Tackling optimal power flow in modern power systems using a new optimization strategy

Abstract

This paper examines the intricate issue of Optimal Power Flow (OPF) optimization concerning the incorporation of renewable energy sources (RESs) into power networks. We present the Boosting Circulatory System Based Optimization (BCSBO) method, a novel modification of the original Circulatory System Based Optimization (CSBO) algorithm. The BCSBO algorithm has innovative movement techniques that markedly improve its exploration and exploitation skills, making it an effective instrument for addressing intricate optimization challenges. The suggested technique is thoroughly assessed utilizing five different objective functions alongside the IEEE 30-bus and IEEE 118-bus systems as test examples. The performance of the BCSBO algorithm is evaluated against many recognized optimization approaches, including CSBO, Moth-Flame Optimization (MFO), Particle Swarm Optimization (PSO), Thermal Exchange Optimization (TEO), and Elephant Herding Optimization (EHO). For the first case with minimizing the fuel cost associated with the thermal power generators, the total cost reported by the BCBSO is obtained as \mathrm{\$}781.8610, which is lower than other algorithms. For the second case, aimed at minimizing the total generating cost while also imposing a fixed carbon tax for thermal units, the derived total cost by the BCBSO is \mathrm{\$}810.7654. For the third case, aimed at minimizing the total cost considering prohibited operating zones of thermal units with RESs, the obtained total cost using the BCBSO is \mathrm{\$}781.9315. For case 4, with network losses included, the value of total costs obtained using the BCBSO is \mathrm{\$}880.4864. The value of total costs considering voltage deviation in case 5 is also obtained as \mathrm{\$}961.4354. For the IEEE 118-bus test system, the total cost is obtained \mathrm{\$}103,415.9315 using the BCBSO. These values reported by the BCBSO are all lower than those obtained by other methods addressed in this paper. The findings highlight the BCSBO algorithm’s potential as a crucial tool for enhancing power systems with renewable energies.

Graphical abstract

Keywords

optimal power flow (OPF) / optimization / boosting circulatory system based optimization / solar energy / wind energy

1 Introduction

Optimal power flow (OPF) is a useful method for both designing and managing power systems. Since its conception in 1962, the OPF has sparked great interest among academics in electrical network optimization (Esmaeel Nezhad et al., 2022; Ahmadi et al., 2014; Muttaqi et al., 2014). At the core of the OPF challenge is a critical mission: to reveal the essential control settings that beautifully harmonize a symphony of chosen goal functions. These goals, similar to the notes of a grand orchestration, often include the beautiful reduction of fuel costs, pollution expenditures, power losses, and the steady maintenance of an optimum voltage profile. All of this, of course, takes place under the watchful gaze of a complex dance of physical and operational constraints. The decision parameters, akin to the skilled dancers of this performance, encompass a diverse cast of characters: the real power outputs at the generator buses (with the exception of the master slack bus), the voltage magnitudes gracing every generator node, the artful tap changers within transformers, and the subtle shunt compensators. Together, these elements create a sublime choreography that seeks to attain the apex of operational efficiency and elegance in the realm of power systems (Razavi et al., 2018, Charwand et al., 2016). To tackle OPF problems, numerous mathematical programming methods have been employed, including interior-point methods, Newton-based techniques, linear programming, and nonlinear programming. Nevertheless, traditional techniques employed to tackle the OPF issue lean upon straightforward and differentiable objective functions. Additionally, these methodologies exclusively contemplate thermal power sources as the sole energy providers. In truth, within the sphere of contemporary power systems, the OPF quandary frequently assumes a nonlinear and possibly non-differentiable nature. As a result, it presents a substantial hurdle for optimization methodologies, particularly those of the conventional ilk. In response to these restrictions, heuristic strategies have surfaced as alternative avenues for addressing the OPF puzzle. These methods proffer the advantage of procuring nearly optimal solutions, regardless of the differentiability aspect of the problem.

The conventional OPF problem primarily revolves around power stations that utilize fossil fuels. In the past 20 years, renewable energy sources (RESs) have experienced significant growth, influenced by various factors. These factors encompass increased energy demand, the necessity to reduce greenhouse gas emissions such as NOx and SOx, the progress of deregulation and liberalization in electricity markets, and the appealing cost-efficiency of renewable energy (Javadi et al., 2021; Esmaeel Nezhad et al., 2025). Clearly, wind and solar energy stand out as the most promising alternatives to fossil fuels for power generation. The swift growth in the adoption of RESs can be attributed to the progress in wind power technology (WT) and solar photovoltaic (PV) systems, leading to decreased installation expenses (Esmaeel Nezhad et al., 2024a, Nezhad et al., 2022, Esmaeel Nezhad et al., 2024b)

Over time, heuristic optimization algorithms have emerged as effective solutions to the various optimization problems in various fields of science in the real world like OPF (Naderi et al., 2021) and optimal reactive power dispatch (ORPD) (Saddique et al., 2020; Ghasemi et al., 2022a) optimization problems in the power systems, taking inspiration from natural events, animal behaviors, and evolutionary concepts. A number of these metaheuristics and optimization algorithms include: liver cancer algorithm (LCA) (Houssein et al., 2023), parrot optimizer (PO) (Lian et al., 2024), slime mold algorithm (SMA) (Li et al., 2020), lungs performance-based optimization (LPO) (Ghasemi et al., 2024b), moth search algorithm (MSA) (Mohamed et al., 2017), hunger games search (HGS) (Yang et al., 2021), geyser inspired Algorithm (GEA) (Ghasemi et al., 2023b), and rime optimization algorithm (RIME) (Su et al., 2023). These algorithms give realistic and effective problem-solving solutions for various OPF problems with different test functions in electrical networks of different sizes. For example, PSOGSA, which combines Particle Swarm Optimization (PSO) and Gravitational Search Algorithm (GSA), has been developed for solving OPF, including FACTS devices and RESs (Hassan et al., 2021), or a new combining Firefly-Bat Algorithm (HFBA-COFS) (Chen et al., 2019a). Optimization studies were conducted to demonstrate the efficacy of the CPSOGSA approach in comparison to other strategies, such as the moth swarm algorithm (MSA), gray wolf optimizer (GWO), and whale optimization algorithm (WOA). Ullah and associates used a hybrid methodology known as PPSOGSA, which blends GSA with phasor PSO (PPSO). In electrical power networks with solar and wind energy sources, this method was used to resolve the OPF problem (Ullah et al., 2019). The use of social spider optimization (SSO) algorithms is a further method for resolving the OPF problem (Nguyen 2019), and the developed GWO (DGWO) (Abdo et al., 2018). In a separate study, Salkuti presented an optimizer to multi-objective OPF (MOOPF) under different operational conditions by the glowworm swarm optimization (GSO) technique and has been tested on the wind energy-integrated IEEE 30-bus and 300-bus electrical networks (Salkuti 2019). There are different algorithms used so far such as Gorilla troops algorithm (GTO) applied to different OPF problems considering uncertainty of RESs in Adrar’s isolated power network (Mouassa et al., 2023), INFO algorithm for OPF in hybrid IEEE 30-bus and IEEE 57-bus power networks with RESs (Belagra et al., 2024), a new slim mold optimizer (SMO) used to solve OPF in Algerian electricity network (Mouassa et al., 2022), as well as the self-adaptive bonobo optimizer (SaBO) (Kouadri et al., 2024) and a new enhanced Kepler optimization algorithm (EKOA) (Abid et al., 2024a) for OPF with RESs in the IEEE 30-bus and the Algerian DZA 114-bus power networks, respectively. Also, dwarf mongoose optimizer (DMO) (Mouassa et al., 2024), turbulent flow in water-based optimization (TFWO) (Sarhan et al., 2022; Zahedibialvaei et al., 2023; Alghamdi 2023; Ghasemi et al., 2020), artificial bee colony (ABC) (Mouassa and Bouktir 2015), non-dominated sorting KOA (NSKOA) (Abid et al., 2024b), and whale migration algorithm (WMA) (Ghasemi et al., 2024a) have been deployed for solving various OPF problems in different networks. Additionally, a recently introduced enhanced Honey Bee Mating search algorithm (MHBMO) algorithm, as presented in reference (Niknam et al., 2011), is designed to address the dynamic OPF (DOPF) problem within power systems. This algorithm considers the valve-point effects and has been tested on the 14, 30, and 118-bus electrical networks. The results demonstrated the effectiveness of MHBMO in solving DOPF.

The algorithm’s performance was thoroughly assessed on both the Indian Utility and IEEE 30-bus test systems across a diverse set of optimization functions (Kathiravan and Kumudini Devi 2017) and the DEPSO algorithm (Duman et al., 2020). In (Sarda et al., 2023), a hybrid optimization method named the Cross Entropy-Cuckoo optimization method (CE-CSA) made its debut.

Furthermore, their endeavors extended to deploying the Constrained Multi-Objective Population External Optimization method (CMOPEO) in grappling with the complex OPF issue entwined with wind and solar energy systems. This advanced algorithm underwent rigorous examination within the confines of the IEEE 30-bus test system, subject to an array of diverse test scenarios, as documented in the reference (Chen et al., 2019b). In a parallel line of inquiry, the Chaotic Bonobo Optimizer (CBO) (Hassan et al., 2022) diligently contemplated optimizing three distinct objective functions: minimizing total operating cost, emissions, and power losses. As proposed, the bird swarm algorithm (BSA) by (Ahmad et al., 2021) consistently achieved more stable and accurate outcomes compared to other algorithms.

A novel adaptive Gaussian TLBO (AGTLBO), presented in reference (Alanazi et al., 2022), successfully addressed the OPF problem and significantly enhanced the performance of the conventional TLBO. The results revealed that the AGTLBO exhibited superior efficiency and effectiveness compared to some new approaches mentioned in the existing paper. To grapple with the multifaceted multi-objective challenge, a Multi-objective Electromagnetism-like Algorithm (MOELA) was meticulously crafted, grounded in the principles of Pareto dominance and an external archive strategy. This ingeniously devised approach underwent a rigorous assessment of the well-established IEEE 30-bus electrical network. Another approach worth mentioning is a new TLBO algorithm enhanced with Lévy mutation (LTLBO) (Ghasemi et al., 2015). The efficacy of this approach underwent a comprehensive and systematic investigation involving in-depth scrutiny and rigorous evaluation. This analysis encompassed assessments on the canonical IEEE 30-bus and IEEE 57-bus electrical networks and entailed the consideration of a diverse array of objective functions. It has also been compared to algorithms described in previous studies.

Recently, Ghasemi and his colleagues have designed and successfully proposed a multi-layer intelligent optimization algorithm called circulatory system-based optimization (CSBO) algorithm in 2022 (Ghasemi et al., 2022b). So, in this article, an advanced, improved, and powerful version of this algorithm called the boosting CSBO (BCSBO) algorithm is used to solve and optimize various problems of OPF in a modified standard power system, the IEEE 30-bus network (Biswas et al., 2018). To show the optimization strength of this new effective optimization strategy in solving different OPF problems, we have compared its performance and the obtained simulation results with several powerful modern and new algorithms including particle swarm optimization (PSO), moth-flame optimization (MFO) (Mirjalili 2015), thermal exchange optimization (TEO) (Kaveh and Dadras 2017), and elephant herding optimization (EHO) (Wang et al., 2016) as well as the original version of the CSBO (Ghasemi et al., 2022b).

The suggested new BCSBO-inspired meta-heuristic has shown outstanding performance, especially in search spaces with complex structures and various local solution traps. The BCSBO algorithm has been proven to outperform its competitors through solution studies involving different objective functions and scenarios. The BCSBO has been applied to address OPF, which consists in incorporating RESs into the solution.

The rest of this paper is structured as follows. In Section 2, we provide the formulation of the OPF problem, explicitly focusing on integrating wind and solar energy systems. Section 3 introduces the power models for wind and solar energy. Section 4 is divided into two sub-sections, where we discuss the CSBO and BCSBO algorithms. Additionally, in Section 5, we elucidate the experimental configurations and the benchmark criteria considered during the experimental studies. In the paper’s final section, specifically Section 6, we present the study’s conclusions and discuss potential future directions for further research.

2 Renewable energy-based OPF modeling

The fundamental objective of OPF is to determine the optimal settings for the control variables, with the overarching goal of minimizing a defined objective function. Concurrently, this optimization endeavor must ensure the fulfillment of both inequality and equality limits. Conventionally the mathematical formulation of this optimization challenge is conventionally articulated as follows (Guvenc et al., 2021):

(1)\mathrm{m}\mathrm{i}\mathrm{n}{f}_{\mathrm{o}\mathrm{b}\mathrm{j}}\left(G,H\right),

subject to:

(2)\{\begin{array}{c}a\left(G,H\right)=0\\ b\left(G,H\right)\le 0\end{array}.

In the provided expression, f_obj (G,H) denotes the OPF problem, while G and H respectively signify the sets of dependent and independent parameters. Furthermore, a(G,H) and b(G,H) represent the equality and inequality limits associated with the problem at hand.

2.1 Dependent variables

G parameters have been given in the below:

(3)G=\left[\begin{array}{c}{P}_{T{h}_1},V_{L_1},...,V_{L_{NPQ}},Q_{T{h}_1},...,Q_{T{h}_{NTHG}},Q_{W{S}_1},...,Q_{W{S}_{NW}},\\ Q_{P{V}_1},...,Q_{P{V}_{NPV}},S_{L_1},...,S_{L_{NTL}}\end{array}\right].

The location of the operational energy produced by the main power generator is denoted as the slack generator, P_{T{h}_1}. V_L represents the voltage measurements of the PQ buses, which help determine the reactive power generated by traditional thermal units Q_{Th}, wind power sources Q_{WS}, and the solar system Q_{\mathrm{P}\mathrm{V}}. Additionally, S_L denotes the overall power transmitted through the lines. In the test power system, NPQ, NTHG, NW, NPV, and NTL represent the quantities of PQ buses, thermal generating units, wind farms, solar energy systems, and transmission lines respectively.

2.2 H parameters

The formulation for the H parameters of OPF has been given as follows.

(4)\begin{array}{c}H=[P_{T{h}_2},...,P_{T{h}_{NTHG}},P_{W{S}_1},...,P_{W{S}_{NW}},P_{PV{S}_1},...,\\ P_{PV{S}_{NPV}},V_{G_1},...,V_{G_{NG}}].\end{array}

In this specific engineering context, P_Th symbolizes the active power output from the TH units, excluding the slack generator. Moreover, P_WS and P_PVS represent the active power contributions generated by WT and PV sources (RESs), respectively. Additionally, V_G corresponds to the voltage magnitudes linked to all generator buses, encompassing the generation units. Lastly, NG denotes the numerical count of generator buses, encompassing all the generation units.

2.3 Test system

The specifications and parameters associated with the IEEE 30-bus test system, which includes provisions for wind and solar energy systems, are meticulously detailed in Table 1.

2.3.1 Cost model of the TH generators

The overall expenditure on fuel for all power generation units is expressed using a polynomial quadratic function, as presented in Eq. (5), which depends on the active power output generated. Equation (6) provides the fuel cost model for TH resources taking into account valve-point effects. In this equation, the coefficients m_i, n_i, and o_i describe the costs associated with the ith TH unit, while the coefficients p_i and r_i represent the impact of VPL effects.

(5)CF\left(P_{Th}\right)=\sum _{i=1}^{NTHG}\left(m_i+n_i{P}_{T{h}_i}+o_i{P}_{T{h}_i}^2\right),

(6)C{F}_1\left(P_m\right)=\sum _{i=1}^{NTHG}\left(\begin{array}{c}{m}_i+n_i{P}_{m_i}+o_i{P}_{m_i}^2\\ +\left|p_i\times \sin \left(r_i\times \left(P_{m_i}^{min}-P_{m_i}\right)\right)\right|\end{array}\right).

2.3.2 Emission and carbon tax model

TH plants that utilize different types of fossil fuels emit various harmful pollutants. Recently, one of the main objectives of the OPF is to optimize these emissions while ensuring that the generated power is sufficient to meet the electrical power system’s demand. This emission is mathematically represented in Eq. (7) (Guvenc et al., 2021).

In light of the growing apprehension surrounding global climate change, a carbon tax model has been factored into the analysis. The computation of the emission cost entails the summation of the carbon tax value to the total emission value, as succinctly defined in Eq. (8). In this equation, F_E signifies the total emission, C_E denotes the emission cost, and C_tax stands for the tax values, respectively (Biswas et al., 2018).

(7)F_E=\sum _{i=1}^{NTHG}\left(\begin{array}{c}\left({\sigma }_i+{\beta }_i{P}_{T{h}_i}+{\tau }_i{P}_{T{h}_i}^2\right)\times 0.01+\\ {\omega }_i{e}^{\left({\mu }_i{P}_{T{h}_i}\right)}\end{array}\right),

(8)C_E=F_E\times C_{tax}.

In the provided context, the symbols σ_i, β_i, τ_i, ω_i, and μ_i correspond to the emission coefficients associated with the ith TH unit.

2.3.3 Forbidden Operational Regions (POZs)

For a TH unit utilizing fossil fuels, the presence of POZs can be delineated in Eq. (9):

(9)\begin{array}{l}{P}_{T{h}_{i,min}}⩽P_{T{h}_i}⩽P_{T{h}_{i,1}}^L\\ P_{T{h}_{1,y-1}}^U⩽P_{T{h}_i}⩽P_{T{h}_{i,y}}^L\\ P_{T{h}_{i,vi}}^U⩽P_{T{h}_i}⩽P_{T{h}_{i,max}}^L\end{array}y=2,3,4,...,v_i.

In the given equation, “vi ” denotes the total count of POZs, “y” represents the specific POZ under consideration, and P_{T{h}_i,y-1}^U and P_{T{h}_i,y}^L correspond to the upper and lower limits of the (y−1)th POZ associated with the i-th unit.

Figure 1 graphically illustrates the characteristic curves for fuel cost, delineating the impact of VPEs and the presence of POZs within the thermal generator system.

2.3.4 Direct cost for PVs and WTs

The direct cost model for a wind unit can be concisely formulated as a linear function contingent upon the scheduled power output (Biswas et al., 2018). In this equation, the direct cost function for wind power is denoted as “D{C}_{WP,i},” while the direct cost coefficient is represented by “wsh,i.” The scheduled power output from the wind power system is symbolized as P_{WS,i}.

(10)D{C}_{WP,i}=C{F}_{WP,i}{P}_{WS,i}=wsh,i\times P_{WS,i}.

Similarly, the direct cost model for a PV unit has been described by Eq. (11). In the context of the PV system, the direct cost model can be succinctly characterized. The elements within this model encompass the direct cost function for PV power, designated as “D{C}_{PV,i}” the direct cost coefficient denoted by “pvsh,i” and the scheduled power output of the PV, represented as “{P_{PVS}}_{,i}” (Guvenc et al., 2021).

(11)D{C}_{PV,i}=C{F}_{PV,i}\left({P_{PVS}}_{,i}\right)=pvsh,i\times {P_{PVS}}_{,i}.

2.3.5 Stochastic cost models for PV and WT units

Moving on to uncertainty cost models, the situations of overestimation and underestimation in renewable energy sources have been considered as stochastic cost models for both PV and WT units. Equations (12) and (13) outline the stochastic cost models for WT (Guvenc et al., 2021).

(12)\begin{array}{rl}O{C}_{WP,i}=& C_{Ow,i}\left(P_{WS,i}-P_{wav,i}\right)\\ =& C_{Ow,i}{\int }_0^{P_{WS,i}}\left(P_{WS,i}-p_{w,i}\right)f_w\left(p_{w,i}\right)\mathrm{d}{p}_{w,i},\end{array}

(13)\begin{array}{rl}U{C}_{WP,i}=& C_{Uw,i}\left(P_{wav,i}-P_{WS,i}\right)\\ =& C_{Uw,i}{\int }_{P_{WS,i}}^{P_{wr,i}}\left(p_{w,i}-P_{WS,i}\right)f_w\left(p_{w,i}\right)\mathrm{d}{p}_{w,i}.\end{array}

In Eqs. (12) and (13), the overestimation and underestimation cost values are represented by O{C}_{WP,i} and U{C}_{WP,i}, respectively. Additionally, C_{Ow,i} and C_{Uw,i} show the uncertainty cost coefficients while P_{wr,i} and P_{WS,i} show the rated and available active energy of the specific WT. For the models of the PV unit, the mathematical model was obtained using the approach method described in Refs. (Guvenc et al., 2021). The overestimation and underestimation situations are expressed as follows:

(14)

(15)

where P_{PVav,i} indicates the available power of the ith PV, C_{Opv,i} and C_{Upv,i} represent the uncertainty cost coefficients, and O{C}_{PV,i} and U{C}_{PV,i} describe the over- and underestimation cost values.

2.4 Objective functions

2.4.1 VPEs considering fuel cost objective function

Eqation (16) represents the modeled cost of OPF. This function encompasses the cost values associated with the VPEs of the TH units, as well as the cost values of the PV and WT units.

(16)\begin{array}{rl}{F}_{obj1}=& C{F}_1\left(P_{th}\right)+\sum _{i=1}^{NW}\left(D{C}_{WP,i}+O{C}_{WP,i}+U{C}_{WP,i}\right)\\ & +\sum _{i=1}^{NPV}\left(D{C}_{PV,i}+O{C}_{PV,i}+U{C}_{PV,i}\right).\end{array}

2.4.2 Objective functions considering tax and emission

The defined objective function for the formulated OPF is precisely outlined in Eq. (17)

(17)F_{obj2}=F_{obj1}+C_E.

2.4.3 Objective function considering POZs

Here, the conventional OPF objective function associated with TH units featuring POZs has been introduced as the designated OPF function, as expressed in Eq. (18).

(18)\begin{array}{rl}{F}_{obj3}=& CF\left(P_{th}\right)+\sum _{i=1}^{NW}\left(D{C}_{WP,i}+O{C}_{WP,i}+U{C}_{WP,i}\right)\\ & +\sum _{i=1}^{NPV}\left(D{C}_{PV,i}+O{C}_{PV,i}+U{C}_{PV,i}\right).\end{array}

2.4.4 Network losses

The minimizing of the network losses of the electrical network is considered as follows:

(19)F_{obj4}=P_{loss}=\sum _{s=1}^{NTL}{G}_{s(\mathit{i}\mathit{j})}\left(V_i^2+V_j^2-2V_i{V}_j\cos \left({\delta }_{ij}\right)\right).

In the given equation, “G_{s(\mathit{i}\mathit{j})}” corresponds to the conductance of the s-th transmission line linking buses i and j, while {\delta }_{ij} quantifies the voltage angle differential between buses i and j.

2.4.5 Voltage deviation (VD)

The proposed OPF problem utilizes a calculation, as demonstrated in Eq. (20), to determine the voltage deviation value of contemporary electrical grids.

(20)F_{obj5}=VD=\sum _{i=1}^{NPQ}\left|V_{L_i}-1\right|.

2.5 OPF constraints

2.5.1 Equality constraints

The equality constraints inherent to the formulated OPF can be aptly expressed as follows:

(21)P_{Gi}-P_{Di}-V_i\sum _{j=1}^{N_{\mathrm{b}\mathrm{u}\mathrm{s}}}{V}_j\left(G_{ij}\cos \left({\delta }_i-{\delta }_j\right)+B_{ij}\sin \left({\delta }_i-{\delta }_j\right)\right)=0,

(22)\begin{array}{c}{Q}_{Gi}+Q_{SHi}-Q_{Di}-V_i\sum _{j=1}^{Nbus}{V}_j(G_{ij}\sin ({\delta }_i-{\delta }_j)\\ -B_{ij}\cos ({\delta }_i-{\delta }_j))=0.\end{array}

In the context of these equations, various variables and parameters hold significance:

• P_Gi, and P_Di show the active power outputs of the ith generating unit, encompassing thermal, wind, and PV units, as well as the load buses.

• Q_Gi, Q_SHi, and Q_Di characterize the reactive power contributions from the ith generating unit, including thermal, wind, and PV units, shunt VAR compensators, and load buses within the electrical grid.

• N_bus stands for the total count of buses within the power system.

• V_i and V_j represent the voltage magnitudes at the ith and jth buses.

• δ_i−δ_j signifies the angular disparity between the voltage phasor values at the ith and jth buses.

• G_ij and B_ij describe the conductance and susceptance attributes governing the transmission line connecting the ith and jth buses.

2.5.2 Inequality limits

1) Generator limits

The boundaries for the permissible ranges of active and reactive power, and voltage magnitudes for all the generators are explicitly specified and denoted in Eq. (23).

(23)\{\begin{array}{llll}{P}_{T{h}_i,min}⩽P_{T{h}_i,max}& i=1:NTHG& & \\ P_{W{S}_i,min}⩽P_{W{S}_i}⩽P_{W{S}_i,max}& i=1:NW& & \\ P_{P{V}_i,min}⩽P_{P{V}_i}⩽P_{P{V}_i,max}& i=1:NPV& & \\ Q_{T{h}_i,min}⩽Q_{T{h}_i}⩽Q_{T{h}_i,max}& i=1:NTHG& & \\ Q_{iW{S}_i,min}⩽Q_{W{S}_i}⩽Q_{W{S}_i,max}& i=1:NW& & \\ Q_{P{V}_i,min}⩽Q_{P{V}_i}⩽Q_{P{V}_i,max}& i=1:NPV& & \\ V_{G_i,min}⩽V_{G_i}⩽V_{G_i,max}& i=1:NG& & \end{array}

2) Security limits

The voltage magnitude at each of the PQ-type buses must conform to specified boundaries, and the apparent power flow on each transmission line is subject to constraints related to its maximum capacity. In these equations, “V_{L_i,min}” and “V_{L_i,max}” respectively denote the down and upper limits of the voltage for the ith PQ bus, while “S_{L_i}” and “S_{L_i,max}” represent the apparent power and the maximum apparent power capacity for the ith transmission line.

(24)\{\begin{array}{lll}{V}_{L_i,min}⩽V_{L_i}⩽V_{L_i,max}& i=1:NPQ& \\ S_{L_i}⩽S_{L_i,max}& i=1:NTL& \end{array}.

The fitness function for SCOPF, encompassing thermal, PV, and WT power units, can be succinctly described as presented in Eq. (25). Where λ_VPQ, λ_Pslack, λ_QTH, λ_QWS, λ_QPV, and λ_SL are the penalty coefficients.

(25)\begin{array}{rl}J=& f_{a{b}_r}(G,H)+{\lambda }_{r{p}_0}\sum _{i=1}^{NPO}{(V{L}_i-V{L}_i^{im})}^2+{\lambda }_{p_{i-1}}(P_{m_1}-P_{m_1}^{im}{)}^2\\ & +{\lambda }_{om}\sum _{i=1}^{NTHG}{(Q_{n_i}-Q_{n_i}^{im})}^2+{\lambda }_{ows}\sum _{i=1}^{NW}{(Q_{n{s}_i}-Q_{n{s}_i}^{im})}^2\\ & +{\lambda }_{0PV}\sum _{i=1}^{NPV}{(Q_{P{V}_i}-Q_{P{V}_i}^{in})}^2+{\lambda }_{SL}\sum _{i=1}^{NTL}{(S_{L_i}-S_{L_i}^{in})}^2.\end{array}

3 Wind-PV uncertainty and power models

The distribution for wind speed has been characterized by the Weibull probability density function (PDF), as elucidated in Eq. (26). Within this equation, “vw” signifies the wind speed, while “ξ” and “ψ” represent the shape and scale factors, respectively (Biswas et al., 2017; Guvenc et al., 2021).

(26)f_v\left(v_w\right)=\left(\frac{\xi }{\psi }\right){\left(\frac{v_w}{\psi }\right)}^{\xi -1}\left({\mathrm{e}}^{-{\left(\frac{v_w}{\psi }\right)}^{\xi }}\right).

In Fig. 2, you can see the outcomes of the Weibull fitting for wind frequency distributions. These results were generated using an 8000-iteration Monte Carlo simulation (Guvenc et al., 2021). The identified method for determining the output power of the WT is as follows:

(27)p_w\left({\nu }_w\right)=\{\begin{array}{lll}0& {\nu }_w⩽{\nu }_{w,w}\mathrm{a}\mathrm{n}\mathrm{d}{\nu }_w>{\nu }_{w,w}& \\ \left({\displaystyle \frac{{\nu }_w-{\nu }_{w,w}}{{\nu }_{w,w}-{\nu }_{w,w}}}\right)\ast p_w& {\nu }_{w,w}⩽{\nu }_w⩽{\nu }_{w,w}& \\ p_{w,w}& {\nu }_{w,w}⩽{\nu }_w⩽{\nu }_{w,w}& \\ 0& \nu ⩾{\nu }_{w,w}& \end{array},

where p_wr, v_w,in, v_w,out, and v_w,r show, respectively, the rated power, cut-in, cut-out and rated wind speeds. The energy production of a WT system is partitioned into discrete intervals contingent upon wind speeds, as delineated in Eq. (27). The associated probability values for these intervals are meticulously elaborated between Eqs. (28) and (30).

(28)f_{-}\left(p_{-}\right)\left\{p_{-}=0\right\}=1-\exp \left(-{\left(\frac{{\nu }_{-\mathrm{i}\mathrm{n}}}{\psi }\right)}^2\right)+\exp \left(-{\left(\frac{v_{-\mathrm{i}\mathrm{n}}}{\psi }\right)}^2\right),

(29)f_w(p_w)\{p_w=p_w\}=\exp \left(-{\left(\frac{{\nu }_{w,x}}{\psi }\right)}^2\right)-\exp \left(-{\left(\frac{{\nu }_{w,\mathrm{o}\mathrm{u}\mathrm{t}}}{\psi }\right)}^2\right),

(30)\begin{array}{rl}{f}_w\left(p_w\right)=& \left[\frac{\xi \left({\nu }_{w,\mathrm{r}}-{\nu }_{w,\mathrm{i}\mathrm{n}}\right)}{{\psi }^{\xi }{p}_{wr}}\right]\times {\left({\nu }_{w,\mathrm{i}\mathrm{n}}+\left(\frac{p_w}{p_{wr}}\right)\left({\nu }_{w,\mathrm{r}}-{\nu }_{w,\mathrm{i}\mathrm{n}}\right)\right)}^{\xi -1}\\ \\ & \times \exp \left(-{\left(\frac{{\nu }_{w,\mathrm{i}\mathrm{n}}+\left({\displaystyle \frac{p_w}{p_{wr}}}\right)\left({\nu }_{w,\mathrm{r}}-{\nu }_{w,\mathrm{i}\mathrm{n}}\right)}{\psi }\right)}^{\xi }\right).\end{array}

The power production of a wind farm is partitioned into discrete intervals contingent upon wind speeds, as delineated in Eq. (27). The associated probability values for these intervals are meticulously elaborated between Eqs. (28) and (30), respectively (Guvenc et al., 2021).

To express the power output of the PV energy systems in relation to solar irradiation, the Lognormal PDF was utilized. The mathematical representation of the probabilistic model and output power of the solar system can be observed in Eqs. (31) and (32) (Guvenc et al., 2021).

(31)f_{G_{PV}}\left(G_{PV}\right)=\frac{1}{G_{PV}\mathrm{\Omega }\sqrt{2\pi }}\exp \left(\frac{-{\left(\ln G_{PV}-\xi \right)}^2}{2{\mathrm{\Omega }}^2}\right)\mathrm{f}\mathrm{o}\mathrm{r}{G}_{PV}>0.

In Fig. 3, you can observe the frequency distribution and lognormal probability of solar irradiation, which were obtained through an 8000-iteration Monte Carlo simulation.

(32)

The Lognormal PDF is characterized by the mean (ζ) and standard deviation (Ω) values, as illustrated in Table 2. In this context, G_PV, G_{P{V}_{\mathrm{s}\mathrm{t}\mathrm{d}}} and P_{P{V}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}}} refer to the probability value, standard solar irradiance, and rated power of the solar system, respectively. At bus 13, these values are defined as 800 W/m² and 50 MW. Additionally, the R_C value is set at 120 W/m². Figure 4 displays a histogram representing the stochastic output power of PV. The line on the graph represents the intended power that PV is designed to transmit to the system. It can be very vital to note that the planned PV power is a variable quantity.

4 Improved meta-heuristic

4.1 The original meta-heuristic

4.1.1 CSBO algorithm

Initially, the CSBO algorithm, similar to other meta-heuristic algorithms, commences by creating an initial population known as blood masses BM_i. This population is tailored to address a specific problem and consists of dimensions D and Npop. The values of the problem parameters are randomly generated within the minimum B{M}_{min} and maximum B{M}_{max}values, similar to Eq. (33). This initial population, akin to blood particles or masses within the body, fulfills a crucial role as previously mentioned.

(33)\begin{array}{c}B{M}_i=B{M}_{min}+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(1,D\right)\times \left(B{M}_{max}-B{M}_{min}\right)\\ i=1:Npop.\end{array}

4.1.2 Movement of blood mass in the veins

In the circulatory system, each blood mass flowing through the veins, referred to as BM_i, is influenced by an external force or pressure. As it moves, the mass always seeks to find a path with more favorable conditions, aiming to minimize its objective function (the amount of force or pressure it experiences). The phenomenon of clogged arteries in the heart can be understood as the entrapment of blood masses in local optimal solutions. However, in order to avoid such situations in reality, it is crucial for the body to continuously optimize the circulatory process. This particular stage of the circulatory cycle is accurately represented by modeling the positions of particles and their corresponding objective function values.

(34)\begin{array}{rl}B{M}_i^{\mathrm{n}\mathrm{e}\mathrm{w}}=& B{M}_i+K_{i1}\times p_i\times \left(B{M}_i-B{M}_1\right)+K_{23}\times p_i\\ & \times \left(B{M}_3-B{M}_2\right).\end{array}

(35)K_{ij}=\frac{F\left(B{M}_j\right)-F\left(B{M}_i\right)}{\left|F\left(B{M}_j\right)-F\left(B{M}_i\right)\right|+\epsilon }.

4.1.3 Population or blood mass flow in pulmonary circulation

As previously stated, the pulmonary system is responsible for handling deoxygenated blood, comparable to a lesser population in the context of optimization. Specifically, within the CSBO framework, during each iteration, the population undergoes sorting, and a certain number NR of the most vulnerable individuals enter the pulmonary circulation. These individuals are then directed toward the lungs to acquire oxygen.

(36)B{M}_i^{new}=B{M}_i+\left(\frac{randn}{it}\right)\times randc\left(1,D\right),i=1:NR.

In the equation above, the term “randn” represents a random number sampled from a normal distribution, “it” signifies the current iteration within the algorithm, “randc” designates a random vector originating from the Cauchy probability distribution, and “D” stands for the dimensionality of the optimization problem. Additionally, the pulmonary circulation modifies the “p” value for this population in the following manner:

(37)p_i=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(1,D\right),i=1:NR.

4.1.4 Population or blood mass flow in systematic circulation

As previously noted, a specific count of the weakest sorted population members, denoted as NR, are introduced into the pulmonary circulation. Concurrently, the remaining portion of the population, represented by NL (NL = Npop - NR), and possessing superior fitness values, are channeled into the systemic circulation. These individuals receive a fresh allocation to circulate throughout the system, as illustrated in the model below:

(38)\begin{array}{rl}& \mathrm{f}\mathrm{o}\mathrm{r}j=1:D\\ & ifrand>0.9\\ & B{M}_{i,j}^{\mathrm{n}\mathrm{e}\mathrm{w}}=B{M}_{1,j}+p_i\ast \left(B{M}_{3,j}-B{M}_{2,j}\right)\\ & \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\\ & B{M}_{i,j}^{\mathrm{n}\mathrm{e}\mathrm{w}}=B{M}_{i,j}\\ & \mathrm{e}\mathrm{n}\mathrm{d}\\ & \mathrm{e}\mathrm{n}\mathrm{d}\\ & i=1:NL\end{array}.

The systemic circulation likewise adjusts the “p” value for this subgroup of the population in the subsequent manner:

(39)p_i=\frac{F\left(B{M}_i\right)-F_{\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}}{F_{\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}}-F_{\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}},i=1:NL.

In this context, “F_{\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}” and “F_{\mathrm{B}\mathrm{e}\mathrm{s}\mathrm{t}}” respectively represent the most unfavorable and optimal values of the cost function achieved up to the current iteration. The optimization procedure will continue for the predetermined number of iterations. Much like other meta-heuristic algorithms, each member within the population will adopt the novel position solely if it results in an improved objective function value. The pseudo-code for the CSBO has been succinctly encapsulated in Algorithm 1, as depicted in Table 3, also, to further understand the framework of the proposed extended optimization strategy and optimization code, the flowchart of this proposed strategy is presented in Fig. 5.

4.2 Boosting circulatory system-based optimization

The original CSBO algorithm faced challenges in quickly converging and avoiding local optima, especially for complex problems like economic load dispatch. To address these issues and enhance the algorithm’s performance, we introduced improved strategies outlined in Eqs. (40) and (41). Through experiments and simulations, we consistently observed that our proposed BCSBO algorithm outperformed the original CSBO algorithm. Specifically, we incorporated additional vectors in the blood movement and population flow phase in systematic circulation, i.e., K_{54}\times p_i\times \left(B{M}_5-B{M}_4\right) and p_i\times \left(B{m}_{5,j}-B{m}_{4,j}\right), which significantly improved the algorithm’s effectiveness. The results demonstrate the positive impact of these auxiliary vectors on enhancing the original algorithm.

Improvements made on CSBO to achieve a power optimizer (BCSBO):

A new boosting movement of blood mass in the veins strategy:

(40)\begin{array}{rl}B{M}_i^{\mathrm{n}\mathrm{e}\mathrm{w}}=& B{M}_i+K_{i1}\times p_i\times \left(B{M}_i-B{M}_1\right)+K_{23}\times p_i\\ & \times \left(B{M}_3-B{M}_2\right)+K_{54}\times p_i\times \left(B{M}_5-B{M}_4\right).\end{array}

The proposed boosting population flow in systematic circulation strategy:

(41)\mathit{B}{\mathit{m}}_{\mathit{i},\mathit{j}}^{\mathrm{n}\mathrm{e}\mathrm{w}}=\mathit{B}{\mathit{m}}_{1,\mathit{j}}+{\mathit{p}}_{\mathit{i}}\times \left(\mathit{B}{\mathit{m}}_{3,\mathit{j}}-\mathit{B}{\mathit{m}}_{2,\mathit{j}}\right)+{\mathit{p}}_{\mathit{i}}\times \left(\mathit{B}{\mathit{m}}_{5,\mathit{j}}-\mathit{B}{\mathit{m}}_{4,\mathit{j}}\right).

5 Solving OPF by the proposed meta-heuristic method

5.1 OPF in the IEEE 30-bus test system

We have done a study on the fundamental CSBO, EHO, TEO, MFO, PSO, and BCSBO techniques using the IEEE 30-bus test system, as illustrated in Fig. 6. Our objective was to address the OPF problem in the presence of wind and solar energy systems. Additionally, we have obtained the control parameters for the optimization algorithms from relevant references, which are presented in Table 4. The system parameters for the IEEE 30-bus test system were obtained from the following references: (Guvenc et al., 2021; Biswas et al., 2017; Mohamed et al., 2017), and are provided in Table 5.

Table 6 displays the cost coefficients for overestimation and underestimation of wind and solar power systems. To calculate the load flow equations for the OPF, which encompasses thermal, wind, and solar generating systems, MATPOWER (Zimmerman et al., 2011) was employed. To statistically assess the simulation results obtained, all optimization algorithms were executed 30 times for each test case of the proposed OPF problem. The simulation studies were conducted based on the test cases outlined below.

5.2 Case 1: Optimizing the fuel cost with VPEs with RESs

In this particular scenario, the aim is to minimize the fuel cost associated with the thermal power generators. Table 7 provides a detailed comparison of control parameters and their respective optimum values attained by six algorithms: EHO, TEO, MFO, PSO, CSBO, and the suggested BCSBO technique. The main objective is to reduce gasoline expenses while complying with all issue limits. The table presents the optimal, suboptimal (Max), mean (Mean), and standard deviation (Std.) values derived from 30 different executions of each method, offering a comprehensive statistical evaluation. The findings underscore the BCSBO method’s superior performance in attaining the lowest objective function value relative to all other algorithms. This supremacy reflects the algorithm’s ability to efficiently explore and utilize the search space. Moreover, the BCSBO algorithm exhibits reliable performance, as shown by its minimal standard deviation and its capacity to avoid constraint breaches. The primary benefit of the proposed BCSBO technique is its capacity to maintain a balance between exploration and exploitation, shown by its smooth convergence curve toward the ideal solution, seen in Fig. 7. In contrast to other methods that may demonstrate oscillations or slower convergence rates, the BCSBO approach attains rapid convergence to a high-quality solution with minimum processing resources.

5.3 Case 2: Optimizing the total cost with emission and carbon tax

The main objective of this particular case is to optimize the overall generating cost while also imposing a fixed carbon tax (C_tax) on the TH units due to their CO2 emissions, as mentioned in Eq. (8). The designated forced carbon tax is set at 20 ($/ton) (Biswas et al., 2017). Table 8 provides a detailed comparison of the control parameters and optimum values identified by many sophisticated optimization techniques, including EHO, TEO, MFO, PSO, CSBO, and the suggested BCSBO approach. This table elucidates the efficacy of different algorithms in tackling the OPF issue, emphasizing cost reduction while complying with system restrictions. Table 8 highlights a significant finding: all algorithms, including the suggested BCSBO approach, attained the minimum overall cost of 810.7654 ($/h). This exceptional result underscores the competitive efficacy of the BCSBO algorithm. What distinguishes the BCSBO technique is its exceptional ability to optimize efficiently while ensuring solution feasibility under established restrictions. A further notable aspect of the BCSBO algorithm’s performance is shown in Fig. 8, which demonstrates the method’s convergence behavior. The quick and seamless convergence curve demonstrates the algorithm’s efficacy in attaining the ideal answer. This swift convergence illustrates the algorithm’s computing efficiency and highlights its resilience in balancing exploration and exploitation throughout the optimization phase. The BCSBO method’s constant attainment of high-quality solutions, shown via its performance in several iterations, establishes it as a dependable and potent instrument for tackling intricate OPF challenges. Its efficiency, along with its adaptation to varied system circumstances, guarantees its relevance in numerous energy optimization contexts. The results shown in Table 8, together with the convergence analysis in Fig. 8, unequivocally demonstrate that the BCSBO approach is the most efficient optimizer for the OPF issue in this research. Its capacity to provide optimum solutions with efficiency and reliability highlights its potential for further applications in energy system optimization.

5.4 Case 3: Optimizing the total cost considering POZs of TH units with RESs

Table 9 presents a comprehensive study of the control parameters and results associated with total cost minimization, including the impact of restricted operating zones (POZs) for thermal units and the incorporation of wind-PV energy systems. The findings were obtained utilizing the suggested BCSBO algorithm in conjunction with several alternative methodologies, enabling a comprehensive comparison across 50 separate trials. Table 9 unequivocally illustrates the BCSBO algorithm’s enhanced efficacy in minimizing power losses and optimizing expenses. The BCSBO approach specifically achieved a minimal power loss of 781.9315 $/h, surpassing all rival algorithms. This underscores the algorithm’s capacity to adeptly manage the difficulties posed by the POZ limitations and the incorporation of renewable energy sources, demonstrating its efficacy in practical energy optimization contexts. Besides its quantitative benefits, the convergence characteristics of the BCSBO algorithm are shown in Fig. 9. The convergence curve of the BCSBO approach demonstrates a notable superiority over other algorithms, characterized by its swift advancement toward the ideal solution. This behavior highlights the algorithm’s efficacy in balancing exploration and exploitation, facilitating a seamless and expedited convergence process. The incorporation of POZ limitations and renewable energy systems adds considerable complexity to the optimization issue, necessitating resilient and adaptable optimization methodologies. The BCSBO algorithm’s consistent performance over several separate executions reinforces its dependability and robustness in tackling these difficulties. The findings in Table 9, corroborated by the convergence traits shown in Fig. 9, confirm that the BCSBO algorithm is an exceptionally efficient and dependable approach for optimizing energy systems with complex constraints. Its capacity to reduce costs and power losses while ensuring fast convergence makes it an essential instrument for addressing intricate power system optimization challenges.

5.5 Case 4: Considering network losses

The results for the decision parameters and total cost obtained from the innovative BCSBO algorithm for case 4 can be found in Table 10. This table also displays the optimal outcomes after conducting 30 independent implementations, along with the time taken to achieve the best result using the proposed BCSBO algorithm and other methods for case 4. Table 10 clearly demonstrates that the proposed BCSBO method attains the best value of (2.0741 MW), which is lower than the values obtained by other methods. Moreover, the standard deviation achieved by the proposed BCSBO algorithm (0.20) is significantly smaller compared to both the original CSBO algorithm (0.46) and the other methods. Additionally, Fig. 10 illustrates the convergence characteristic of the proposed BCSBO and other optimizers for this particular case. The BCSBO algorithm exhibits smoother convergence curves when compared to alternative algorithms.

5.6 Case 5: Optimizing V.D

Table 11 presents a comprehensive study of the decision parameters and voltage deviation (V.D.) derived from the IEEE 30-bus system including RESs using the proposed BCSBO algorithm. The table presents the findings from 30 separate simulation runs, contrasting the BCSBO approach with other optimization strategies. Table 11 illustrates the advantages of the BCSBO algorithm in reducing V.D. and attaining optimum system performance. The BCSBO method has superior results for essential statistical metrics, including standard deviation (Std.), indicating its consistency and resilience throughout several iterations. These results highlight the method’s efficacy in tackling the issues presented by the OPF problem in systems with integrated RESs. Additionally, Fig. 11 depicts the convergence characteristics of the proposed BCSBO algorithm in relation to other approaches. The convergence behavior of BCSBO demonstrates a swift and seamless progression toward the optimum solution, highlighting its efficacy in traversing the intricate optimization terrain of the OPF problem. The stability and rapid convergence further confirm the robustness of the BCSBO algorithm in harmonizing exploration and exploitation throughout the optimization process. The use of renewable energy sources in the IEEE 30-bus system brings complications, including intermittent power production and integration limits, necessitating that optimization methods be both adaptable and efficient. The findings indicate that the BCSBO approach not only achieves reduced voltage variations but also sustains consistent and steady performance, even under adverse settings. The results from Table 11, corroborated by the convergence trends in Fig. 11, decisively affirm the BCSBO algorithm as a highly efficient and resilient optimizer for addressing the OPF issue in power systems with renewable energy sources. Its capacity to get optimal results while facilitating fast convergence makes it an essential instrument for improving the efficiency and dependability of contemporary power networks.

5.7 Case 6: OPF in IEEE 118-bus test system

In the following simulation, we have implemented the optimization methods in the previous section, CSBO, EHO, TEO, MFO, PSO, and BCSBO techniques, on the IEEE 118-bus test system to test and demonstrate the performance and efficiency of the proposed improved metaheuristic for the various OPF problems (Duman et al., 2020; Ghasemi et al., 2023a). The best choice for the various control parameters of the algorithms used is the same as in the previous section, with the difference that the selected population for the algorithms is 75 and the number of iterations of the algorithms is 2000. The system parameters for the IEEE 118-bus test system were obtained from the following references (Duman et al., 2020; Ghasemi et al., 2023a).

Nine transformers, two reactors, 12 shunt capacitors, 50-four generators, and 186 branches make up the test network under analysis (Duman et al., 2020; Ghasemi et al., 2023a). The reactive power injections of 12 capacitors, nine transformer tap settings, the active power output of 54 generators, and bus voltage levels are among the 129 parameters that are optimized. The reactive power provided by the capacitors spans from 0 to 30 Mvar. All buses have their voltage levels kept between 0.94 and 1.06 per unit, transformer taps are calibrated between 0.90 and 1.10 per unit (Duman et al., 2020). This system integrates RESs across many busses. Solar power units are situated at buses 6, 15, and 34, and wind turbines are strategically located at busses 18, 32, 36, 55, 104, and 110 (Alghamdi and Zohdy, 2024). The optimal outcomes attained by several algorithms in this investigation are shown in Table 12. An examination of this table clearly indicates that the suggested technique exhibits enhanced optimization performance as the dimensions of the test network expand, unlike the majority of other methods. Consequently, it can be said that the strategy described in this paper is a successful and significant approach for future research initiatives. In addition, convergence trajectories of different methods for the IEEE 118-bus test system have been shown in Fig. 12. From this figure, it can be seen that, with increasing complexity and dimension of the OPF problem under consideration, the proposed BCSBO method still has a good convergence characteristic and is very competitive, especially compared to the initial optimization CSBO method.

5.8 Discussion

The performance of the BCSBO algorithm is evaluated against various meta-heuristic approaches under the same optimization circumstances, with results shown in Tables 7 to 12. Although the numerical findings indicate that BCSBO excels in attaining lower objective function values and quicker convergence in the majority of instances, a more profound comprehension of its behavior elucidates the underlying causes for these outcomes.

5.8.1 Efficiency of blood mass movement in Veins

The algorithm’s capacity to model the dynamic flow of blood masses under pressure guarantees a thorough investigation of the search space. This approach avoids premature convergence by avoiding local optima, which is a typical difficulty in meta-heuristic optimization. The unpredictability generated by Eqs. (34) and (41) substantially enhances the exploration of the solution space.

5.8.2 Function of pulmonary circulation in avoiding local optima

The most vulnerable people experience significant random disturbances via the pulmonary circulation mechanism. This phase not only diversifies the population but also improves the algorithm’s capacity to recover from unfavorable conditions. This method elucidates why BCSBO surpasses other algorithms in high-dimensional or intricate issues, as seen by test cases like the IEEE 30-bus and IEEE 118-bus systems. The random variation in these phases helps to explore uncharted portions of the solution space and hence increases the resilience of the method.

5.8.3 Function of systemic circulation in exploitation

The methodical circulation guarantees the refinement of the most promising solutions, hence enhancing convergence speed and accuracy. By establishing a balance between exploration and exploitation, the algorithm efficiently constricts the search area during the latter phases of optimization. This characteristic is apparent in the accelerated convergence patterns of BCSBO relative to other algorithms, as seen in Figs. 7 and 12. The optimization framework’s capacity to refine solutions throughout the systemic circulation phase facilitates superior outcomes with less computing expenditure.

5.8.4 Tackling the OPF challenge with renewable energy sources

The integration of RESs into OPF optimization presents considerable challenges owing to the intrinsic fluctuation and uncertainty associated with renewable power supply. BCSBO solves this difficulty by constantly modifying its search method using the circulatory system-inspired model, guaranteeing that optimum solutions may be identified even in complicated systems. The algorithm’s adaptability and efficacy are essential for addressing OPF issues related to RESs, where optimization must reconcile renewable energy with grid stability.

5.8.5 Scalability trade-offs

The BCSBO method demonstrates superior performance in smaller-scale systems; nevertheless, its computational expense somewhat escalates in bigger systems owing to the intricacies involved in simulating several circulatory phases. This phenomenon is seen in the IEEE 118-bus system, where the temporal complexity surpasses that of more straightforward algorithms like as PSO and MFO. The enhanced accuracy and better convergence properties in addressing OPF issues with RESs validate this trade-off in situations when solution quality is paramount. The BCSBO’s capacity to cope with large-scale difficulties while retaining high performance underscores its promise for real-world applications, notably in integrating RESs into power grids.

5.8.6 Comparison with alternative optimization methods

The findings from the IEEE 30-bus and IEEE 118-bus systems indicate that BCSBO surpasses conventional algorithms, including EHO, TEO, MFO, PSO, and its precursor, CSBO. The comparisons were conducted using five distinct objective functions, and BCSBO shown better performance regarding solution quality and computing efficiency. The efficacy of BCSBO is especially remarkable in addressing the complexity posed by RESs, underscoring its promise as a viable solution for contemporary OPF challenges.

BCSBO offers a unique ability to balance exploration and exploitation by integrating biological factors like pulmonary and systemic circulation into the optimization process. This dual approach explains why it outperforms traditional algorithms in tackling a range of optimization problems, particularly those involving the integration of renewable energy sources into power networks.

The BCSBO method, although exhibiting robust performance in several optimization situations, has drawbacks characteristic of meta-heuristic algorithms. These encompass:

• Selection of control parameters

BCSBO necessitates the meticulous selection of certain control parameters by the user. The algorithm’s effectiveness may be contingent upon the values of these parameters, necessitating experimentation or previous knowledge to identify the ideal parameter configuration for each task.

• Scalability in high-dimensional situations

Although BCSBO is proficient for small to medium-sized issues, similar to other meta-heuristic algorithms, it may encounter difficulties when addressing very complicated or high-dimensional situations. The approach may need more improvements, including hybridization with other techniques, to improve its scalability and get superior performance in certain instances.

• Requirement for additional enhancements in specific situations

In cases of especially complex optimization problems, such as those with extensive search spaces or highly nonlinear systems, BCSBO may not consistently provide the most optimum solutions. Ongoing study and enhancement are essential to modify the algorithm for these more complex conditions.

These limits show that although BCSBO is a viable strategy for many issues, there may be cases where more study or hybridization with other approaches would be useful to maximize performance.

6 Conclusions

Addressing the OPF problem with RESs poses a significant challenge for various algorithms, particularly in complex systems. This paper introduces the BCSBO algorithm as a solution for OPF with RESs, considering five different objective functions. The effectiveness of the suggested BCSBO technique is evaluated using the IEEE 30 test and the IEEE 118-bus test systems and compared against EHO, TEO, MFO, PSO, and CSBO algorithms. The simulation results demonstrate that suggested CSBO is an effective and powerful method for addressing the OPF problem with RESs in power systems. In future research, the BCSBO algorithm will be further utilized to solve the OPF, incorporating FACTs devices. The suggested BCSBO algorithm has considerable promise for future study and applications in many scientific and technical domains. It is adaptable for many optimization problems, including function optimization, mechanical engineering issues, and medicinal applications like for learning machine (Jin et al., 2019), for diagnosis of brain disease (Fei et al., 2020), and prediction of recurrent spontaneous abortion (Shi et al., 2022). Furthermore, expansions of the technique may be investigated, including the creation of fuzzy, binary, and multi-objective variations. Furthermore, the BCSBO algorithm may be integrated with other sophisticated optimization methods to augment its performance and efficiency, hence expanding the horizons of optimization study. These instructions underscore the flexibility and adaptability of the BCSBO algorithm, enabling new applications and progress across several fields.

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