.14	1011.4	5665.2701	0.5456
7	80	147.04	93.68	137.49	99.87	197.45	99.25	237.15	1011.9	6115.7267	0.5474
8	90	172.16	99.6	138.22	95.92	224.51	99.3	306.9	1136.6	7498.4217	0.7289
9	90	162.49	98.49	137.57	99.31	218.29	98.27	322.03	1136.5	7416.2094	0.7314
10	90	165.57	99.5	139.94	97.77	222.32	99.76	312.5	1137.4	7486.1646	0.7274
11	100	189.62	99.97	139.91	98.09	293.16	99.26	344.39	1264.4	10217.2251	0.9608
12	110	223.35	99.88	139.28	99.95	322.15	97.58	408.62	1390.8	12079.2165	1.2407
13	110	218.12	93.51	139.35	98.11	334.03	97.91	409.33	1390.4	12530.0208	1.2559
14	100	194.91	98.89	138.66	99.79	281.85	98.8	350.93	1263.8	9860.2829	0.9594
15	100	189.62	99.97	139.91	98.09	293.16	99.26	344.39	1264.4	10217.2251	0.9608
16	90	172.16	99.6	138.22	95.92	224.51	99.3	306.9	1136.6	7498.4217	0.7289
17	90	165.57	99.5	139.94	97.77	222.32	99.76	312.5	1137.4	7486.1646	0.7274
18	110	223.35	99.88	139.28	99.95	322.15	97.58	408.62	1390.8	12079.2165	1.2407
19	110	221.64	98.86	139.89	98.29	330.42	99.35	402.83	1391.3	12362.4411	1.243
20	110	218.12	93.51	139.35	98.11	334.03	97.91	409.33	1390.4	12530.0208	1.2559
21	110	228.31	98.92	136.73	96.38	328.42	98.65	405	1392.4	12283.6933	1.2546
22	90	165.57	99.5	139.94	97.77	222.32	99.76	312.5	1137.4	7486.1646	0.7274
23	80	147.04	93.68	137.49	99.87	197.45	99.25	237.15	1011.9	6115.7267	0.5474
24	80	147.04	93.68	137.49	99.87	197.45	99.25	237.15	1011.9	6115.7267	0.5474
Average										8147.7	0.7871

The authors have examined a multi-objective function Case 3 that concurrently minimizes generating cost and emission by recording the same for a full 24-hour window in order to support the superiority and adaptability of the suggested AHA (refer Table 9). Critical examination reveals that the considered algorithms have successfully been able to minimizes the objective function to 5257.8 ( $\mathrm{\$}/\mathrm{h}$ ) and 1.1815 ( $\mathrm{t}/\mathrm{h}$ ), respectively.

Table 9.

Various case-studies investigated in this article for IEEE 118-bus system.

Case	Singleobjective	Multi-objective	Considered objectives	Constraints	Testsystem
10	$✓$		Total generation cost minimization incorporating valve point effects for thermal, hydro, and wind–PV energy	.5 Equality and inequality	.5 IEEE 118-bus system
11	$✓$		Emission minimization incorporating wind, PV and hydro energy
12		$✓$	Simultaneous minimization of cost and emission
13	$✓$		Minimization of stability index for thermal, hydro, and wind–PV energy

The statistical representation of the computed results have been depicted in Table 10. Each of the 30 trials used to obtain the statistics results are shown in Table 10. The comparison has been carried out to justify the robustness of the proposed AHA techniques in comparison to other well established heuristic approach. From the referred table (Table 10), in Case 1 is been observed that there is reduction of 0.27% and 0.33% in the generation cost by applying the AHA with respect to CRO and GWO. A similar reduction of 1.74% and 3.078% in the emmision dispatch has been depicted in Table 10. While implementing multi-objective function also, there is considerable reduction of 0.25% and 0.35% in the fitness function while incorporating our proposed AHA technique in comparison to CRO and GWO techniques. Also, since the standard deviation (shown in Table 10) is the lowest for Cases 1 through 3, where AHA is used, it can also be said that AHA has the best degree of accuracy. Additionally, box plots are generated using this statistical data in Figures 6 and 7. The box plot makes it easier to compare vast amounts of data. The lowest, maximum, and average values are significantly extremely closed when AHA is used in place of CRO and GWO, demonstrating the robustness of the suggested AHA approach. Table 11 presents the results of the one-way ANOVA test for Case 3, which further validates the performance superiority of the suggested AHA. The ANOVA findings for Case 3 are shown graphically in Figure 8.

Figure 6.

Box plot for Case 1.

Figure 7.

Box plot curve for Case 2.

Figure 8.

ANOVA plots corresponds to Case 9.

Table 10.

An overview of the updated IEEE 118-bus system 3 under consideration.

Element	Size	Description
Bus	118	Taher et al. (2019)
Transmission line	186	Taher et al. (2019)
Thermal generator	54	Swing Buses:69.
Wind generator	1	Buses:12.
Solar PV unit (SPV)	1	Bus:15
Hydro	1	Bus:18
On load tap changer	19	$TC{T}_{8,5}$ , $TC{T}_{26,25}$ , $TC{T}_{30,17}$ , $TC{T}_{38,37}$ , $TC{T}_{63,59}$ , $TC{T}_{64,61}$ , $TC{T}_{65,66}$ , $TC{T}_{68,69}$ , and $TC{T}_{81,80}$
Optimization parameter	131	$PG$ :57; $VG$ :57; $TCT$ :9; $SC$ :13
Load capacity		4242.0 MW, 1439.0 MVAr
Load voltage limits	64	[0.94–1.06] $\mathrm{p}.\mathrm{u}.$
Shunt device	12	$S{C}_{34}$ , $S{C}_{44}$ , $S{C}_{45}$ , $S{C}_{46}$ , $S{C}_{48}$ , $S{C}_{74}$ , $S{C}_{79}$ , $S{C}_{82}$ , $S{C}_{83}$ , $S{C}_{105}$ , $S{C}_{107}$ , and $S{C}_{110}$

Table 11.

Ideal control variable configurations for the POPF’s IEEE 118-bus system to account for variations in demand during a typical day.

Generation(MW)	Hour wise generation schedule
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
$\mathrm{\%}Load$	70	70	70	70	75	80	80	90	90	90	100	110	110	100	100	90	90	110	110	110	110	90	80	70
$P_{TG1}$	11.2405	66.052	80.7048	20.7852	86.0481	66.3023	24.6091	59.6969	53.2103	95.714	44.3806	49.3605	99.0786	61.0994	12.8736	7.064	63.5261	39.2727	81.0854	75.4577	80.736	84.5129	70.399	23.1641
$P_{TG4}$	68.7443	99.1294	22.773	71.0743	23.617	84.744	99.1994	43.6422	30.3425	76.664	0.5419	28.4966	9.9146	27.5994	80.969	61.8629	51.0692	94.4458	58.6813	14.4274	5.0294	17.4042	46.1404	28.1596
$P_{TG6}$	4.585	9.9988	48.7872	1.1815	65.1911	59.7868	62.2884	96.9346	30.5106	91.5151	69.825	34.195	62.7182	1.4366	60.4522	59.3976	31.3372	13.9972	4.3046	65.7777	91.1477	26.0399	14.4993	5.48
$P_{TG8}$	94.9439	39.6687	54.4371	61.1403	80.9454	10.8646	7.8583	47.4907	20.7008	79.4303	61.5531	56.7655	75.4555	86.2749	82.7519	70.4619	2.9545	34.145	2.9184	0.9496	29.9844	32.5238	53.4304	57.7485
$P_{TG10}$	527.579	395.7002	372.5184	290.2117	170.5361	232.1711	261.3647	446.4796	86.1653	338.4409	348.9526	66.8574	354.2499	180.6367	371.8724	482.517	300.5099	460.3331	363.1386	166.5688	38.9802	268.8059	128.6999	184.3057
$P_{TG12}$	64.2136	153.626	12.5585	104.3984	0.8007	108.2759	85.1436	32.1959	59.1629	103.3422	163.0721	184.1288	184.3576	148.0993	110.3292	117.6662	184.8117	35.0878	183.2498	81.7451	158.734	21.8976	42.2202	159.9522
$P_{TG15}$	14.9827	41.9514	69.3023	49.6773	6.6785	17.7822	99.1713	92.6753	67.542	51.6976	95.1531	46.0611	52.9481	12.3539	36.4998	21.9039	92.7304	98.3799	12.5804	67.309	26.2602	94.4422	58.7177	30.233
$P_{TG18}$	77.3466	29.3041	24.6861	48.5717	60.1355	94.3721	99.6231	66.1076	16.3245	53.0826	59.1965	77.0099	94.7768	70.3772	63.4729	76.6693	39.9017	82.5833	59.5873	66.3536	25.847	18.2583	9.9517	47.1087
$P_{TG19}$	9.0768	78.6615	50.2496	55.8265	11.7558	26.2843	22.2062	72.7168	54.7595	29.6428	28.3514	62.0743	86.0139	44.4697	99.6978	55.6299	83.2538	33.4084	13.2926	70.4138	92.4769	71.3038	28.4559	64.1512
$P_{TG24}$	80.2027	42.9486	42.9443	20.3746	75.3641	7.8307	76.2734	0.2993	54.1252	61.5437	70.552	68.0743	94.442	33.965	33.9596	13.6879	65.4077	25.197	33.878	66.2009	63.3795	19.1087	34.2598	56.5579
$P_{TG25}$	245.789	306.5803	314.2689	283.8421	44.8094	50.2344	8.8856	155.0139	39.9385	57.6954	155.5266	202.6465	78.067	260.3389	107.1129	187.2369	248.9901	294.2652	138.5743	289.7533	290.0915	301.7905	140.5847	315.3144
$P_{TG26}$	109.64	386.1443	145.1105	314.6657	225.5441	163.0855	101.5668	411.1951	244.5956	164.8874	336.777	370.9978	206.3422	226.611	103.0712	289.8494	116.0269	388.1883	184.8297	284.3572	355.3891	315.5295	75.7144	303.6927
$P_{TG27}$	54.7096	25.4044	27.3096	29.2744	6.1613	33.7783	40.053	55.0168	10.0506	57.0656	78.0829	95.3277	86.5074	84.3366	76.4784	67.9608	17.095	11.4347	94.6117	12.2992	60.2011	34.5874	17.3249	55.2382
$P_{TG31}$	8.7355	89.0566	74.9021	85.1593	45.4539	13.8698	24.9533	17.8953	71.3394	73.2156	28.8661	47.6169	57.5313	70.8925	61.5543	72.2352	40.6418	15.7279	102.8201	36.2319	82.3495	14.6818	46.5648	17.0787
$P_{TG32}$	90.9368	63.0839	93.4263	96.6682	13.1993	86.9823	18.0996	28.165	13.4361	73.5139	84.2769	41.4615	89.8967	68.4677	46.1009	64.3322	10.3255	77.0674	43.6451	16.7424	73.6974	83.8245	9.7951	89.2884
$P_{TG34}$	73.161	15.6452	8.5499	19.8039	36.2675	11.261	49.2039	11.4828	99.6698	65.6025	34.2373	60.2349	79.9412	84.1243	8.7787	79.4226	96.6483	85.0342	94.5825	58.0704	24.7113	79.647	88.8307	75.9298
$P_{TG36}$	13.9725	19.6704	16.2193	20.4421	56.2807	93.6504	85.6964	58.0239	3.2452	71.0306	51.2355	88.4523	49.1009	52.8918	62.6498	78.1016	97.0582	81.9757	42.7095	96.7262	85.8282	52.5594	34.6312	55.2566
$P_{TG40}$	46.0971	85.2444	12.8909	65.6111	62.2081	7.2951	32.8411	72.0171	80.1402	67.1863	41.3126	74.0073	66.7565	18.1992	46.4599	88.958	24.6786	1.0416	99.0869	32.5447	25.631	89.8789	10.6745	71.6009
$P_{TG42}$	6.7372	83.1832	57.3926	5.1186	91.9364	86.831	12.327	27.0168	54.3803	91.9242	10.6136	17.1652	98.2142	65.346	7.3477	91.9217	73.6275	28.1894	30.9188	6.8361	39.929	68.1942	88.9696	84.9559
$P_{TG46}$	105.2141	58.6131	14.4383	60.7784	41.8761	92.8043	48.883	39.6485	111.3406	75.8603	8.0837	20.4047	89.7302	91.0541	31.6083	115.7198	75.9426	58.4833	31.2678	19.2277	38.3526	117.3652	107.6165	114.9821
$P_{TG49}$	231.0683	195.093	92.9914	200.206	243.6252	138.6275	289.5275	77.6784	98.6846	37.7802	182.5392	75.644	104.1956	293.6356	172.1957	165.4211	5.1465	127.6342	153.1981	75.7281	107.693	117.5	263.8428	136.7198
$P_{TG54}$	17.6624	134.4995	133.3043	52.1123	62.2419	143.1099	65.8207	106.1601	78.3233	83.1279	129.5866	102.0057	89.2795	76.7994	60.4164	120.536	119.7207	13.1645	15.7282	4.1376	133.4702	9.4654	96.3849	80.2503
$P_{TG55}$	99.8928	43.4587	92.4744	94.1399	68.0742	49.77	37.9792	91.9205	73.4803	21.6977	86.5558	82.5111	80.3784	38.9128	35.1605	21.4577	50.4771	25.5575	17.7972	26.6177	55.7618	29.718	23.8959	24.6772
$P_{TG56}$	34.7732	19.3046	49.9432	10.2176	82.6112	26.8991	51.9972	69.719	67.0279	59.2571	71.3912	68.7857	3.5847	8.9289	83.6928	33.2845	99.6497	61.9684	80.6658	3.3429	71.1876	96.8077	76.1264	62.0243
$P_{TG59}$	118.7359	80.7541	53.4537	227.1397	107.1978	86.3841	154.0033	171.4008	215.6204	116.3099	42.6511	78.3865	164.6983	207.3083	208.0621	39.9895	139.0022	45.5679	126.591	217.8472	246.9359	106.212	47.0876	84.788
$P_{TG61}$	46.046	21.4695	157.4135	188.8955	35.1952	19.0452	168.7083	156.5291	40.1689	153.9894	48.9164	15.2701	128.2025	140.981	239.724	252.1678	64.8631	86.5847	107.3682	83.6952	248.3606	245.0176	159.3217	112.9497
$P_{TG62}$	2.9103	94.0659	77.1696	75.1995	4.6269	30.7068	94.3029	21.4219	77.5601	8.9473	3.8311	46.059	89.2312	26.0005	0.5458	36.553	86.056	23.2377	30.2515	93.2667	55.791	95.323	93.174	73.4977
$P_{TG65}$	195.7357	114.4314	11.6195	144.0921	79.6259	238.2045	21.8439	387.2427	469.6599	327.5026	356.4796	284.3207	350.7492	452.496	349.197	196.0042	29.5504	91.8523	421.8545	319.5727	335.5547	486.4733	29.8754	329.9499
$P_{TG66}$	69.2012	207.8736	350.5055	101.1464	242.3215	365.6944	293.977	385.1622	365.8691	249.9472	80.8663	254.1989	481.0067	223.0817	382.5161	452.4462	366.9754	351.0876	437.9484	318.135	247.3391	117.5038	338.4799	475.6036
$P_{TG69}$	44.2389	93.6067	64.6596	54.3518	61.9589	19.2919	62.8526	2.9924	13.9337	21.4211	55.0803	43.163	49.3683	24.815	73.1806	99.5762	13.898	30.5338	92.9316	60.9192	51.0343	87.5738	89.4736	66.0588
$P_{TG70}$	35.8318	75.1917	40.8826	96.1587	64.1782	25.5297	39.059	89.1789	98.719	98.34	13.1302	53.213	25.8198	51.5626	69.8929	4.5294	30.721	51.2041	2.207	55.5728	24.9237	43.1608	19.5067	3.3109
$P_{TG72}$	4.3683	93.5667	82.9391	65.2989	9.2046	17.1739	61.46	98.667	80.3476	16.2383	46.1225	78.0719	35.0993	66.7673	84.7345	53.0542	1.1798	58.8685	96.6755	88.8356	19.8027	32.8788	90.2497	83.0773
$P_{TG73}$	86.0285	78.1135	36.3659	37.7664	47.5335	85.5457	75.0334	4.7102	61.6038	23.1009	91.8756	94.1157	40.2786	14.7639	44.4532	11.5163	65.7154	10.2312	84.2968	99.109	65.6799	69.3523	48.0677	65.37
$P_{TG74}$	77.6849	2.7666	75.1436	50.3147	49.7621	8.8621	80.6496	91.4817	63.1457	41.7733	68.1041	22.3236	68.3733	56.9395	22.5437	11.7934	58.1179	17.194	40.2337	51.1866	14.1127	8.1621	34.9151	80.9957
$P_{TG76}$	20.2813	28.0077	88.6031	70.8727	89.3437	62.4264	79.9707	31.2405	93.0004	74.2548	72.6198	52.7708	17.7576	71.0787	30.9857	50.77	39.2744	42.6223	21.4615	63.9414	70.8597	62.4016	36.4052	61.1916
$P_{TG77}$	95.5214	96.3895	42.8159	39.3562	31.0212	82.7371	62.2802	72.7	83.8469	32.425	54.9197	77.9284	59.3497	9.8042	16.3704	46.5065	40.2705	81.4074	80.8974	94.8912	72.3318	57.0825	77.5457	12.0557
$P_{TG80}$	37.4613	517.2836	210.6282	420.954	567.5043	494.8779	178.0363	130.1323	454.9121	282.6562	511.2422	258.51	424.5601	169.384	486.9667	495.7761	401.9213	490.1709	92.6791	44.5141	272.5405	322.2267	407.9818	426.1519
$P_{TG85}$	67.7971	15.8176	25.5016	25.7308	16.9325	29.4361	93.7314	55.591	53.8349	37.8102	90.0552	29.99	31.6371	91.0294	94.3297	69.0468	27.4826	58.5644	54.3807	49.413	57.366	32.4919	91.6704	9.9068
$P_{TG87}$	99.2068	45.3385	25.9006	80.89	89.9024	89.4206	80.919	4.8031	24.8167	45.0362	13.4034	92.2528	6.8842	94.0201	43.9429	68.256	103.7407	78.441	18.1612	12.3857	14.3491	2.8363	43.6979	7.7696
$P_{TG89}$	702.5422	125.5069	68.2311	439.2687	123.4257	269.2258	522.0046	5.4848	114.1633	419.4395	254.8331	607.6195	605.7701	699.2078	652.3241	463.6627	657.5614	588.1715	110.6766	607.3403	490.6141	331.8993	568.664	515.6556
$P_{TG90}$	13.3559	46.0114	56.6987	54.7587	94.2727	40.5239	1.0902	39.6054	48.31	88.8244	80.9171	33.9216	78.8091	95.7408	64.5963	41.1916	85.8933	45.006	34.0847	44.1557	67.5743	48.8868	64.989	51.0585
$P_{TG91}$	22.4969	55.1971	20.8826	38.2554	19.2731	20.0288	69.8932	17.4924	1.1877	99.8879	49.7499	28.5597	99.2442	44.9563	59.0866	39.3002	99.0383	6.76	54.5024	14.8873	48.8105	17.4697	40.7774	11.2697
$P_{TG92}$	80.0882	9.9831	19.2151	90.8125	18.0722	89.9248	95.2978	57.3468	8.689	23.4511	81.6624	2.3535	53.9445	28.1648	66.3392	46.4734	87.4064	60.2768	9.161	73.1159	54.2026	92.379	33.447	18.3017
$P_{TG99}$	54.5468	62.6834	42.5292	97.2445	82.4622	38.6859	76.0779	9.5668	39.6937	66.3759	98.6012	12.6616	73.0068	99.158	92.4386	97.977	29.7676	63.3183	58.6737	65.7472	47.6207	77.3813	48.1779	83.4862
$P_{TG10}$	041.149	199.3239	330.6888	123.011	274.0577	222.5198	98.4548	322.1982	289.7725	42.5284	227.7209	171.6627	342.0886	231.3608	18.2774	66.5627	312.432	263.4813	297.0299	22.4879	29.5768	255.0954	164.1525	127.2696
$P_{TG13}$	08.958	14.9106	14.6485	106.1644	73.9904	105.3412	112.1873	51.326	114.9436	14.5594	113.3576	125.6172	132.8988	64.2364	34.2155	135.0174	137.8604	56.9041	3.4702	85.2705	87.3136	118.2176	134.7102	135.3933
$P_{TG104}$	86.0992	67.0081	78.2819	18.4355	62.2189	77.885	39.7707	57.8808	81.257	43.5951	84.1238	31.5148	33.2866	79.7096	29.9372	96.7035	68.069	61.2073	86.1441	64.0886	92.662	77.6505	20.9054	98.2444
$P_{TG105}$	80.2531	89.9535	83.3919	17.3689	83.8346	31.4967	21.5645	16.559	12.0384	66.1424	49.8963	62.2818	86.2035	16.3601	79.0657	55.9467	79.2414	51.7713	35.829	84.6152	74.0343	35.6088	23.5949	9.1943
$P_{TG107}$	38.1086	56.5191	41.6541	18.4942	55.6702	38.6426	96.5993	8.6968	34.6126	53.5883	22.1412	69.2219	52.1874	42.4257	34.5036	13.5879	89.6668	22.9589	20.8335	63.4645	52.7992	29.6997	9.5329	21.5791
$P_{TG110}$	44.5748	99.975	39.5226	79.2291	75.2762	91.0767	26.7849	13.7904	8.2347	49.7506	96.2697	96.7873	76.8041	46.6872	18.341	80.1637	42.5118	93.6511	57.187	51.6705	43.6194	67.2165	18.6403	69.3555
$P_{TG111}$	101.7111	48.8346	35.7078	22.0695	45.0772	48.6797	51.8225	88.478	96.081	22.9179	70.1694	109.6387	115.735	83.2519	28.5706	12.0052	88.1863	104.1644	123.5438	83.7578	23.6473	74.027	7.9075	15.5877
$P_{TG112}$	39.1408	41.1965	37.7596	39.6618	57.5628	83.8925	72.9235	22.3419	81.7608	54.7157	78.8783	87.0373	92.8023	57.3537	50.3766	82.8452	3.3363	36.6453	26.4532	6.9024	87.622	4.3604	72.497	98.0946
$P_{TG113}$	90.5235	33.6984	62.6827	48.9496	70.4278	47.4241	75.4042	21.0982	2.073	77.3577	24.0908	44.0037	55.4892	13.2807	1.8369	45.728	60.3461	35.7714	96.0413	37.0362	66.9515	96.4218	80.8089	26.141
$P_{TG116}$	358.0749	465.8836	679.3527	357.1592	745.3082	344.026	505.2721	160.694	80.7865	391.8555	414.5797	279.8054	81.7077	659.8407	443.74	265.6999	246.8658	465.0066	169.9792	628.5199	95.4691	758.955	682.3782	562.3381

Observation of IEEE 118-bus system

Additionally, the IEEE 118-bus test system has been taken into consideration for validation in order to show the resilience and broader application of the suggested optimization methodology. Four cases have been taken into consideration in this research project, as shown in Table 9. These comprise two different objective functions: emission cost minimization (Case 11) and a single objective function that minimizes overall fuel cost with valve point effects (Case 10). Moreover, a multi-objective function (Case 12) that simultaneously minimizes generation cost and emission, followed by a voltage stability index at Case 13 is considered. The study includes the voltage stability index (VSI) to evaluate the system’s stability margin under various loading and renewable penetration conditions, in addition to the standard goals of generation cost minimization, emission reduction, and the combined multi-objective formulation. The different data for the IEEE 118-bus system for the POPF problem under examination are displayed in Table 10. The system consists of 118 buses and 80 branches. In the IEEE 118-bus system under consideration, 12 compensating devices have been commissioned, and a total of 19 tap changing transformers located at branches 8-5, 26-25, 30-17, 38-37, 63-59, 64-61, 65-66, 68-69, and 81-80 are used. Here 4242.0 MW, 1439.0 MVAr load has been set as the load demand. The efficacy and dependability of the suggested AHA-based strategy in large-scale and intricate power systems are confirmed by this comprehensive investigation. The parameters for the IEEE 118-bus has been illustrated in Table 11 and the computed outcomes for IEEE 118-bus for three different cases has been shown in Table 12. A thorough probabilistic modeling methodology for RESs, particularly solar and WP, has been incorporated into the sensitivity analysis of the IEEE-57 bus system under Case 7 in Table 13. The research assesses how stochastic fluctuations in renewable energy affect stability indices, operational costs, and system performance. The study illustrates how uncertainties in wind speed and solar irradiance affect the overall system behavior by evaluating several probabilistic scenarios. This shows how the suggested optimization approach can maintain dependable and effective operation under varying renewable conditions.

Table 12.

Simulation results of POPF for Cases 10, 11, 12, and 13 (IEEE 118-bus test system).

	Case 10	Case 11	Case 12	Case 13
Fuel cost ($/day):	8319066.207	11769807.84	10269012.61	10113456.9
Wind cost ($/day):	1152.35	1152.35	1152.35	1152.35
Solar cost ($/day):	1963.2885	1963.2885	1963.2885	1963.2885
Total cost ($/day):	8322181.846	11772923.48	10272128.25	10116572.54
Emission (t/day):	314.6943	251.413	289.4229	302.5643
Stability index:	NA	NA	NA	0.897

Table 13.

Sensitivity analysis of the IEEE 57-bus system with probabilistic modeling of renewable energy sources (wind and solar) considering case 7.

Weibull PDF parameters	Input parametervariation (%)	Average objectivecost ($/h)	Output variation (%)	Weibull PDF parameters	Input parametervariation (%)	Average objectivecost ($/h)	Output variation (%)
$\alpha$ =10; $\beta$ =2	0%	4808	0	$\zeta$ =6; $xi$ =0.6	0%	4808	0
$\alpha$ =9; $\beta$ =1.8	$-$ 10%	4796.742	0.2347	$\zeta$ =5; $xi$ =0.54	$-$ 10%	4782.345	0.5365
$\alpha$ =8; $\beta$ =1.6	$-$ 20%	4775.443	0.6818	$\zeta$ =4.8; $xi$ =0.48	$-$ 20%	4767.453	0.8505
$\alpha$ =6; $\beta$ =1.2	$-$ 40%	4753.783	1.1405	$\zeta$ =3.6; $xi$ =0.36	$-$ 40%	4743.356	1.3628

Conclusion and scope of future work

In order to build a conventional IEEE 57-bus system, the present research aims to address the POPF problem, which entails combining wind farms, solar plants, hydro power generation in different combinations. For any Test System, fuel cost minimization and emission minimization has been considered the main function. Beyond the particular test methods under consideration, the study’s conclusions have wider ramifications. Reliable and effective probabilistic optimization frameworks are becoming more and more important as power systems throughout the world shift toward greater proportions of renewable energy to satisfy carbon reduction targets and lessen reliance on fossil fuels. Large linked grids, microgrids, and hybrid renewable energy parks in various geographic locations can all benefit from the scalable and flexible solution offered by the suggested AHA-based POPF strategy. The approach promotes increased operational dependability, cost effectiveness, and lower environmental emissions by skillfully controlling the risks related to wind, solar, and hydropower generating. This immediately supports global sustainability goals, such as low-carbon electricity markets and cleaner energy transitions. Therefore, system operators, legislators, and energy planners seeking resilient and ecologically conscious power system operation can benefit from the research’s findings. The study’s primary findings are as follows:

For the first time, the POPF problem is solved using a novel meta-heuristic method known as the artificial hummingbird algorithm (AHA). Three distinct test systems are taken into consideration in the study.

In the first test scenario, the suggested AHA is used to assess a traditional POPF problem with solely thermal generators in order to optimize overall generating cost and emissions. The suggested AHA optimization strategy works better than the other approaches this study looked at, showing increased dependability and efficiency.

RESs, such as wind, solar PV, and hydroelectric power, partially replace the thermal generators in the second test system. The AHA does not produce adequate results in this arrangement.

With the third test system—where RESs are introduced alongside the original number of thermal generators—the most advantageous outcomes are attained in terms of both generating cost and emissions.

According to the simulation findings, the suggested AHA reduces the computing load while managing the POPF problem’s nonlinearity. When RESs are connected with the IEEE 57-system, the generating cost and emissions are reduced, as shown in Figures 12 and 14.

The statistical approach and ANOVA plot clearly justifies the robustness and efficiency of the proposed AHA techniques.

Thus, it can be said that the AHA has potential for use in future research applications and is well suited for large-scale power systems.

With comprehensive uncertainty modeling, penalty-based cost treatment, and statistical robustness evaluation, this study is the first to show AHA for a multi-renewable POPF issue, providing better cost and emission results than previous optimization techniques. Nonetheless, the following research might be expanded upon:

The proposed method might be applied in a real-world setting.

Future research may also look into integrating biomass, EV, and tidal energy into the system.

As the intended system is expanded, further nonlinearities could appear.