Optimal sizing of hybrid renewable energy systems relying on the black winged kite algorithm for performance evaluation

Abstract

The hybrid photovoltaic (PV) and wind turbine (WT) system combined with a battery (BT) have emerged as an effective renewable energy solution. As the adoption of hybrid systems gain popularity, determining the optimal system size plays a vital role for achieving cost-effectiveness. This study employs a recently developed Black-Winged Kite Algorithm (BKA) to optimize the size of a PV/WT/BT hybrid system to continuously distribute the electrical power to an educational institution sited in Puri, Odisha, India, with the goal of minimizing the total annual cost (TAC) while considering least levelized cost of electricity (LCOE). To ensure the system reliability, the maximum loss of power supply probability (LPSP) is considered 5%. To validate the performance, the BKA algorithm is compared against four prominent intelligence algorithms, including Harris Hawks Optimisation (HHO), Sine Cosine Algorithm (SCA), White Shark Optimisation (WSO), Arithmetic Optimisation Algorithm (AOA), and Snake Optimiser (SO). The examination focused on estimating the optimum size of the suggested off-grid hybrid system in the context of statistical outcome scenarios. The experimental outcomes demonstrated that the suggested BKA approach in optimization of renewable energy system provided the best results compared to HHO, SCA, WSO, and AOA with a TAC of $7105.23 and LCOE of $0.1874 per kWh at LPSP of 5%, achieving an optimal configuration of 60.4722 kW PV, 23.8337 kW WT, and 13.6159 kWh BT. Additionally, the convergence study reveals that BKA has better convergence and convergence than the other four intelligence algorithms by achieving the optimum solution. As a result, the PV/WT/BT combination evolved as a more realistic choice in designing a reliable and costless hybrid system for satisfying load demand in regional areas.

Introduction

In the current scenario, the population of many countries throughout the world have progressively grown, which has resulted in an increasing need for electric power for both the industrial and home sectors. For this reason, there is a pressing need to increase the amount of electricity that is being produced in order to keep the fast-growing demand under control. Worldwide, there has been a transition in focus from the use of fossil fuels to the adoption of renewable energy sources. The transition has been motivated by the awareness that using energy from fossil fuels has harmful impacts on the environment such as pollution, emission of greenhouse gases releases, and climate changes¹. Renewable energy sources (RES), while environmentally friendly and plentiful, depend on their geographical location and demonstrate variability². Presently, there is a global trend in the direction of the expansion of renewable energy sources³. Therefore, there exist an increasing demand for the development of renewable and eco-friendly renewable sources of energy. The most renowned RESs such as wind, solar power, small hydroelectric plants, cogeneration bagasse, and biomass, have been shown to make a major contribute to the growth of developing nations⁴. The use of wind and solar energies as the principal sources of clean renewable energy has generated significant interest and research in electrification initiatives globally. Recent advancements in technology have augmented the energy output of renewable energy systems⁵. The renewable power industry has experienced a significant rise in investment and regulatory initiatives that encourage the successful utilisation of solar as well as biomass sources of energy. The majority of individuals reside in rural and isolated areas that lack grid connectivity, and the expense of extending the grid to these locations is prohibitively large⁶. Consequently, decentralised environmentally friendly source of energy that consist of biomass, wind, tiny hydroelectric facilities, and solar panels (PV), together with the amalgamation of diverse sources, can be utilised to electrify these areas⁷. Recent advancements in mathematical models have enhanced planning for such decentralized solutions, assisting optimize rural electrification plans⁸. Renewable sources of power possess a fundamental drawback because to their intrinsic variability. Unlike the predictable nature and controllability of energy derived from fossil fuels production, renewable energy generation depends on environmental factors and may exhibit less consistency^9,10. To mitigate this difficulty, hybrid renewable energy systems that amalgamate several energy sources may regulate the strengths of one to compensate for the deficiencies of another. However, the advancement of hybrid renewable energy systems (HRES) has emerged as a formidable challenge because of the unpredictable characteristics of energy derived from clean energy sources and several facets associated with system design, encompassing both technical and economic feasibility^11–13. HRES representations are intended for off-grid applications in distant regions where traditional fossil fuels are unfeasible, although they may also operate in grid-connected mode, so augmenting worldwide protection, economic efficiency, quality, dependability, and perhaps upgrading social indicators^14,15. The exploration of hybrid power systems is highly complex owing to the diversity of generating systems involved and necessitates comprehensive examination. The main purpose of renewable-based hybrid operating systems is to optimise operational circumstances, undertake economic assessments, and improve performance to satisfy all physical and technological specifications. An ideal scaling strategy guarantees superior performance and low expenditure while effectively utilising system components¹⁶. When the number of factors and variables in HRES escalates, optimisation approaches are essential for scaling the system components. Despite several optimisation strategies explored before to modify various components of HRES, an effective approach for estimating optimal sizing to identify dependable and cost-effective systems is rarely discovered.

Literature review

The optimization of HRES has been widely studied using various techniques, including software tool deterministic method and advanced metaheuristic algorithms. The software simulation tools¹⁷ such as Homer, iHOGA, RETScreen, HYBRID2, INSEL, HybSim, SOMES, and many others, which are easy and extensively used due to their commonality and effectiveness. However, the primary limitation of this tools is they do not permit users to modify modelling equations, sizing procedures, or algorithms; they also limit access to calculations¹⁸. Conversely, the deterministic approaches¹⁹ such as linear programming, iterative method, graphical method, probabilistic method and analytical methods which shown more effective than software tolls. However, these methods have several flaws including trapped in local optimal scenario and suffer to determine the global optima. The last one is metaheuristic methods derived from natural behaviour have been identified as promising method and have the ability to discover optimum solutions and prevent them from getting stuck in local optima, while superior to deterministic approaches in determining the best optimal solutions. Metaheuristic optimization methods are used to efficiently explore intricate search spaces and discover nearly optimum solutions for HRES design problems. Despite this, the NFL (No-Free-Lunch) theorem posits that the best possible outcome may be attained for a certain cost function utilizing a specific meta-heuristic method²⁰. However, it is crucial to acknowledge that the exactly same method may produce poor outcomes when implemented with diverse cost functions^21,22. These reasons have prompted researchers to investigate novel metaheuristic approaches for solving the HRES sizing issue. Numerous researchers have endeavoured to establish an appropriate design for the hybrid energy system. Some researchers concentrated on integrating various energy sources in the sizing of energy systems, while others emphasized diverse design methodologies and their goals. Several studies that have been done in this area are reviewed below. Fayza S. Mahmoud et al.²³ integrate salp swarm algorithm, grey wolf algorithm, and improved grey wolf optimizer to determine the best size of a hybrid system including WT/PV/DG/BS, aimed at reducing energy cost and loss of power supply probability (LPSP). Moghaddam s. et al.²⁴ presented a hybrid photovoltaic/wind/battery system to improve the load supply dependability while diminishing NPC. They introduced improved crow search algorithm in system design for Zanjan city in Iran. Hongxing et al.²⁵ introduces an optimum design for solar-wind-battery hybrid system aimed at minimizing the total annual energy cost while considering reliability constraint associated with the probability of unsupplied load energy, employing a genetic algorithm (GA). Khan et al.²⁶ introduced two enhanced evolutionary sizing algorithms, along with three metaheuristic algorithm such as teaching learning-based algorithm, enhanced differential evolutionary algorithm, and salp swarm algorithm for optimizing the size of PV/WT/BT hybrid system in a standalone set up, with the goal of minimizing the total annual cost while considering a reliability constraint. Marcel et al.²⁷ implemented an iterative approach to optimize the size of a suggested hybrid PV/WT/BT renewable energy system, aiming to achieve the lowest annualized cost for a Tanzanian village, addressing the issue of electricity availability in developing countries. Abbassi et al.²⁸ projected a modified moth flame optimizer as an innovative size optimization approach for photovoltaic-wind hybrid off-grid systems integrated with a hybrid energy storage system featuring two distinct dynamics. The aim is to enhance reliability by reducing electricity costs, maximizing the use of renewable energy sources, and decreasing the Loss of Power Supply Probability (LPSP). Ayan and Toylan²⁹ modelled a hybrid system combining PV, BT, and WT configuration, optimized for the load demand of a school building in Turkey. They utilized HOMER software tool and the artificial bee colony optimizer to achieve maximum energy generation at minimal cost across different LPSP values. Naderipour et al.³⁰ projected an optimum hybrid renewable energy system incorporating photovoltaic, wind, and battery technologies to reduce total cost. This design incorporates loss of load probability as a reliability constraint, specifically designed for a city in Iran, while also taking into account the components’ outage rates. Arabi-Nowdeh et al.³¹ used spotted hyena optimizer to benchmark the system size and reliability for a hybrid PV/ WT/BT system operating both off-grid and on-grid modes The study reveal that SHO outperformed PSO in cost minimization and system efficiency in the on-grid mode, which lowered total net present cost and enhanced dependability. Mahmoudi³² et al. introduced a multi-objective optimization strategy employing gravitational search algorithm and non-dominated sorting approach. This approach aims to optimize hybrid renewable energy systems by lowering power supply loss, minimizing costs and reducing CO₂ emissions, and boosting the renewable energy fraction. Sarangi et al.³³ integrate EOBMGO to determine the optimal configuration of an off-grid hybrid system, with the goal of minimizing the total annual cost. Furthermore, Table 1 provides an additional overview of research conducted on the size of HRES, focusing on the configuration of system components, optimisation techniques, and design objectives.

Table 1.

List of several studies that optimised HRES using various optimization methods.

SI.no	Algorithm	Hybrid System	Location	Objective function
1.	NSGA II34	PV-WT-BT	Kent, UK	Minimize the total system cost and maximize system reliability
2.	PSO35	PV-SB, WT-BS, PV-WT-BS	China	Minimize the cost of energy
3.	BA36	PV-WT-BT	Tunisia	Minimize annual cost
4.	FA37	PV-WT-SB	India	Minimize the electricity cost
5.	GWO38	PV-DG-BT	Algeria	Minimize total cost
6.	CSA39	PV-WT-DG-BT	Saudi Arabia	Minimize the overall cost
7.	GO40	PV-WT-DG-BT	Nigeria	Minimize the cost of energy and maximize system reliability
8.	ABC41	PV-WT-BS	Italy	Maximize the energy saving benefits
9.	TLBO42	PV-BT	Iran	Minimize NPC and COE
10.	DFA43	PV-WT-Fuell cell-BT	India	Control Power flow
11.	TSA44	PV-WT-BT-DG	Abu-Monqar, Egypt	Minimize energy cost and satisfying reliability index
12.	SSO45	PV-WT-DG-BT	Saudi Arbia, Egypt	Minimize the cost of energy
13.	MBA46	PV-WT-Fuell cell	Egypt	Minimize annual cost
14.	HHA47	PV-WT-DG-BT	Saudi Arabia’s Northern region	Annualized system cost
15.	RIME48	PV-WT-BT	Egyptian city of Siwa, Egypt	Minimize LPSP
16.	HBBCO49	PV-WT-BT	Qazvin, Iran	Total present cost (TPC)

Depending on the various algorithms applied to address the design challenges of a hybrid system, each with its some advantages and disadvantages. Below we provided examples of some commonly used algorithms. The genetic algorithm is highly effective for optimizing problems involving discrete quantities. However, in scenarios where the search space is comparatively limited, the genetic algorithm exhibits a slower performance in relation to certain other equivalent techniques⁵⁰. Although PSO has shown several advantages such as a high convergence rate, ability to share information among the individuals, and strong robustness to local optima trapping, it also has disadvantage including premature convergence and poor diversity⁵¹. The ABC and CSA demonstrate effective performance in addressing optimization challenges, delivering high computational speed and power, However, when dealing with large scale problems, these methods may get stuck in local optima region and struggle to reach at optimum solution²⁴. The GWO has advantages because it has lesser parameters, simple structure, and is easy to implementation. On the other hand, GWO is slow to converge, poor solution accuracy, and is prone to being stuck in the local optimal⁵². The TLBO algorithm has gained popularity for its robust global search ability and stability, but it suffers from slow convergence, is prone to local optima, and imbalance between exploration⁵³. The FA algorithm demonstrates efficiency and ease of execution. Still, it has problems with poor solution speed and a tendency to converge in local optima and fails to balance exploration and exploitation⁵⁴.

After performing a comprehensive literature study, it is observed that competition is about adopting a technique characterized by superior convergence accuracy and speed to obtain the global optimum. Conversely, an algorithm may successfully solve one optimisation problem yet fail to address another. As a result, selecting an effective algorithm with high precision and convergence speed is critical for reaching the ideal size of hybrid systems. Furthermore, having a nonzero reliability constraint can result in considerable cost savings while assuring a sufficient energy supply.

Motivation and contribution

Based on the literature review, meta-heuristic algorithms are more effective and reliability in solving the optimization challenges of HRES design. Moreover, the No-Free-Lunch (NFL) theorem signifies that no single algorithm can effectively address all optimization problems, highlighting the necessity for new approaches. Hence motivating by the above factors this work employs a newly developed Black-Winged Kite algorithm to design an optimal off-grid HRES configuration with the least annual cost and levelized cost of electricity. The current study endeavours to tackle a specific load requirement for an educational system located in Puri, Odisha, India. To generate the continuous power supply, the current research performed a techno-economic evaluation of the HRES system, which includes photovoltaic (PV), wind, and battery. The proficiency of the applied BKA algorithm is verified by comparing it with four prominent intelligence algorithms: the Arithmetic Optimisation Algorithm (AOA), Harish Hawk’s Optimiser (HHO), White Shark Optimiser (WSO), and Snake Optimiser (SO). The primary goal was to determine optimal size HRES, focusing on achieving an effective convergence speed and minimising the system’s annual cost.

The following summarized the significant contributions of this research work.

• Recently developed BKA algorithm is employed first time to benchmark the sizing of the suggested HRES by minimizing the total annual cost of a system with the intended loss of power supply.

• Technical and economical estimations have been conducted for the projected HRES integrating photovoltaic, wind turbine and battery (PV/WT/BT) to generate electricity for an educational system of puri, Odisha, India.

• To demonstrate the effectiveness of BKA algorithm it is compared against four prominent algorithms in terms of the various statistical findings, including best and worst solutions, convergence rate, mean, standard deviation, crest, variance, and discrepancies near the global solution.

• The experimental findings demonstrated that the projected BKA algorithm outperforms the compared intelligence algorithms in determining the optimal size of HRES.

The remaining sections of the paper are summarised in the following fashion: “Black-Winged kite algorithm” section delivers a comprehensive explanation of the BKA algorithm. “Modeling and methodology of HRES” section outlines the detailed mathematical frameworks of the system components of HRES. “Formulation of objective function” section delineates the construction of the optimisation problem. “Simulation analysis of HRES” section delineates the simulation analysis of HRES. Lastly, “Conclusion” section concludes the study and proposes possible directions for further investigation.

Black-Winged kite algorithm

This study applies an optimization method called Black-winged Kite Algorithm (BKA) for the first time to determine the optimal size of the renewable system with minimum cost. The BKA was primarily presented by the researchers Wang et al. in 2024⁵⁵ and is a swarm-based optimization technique encouraged by the hunting tactics and migration strategy of black-winged kites. This diminutive bird species is distinguished by its blue-grey upper body part and underparts appearing white. Its prey is small animals, such as birds, reptiles, and beetles. They are well-known for their powerful hovering ability and excellent hunting performance. The BKA models its optimization process through two key stages: the attacking behavior, which simulates their hunting systems, and the migration behavior, which reflects their movement patterns. These features serve as a source of inspiration for the BKA.

Mathematical model

This section captures the essence of the Black-winged kite’s migratory and attacking behaviors, skilfully modeling them to shape the BKA algorithm and simulate its optimization prowess.

Initialization of population

In the BKA, population initialization begins with the help of random initiating solutions. The initial position each black-winged kite (bk) is produced by providing the Eq. (1).

1

Here, and designate the lower and upper borders of the search domain, respectively, while represents a random value. The following matrix () embodies the current location of each black-winged kites (bk).

2

In the Eq. (2), denotes the dimension of black-winged kites. The symbols and signify the size of individuals and dimensionality of the search domain, respectively.

Attacking behaviour

Black-winged kites are highly skilled at hunting little grassland creatures and bugs. Black-winged kites change their wing and tail angles in response to wind velocity while in flight. They float silently to monitor prey species and perform rapid dives to capture it. This approach encompasses various attack patterns for global search and exploration. Figure 1 shows the kite floating through the air, while Fig. 2 displays the kite in attack mode for their prey. The mathematical model for their attack behavior is specified below:

3

4

where and denotes the black-winged kites updated position and current position at and iterations respectively. Symbols, and denote the randomly selected number belongs to [0,1] and constant value 0.9, respectively. and represent the total and current iteration, respectively.

Fig. 1.

Black-winged kites floating in the air and waiting for attack.

Fig. 2.

Searching and attacking for potential prey while floating in the air.

Migration behavior

The intricate migratory behavior of birds can be affected by multiple factors related to the environment, especially food availability and climate over time. In response to seasonal variations, many birds migrate from northern to southern parts to find better living conditions and food sources. Leaders typically command the migration process, and the navigational abilities of these individuals are essential to the team’s achievements. The mathematical design for this strategy is outlined below:

5

6

Here, denotes the leader of black-winged kites, epitomizes the fitness value of individuals, while refers to the fitness value of arbitrary black-winged kites. characterizes the Cauchy mutation shown in Eqs. (7) and (8):

7

When the values of and , the above formula is:

8

Modeling and methodology of HRES

The HRES which operates off-grid including photovoltaic panel, wind turbine, and battery is specially designed to deliver a consistently reliable and sustainable energy source in remote or off-grid regions. By utilising the photovoltaic phenomenon, PV systems are able to directly transform sunlight into electrical energy. They generate direct current (DC) power, which may either be utilised immediately to deliver the load or preserved in a battery bank for further usage. Wind turbines are a supplementary source of power, and they are particularly useful in situations when solar energy is not sufficient. These panels are a useful addition to photovoltaic panels since they generate electricity regardless of the weather conditions, ensuring a more consistent and uninterrupted supply of energy. The extra electrical energy that is created by the photovoltaic panels and wind turbines is preserved in the battery bank. Whenever the requirement for energy is surpassing than the capacity of the generation system, this stored energy can be utilised. Direct current (DC) power that is generated from PV and wind turbines, which is stored in batteries, is converted into AC (alternating current) power by the inverter. This type of electricity is typically utilised by the majority of equipment. The incorporation of DC-generated renewable energy into AC-based electrical demands is made possible as a result of this function. The term “load” refers to the many electrical devices and systems that are able to benefit from the power that is produced by the HRES. Figure 3 depicted the off-grid HRES that consists of various components including wind, solar, battery, and other electrical devices.

Fig. 3.

Graphical design of off-grid HRES.

Framework of PV model

Radiation from the sun has emerged as the major source of energy for a solar-powered system, and it is abundantly accessible in the surrounding atmosphere. However, the probable source of solar radiation differs depending on the region being considered. To effectively replicate the functioning associated with solar energy generation, it is crucial to collect sufficient solar radiation readings to the research area throughout a 1-year time frame, with observation recorded over one-hour gaps. Throughout the year of 2020, solar radiation statistical data were obtained via the NSRDB Power Data Access Viewer. The output () of a photovoltaic system is mathematically defined by Eq. (9) ^56,57:

9

Here represents the rated capacity of the solar panel in , signifies the radiation of solar in (W/m²). stand for the number of solar panels. represents the PV directing factor (). denotes the cell temperature under reference operating circumstances, presents the temperature coefficient (%/⁰C), is the corresponding solar radiation (W/m²), and stands for ambient temperature is expressed in (⁰C).

Framework of wind turbine model

Wind turbines are devices that convert wind-generated energy into the electrical energy through the motion of rotation. The wind speed at a specific hub height can be estimated by deploying the power law formula, which is given by Eq. (10) ^{58 48}.

10

The following equation serves as a tool for performing a mathematical calculation to determine the power output from a wind turbine¹⁸.

11

Here, and and symbolize the wind speed at specific hub-heights and , respectively. Symbol is a constant number whose value is 0.14. denotes the rated capacity of wind turbine. V(t) epitomizes the speed of wind (m/s), and designate the cut-in and cut-out speed, respectively, signifies the rated wind speed, epitomizes output power (kW).

Framework of battery modelling

Battery storage systems enable incorporation of intermittent sources of renewable energy and endorse more effective utilisation of renewable source of energy. The systems that store energy are designed to regulate the equilibrium between output and demand, thereby improving the reliability as well as stability of renewable energy sources. Multiple types of storage equipment have been employed, including batteries, supercapacitors, fuel cells, and flywheels. The present research work integrates a cost-effective battery bank as a device to store energy, which constitutes a substantial share of the overall cost of the system. Operational strategy of the battery significantly influences its overall lifespan and effectiveness. It is essential that the battery’s charge and discharge states remain within the established upper and lower boundaries. The charging process of the battery initiates during the entire power produced via natural sources (solar and wind) surpasses the load requirement. In the absence of alternative power sources, the battery initiates the discharge of its stored energy to satisfy the requirement. Details mathematical calculation of its state of charge can be performed deploying the Eq. (12) and Eq. (13) ³⁷.

12

13

Here, and epitomizes the energy that is preserved at hour t and , respectively. signifies the generation of total energy through renewable sources, represents the power required at time , and and denote the proficiency of inverter and battery, respectively.

Power converter

For the purpose of ensuring that the flow of energy is maintained in a balanced manner, a hybrid configuration requires the utilisation of a power converter that is composed of AC as well as DC operational factor. There are two ways of power translation (DC/AC and AC/DC) that operate via the load’s requisite frequency. The converter rating corresponding to peak load demand is expressed by Eq. (14) ¹⁶:

14

Here, embodies size of the converter. epitomizes the peak power load demand (in kW). characterizes the efficacy of the converter.

Formulation of objective function

The primary objective function of this study aims to decrease the TAC of a hybrid system by figuring out the optimum size of the HRES factors. There are four factors that are associated with TAC of the system: maintenance cost, salvage values, capital cost, and replacement cost. Following describes how to minimise the objective function:

where, TAC signifies the total annual cost and is calculated by using Eq. (15) ⁵⁹.

15

Here, —Number of factors, —Capital cost of the various factors, —Replacing cost of the various factors, CRF—Capacity recovery factor, —Single Payment Present Worth, –Capacity recovery factor, —Components’ operating and maintenance costs, —Salvage cost.

The LCOE (levelized cost of electricity) of the system and CRF can be estimated by executing the Eqs. (16) and (17) ^60,61.

16

where denotes the total amount of electricity served in a year.

17

Here, denotes the nominal discount rate, signifies the inflation rate, represents the real discount rate, and symbolizes the system lifetime.

Since, the system that has been designed is a hybrid electrical system, reliability plays an essential factor. This study considers LPSP (loss of power supply probability) to act as reliability constraint. For LPSP = 0 indicates that the installed HRES will always generate sufficient amount of energy to fulfil the recruitment. Without it, the system will lose load and become less reliable. The expression for the LPSP index is outlined by Eq. (18) ^56,62:

18

Here indicates the renewable power from wind and solar, symbolizes load demand, embodies the amount of energy retained inside the battery cell at time (t-1), and epitomizes minimum charge of the battery cell.

Simulation analysis of HRES

In this study the selected research site is puri location which is located in the eastern part of Odisha, India, on the Bay of Bengal River. The geographical coordinates of puri lie around the 19.9965 latitudes and 85.8486 longitudes. The NSRDB power data access viewer internet database for 2020 was adopted for getting temperature, speed of wind, and solar radiation data for the chosen location. At this specific location, there is a substantial amount of access to wind and solar resources throughout the whole year. The data that has been collected for the location is displayed in Figs. 4, 5, and 6, which includes information on the ambient temperature, wind speed, and solar radiation. The load requirements are the primary factor that determines the best possible configuration of the HRES, which symbolises the amount of energy that is consumed over the course of a year. Data on the amount of electricity used by educational institutions in Puri are utilised in this work. A representation of the projected daily load profile throughout the year is depicted in Fig. 7. As per the energy consumptions that were acquired for educational loads, the system is recommended for a load requirement of 37,909.4 kWh per year. The purpose of the present study is to generate research findings for the first time in the chosen region, with the goal of achieving minimum energy costs. Table 2 presents a detailed summary of the economic and technical specifications of the various components utilised in this research. The current study operates under the premise that the project will span an estimated time frame of 20 years.

Fig. 4.

Graphical representation solar radiation (w/m²).

Fig. 5.

Graphical representation of wind speed (m/s).

Fig. 6.

Graphical representation ambient temperature (°C).

Fig. 7.

Graphical representation yearly load profile (kW).

Table 2.

Summary of the various HRES components and their respective parameter values ^16,63–68.

Component	Parameter	Value
Solar	Unit capital cost ()	220 $/kw
	Reference solar radiation ()	1000 w/m2
	Temperature coefficient ()	− 0.39%/°C
	Rated power ()	1 kw
	Lifetime	20 years
	Reference temperature ()	25 °C
	Operator and Maintenance cost ()	10 $/kw year
Wind Turbine	Unit capital cost ()	240 $/kw
	Rated Power	1 kw
	Shear component ()	0.14 dimensionless
	Cut in speed ()	3 m/s
	Hub height ()	50 m
	Rated wind speed ()	11 m/s
	Cut out speed ()	20 m/s
	Operator and maintenance cost ()	10 $/kw year
	Hub height ()	50 m
	Life time	20 years
Battery	Capital cost ()	250 $/kw
	Replacement cost ()	250 $/kw
	Rated capacity	1 kw
	Battery type	Lead acid
	Charging efficiency (	90%
	Discharging efficiency (	100%
	Life time	5 years
Converter	Capital cost ()	3367$/kw
	Life time	20 years
	Efficiency ()	95%
	Operator and maintenance cost ()	20 years
	Rated power	1 kw

Result analysis of PV-WT-BT off-grid system

In this research work MATLAB 2022b program has been integrated to experiment the BKA algorithm efficiency for solving optimal sizing of HRES. To investigate the superiority and reliability of the consider BKA algorithm and to figure out the optimal size for HRES, four prominent intelligence algorithms including Arithmetic Optimization Algorithm (AOA)⁶⁹ Harish Hawk’s Optimizer (HHO)⁷⁰, White Shark Optimizer (WSO)⁷¹, and Snake Optimizer (SO)⁷² are selected for the comparison purposes. To ensure that all considered algorithms are treated fairly, each intelligence algorithms are runed 25 independent trials with a maximum of fifty iterations and 30 population sizes.

This study is performed for the LPSP (5%) to make sure the reliability of the suggested HRES system.

Off-grid HRES systems provide access to electricity in town and village areas where it may not be feasible to find grid infrastructure. Off-grid systems offer significant advantages in the context of environmental sustainability and the effectiveness of costs, since they can be designed and enhanced to satisfy specific energy requirements, thereby providing versatile in the system design and configuration. The dependability of the source of electrical power is crucial for a completely off-grid energy system, necessitating the integration of alternative solutions to guarantee continuous energy availability. This study employs an off-grid HRES that integrates PV, WT, and BT to mitigate dependability issues. When renewable energy production goes over load demand, excess energy is saved in battery cells until they reach their rated capacity. On the contrary, when renewable energy generation is inadequate, the batteries drain to provide the demand. Photovoltaic panels may create significant amounts of power during the daytime, but they are incapable of producing electricity at night. To mitigate this constraint, wind turbines and solar power panels are connected, as their synergistic functioning guarantees a constant energy supply throughout the day and night.

Table 3 displays the solutions provided by the algorithms for the sizing problem, taking into account the influence of the solar derating factor (0.33). The outcome metrics presented in Table 3 demonstrate the efficiency of the developed hybrid PV/WT/BT (HPWB) system in attaining optimal performance. The experimental outcomes of compared algorithm in Table 3 for different reliability index measurements showed the total annual cost with LPSP 5%. These outcomes demonstration that the lowest annual cost needed for the HRES, employing the BKA intelligence method and 60.4722 kW solar panel, 23.8337 kW turbines for wind power, and 13.6159 kWh batteries, is $ 7105.2316, with a cost of electricity of 0.1874 $/kWh. As compared to the AOA, which has displayed unsatisfactory overall performance, the BKA has delivered a notable enhancement in its statical performance, evidenced by a 22.1144% reduction in TAC and a 1.0037% drop in LCOE. Hence, from the statistical outcomes, it is pointed out that the BKA algorithm generates superior outcomes against the other intelligence algorithms. The LCOE calculated by the BKA indicates that the suggested method provides power to off-grid areas at a cost-effective rate. Consequently, while evaluating the comprehensive study of economic variables and reliability, the results attained by the BKA may be regarded as the most advantageous combination.

Table 3.

Findings of the optimal sizing problem of HRES obtained by the MAs for LPSP (5%).

HPWB	LPSP = 5%
	Algorithms	AOA	HHO	WSO	SO	BKA
Best		5.648441	65.7896	58.3964	60.0286	60.4722
		94.48363	13.8249	43.1098	25.1086	23.8337
		95.72957	11.7963	14.2936	14.3024	13.6159
	TAC	9122.6477	7176.6299	7550.6970	7144.6137	7105.2316
	LCOE	0.2406434	0.1893	0.1992	0.1885	0.1874
	LPSP	3.8172489	5.0000	3.0177	4.7170	4.9998
Worst		18.558278	11.8773	16.8846	35.3932	39.6468
		481.44172	55.9792	452.9902	44.6921	79.6463
		7.69638	172.1823	128.2925	53.9867	27.6054
	TAC	17,051.354	13,042.0396	23,174.8926	7840.3504	7746.6703
	LCOE	0.4497922	0.3440	0.6113	0.2068	0.2043
	LPSP	3.9171825	4.5387	0.0000	4.6644	4.9948

Significant values are in [bold].

Convergence analysis

Convergence analysis is one of the important statistical tools for assessing the accuracy and reliability of numerical approaches or algorithms utilised across many disciplines. It requires the analysis of convergence curves, which are essential for evaluating the convergence rate of the suggested method against various intelligence algorithms as well as exploring techniques to get the optimum solution. To determine the proficiency of the BKA method, it is compared against prominent intelligence algorithms including AOA, HHO, WSO, and SO. Figure 8 depicts the convergence curve of various intelligence algorithm for PV/WT/BT configuration. Figure 8 illustrates that the BKA attains the better convergence accuracy after twenty iteration and reached the optimum value before the against algorithms. For algorithm WSO the convergence rate is very poor and attained constant values at every iteration. However, if the convergence curve of HHO attained faster at initial iteration but at the iteration increases it has shown poor performance. Hence, in comparison to existing intelligence algorithms, the suggested hybrid system has been replicated more rapidly, owing to the efficacy of the BKA approach.

Fig. 8.

Convergence curve of BKA and other MAs.

Statistical analysis

For the purpose of confirming the optimum conclusions of the algorithm, the statistical analysis for evaluation of data is important. Over the course of 25 runs, a number of statistical assessments, including the mean, standard deviation, variance, and crest factor, for each examined algorithm have been compiled and outlined in Table 4. The outcomes shown in Table 4 demonstrate that BKA algorithms delivers superior performance over all compared algorithms. The statistical examination of the TAC solution reveals that the results of the BKA intelligence algorithm have a low mean, standard deviation, variance and crest value in compared to the results of the AOA, HHO, WSO, and SO algorithms. BKA, on the other hand, has a low standard deviation and more consistently finds a higher value for the objective function. This is in contrast to WSO, which has a big standard deviation and shows that locating the optimal solution may include local optimum positions. Figure 9 illustrates the convergence process and the fluctuation in annual costs across 25 independent trails. An examination of convergence qualities reveals that BKA has superior exploratory capabilities compared to the AOA, HHO, WSO, and SO algorithms. Minimising the standard deviation yields global solutions, distinguished by rapid convergence and less oscillations. A high standard deviation induces more fluctuations around a stable state, hence entrapping local solutions. Moreover, a box plot comparison has been presented in Fig. 10, that further illustrates the efficacy of the projected BKA method in comparison to existing intelligence algorithms for the HRES design issue. Upon doing a box plot analysis, it is obvious that the BKA has a median value below the average, accompanied by a reduced interquartile range. These data demonstrate that BKA displays a strong and reliable performance in the HRES design issue.

Table 4.

Statistical outcomes of optimal sizing problem obtained by the MAs for LPSP (5%).

HPWB	LPSP = 5% Algorithms
	Measure	AOA	HHO	WSO	SO	BKA
	Mean	1.0824E+04	8.6072E+03	1.4764E+04	7.3301E+03	7.2846E+03
	Std	2.0437E+03	1.4437E+03	4.1158E+03	2.2015E+02	1.7969E+02
	Variance	4.1766E+06	2.0843E+06	1.6940E+07	4.8467E+04	3.2287E+04
	Crest	1.5490	1.4952	1.5142	1.0692	1.0631

Fig. 9.

Variation of total annual cost of BKA and other MAs.

Fig. 10.

Boxplot graph for BKA and other MAs.

Conclusion

In this study, the recently designed black-winged kite algorithm (BKA) has been employed to optimize the sizing for an off-grid hybrid renewable energy system (HRES) comprising photovoltaic panel (PV), wind turbine (WT), and battery (BT). The system has been modelled to supply electricity to an educational institute in Puri, Odisha, India, with the goal of diminishing the total annual cost. To statistically benchmark the proficiency of the BKA intelligence algorithm, five key performance metrics, such as best solution, worst solution, standard deviation, mean, variance, and crest, have been calculated and compared with four prominent intelligence algorithms, such as Harris Hawks Optimisation (HHO), Sine Cosine Algorithm (SCA), White Shark Optimisation (WSO), Arithmetic Optimisation Algorithm (AOA), and Snake Optimiser (SO). The statistical results demonstrated that the newly projected BKA method outperformed the comparative intelligence algorithms, establishing the best HRES configuration with the lowest annual cost of $7105.23 and exhibiting a superior convergence rate. The low standard deviation of the BKA also indicated reduced oscillations around the optimal solution, underscoring its robustness and proficiency. In contrast, the WSO and SO algorithms displayed higher standard deviations, suggesting a greater tendency to become trapped in local optima and exhibit increased oscillations around the desired solution. Finally, the statistical results demonstrate the effectiveness and reliability of the suggested approach based on BKA in designing a cost effective and reliable hybrid system. However, this study primarily focusses on a hybrid PV/WT/Battery configuration. In future studies applying the suggested algorithm to optimize other HRESs with various configurations and analysing the outcomes can provide comprehensive an additional guide for energy engineers. Furthermore, the study of optimal sizing through hybrid optimization method paves the way for future research, addressing the complexities and challenges associated with hybrid systems.

Abbreviations

HRES

Hybrid renewable energy systems

BKA

Black-winged kite algorithm

RES

Renewable energy sources

NFL

No-free-lunch theorem

PV

Solar panel

WT

Wind turbine

BT

Battery

NSGAII

Non-dominated sorting genetic algorithm II

DG

Diesel generator

TAC

Total annual cost

LPSP

Loss of power supply probability

BS

Battery storage

PSO

Particle swarm optimizer

BA

Bat algorithm

FA

Firefly algorithm

GWO

Grey wolf optimizer

CSA

Cuckoo search optimizer

GO

Grasshopper optimizer

ABC

Artificial bee colony

TLBO

Teaching learning-based algorithm

DFA

Dragon fly algorithm

TSA

Tunicate swarm optimizer

SSO

Social spider optimizer

MBA

Mine blast algorithm

HHA

Harish hawk optimizer

RA

Rime algorithm

HBBCO

Hybrid big-bang crunch optimizer

EBMGO

Evolved mountain gazelle optimizer

SO

Snake optimizer

WSO

White shark optimizer

Number of solar panels

Radiation of solar

Cell temperature under reference operating circumstances

Temperature coefficient

Reference solar radiation

CRF

Capacity recovery factor

Ambient temperature

Rated wind speed

Output power

Proficiency of inverter

Proficiency of battery

Peak power load demand

Efficacy of the converter

Capital cost of the various factors

Replacing cost of the various factors

Salvage cost

Wind speed at specific hub-height

Wind speed at specific hub-height

SPPW

Single payment present worth

Components’ operating and maintenance costs

Number of factors

Total amount of electricity served in a year

Cut-in speed

Cut-out speed

R Rated capacity of wind turbine

LCOE

Levelized cost of electricity

NPC

Net present cost

Rated capacity

Author contributions

Sarada Mohapatra: Investigation, Formal analysis, Data curation, Conceptualization, Methodology, Writing-original draft. Deepa K: Visualization, Conceptualization, Methodology, Validation, Supervision, Writing-review. Porselvi T: Validation, Project administration, Visualization, review & editing. S.V. Tresa Sangeetha: Validation, Project administration, Visualization, review & editing. R.A.: Validation, Project administration, Visualization, review & editing. A.E.: Validation, Project administration, Visualization, review & editing.

Funding

This research was funded by Taif University, Taif, Saudi Arabia project number (TU-DSPP-2024-17).

Data availability

All data generated or analyzed during this study are included in this article.

Declarations

Competing interests

The authors declare no competing interests.

Ethical approval

This article contains no studies with human participants or animals performed by authors.