Solution of optimal power flow problem in modern power systems using circulatory system-based optimization approach

Abstract

To address the growing complexity and stochastic behavior of modern power grids, a comprehensive strategy is proposed in this research to enhance the performance of a large-scale transmission network connected to a hybrid conventional and renewable generator system. While various methods have been applied to the Optimal Power Flow (OPF) problem, few effectively balance economic, technical, and environmental constraints under high-penetration renewable scenarios. This paper introduces a novel application of Circulatory System-Based Optimization (CSBO) on the adjusted IEEE 30 and 57 bus frameworks, with three scenarios taken into consideration for each system, starting with the minimization of the overall generation costs, followed by the reduction of the real power outages, and lastly, the minimization of generation costs including pollution effects. The proposed algorithm is compared alongside Mountain Gazelle Optimization (MGO), Artificial Rabbits Optimization (ARO), and Dwarf Mongoose Optimization Algorithm (DMOA), besides other algorithms highlighted in previous studies. The outcomes demonstrate the efficacy and superiority of the suggested CSBO algorithm over competing methods in terms of quality-feasible solutions and computational time required. For instance, the total generation costs were minimized to $787.359/h for IEEE 30-bus with a CPU time of 295.375 s, which translates to an 11% time frame reduction in comparison to MGO. Nevertheless, each metaheuristic technique presents distinct advantages in OPF with the presence of renewables: DMOA excels in broad search capability, MGO balances exploration with exploitation, ARO attains a fast convergence rate, and CSBO gives high precision and offers low computational time, making these algorithms vital for managing the uncertainties of Renewable Energy Resources (RERs).

Keywords

transmission network optimal power flow metaheuristic algorithm renewable energy sources circulatory system-based optimization computational time

Introduction

Motivation

The share of RERs in the energy mix has increased exponentially in recent years due to massive investments pouring into this sector and backed by government policies that aim to overcome our dependence on fossil fuels, safeguard the environment and public health, and ensure the security of supply and energy independence (Fellahi et al., 2024). Conventional power networks are dominated by thermal plants as the primary source of energy, and despite being a stable source of electricity, they release harmful emissions that contribute significantly to rising global temperatures and shifts in climate patterns, which leads to severe weather phenomena and intense heat conditions, flash floods, cyclones, and tornadoes, causing devastating damage economically and loss of lives. On the other hand, renewables hold the advantage of being clean, sustainable, and eco-friendly, but they are also unstable, unpredictable, and difficult to monitor (Syama et al., 2024), which may lead to interruptions in the electricity supply. An example regarding this matter is the unprecedented power outage that occurred across the Iberian Peninsula on April 28, 2025, and lasted for a couple of hours. While analysts suggested that the blackout was triggered by a voltage surge followed by a series of disconnections that shut down the entire grid. The official investigation has yet to conclude the root cause of the blackout. This incident exposed the vulnerability of electrical networks to withstand high penetration of solar and wind; it also represents an opportunity to upgrade the transmission infrastructure to accommodate the fluctuations of RERs and improve the grid’s stability and resilience (Goiana-da-Silva et al., 2025).

The intermittent nature of RERs was and still is the main obstacle preventing a worldwide radical shift towards renewables. To solve this problem, OPF is an essential aspect that needs to be addressed. It is utilized to enhance the grid’s power flow by reducing designated target functions while ensuring satisfactory power system performance. The integration of stochastic renewable sources into traditional grids significantly complicates the OPF problem, requiring a switch from traditional calculus-based methods to more robust metaheuristic approaches that can handle non-linear constraints to increase the contribution of wind, solar, and other renewable sources while also balancing the supply and demand and lowering the electricity bill (Fellahi et al., 2025). Researchers have been using classical methods for quite some time to tackle real-world optimization problems. These traditional techniques have fallen short against large and complicated situations, so scientists shifted their focus towards metaheuristic algorithms as an effective tool to identify near-optimal solutions in complex and intractable problems during a short period of time. These new heuristics are nature-inspired and can be effectively implemented in various domains and fields, including power system optimization and stochastic renewable energy integration into high-voltage transmission lines (Dokeroglu et al., 2019).

Related investigation

Recent studies conducted on OPF optimization were predominantly based on metaheuristic algorithms with thermal, wind, and solar as the common sources; for instance, Bird Swarm Algorithm (BSA) is proposed in Ahmed et al., 2021 to handle the OPF issue while taking into consideration load uncertainty and stochastic RERs’ power output; White Sharks Algorithm (WSA) (Ali et al., 2022) is implemented based on a single objective function to minimize the overall generation costs; Slime Mould Algorithm (SMA) (Mouassa et al., 2022) is adopted for a large-scale restricted optimization problem with non-linear characteristics to lower the operating costs of the grid. Authors in Mouassa et al. (2024) provided an alternative to stochastic OPF challenges while preserving the system’s stability using Dwarf Mongoose Optimizer (DMO). Self-adaptive Bonobo Optimizer (SaBO) (Kouadri et al., 2024) is employed on the altered IEEE 30-bus network and the Algerian 114-bus networks to guarantee an economical, stable, and dependable system; and the Giant Trevally Optimization (GTO) (Hashish et al., 2023) is recommended to tackle the uncertainty-based OPF problem, taking into account the ambiguity of RESs. Mayfly algorithm (Zhu et al., 2024) is utilized to identify the ideal settings of the generating plants, including voltage magnitudes. Grey Wolf Optimizer (GWO) (Krishna Reddy et al., 2024), Jellyfish Search Optimization (JSO) (Farhat et al., 2021), Salp Swarm Algorithm (SSA) (El-Fergany and Hasanien, 2020), Multi-Objective Search Group Algorithm (MOSGA) (Huy et al., 2022), Non-dominated Sorting Colliding Bodies Optimization (NSCBO) (Pulluri et al., 2024), and Teaching-Learning-Based Optimization (TLBO) (Sulaiman, Mustaffa and Mohd Rashid, 2021) are some of the many techniques implemented in the same context with differences regarding the objective functions and the type of network tested.

A modified and improved algorithm can have better performance than the original one with near-optimum results, for example, Khan, Wang, Habib, et al. (2024a) introduced an Improved Liver Cancer Algorithm (ILCA) for a stochastic OPF framework Khan, Wang, Jamal, et al. (2024b) utilized a modified Artificial Rabbit Optimizer (ARO) to enhance solution accuracy. Furthermore, researchers have explored and developed Runge-Kutta optimizers to manage the integration of TCSC and renewable sources (Ebeed et al., 2023). Adaptive Lightning Attachment Procedure Optimizer (ALAPO) has shown promising outcomes in addressing the uncertainties inherent in modern power systems (Adhikari et al., 2023). Enhanced COVID Optimization Algorithm (COVIDOA) (Albaaj and Kaplan, 2025) introduces changes to the frameshifting technique and mutation phases of the original coronavirus algorithm to enhance its search efficiency. Improved Equilibrium Optimizer (IEO) (Nguyen et al., 2022) where the exponential value is substituted by a function that is independent of iteration count. This adjustment to the IEO method enhances its exploration capability in comparison to EO. Improved Particle Swarm Algorithm (IPSO) (Ahmed, Osman and Korovkin, 2021) is a novel variant of the PSO method known as time-dependent non-linear acceleration coefficient, proposed as a parameter change strategy that significantly improves the algorithm’s performance. Other examples for this matter include Modified Genetic Algorithm (MGA) (Li et al., 2022), Modified Rao-2 (MRao-2) algorithm (Hassan et al., 2021), Modified Turbulent Water Flow-based Optimization (MTWFO) (Alghamdi, 2022b), Mutation-based PSO approach (Samakpong et al., 2022), Modified Levy Interior Search Algorithm (LISA) (Karthik et al., 2021), and Improved Levy Coyote Optimization Algorithm (LCOA) (Kaymaz et al., 2021).

The hybridization approach consists of two or more metaheuristic algorithms merged and cultivated together; this concept makes it possible to benefit from the overall advantages of optimization methods (Bouaouda and Sayouti, 2022); for instance, a multi-population gorilla troop optimizer with chaotic-quasi-oppositional-phasor logic (Jamal et al., 2024) was proposed to solve the OPF problem for IEEE 30 and 57 bus systems. Hybrid Artificial Ecosystem-based Optimization and Chaos Game Optimization (AEO-CGO) (Hassan et al., 2023), where the validation of the modified optimizer starts by evaluating its performance on widely recognized benchmark optimization functions, showing that it outperforms CGO and AEO, Hybrid Firefly and JAYA (HFAJAYA) (Alghamdi, 2022a). This problem entails optimizing variables like online capacity, generator output, power stability, and bus voltage to minimize production costs. Hybrid Particle Swarm Optimization with Grey Wolf Optimization (PSO-GWO) (Riaz et al., 2021). This approach incorporates a hybrid inequality constraint handling mechanism that maintains only feasible solutions without modifying the original objective function. Gaussian Bare-bones Levy and Circulatory System-Based Optimization (GBLCSBO) (Ghasemi et al., 2023). This method seeks to improve solution accuracy by augmenting solution diversity via an optimization process. Other algorithms include Artificial Hummingbird and Manta Ray Foraging Optimization (AHMRFO) (Hassan et al., 2024) and hybrid spotted hyena algorithm, quadratic approximation operator, and grasshopper optimization (Suresh Babu et al., 2025).

This paper’s power system is made up of a unique combination of energy resources. While wind and solar are the common duo in RERs, the insertion of hydropower assimilates the advantage of being consistently available. This hydroelectric unit is relatively small and is incorporated with one of the solar stations. Studies have already addressed this system, and the outcomes were favorable among academia; for example, multi-objective evolutionary algorithm based on decomposition (Biswas et al., 2018) where an economic environmental bi-objective non-linear framework aims to penetrate more RESs into the grid, taking into consideration the power network security, transmission lines capacity, and voltage bus limits. In the second example, the authors implemented Barnacles Mating Optimizer (BMO) algorithm on the altered IEEE 30 and 57 bus networks in the presence of Solar Photovoltaic (SPV), wind, and SPV-hydro units alongside traditional generators to lower the total generating costs, the active power losses, and the overall emissions (Sulaiman and Mustaffa, 2021). Another study was conducted using the same system and objective functions with a different optimization method, which is weighted means of vectors INFO. The results illustrated INFO’s ability to balance exploration and exploitation while avoiding premature convergence (Belagra et al., 2023). Despite the relative success of these algorithms, a significant gap remains in achieving high-precision results for large-scale hybrid networks (like the IEEE 57-bus) without incurring prohibitive computational costs or falling into local optima. Consequently, there is a critical need for optimization frameworks that can better navigate the high-dimensional search space of modern grids. This study addresses this need by introducing CSBO and leveraging its unique dual-flow dynamics to ensure robust convergence where traditional methods falter.

Contribution and organization

This research analyzes the implementation of a bio-inspired algorithm to solve the OPF problem in a large-scale system associated with a combination of thermal, wind, Solar Photovoltaic (SPV), and integrated SPV-small hydro units. CSBO was selected for this study due to its unique ability to balance exploration and exploitation through simulated pressure-flow dynamics, which addresses the convergence limitations of traditional metaheuristics in high-dimensional OPF problems. The suggested method is tested alongside Mountain Gazelle Optimization (MGO), Artificial Rabbits Optimization (ARO), Dwarf Mongoose Optimization Algorithm (DMOA), and six other methods in previous studies: Mouth Flame Optimization (MFO), Barnacles Mating Optimizer (BMO), Gorilla Troops Optimizer (GTO), Artificial Ecosystem-based Optimizer (AEO), weighted mean of vectors INFO, and Particle Swarm Optimization (PSO). The novelty of this study lies in the first-time application of CSBO to the multi-objective OPF problem. Unlike existing methods, CSBO’s unique pressure-flow logic provides a dual mechanism for exploration and exploitation that is uniquely suited for the high-dimensional challenges of hybrid energy systems. The primary contributions and originality of this paper can be summarized pointwise as below:

• Building upon previous and current state-of-the-art research to address the problems facing the electrical grid with the ever-increasing penetration of RESs.

• Offering CSBO method as an alternative to solve the OPF problem with its ability to withstand a vast, interconnected, and complex high-voltage system.

• Development of a comprehensive multi-objective optimization framework that simultaneously minimizes Total Generation Cost (TGC), Transmission Power Losses, and Environmental Emissions within a unified stochastic environment.

• Analyzing the outcomes of CSBO implementation regarding the power, voltage, and security constraints, besides the computational time needed for the simulation.

• A comparative analysis of the proposed algorithm with previous techniques is illustrated to show the superiority of CSBO over competing methods.

The core problem addressed in this research is the optimization of large-scale transmission networks under the uncertainty of hybrid energy sources. To this end, the study evaluates the CSBO algorithm across three distinct scenarios. The structure of the subsequent sections is as follows: Section two indicates the statement of the problem and the restrictions regarding the OPF problem, the target functions of each scenario, and mathematical models of all kinds of generators present in this system. The second section is dedicated to the optimization method (CSBO) and other algorithms, and the third section highlights the results obtained after the simulation with extensive analysis and interpretation, and finally, the conclusion in the fifth section.

Problem statement

Solving the OPF problem requires determining the control variable values to minimize a certain target function while maintaining security restrictions in check. The OPF problem is represented as (Abid et al., 2024b):

M i n f (x, y)

(1)

L i a b l e t o : {\begin{cases} g (x, y) = 0 \\ h (x, y) \leq 0 \end{cases}

(2)

where

f (x, y)

is the target function,

x

and

y

represent, respectively, the vectors of control and state variables,

g (x, y)

and

h (x, y)

denote, respectively, the equality and inequality restrictions. The variables under control are the active power and the voltage of the generators. While the state variables are the reactive power of generators, the magnitude of the voltage at load buses, and the apparent power of the lines.

Objective function (OF)

The target functions in this research are divided into three scenarios: reduction of overall generation costs, reduction of active power losses (APLs), and reduction of generation costs, including emission effects.

OF 1: Reduction of the total generation costs (TGCs)

The minimization of the TGCs requires reducing the price of each generator in the system to the absolute lowest cost as follows:

A. The price of thermal generators (TGs)

The cost of a TG’s fuel can be expressed as (Mouassa et al., 2023):

C_{T} = \sum_{i = 1}^{N_{T}} (m_{i} + n_{i} P_{T, i} + z_{i} P_{T, i}^{2}) ($ / h)

(3)

where

N_{T}

indicates the number of TGs,

m_{i}, n_{i}

and

z_{i}

denote the cost coefficients corresponding to the

i

-th TG producing power output

P_{T, i}

To calculate the cost function of a TG precisely, the effect of the valve point must be included as a sinusoidal function, added to the previous equation, and written again as follows (Sulaiman, Mustaffa and Mohd Rashid, 2021):

C_{T} = \sum_{i = 1}^{N_{T}} m_{i} + n_{i} P_{T, i} + z_{i} P_{T, i}^{2} + | d_{i} \times \sin (e_{i} (P_{T, i}^{m i n} - P_{T, i})) | ($ / h)

(4)

where

d_{i}

and

e_{i}

denote the effect of the valve point’s coefficients,

P_{T, i}^{\min}

is the minimum power related to the

i

-th TG.

B. The price of the wind turbine generator (WTG)

The direct price of the WTG in relation to its scheduled power is expressed as (Mouassa et al., 2024):

C_{w} (P_{w s}) = g . P_{w s}

(5)

where

C_{w}

represents the direct fee of the WTG,

P_{w s}

and

g

denote the scheduled power and the direct cost coefficient of the WTG, respectively.

To evaluate the uncertainties in wind power, two scenarios are distinguished. The first is when the power delivered from the WTG is higher than expected; in this case, the wind source is underestimated, and the surplus may be wasted. To cope with this problem, the power network operator must decrease the output from TGs; otherwise, they must incur a penalty fee associated with the excess amount.

The penalty fee of the WTG is expressed by (Farhat et al., 2021):

C_{P w} (P_{w a} - P_{w s}) = K_{P w} (P_{w a} - P_{w s}) = K_{P w} \int_{P_{w s}}^{P_{w r}} (P_{w a} - P_{w s}) F_{w} (P_{w}) d P_{w}

(6)

where

P_{w a}

and

P_{w r}

are the accessible and rated powers from the WTG, respectively.

K_{P w}

denotes the penalty fee coefficient of the WTG, and

F_{w} (P_{w})

is the Probability Density Function (PDF) of the power output from the WTG.

The second circumstance happens when the power supplied from the WTG is less than expected; in this case, the wind source is overestimated. To cope with this problem, the power network operator must ensure a spinning reserve to avoid interrupting the power network and provide the electricity needed for the consumers.

The reserve fee of the WTG is expressed by (Altun et al., 2024):

C_{R w} (P_{w s} - P_{w a}) = K_{R w} (P_{s w} - P_{w a}) = K_{R w} \int_{0}^{P_{w s}} (P_{s w} - P_{w}) F_{w} (P_{w}) d P_{w}

(7)

where

K_{R w}

is the reserve fee coefficient of the WTG.

The cost of the WTG is formulated as (Alghamdi, 2022a):

C_{W} = C_{w, j} (P_{w s}) + C_{R w} (P_{w s} - P_{w a}) + C_{P w} (P_{w a} - P_{w s})

(8)

where

C_{R w}

and

C_{P w}

denote the reserve and penalty costs of the WTG.

C. The price of the SPV generator

The direct price of the SPV unit in relation to its scheduled power is expressed by (Suman and Meena, 2021):

C_{S} (P_{S s}) = h . P_{S s}

(9)

where

C_{S}

represents the direct fee of the SPV unit,

P_{S s}

and

h

denote the scheduled power and the direct cost coefficient of the SPV unit, respectively.

Like wind, the SPV station provides uncertain output, and two scenarios are taken into consideration. The first one is underestimation; it is modeled using lognormal PDF.

The penalty fee of the SPV unit is expressed by (Ali et al., 2022):

C_{P s} (P_{S a} - P_{S s}) = K_{P s} (P_{S a} - P_{S s}) = K_{P s} . F_{S} (P_{S a} > P_{S s}) [E (P_{S a} > P_{S s}) - P_{S s}]

(10)

where

P_{S a}

is the available power from the SPV unit,

K_{P s}

refers to the penalty cost coefficient of the same unit,

F_{S} (P_{S a} > P_{S s})

is the probable surplus delivered from the SPV unit compared to the scheduled power

P_{S s}

E (P_{S a} > P_{S s})

represents the expected surplus from the SPV power above

P_{S s}

The second one is overestimation; the reserve cost is evaluated for the SPV unit using the following expression (Hassan et al., 2024):

C_{R s} (P_{S s} - P_{S a}) = K_{R s} (P_{S s} - P_{S a}) = K_{R s} . F_{S} (P_{S a} < P_{S s}) [P_{S s} - E (P_{S a} < P_{S s})]

(11)

where

K_{R s}

is the reserve fee coefficient of the SPV unit,

F_{S} (P_{S a} < P_{S s})

denotes the possible shortage of the SPV power that may occur compared with the scheduled power

P_{S s}

, while

E (P_{S a} < P_{S s})

is the expected SPV power produced below

P_{S s}

The price of the SPV generator is formulated as (Huy et al., 2022):

C_{S} = C_{S} (P_{S s}) + C_{R s} (P_{S s} - P_{S a}) + C_{P s} (P_{S a} - P_{S s}) ($ / h)

(12)

where

C_{R s}

and

C_{P s}

denote the reserve and penalty fees of the SPV unit.

D. The price of the SPV unit with a small hydropower generator

The system also includes a combination of an SPV unit with a small hydropower generator. The direct cost of the latter is (Belagra et al., 2023):

C_{S H} (P_{S H}) = C (P_{S} + P_{H}) = g_{S} P_{S} + h_{H} P_{H}

(13)

where

C_{S H}

denotes the direct price of the SPV-hydro unit,

P_{S H}

indicates the scheduled power of the solar hydro generator,

P_{S}

and

P_{H}

are the outputs of the SPV and hydroelectric generators, respectively.

g_{S}

and

h_{H}

represent the direct cost coefficients corresponding to the SPV and hydro generators, respectively.

The penalty fee of the combined SPV-hydro station is formulated as (Sulaiman and Mustaffa, 2021):

C_{P S H} (P_{S H a} - P_{S H}) = K_{P S H} (P_{S H a} - P_{S H}) = K_{P S H} . F_{S H} (P_{S H a} > P_{S H}) [E (P_{S H a} > P_{S H}) - P_{S H})]

(14)

where

K_{P S H}

represents the penalty fee coefficient for the combined SPV-hydro unit,

P_{S H a}

indicates the available power from the same unit,

F_{S H} (P_{S H a} > P_{S H})

denotes the probability of the SPV surplus being greater than the scheduled power

P_{S H}

E (P_{S H a} > P_{S H})

refers to the expected SPV-hydro power above

P_{S H}

The reserve cost of the combined SPV-hydro generator is expressed by (Biswas et al., 2018):

C_{R S H} (P_{S H} - P_{S H a}) = K_{R S H} (P_{S H} - P_{S H a}) = K_{R S H} . F_{S H} (P_{S H a} < P_{S H}) [P_{S H} - E (P_{S H a} < P_{S H})]

(15)

where

K_{R S H}

represents the reserve fee coefficient for the combined SPV-hydro unit,

F_{S H} (P_{S H a} < P_{S H})

denotes the probability of the SPV-hydro shortage being lower power than the scheduled one

P_{S H a}

E (P_{S H a} < P_{S H})

refers to the expected SPV-hydro power below

P_{S H}

The cost of the SPV-hydro generator is formulated as:

C_{S H} = C_{S H} (P_{S H}) + C_{R S H} (P_{S H} - P_{S H a}) + C_{P S H} (P_{S H a} - P_{S H})

(16)

where

C_{R S H}

and

C_{P S H}

are the reserve and penalty fees of the SPV-hydro generator, respectively.

The first objective function is the combined costs of the TGs ( $C_{T}$ ), the WTG ( $C_{W}$ ), the SPV ( $C_{S}$ ) and the SPV-hydro ( $C_{S H}$ ) generator; it can be articulated as:

F_{1} = T G C s = C_{T} + C_{W} + C_{S} + C_{S H}

(17)

OF 2: Reduction of active power losses (APLs)

The APLs illustrate the second objective function, which is modeled as (AlRashidi and El-Hawary, 2007):

F_{2} = A P L s = \sum_{k = 1}^{N_{L}} G_{k} [V_{i}^{2} + V_{j}^{2} - 2 V_{i} . V_{j} \cos (σ_{i} - σ_{j})]

(18)

where

N_{L}

is the number of transmission lines,

G_{k}

is the conductivity of line

k

V_{i}

V_{j}

and

σ_{i}

σ_{j}

denote the voltage magnitudes and the angles at buses

i

and

j

, respectively.

OF 3: Minimization of the TGCs, including emission effects

Many harmful emissions are generated from conventional power plants, like CO₂, SO₂, and NOx. These gases rise with the increase in the power produced. They are calculated in t/h and expressed as (Ali et al., 2022):

E = \sum_{i = 1}^{N_{T}} [(m_{i} + n_{i} P_{T, i} + z_{i} P_{T, i}^{2}) \times 0.01 + ω_{i} e^{(P_{T, i} μ_{i})}]

(19)

where

N_{T}

is the number of TGs,

P_{T, i}

denotes the power output from the

i

-th TG.

m_{i}, n_{i}, z_{i}, ω_{i}

and

μ_{i}

represent the emission coefficients of the

i

-th TG.

To reduce greenhouse gases in the atmosphere, a carbon tax is imposed to put pressure on power companies to reduce their carbon footprint.

The emission cost $C_{E}$ ($/h) is presented as:

C_{E} = C_{T} . E

(20)

where

C_{T}

is the carbon tax (per unit).

The third objective function is presented as the generation cost $F_{1}$ added to the emission cost $C_{E}$ and is expressed as follows:

F_{3} = F_{1} + C_{E}

(21)

Constraints

Solving the stochastic OPF objectives includes various equality and inequality constraints related to each power generator. These restrictions should not be violated, and the values obtained must be within the limits and bounds indicated as follows:

Equality restrictions

Equality restrictions ensure that the power produced on one side equals the demand and losses on the other. It is expressed using the following equations (El-Fergany and Hasanien, 2020):

P_{G i} - P_{D i} = V_{i} \sum_{j = 1}^{N B} V_{j} [G_{i, j} \cos (σ_{i, j}) + B_{i, j} \sin (σ_{i, j})]

(22)

Q_{G i} - Q_{D i} = V_{i} \sum_{j = 1}^{N B} V_{j} [G_{i, j} \sin (σ_{i, j}) - B_{i, j} \cos (σ_{i, j})]

(23)

where

P_{G i}

and

Q_{G i}

are the real and reactive powers generated, respectively, at bus

i

P_{D i}

and

Q_{D i}

represent the demand real and reactive powers, respectively, at bus

i

G_{j, i}

and

B_{j, i}

denote the conductance and susceptance, respectively.

Inequality constraints

Inequality restrictions are the limitations of elements within the power network, besides the security limits associated with the lines and the buses.

The active and reactive power outputs and the voltage limits are restricted by specific upper and lower limits as follows (Abid et al., 2024a):

P_{T, i}^{m i {n_{T, i}}_{T, i}^{\max}} i = 1, 2, 3 . . ., N_{T}

(24)

Q_{T, i}^{m i {n_{T, i}}_{T, i}^{\max}} i = 1, 2, 3, . . ., N_{T}

(25)

P_{W}^{m i {n_{W}}_{W}^{\max}}

(26)

Q_{W}^{m i {n_{W}}_{W}^{\max}}

(27)

P_{S}^{m i {n_{S}}_{S}^{\max}}

(28)

Q_{S}^{m i {n_{S}}_{S}^{\max}}

(29)

P_{S H}^{m i {n_{S H}}_{S H}^{\max}}

(30)

Q_{S H}^{m i {n_{S H}}_{S H}^{\max}}

(31)

Equations (24) and (25) represent the limits of the active and reactive powers from TGs, respectively. Equations (26), (28), and (30) refer to the limits of the active powers from wind, SPV, and SPV-Hydro generators, respectively. Equations (27), (29), and (31) denote the limits of the reactive powers from wind, SPV, and SPV-Hydro generators, respectively.

The security restrictions are represented as (Karthik et al., 2021):

V_{T, i}^{m i {n_{T, i}}_{T, i}^{\max}} i = 1, 2, 3, . . ., N_{T}

(32)

V_{l, i}^{m i {n_{l, i}}_{l, i}^{\max}} i = 1, 2, 3, . . ., N_{l}

(33)

S_{L, i} \leq S_{L, i}^{\max} i = 1, 2, 3, . . ., N_{L}

(34)

where

V_{l, i}^{\min}

and

V_{l, i}^{\max}

indicate the voltage magnitude’s lowest and highest bounds, respectively.

N_{L}

and

N_{l}

are the numbers of lines and buses, respectively.

S_{L, i}

and

S_{L, i}^{\max}

refer to the apparent power and its highest boundary for the line

i

, respectively.

For TGs, it is important to add the Prohibited Operating Zones (POZs) for more precise modeling as follows:

P_{T, i}^{\min p o z, j} \leq P_{T}^{i} \leq P_{T, i}^{\max p o z, j}

(35)

where

P_{T, i}^{\min p o z, j}

and

P_{T, i}^{\max p o z, j}

represent the minimum and maximum limits of the

j

-th POZ for the

i

-th TG.

Modeling the uncertainty of renewables

To model the stochastic nature of RESs, we use the Weibull PDF, lognormal PDF, and Gumbel distribution for the WTG, SPV generator, and hydropower outputs, respectively, as follows:

Stochastic wind power

The wind distribution is approximated by employing the Weibull PDF as follows (Hasanien et al., 2023):

F (V) = (\frac{S}{L}) {(\frac{V}{L})}^{(S - 1)} \exp - {(\frac{V}{L})}^{S}

(36)

where

V

is the speed of the wind in m/s,

S

and

L

represent shape and scale factors, respectively.

The Weibull distribution mean is expressed as (Nguyen et al., 2022):

M = L . Γ (1 + S^{- 1})

(37)

The gamma function is represented by (Kaymaz et al., 2021):

Γ = \int_{0}^{\infty} e^{- t} t^{x - 1} d t

(38)

The output of the WTG is influenced by the speed of the wind; thus, the power delivered is expressed as (Huy et al., 2023):

P_{w} (V) = {\begin{cases} 0 f o r V 〈 V_{a} V 〉 V_{d} \\ P_{w r} (\frac{V - V_{a}}{V_{r} - V_{a}}) f o r V_{a} \leq V \leq V_{r} \\ P_{w r} f o r V_{r} \leq V \leq V_{d} \end{cases}

(39)

where

P_{w r}

represents the rated output of the WTG,

V_{a}

and

V_{d}

are the activation and deactivation wind speeds, respectively.

V_{r}

is the rated wind speed.

According to equation (39), when $V$ is higher than $V_{d}$ but lower than $V_{a}$ , the power delivered is zero. The rated power $P_{w r}$ is delivered when $V$ falls between $V_{r}$ and $V_{d}$ . To distinguish these regions, the probabilities of the WTG model are represented as (Hassan et al., 2023):

F (P_{w}) {P_{w} = 0} = 1 - \exp [- {(\frac{V_{a}}{L})}^{S}] + \exp [- {(\frac{V_{d}}{L})}^{S}]

(40)

F (P_{w}) {P_{w} = P_{w r}} = 1 - \exp [- {(\frac{V_{r}}{L})}^{S}] + \exp [- {(\frac{V_{d}}{L})}^{S}]

(41)

The mathematical model of wind energy’s probability is formulated as follows (Sallam et al., 2024):

F (P_{w}) = \frac{S (V_{r} - V_{a})}{L^{S} . P_{w r}} {[V_{a} + \frac{P_{w}}{P_{w r}} (V_{r} - V_{a})]}^{S - 1} \exp [- {(\frac{V_{a} + \frac{P_{w}}{P_{w r}} (V_{r} - V_{a})}{L})}^{S}]

(42)

Stochastic solar power

The lognormal PDF of the solar irradiance $G$ is expressed by (Riaz et al., 2021):

F_{G} = \frac{1}{G μ \sqrt{2 π}} \exp [\frac{- {(\ln x - σ)}^{2}}{2 μ}] G > 0

(43)

where

μ

and

σ

represent the mean and typical deviation, respectively.

The conversion of solar irradiance into power is modeled as follows (Daqaq et al., 2022):

P_{s r} = {\begin{cases} P_{s r} (\frac{G^{2}}{G_{s r} G_{c}}) for 0 \leq G \leq G_{c} \\ P_{s r} (\frac{G^{2}}{G_{s r}}) for G \geq G_{c} \end{cases}

(44)

where

P_{s r}

represents the SPV’s rated output power,

G_{s r}

denotes the irradiance in the standard conditions, and

G_{c}

is the irradiance at a certain point.

Hydropower

Unlike wind and solar, hydroelectricity is a relatively stable source of energy; it is modeled using the Gumbel distribution as follows (Biswas et al., 2018):

F (Q) = \frac{1}{φ} \exp (\frac{Q - δ}{φ}) \exp [- \exp (\frac{Q - δ}{φ})]

(45)

where

Q

represents the river discharge rate,

φ

and

δ

are the scale and location parameters, respectively.

The power generated from the hydropower unit is formulated as:

P_{H} (Q) = λ . ρ . g . Q . H

(46)

where

λ

denotes the WTG efficiency (equal to 0.85),

ρ

is the density of the water (equal to 1000

k g / m^{3}

g

represents the acceleration resulting from gravity (equal to 9.81

m / s^{2}

H

refers to the effective pressure head (equal to 25 m).

Capacity determination of RESs

To illustrate how the capacity of RESs is determined, it is important to define the operating strategy of each flexible source under a reasonable capacity limit. For instance, in the adjusted IEEE 30 bus network, the WTG linked to bus 5 has a total of 25 wind turbines, each with a rated power equal to 3 MW. The overall capacity of this WTG is 75 MW; the energy output from the turbines is different according to the wind speed. Then in the case of TGC minimization, the augmented objective function involves direct, over- and underestimation costs on RESs. In this situation, we have either overestimation or underestimation of the scheduled power from renewable units. More precisely, due to the intermittence of flexible resources, the power produced can surpass the scheduled power, which leads to an underestimation of the available amount. Otherwise, the power produced is less than the scheduled power, which leads to an overestimation of the power output. It’s worth noting that the objective is to reduce the TGC while the output from RESs is optimal and does not exceed its maximum capacity limit to obtain a better operation point for the energy system, resulting in a minimum outcome for the TGC.

The target function of the OPF in this section takes into consideration the direct penalty and reserve costs of the RESs, including the production fees of the TGs. To this end, the operator of the system must provide a spinning reserve to ensure an uninterrupted electricity supply. The price of committing the reserve of the generation units to meet an overestimated amount is defined as the “reserve fee.” The underestimation is contrary to the overestimation scenario; at this point, the situation we tend to avoid is producing an energy surplus that will probably end up being wasted. In this case we reduce the power output from conventional units (thermal units), thus stabilizing the system with the amount of scheduled power from RESs and obtaining better values of control variables and minimum results regarding TGCs.

Metaheuristic optimization algorithms (MOAs)

Broadly speaking, a metaheuristic is viewed as a general algorithmic framework that can be applied to address a variety of optimization problems, requiring only minor adjustments to suit specific cases. MOAs are relatively easy to implement. They go through the search space to identify a satisfactory solution using different techniques. Exploration and exploitation are the primary mechanisms utilized in the search process. Nature-based MOAs can be divided into three main categories: swarm-based algorithms, bio-inspired algorithms, and chemistry/physics-based algorithms. This classification differs from one source to another (Abdel-Basset et al., 2018).

Proposed method: Circulatory system-based optimization (CSBO)

CSBO is a biologically inspired computational method developed to tackle complex optimization problems. This algorithm draws its conceptual foundation from the human circulatory system, particularly the dynamic behavior of blood flow through two primary circuits: the pulmonary and systemic circulations. By mimicking the regulatory mechanisms, flow patterns, and feedback processes within these biological networks, CSBO aims to efficiently search and utilize the solution space, making it an effective element for solving different optimization tasks in engineering, science, and beyond (Kanouni et al., 2025).

The human circulatory system

The body’s circulatory system is crucial for survival, as it ensures a steady supply of oxygen, nutrients, and waste removal to and from cells. This system comprises two circuits: pulmonary circulation, which transfers blood from the heart and lungs and vice versa, and systemic circulation, which delivers blood to all other body tissues. The heart, as the central organ, drives this essential process. The heart’s left and right ventricles collaborate to pump blood through rhythmic muscular contractions. Veins carry blood with low amounts of oxygen to the lungs through the pulmonary artery, where it absorbs oxygen and becomes oxygenated. This blood with a high amount of oxygen is then transported from the heart and distributed throughout the body to supply organs and tissues with oxygen (Kim, 2022).

Mathematical model of CSBO

CSBO is a bio-inspired algorithm. The optimizer draws inspiration from the blood vessel’s function in the human body, emulating both pulmonary and systemic circulation to carry out optimal tasks. Its straightforward design, ease of use, and absence of user-defined parameters are significant benefits (Bakır, 2024). The selection of the CSBO algorithm is justified by its unique hierarchical search structure. Unlike standard metaheuristics, CSBO mimics the dual-flow dynamics of the circulatory system, which provides a superior balance between global exploration and local exploitation. This is particularly advantageous for the OPF problem, where the search space is heavily constrained and non-linear. The algorithm’s ability to maintain population diversity through its flow-based position updating helps avoid the premature convergence issues often found in traditional methods like PSO. CSBO algorithm was initiated in 2022 by Ghasemi et al. and can be divided into four phases (Ghasemi et al., 2022).

A. Initialization phase

The initial population represents blood masses or particles; they range between minimum and maximum values as follows:

B m_{i} = B {m m i n m a x}_{\min}

(47)

where

B m_{i}

represents

i

-th blood mass,

D

refers to the number of dimensions.

B. The vein’s blood flow

The $i$ -th blood particle moves according to the pressure or force applied where conditions are favorable; as a result, the objective function value (force or pressure) diminishes. Clogged arteries can be modeled as trapped in a locally best solution, a scenario we aim to avoid, just as in real-life circulation. As the body keeps functioning, the algorithm persistently progresses the optimization task. This phase is portrayed by modeling the particle’s position and its relative target function values:

B m_{i}^{n e w} = B m_{i} + K_{i 1} \times P_{i} \times (B m_{i} - B m_{1}) + K_{23} \times P_{i} \times (B m_{3} - B m_{2})

(48)

K_{i, j} = \frac{F (B m_{j}) - F (B m_{i})}{| F (B m_{j}) - F (B m_{i}) | + ψ} = {\begin{cases} 1 i f F (B m_{i}) < F (B m_{j}) \\ - 1 i f F (B m_{i}) > F (B m_{j}) \\ 0 i f F (B m_{i}) = F (B m_{j}) \end{cases}

(49)

where

K_{i, j}

denotes the direction in which the blood mass

i

moves

B m_{i}

P_{i}

represents a value that ranges between 0 and 1; it defines the displacements and shifts toward an improved value with each cycle.

C. Blood flow in the pulmonary circulation

Pulmonary circulation handles deoxygenated blood, which mimics the weaker portion of the population. In CSBO, the population is coordinated in all iterations, and a specified number (NR) of the weakest individuals is sent into the pulmonary circulation, representing their transition to the lungs to acquire oxygen.

B m_{i}^{N e w} = B m_{i} + (\frac{R a n d N}{i t e r}) * R a n d C (1, D) i = 1 : NR

(50)

where

R a n d N

is a random number,

i t e r

is the number of iterations,

R a n d C

presents a random number from Cauchy probability distribution, and

D

indicates the dimension number of the optimization task.

For the current population, $P_{i}$ changes its value according to the circulation in the pulmonary system as follows:

P_{i} = R a n d (1, D) i = 1 : NR

(5)

D. Blood flow in the systemic circulation

The remaining population $(N L = N p o p - N R)$ with an improved fitness value enters the systemic circulation with an updated amount to flow throughout the body, as follows:

B m_{i, j}^{N e w} = B m_{1, j} + P_{i} . (B m_{3, j} - B m_{2, j})

(52)

For this population, $P_{i}$ is corrected as well by the systemic circulation as described below:

P_{i} = \frac{F (B m_{i}) - F_{W o r s t}}{F_{B e s t} - F_{W o r s t}} i = 1 : N L

(53)

where

F_{B e s t}

and

F_{W o r s t}

represent the top and bottom fitness functions for the current iteration.

The optimization cycle continues for a predetermined number of iterations. As with other algorithms, each population member adopts a new position if it achieves an improved fitness function value.

The mathematical framework of CSBO mimics the efficiency of a closed-loop circulatory system. In the context of the IEEE 30 and 57 bus systems, this allows for a more granular search of the feasible region, ensuring that constraints related to renewable energy volatility are handled with higher precision.

The flowchart and pseudocode of CSBO algorithm are highlighted in Figures 1 and 2, respectively.

Figure 1.

Flowchart of CSBO (Kanouni et al., 2025).

Figure 2.

CSBO algorithm pseudocode (Ghasemi et al., 2022).

Mountain gazelle optimizer (MGO)

Drawing inspiration from the social behavior and hierarchical structure of wild gazelles that live in the mountains, MGO is an evolutionary metaheuristic technique that uses mathematical formulations of the communal and hierarchical order of mountain gazelles to create an optimization algorithm. This optimizer utilizes four primary components of bachelor males, maternal herds, solitary territorial males, and migration to investigate and locate food supplies. MGO preserves its search potential and performs well as the magnitude of optimization issues increases (Abdollahzadeh et al., 2022).

Artificial Rabbits Optimization (ARO)

ARO is inspired by the survival behaviors of wild rabbits such as deflection, foraging, and random concealing. Deflection foraging consists of feeding near the nests of other rabbits, which helps keep the rabbit’s own nest concealed from predators. In contrast, the random hiding strategy allows a rabbit to select one of its burrows at random for hiding, reducing the chances of being caught by predators. Additionally, when a rabbit’s energy levels drop, it tends to shift from detour foraging to the random hiding strategy. This algorithm has been validated, and the experiment’s results indicate its feasibility (Wang et al., 2022).

Dwarf Mongoose Optimization Algorithm (DMOA)

DMOA is a nature-inspired metaheuristic technique that simulates the foraging behavior of dwarf mongooses using their compensatory behavioral adaptations. The limited way of capturing prey has notably influenced the communal conduct and ecological adjustments of mongooses, driving them to adapt in ways that support effective family nourishment. These behavioral adaptations include changes in prey size, how they use their environment, the size of their groups, and their practices around sharing food. DMO can identify the global optimal solutions to various optimization problems (Agushaka et al., 2022).

Results and discussion

IEEE 30-bus modified network

To demonstrate the efficiency of the CSBO method in solving the stochastic OPF problem, a power system comprising a mix of conventional and renewable energy resources is considered. The system consists of three thermal generators, one wind farm, one solar photovoltaic (SPV) unit, and one solar photovoltaic–hydro (SPV-hydro) station. These units are linked to an IEEE 30-bus network illustrated in Figure 1. The initial demand for real and reactive powers is 283.4 MW and 126.2 MVAR, respectively. Three scenarios are taken into consideration:

• Scenario 1: Minimization of the total generation costs (TGCs)

• Scenario 2: Reduction of the real power outages

• Scenario 3: Reduction of the production price, including emission effects

The suggested CSBO method was implemented using MATLAB 2021a alongside MATPOWER package on a laptop computer characterized by an Intel Core i5 microprocessor with an execution speed of 2.6 GHz and a RAM memory of 4 GB. The parameters concerning the setting of the algorithm consist of a maximum iteration of 300 and a population of 30 individuals. This number is obtained after a trial of population sizes like 20, 30, 40, and 70 as an empirical test on the performance of the CSBO technique (results not reported herein). The outcomes show that 30 is the most favorable for all case studies because when we decrease the number of individuals (under 30), the algorithm struggles with premature convergence, and the more we increase the number of individuals (over 30), the longer it takes to finish the simulation with a huge computational effort required and no improvement in the results obtained. For that reason, we focused on solving the technical problem of the OPF at a fixed population size of 30 for all tested algorithms for a fair comparison.

The cost and emission coefficients for the thermal generators presented in Table 1 are adopted from Huy et al. (2022) to represent realistic operational characteristics and environmental constraints. Similarly, the structural parameters of the modified IEEE 30-bus system in Table 2, including the allocation of renewable energy sources (wind at bus 5, SPV at bus 11, and SPV-hydro at bus 13), follow the standardized configuration established in Hassan et al. (2023). The suitability of these parameters lies in their widespread acceptance as benchmark cases, allowing for a rigorous and transparent comparison between the proposed CSBO algorithm and existing state-of-the-art metaheuristics.

Table 1.

Coefficient fees and emissions of TGs (Huy et al., 2022).

	TG1 (bus 1)	TG2 (bus 2)	TG3 (bus 8)
A	0	0	0
B	2	1.75	3.25
C	0.38	0.0175	0.0083
D	18	16	12
E	0.037	0.38	0.045
$α$	0.04091	0.02543	0.05326
$β$	−0.05554	−0.06047	−0.03550
$γ$	0.06490	0.05638	0.03380
$ω$	0.0002	0.0005	0.0020
$μ$	6.667	3.333	2.000

Table 2.

Parameters of the IEEE 30 bus modified network (Hassan et al., 2023).

Item	No.
Generators/transmission lines	6/41
Buses/load buses	30/24
Thermal units	3 (Bus 1, 2, and 8)
Wind generator/SPV generator	Bus 5/bus 11
SPV-hydro station	Bus 13
Control variables	12

The limits of the control and state variables of each generator must not be violated to ensure stability and security of the power system; these boundaries are displayed in Tables 3 and 4, respectively.

Renewable generators are defined by their output power and their location in addition to other parameters highlighted in Table 5.

Table 3.

Lower and higher limits of the control variables.

Variable	$P_{T 1}$	$P_{T 2}$	$P_{T 3}$	$P_{W}$	$P_{S}$	$P_{S H}$	$P_{g, i}$
Minimum	50	20	10	0	0	0	0.95
Maximum	140	80	35	75	60	50	1.1

Table 4.

Lower and higher limits of the state variables.

Variable	$Q_{T 1}$	$Q_{T 2}$	$Q_{T 3}$	$Q_{W}$	$Q_{S}$	$Q_{S H}$
Minimum	−20	−20	−15	−30	−25	−20
Maximum	150	60	40	35	30	25

Table 5.

Parameters of the renewable units.

Wind farm
Bus	No. of turbines	Estimated power (MW)	Weibull PDF settings	Weibull mean
5	25	75	S = 2, L = 9	V = 7.796

SPV unit
Bus	Rated power (MW)	Lognormal PDF settings	Lognormal mean
11	50	$μ$ = 6, $σ$ = 0.6	G = 486 W/m²

SPV-hydro station
Bus	SPV rated power (MW)	Lognormal PDF settings	Lognormal mean
13	45	$μ$ = 6, $σ$ = 0.6	G = 486 W/m²

SPV-hydro station
Bus	Hydro rated power (MW)	Gumbel PDF settings
13	5	$ρ$ = 15, $τ$ = 1.2

To ensure the practical relevance of the cost estimations, the coefficients used for the thermal units and the maintenance costs for renewable sources were cross verified against the International Renewable Energy Agency (IRENA) Levelized Cost of Energy (LCOE) benchmarks. The integration of SPV-hydro and wind generation at buses 5, 11, and 13 utilizes stochastic cost functions that simulate the real-world variability of energy prices and fuel consumption, bridging the gap between theoretical benchmark systems and practical grid operation (Figure 3).

Figure 3.

Altered IEEE 30 bus test network (Biswas et al., 2018).

Weibull fitting curve and the wind distribution frequency at bus 5 are defined using the Monte Carlo method after executing 8000 samples, as shown in Figure 4. In a similar way, the SPV unit power output is obtained using the same method with the same number of patterns (Hassan et al., 2023). The lognormal PDF curve and the solar irradiance distribution at buses 11 and 13 are displayed in Figures 5 and 6, respectively. The real power output from the solar unit at bus 13 is presented in Figure 7, showcasing the matching of the available active power with the distribution of the solar irradiance.

Figure 4.

Wind speed distribution for the wind generator at bus 5.

Figure 5.

Solar irradiance distribution for the SPV unit at bus 11.

Figure 6.

Solar irradiance distribution for the SPV unit at bus 13.

Figure 7.

The active power from the SPV unit at bus 13.

Figure 8 refers to the Gumbel fitting curve and the frequency distribution of the river stream for the hydropower station using the Monte Carlo method. The histograms in Figures 9 and 10 represent the accessible solar and hydropower from the designated site at bus 13 and for the SPV unit and the hydro station, respectively. Figure 11 displays the real power delivered from the combined SPV-hydro generator. It is worth noting that the following scenarios are all conducted using the Weibull PDF, the lognormal PDF, and the Gumbel fitting for the wind, solar, and hydropower, respectively.

Figure 8.

River influx rate for the hydroelectric station at bus 13.

Figure 9.

Accessible solar power from the location and for the SPV unit at bus 13.

Figure 10.

Accessible hydropower from the location and for the hydro station at bus 13.

Figure 11.

The active power for the SPV-hydro station at bus 13.

• Case 1: Reduction of the overall generation costs

Power companies are required to provide a competitive price of electricity per kilowatt for consumers; this can only be achieved by reducing the production cost. The primary target function in the first case is to minimize the overall fees of the thermal and renewable units. Table 6 shows the detailed optimal results regarding the real and reactive powers, the voltage magnitudes, and the total price of thermal, wind, SPV, and SPV-hydro generators combined. With respect to each algorithm after 30 runs of simulation. Both the state and control variables are maintained within the predefined boundaries for all the algorithms. CSBO outperformed all previous methods by reducing the total generating costs to 787.3598 $/h compared with 787.8765 $/h for MGO, 788.403 $/h for ARO, and 787.832 $/h for DMOA, which ranked second.

Table 6.

Detailed optimal results for case 1.

Variable	MFO (Sulaiman and Mustaffa, 2021)	BMO (Sulaiman and Mustaffa, 2021)	GTO (Belagra et al., 2023)	AEO (Belagra et al., 2023)	INFO (Belagra et al., 2023)	PSO (Sulaiman and Mustaffa, 2021)	DMOA	ARO	MGO	CSBO
$P_{T 1}$ (MW)	134.907	134.912	134.910	134.910	134.930	134.907	134.94	134.91	135.33	135.23
$P_{T 2}$ (MW)	40.004	40.00	42.580	40.450	41.990	40.00	38.20	38.17	39.72	39.620
$P_{W}$ (MW)	46.202	46.937	48.180	47.550	48.200	46.429	46.66	48.51	46.89	46.800
$P_{T 3}$ (MW)	10.00	10.00	10.00	10.01	10.01	10.00	10.09	10.07	10.06	10.050
$P_{S}$ (MW)	44.727	43.799	39.660	41.540	40.090	44.770	41.90	40.14	41.20	40.660
$P_{S H}$ (MW)	13.304	13.483	13.970	14.790	14.060	13.032	17.42	17.37	17.66	16.850
$V_{1}$ (p.u)	1.083	1.083	1.077	1.076	1.075	1.083	1.078	1.077	1.077	1.086
$V_{2}$ (p.u)	1.068	1.069	1.064	1.060	1.061	1.068	1.062	1.062	1.060	1.068
$V_{5}$ (p.u)	1.046	1.047	1.039	1.037	1.041	1.046	1.041	1.038	1.038	1.041
$V_{8}$ (p.u)	1.048	1.049	1.040	1.041	1.041	1.048	1.042	1.041	1.040	1.045
$V_{11}$ (p.u)	1.100	1.100	1.098	1.100	1.099	1.100	1.096	1.099	1.087	1.096
$V_{13}$ (p.u)	1.053	1.052	1.052	1.067	1.061	1.052	1.048	1.060	1.064	1.053
$Q_{T 1}$ (MVAR)	−0.456	−0.735	1.350	4.880	0.350	−0.2965	7.44	3.44	6.41	12.530
$Q_{T 2}$ (MVAR)	14.467	14.582	25.810	15.880	19.740	14.663	16.90	21.30	16.87	17.560
$Q_{W}$ (MVAR)	23.250	23.729	24.110	24.790	27.660	23.2576	27.01	23.19	25.35	22.350
$Q_{T 3}$ (MVAR)	37.050	37.064	38.380	38.830	38.840	36.860	39.93	39.00	39.73	39.530
$Q_{S}$ (MVAR)	28.608	28.425	30.00	29.960	29.800	28.578	29.62	29.74	25.83	28.140
$Q_{S H}$ (MVAR)	14.885	14.477	17.040	22.460	20.300	14.722	15.52	19.40	22.05	15.890
$C o s t$ ($/h)	789.527	789.124	789.223	789.318	788.941	789.484	787.832	788.403	787.876	787.359

The outcomes regarding previous studies were the following: $789.124/h for BMO, $789.223/h for GTO, $789.318/h for AEO, $788.941/h for INFO, and $789.484/h for PSO, and the worst value is acquired by MFO, which generated 789.5271 $/h, with a difference of 2.1673 $/h in comparison with CSBO. This reduction will result in $52.0152 and $18,985.548 savings per day and year, respectively. According to Table 7, CSBO had the lowest minimum, maximum, and average generating costs at $787.36/h, $788.05/h, and $787.70/h, respectively. While the lowest standard deviation was obtained by DMOA at 0.298, followed by MGO at 0.396, CSBO at 0.487, and ARO at 0.566. The shortest Central Processing Unit (CPU) was obtained by CSBO in a period of 295.375 s compared with 331.599 s, 555.860 s, and 981.197 s for MGO, ARO, and DMOA, respectively. Figure 12 illustrates the convergence curves of the four methods put forward where CSBO reached the lowest value possible regarding the TGCs with a slow but steady convergence.

Table 7.

Results obtained from the studied methods for case 1.

Algorithms	Min	Max	Average	Standard	Time (s)
DMOA	787.832	788.253	788.043	0.298	981.197
ARO	788.403	789.204	788.803	0.566	555.860
MGO	787.87	788.43	788.15	0.396	331.599
CSBO	787.36	788.05	787.70	0.487	295.375

Figure 12.

Convergence curves of the optimization methods for case 1.

• Case 2: Reduction of the real power losses

The active power losses that occur in overhead transmission lines are a significant issue that must be resolved within the integrated energy system to ensure the security and stability of the grid. The minimization of APLs is the main objective function in the second case. Table 8 highlights the best results of the real and reactive powers, the voltage magnitude values, and the overall power loss from the combined thermal-renewable power system after executing 30 independent runs. The outcomes reveal no violations of the limits previously mentioned regarding the control and state variables. The convergence features of each method are portrayed in Figure 13, showing a clear prevalence of CSBO over MGO, ARO, and DMOA.

Table 8.

Detailed optimal results for case 2.

Variable	MFO (Sulaiman and Mustaffa, 2021)	BMO (Sulaiman and Mustaffa, 2021)	GTO (Belagra et al., 2023)	AEO (Belagra et al., 2023)	INFO (Belagra et al., 2023)	PSO (Sulaiman and Mustaffa, 2021)	DMOA	ARO	MGO	CSBO
$P_{T 1}$ (MW)	50.00	50.00	50.00	50.02	50.01	50.00	49.35	50.02	49.99	49.59
$P_{T 2}$ (MW)	30.00	30.00	25.00	29.80	25.16	40.00	27.18	30.60	28.56	22.22
$P_{W}$ (MW)	75.00	75.00	75.00	74.95	74.99	75.00	74.90	74.99	74.98	74.98
$P_{T 3}$ (MW)	35.00	35.00	35.00	34.91	34.99	35.00	34.61	34.98	34.32	34.85
$P_{S}$ (MW)	50.00	50.00	50.00	59.86	59.99	50.00	59.87	59.94	59.79	59.95
$P_{S H}$ (MW)	45.00	45.566	50.00	35.98	40.35	35.586	39.61	34.98	37.90	43.91
$V_{1}$ (p.u)	1.059	1.059	1.061	1.058	1.060	1.061	1.061	1.064	1.061	1.061
$V_{2}$ (p.u)	1.054	1.054	1.055	1.053	1.055	1.056	1.057	1.057	1.056	1.054
$V_{5}$ (p.u)	1.044	1.044	1.049	1.040	1.045	1.045	1.044	1.045	1.043	1.044
$V_{8}$ (p.u)	1.049	1.049	1.046	1.045	1.049	1.049	1.047	1.048	1.048	1.049
$V_{11}$ (p.u)	1.100	1.100	1.085	1.092	1.096	1.100	1.085	1.094	1.084	1.093
$V_{13}$ (p.u)	1.060	1.060	1.066	1.056	1.063	1.057	1.061	1.053	1.066	1.066
$Q_{T 1}$ (MVAR)	−4.806	−4.805	−1.04	0.38	−0.89	−4.658	−3.46	4.73	−1.20	1.89
$Q_{T 2}$ (MVAR)	6.944	6.943	15.78	17.16	14.43	7.315	23.49	14.14	18.67	12.15
$Q_{W}$ (MVAR)	20.594	20.596	24.430	21.70	24.34	20.714	22.05	22.66	22.10	24.14
$Q_{T 3}$ (MVAR)	36.291	36.291	40.00	39.230	39.45	36.624	38.77	39.39	39.61	39.10
$Q_{S}$ (MVAR)	29.888	29.292	25.00	29.960	29.98	29.831	26.48	29.61	25.86	28.91
$Q_{S H}$ (MVAR)	18.607	18.604	22.010	19.350	20.41	17.148	20.28	16.74	22.48	21.78
$P_{L o s s}$ (MW)	2.1669	2.1669	2.2212	2.1244	2.0938	2.1868	2.1243	2.1140	2.1335	2.1017

Figure 13.

Convergence curves of the optimization methods for case 2.

Table 9 indicates the superiority of CSBO over the tested methods. For instance, MGO generated 2.1335 MW of APLs, which is 0.0318 MW higher than CSBO. The latter also had the best medium, maximum, minimum, and standard deviation values and the shortest computational time, taking only 131.765 s, far ahead of the other algorithms, like DMOA, which took the longest period at 823.009 s. The reduction in power losses was 2.1017 MW for CSBO, ranking second right behind INFO, which ranked first, generating 2.0938 MW losses. MFO, BMO, GTO, AEO, and PSO produced 2.1669 MW, 2.1669 MW, 2.2212 MW, 2.1244 MW, and 2.1868 MW, respectively.

Table 9.

Results obtained from the studied methods for case 2.

Algorithms	Min	Max	Average	Standard	Time (s)
DMOA	2.1243	2.1454	2.1349	0.015	823.009
ARO	2.1140	2.1339	2.1240	0.014	631.073
MGO	2.1335	2.1555	2.1445	0.016	418.298
CSBO	2.1017	2.1169	2.1093	0.010	131.765

• Case 3: Minimization of generation costs, including emission effects

Thermal power plants are responsible for releasing harmful emissions, like CO₂, SO₂, and NOx. These gases rise with the increase in the power produced. To tackle this problem, a carbon tax is imposed to put pressure on companies to reduce their carbon footprint and increase the share of renewables. In this scenario, the reduction of the combined cost-emission is the target function. Table 10 represents the statistical outcomes derived from 30 simulation runs. The state and control variables were all maintained within the identified bounds. CSBO had the lowest generation cost, including emission effects, among all competing methods at 818.605 $/h compared with 820.807 $/h for MFO, 820.485 $/h for BMO, 820.581 $/h for AEO, 820.559 $/h for INFO, 820.806 $/h for PSO, 819.153 $/h for DMOA, 819.008 $/h for ARO, and 819.225 $/h for MGO. The worst outcome was obtained by GTO at 821.749 $/h, and by comparing CSBO with GTO, the cost savings are 3.144 $/h, 75.456 $/day, and 27,541.44 $/year.

Table 10.

Detailed optimal results obtained for case 3.

Variable	MFO (Sulaiman and Mustaffa, 2021)	BMO (Sulaiman and Mustaffa, 2021)	GTO (Belagra et al., 2023)	AEO (Belagra et al., 2023)	INFO (Belagra et al., 2023)	PSO (Sulaiman and Mustaffa, 2021)	DMOA	ARO	MGO	CSBO
$P_{T 1}$ (MW)	126.593	126.693	126.42	127.21	126.63	126.59	126.54	125.46	126.89	126.200
$P_{T 2}$ (MW)	44.221	44.232	44.43	46.88	44.35	44.221	44.30	43.13	46.010	44.090
$P_{W}$ (MW)	48.509	48.492	48.53	49.76	48.66	48.50	48.92	47.62	49.600	48.720
$P_{T 3}$ (MW)	10.00	10.00	10.00	10.00	10.01	10.00	10.01	10.00	10.01	10.050
$P_{S}$ (MW)	45.026	45.026	45.33	41.38	45.52	45.026	40.46	45.09	38.880	41.880
$P_{S H}$ (MW)	14.452	14.453	14.17	13.75	13.71	14.453	18.67	17.52	17.580	17.930
$V_{1}$ (p.u)	1.081	1.0816	1.076	1.078	1.076	1.082	1.074	1.075	1.076	1.075
$V_{2}$ (p.u)	1.068	1.068	1.062	1.063	1.061	1.068	1.061	1.060	1.062	1.064
$V_{5}$ (p.u)	1.046	1.047	1.039	1.039	1.041	1.046	1.039	1.037	1.038	1.044
$V_{8}$ (p.u)	1.048	1.049	1.041	1.040	1.041	1.049	1.042	1.041	1.040	1.042
$V_{11}$ (p.u)	1.100	1.100	1.097	1.097	1.098	1.10	1.100	1.099	1.093	1.094
$V_{13}$ (p.u)	1.053	1.054	1.053	1.051	1.058	1.054	1.060	1.068	1.063	1.064
$Q_{T 1}$ (MVAR)	−0.994	−1.105	4.93	5.08	5.01	−0.969	1.77	4.57	3.390	−1.300
$Q_{T 2}$ (MVAR)	13.742	13.839	19.25	22.26	15.50	13.745	19.02	16.87	21.340	22.900
$Q_{W}$ (MVAR)	23.002	23.013	25.20	24.29	27.24	22.988	25.30	23.61	23.420	27.830
$Q_{T 3}$ (MVAR)	37.131	36.984	39.24	37.53	39.20	37.135	39.57	38.18	38.040	37.110
$Q_{S}$ (MVAR)	28.635	28.515	30.00	29.98	29.84	28.631	30.00	29.82	27.760	27.700
$Q_{S H}$ (MVAR)	15.099	15.216	17.51	16.76	19.32	15.095	19.54	22.70	21.180	20.840
$C + E$	820.807	820.485	821.749	820.581	820.559	820.806	819.153	819.008	819.225	818.605

According to Table 11, CSBO had the shortest execution time at 167.191 s, followed by ARO at 372.565 s, MGO at 519.560 s, and BMOA at 554.905 s. The combined output power from wind, solar, and solar hydro units has indeed increased from 104.31 MW in the first case (Table 6) to 108.53 MW in this case, with 4.22 MW more clean energy delivered to the grid, resulting in a higher penetration of renewable generators after imposing the carbon tax. In Figure 14, the convergence curves confirm the superiority of CSBO over MGO, ARO, and DMOA. Figure 15 visualizes the voltage magnitudes of load buses using the CSBO algorithm for cases 1, 2, and 3; again, the voltage profile in each scenario didn’t surpass the upper and lower limits.

Table 11.

Results obtained from the studied methods for case 3.

Algorithms	Min	Max	Average	Standard	Time (s)
DMOA	819.15	819.45	819.30	0.212	554.905
ARO	819.01	819.19	819.10	0.127	372.565
MGO	819.22	819.44	819.33	0.156	519.560
CSBO	818.61	819.88	819.25	0.898	167.191

Figure 14.

Convergence curves of the optimization methods for case 3.

Figure 15.

Voltage profiles of load buses using CSBO algorithm for cases 1-3.

IEEE 57 bus modified network

To further enrich this study, a second system is utilized to reinforce the validity of CSBO. The altered IEEE 57 bus network is adjusted to pave the way for renewable units to be connected as shown in Figure 16; it consists of 41 load buses besides seven other buses linked to generators. Four of them are thermal plants on buses 1, 2, 3, and 8; a WTG at bus 12; an SPV unit at bus 9; and an SPV-hydro generator at bus 6. The number of lines is 88, and the total number of control variables is 14. The initial demand for active power equals 1250.80 MW, while the reactive power is around 336.40 MVAR. The control and state variable limits are given in Table 11.

Figure 16.

Altered IEEE 57 bus test network (Sulaiman and Mustaffa, 2021).

This system shows significant voltage drops (violations of reactive generators installed at busbars #2 and #9), especially when RESs are integrated, which makes it more difficult to guarantee the validity of the solutions. For example, in Rizvi et al. (2020), the authors examine how reactive power limits affect generator buses during load increase. This study directly shows that a violation of reactive power capability at bus 9 (and later bus 2 under high load) leads to significant voltage drops. Therefore, buses 2 and 9 are intentionally stressed in simulations to test the algorithms’ ability to restore voltage stability through VAR optimization or voltage control coordination.

The limits of the control and state variables are shown in Tables 12 and 13, respectively.

Table 12.

Lower and higher limits of the control variables.

Variable	$P_{T 1}$	$P_{T 2}$	$P_{T 3}$	$P_{W}$	$P_{S}$	$P_{S H}$	$P_{g, i}$
Minimum	0	0	0	0	0	0	0
Maximum	575.88	100	140	550	220	100	210

Table 13.

Lower and higher limits of the state variables.

Variable	$Q_{T 1}$	$Q_{T 2}$	$Q_{T 3}$	$Q_{W}$	$Q_{S}$	$Q_{S H}$
Minimum	−140	−17	−10	−140	−3	−8
Maximum	200	50	60	200	9	25

• Case 4: Reduction of the TGCs

Case four consists of reducing the overall generating costs in the adjusted IEEE 57-bus network. Table 14 highlights the optimal results regarding the state and control variables, where none of them surpassed the limits provided previously, and they were all maintained within the predefined boundaries after 30 runs of simulation. The convergence curves of CSBO, MGO, ARO, and DMOA are shown in Figure 17, where at the end of the simulation, CSBO happens to have the lowest generation cost. The TGCs were 5257.941 $/h for CSBO, while the worst outcome was acquired by PSO at 5417.538 $/h, with a difference of 160.812 $/h and a 2.95% decrease in total generation costs in CSBO’s favor, resulting in 3859.488 $/day and 1,408,713.12 $/year in cost savings. Other algorithms performed well in this scenario, with TGCs not far behind CSBO; for instance, INFO 5259.204 $/h, GTO 5260.00 $/h, AEO 5260.249 $/h, and MGO 5263.057 $/h.

Table 14.

Detailed optimal results obtained for case 4.

Variable	MFO (Sulaiman and Mustaffa, 2021)	PSO (Sulaiman and Mustaffa, 2021)	AEO (Belagra et al., 2023)	INFO (Belagra et al., 2023)	BMO (Sulaiman and Mustaffa, 2021)	GTO (Belagra et al., 2023)	DMOA	ARO	MGO	CSBO
$P_{T 1}$ (MW)	554.293	500.482	556.88	556.81	555.57	556.35	563.70	560.97	563.83	556.89
$P_{T 2}$ (MW)	100.00	100.00	100.00	100.00	100.00	100.00	96.79	96.91	95.86	100.00
$P_{T 3}$ (MW)	76.624	140.00	76.620	76.610	76.624	76.620	76.92	76.35	76.80	75.95
$P_{S H}$ (MW)	100.00	100.00	100.00	100.00	100.00	100.00	99.90	99.19	98.89	99.45
$P_{T 4}$ (MW)	54.476	38.486	51.23	51.230	51.271	51.68	50.37	53.78	50.67	52.75
$P_{S}$ (MW)	200.00	200.00	200.00	200.00	200.00	200.00	199.30	199.17	200.00	200.00
$P_{W}$ (MW)	210.00	210.00	210.00	210.00	210.00	210.00	209.63	209.85	209.92	210.00
$V_{1}$ (p.u)	1.100	1.100	1.051	1.067	1.100	1.059	1.041	1.033	1.042	1.057
$V_{2}$ (p.u)	0.950	1.100	1.043	1.055	1.100	1.048	1.025	1.022	1.027	1.047
$V_{3}$ (p.u)	1.100	0.950	1.020	1.019	0.950	1.014	1.000	0.995	1.007	1.019
$V_{6}$ (p.u)	0.950	1.100	1.014	1.015	1.024	1.012	1.012	1.005	1.004	1.010
$V_{8}$ (p.u)	1.052	1.015	1.013	1.014	1.021	1.016	1.025	1.015	1.021	1.023
$V_{9}$ (p.u)	1.100	1.100	0.985	0.984	0.950	0.987	0.995	0.994	0.994	0.991
$V_{12}$ (p.u)	0.958	1.100	0.975	0.968	0.980	0.976	0.986	1.002	0.984	0.967
$Q_{T 1}$ (MVAR)	200.00	92.109	98.26	143.76	200.00	125.02	109.28	68.10	109.43	124.62
$Q_{T 2}$ (MVAR)	−17.00	50.00	50.00	49.990	50.00	49.98	25.16	43.56	18.56	45.39
$Q_{T 3}$ (MVAR)	60.00	−10.00	48.70	27.890	−10.00	21.83	11.02	6.36	38.21	42.35
$Q_{S H}$ (MVAR)	−8.00	25.00	24.22	24.96	25.00	21.95	22.88	23.38	4.85	6.70
$Q_{T 4}$ (MVAR)	179.964	18.256	109.32	113.89	106.249	117.75	132.42	108.15	130.77	136.16
$Q_{S}$ (MVAR)	9.00	9.00	−1.480	−1.690	−3.00	−1.63	4.51	5.17	8.32	5.64
$Q_{W}$ (MVAR)	−46.269	155.00	47.28	16.80	−1.4767	41.06	79.05	128.35	71.85	16.65
$C o s t$ ($/h)	5316.14	5417.53	5260.24	5259.20	5300.45	5260.00	5267.83	5270.29	5263.05	5257.94

Figure 17.

Convergence curves of the optimization methods for case 4.

• Case 5: Minimization of the APLs

With a larger system, the IEEE 57 bus network will certainly have more power losses than the previous one (IEEE 30 bus); in this regard, it is necessary to reduce the real power losses using an adequate method. The best results acquired for the fifth scenario within 30 independent runs of simulation are indicated in Table 15, with CSBO achieving the best results for APLs at 19.6804 MW. The control and state variable values are all maintained within the specific limits. The worst result was obtained by PSO at 21.3621 MW; that’s 1.6913 MW more real power losses produced in comparison with CSBO, which represents a 7.72% decrease in APLs. Algorithms like INFO, AEO, and GTO performed well with power losses around 19.7040 MW, 19.7633 MW, and 19.7703 MW, respectively. Figure 18 portrays the convergence curve of CSBO alongside DMOA, ARO, and MGO, which generated 20.942 MW, 20.0659 MW, and 20.448 MW, respectively.

Table 15.

Detailed optimal results obtained for case 5.

Variable	MFO (Sulaiman and Mustaffa, 2021)	PSO (Sulaiman and Mustaffa, 2021)	AEO (Belagra et al., 2023)	INFO (Belagra et al., 2023)	BMO (Sulaiman and Mustaffa, 2021)	GTO (Belagra et al., 2023)	DMOA	ARO	MGO	CSBO
$P_{T 1}$ (MW)	259.83	273.41	298.32	302.96	262.68	293.00	301.60	292.63	263.43	291.90
$P_{T 2}$ (MW)	39.25	34.07	0.99	3.32	39.12	7.57	15.89	2.11	35.32	25.81
$P_{T 3}$ (MW)	140.00	140.00	139.98	139.93	140.00	140.00	122.16	139.24	130.02	137.80
$P_{S H}$ (MW)	100.00	100.00	100.00	100.00	100.00	100.00	87.30	96.22	67.57	96.37
$P_{T 4}$ (MW)	323.02	314.67	321.26	314.31	319.77	320.00	340.88	331.09	364.93	308.23
$P_{S}$ (MW)	200.00	200.00	200.00	200.00	200.00	200.00	197.81	199.88	200.00	200.00
$P_{W}$ (MW)	210.00	210.00	210.00	209.99	210.00	210.00	206.10	209.70	209.99	210.00
$V_{1}$ (p.u)	1.050	1.051	1.021	1.023	1.045	1.021	1.031	1.009	1.013	1.041
$V_{2}$ (p.u)	1.100	1.100	1.014	1.015	1.100	1.013	1.017	1.002	1.010	1.033
$V_{3}$ (p.u)	1.100	1.100	1.016	1.017	1.100	1.010	0.999	1.005	1.014	1.031
$V_{6}$ (p.u)	1.100	1.100	1.025	1.022	1.100	1.021	1.003	1.020	1.033	1.026
$V_{8}$ (p.u)	1.030	1.032	1.038	1.029	1.032	1.038	1.015	1.037	1.043	1.027
$V_{9}$ (p.u)	1.099	0.950	1.003	0.999	0.950	1.005	0.992	1.005	1.004	1.004
$V_{12}$ (p.u)	0.950	0.950	0.977	0.979	0.957	0.981	0.996	0.991	0.979	0.995
$Q_{T 1}$ (MVAR)	123.76	123.74	61.36	62.36	105.81	63.65	96.21	37.87	45.43	72.53
$Q_{T 2}$ (MVAR)	50.00	50.00	49.91	48.10	50.00	50.00	43.08	47.52	47.32	45.54
$Q_{T 3}$ (MVAR)	60.00	60.00	31.10	35.75	60.00	17.62	−6.55	17.51	30.05	39.49
$Q_{S H}$ (MVAR)	25.00	25.00	0.120	2.810	25.00	−4.140	−2.78	−1.13	18.68	3.14
$Q_{T 4}$ (MVAR)	60.39	71.64	87.95	72.39	66.64	91.720	59.43	86.81	87.98	49.50
$Q_{S}$ (MVAR)	9.00	−3.00	9.00	8.92	−3.00	9.00	−2.17	5.62	−0.39	4.93
$Q_{W}$ (MVAR)	−43.207	−42.051	42.130	51.220	−21.547	53.930	100.75	89.32	55.63	61.43
$P_{L o s s}$ (MW)	21.303	21.362	19.763	19.704	20.785	19.770	20.942	20.065	20.448	19.680

Figure 18.

Convergence curves of the optimization methods for case 5.

• Case 6: Reduction of TGCs, including emission effects

The reduction of energy production costs and emission effects is the target function in the sixth case. Table 16 denotes the optimal results of the real and reactive powers of each unit and their respective voltage magnitudes. CSBO had the best results, with the lowest generation cost, including emissions, of all the algorithms tested at 5293.93 $/h compared with 5332.379 $/h obtained by MFO, which ranked last, resulting in a difference of 39.9971 $/h, with cost savings reaching 959.93 $/day and 350,374.596 $/year. Other well-performing algorithms include ARO at $5294.24/h, MGO at $5294.774/h, INFO at $5295.859/h, AEO at $5298.192/h, and GTO at $5299.694/h. Figure 19 represents the curves of convergence of the tested algorithms, where CSBO obtained the best outcomes. Moreover, there are no violations of state and control variables; this is evident in Figure 20, which illustrates the voltage levels at load buses using the CSBO algorithm for cases 4, 5, and 6, thus indicating that the voltage magnitudes were all within the predefined bounds.

Table 16.

Detailed optimal results obtained for case 6.

Variable	MFO (Sulaiman and Mustaffa, 2021)	PSO (Sulaiman and Mustaffa, 2021)	AEO (Belagra et al., 2023)	INFO (Belagra et al., 2023)	BMO (Sulaiman and Mustaffa, 2021)	GTO (Belagra et al., 2023)	DMOA	ARO	MGO	CSBO
$P_{T 1}$ (MW)	553.316	554.165	556.410	553.970	551.890	554.410	565.76	557.45	557.73	555.12
$P_{T 2}$ (MW)	100.00	100.00	99.170	100.00	100.00	100.00	94.47	99.89	100.00	99.67
$P_{T 3}$ (MW)	76.624	76.624	76.630	76.550	76.624	76.63	74.06	76.61	76.47	73.80
$P_{S H}$ (MW)	100.00	100.00	100.00	100.00	100.00	100.00	99.70	99.21	99.98	99.92
$P_{T 4}$ (MW)	52.552	51.776	52.430	53.670	52.193	53.440	51.55	51.83	51.57	56.04
$P_{S}$ (MW)	200.00	200.00	200.00	200.00	200.00	200.00	199.83	199.99	200.00	199.91
$P_{W}$ (MW)	210	210	210.00	210.00	210	210.00	209.80	209.97	208.93	209.89
$V_{1}$ (p.u)	1.078	1.100	1.067	1.065	1.090	1.046	1.051	1.063	1.066	1.062
$V_{2}$ (p.u)	1.100	0.950	1.056	1.053	1.100	1.038	1.039	1.049	1.051	1.049
$V_{3}$ (p.u)	1.036	1.100	1.020	1.019	0.961	1.016	1.010	1.009	1.012	1.016
$V_{6}$ (p.u)	1.100	1.100	1.008	1.014	1.100	1.012	1.005	1.008	1.009	1.013
$V_{8}$ (p.u)	1.011	1.001	1.018	1.013	1.044	1.012	1.017	1.019	1.013	1.019
$V_{9}$ (p.u)	1.019	1.100	0.988	0.985	0.950	0.989	0.991	0.990	0.987	0.988
$V_{12}$ (p.u)	0.985	0.985	0.964	0.968	1.100	0.979	0.981	0.975	0.975	0.970
$Q_{T 1}$ (MVAR)	130.90	200.00	146.510	138.29	65.396	87.78	110.87	143.23	151.96	142.95
$Q_{T 2}$ (MVAR)	50.00	−17.00	49.94	49.62	50.00	48.33	43.63	47.40	40.68	39.82
$Q_{T 3}$ (MVAR)	42.54	60.00	34.95	30.31	−10.00	45.12	28.08	7.67	13.15	28.06
$Q_{S H}$ (MVAR)	25.00	25.00	5.78	24.98	25.00	24.93	9.64	13.81	19.34	18.44
$Q_{T 4}$ (MVAR)	78.667	61.374	126.76	109.10	63.461	101.07	120.36	127.04	111.67	123.46
$Q_{S}$ (MVAR)	9.00	9.00	8.80	3.02	−3.00	9.00	9.09	4.15	5.35	0.62
$Q_{W}$ (MVAR)	27.217	26.599	3.090	18.530	155.00	59.210	56.97	33.86	33.78	21.31
$C + E$	5332.37	5332.05	5298.19	5295.85	5320.85	5299.69	5297.38	5294.24	5294.77	5293.93

Figure 19.

Convergence curves of the optimization methods for case 6.

Figure 20.

Voltage levels at load buses using CSBO algorithm for cases 4–6.

The execution time (CPU) is a useful way to compare how well different algorithms perform, with the simulation time for each method shown in the last column of Tables 7, 9, and 11 for cases 1, 2, and 3, where the proposed algorithm (CSBO) was the quickest and took the least time to reach near optimum results in comparison to other methods tested. Moreover, due to the complexity of the task, we use execution time as a metric of assessment.

Conclusion

The main objective of this work is to examine an overall pathway to optimize the conducting of energy networks with the participation of different power generators. A bio-inspired metaheuristic algorithm with the name of Circulatory System-Based Optimization (CSBO) was introduced and compared with other methods like MGO, ARO, and DMOA to process the OPF problem of a system connected to thermal, solar photovoltaic, wind, and joint SPV-hydroelectric generators. Two test systems were selected: the altered and adjusted IEEE 30 and 57 bus networks. To show the validity of the CSBO method, three target functions were adopted for each system: first, the reduction of the overall generation costs; second, the reduction of power losses; and lastly, the lowering of the total operating fees and emission effects simultaneously.

The novelty of this research lies in the successful implementation of CSBO algorithm as a robust alternative for managing the multi-objective complexities of hybrid energy systems. Unlike traditional metaheuristics, the unique dual-mode pressure-flow mechanics of CSBO allow it to bridge the research gap identified in existing literature, specifically the need for high-precision convergence in large-scale transmission networks. By effectively balancing technical, economic, and environmental objectives while maintaining low computational costs.

The findings revealed a clear superiority of the suggested framework over all previous ones conducted in the same circumstances; the TGCs, the APLs, and the combined cost-emissions were all reduced to minimum values in both test systems compared with previous competing methods like MFO, PSO, AEO, INFO, BMO, and GTO. This research showcased the astonishing results of the CSBO algorithm’s resilience, accuracy, and ability to balance exploration and exploitation while avoiding premature convergence, underscoring its applicability in multi-objective function situations and various power systems. This is evident with the reduction of the computational time (CPU).

In a broader global context, the findings offer significant implications for the transition toward decarbonized energy systems. By managing the stochastic nature of solar and wind energy, the CSBO-based approach enables grid operators to integrate higher shares of renewables without compromising system reliability. The reduction in total generation costs and power losses translates to lower operational expenditures for utilities, which can ultimately reduce energy prices for consumers. Furthermore, the ability to minimize environmental emissions directly supports international climate goals, offering a scalable decision-making tool for policymakers and engineers working to modernize aging infrastructure into resilient, intelligent microgrids. Future research will focus on the real-time implementation of the CSBO algorithm using Hardware-in-the-Loop (HIL) simulation to further validate its performance under physical hardware constraints and communication delays. CSBO method could also be modified, improved, or hybridized to obtain an even better-performing algorithm and utilize it for a wide range of purposes like distribution networks, microgrids, EV charging infrastructure, and energy storage units.

Footnotes

Acknowledgements

The authors extend their appreciation to the Ministry of Higher Education and Scientific Research, Algeria, for supporting this work through the project PRFU, under Grant No. A01L07UN100120230001. The first author would also like to express his deepest gratitude to Dr Souhil MOUASSA for his outstanding guidance, continuous support, and valuable contributions throughout the completion of this research work.

ORCID iDs

Souhil Mouassa

Hacene Mellah

Ethical considerations

All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

Credit statement (Author contribution)

Anwar Fellahi: Conceptualization, methodology, formal analysis, investigation, data curation, visualization, writing original draft, writing, review, and editing.

Souhil Mouassa: Supervision, conceptualization support, software development, methodology guidance, writing, resources, review, and editing.

Abdelghani Yahiou: Validation, review, and editing.

Hacene Mellah: Supervision, validation, resources, review, and editing.

All authors have read and approved the final manuscript.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Higher Education and Scientific Research, Algeria, for supporting this work through the project PRFU, under Grant No. A01L07UN100120230001.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.*

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