Optimal power flow of hybrid wind/solar/thermal energy integrated power systems considering renewable energy uncertainty via an enhanced weighted mean of vectors algorithm
Ahmed H. A. Adam, Salah Kamel, Mohamed H. Hassan, Ghazally I. Y. Mustafa
TL;DR
This paper introduces a new optimization algorithm to manage energy systems with wind, solar, and thermal power, improving efficiency and reducing costs despite renewable energy uncertainty.
Contribution
A novel ARINFO algorithm is proposed, combining artificial rabbits optimization with a weighted mean of vectors approach to solve optimal power flow under uncertainty.
Findings
ARINFO achieved 1st rank on CEC-2017 and CEC-2022 test suites.
ARINFO reduced generation cost to 781.1538 $/h and emissions to 0.0922140 t/h in the IEEE 30-bus system.
ARINFO attained a total cost of 20193.270 $/h in the IEEE 57-bus system, demonstrating scalability.
Abstract
The rising global energy demand, along with the growth of electric power transmission and distribution systems, has intensified the need to incorporate renewable energy sources to foster sustainable development. However, achieving optimal operation within such systems poses significant challenges due to the stochastic nature of renewable energy generation. As a result, the optimal power flow (OPF) problem becomes increasingly complex when addressing the inherent uncertainty of renewable inputs. This study presents a new approach to addressing the OPF problem through the implementation of a hybrid Weighted Mean of Vectors Optimization Algorithm (INFO) based on artificial rabbits optimization (ARO) called ARINFO technique. The proposed ARINFO algorithm aims to reach an exploration-exploitation balance to improve search efficiency. To effectively manage the uncertainty associated with…
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Fig 28- —School of Mechanical and Electrical Engineering, Quanzhou University of Information Engineering
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Taxonomy
TopicsOptimal Power Flow Distribution · Electric Power System Optimization · Energy Load and Power Forecasting
1. Introduction
1.1. Motivation
The concept of optimal power flow (OPF) was first introduced by Carpentier in 1962. Since that time, a multitude of methodologies have been developed to tackle the OPF problem, which is essential for minimizing power losses, enhancing voltage stability, reducing generation costs, and curbing greenhouse gas emissions. The OPF problem is typically subject to a range of physical and operational constraints, including generator capability limits, voltage bounds at buses, transmission line capacities, and permissible power flow through cables. These constraints significantly complicate the optimization process, especially in large-scale power systems, necessitating careful strategies to ensure that system parameters remain within feasible limits.
Traditionally, OPF formulations have concentrated on conventional generation sources reliant on fossil fuels, resulting in a highly complex optimization landscape characterized by mixed-integer, nonlinear, and non-convex properties. As the integration of renewable energy sources into modern power grids continues to grow, it is essential to incorporate the inherent uncertainty associated with renewables into OPF analysis, particularly during both planning and operational phases.
To address these challenges, various classical optimization techniques have been employed, including quadratic programming, nonlinear programming, mixed-integer linear programming, and interior-point methods. These approaches are appreciated for their rapid convergence rates and ability to generate optimal solutions, and many have been successfully implemented in real-world applications. However, a common limitation of these methods is their dependence on the linearization of the objective function, which may diminish their effectiveness in managing complex, nonlinear systems.
As an alternative, heuristic optimization algorithms have been proposed to overcome the limitations of traditional methods. These approaches do not require gradient information or function linearization and have demonstrated promise in solving intricate OPF problems characterized by high-dimensional and non-convex search spaces.
1.2. Literature review
A diverse array of meta-heuristic algorithms has been investigated in the literature to address various formulations of the optimal power flow (OPF) problem. One of the earlier approaches employed a sequential genetic algorithm (GA) combined with a simple genetic algorithm (SGA) aimed at effectively optimizing control variables while adhering to operational constraints within the system [1]. Another significant study introduced a robust Tabu Search-based method, which was validated on the IEEE 30-bus system, showcasing both reliability and computational efficiency [2]. Differential evolution (DE) has also garnered considerable attention due to its rapid convergence capabilities, making it particularly well-suited for OPF scenarios characterized by complex decision variables. However, a consistent challenge associated with DE is its tendency to converge on local optima rather than exploring the global solution space [3].
Despite the widespread application of heuristic techniques to solve OPF problems, many of these methods encounter limitations that hinder their practical effectiveness. For instance, particle swarm optimization (PSO) is valued for its ease of implementation and quick convergence; however, it is susceptible to premature convergence and may perform poorly in high-dimensional scenarios or under stringent constraints [4]. Simulated annealing (SA), while capable of escaping local optima, is highly reliant on the design of the cooling schedule and can display slow convergence rates [5]. Although ant colony optimization (ACO) proves effective for discrete problem domains, it generally converges slowly and may stagnate without adequate diversification strategies [6]. While differential evolution (DE) possesses a strong exploratory mechanism, its success heavily depends on carefully tuning mutation and scaling parameters [7]. The artificial bee colony (ABC) algorithm tends to lack sufficient exploitation strength, particularly during the final stages of optimization [8]. Additionally, newer bio-inspired methodologies such as the firefly algorithm (FA) and whale optimization algorithm (WOA) show promise, yet they often struggle with slow convergence and stagnation issues when faced with complex, multimodal search landscapes [9–11].
Numerous studies have proposed enhancements to metaheuristic optimization techniques to address challenges such as early convergence and solution quality in solving the OPF problem. A notable instance is a modified Runge-Kutta optimization algorithm introduced in [12], designed to enhance the integration of renewable energy sources (RES) into power systems. This approach accounts for the inherent intermittency and stochastic behavior of RES, particularly in the context of flexible AC transmission systems (FACTS). The study focused on minimizing three objective functions: total expected power losses (TEPL), total expected voltage deviations (TEVD), and total expected voltage stability (TEVS), with performance evaluated on the IEEE 57-bus system. In [13], a modified turbulent flow of water optimization (MTFWO) algorithm was employed to address the OPF problem within systems that include wind turbines (WT) and photovoltaic (PV) units. This approach considers the output power from renewable energy sources as a dependent variable while managing voltage levels to ensure operational balance. Simulations were conducted on the IEEE 30-bus system. The research presented in [14] introduced the white shark optimizer (WSO) as a solution for the OPF problem, aimed at minimizing fuel costs while adhering to system constraints. The integration of solar and wind energy alongside conventional thermal units was modeled using lognormal and Weibull probability distribution functions (PDFs), respectively. The algorithm’s effectiveness was illustrated through its application to the IEEE 30-bus benchmark.
A hybrid decomposition-based multi-objective evolutionary algorithm was introduced in [15] to address the OPF problem while considering WT and PV output uncertainties. The optimization aims to minimize total generation cost (TC), active power losses (APL), emissions (TE), and voltage deviation. The approach utilized a novel constraint-handling strategy along with Monte Carlo simulation for effective uncertainty modeling. The methodology was validated on the IEEE 30-bus, 57-bus, and 118-bus systems. In [16], the chaos game optimization (CGO) algorithm was utilized to tackle the OPF problem through the integration of RES and flexible AC transmission system (FACTS) devices. This method employs power generated by RES as state variables, while bus voltages are treated as control variables, with output predictions modeled using Weibull and lognormal PDFs. Validation of this approach was conducted using the IEEE 30-bus test system. Additionally, the enhanced hunter-prey optimization (EHPO) algorithm, proposed in [17], enhances exploration and exploitation capabilities for OPF challenges involving WT and FACTS integration. This algorithm incorporates random mutation and adaptive strategies to balance the global and local search phases effectively.
Moreover, in [18], a hybrid optimization framework that combines artificial ecosystem-based optimization (AEO) and chaos game optimization (CGO), known as ACGO, was developed to tackle the OPF problem with integrated RES. This method considers the uncertainties associated with wind and solar generation, employing Weibull and lognormal probability distributions. The proposed approach was validated using both the IEEE 30-bus and 57-bus systems, showcasing its effectiveness in managing OPF under uncertain conditions.
Recent advancements in addressing the stochastic optimal power flow problem have highlighted the integration of RESs, such as wind and photovoltaic (PV) systems, with modern optimization techniques. For instance, a modified runge-kutta optimizer was introduced in [12] to effectively manage uncertainties in power systems that incorporate wind, PV, and FACTS like TCSC. This study showcased significant improvements in minimizing power losses and voltage deviations using the IEEE 57-bus system under stochastic conditions. Additionally, an enhanced version of the gorilla troops optimizer, as described in [19], utilized chaotic-quasi-oppositional learning and multi-population strategies to improve the balance between exploration and exploitation in optimal power flow problems. This method exhibited robust convergence and solution quality across various objectives, including fuel cost optimization and voltage stability.
Collectively, these studies underscore a trend towards biologically inspired, hybrid metaheuristic strategies that effectively tackle the nonlinear and non-convex challenges inherent in SOPF problems involving integrated renewable energy and uncertainty modeling. These insights not only enhance the current methodological landscape but also reinforce the relevance of the proposed ARINFO algorithm.
The increasing integration of renewable energy sources into power grids poses a significant challenge in maintaining the stability and efficiency of modern electrical systems. Sources like wind and solar introduce substantial variability and uncertainty into power flow, complicating the optimal operation of power systems. Consequently, traditional methods for solving the OPF problem must be adapted to account for these uncertainties. Despite advancements in metaheuristic applications for OPF issues, effectively managing the inherent uncertainties of RES, such as wind and solar, is a considerable challenge. The stochastic nature of RES generation leads to complexities associated with both overestimation and underestimation of power, making it essential to include reserve and penalty costs in the optimization model. Additionally, traditional algorithms often struggle to maintain a balanced equilibrium between exploration and exploitation, resulting in premature convergence or suboptimal solutions in high-dimensional, non-convex search spaces.
This study addresses the OPF problem within the framework of hybrid energy systems that combine renewable sources with conventional thermal units. The objective is to optimize overall power generation while minimizing both costs and emissions. By incorporating uncertainty modeling for renewable generation, this research proposes a novel optimization algorithm to manage hybrid power systems’ inherent complexities. In particular, the study aims to deliver a robust and efficient technique for solving the OPF problem while explicitly considering uncertainties in both renewable generation and load demand.
1.3. Importance of investigation
As highlighted by previous studies, various optimization techniques have been effectively employed to tackle engineering challenges, particularly in OPF problems. These studies underscore the success of such methods and the ongoing necessity of developing more robust and efficient optimization strategies tailored to specific OPF scenarios. Moreover, the No-Free-Lunch (NFL) theorem [20] illustrates that no single metaheuristic algorithm can universally solve all optimization problems. This theoretical constraint reinforces the importance of designing novel algorithms.
To achieve the global optimum while avoiding the pitfalls of local optima, various strategies can be employed. Effective algorithmic planning that promotes intelligent scheduling of transitions between exploration and exploitation phases, along with establishing an appropriate equilibrium between these phases, is among the viable approaches. Adequate scheduling and balancing can mitigate the accuracy issue to a degree; nevertheless, the most potent method involves integrating the most appropriate variety throughout the solution process. Therefore, hybridization is regarded as a robust strategy that raises the necessary diversity in the quest for the global optimum, thereby minimizing the likelihood of becoming entrenched at a local optimum point [21]. As a result, one of the objectives of this study is to introduce a hybrid algorithm designed to enhance solution accuracy while maintaining high solution diversity throughout the entire solution process.
Based on these insights, this study introduces a novel optimization method, ARINFO, which enhances the weighted mean of vectors (INFO) algorithm. Utilizing INFO’s robust global search capabilities in conjunction with ARO’s effective local refinement, the hybrid approach adeptly balances exploration and exploitation. This combined framework harnesses the advantages of both algorithms, enhancing both extensive exploration and rapid local refinement abilities. During the global exploration phase, INFO plays a crucial role, while ARO is activated during the local refinement phase. At the beginning of each iteration, an update to the population is executed using INFO. Subsequently, in the middle of the iteration, local refinement occurs in the vicinity of the optimal solution with ARO, and finally, at the conclusion of the iteration, the optimal solution is revised. Consequently, the search remains expansive, while refinement is concentrated around the optimal solution. This methodology ensures that INFO’s global exploration capability significantly boosts the likelihood of discovering superior solutions across a vast solution space. Furthermore, ARO’s local search capability facilitates a more rapid enhancement of the solution. The integration of these two algorithms results in enhanced solution quality and reduced optimization duration. This adaptive framework enables the execution of both global and local searches, allowing for adjustments to evolving optimization conditions. The proposed approach aims to reduce power losses, improve voltage stability, lower generation costs, and mitigate gas emissions. The choice of the underlying algorithm is informed by its proven effectiveness in addressing complex mathematical and engineering design challenges across various domains.
1.4. Main contributions
While the original INFO optimizer demonstrates strong convergence speed and promising performance through its mean-based update rule, vector combination, and local search strategies, it also has several limitations that an enhanced version can address [22]. Notably, INFO’s dependence on the weighted mean and fixed procedures may result in premature convergence in complex or multimodal problems due to inadequate mechanisms for maintaining population diversity over time. Furthermore, although INFO includes a local search step, its structure is relatively simplistic and deterministic, which may hinder its effectiveness in fine-tuning solutions or escaping difficult local optima. Additionally, INFO does not incorporate adaptive control or feedback mechanisms to dynamically balance exploration and exploitation throughout the optimization process, which can diminish its robustness across varied problem landscapes [23]. These shortcomings lay the groundwork for enhancing INFO by introducing mechanisms that promote adaptability, diversity maintenance, and improved local refinement, thereby representing the core contributions of the proposed work.
This study makes several significant contributions to the field of OPF optimization:
A novel and enhanced variant of the ARINFO algorithm is introduced. This improved algorithm tackles OPF challenges by minimizing power losses, enhancing voltage stability, reducing generation costs, and limiting greenhouse gas emissions.The effectiveness of the ARINFO algorithm is rigorously validated through a comprehensive evaluation using 23 standard benchmark functions and 12 functions from the CEC-2022 test suite.A thorough performance assessment of ARINFO is conducted, employing statistical measures, analyzing convergence behavior, performing boxplot comparisons, and executing qualitative evaluations.The results generated by ARINFO across all test cases are compared against nine well-established metaheuristic algorithms, including the original INFO algorithm [23], artificial hummingbird algorithm (AHA) [24], artificial rabbits optimization (ARO) [25], whale optimization algorithm (WOA) [26], spider wasp optimizer (SWO) [27], particle swarm optimization (PSO) [28], moth–flame optimization (MFO) [29], seagull optimization algorithm (SOA) [30], and sine cosine algorithm (SCA) [31].Furthermore, the IEEE 30-bus system is modified to include renewable energy sources, specifically wind and solar units. ARINFO, in conjunction with AHA, ARO, PSO, SOA, MFO, and the original INFO algorithm, is employed to address the OPF problem under four distinct objective functions.Practical case studies are undertaken that consider generation and demand uncertainty, as well as ramp rate constraints for thermal power units. The outcomes from ARINFO are compared with those from alternative methods within these scenarios to assess its robustness and applicability in real-world situations.A modified IEEE 57-bus test system and an IEEE 118-bus test system are utilized to evaluate the proposed method’s scalability and generalizability.
The findings provide strong evidence that the proposed ARINFO algorithm exhibits superior performance compared to the conventional INFO algorithm in addressing the OPF problem, particularly in terms of solution quality and convergence behavior.
1.5. Structure of the paper
The remainder of this paper is organized as follows: Section 2 introduces the formulation of the OPF problem. Section 3 examines the constraints associated with the OPF model. Section 4 delves into various objective functions pertinent to OPF optimization. Section 5 details the development of the proposed ARINFO optimization framework. Section 6 presents simulation results along with a critical discussion. Section 7 illustrates the practical implementation of the ARINFO framework within the context of OPF. The final section provides concluding remarks and offers suggestions for future research directions.
2. Problem formulation of optimal power flow
The OPF problem is a mathematical optimization framework aimed at identifying the most efficient operating conditions for power system components while adhering to all relevant constraints. Generally, the OPF seeks to optimize a specific objective function, such as minimizing generation costs or power losses, subject to equality and inequality constraints that reflect physical and operational limits. In essence, the OPF problem can be formulated in the following mathematical structure [14]:
Subject to
In this context, F represents the objective function, x signifies the vector of state variables, and u corresponds to the vector of control variables. The functions gi and hj denote the equality and inequality constraints, respectively, while m and n represent the total numbers of equality and inequality constraints in the system [14].
The state variables, also referred to as dependent variables, can be represented in vector form as follows:
where PG1 represents the active power output of the slack bus, while VL denotes the voltage magnitude at the load buses. NPQ is the total number of load buses, and QG refers to the reactive power outputs of the generators, with NPV indicating the number of generator buses. STL signifies the apparent power flow through the transmission lines, and NTL represents the total number of transmission lines.
The control variables, also referred to as independent variables, can be organized in a vector format as follows:
Here, PG represents the active power output from the generators, while NG indicates the total count of generators. VG refers to the voltage magnitude at the generator buses. QC signifies the reactive power contributed by shunt compensators, with NC denoting the total number of these compensators. T represents the tap settings of transformers, and NT corresponds to the overall number of transformers within the system.
This section presents a comprehensive analysis of the production costs associated with each power source in the IEEE 30-bus system. As the production cost methodology for the IEEE 57-bus system is the same, it will not be detailed here.
2.1. Thermal power generation cost
The costs associated with thermal power generation (TPG) are assessed using equation (5), which accounts for the valve-point effect of thermal units to improve cost estimation accuracy [32].
where PTGi denotes the output power of the i^th^ TPG. The coefficients ai, bi, and ci are associated cost coefficients. Additionally, di and ei represent the valve-point effect coefficients linked to each TPG. NTG indicates the total number of thermal generators in the system, while PTGi^min^ marks the minimum generation limit for the i^th^ TPG. In Appendix A, Table A1 in S1 File provides the cost coefficients related to TPGs [14].
2.2. Cost components of wind power generation
Unlike TPG, wind power generation (WPG) is characterized by considerable variability and uncertainty. As a result, the costs associated with wind energy production are calculated using a distinct methodology, as detailed below.
a. Direct Cost Component
The direct cost of power generated by WPG, in relation to its scheded output, is determined using the following expression:
where Pswj refers to the scheduled power output of the j^th^ wind power plant, while hwj indicates its corresponding direct cost coefficient.
b. Uncertain Cost Component
Due to the inherent variability of WPG, actual output may often fall short of the scheduled values, leading to an overestimation of power availability. In these situations, the ISO is required to compensate for the shortfall by utilizing adequate spinning reserves [33]. The costs associated with maintaining these reserves are quantified as follows:
In this context, Kreswj refers to the reserve cost coefficient associated with the j^th^ wind power plant, while Pava.wj represents the actual power output produced by that unit. The probability density function (PDF) for the wind power output of the j^th^ wind plant is denoted by fwind.j.
On the other hand, the actual power output from a WPG may surpass its scheduled value, a situation known as underestimation. In these instances, the ISO faces a penalty cost for the excess generation, calculated as follows:
In this context, Kpenwj denotes the penalty cost coefficient, while Prated.wj represents the rated power of the j^th^ wind power plant.
2.3. Probabilistic characterization of wind power generation
This section discusses the probabilistic modeling of power output from wind plants, represented by the term fwind.j(Pwind.j) in (7) and (8). The Weibull PDF is commonly used to characterize wind speed distributions [32]. Consequently, the probability of wind speed is evaluated using the Weibull distribution, as defined by the following expression:
where Wdv represents the wind speed measured in (m/s), while k and c signify the shape and scale parameters of the Weibull distribution, respectively. The mean wind speed corresponding to the Weibull probability density function is calculated using the following expression:
In this expression, Γ denotes the gamma function, which is defined mathematically as follows:
This study involves modifications to the IEEE 30-bus test system, which include replacing two thermal generators situated at buses 5 and 11 with wind turbine generators. Additionally, the thermal generator at bus 13 has been substituted with a solar photovoltaic (PV) unit. In Appendix A, the parameters for the Weibull distribution, specifically the scale (c) and shape (k) values used for modeling wind speed, are presented in Table A2 in S1 File, while the wind frequency distribution, obtained through an 8000-iteration Monte Carlo simulation, is depicted in Fig A1 in S1 File, showcasing the precision of the Weibull fit [32].
2.4. Power models for wind turbine generators
As previously mentioned, the modified IEEE 30-bus power system includes two WPGs. WPG1, situated at bus 5, has a rated active power output of 75 MW, while WPG2, connected at bus 11, has a capacity of 60 MW. The relationship between wind speed and the resulting power output of these generators is described by the following equation [14]:
where Wdvin represents the cut-in wind speed, Wdvout denotes the cut-out wind speed, and Wdvr indicates the rated wind speed. Prated.w corresponds to the rated power output of the wind turbine. The probability of the wind power output falling within a specific discrete range can be calculated using the following expression:
For the continuous operating range, the probability of the wind farm’s power output can be assessed with the following equation:
2.5. Cost components of solar power generation
The total cost of electricity generated by a solar photovoltaic (PV) system consists of two primary components: the direct generation cost and the costs related to output uncertainty. This cost structure is similar to that employed for wind energy systems [23].
a. Direct Cost Component
The direct cost of solar energy production is calculated using the following equation:
where Pssk represents the scheduled power output of the k^th^ solar power plant, while gsk denotes the direct cost coefficient linked to that unit.
b. Uncertain Cost Component
Similar to wind power systems, the cost analysis of solar power generation considers both scenarios of overestimation and underestimation [34]. A reserve cost is incurred when the predicted solar output surpasses the actual generation. This reserve cost linked to overestimation is determined using the following expression:
In this formulation, Kressk denotes the reserve cost coefficient for the kth solar power plant, while Pava.sk represents its available power output. The term fsolar.k(Pava.sk < Pssk) captures the probability that the plant’s actual output falls short of the scheduled value Pssk. Meanwhile, Ex(Pava.sk < Pssk) refers to the expected magnitude of this shortfall [35]. The penalty incurred for underestimating solar power production is quantified by the following equation:
where Kpensk represents the penalty cost coefficient associated with the k^th^ solar plant. The term fsolar.k(Pava.sk > Pssk) indicates the probability that the actual solar power output surpasses the scheduled generation. Additionally, Ex(Pava.sk < Pssk) denotes the expected surplus output from the k^th^ solar plant [36,37].
2.6. Probabilistic characterization of solar power plants
The power output of a solar photovoltaic generator is mainly determined by solar irradiance (I), which is often modeled using a lognormal PDF. Consequently, the statistical behavior of solar irradiance can be represented by the following lognormal distribution :
Let fI(I) denote the probability density function of solar irradiance. This irradiance is modeled using a lognormal distribution, which is defined by its mean μ and standard deviation σ. The mean value of the lognormal distribution, represented as Mlgn, is calculated using the following expression:
In Appendix A, Fig A2 in S1 File illustrates the frequency distribution alongside the lognormal probability curve for solar irradiance based on an 8000-iteration Monte Carlo simulation. The specific parameter values utilized in the lognormal probability density function for this analysis are detailed in Table A2 in Appendix A in S1 File.
2.7. Power models for solar photovoltaic systems
The electrical power produced by solar photovoltaic systems is directly affected by the intensity of solar irradiance (I). Consequently, the relationship between solar radiation and the output power of a photovoltaic system can be mathematically expressed as follows [32]:
In this expression, Istd signifies the standard solar irradiance level, which is typically established at 800 W/m², while Tc represents a specific threshold irradiance value set at 120 W/m². The term Psolarr refers to the rated capacity of the solar photovoltaic system.
The reserve cost linked to solar power, as detailed in equation (17), can be reformulated by integrating the probability distribution of solar power availability as follows:
where Psn− represents the shortfall in solar power, defined as the amount of actual output that falls below the scheduled generation level. This is depicted on the left-hand side of the distribution curve for scheduled solar power output (Pssk) in Fig A3 (Appendix A) in S1 File. The notation fsn− refers to the relative frequency associated with instances of this shortfall, while N^−^ denotes the number of discrete intervals (bins) within this underperformance region [38].
Furthermore, the penalty cost expression previously introduced in equation (18) can be reformulated as follows:
The variable Psn+ denotes the surplus solar power, which refers to the amount of generated power that exceeds the scheduled output. This condition is illustrated in the right-hand section of the distribution curve for expected solar power output (Pssk) shown in Fig A3(Appendix A) in S1 File. The variable fsn+ represents the relative frequency of these surplus occurrences, while N+ indicates the number of discrete intervals (bins) within this region of excess generation.
2.8. Emission
Electricity generation from conventional fossil fuel sources significantly contributes to greenhouse gas emissions. Among the most harmful byproducts are sulfur oxides (SOₓ) and nitrogen oxides (NOₓ) [12]. The levels of these pollutants typically correlate with the amount of electrical power produced by thermal power plants. This relationship, which quantifies emissions in (t/h) as a function of generated output in (p.u. MW), is mathematically represented in equation (24):
In this formulation, the coefficients αi, βi, γi, ωi, and μi represent the emission parameters associated with TPGs. The specific values assigned to these coefficients are detailed in Table 1A (Appendix A in S1 File) and align with the data referenced in [24]. The total emissions, measured in metric tonnes, are then converted into an economic cost, referred to as EC (in /tonne*.
3. Problem constraints
The optimal power flow problem is governed by a set of operational constraints that are essential for ensuring the system operates effectively. These constraints are classified into two primary categories: equality and inequality.
3.1. Equality constraints
Equality constraints are primarily designed to maintain a power balance within the system. They stipulate that the total generated active and reactive power must equal the sum of consumed power (including both active and reactive power) as well as the losses within the transmission system [39].
In this context, NB signifies the total number of buses within the power system. The variables PD and QD correspond to the active and reactive components of the load demand, respectively, while PG and QG represent the active and reactive power generated. The voltage angle difference between buses i and j is denoted by δij, with Bij and Gij representing the susceptance and conductance of the transmission line that connects the two buses.
3.2. Inequality constraints
Inequality constraints are essential for enforcing the operational limits of power system components and ensuring the overall security of the network. These constraints define the permissible operating ranges for generation units, transmission lines, and load buses. They can be systematically categorized as follows [14]:
a. Generator Restrictions
Each generator’s operation within the power system is restricted by established lower and upper limits. These constraints pertain to active power output, reactive power output, and generator voltage levels, as outlined in equations (28), (29), and (30), respectively. In this context, NTG represents the total number of generators in the network [12].
In addition, the generation limits for renewable energy sources, specifically wind turbines and solar photovoltaic (PV) units, are defined by equations (31) through (34). These equations outline the permissible ranges for their active and reactive power outputs.
b. Transformer Restrictions
c. Shunt Compensator Restrictions
where NTG, NT, and NC represent the total number of power generators, power transformers, and shunt compensators, respectively.
d. Ramp Rate Restrictions for TGs
Within the framework of optimal power flow analysis, ramp-rate constraints for thermal generators are imposed to restrict the permissible rate of change in power output between successive time intervals, expressed formally as follows:
where P^0^TGi refers to the output power of the i^th^ thermal generator during the previous time interval. The parameters Uri and Dri represent the upper and lower ramp-rate limits, respectively, which dictate the permissible changes in output for that particular generator [40,41].
e. Transmission Line Capacity Constraints
Power flow through transmission lines must remain within established thermal and operational limits to ensure system stability and safety. This requirement is mathematically expressed in equation (39), where NL indicates the total number of transmission lines in the electrical network.
f. Load Bus Voltage Restrictions
The voltage levels at load buses are subject to predefined lower and upper bounds to ensure stable and secure system operation. These limits can be expressed as:
where NLB represents the total number of load buses in the network.
An additional critical metric associated with load buses is the voltage deviation, which quantifies the deviation of each load bus voltage from the nominal value (typically 1.0 p.u.). It is calculated using the following expression:
This parameter serves as an indicator of voltage quality across the system, with lower values signifying better adherence to the nominal voltage level.
4. OPF objective functions
The OPF framework presented in this study aims to optimize various performance metrics, focusing on the minimization of total generation costs (both with and without the application of carbon emission taxes), the reduction of greenhouse gas emissions, and the minimization of active power losses. These objectives are approached through distinct mathematical formulations.
4.1. Case 1: Minimization of Total Generation Cost Without Carbon Tax (F1)
The objective function F1 reflects the total production cost without carbon taxation. This is formulated by aggregating the cost components outlined in the “OPF problem” section. Consequently, the expression for F1 is defined as follows:
4.2. Case 2: Minimization of Total Generation Cost with Carbon Tax (F2)
The objective function F2 enhances the cost formulation presented in F1 by integrating the costs associated with carbon emissions, as defined in equation (25). Therefore, the total cost under carbon taxation is represented by the following expression:
4.3. Case 3: Minimization of Carbon Emissions (F3)
The objective function F3 concentrates on decreasing the total carbon emissions produced by thermal power units. It is formulated according to the emission model detailed in equation (24) and can be mathematically expressed as follows:
4.4. Case 4: Minimization of Active Power Losses
The objective function in this scenario, denoted as F4, aims to minimize active power losses within the power system. These losses are quantified using equation (45), which models power dissipation across transmission lines.
In this formulation, δij represents the voltage angle difference between buses i and j, NL is the total number of transmission lines, and Gij denotes the conductance of the line connecting the two buses. Based on this, the objective function F4 is defined as:
5. Development of the Proposed ARINFO optimization framework
5.1. Overview of the Weighted Mean of Vectors Algorithm (INFO)
The INFO algorithm, introduced in [42], offers a novel optimization framework based on a modified version of the weighted mean of vectors method. As a recently developed metaheuristic, INFO utilizes a vector-based update mechanism along with a local search strategy to navigate the solution space effectively [22]. The algorithm proceeds through the following sequential phases:
Updating rule stageVector combinationLocal search execution
- ** Initialization phase **
The initial population for the INFO algorithm is generated randomly and is represented as follows:
Here, Np denotes the population size, while D represents the dimensionality of the optimization problem. Two key parameters play a crucial role in the vector updating mechanism within INFO: the scaling factor (σ), which adjusts the magnitude of the weighted mean vector, and the weighted mean factor (δ), which enhances the influence of the generated vector during the update process.
ii. Updating rule stage
During this phase, weighted mean vectors are computed using randomly selected differential vectors. The MeanRule, a mean-based updating mechanism, is applied to adjust the positions of solution vectors. This rule incorporates information from the worst, a randomly chosen better (selected from the top five performing solutions), and the best solutions within the population. The MeanRule formulation is expressed as follows [22]:
In which
where W1, W2, and W3 can be calculated as follows:
In this context, f(x) represents the objective function, while a1, a2, and a3 are randomly selected integers within the range [1, Np]. The parameter ε denotes a small positive constant. The variables xbs, xbt, and xws refer to the best, better, and worst solutions, respectively, within the g^th^ generation.
The scale factor δ, as indicated in equation (59), modulates the influence of the weighted vector. Furthermore, the parameter β is a dynamic variable that evolves according to an exponential function, as outlined in equation (60).
Maxg represents the maximum number of iterations. Within the INFO algorithm, a convergence acceleration (CA) parameter is utilized to enhance global search performance by more effectively directing the population toward promising areas of the solution space. The CA parameter is mathematically defined as follows:
Here, randn denotes a random variable drawn from a standard normal distribution. The newly generated vector is defined as follows:
The proposed update mechanism, which incorporates the variables xbs, xbt, xl^g^, and xa1^g^, is formulated according to the following scheme:
In this formulation, z1l^g^ and z2l^g^ represent the newly generated vectors for the g^th^ generation, with σ indicating the vector scaling factor, as described in equation (64). It is worth noting that the parameter α within equation (64) can be dynamically adjusted in accordance with the exponential function defined in equation (65).
The parameters c and d represent constant values, which are explicitly defined as 2 and 4 in this context.
iii. Vector combining stage
To enhance population diversity, the two vectors computed in the previous section (z1l^g^ and z2l^g^) are combined with vector xl^g^ when the condition rand < 0.5 is met. This process generates the new vector ul^g^ as described in equations (66–68). This operator encourages local search, resulting in the creation of a new vector.
if rand < 0.5
if rand < 0.5
else
end
else
end
The combined vector ul^g^ is generated in the g^th^ iteration through vector fusion, where the scaling factor μ is determined stochastically as 0.05 multiplied by a uniformly distributed random variable (μ = 0.05 × randn).
iv. Local search stage
A local search operator has been introduced to enhance the local search capabilities of the INFO algorithm and to prevent convergence to suboptimal solutions. This operator utilizes the global best position (x^g^best) along with a mean-based weighting mechanism, as detailed in equations (69–72). When the condition r < 0.5 is satisfied (where r is a uniformly distributed random variable within the interval [0,1]), the operator generates a new solution vector in the vicinity of x^g^best.
if rand < 0.5
if rand < 0.5
else
end
end
in which
ϕ denotes a randomly generated number within the interval [0, 1]. The term xmd refers to a candidate solution created by a random combination of xavg, xbt, and xbs, which enhances the stochastic characteristics of the algorithm and encourages a more extensive exploration of the solution space. Furthermore, the random variables v1 and v2 are defined as follows:
In this context, p represents a randomly selected value within the interval [0, 1]. The random variables v1 and v2 may enhance the impact of the best-known position within the solution vector. In summary, the comprehensive structure of the proposed INFO algorithm is outlined in Algorithm 1, and its procedural flow is visually illustrated in Fig 1 [2].
The flowchart of the INFO algorithm.
Algorithm 1. The Pseudo-code of the INFO algorithm [38]
1: STEP 1. Initialization
2: Set parameters Np and Maxg
3: Produce an initial population P^0^ =
4: Calculate the objective function value of each vector f( ), i = 1, ..., Np
5: Determine the optimal vector xbs
6: STEP 2. for g = 1 to Maxg do
7: for i = 1 to Np do
Select randomly a ≠ b ≠ c ≠ i inside the range [1, Np]
► Updating rule stage
8: Calculate the vectors and by equation (63)
► Vector combining stage
9: Compute the vector using equations (66-68)
► Local search stage
10: Compute the local search operator using equations (69-73)
11: Compute the objective function value f( )
12: if then
13: Otherwise
14: end for
15: Update the optimal vector (xbs)
16: end for
17: STEP 3. Return Vector as the final solution.
5.2. Overview of the Artificial Rabbits Optimization (ARO)
The original ARO mimics the foraging and hiding tactics of actual rabbits, as well as their energy shrink, leading to transiting between these tactics [25].
a) Detour foraging (exploration)
In detour foraging behavior of ARO, each individual in the search space tends to update its location towards the other search individual chosen randomly from the group and add a perturbation. The following equation describe the mathematical model of the detour foraging:
b) Random hiding (exploitation)
To escape from predators, a rabbit commonly digs some holes nearby its nest for hiding. This equation is given in this regard as:
The rabbits to be survive need to find a safe residence to hide. So, they select randomly a hole from their holes for hiding to escape from getting caught. this random hiding tactic is modeled as below:
After detour foraging or p random hiding is reached, the position update of the i^th^ rabbit is:
c) Energy shrink (switch from exploration to exploitation)
An energy factor is considered to model the switch from exploration to exploitation phases. The energy factor in this algorithm can be given as follows:
where, is the index of the best solution.
5.2. Modified ARINFO Algorithm
The flowchart of the proposed ARINFO algorithm is presented in Fig 2. Moreover, Algorithm 2 describes the ARINFO algorithm’s pseudocode. The proposed ARINFO algorithm offers several advantages in addressing complex optimization problems. Its primary strength lies in achieving an effective exploration-exploitation balance, which enhances search efficiency and allows the algorithm to navigate diverse solution spaces while converging on optimal solutions. By integrating ARO with INFO, ARINFO demonstrates improved robustness and adaptability across various problem domains. Additionally, it can efficiently handle high-dimensional datasets, making it suitable for real-world applications requiring precise optimization. However, ARINFO may face challenges due to its computational complexity, which can increase with larger datasets or intricate problem structures, potentially slowing convergence.
Flowchart of the ARINFO algorithm.
Algorithm 2: Pseudocode of the ARINFO algorithm
Set the control parameter (Dimension of problem (d), maximum iteration, population size), and h
Initialize the population randomly
Evaluate the fitness of the new solution
Obtain the best solution
While t ≤ T
%INFO
for i = 1: Np
Select randomly a ≠ b ≠ c ≠ i inside the range [1, Np]
Updating rule stage
Calculate the vectors and by equation (63)
Vector combining stage
Compute the vector using equations (66–68)
Local search stage
Compute the local search operator using equations (69–73)
Compute the objective function value
Update the optimal vector (xbs)
end for
%ARO
Calculate the energy factor A
If
Choose a rabbit randomly and perform detour foraging
Else If
Generate burrows and randomly pick one as hiding and perform random hiding
End If
Check the limits of the new locations and evaluate the fitness values
Find the new solution if the fitness is better
t = t + 1
End while
Output the best solution
6. Simulation results and discussion
The OPF problem is addressed through the proposed ARINFO algorithm. To evaluate its performance and computational efficiency, ARINFO is tested on the benchmark suites CEC-2017 and CEC-2022 [43–45]. The results are then compared with several established optimization algorithms, including INFO, AHA, ARO, WOA, SWO, PSO, MFO, SOA, and SCA. To ensure consistency and fairness in the comparisons, all competing algorithms were configured using the parameter settings outlined in their original publications. ARINFO was implemented in MATLAB 2024a and executed on a system equipped with an Intel Core i7 processor (2.10 GHz) and 32 GB of RAM. Detailed numerical results are presented in the following sections.
a. Exploration vs Exploitation
Examining the balance between the exploration and exploitation phases of the ARINFO can provide significant insights for tackling practical optimization issues. To facilitate a comprehensive assessment of the ARINFO, this research investigates the algorithm’s capabilities in exploration and exploitation, as informed by the five CEC-2022 benchmark functions analyzed. Fig 3 depicts the equilibrium between the exploratory and exploitative characteristics of the ARINFO, INFO, and ARO techniques. In this figure, the red curve denotes the algorithm’s exploratory inclinations, while the blue curve emphasizes its exploitative capabilities. From this figure, the proposed ARINFO demonstrates a better balance between exploration and exploitation, which typically leads to stronger global search ability and improved optimization performance.
Exploration vs exploitation analysis of ARINFO, INFO, and ARO algorithms.
b. Qualitative Analysis
Qualitative analysis serves to evaluate the robustness of the proposed ARINFO algorithm by examining the behavior of search agents and the evolution of the objective function throughout the optimization process. Fig. B1 (Appendix B) in S1 File illustrates qualitative outcomes for a representative set of test functions: three unimodal functions (F1, F4, and F7), three multimodal functions (F8, F10, and F12), and three composite functions (F14, F18, and F21) [46].
In the first column of Fi B1, a three-dimensional visualization of each fitness landscape provides insights into the shape and complexity of the objective function. The second column presents the progression of search agents across iterations, demonstrating that ARINFO effectively guides the population toward regions of lower objective values. The third and fourth columns, respectively, depict the average fitness values over time and the convergence trajectory of the algorithm. Analysis of these plots reveals significant fluctuations in the early stages of the search, indicative of extensive exploration. These fluctuations diminish later, reflecting a transition to exploitation and convergence toward optimal or near-optimal solutions. This observed behavior confirms the algorithm’s robustness and capacity to balance exploration and exploitation effectively.
c. Statistical Analysis
The statistical performance of the proposed ARINFO algorithm, alongside INFO [23], AHA [24], ARO [25], WOA [26], SWO [27], PSO [28], MFO [29], SOA [30], and SCA [31], is presented in Tables B1 and B3 (Appendix B) in S1 File for the CEC 2017 and CEC 2022 benchmark functions, respectively. In these tables, the best-performing results are highlighted in bold red for clarity. Based on the results shown in Table B1 (Appendix B) in S1 File, ARINFO outperforms the competing algorithms on several benchmark functions, including F1 to F4, F6, F12, and F15 to F23. Its performance is comparable to other leading algorithms on functions F9 to F11 and F14. However, ARINFO performs comparatively less on functions F5, F7, F8, and F13. The average rank and overall ranking for the CEC 2017 benchmarks are summarized in Table B2 in S1 File, which confirms the superior performance of ARINFO relative to the other optimization techniques.
Similarly, the statistical analysis in Table B3 in S1 File demonstrates ARINFO’s leading performance on most functions of the CEC 2022 suite, specifically CEC01, CEC03 to CEC07, and CEC10 to CEC11. For CEC08, ARINFO performs on par with competing algorithms, while it outperforms CEC02, CEC09, and CEC12. The corresponding average rank and overall rankings for the CEC 2022 benchmark functions are provided in Table B4, further underscoring ARINFO’s strong comparative performance.
d. Convergence Behavior
The convergence trends for the benchmark functions CEC-2017 and CEC-2022 are depicted in Figs B2 and B3 (Appendix B) in S1 File, respectively. As illustrated in these figures, the proposed ARINFO algorithm demonstrates superior convergence characteristics for the majority of test functions, achieving faster convergence rates compared to other algorithms such as INFO, AHA, ARO, WOA, SWO, PSO, MFO, SOA, and SCA.
However, there are specific instances, such as with functions F7, F8, and F13, where algorithms like SOA and ARO exhibit more favorable convergence dynamics. Additionally, PSO performs best for function CEC09. Despite these exceptions, ARINFO consistently converges more efficiently in most scenarios, requiring fewer iterations to attain optimal or near-optimal solutions. These findings underscore the robustness and effectiveness of the algorithm in addressing a diverse array of complex optimization problems.
e. Boxplot Analysis
The boxplot representations comparing the ARINFO algorithm with competing methods on the CEC-2017 and CEC-2022 benchmark functions are illustrated in Figs B4 and B5 (Appendix B). In these plots, the whiskers represent the minimum and maximum values achieved by each algorithm. A narrower box indicates greater consistency and robustness in the results. The ARINFO algorithm exhibits remarkable statistical stability, with no outliers detected across more than thirteen functions within the CEC-2017 suite, including F1, F2, F3, F4, F9, F10, F11, F14, F15, F17, F18, F22, and F23. Similarly, it shows strong performance on key functions of the CEC-2022 suite, particularly CEC01, CEC05, CEC09, and CEC10.
Overall, the boxplot distributions reveal that ARINFO consistently achieves lower objective values and demonstrates tighter performance variability compared to its competitors. This confirms its superior reliability and effectiveness in addressing complex optimization challenges.
f. p-Value-Based Statistical Analysis
The statistical validity of the performance differences between the proposed ARINFO algorithm and its competitors is evaluated through the Wilcoxon rank-sum test. This non-parametric test facilitates pairwise comparisons between optimization methods by analyzing their result distributions. A significant difference is indicated by a p-value below 0.05, while a p-value exceeding this threshold suggests no statistically significant distinction. Table B5 (Appendix B) in S1 File displays the p-values for comparisons between ARINFO and various algorithms (INFO, AHA, ARO, WOA, SWO, PSO, MFO, SOA, and SCA) on the CEC-2017 benchmark functions. The majority of these p-values are below 0.05, indicating substantial performance differences in favor of ARINFO. Instances where p-values exceed 0.05 are highlighted in bold, signifying that in those cases, ARINFO’s performance is statistically comparable to that of other methods.
Likewise, Table B6 (Appendix B) in S1 File presents the p-values for comparisons on the CEC-2022 suite. Once again, most values fall below the significance threshold, affirming ARINFO’s statistically superior results in the majority of benchmark scenarios. Bold entries indicate non-significant differences (p-value > 0.05). Additionally, “NaN” values reflect identical outcomes between ARINFO and another algorithm, rendering statistical comparison unfeasible.
Furthermore, the results from the benchmark functions CEC-2017 and CEC-2022, as outlined in Tables B7 and B8 (Appendix B) in S1 File, were subjected to statistical analysis using ANOVA, the Friedman test, and the Kruskal-Wallis test. The findings consistently demonstrate that the proposed ARINFO algorithm outperforms the nine alternative optimization methods, as indicated by p-values below the 0.05 significance threshold, thereby affirming its superior effectiveness.
7. Application of ARINFO to the Optimal Power Flow Problem
This section provides a thorough evaluation of the performance of the ARINFO algorithm in solving the OPF problem. It compares ARINFO to several established metaheuristic algorithms, including ARO, AHA, PSO, SOA, MFO, and the standard INFO. The analysis utilizes multiple realistic case studies to validate the effectiveness and robustness of ARINFO in tackling OPF challenges.
The IEEE 30-bus test system has been modified to integrate renewable energy sources. Specifically, the thermal generators at buses 5 and 11 have been replaced with wind power generators, while the thermal unit at bus 13 has been substituted with a solar photovoltaic (PV) system. A similar modification has been applied to the IEEE 57-bus system, where thermal generators at buses 2 and 6 are now wind farms, and the generator at bus 9 has been transformed into a solar PV plant. Detailed specifications of the integrated wind and solar units for both systems are outlined in Table A2 in Appendix A in S1 File and Table 1.
Table 1: Wind and solar plants data- IEEE −57 bus system [23].
In the IEEE 30-bus system, the control variables comprise the scheduled active power outputs of all thermal generators (excluding the slack bus at bus 1), as well as the scheduled outputs of the two wind farms and the solar PV plant. Additionally, the voltage magnitudes at all generator buses are included as control parameters. In the configuration of the IEEE 57-bus system, the set of control variables is expanded to encompass the reactive power injections from shunt compensators and the tap positions of transformer branches. Case studies 1–8 are based on the modified IEEE 30-bus system, while Cases 9 and 10 focus on the IEEE 57-bus system. All simulation experiments are conducted using MATLAB, ensuring a realistic modeling of system dynamics and optimization under practical constraints.
7.1. Case 1: Minimizing the Total Cost
This section addresses the optimization problem of the modified IEEE 30-bus power system, focusing on minimizing the total generation cost while adhering to system constraints outlined in equation (42). The ARINFO algorithm achieves this goal with a termination criterion of 300 maximum iterations per run across 30 independent trials. The statistical results indicate that ARINFO consistently produces outcomes close to the optimal solution. The recorded total cost values ranged from a best of 781.1538/h, yielding a mean value of 781.8054$/h and a standard deviation of 0.1553. These findings reflect both the high quality and robustness of the solutions. The average computational time per run was approximately 581.4 sec.
For further analysis, three representative runs, the 15th, 19th, and 27th, were selected. As illustrated in Fig 4, the corresponding generation costs for these runs were 781.3501, 781.1538, and 781.4528$/h, respectively, demonstrating the algorithm’s ability to achieve near-optimal solutions consistently.
Convergence curves of the ARINFO algorithm for the three best-performing runs.
A crucial component of the OPF analysis is maintaining voltage levels at load buses within acceptable operational limits. In the context of the IEEE 30-bus system, these voltages must be constrained within the range of 0.95 to 1.05(p.u.). Proper regulation of load bus voltages is essential to ensure system stability and reliability. Fig 5 illustrates the voltage profiles of the load buses across the initial three iterations utilizing the proposed INFO algorithm. As shown in Table 2 and Fig 6, the voltages at all load buses remain comfortably within the established safety margins. Furthermore, the voltage levels at the generator buses are consistently maintained within their specified upper and lower thresholds.
Table 2: Simulation results for Case 1.
Voltage profile of load buses.
Voltage profiles of generator buses.
Case 1 focuses on minimizing the total generation cost across all power sources within the modified IEEE 30-bus power system, as defined by equation (42). The parameters of the PDF utilized in this scenario are outlined in Table 2, while the relevant cost coefficients are detailed in Table A1 in Appendix A. The optimization results obtained using the proposed ARINFO algorithm are presented in Table 2, alongside comparisons with other optimization methods such as AHA, ARO, PSO, SOA, MFO, and the standard INFO. The control variables for this case include the voltage magnitudes at all generator buses and the scheduled active power outputs of all generators except TPG1, which acts as the slack bus. The operational limits for these control variables conform to the specifications provided in [42]. In addition to the objective function values shown in Table 2, the total voltage deviation across the load buses, calculated using equation (42), and the overall system power loss determined by equation (45) are also documented.
The results highlight the exceptional performance of the ARINFO algorithm, which achieved the lowest total generation cost of 781.1538$/h, demonstrated the shortest computational runtime of 581.4 sec, and exhibited the fastest convergence rate, as illustrated in Fig 7.
Convergence performance comparison for Case 1.
7.2 Case 2: Optimization of Total Cost with Varying Reserve Cost Coefficients
In this case, the same system parameters from Case 1 are retained, with the sole exception being the modification of the reserve cost coefficient (RCC) to evaluate its effect on optimal power generation scheduling. To explore the impact of different RCC values, three subcases are examined: Case 2a with RCC = 5, Case 2b with RCC = 6, and Case 2c with RCC = 7. The penalty cost coefficient (PCC) for all RESs remains constant at 1.5, consistent with Case 1 [14].
For each subcase, the optimal power output schedule for all generators is established and illustrated in Fig 8. As anticipated, increasing the reserve cost coefficient leads to a decrease in the scheduled output from wind and solar power sources. This outcome highlights the system’s strategy to minimize potential reserve costs linked to power overestimation by reducing dependence on variable renewable generation. Consequently, thermal power units are scheduled to generate more electricity to offset the diminished contribution from RESs.
Optimal scheduled active power outputs of all generators under different reserve cost coefficients.
While this adjustment alleviates production costs related to renewables, it results in higher costs for thermal generation. Overall, the total generation cost rises with increasing reserve cost coefficients, as depicted in Fig 9.
Impact of varying reserve cost coefficients on individual cost components.
7.3. Case 3: Optimization of Total Cost with Varying Penalty Cost Coefficients
In this case study, all system parameters are consistent with those in Case 1, except the penalty cost coefficients (PCCs). To examine the effects of varying penalty costs, we conducted three subcases applying different PCC values to both wind and solar power sources: Case 3a with PCC = 3, Case 3b with PCC = 4, and Case 3c with PCC = 5. The RCC for all RESs remains constant at 3, aligning with its value in Case 1 [14].
The optimal power schedules for all generators across these subcases are displayed in Fig 10. As anticipated, an increase in the PCC results in a corresponding rise in the scheduled power output from RESs. This adjustment aims to mitigate the risk of incurring high penalties associated with underestimating renewable generation. Consequently, the scheduled output from thermal generators is reduced to accommodate the greater share of renewable energy in the mix.
Optimal scheduled active power outputs of all generators under different penalty cost coefficients.
These scheduling adjustments have a direct impact on the cost structure, as shown in Fig 11. While production costs for wind and solar generation increase, thermal generation costs decline. Nevertheless, the total generation cost exhibits an upward trend across the subcases due to the heightened reliance on renewables, which results in higher penalty costs.
Impact of varying penalty cost coefficients on individual cost components.
The reactive power outputs for Cases 1, 2, and 3 are depicted in Fig 12. The data indicate that both TPG3 and WPG2 consistently exceeded their maximum reactive power thresholds. This finding emphasizes the critical importance of incorporating reactive power constraints within the optimization process to maintain system feasibility and ensure reliable performance.
Scheduled reactive power outputs of all generators in Cases 1, 2, and 3.
Fig 13 presents the voltage magnitudes at the generator buses for the same scenarios. All observed voltage levels fall within the permissible range of 0.95 to 1.1 per unit, thereby confirming compliance with established voltage standards across all evaluated cases.
Voltage profiles of generator buses in Cases 1, 2, and 3.
7.4. Case 4: Minimization of Total Generation Cost with Carbon Tax
In this scenario, the ARINFO algorithm is utilized to assess the impact of implementing a carbon tax on emissions from thermal power generation. The goal is to minimize the total generation cost under a carbon taxation policy, as articulated in equation (43). All system parameters remain consistent with those outlined in Tables 2 and 11, except for the carbon tax rate, which is $20 per tonne of CO₂ emissions.
Table 11: Optimal settings of control variables for Case 11 using the ARINFO algorithm.
In contrast to case 1, where no emissions penalty was applied, this scenario illustrates a more significant integration of RES within the optimal generation schedule. The degree of RES penetration is predominantly driven by the emission levels of thermal units and the magnitude of the carbon tax. This relationship is clearly reflected in the simulation results presented in Table 3.
Table 3: Simulation outcomes for Case 4.
The convergence behavior of the ARINFO algorithm, alongside competing algorithms, is illustrated in Fig 14, showcasing ARINFO’s superior performance in minimizing total generation costs. The algorithm achieves a minimum cost of 808.7305$/h with a commendably low computational time of 373.517sec.
Convergence performance comparison for Case 4.
7.5. Case 5: Minimizing Carbon Emissions
This scenario centers exclusively on minimizing greenhouse gas emissions by applying the ARINFO algorithm, as outlined in equation (44). The system configuration comprises three thermal generation units identified as significant contributors to carbon emissions. In contrast to previous cases, the objective in this instance focuses solely on emission reduction, excluding cost considerations.
As shown in Table 4, this environmentally focused strategy substantially decreases total emissions compared to case 1, though it does incur higher generation costs. The ARINFO algorithm is exceptionally effective in this context, surpassing not only its predecessor INFO but also other benchmark optimization methods in terms of emission reduction and convergence efficiency, as illustrated in Fig 15.
Table 4: Simulation outcomes for Case 5.
Convergence performance comparison for Case 5.
7.6. Case 6: Minimization of Active Power Losses
A primary objective in OPF analysis is the reduction of active power losses. This case specifically focuses on that goal, adhering to the formulation outlined in equation (46). The findings from this analysis are summarized in Table 5 and illustrated in Fig 16. Among the algorithms assessed, the ARINFO method achieved the lowest power loss and demonstrated the fastest convergence rate, outperforming both the original INFO method and other comparative techniques. Furthermore, Fig 17 presents the voltage profiles of the load buses in the IEEE 30-bus system for cases 1, 4, 5, and 6, employing the ARINFO algorithm. This confirms that all bus voltages remain within the acceptable range of 0.95 to 1.05 p.u., thus fulfilling standard voltage stability criteria.
Table 5: Simulation outcomes for Case 6.
Convergence performance comparison for Case 6.
Load bus voltage profiles of the IEEE 30-bus power system in Cases 1, 4, 5, and 6 using the ARINFO optimization algorithm.
7.7. Case 7: Incorporating Thermal Generator Ramp-Rate Constraints
In this scenario, the objective remains the minimization of total generation costs, as in case 1. However, unlike case 1, this setup introduces ramp-rate limitations for TPGs. The previous hour’s output of each TPG, along with their respective ramp-rate limits, is detailed in Table A1 in Appendix A. Table 6 summarizes the optimal results under these constraints, while Fig 18 illustrates the convergence behaviors of the various algorithms tested.
Table 6: Simulation outcomes for Case 7.
Comparative convergence performance for Case 7.
The simulation results clearly indicate that applying ramp-rate limits on TPGs leads to an increase in total generation costs. This increment occurs because the operating points of TPGs are adjusted away from their economically optimal positions to meet the ramping constraints. Nevertheless, the ARINFO algorithm demonstrated its superiority, achieving the lowest generation cost of 798.3816 $/h, with rapid convergence and a modest computational time of 385.846 sec, surpassing the performance of the other evaluated techniques.
Fig 19 presents a comparative analysis of total generation costs for cases 1, 4, and 7 utilizing the ARINFO approach. The findings highlight that the implementation of ramp-rate constraints, along with the introduction of a carbon tax on thermal generators, leads to an increase in overall generation costs.
Comparative analysis of total generation costs in Cases 1, 4, and 7.
7.8. Case 8: Load Demand Uncertainty Analysis
This scenario assesses the performance of the ARINFO algorithm in the context of load demand uncertainty. A load-based scenario analysis is conducted to investigate the OPF under various system loading conditions. The uncertainty in demand is modeled using a normal PDF, characterized by a mean load (ld) of 70 and a standard deviation (rd) of 10, as referenced in [47]. Four distinct loading scenarios are defined, each with specific probabilities and mean values, as outlined in Table 7.
Table 7: Mean values and probabilities of different loading scenarios [47].
The simulation results for these scenarios, presented in Table 8, reveal a clear trend: as system loading increases, so does the total generation cost, as depicted in Fig 20. Moreover, both total power losses (Ploss) and carbon emissions exhibit an upward trend with rising demand while voltage deviation (Vd) decreases, as illustrated in Fig 21. Importantly, all system operational constraints remain satisfied across all scenarios, with load bus voltages consistently maintained within the acceptable range of 0.95 to 1.05 p.u., as shown in Fig 22. These findings confirm the robustness of the ARINFO algorithm in managing load uncertainty while ensuring system reliability and performance.
Table 8: Simulation outcomes for Case 8.
Total cost under versus loading scenarios.
Impact of loading level on voltage deviation, power losses, and carbon emissions.
Voltages of the load buses during the 4 loading scenarios.
7.9. Case 9: Minimizing Total Generation Cost in the IEEE-57 Bus System
This scenario examines the effectiveness of the ARINFO algorithm in tackling the OPF challenge within the intricate IEEE 57-bus network, with a primary focus on minimizing total generation costs. The performance of ARINFO is compared against several other well-known metaheuristic methods, including ARO, AHA, PSO, SOA, MFO, and the original INFO, all under uniform simulation conditions. The objective function and constraints are consistent with those utilized in the IEEE 30-bus system, as outlined in equation (42).
The enhanced configuration of the IEEE 57-bus network features four thermal generation units at buses 1 (swing), 3, 8, and 12, two wind power plants at buses 2 and 6, and a solar PV unit at bus 9. The cost and emission coefficients for the thermal units are detailed in Table 11, while the parameters for the Weibull and lognormal PDFs are listed in Table A2 in Appendix A in S1 File. The network’s total active and reactive load demands are 1250.8 MW and 336.4 MVAR, respectively.
The simulation results, summarized in Table 9, illustrate that ARINFO excels in both cost minimization and convergence performance, as shown in Fig 23. Additionally, the voltage levels at all load buses remain within acceptable limits, as displayed in Fig 24, further confirming the operational viability and robustness of the proposed solution.
Table 9: Simulation outcomes for Case 9.
Comparative convergence performance for Case 9.
Voltages of the load buses of the modified IEEE-57 bus power system.
7.10. Case 10: Minimization of Total Generation Cost with Carbon Tax in the IEEE 57-Bus System
This scenario introduces a more complex objective function aimed at minimizing the overall generation cost while incorporating a carbon tax on emissions from thermal generators in the IEEE 57-bus power system. The objective function follows the formulation described in equation (43), consistent with prior scenarios. All system parameters remain identical to those in case 9, with the carbon tax rate set at 20$ per tonne of emissions.
The simulation is executed over 30 independent runs, each comprising 300 iterations. The results, detailed in Table 10, indicate that the ARINFO algorithm effectively achieves the lowest total generation cost while maintaining efficient convergence behavior. Comparative convergence trajectories between ARINFO and competing optimization methods, including the original INFO algorithm, are presented in Fig 25, highlighting ARINFO’s superior performance in both cost reduction and convergence speed. Additionally, the voltage magnitudes of all load buses remain within the prescribed operational limits, as confirmed by the profiles in Fig 26.
Table 10: Simulation outcomes for Case 10.
Comparative convergence performance for Case 10.
Voltages of the load buses of the modified IEEE-57 bus power system.
7.11. Case 11: Optimization of Fuel Cost for the Large-Scale Test System
To show the effectiveness of the proposed ARINFO for the large-scale test system, it is tested using the IEEE 118-bus system. The line and bus data of the IEEE 118-bus system can be found in [48]. The IEEE 118-bus system has 54 generating units. Bus 69 is chosen as the slack bus. The voltage magnitude limits of all buses are taken between 0.95 p.u. and 1.06 p.u. Also, the tap setting for tap changing transformers is varying between 0.9 p.u. To 1.1 p.u. Moreover, the limit of VAR compensators is assumed to vary between 0 and 0.3 p.u.
In this case, the ARINFO is applied to solve the OPF problem considering the fuel cost for the IEEE 118-bus system. This system is employed in this paper to test the scalability of the proposed ARINFO and prove its ability to solve large-scale systems. The obtained results of ARINFO are compared with INFO and ARO in Table 11. The results in Table 11 proves the superiority of the proposed ARINFO over other methods in solving the OPF problem for the large-scale system. The ARINFO’s objective function (136606.5 $/h) is better than the objective function of other methods without any violation of the constraints. The voltage magnitudes of all buses of the ARINFO are within the minimum and maximum limits as shown in Fig 27. In addition, the ARINFO has smooth and fast convergence characteristics in comparison with other methods, as is clear in Fig 28.
Voltages of the load buses of the IEEE-118 bus power system.
Comparative convergence performance for Case 11.
8. Conclusion
This study presents ARINFO, an enhanced variant of the INFO optimization algorithm, specifically designed to tackle the complexities of the OPF problem amidst the uncertainties that arise from integrating renewable energy sources. ARINFO employs stochastic models to capture the variability of renewable energy outputs, utilizing Weibull and lognormal distributions for wind and solar generation, respectively. Through thorough validation against benchmark functions from the CEC-2017 and CEC-2022 suites, along with a comparative analysis of nine established metaheuristic algorithms, ARINFO showcases superior solution quality, accelerated convergence, and increased robustness across various OPF objectives, including minimizing generation costs (with and without carbon taxation), reducing emissions, and minimizing active power losses.
Furthermore, the study considers the effects of varying reserve and penalty costs, different load demand scenarios, and ramp rate limits for thermal generators, thereby providing a comprehensive framework for managing the inherent uncertainties present in modern power systems.
8.1. Broader implications and global context
The findings of this research hold significant implications for the global energy sector’s transition towards sustainability and resilience. The ARINFO algorithm’s capacity to manage renewable energy uncertainty directly addresses the challenges of integrating higher proportions of renewable energy into existing grids. By modeling the costs of overestimation and underestimation, ARINFO enables grid operators to more effectively accommodate intermittent renewable resources, such as wind and solar, without compromising system stability. This capability is crucial for nations aiming to achieve global carbon neutrality goals.
Moreover, ARINFO facilitates the development of resilient and adaptive power systems. Its ability to optimize power flow under varying conditions, whether driven by fluctuating demand or renewable generation, enhances the economic efficiency of power structures. This optimization can lead to considerable savings for utility companies, ultimately lowering electricity costs for consumers while improving the competitiveness of industries on a global scale. Such advancements are particularly advantageous for developing nations rapidly expanding their power infrastructure and seeking to incorporate renewable energy sources into their grids.
The framework’s multi-objective capabilities also contribute directly to climate change mitigation efforts. By balancing cost reduction with emission minimization, ARINFO equips policymakers and system planners with a valuable tool to aid the transition towards cleaner, more sustainable energy solutions. Notably, ARINFO’s effectiveness in optimizing scenarios involving carbon taxation underscores its potential to steer operational strategies that penalize carbon emissions while promoting cleaner energy sources.
Finally, the scalability of ARINFO, as demonstrated through its application to both medium-sized and larger IEEE test systems, indicates its suitability for real-world, large-scale applications in national and regional grids. This scalability is essential for modern power systems, especially in regions rapidly transitioning to renewable energy sources.
8.2. Future work
Future research could build upon this study by incorporating energy storage systems to manage the variability of renewables better, investigating real-time OPF scenarios, and applying ARINFO to larger hybrid grid models. Furthermore, the integration of adaptive control mechanisms or the coupling of ARINFO with deep learning models has the potential to enhance predictive capabilities and elevate the performance of next-generation power systems. Also, it is important to consider more detailed emission formulations incorporating startup/shutdown processes, ramping limits, and additional environmental constraints to further enhance the practical realism of the proposed approach in future works.
Supporting information
S1 FileSupplementary Material.Appendix A.Table A1 Cost coefficients of TPGs. Table A2 PDF parameters for wind and solar PV power stations. Fig A1 Weibull PDF distribution of wind speed for WPGs. Fig A2 Lognormal PDF of solar irradiance distribution for solar PV. Fig A3. Actual power distribution of the solar photovoltaic generator Appendix B. Fig B1 Qualitative results (3D-view, search history, average objective function, convergence curve). Table B1 Statistical results of competitive techniques for the CEC-2017 test suite. Table B2 Average ranks and overall rankings on CEC-2017. Table B3 Statistical results of competitive techniques for the CEC-2022 test suite. Table B4 Average ranks and overall rankings on CEC-2022. Fig B2 Convergence curves for CEC-2017. Fig B3 Convergence curves for CEC-2022. Fig B4 Box plots for CEC-2017. Fig B5 Box plots for CEC-2022. Table B5 Wilcoxon rank-sum test results for CEC-2017. Table B6 Wilcoxon rank-sum test results for CEC-2022. Table B7 P-value-based statistical metrics for CEC-2017. Table B8 P-value-based statistical metrics for CEC-2022.(DOCX)
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