Optimal power dispatch in hybrid power system for medium- and large-scale practical power systems using self-adaptive bonobo optimizer algorithm

Abstract

Incorporating renewable energy sources (RESs) introduces a notable amount of uncertainty in the optimal planning and operation of electrical power grids. Under these circumstances, this paper proposes the application of a recently introduced metaheuristic optimization technique to solve the stochastic optimal power flow (OPF) problem involving wind and solar power sources. The self-adaptive bonobo optimizer (SaBO) is used to minimize three distinct objective functions: (i) Total generation cost (TGC) minimization, including both thermal and wind/solar generation costs, (ii) Power loss minimization, (iii) Combined generation cost and emissions effect minimization. The costs associated with the stochastic generation of wind and solar power included direct costs, reserves and penalty costs from the overestimation and underestimation of available wind and solar power, respectively. The performance of the proposed algorithm is evaluated on two power systems: the modified IEEE 30-bus and the Algerian DZA 114-bus test systems. To demonstrate the efficacy of the SaBO, the obtained results have been compared with those obtained from the Kepler optimization algorithm (KOA) and other recently published optimizers under the same case studies and constraints. The comparative results clearly show the superiority of the SaBO algorithm over all other well-known optimization algorithms provided in the literature for solving the OPF problem. This is evidenced by minimizing total generation costs of 781.2363 $/h for the modified IEEE 30-bus and 16,706.1630 $/h for the Algerian DZA-114-bus system. Furthermore, the integration of RES led to a notable 2.33% and 11.67% reduction in total generation cost for the IEEE 30-bus and Algerian DZA 114-bus systems, respectively, compared to their initial configurations without RESs. The promising findings highlight the powerful of the optimizer to solve non-linear and complex optimization problems in power systems.

Keywords

Metaheuristic optimization technique optimal power flow self-adaptive bonobo optimizer renewable energy sources wind and solar generation combined generation cost and emissions Kepler optimization algorithm

Introduction

Carpentier (1962) introduced the Optimal Power Flow (OPF), as an efficient and essential tool for the control and operational planning of electric power systems. The OPF has been widely applied in power systems and can be characterized as a large-scale optimization problem that involves non-linear objective functions and constraints. The OPF process aims to find the optimal solution that minimizes a desired objective function. This involves adjusting the control variables to meet diverse power flow constraints, including both equality and inequality constraints (Kouadri et al., 2020a).

Typically, various objective functions associated with the electrical power system are employed as the optimization objective. The most common objective functions include minimizing fuel cost, power transmission loss, gas emissions effect, voltage stability index, and voltage deviation, … etc. (Ali et al., 2022; Huy et al., 2022; Mouassa et al., 2022),. In this context, the adjustable control variables for achieving the desired system operation include the active power output of generators, generators voltage, transformers’ tap setting, and the reactive power output from compensators as well as FACTS devices (Ali et al., 2022; Mouassa et al., 2022).

The study of classical OPF problems focuses on conventional thermal generators powered by fossil fuels. However, given the significant growth of RESs, the OPF analysis becomes necessary to solve the complexities of the power system network caused by the intermittence and uncertainty associated with RESs (Biswas et al., 2017). In this context, the OPF problem’s computational complexity increases considerably and becomes more challenging (Li et al., 2020). To address these complexities, it is essential to formulate the problem that incorporating stochastic RESs like wind turbines (WT) and photovoltaic (PV) into the optimization process for the effective planning and operation of power systems. Several metaheuristic optimization techniques designed recently to effectively address this stochastic problem. These methods are widely used to overcome the limitations of conventional techniques, including aspects, such as robustness, accuracy, and performance convergence (Sulaiman and Mustaffa, 2021). Recently, there has been a notable increase in the attention given by researchers to the use of metaheuristic methods to solve issues related to the integration of stochastic RESs.

In the literature, Ali et al. (2022) proposed a white shark optimizer (WSO) algorithm to solve the OPF problem incorporating conventional thermal power generators (TPGs) and renewable energy generators (REGs). These later include solar power generators (SPGs) and wind power generators (WPGs). In this research work, the authors utilized lognormal and Weibull probability distribution functions (PDFs) to predict the output power of solar PV and wind generators. The results clearly show that WSO outperforms the Northern Goshawk Optimizer (NGO), Smell Agent Optimizer (SAO) and the Pelican Optimization Algorithm (POA). Moreover, the application of the suggested model on a large-scale system has not been carried out yet. The authors of Li et al. (2022) examined the multi-objective optimization problem considering stochastic solar and wind power in the modified IEEE 30-bus and IEEE 57-bus systems. In this context, two multi-objective algorithms called adaptive crossover non-dominated sorting differential evolution (ACNSDE) and non-dominated sorting genetic algorithm (NSGA-II) have been proposed to address this issue. Results from simulations suggest that NSGA-II outperforms ACNSDE in terms of performance when considering four different optimization objectives: namely generation cost, active power transmission loss, deviation of voltage, and emissions effects. In Sulaiman and Mustaffa (2023), a technique based on the improved salp swarm algorithm (iSSA) was proposed to address the OPF issues in a solar-thermal power system. The simulation was conducted on the IEEE 30-bus system, evaluating three different objectives: (1) Minimize power transmission loss, (2) Minimize generation cost, and (3) Minimize of combined generation cost considering emission effects. The proposed iSSA algorithm demonstrated remarkable efficiency compared to SSA and the different existing algorithms outlined in the literature. Mouassa et al. (2022) proposed the application of the slime mold algorithm (SMA) to solve a large-scale constrained optimization problem considering the stochastic behavior of solar and wind sources. The effectiveness of the SMA was evaluated through testing on modified IEEE 30-bus and real Algerian DZA 114-bus systems. The numerical results clearly show that the suggested SMA outperforms various other techniques available in the literature for addressing this problem involving renewable energy integration. In attempting the same issue, Nguyen et al. (2022) introduced a novel approach to solving the OPF problem with considering both solar and wind power, by employing an improved equilibrium optimizer (IEO) algorithm. The IEO approach proved its effectiveness in both small and large-scale systems against other existing techniques mentioned in this research.

Other notable uses of metaheuristic techniques for OPF problems with uncertain RESs include a Multi-Objective Search Group Algorithm (MOSGA) (Huy et al., 2022), Modified JAYA Algorithm (Elattar and ElSayed, 2019), Grey Wolf Optimization (GWO) (Khan et al., 2020), Multi-Objective Evolutionary Algorithm (MOEA) (Avvari and D. M, 2022), Modified Moth Swarm Algorithm (MMSA) (Elattar, 2019), Multi-Objective Horse Herd Optimization (MOHHO) (Ida Evangeline and Rathika, 2022), Flow Direction Algorithm (FDA) (Maheshwari et al., 2023), Gaussian bare-bones Levy-flight firefly algorithm (GBLFA) (Alghamdi, 2022), Jellyfish Search Optimizer (JS) (Farhat et al., 2021), Bird Swarm Algorithm (BSA) (Ahmad et al., 2021), Slime Mold Algorithm (SMA) (Kouadri et al., 2020b), Kepler Optimization Algorithm (KOA) (Abid et al., 2024), Adaptive Lightning Attachment Procedure Optimizer (ALAPO) (Adhikari et al., 2023), Circle search algorithm (CSA) (Shaheen et al., 2022), Equilibrium optimization (EO) (Amroune, 2022), Chaotic African Vultures Optimization Algorithm (CAVOA) (Mohamed et al., 2024), and Moth Flame Optimization (MFO) (Pandya and Jariwala, 2022).

The following summarizes the contribution and organization of the paper:

To the best of the authors’ knowledge, there is no existing documentation in the literature on the application of the self-adaptive bonobo optimizer (SaBO) to address the OPF problem under the uncertain power outputs from wind and solar generators.

In this paper, a novel physics-based metaheuristic optimization SaBO algorithm was designed to tackle the stochastic OPF problem incorporating wind and solar power generators. This algorithm was developed in 2023 by Das et al. (2023). The uncertainties associated with wind speed and solar irradiance are modeled using lognormal and Weibull probability distribution functions (PDFs), respectively.

To evaluate the proposed SaBO, four case studies are conducted on the modified IEEE 30-bus and practical Algerian 114-bus power systems connected to wind and power generators. The achieved results from the proposed algorithm are compared with other well-known optimization algorithms in the literature. Through these comparisons, the efficiency and performance of SaBO in addressing the stochastic OPF problem in these diverse power systems considering stochastic wind and solar power are thoroughly evaluated.

The simulation results show that SaBO can achieve a better fitness value in optimizing the different OPF objective functions involving minimization of the total generation cost, power loss and combined generation cost and emissions effect. In addition, a statistical analysis of SaBO and KOA algorithms using a diagram known as a boxplot has been performed for both power systems.

The remaining sections of the paper are structured as follows: Section 2 presents the formulation of the stochastic OPF model including wind and solar power. In Section 3, the paper provides a comprehensive explanation of the SaBO. Section 4 is dedicated to discussing the various numerical results. The obtained solutions from applying the SaBO to solve the stochastic optimization problem are presented and critically examined. The conclusion of this work is presented in Section 5.

Problem formulation

The OPF aims to minimize the specified power system objective functions while respecting several optimization constraints. The different objective functions that are to be optimized in this work are detailed in the following section:

Cost for thermal power units

The cost associated with conventional TPGs may be calculated using the formula below:

C_{T h P G} = \sum_{i = 1}^{N_{T h G}} a_{i} + b_{i} P_{{T h G}_{i}} + c_{i} P_{{T h G}_{i}}^{2}

(1)

where a_i, b_i, and c_i denote the cost coefficients of the i-th conventional TPGs. N_ThG is the total number of conventional TPGs-producing power output P_ThG.

This study incorporates the valve-point loading effect for more precise and realistic modeling of the function of fuel cost. Equation (2) represents the modified cost function with valve-point loading.

C_{T h P G} = \sum_{i = 1}^{N_{T h G}} a_{i} + b_{i} P_{{T h G}_{i}} + c_{i} P_{{T h G}_{i}}^{2} + | d_{i} * \sin (e_{i} (P_{{T h G}_{i}}^{\min} - P_{{T h P G}_{i}})) |

(2)

where d and e denote the cost coefficients of the ith TPG under the loading effect. $P_{ThP G_{i}}^{\min}$ is the minimum output of thermal power generators.

Cost of wind and solar PV power

The cost associated with power generation from both wind and solar photovoltaic (PV) sources includes the following different components: the direct cost related to the scheduled output power, the penalty cost associated with underestimation of output power, and the reserve cost associated with overestimation.

Direct cost of wind and solar photovoltaic power

The direct costs related with jth wind and kth solar PV plants in terms of the scheduled output power are expressed as follows (Biswas et al., 2017):

C_{w, j} (P_{w s, j}) = d_{w, j} * P_{w s, j}

(3)

C_{s, k} (P_{s s, k}) = d_{s, k} * P_{s s, k}

(4)

where d_w and d_s represent the direct cost coefficients attached to jth wind and kth solar power generators, respectively. P_ws and P_ss denote the planned power output from these respective generators.

Uncertainty cost evaluation of wind power

Considering uncertainties, the following two scenarios are possible:

(1) In the case where the actual output of WPG power is lower than the initial estimate value, the operator must maintain a spinning reserve to ensure a steady electricity supply to consumers. This case is referred to as the overestimation. The cost associated with maintaining a spinning reserve is referred to as reserve cost (Panda and Tripathy, 2015), and can be expressed as follows (Biswas et al., 2017):

C_{R w . j} (P_{w s, j} - P_{w a v, j}) = k_{R w, j} (P_{w s, j} - P_{w a v, j}) = k_{R w} \int_{0}^{P_{w s, j}} (P_{w s, j} - P_{w, j}) f_{w} (P_{w, j}) {d P}_{w, j}

(5)

where K_Rw presents the coefficient of reserve cost associated with jth WPGs, P_wav is the actual wind power output from the same generators and f_w (P_w) denotes the wind power PDF for jth WPGs.

(2) Contrary to the first case, when the actual power output of WPGs surpasses the initial estimate value, the excess power becomes unusable and is wasted. This situation is termed the underestimation of output power, necessitating the independent system operator (ISO) to pay a penalty cost against each surplus amount of power. This cost can be expressed as follows (Biswas et al., 2017).

C_{P w . j} (P_{w a v, j} - P_{w s, j}) = k_{P w, j} (P_{w a v, j} - P_{w s, j}) = k_{P w, j} \int_{P_{w s, j}}^{P_{w r, j}} (P_{w, j} - P_{w s, j}) f_{w} (P_{w, j}) {d P}_{w, j}

(6)

where K_Pw presents the coefficient of penalty cost associated with jth WPGs and P_wr denotes the rated power generated by WPGs.

The total wind power cost can be given as:

C_{w i n d} (P_{w s}) = \sum_{i = 1}^{N_{W G}} [d_{w, j} * P_{w s, j} + k_{R w, j} (P_{w s, j} - P_{w a v, j}) + k_{P w, j} (P_{w a v, j} - P_{w s, j})]

(7)

Uncertainty cost evaluation of solar power

Similarly, to the wind power plant, the solar PV plant also experiences intermittent and uncertain variations in its power output. The method employed to handle both overestimation and underestimation of solar power output aligns with that applied for wind output power. To simplify the calculation process, the model of reserve and penalty cost for solar PV power is formulated based on the concept introduced in Shi et al. (2012). This approach is adopted due to the distinct nature of solar radiation, which follows a lognormal PDF (Chang, 2010).

(1) Reserve cost for the kth solar PV power generators is given by (Biswas et al., 2017):

C_{R s . k} (P_{s s, k} - P_{s a v, k}) = k_{R s, k} (P_{s s, k} - P_{s a v, k}) = k_{R s, k} \times f_{s} (P_{s a v, k} < P_{s s, k}) \times [P_{s s, k} - E (P_{s a v, k} < P_{s s, k})]

(8)

where K_Rs presents the coefficient of reserve cost attached with kth SPGs, and P_sav is the actual available solar power output. f_s(P_sav,k < P_ss,k) signifies the probability of experiencing a shortage in solar PV power compared to the scheduled power P_ss,k, while E(Psav,k < Pss,k) represents the expected occurrence of solar power below P_ss,k.

(2) Penalty cost for the kth solar PV power generators is given by (Biswas et al., 2017):

C_{P s . k} (P_{s a v, k} - P_{s s, k}) = k_{P s, k} (P_{s a v, k} - P_{s s, k}) = k_{P s, k} \times f_{s} (P_{s a v, k} > P_{s s, k}) \times [E (P_{s a v, k} > P_{s s, k}) - P_{s s, k}]

(9)

where K_Ps presents the coefficient of penalty cost attached with kth SPGs. f_s(P_sav,k > P_ss,k) is the probability of excess solar PV power, while E(P_sav,k > P_ss,k) represents the expected occurrence of solar power exceeding the scheduled power.

The total solar power cost can be given as:

C_{s o l a r} (P_{s s}) = \sum_{i = 1}^{N_{S G}} [d_{s, k} * P_{s s, k} + k_{Rs, k} (P_{s s, k} - P_{s a v, k}) + k_{P s, k} (P_{s a v, k} - P_{s s, k})]

(10)

Objective function

Three OPF objective functions in this work are formulated with incorporating of RESs into the conventional power grid. These optimization objectives can be presented as:

First objective function (F1) : In this objective function, OPF is formulated with the presence of both wind and solar power in order to minimizes the generation cost. In this case, the emission cost is not included. Mathematically can be expressed as:

F_{1} = C_{T h P G} + C_{w i n d} + C_{s o l a r}

(11)

where C_ThPG is the function of conventional TPG cost. C_wind and C_solar represent the functions of wind and solar power costs, respectively.

Second objective function (F2) : In the second, the total active power transmission loss (P_loss) is formulated to minimize. Thus, P_loss can be expressed as:

F_{2} = P_{l o s s} = \sum_{i = 1}^{nl} \sum_{j \neq i}^{nl} (G_{i, j} V_{i}^{2} + V_{j}^{2} - 2 V_{i} V_{j} \cos (δ_{i, j}))

(12)

where G_i,j is is the transfer conductance and δ_i,j = δ_i − δ_j signifies the difference in voltage angles between buses i and j.

Third objective function (F3) : the third objective function aims to minimize the first objective function with including the emission cost in the OPF formulation.

F_{3} = F_{1} + C_{tax} \times E

(13)

where C_tax denote the carbon tax, with a specified value set at 20 ($/h) and E represents the emission, calculated according to the following equation:

E = \sum_{i = 1}^{N_{T h G}} [(α_{i} + β_{i} P_{Th G_{i}} + γ_{i} P_{Th G_{i}}^{2}) \times 0.01 + w_{i} e^{(μ_{i} P_{Th G_{i}})}]

(14)

here α_i, β_i, γ_i, w_i, and μ_i denote the emission coefficients of the ith TPG.

Constraints

Equality-constraints : These constraints in the OPF model express the balance of power flow, signifying that the total power generated by generators should equal the combined values of load demands and power losses of both active and reactive power flow. The formulation of these OPF constraints is as follows:

P_{G_{i}} - P_{d_{i}} = V_{i} \sum_{j = 1}^{N} V_{j} (g_{ij} \cos δ_{ij} + z_{ij} \sin δ_{ij})

(15)

Q_{G_{i}} - Q_{d_{i}} = V_{i} \sum_{j = 1}^{N} V_{j} (g_{ij} \sin δ_{ij} + z_{ij} \cos δ_{ij})

(16)

Inequality-constraints : These constraints in the OPF model express the operation limits of the electric power system, categorized as follows:

(a) Generator constraints:

P_{T h P G, i}^{m i n} \leq P_{T h P G, i} \leq P_{T h P G, i}^{m a x}, i = 1, \dots ., N_{T h G}

(17)

P_{w s, j}^{m i n} \leq P_{w s, j} \leq P_{w s, j}^{m a x}, j = 1, \dots ., N_{W G}

(18)

P_{s s, k}^{m i n} \leq P_{s s, k} \leq P_{s s, k}^{m a x}, k = 1, \dots ., N_{S G}

(19)

Q_{T h G, i}^{m i n} \leq Q_{T h G, i} \leq Q_{T h G, i}^{m a x}, i = 1, \dots ., N_{T h G}

(20)

Q_{w s, j}^{m i n} \leq Q_{w s, j} \leq Q_{w s, j}^{m a x}, j = 1, \dots ., N_{W G}

(21)

Q_{s s, k}^{m i n} \leq Q_{s s, k} \leq Q_{s s, k}^{m a x}, k = 1, \dots ., N_{S G}

(22)

Q_{C, k}^{m i n} \leq Q_{C, k} \leq Q_{C, k}^{m a x}, i = 1, \dots ., N_{C}

(23)

V_{Gi}^{m i n} \leq V_{Gi} \leq V_{Gi}^{m a x}, i = 1, \dots ., N_{G}

(24)

(b) Security constraints:

\begin{matrix} V_{L, m}^{m i n} \leq V_{L, m} \leq V_{L, m}^{m a x}, & m = 1, \dots ., N_{L B} \end{matrix}

(25)

S_{l} \leq S_{l}^{m a x}, l = 1, \dots ., N_{L i n e}

(26)

T_{h}^{m i n} \leq T_{h} \leq T_{h}^{m a x}, h = 1, \dots ., N_{T}

(27)

The active power constraints of ThPGs, WPGs, and SPGs are represented, respectively, by equations (17)–(19). As well, the reactive power constraints of the same generators and shunt reactive power sources are represented by equations (20)–(23). N_G being the number of generators, while N_C is the number of shunt capacitors. Equation (24) is the voltage constraint of generator buses. The security constraints involve limitations on both load bus voltages (PQ) and transmission line capacity, as expressed in equations (25) and (26), respectively. Furthermore, the limitation imposed on tap-changing transformers is given by equation (27). N_LB is the number of load busses, N_line is the number of transmission lines, as well as N_T denotes the number of transformers.

Stochastic wind/solar power and uncertainty models

The following sections present a detailed description of stochastic wind and solar power modeling.

Wind and solar power models

Due to the stochastic nature of wind speed, its probability distribution is determined using the Weibull PDF, mathematically expressed as (Biswas et al., 2017):

f_{v} (v) = (\frac{k}{c}) + {(\frac{v}{c})}^{(k - 1)} e^{- {(v / v c c)}^{k}} f o r 0 < v < \infty

(28)

where v represents the speed of wind (m/s). c and k are the scale and shape factors, respectively

The mean of Weibull distribution is given by:

M_{w b l} = c * Γ (1 + k^{- 1})

(29)

The gamma function Γ is expressed as follows:

Γ (x) = \int_{0}^{\infty} ex p^{- t} t^{x - 1} d t

(30)

The output wind power of WPGs can be represented in four regions based on the corresponding wind speeds, as indicated in the following equation (6):

P_{w} (v) = {\begin{matrix} 0 & i f v < v_{i n} a n d v > v_{o u t} \\ P_{w r} (\frac{v - v_{i n}}{v_{r a t e d} - v_{n}}) & i f v_{i n} \leq v \leq v_{v_{r a t e d}} \\ P_{w r} & i f v_{r a t e d} \leq v \leq v_{o u t} \end{matrix}

(31)

where v_in corresponds to the speed of wind at which the wind turbine commences to produce power, v_out corresponds to the speed of wind at which the wind turbine disconnects, and v_rated denotes the speed of wind at which the output mechanical power reaches the rated power. P_wr represents the rated output power of the wind generator.

The solar PV generator’s output is influenced by the distribution of solar irradiance (G). The probability distribution of G following lognormal PDF, which is expressed as follows:

f_{G} (G) = \frac{1}{G σ \sqrt{2}} e^{- {(\ln G - μ)}^{2} / - {(\ln G - μ)}^{2} 2 σ^{2} 2 σ^{2}}

(32)

here, σ denotes the standard deviation, and µ stands for the mean of the lognormal PDF, which is given as follows:

M_{\log} = e^{(μ + (σ^{2} / σ^{2} 2 2))}

(33)

Based on G, the solar PV-generated power could be expressed as follows:

P_{S} (G) = {\begin{matrix} P_{s r} (\frac{G^{2}}{G_{S} R_{C}}) f o r 0 \leq G \leq R_{C} \\ P_{s r} (\frac{G^{2}}{G_{S}}) f o r G \geq R_{C} \end{matrix}

(34)

Here, P_sr represents the rated power generated by the solar PV plants, G_S denotes the standard solar irradiance (STC), and R_C signifies a specific irradiance point.

Wind power probability model

The variability of wind, characterized by its intermittent and uncertain nature, leads to a discrete availability of wind power output, as illustrated in equation (31). According to this equation, it can be noted that the available wind power output is variable in some regions of wind speed.

The first condition involves the turbine power generated being zero when the speed of wind (v) is below the cut-in speed (v_in) or exceeds the cut-out speed (v_out). Whereas the turbine produces rated power P_wr when v is between rated wind speed (v_rated) and v_out. The probabilities of wind power generated for each discrete zone are expressed as follows (Biswas et al., 2017):

f_{w} (P_{w}) {P_{w} = 0} = 1 - e x p (- {(\frac{v_{c u t - i n}}{c})}^{k}) + e x p (- ({(\frac{v_{c u t - o f f}}{c})}^{k}))

(35)

f_{w} (P_{w}) {P_{w} = P_{w r}} = e x p (- {(\frac{v_{r a t e d}}{c})}^{k}) - e x p (- {(\frac{v_{c u t - o f f}}{c})}^{k})

(36)

However, the power generated by the wind turbine in the continuous portion is between the v_in and v_rated of wind, which can be explained as follows:

f_{w} (P_{w}) = \frac{k (v_{r a t e d} - v_{i n})}{λ^{k} P_{w r}} {(v_{i n} + \frac{P_{w}}{P_{w r}} (v_{r a t e d} - v_{i n}))}^{k - 1} * e x p (- {(\frac{v_{i n} + \frac{P_{w}}{P_{w r}} (v_{r a t e d} - v_{i n})}{λ})}^{k})

(37)

Self-adaptive bonobo optimizer (SaBO)

Recently, a new stochastic optimization technique named Self-adaptive Bonobo Optimizer (SaBO) was developed in 2023 by Das et al. (2023). It draws inspiration from both the mating strategies and social behavior of bonobos. The main feature of SaBO is the ability to update its search parameters based on the repulsion-based learning method. It has four fundamental mating strategies: consortship mating, extra-group mating, promiscuous mating, and restricted mating, similar to the Bonobo Optimizer. Nevertheless, the proposed SaBO introduces various modifications aimed at enhancing its performance. In addition to the existing population, SaBO incorporates three extra-memorized populations, with their members actively contributing to the maturation process, producing new bonobos. SaBO autonomously adjusts the primary controlling parameters, namely phase probability (P_P) and sharing coefficient (S_C), by implementing repulsion-based learning for iterative updates. This learning method depends entirely on feedback obtained from the search process. The two main controlling parameters, P_P and S_C, can vary in value in the range of [0-1]. In every iteration, N values of P_P or S_C are assigned to N solutions within the current population, with the expectation that these parameter values are different from one another. First, a normal distribution with a mean of 0.5 and a standard deviation of σ is used to generate N number of parameters. The parameter σ is subject to variation within its minimum and maximum values, with the initial value fixed at its maximum value. The fundamental process of updating P_P through the repulsion technique according to equation (38) is detailed in Das et al. (2023) as well as depicted in Figure 1.

{P_{P}}_{b e t t e r}^{m o d i f i e d} = P_{P b e t t e r} \pm \frac{σ \times P_{P b e t t e r} \times P_{P w o r s e}}{e^{{(P_{P b e t t e r} - P_{Pworse})}^{2}}}

(38)

Mating strategies

The SaBO algorithm utilizes four distinct mating strategies, incorporating concepts from both the positive phase (PP) and negative phase (NP), to generate new bonobos. During each iteration, the solutions are updated, with the selection between PP or NP determined by the probability associated with the P_P. Moreover, when a mating strategy, either its restrictive or promiscuous, is applied with a probability of 0.5, new bonobos are generated through the PP. On the other hand, In NP, a new bonobo is generated by either following the extra-group or the consortship mating strategy. Thus, the mating strategy of bonobos is determined by the P_P parameter, initially set at 0.5 and changing with each iteration based on the new values obtained from equation (38). Detailed explanations of various mating strategies employed to generate new bonobos are presented in the following section:

Promiscuous and restrictive mating strategies

During the positive phase (NP), the generation of the new bonobo through promiscuous and restrictive mating strategies follows equations (39) and (40), respectively.

\begin{matrix} {n e w_b o n o b o}_{i} = {b o n o b o}_{i} + S c_{i} \times \\ [({b o n o b o}_{p} - {b o n o b o}_{i}) + ({b o n o b o}_{k 1} - oldpo p_{k 2}) + (oldpo p_{k 3} - {b a d p o p}_{k 4})] \end{matrix}

(39)

\begin{matrix} n e w_{b o n o b o}_{i} = {b o n o b o}_{i} + S c_{i} \times \\ [({b o n o b o}_{p} - {b o n o b o}_{i}) + ({b o n o b o}_{k 1} - oldpo p_{k 2}) + ({b o n o b o}_{k 2} - {b a d p o p}_{k 4})] \end{matrix}

(40)

Here, new_bonobo_i represents the ith-new solution, while, bonobo_i, bonobo_p, bonobo_k1 bonobo_k2 respectively denote the ith, pth, k₁th and k₂th-solutions of the current population. oldpop_k2 and oldpop_k3 are the k₂th and k₃th-solutions of oldpop population, respectively. Similarly, badpop_k4 denotes the k₄th-solution of badpop population. The parameter of sci is the ith-sharing coefficient. The different numbers k1, k2, k3, and k4 are randomly selected from the range (1, N), and they are not equal to either i or p.

Extra-group and consortship mating strategies

On the contrary to PP, the search process in the NP focuses on exploring previously undiscovered regions. This exploration is accomplished by implementing either the extra-group or consortship mating strategies.

In the extra-group mating strategy, the solution is updated if the generated random number is equal to or below the value of extra-group mating occurrence probability (Pxgm), following equations (41)–(43). The parameter P_xgm is variable and ranges from $[P_{x g m}^{\min}, P_{x g m}^{\max}]$ .

I_{2} = e^{(r_{1} - \frac{1}{r_{1}})}

(41)

n e w_{b o n o b o}_{j}^{i} = {b o n o b o}_{j}^{i} + I_{2} \times (V_{\max, j} - {b o n o b o}_{j}^{i}), i f r_{2} \leq 0.5

(42)

n e w_{b o n o b o}_{j}^{i} = {b o n o b o}_{j}^{i} - I_{2} \times ({b o n o b o}_{j}^{i} - V_{\min, j}), i f r_{2} > 0.5

(43)

where I₂ is an intermediate parameter, while r₁ and r₂ represent two randomly chosen numbers from the range (0, 1). The variables new_bonobo_i^j and bonobo_i^j respectively refer to the jth-variables of the generated solution and the ith-bonobo of the existing population.

In the consortship mating strategy, the solution is updated if the generated random number r₂ is larger than the value of Pxgm, following equations (44)–(50).

n e w_{b o n o b o}_{j}^{i} = {b o n o b o}_{j}^{i} + I_{3} \times ({b o n o b o}_{j}^{i} - {b a d p o p}_{j}^{i}), i f r_{3} \leq 0.5

(44)

{n e w_b o n o b o}_{j}^{i} = {b o n o b o}_{j}^{i} + I_{2} \times ({b o n o b o}_{j}^{i} - {b a d p o p}_{j}^{i}), i f r_{4} \leq P_{P_{i}}

(45)

{n e w_b o n o b o}_{j}^{i} = {b o n o b o}_{p}^{j} + I_{5} \times ({b o n o b o}_{p}^{j} - {b a d p o p}_{j}^{i}), o t h e r w i s e

(46)

I_{3} = e^{(r_{1} - \frac{2}{r_{1}})}

(47)

I_{5} = e^{(r_{1} - \frac{I_{4}}{r_{1}})}

(48)

I_{4} = m a x i m u m o f (2, d v \times {t s g s}_{\max})

(49)

d v = \min of (1, \frac{{(\sum_{j = 1}^{d} S D_{j})}_{current population}}{{(\sum_{j = 1}^{d} S D_{j})}_{Initial population}})

(50)

Where; I₃, I₄, and I₅ represent the three intermediate parameters, respectively. The parameter tsgs_max denotes the upper limit for the size of the temporary sub-group.

Modified boundary handling techniques

If the newly generated bonobo exceeds the maximum limit of variables, it is given the maximum limit with a 0.5 probability of occurrence; otherwise, it undergoes an update using either equation (51) or (52) with a 0.5 probability. Likewise, if the newly generated bonobo exceeds the lower variable limit, its value is set to the lower limit; otherwise, it is modified using equations (51) and (52).

n e w_{b o n o b o}_{j}^{i} = r_{5} \times {b o n o b o}_{j}^{i} + (1 - r_{5}) \times {V a r}_{\max}^{j}

(51)

n e w_{b o n o b o}_{j}^{i} = r_{5} \times {b o n o b o}_{j}^{i} + (1 - r_{5}) \times {V a r}_{\min}^{j}

(52)

where r₅ represents a randomly generated number from the range of between 0 and 1.

The flowchart and pseudo-code of the provided SaBO algorithm for addressing the OPF problem are shown in Figure 2 and Table 1, respectively.

Table 1.

The pseudo-code of SaBO.

The pseudo-code of SaBO
Input data, No. of population (N_P), No. of control variables (N), Maximum iteration (Max_iter) Generate initial bonobo population of size N Evaluate the bonobos fitness Sorting of the population through fitness and determining the alpha-bonobo Initialize and memorize old, bad and worse populations while the stopping criterion is not satisfied Choose the p^th-bonobo randomly from good solutions If r ≤ Pp Apply promiscuous or restrictive mating strategy to generate a new bonobo using equations (39) and (40) else If r₆≤ Pxgm Apply extra-group mating strategy to generate a new bonobo using equations (41) and (42) else If r₃≤ 0.5 Apply consortship mating type to create a new bonobo using equation (44) else If r₄≤ P_P,i Apply consortship mating type to create a new bonobo using equation (45) else Apply consortship mating type to create a new bonobo using equation (46) end end end end Set variable bounding conditions Evaluate the fitness of solutions and determine alpha-bonobo Use repulsion-based learning to update SaBO parameters such as P_P and sc Sorte of current-worse mixed populations based on fitness Select N new best bonobos as the current population from the population of size 2N

The pseudo-code of SaBO

Input data, No. of population (N_P), No. of control variables (N), Maximum iteration (Max_iter)
Generate initial bonobo population of size N
Evaluate the bonobos fitness
Sorting of the population through fitness and determining the alpha-bonobo
Initialize and memorize old, bad and worse populations
while the stopping criterion is not satisfied
Choose the p^th-bonobo randomly from good solutions
If r ≤ Pp
Apply promiscuous or restrictive mating strategy to generate a new bonobo using equations (39) and (40)
else
If r₆≤ Pxgm
Apply extra-group mating strategy to generate a new bonobo using equations (41) and (42)
else
If r₃≤ 0.5
Apply consortship mating type to create a new bonobo using equation (44)
else
If r₄≤ P_P,i
Apply consortship mating type to create a new bonobo using equation (45)
else
Apply consortship mating type to create a new bonobo using equation (46)
end
end
end
end
Set variable bounding conditions
Evaluate the fitness of solutions and determine alpha-bonobo
Use repulsion-based learning to update SaBO parameters such as P_P and sc
Sorte of current-worse mixed populations based on fitness
Select N new best bonobos as the current population from the population of size 2N

Simulation and results

This study suggests using the SaBO algorithm to address stochastic OPF problems involving the integration of WPGs and SPGs plants. The SaBO algorithm was implemented on both the modified IEEE 30-bus and the Algerian electricity DZA 114-bus power systems. In this context, several case studies based on objective functions are performed. The simulations are executed using the MATLAB R2021a programing language on a personal computer equipped with an Intel^® Core™ i5-7300U CPU @ 2.60GHz and 8 GB of RAM. To establish the superiority of the proposed algorithm and KOA, several defining characteristics are considered:

Control variables: 12 for IEEE 30-bus and 30 for DZA 114-bus;

Population size: 30 for IEEE 30-bus and 60 for DZA 114-bus;

Maximum iterations: 300 for IEEE 30-bus and 400 for DZA 114-bus;

Independent trial runs: 40 runs for IEEE 30-bus and 30 runs for DZA 114-bus;

Application on modified IEEE 30-bus test system

The application of the suggested SaBO in the first part is applied on the modified IEEE 30-bus power. Three distinct cases are examined in order to verify the effectiveness of both the SaBO and the selected KOA algorithm. This modified system consists of three conventional ThPGs located at buses No. 1, 2, and 8, two WPG farms at buses No. 5 and 11, and one SPG plant at bus No. 13. All Comprehensive details regarding this power system can be found in Sulaiman and Mustaffa (2021) and Table 2. Table 3 provides all PDF parameters and details for both wind and solar power plants. In addition, the coefficients of direct cost for wind and solar power are d_w,₁ = 1.6, d_w,₂ = 1.75, and d_s,₁ = 1.6, respectively. The penalty cost coefficients associated with underutilizing the excess of wind and solar power available are K_Pw,₁ = 1.5, K_Pw,₂ = 1.5, and K_Ps,₁ = 1.5, respectively. As well as, the reserve cost coefficients due to the overestimation of the same sources are assumed K_Rw,₁ = 3, K_Rw,₂ = 3, and K_Rs,₁ = 3, respectively.

Table 2.

Thermal generator coefficients of the modified IEEE 30-bus system.

No. of generator	Generator bus	Fuel cost coefficients					Emission coefficients
No. of generator	Generator bus	a	b	c	d	e	α	β	γ	w	μ
ThG1	Bus 1	0	2	0.00375	18	0.037	409.1	−555.4	649	0.0002	6.667
ThG2	Bus 2	0	1.75	0.0175	16	0.038	254.3	−604.7	563.8	0.0005	3.333
ThG3	Bus 8	0	3.25	0.00834	12	0.045	532.6	−355	388	0.002	2

Table 3.

PDF parameters of two WPG farms and one SPG plant.

Two WPG farms					One SPG plant
Wind farm	No. of turbines	Rated power P_wr (MW)	Weibull PDF parameters	Weibull mean	Rated power, P_sr (MW)	Lognormal PDF parameters	Lognormal mean
Bus 5	25	75	k = 2	v = 7.976 m/s	50 #bus13	µ = 6	G = 483 W/m²
Bus 5	25	75	c = 9	v = 7.976 m/s		σ = 0.6
Bus11	20	60	k = 2	v = 8.862 m/s
Bus11	20	60	c = 10	v = 8.862 m/s

Three different case studies are conducted on the modified IEEE-30 bus system. In the first case, the primary objective is to minimize the total generation cost, followed by the second case, which is dedicated to reducing the active power transmission loss. The last case is focuses on optimizing the combined generation cost with the inclusion of a carbon-emission tax. The results obtained for each case are presented and discussed in the following section.

First case: Minimization of total generation cost (TGC).

Second case: Minimization of active power loss (Ploss).

Third case: Minimization of combined generation cost and emission (CGCE).

Case 1: Minimization of total generation cost

In the first case, SaBO is applied to minimize the first objective function that includes the TGC. The comparison of results involving the proposed algorithm and the KOA algorithm is presented in Table 4. Figures 3 –5 illustrate the best Weibull fitting and wind output distribution achieved at buses 5 and 11 through 8000 scenarios of Monte Carlo. The Weibull and lognormal PDF parameters utilized in the simulation are the same as mentioned in Table 3. Additionally, Figure 6 illustrates the solar power output distribution at bus No. 13.

Table 4.

Optimal results via SaBO and KOA for modified IEEE 30-bus system: Cases 1–3.

Control variables	Limits		Case 1		Case 2		Case 3
Control variables	Min	Max	KOA	SaBO	KOA	SaBO	KOA	SaBO
P_TG1	50	140	134.920	134.91	50.010	50.0000	123.60	123.68
P_TG2	20	80	28.1398	29.04808	25.3499	24.97237	33.0765	33.4878
P_WG1	0	75	43.4198	44.02894	75.0000	75.0000	46.1000	46.2486
P_TG3	10	35	10.0000	10.0000	34.9940	34.99978	10.0000	10.0000
P_WG1	0	60	36.7311	37.1686	59.9995	60.0000	38.8087	38.9421
P_SG1	0	50	35.9024	33.9372	40.0449	40.4236	37.0199	36.2405
V₁	0.95	1.1	1.0719	1.07215	1.0584	1.05785	1.0704	1.0705
V₂	0.95	1.1	1.0571	1.0571	1.0525	1.052427	1.0569	1.0569
V₅	0.95	1.1	1.0336	1.0349	1.0435	1.043161	1.0355	1.0356
V₈	0.95	1.1	1.0385	1.0390	1.0490	1.049226	1.0395	1.0396
V₁₁	0.95	1.1	1.1000	1.1000	1.0984	1.098864	1.1000	1.1000
V₁₃	0.95	1.1	1.0566	1.0541	1.0592	1.060322	1.0557	1.05524
Q_TG1	−20	150	−2.54570	−1.8849	−4.2261	−5.4340	−2.7620	−2.5556
Q_TG2	−20	60	14.35749	12.7401	5.3784	6.3169	12.0922	11.7988
Q_WG1	−30	35	1.76324	2.6949	0.4156	0.0433	2.4066	2.4499
Q_TG3	−15	40	31.91481	32.7698	35.1533	35.3977	32.7296	32.9242
Q_WG2	−25	30	29.85686	29.8945	29.9197	29.9999	29.8809	29.8873
Q_SG1	−20	25	16.96626	16.0001	17.4819	17.8381	16.5105	16.3498
Total generation cost ($/h)			781.3568	781.2363	880.2709	879.7964	810.0663	809.7452
Emission (t/h)			1.76295	1.76195	0.09887	0.09897	0.89502	0.89945
Power losses (MW)			5.7082	5.6908	1.9963	1.9958	5.2011	5.2021
Voltage deviation (p.u.)			0.48188	0.47550	0.54905	0.55567	0.48459	0.48336
Computation time (s)			266.3273	151.425	192.525	207.4207	144.051	276.301

Table 4 provides the control variables’ bounds and the comprehensive optimal results achieved through the SaBO and KOA algorithms for the three case studies in a 40-time simulation run. In the first case, this table reveals that the outcomes for reactive power generated and control variables obtained via both algorithms are within their limits. Moreover, it can also be seen that the SaBO achieved the lowest generation cost, emission, power loss, and deviation of voltage (VD) with values of 781.2363 $/h, 1.76195 t/h, 5.6908 MW, and 0.47550 p.u., respectively. The TGC of 781.2363 $/h using SaBO include a thermal generation cost of 442.3943 $/h, wind generation cost of 247.9813 $/h and solar generation cost of 90.8606 $/h. The simulation results indicate that the proposed algorithm outperformed the KOA algorithm for the OPF solution. Moreover, the incorporation of wind and solar power generators led to a notable decrease in the TGC to 781.2363 $/h, representing a reduction in the cost of approximately 18.66 $/h (2.33%) compared to the total cost without these sources, which was estimated at 799.8975 $/h. Figure 7 illustrates the convergence curve of generation cost minimization (case 1) through both SaBO and KOA over iterations. The illustration shows the notable superiority and robustness of the SaBO algorithm in achieving a faster convergence rate toward the global optimum compared to KOA.

Table 5 shows a comparison of OPF solutions for case 1 in the modified IEEE 30-bus system with the integration of RESs. Notably, SaBO exhibits superior performance, outperforming all other published algorithms in the literature, except for SMA and DMO algorithms, which achieved a generation cost of 781.0786 and 780.989 $/h, respectively. Additionally, SaBO demonstrates its superiority with a total power loss value of 5.6908 MW, outperforming both SMA (5.7502 MW) and DMO (5.7209 MW) algorithms. The pollutant gas emission in the first case using the SaBO is 1.76195 t/h which is less when contrasted with the outcomes obtained from different optimization techniques. The complete terms corresponding to all the abbreviations are indicated in the table of the appendix.

Table 5.

Comparison of OPF solutions for case 1.

Algorithms		TGC ($/h)	P_loss (MW)	VD (p.u)	Emission (t/h)
SaBO		781.2363	5.6908	0.47550	1.76195
KOA		781.3568	5.7082	0.48188	1.76295
SMA	Mouassa et al. (2022)	781.0786	5.7502	0.74015	NR
WSO	Ali et al. (2022)	781.733	5.80114	0.468569	1.763241
SHADE	Biswas et al. (2017)	782.503	5.770	0.463	1.762
EJADE	Li et al. (2020)	782.2426	NR	NR	NR
BMO	Sulaiman and Mustaffa (2021)	781.6519	NR	NR	NR
IEO	Nguyen et al. (2022)	782.0343	5.9771	NR	0.1603
GWO	Khan et al. (2020)	781.40	5.44	1.05	1.76
MGBLFA	Alghamdi (2022)	781.3930	5.7968	0.45565	1.76242
JS	Farhat et al. (2021)	781.6387	5.773893	0.448284	1.761998
GTO	Mouassa et al. (2023)	781.2626	5.7117	0.4838	1.76209
DMO	Mouassa et al. (2024)	780.989	5.7209	0.46464	1.76298
ARO	Mouassa et al. (2024)	781.2613	5.7112	0.4959	1.7621
GBLCSBO	Ghasemi et al. (2023)	781.8010	5.7653	0.46270	1.76197
SAWGA	Trojovský et al. (2024)	782.0238	5.7866	0.45505	1.76226

Case 2: Minimization of total active power loss

Reducing the active power transmission loss is the objective function in the second case. The optimal results for this case after 30 independent runs are tabulated in Table 4. From this table, the optimization outcomes reveal that using the SaBO algorithm results in a power loss of 1.9958 MW and a generation cost of 879.7964 $/h without any violation of the constraints. These optimal values are notably lower than those achieved by the KOA algorithm, which recorded 1.9963 MW for power loss and 880.2709 $/h for generation cost. Furthermore, the integration of renewable sources resulted in a significant reduction in total power loss to 1.9963 MW when using SaBO. This reflects a decrease in power loss of approximately 31.16% compared to the reference power loss without these sources, which was estimated at 2.9 MW.

The convergence curve for case 2 is depicted in Figure 8. This figure indicates that SaBO rapidly reached convergence, achieving the minimum active power loss after only 70 iterations. In contrast, the KOA algorithm required 200 iterations to reach the optimum.

Table 6 displays the statistical comparison results between SaBO and various algorithms from the literature. It made sense that the proposed SaBO algorithm gives the best value compared to those found by KOA (1.9963 MW), EJADE (2.0478 MW), BMO (2.0646 MW), GWO (2.0616 MW), IEO (2.2211 MW), EKOA (1.9962 MW), GBLCSBO (2.0741 MW), and SAWGA (2.1037 MW). The comparison results prove that the SaBO outperforms all other available algorithms that mentioned in this table.

Table 6.

Comparison of OPF solutions for case 2.

Algorithms		TGC ($/h)	P_loss (MW)	VD (p.u)	Emission (t/h)
SaBO		879.7964	1.9958	0.55567	0.09897
KOA		880.2709	1.9963	0.54905	0.09887
EJADE	Li et al. (2020)	NR	2.0478	NR	NR
BMO	Sulaiman and Mustaffa (2021)	NR	2.0646	NR	NR
GWO	Sulaiman and Mustaffa (2021)	NR	2.0616	NR	NR
IEO	Nguyen et al. (2022)	877.7513	2.2211	NR	0.0938
EKOA	Abid et al. (2024)	880.733	1.9962	0.55507	0.099
GBLCSBO	Ghasemi et al. (2023)	880.6382	2.0741	0.50841	0.09894
SAWGA	Trojovský et al. (2024)	884.7377	2.1037	0.54543	0.09856

Case 3: Minimization of combined generation cost and emission (CGCE)

The combined generation cost with carbon-emission tax is used as the OPF objective function in the third case. The detailed optimal results applied to SaBO and KOA for case 3 are listed in Table 4. The results from this table indicate that the CGCE using SaBO led to 809.7452 $/h with a low computational time of 144.051 seconds, which is less compared to the KOA with the value of 810.0663 $/h. The convergence curve of the CGCE with SaBO and KOA over iterations is depicted in Figure 9. It is worth highlighting that the SaBO achieves convergence to the optimal value in just 50 iterations, demonstrating a notably faster convergence compared to KOA, which necessitates 90 iterations to reach the global optimal fitness.

Table 7 reveals comparison results of OPF solutions using SaBO and different existing techniques in order to minimize the CGCE. This table displays that the optimum value of the CGCE is obtained by SaBO (809.7452 $/h) followed by GWO (809.93 $/h), KOA (810.0663 $/h), … etc. These results make clear that the SaBO outperforms all other published algorithms that are applied to minimize the CGCE for the IEEE 30-bus system.

Table 7.

Comparison of OPF solutions for case 3.

Algorithms		CGCE ($/h)	P_loss (MW)	VD (p.u.)	Emission (t/h)
SaBO		809.7452	5.2021	0.48336	0.89945
KOA		810.0663	5.2011	0.48459	0.89502
SMA	Mouassa et al. (2022)	810.3875	NR	0.4760	0.8986
SHADE	Biswas et al. (2017)	810.346	5.276	0.469	0.891
EJADE	Li et al. (2020)	811.2226	NR	NR	NR
BMO	Sulaiman and Mustaffa (2021)	810.7982	NR	NR	NR
MFO	Sulaiman and Mustaffa (2021)	811.4229	NR	NR	NR
GWO	Khan et al. 2020)	809.93	4.99	1.07	0.86
JS	Farhat et al. (2021)	810.1201	5.276	0.46884	0.89377
EKOA	Abid et al. (2024)	809.89	NR	NR	0.8812
KOA	Abid et al. (2024)	810.1167	NR	NR	0.8836
GBLCSBO	Ghasemi et al. (2023)	810.5507	5.2762	0.47055	0.87348
SAWGA	Trojovský et al. (2024)	810.7448	5.2758	0.47047	0.87627

Figure 10 illustrates the load buses voltage magnitude using SaBO for each of the three cases investigated. Notably, the figure demonstrates that the voltage magnitudes for three cases range within acceptable limits, between a lower and upper limit of 0.95 and 1.05 p.u., respectively.

Table 8 presents the statistical results of SaBO and KOA for three case studies within 30 independent runs of the simulation. The table reveals that SaBO outperformed the KOA algorithm in terms of the minimum, maximum and average results of three different objective functions. As well as the best standard deviation of fitness is given by SaBO compared to KOA.

Table 8.

Statistical results of SaBO and KOA for cases 1–3.

Case No.	Algorithms	Minimum	Maximum	Average	Standard deviation
#Case 1	KOA	781.3568	782.6029	781.9529	3.010E−01
#Case 1	SaBO	781.2363	782.3567	781.7031	2.848E−01
#Case 2	KOA	1.9963	2.0008	1.9978	1.019E−03
#Case 2	SaBO	1.9958	1.9958	1.9958	1.349E−15
#Case 3	KOA	810.0663	811.6401	810.7825	3.582E−01
#Case 3	SaBO	809.7452	811.8389	810.6723	3.465E−01

The boxplot of three cases using SaBO and KOA for 40 times run are shown in Figure 11. This boxplot shows that SaBO achieved the lowest values for minimum, maximum, and average fitness, outperforming KOA, which exhibited the worst fitness in the three cases. Therefore, SaBO demonstrates superior and robust performance when solving the OPF problem considering stochastic RESs compared to the KOA.

Application on modified Algerian power system DZA 114-bus

To demonstrate the effectiveness of SaBO in a practical power system incorporating RESs, the study focuses on the Algerian DZA 114-bus power system. This system includes the integration of two wind generators at buses number 52 and 83. Additionally, a solar plant is integrated into the system at bus No. 109. These locations of RESs are selected as in Mouassa et al. (2022), to compare the results obtained from this configuration with those documented in the same reference. Figure 12 displays the schematic representation of the Algerian electric power system, which consists of 15 generators and 175 transmission lines, including 16 transformers tap ratios positioned between line 160 and line 175. This electrical system supplies 99 loads of 3727 MW + j 2070 MVar. The upper and lower bounds of the 30 control variables that are to be minimized for the Algerian DZA 114 bus system are shown in the results table.

Case 4: Minimization of total generation cost for Algerian DZA 114-bus system

The last case investigated was minimizing the TGC for the Algerian DZA 114-bus power system. The control variables in this case include 30 settings, that is, 15 for active power-generating outputs and 15 for generator voltages. The detailed optimal results achieved in this case using the SaBO and KOA are summarized in Table 9. In this table, all control variables are within acceptable limits. Furthermore, the SaBO algorithm yields better values for total fuel cost, emissions, and active power losses, specifically 16,706.1630 t/h, 1.41045 t/h, and 67.2569 MW, respectively, compared to the optimal solution obtained by the KOA algorithm (16,734.5378 t/h, 1.41337 t/h, and 70.5113 MW). These results suggest that the proposed algorithm is more effective than KOA in addressing the OPF problem including stochastic RESs in real and practical power system. In addition, the integration of wind and solar generators in the Algerian power system significantly reduced the total generation cost to 16,706.1630 $/h compared to the initial cost without these sources of 18,914.105 $/h that was obtained by Kouadri et al. (2020b). This represents a significant cost reduction of approximately 2,207,942 $/h (11.67%).

Table 9.

Comparison of OPF solutions for case 4 in Algerian DZA 114-bus system.

Algorithms		TGC ($/h)	P_loss (MW)	VD (p.u)	Emission (t/h)
SaBO		16,706.1630	67.2569	5.83669	1.41045
KOA		16,734.5378	70.5113	3.54769	1.41337
SMA	Mouassa et al. (2022)	16,693.11	68.5089	5.3316	NR
AEO	Mouassa et al. (2022)	16,767.82	76.5565	3.01914	NR
HGS	Mouassa et al. (2022)	16,794.44	71.5737	3.7598	NR
GTO	Mouassa et al. (2022)	16,795.02	72.1003	3.8629	NR
OPA	Mouassa et al. (2022)	16,811.17	74.0320	3.428	NR
EKOA	Abid et al. (2024)	16,701.77	69.4451	4.6987	NR
KOA	Abid et al. (2024)	16,730.907	72.0597	4.40771	NR

The convergence curves of both SaBO and KOA algorithms are illustrated in Figure 13. It is clear from this figure that the SaBO rapidly converges to the optimal solution within the first quarter of iterations, whereas the KOA algorithm reaches convergence toward the optimal solution at iteration 260.

Table 9 summarizes the comparison results for generation cost, power transmission loss, deviation of voltage, and emission obtained by SaBO and other recently developed algorithms in the literature to illustrate the performance of the SaBO to solve the OPF problem in a larger dimension. From this table, SaBO achieves a minimum TGC of 16,706.1630 $/h. As a result, SaBO outperforms all algorithms that mentioned in Table 10, except the SMA and EKOA algorithms, which have a value of 16,693.11 and 16,701.77 $/h, respectively. Furthermore, SaBO stands out with its superior total power loss value (67.2569 MW) compared to both SMA (68.5089 MW) and EKOA (69.4451 MW) algorithms. The difference between the value yielded by SaBO and the worst value achieved by OPA stands at 105.007 $/h, a significant differential that equates to notable cost savings of around 2520.168 $/day and 919,861.32 $/year.

Table 10.

Optimal results via SaBO and KOA for Algerian DZA 114-bus system: Case 4.

Variables	Limits		Case 4		Variables	Limits		Case 4
Variables	Min	Max	KOA	SaBO	Variables	Min	Max	KOA	SaBO
P_TG4	135	1350	410.390	413.8700	V_TG98	0.9	1.1	1.0867	1.0920
P_TG5	135	1350	410.9851	412.9679	V_TG100	0.9	1.1	1.0969	1.0999
P_TG11	10	100	99.9998	99.9712	V_TG101	0.9	1.1	1.0995	1.0990
P_TG15	30	300	178.6338	176.646	V_SG109	0.9	1.1	1.1000	1.0990
P_TG17	135	1350	405.2193	393.2027	V_TG111	0.9	1.1	1.0512	1.0616
P_TG19	34,5	345	165.6137	168.3079	Q_TG4	−20	400	347.5431	261.6533
P_TG22	34,5	345	164.9992	165.5787	Q_TG5	−20	200	143.1950	196.8938
P_WG52	0	345	345.0000	344.9988	Q_TG11	−50	100	76.9952	97.5738
P_TG80	34,5	345	156.2490	160.8144	P_TG15	0	100	53.7218	78.2621
P_WG83	0	300	300.0000	299.9933	Q_TG17	0	400	394.1644	368.3421
P_TG98	30	300	160.6672	157.9293	Q_TG19	0	60	52.8076	59.7771
P_TG100	60	600	599.9881	599.9899	P_TG22	0	50	40.8260	49.0861
P_TG101	20	200	200.0000	200.0000	Q_WG52	0	50	48.0813	46.3767
P_SG109	0	100	99.7639	99.9924	Q_TG80	0	60	58.4207	59.6749
P_TG111	10	100	99.9984	99.9938	Q_WG83	−50	200	164.5333	167.8279
V_TG4	0.9	1.1	1.0569	1.0904	Q_TG98	0	50	44.9037	48.2941
V_TG5	0.9	1.1	1.0444	1.0841	Q_TG100	0	270	126.4374	108.5647
V_TG11	0.9	1.1	1.0518	1.0979	Q_TG101	−50	200	172.1064	143.3928
V_TG15	0.9	1.1	1.0553	1.0999	Q_SG109	−50	100	32.7603	28.3043
V_TG17	0.9	1.1	1.0827	1.0998	Q_TG111	−50	155	72.0252	74.4804
V_TG19	0.9	1.1	1.0351	1.0627	Generation cost ($/h)			16,734.5378	16,706.1630
V_TG22	0.9	1.1	1.0563	1.0787	Emission (t/h)			1.41337	1.41045
V_WG52	0.9	1.1	1.0808	1.0999	Power loss (MW)			70.5113	67.2569
V_TG80	0.9	1.1	1.0488	1.0574	Voltage deviation (p.u.)			3.54769	5.83669
V_WG83	0.9	1.1	1.0929	1.0992	Computation time (s)			2060.0173	1135.8927

Table 11 presents the minimum, maximum, and average results of the objective function obtained by SaBO and KOA for case 4 from a 30-time run. Notably, the table indicates that SaBO consistently outperformed the KOA algorithm in all obtained results.

Table 11.

Statistical results of SaBO and KOA for Algerian DZA 114-bus system: Case 4.

Case No.	Algorithms	Minimum	Maximum	Average	Standard deviation
Case 4	KOA	16,734.5378	16,779.6009	16,758.6702	1.511E + 01
Case 4	SaBO	16,706.1630	16,738.2864	16,718.0855	7.549E + 00

In Figure 14, the boxplot illustrates the generation cost obtained from 30 runs using SaBO and KOA. The illustration indicates that SaBO consistently achieves lower values for minimum, maximum, and average fitness results in comparison to the KOA algorithm. Furthermore, it can also be seen that SaBO exhibits robust performance and superior accuracy in OPF solutions compared to the KOA.

Conclusion

This paper investigates a recently developed metaheuristic optimization algorithm named self-adaptive bonobo optimizer (SaBO) to address the OPF problem in electric power systems that include thermal, wind and solar power generators. The inherent variability of wind and solar power outputs has been characterized by modeling their uncertainties using the Weibull and lognormal PDFs, respectively. To evaluate the efficacy of the SaBO to solve this stochastic problem on both the modified IEEE 30-bus and Algerian DZA 114-bus power, three OPF objective functions have been examined that is, the minimizations of total generation cost, active power transmission loss, as well as combined generation cost and emission effect. The generation cost function included the thermal fuel cost, the direct costs associated with wind and solar generators, as well as the reserve and penalty costs due to the overestimation and underestimation of available wind and solar power, respectively. The optimal minimization of these objective functions involves adjusting the real output power and voltage magnitudes of different types of power generators. The simulation outcomes of the SaBO are compared with those obtained from the KOA algorithm, as well as with the findings of other state-of-the-art optimization techniques provided in the literature. The comparisons result clearly show that the suggested SaBO outperformed the performance of both the KOA and the majority of the algorithms found in the literature for all case studies. Thus, the SaBO algorithm demonstrates its effectiveness in successfully solving more complex and stochastic optimization problems, even in practical electrical network cases, all the while maintaining the feasibility of the solutions obtained.

Footnotes

Appendix

Abbreviations
ACNSDE	Adaptive Crossover Non-Dominated Sorting Differential Evolution
ARO	Artificial Rabbits Optimization
	I. Artificial Rabbits Optimization
	II. Artificial Rabbits Optimization
BMO	Barnacles Mating Optimizer (BMO)
_CGCE	Combined Generation Cost and Emissions
_DMO	Dwarf Mongoose Optimizer
_GBLCSBO	Gaussian bare-bones Levy circulatory system-based optimization
IEO	Improved Equilibrium Optimizer
ISO	Independent System Operator
ISSA	Improved Salp Swarm Algorithm
KOA	Kepler optimization algorithm
NGO	Northern Goshawk Optimizer
NSGA-II	Non-dominated Sorting Genetic Algorithm
OPF	Optimal Power Flow
_PDFs	Probability Distribution Functions
POA	Pelican Optimization Algorithm
_PV	Photovoltaic
RESs	Renewable Energy Sources
SaBO	Self-adaptive bonobo optimizer
SAO	Smell Agent Optimization
SMA	Slime Mould Algorithm
SPGs	Solar Power Generators
_TGC	Total Generation Cost
TPGs	Thermal Power Generators
WPGs	Wind Power Generators
WSO	White Shark Optimizer
_WT	Wind Turbines

Acknowledgements

The authors deeply appreciate the support of the “Electric Operator System” of Algeria for providing access to the Algerian power system data. I would like to thank Dr. Souhil Mouassa from electrical engineering departement, university of bouira , for his significant contribution, which greatly aided in the completion of this work.

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Ramzi Kouadri

Souhil Mouassa

Francisco Jurado

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