Multi-Criteria Optimization of a Standalone Photovoltaic System in Cyprus (Techno-Economic Analysis)

Abstract

Photovoltaic systems are increasingly recognized as one of the most advanced, efficient, and rapidly developing methods of electricity generation, utilizing the limitless potential of solar radiation while offering environmentally sustainable solutions to contemporary energy challenges. However, despite their clear benefits, issues such as high initial investment costs and relatively low energy efficiency must be carefully addressed during the design phase. Key considerations include the quantity and type of panels, battery capacity and number, environmental conditions, site-specific factors, and the mathematical models and interconnection strategies of system components. This study proposes a two-stage optimization approach for standalone photovoltaic systems, employing three distinct optimization algorithms—NSGA-II, DEMO, and Particle Swarm Optimization—to minimize both the Loss of Load Probability (LLP) and the life cycle cost (LCC). In the second stage, optimal solutions from the Pareto front are evaluated using three multi-criteria decision-making techniques: the hybrid AHP-TOPSIS method, VIKOR, and PROMETHEE. The proposed framework is applied to systems with storage batteries designed for deployment in three Cypriot cities, aiming to meet energy demands of 10, 15, and 20 kWh. The findings reveal a strong correlation between economic and energy performance and the degree of load coverage, with the combination of the DEMO algorithm and the AHP-TOPSIS method emerging as the most effective solution.

1. Introduction

One of the most critical issues concerning the global community is the ongoing quest for renewable energy sources, driven by extensive industrial and urban development combined with existing environmental challenges and the phenomena of climate change. New global conditions make it imperative to utilize environmentally friendly energy forms, such as solar and wind power, which are increasingly becoming pivotal for national energy strategies. Until recently, electric power was primarily generated by fossil fuel-based conventional power plants. As these resources diminish and their environmental toll escalates, the integration of renewable energy sources is no longer optional but essential [1].

Among renewable technologies, photovoltaic (PV) systems stand out as a particularly efficient method of harnessing solar energy. Yet their intermittent nature poses significant limitations, especially in standalone applications where grid access is unavailable. To address this, storage technologies such as batteries are essential to balance energy production and demand, ensuring continuity and reliability in power supply [2,3].

Despite the growing adoption of PV systems, there remains a clear gap in the literature regarding the systematic optimization of autonomous PV systems that simultaneously address system reliability and economic performance. Particularly lacking are methodologies that integrate advanced metaheuristic algorithms with multi-criteria decision-making (MCDM) techniques to optimize system design under real-world conditions and diverse load demands.

The primary objective of this research is to design and optimize a cost-effective and technically reliable standalone photovoltaic system that can meet daily energy demands with minimal interruption. Specifically, the study aims to minimize both the Loss of Load Probability (LLP)—which measures the system’s reliability—and the life cycle cost (LCC)—which reflects its economic efficiency. This dual-objective optimization is carried out using genetic algorithms in combination with Multi-Criteria Decision-Making (MCDM) methods, focusing on two main decision variables: battery storage capacity and the number of parallel photovoltaic units. By applying this approach to real-world case studies in Cyprus, the research seeks to provide a replicable, scalable, and location-sensitive framework that bridges the gap between reliability—focused design and cost—and effective implementation in standalone renewable energy systems.

This research directly addresses this gap by proposing a novel, integrated optimization framework for standalone photovoltaic systems, combining genetic algorithms (GAs) with MCDM methods to simultaneously optimize two conflicting objective functions: Loss of Load Probability (LLP) and life cycle cost (LCC). The decision variables considered are the battery storage capacity and the number of parallel PV units, allowing for a robust analysis of both energy reliability and cost effectiveness.

The proposed methodology is applied in three cities in Cyprus—Nicosia, Larnaca, and Limassol—under varying energy load profiles. By identifying optimal system configurations that minimize cost without compromising performance, the study introduces a practical and innovative tool for the design of high-efficiency, cost-effective standalone PV systems. The key innovation lies in the dual-objective optimization approach, which moves beyond traditional sizing techniques by capturing the multi-dimensional trade-offs inherent in renewable energy system design.

In the first chapter of this paper, a general introduction to the study’s orientation is provided. Key issues that have driven the search for and utilization of renewable energy sources are discussed, with an emphasis on photovoltaic technology. A brief outline of the study’s subject and objectives is also presented.

In the second chapter, similar studies and works are reviewed, focusing on those that utilize relevant algorithms and methods to address either similar or entirely different problems.

In the third chapter, the problem formulation is presented, which includes the mathematical model, the decision variables, as well as the objective functions and the methodology for their calculation.

In the fourth chapter, the components of the standalone photovoltaic system to be optimized are presented, along with the governing equations. Additionally, brief descriptions and definitions of the algorithms and multi-criteria methods to be used in the problem are provided, outlining how these will be employed to generate the results.

In the fifth chapter, the case study is conducted, and the data used for the problem are presented in detail.

In the sixth chapter, which constitutes the largest part of the study, the results of the algorithms and methods employed are presented, along with the final solutions selected.

Finally, in the seventh chapter, an evaluation of the obtained results is carried out, while the eighth chapter concludes with the final outcomes, specifically the determination of the decision variables to achieve the optimal result for each case studied.

2. Relative Work

There are numerous studies on the optimization of standalone photovoltaic systems, each differing in modeling and construction. Merei, Berger, and Uwe Sauer conducted the modeling and optimization of a standalone hybrid energy system consisting of photovoltaic panels, a wind turbine, a diesel generator, and batteries for excess energy storage, concluding that battery usage combined with renewable energy sources is both economically and ecologically efficient [4]. Kumari and Geethanjali focused on deriving photovoltaic cell design parameters using an enhanced evolutionary computational method based on an Adaptive Genetic Algorithm (AGA), aiming to find optimal cell parameters by applying multiple criteria, such as minimizing Least Square Error (LSE) and optimizing Pearson’s error (PRO) [5]. Chel, Tiwari, and Chandra evaluated the sizing and cost of components in a standalone photovoltaic system and a building-integrated photovoltaic system, extensively analyzing the life cycle cost (LCC) factor and how it can be optimized [6].

Khatib, Ibrahim, and Mohamed reviewed methodologies for determining the size of a photovoltaic generator and battery in a standalone PV system using various criteria [7], while Muhsen, Khatib, and Haider presented a feasibility study and load sensitivity analysis of a water-pumping PV system with battery storage and a diesel generator, showing that system configuration and the initial state of the storage tank are critical factors to consider [8].

While many examples and studies explore photovoltaic system optimization, this study draws primarily on the work of Muhsen, Nabil, Haider, and Khatib, who optimized a standalone photovoltaic system in Baghdad using an advanced form of the DEMO algorithm. They further applied the hybrid multi-criteria analysis method AHP-TOPSIS to obtain optimal solutions for criteria and decision variables, ultimately enabling the optimal construction of the photovoltaic system [9].

Additional recent studies have expanded the methodological landscape of PV system design, often combining multi-criteria decision making with techno-economic optimization. Zanlorenzi, Szejka, and Cnciglieri Junior [10], Serrano-Gomez, Gill-Garcia, Garcia-Cascales, and Fernandez-Guillamon [11], and Peirow, Astaraei, and Asi [12] all employed MCDM techniques—such as AHP and TOPSIS—to evaluate and select optimal configurations for standalone or hybrid PV systems, addressing both technical and economic performance while tailoring solutions to specific environments like microgrids, isolated regions, and hospital infrastructure. Similarly, Rahman, Abu Ther, and Kabir [13] used fuzzy AHP and fuzzy TOPSIS in an industrial product portfolio context, showcasing the versatility of these tools beyond energy systems. In parallel, Ahmed, Basit, Abid, Harron, Kakar, Ullah, and Khan [14], Jahed, Abbaspour and Ahmadi [15], and Nirbheram, Mahesh and Bhimaraju [16] focused on the techno-economic viability of standalone or hybrid PV systems, with Ahmed and colleagues analyzing cost-effective solar energy access in rural Pakistan, Jahed et al. evaluating different battery types in a 3 MW off-grid PV–wind hybrid system, and Nirbheram et al. incorporating long-term degradation effects to improve sizing accuracy and system reliability. These studies emphasize the growing emphasis on both economic feasibility and lifecycle performance in optimizing standalone renewable systems.

Despite the breadth of existing research, there remains a noticeable gap in studies that holistically integrate multiple optimization techniques and decision-making methods in the design of standalone photovoltaic systems tailored to diverse load scenarios and specific geographic conditions. Most prior studies focus on single or limited optimization strategies, often without incorporating a combination of technical, economic, and environmental criteria in a unified framework. Furthermore, there is a limited number of comparative studies that evaluate system performance across multiple urban regions using standardized criteria. This gap highlights the necessity for more extensive research that leverages a multifaceted methodological approach, enabling deeper insight into the complex trade-offs and synergies involved in system optimization. The novelty of this study lies in addressing these shortcomings by introducing a comprehensive and comparative analysis using advanced optimization and decision-making tools.

This study combines several different methods from those presented in the previous works, integrating three distinct optimization algorithms and three multi-criteria decision-making methods. The use of this variety of approaches allows for a comprehensive analysis of the system from multiple perspectives, ensuring the most efficient and sustainable solutions for the design and operation of a standalone photovoltaic system. Additionally, the combination of these algorithms and methods enables better handling of fluctuations and uncertainties that may arise from varying operating conditions, such as the availability of renewable energy sources and load variations. In this way, the complexity of the problem is addressed with flexibility, ensuring both the economic and environmental efficiency of the system under real-world conditions.

3. Problem Formulation

To design a standalone photovoltaic system, precise sizing of all components, including storage units, is essential. For this purpose, it is crucial to define both energy and economic parameters that represent key aspects of the system, forming the basis for solving the optimization and construction problem of the photovoltaic system. Examples of these parameters include the Loss of Load Probability (LLP), Loss of Power Supply Probability (LPSP), Equivalent Loss Factor (ELF), Loss of Load Expected (LOLE), Levelized Cost of Energy (LCOE), Life Cycle Emissions (LCEs), life cycle cost (LCC), Net Present Value (NPV), and Annualized Cost of System (ACS), among others.

In this study, emphasis is placed on the LLP and LCC parameters, aiming to minimize them. By using decision variables such as the Number of Parallel PV Modules (N_pm) and the Total Battery Bank Capacity (C_bat), the method for calculating the criteria, or objective functions, LLP and LCC are established, ultimately identifying the optimal solution between them.

Various methods and algorithms can be employed to determine the optimal solutions for the two objective functions. In this study, three of the most widely used optimization algorithms for multi-criteria problems—NSGA-II, DEMO, and Particle Swarm Optimization (PSO)—are proposed and applied, along with three multi-criteria decision-making methods to select the best solutions generated by these algorithms. These methods are the hybrid AHP-TOPSIS, VIKOR, and PROMETHEE. Thus, the optimal values for the LLP and LCC criteria are determined, along with the corresponding values of N_pm and C_bat, ultimately enabling the construction of the photovoltaic system.

3.1. Mathematical Model

3.1.1. Decision Variables

For the design and optimization of the standalone photovoltaic system, the selected decision variables were the number of parallel modules (Npm) and the battery capacity (Cbat) to be integrated into the system. These variables are subject to system constraints and are optimized to minimize two objective functions: Loss of Load Probability (LLP) and life cycle cost (LCC). To define appropriate search bounds for the optimization algorithm, we first calculate the number of series-connected PV modules (Nsm) and batteries (Nsb), using the system voltage requirements:

$$\mathrm{N}\mathrm{s}\mathrm{m}={\displaystyle \frac{\mathrm{V}\mathrm{d}\mathrm{c}}{\mathrm{V}\mathrm{m}\mathrm{p}}}$$

(1)

$$\mathrm{N}\mathrm{s}\mathrm{b}={\displaystyle \frac{\mathrm{V}\mathrm{d}\mathrm{c}}{\mathrm{V}\mathrm{b}}}$$

(2)

where V_mp is the nominal voltage of the photovoltaic module and V_b is the nominal voltage of the battery.

Next, we estimate the lower boundary of the number of parallel PV modules (Npm) using the equation:

$$P={\displaystyle \frac{El}{n_{pv}\ast n_w\ast n_{inv}\ast A_{pv}\ast PSH}}{S}_f$$

(3)

where P is the estimated value of the total number of modules, El represents the average daily load energy, n_pv is the efficiency of the photovoltaic panel, n_w is the efficiency of the wiring, n_inv is the inverter efficiency, and A_pv is the total surface area of the photovoltaic module in square meters (m²). PSH (Peak Sun Hours) indicates the daily hours during which solar irradiance of 1000 W/m² is available, and S_f is a safety factor determined by the system designer. S_f accounts for photovoltaic module losses, such as reduced efficiency due to high temperatures, dust accumulation, or shading. A typical value for S_f is 1.5, although this can vary depending on the system [9].

For calculating the boundaries of battery capacity, we consider daily energy production and energy needs. Specifically, the required battery capacity is calculated as the difference between daily energy demand and minimum daily energy production, multiplied by the battery’s Depth of Discharge (DOD). The effective battery capacity is the required capacity divided by the DOD. This value is converted to Wh and used to define a feasible search space, with the upper limit set to five times the minimum required capacity to allow design flexibility under different load profiles and solar conditions.

These model-derived expressions do not prescribe fixed system parameters but rather define the feasible solution space for the optimization process. The genetic algorithm, in combination with MCDM techniques, is applied within this space to identify system configurations that optimally balance technical reliability (LLP) and economic viability (LCC), based on real-world environmental conditions and load demands.

3.1.2. Objective Functions

• Loss of Load Probability (LLP)

To ensure the reliability of a photovoltaic system in meeting load demand at any given time, the most commonly used parameter is the Loss of Load Probability (LLP). LLP is defined as the ratio between energy deficit and energy demand for a specific load over the entire operational period of the installation, as expressed in the equation below [17].

$$\mathrm{L}\mathrm{L}\mathrm{P}={\displaystyle \frac{\int \mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}{\mathrm{E}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y} \mathrm{d}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}}}$$

(4)

After extensive research and experimentation with various methods to identify the most suitable and reliable approach for calculating LLP, as well as for determining the power output of the photovoltaic array and the daily and hourly energy deficit, it was found that no universally accepted method exists for this purpose. Specifically, estimating whether the energy generated by the system is sufficient to meet demand depends on the solar irradiance and temperature experienced by the photovoltaic system. Consequently, there are no specific mathematical equations that comprehensively link all these parameters.

• Life Cycle Cost (LCC)

Life cycle cost (LCC) analysis is conducted to estimate the cost per unit of electricity generated by the photovoltaic system. It is defined as the present value of all expenses associated with the system over its lifespan [6]. The LCC is the sum of four costs: initial cost, maintenance cost, replacement cost, and residual value.

Initial Cost (C_CAP): This includes the initial cost of all system components (panels, batteries, inverters, charge controllers), as well as installation costs. The initial cost is already at its present value, but other costs depend on their future values [6].

Maintenance Cost (C_M): This cost is incurred annually over the system’s entire lifespan. The initial annual maintenance cost for each component is calculated as follows:

First, the annual maintenance cost of photovoltaic modules is calculated as the product of the modules’ annual maintenance rate, the total number of parallel modules Npm, and the power per module. Similarly, the battery’s annual maintenance cost is calculated by multiplying its annual maintenance rate by its total capacity Cb.

For each passing year, a discount factor is calculated as follows:

$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}.\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}={\displaystyle \frac{1}{\left({\left(1+\mathrm{d}\mathrm{r}\right)}^{\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}\right)}}$$

(5)

where dr is the discount rate [18], chosen to be 3.5% [19], slightly above a typical rate for Europe. This factor is then multiplied by the annual maintenance cost of each component, yielding the discounted maintenance cost for each year. Thus, as each year passes, the initial maintenance cost is reduced.

Finally, the annual costs are summed to calculate the total maintenance cost (CM) of the system components over the years.

Replacement Cost (C_REP): This is also treated as a recurring cost. Since components do not need to be replaced every year, replacement costs are calculated similarly to maintenance costs, where the initial cost of each component needing replacement is multiplied by the discount factor for the replacement year. These replacement costs are then summed for all replacement years to yield the total replacement cost (CREP) of components requiring replacement during the system’s life.

Residual Value (C_S): This is the revenue generated from selling the “scrap” of the system at the end of its lifespan. This value is assumed to be 0.14% of the total present value of the system.

Thus, the total life cycle cost is calculated as follows [8]:

LCC = C_CAP + C_M + C_REP − C_S

(6)

4. Materials and Methods

4.1. Network Elements

4.1.1. Photovoltaic Panel

When modeling a photovoltaic array in a photovoltaic system, the most commonly used model is the single-diode photovoltaic cell model, as it provides a balance between simplicity and accuracy. The equation governing the I-V characteristics of the module is as follows:

$$\mathrm{I}={\mathrm{I}}_{\mathrm{p}\mathrm{h}}-{\mathrm{I}}_{\mathrm{o}}\left[\exp \left({\displaystyle \frac{\mathrm{q}\left(\mathrm{V}+{\mathrm{R}}_{\mathrm{s}}\mathrm{I}\right)}{\mathrm{a}\mathrm{K}\mathrm{T}}}\right)-1\right]-{\displaystyle \frac{\mathrm{V}+\mathrm{I}{\mathrm{R}}_{\mathrm{s}}}{\mathrm{R}\mathrm{p}}}$$

(7)

where I represents the module’s output current (A), V is the output voltage (V), Iph is the photocurrent (A), Io is the diode’s saturation current (A), R_s and R_p are the series and parallel resistance values of the photovoltaic module (Ω), respectively, a is the diode’s ideality factor, q is the electron charge, K is the Boltzmann constant, and T is the cell temperature in K [20]. The five parameters I_ph, I_o, R_s, R_p, and a are provided in [21]. Consequently, the output of the photovoltaic array can be determined by:

$${\mathrm{P}}_{\mathrm{p}\mathrm{v}}=\mathrm{I}\mathrm{V}$$

(8)

However, despite the initial assumption that this method would yield reliable results, the findings were not sufficiently accurate in practice to support the experiment’s continuation and the photovoltaic implementation. Therefore, an alternative method was explored for calculating the power output of the photovoltaic array.

By knowing the distribution of solar irradiance on the surface throughout all solar hours of a typical meteorological year, it is possible to calculate the expected hourly and cumulative electrical output of all photovoltaic modules on the surface. Initially, ambient temperature data are used to calculate each module’s hourly temperature based on the following equation:

$${\mathrm{T}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}=\mathrm{T}+{\displaystyle \frac{{\mathrm{T}}_{\mathrm{n}\mathrm{o}\mathrm{c}\mathrm{t}}-20 °\mathrm{C}}{800 \mathrm{W}{\mathrm{m}}^{-2}}}\mathrm{G}$$

(9)

where T represents the ambient temperature in °C, T_NOCT is the nominal operating cell temperature for each photovoltaic cell, and G is the solar irradiance in W/m² [22]. Following the calculation of each module’s voltage–current, the output power of each photovoltaic module is calculated according to:

$${\mathrm{I}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}={\mathrm{I}}_{\mathrm{p}\mathrm{h}}\ast {\displaystyle \frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{r}\mathrm{e}\mathrm{f}}}}$$

(10)

$${\mathrm{V}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}={\mathrm{V}}_{\mathrm{o}\mathrm{c}}+\mathrm{t}\mathrm{e}\mathrm{m}{\mathrm{p}}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}}\ast \left({\mathrm{T}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}-{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)$$

(11)

$${\mathrm{P}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}={\mathrm{V}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}\ast {\mathrm{I}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}\ast {\mathrm{A}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}$$

(12)

where G_ref is the reference irradiance, usually valued at 1000 W/m², V_oc is the module’s open-circuit voltage, T_ref is the reference temperature, typically set to 298.15 K (273.15 + 25 °C), Apv is the total area of each module in m², and temp_coefficient is the temperature coefficient used [23].

Finally, the system’s total output power is obtained by multiplying the output power of each module by the total number of parallel photovoltaic modules to be employed.

$$\mathrm{P}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}={\mathrm{p}}_{\mathrm{p}\mathrm{v}}{,}_{\mathrm{m}}\ast {\mathrm{N}}_{\mathrm{p}\mathrm{m}}$$

(13)

4.1.2. Batteries

For the simulation of a standalone photovoltaic system, lead-acid batteries have been proven to be the most suitable option. Although the applied model disregards the effects of overcharging and deep discharging, it still yields reliable performance. This is primarily attributed to the presence of charge controllers in most photovoltaic systems, which prevent the battery from reaching critically low or excessively high states of charge.

4.1.3. Inverters

Inverters are critical components in systems utilizing renewable energy sources. Their primary function is to convert direct current (DC), generated by photovoltaic panels or stored in batteries, into alternating current (AC), which is then used by electrical appliances and power distribution networks. This conversion is achieved through the use of power switches that control the current’s direction, with a key design focus on providing a stable output voltage despite fluctuations in the DC input or changes in load [24].

An inverter is selected based on its ability to handle the maximum expected power of the AC loads. Therefore, during selection, it is always verified to be at least 20% higher than the total rated power of the required loads [25].

The efficiency of an inverter is defined as the ratio of the produced AC power to the incoming DC power. High-efficiency inverters minimize energy losses, enhancing the overall system efficiency, while high-quality inverters are designed to deliver reliable performance over multiple years, reducing the need for frequent repairs and replacements. Typical efficiencies of modern inverters range between 90% and 98%.

4.1.4. Charge Controllers

The battery charge controller is used in photovoltaic systems to ensure the safe charging of batteries and to prevent overcharging. It also contributes to the long-term lifespan of the batteries. A charge controller’s algorithm reflects the efficiency of the photovoltaic system’s usage and battery charging, extending battery life and enhancing the system’s ability to meet load demands. Additionally, it must be capable of handling the system’s short-circuit current, while its primary role is to maintain the batteries at the highest possible state of charge, protecting them from excessive discharge due to the loads and the photovoltaic system.

Often, the electricity generated by the photovoltaic system exceeds load requirements, particularly during the summer months when solar radiation is high. Therefore, a charge controller must be used to keep the batteries safe from overcharging. This capability must remain consistent despite seasonal changes in load demand, the size of the system, or variations in solar radiation [25]. Figure 1 illustrates the representation of a standalone photovoltaic system with a battery.

4.2. Optimization Algorithms

4.2.1. Genetic Algorithms [26]

Genetic algorithms (GAs) were inspired by the theory of evolution, which explains the origin of species. In nature, stronger species have a higher probability of passing their genes to future generations through reproduction, while weaker species face the risk of extinction. Over time, species with the optimal combination of genes become dominant within their population and environment. Occasionally, during the slow and prolonged process of evolution, random changes can occur in genes. If these changes provide additional advantages in the struggle for survival, new species evolve and prevail. Conversely, if the resulting gene combinations are unsuccessful, they are eliminated, and the species’ chances of survival diminish.

In GA terminology, each possible solution is represented by an individual. Each individual consists of discrete units called genes, where each gene controls one or more characteristics of the individual. The collection of possible solutions in the genetic algorithm, or the group of individuals, is referred to as the population. The population is typically initialized randomly, and as the search progresses, it increasingly includes better solutions, eventually converging to one optimal solution or a set of optimal solutions [27]. The most fundamental methodology of genetic algorithms (GAs) consists of two main steps: mutation and crossover. During mutation, random changes are introduced to the characteristics of individuals, reintroducing genetic diversity into the population. Most of the time, the mutation rate—the probability of changing the characteristics of a gene, and thus the individual—is very low. Consequently, the newly generated individual will not differ significantly from the original. In the crossover process, two individuals, called parents, combine to form new individuals, known as offspring. Parent selection is random and drawn from the existing population. Through the repeated application of the crossover operator, the genes of high-performing individuals tend to appear more frequently within the population, eventually leading to an overall optimal solution [27].

4.2.2. Non-Dominated Sorting Algorithm II (NSGA-II)

The initial version of the Non-dominated Sorting Genetic Algorithm (NSGA) proved effective in later stages of problem solving due to its high computational complexity, reliance on distribution parameters, and non-elitist approach. Consequently, a second version, known as NSGA-II, was developed to address these challenges [28].

NSGA-II uses a “crowded comparison operator”, which is parameter independent, to maintain diversity. The methodology is straightforward: in the elitist approach, parent and offspring solutions are combined to form a population of double the original size. Then, candidates for the Pareto front are selected one by one based on the fast non-dominated sorting technique. However, complexity arises when the selection of individuals from a specific front does not require all solutions to be included. In these cases, the crowded comparison operator is used to select the best, most diverse individuals.

The first step in this algorithm is the creation of an initial population. The real numbers corresponding to design variables are converted into binary numbers, with each binary sequence referred to as an individual. The initial population is generated randomly, and each individual’s fitness is evaluated based on the objective functions. Then, through the processes of mutation and crossover, new populations are generated and combined. Selection and reduction of individuals are conducted using Pareto ranking, ultimately producing a new generation of individuals derived from the previous generation. This process is repeated until the objective function reaches a predefined value. A unique feature of NSGA-II is its selection method, where each individual’s dominance is checked to determine if it belongs to the Pareto front. The second Pareto front is formed by individuals that are dominated. This process continues until all individuals are placed in different Pareto fronts with various rankings. Additionally, individuals are ranked based on their distance from the nearest neighbor to ensure uniformity across the Pareto front.

NSGA-II represents a significant improvement over the original NSGA and has been widely used to solve multi-criteria optimization problems. Its efficiency and ability to maintain diversity make it ideal for a variety of applications across scientific and engineering fields.

4.2.3. Particle Swarm Optimization (PSO)

The Particle Swarm Optimization (PSO) method is rooted in theories of bird flocking, fish schooling, and swarm intelligence in general [29]. It also shares similarities with genetic algorithms and evolutionary strategies. Despite its simplicity, PSO can address many of the same types of problems as genetic algorithms due to its use of memory [30].

In the PSO method, each potential solution is assigned a random velocity, and the potential solutions, referred to as particles, “fly” through the search space. Each particle tracks its coordinates, which are linked to the best solution (fitness) it has achieved up to that point. This value is stored and is called the personal best, or pbest, while another value is also tracked: the best value achieved so far by any particle in the population, known as the global best, or gbest.

The concept of PSO optimization involves changing the velocity, or acceleration, of each particle at each time step in the direction of both pbest and gbest. This acceleration is modified by a random term, with different random values generated for acceleration toward pbest and gbest. Additionally, each particle tracks another variable: the best solution achieved within a local “neighborhood” of particles, known as the local best, or lbest [31].

4.2.4. Differential Evolution for Multi-Objective Optimization (DEMO)

DEMO (Differential Evolution for Multi-objective Optimization) is a simple yet effective algorithm that applies the principles of Differential Evolution (DE) to navigate and search the decision space in multi-objective problems. It initially employs non-dominated sorting and crowding distance, similar to NSGA-II, to select the best solutions for the next generation. Like DE, DEMO operates with a population of solutions, where the first population is randomly initialized and from the first generation onward, the search expands into the region of interest.

At each DEMO step, a new solution is constructed from three existing solutions and one parent solution. This process involves mutation and crossover. Specifically, the mutation scaling factor FFF determines how far the new solution will be from current solutions, with smaller values placing it closer and larger values placing it further. The crossover probability CRCRCR defines the likelihood that the new solution will not inherit variable values from its parent, meaning that smaller CRCRCR values result in a new solution very similar to its parent. The values of both parameters remain constant throughout the algorithm’s operation [32].

4.3. Multi-Criteria Decision-Making (MCDM) Methods

4.3.1. AHP-TOPSIS

For ranking the configurations of the standalone photovoltaic system, the combination of AHP and TOPSIS methods has proven to be highly reliable and effective [8,33,34]. The AHP method is used to determine the appropriate weight for each criterion in the decision matrix, and these weights are then applied within the TOPSIS method to establish the preferred ranking of the optimal configurations. This specific model is discussed in detail in the following subsection.

4.3.2. VIKOR

The VIKOR method focuses on ranking and selecting from a set of alternative solutions characterized by conflicting criteria, identifying a compromise solution based on a specific measure. Assuming that each alternative is evaluated according to each criterion function, the compromise ranking can be presented by comparing the degree of proximity to the ideal alternative. The compromise solution represents a feasible option that is closest to the ideal solution.

4.3.3. PROMETHEE

The PROMETHEE method is an outranking technique for multi-criteria optimization problems, developed by Brans et al. [35] and Brans and Vincke [36]. The input data requirements are similar to those of the VIKOR and TOPSIS methods, but PROMETHEE additionally requires the user to input specific variables based on the chosen preference function.

In PROMETHEE, a preference degree indicates how one action is favored over another. For small deviations between the evaluations of a pair of criteria, the decision maker may assign a slight preference. If the deviation is considered negligible, this can also be modeled within PROMETHEE. Conversely, for larger deviations, the decision maker should assign a strong preference for one action over another. If the deviation exceeds a certain threshold set by the decision maker, an absolute preference for one action over the other is established. This preference degree is represented as a real number, always ranging between 0 and 1.

5. Case Study

The system under study is a standalone photovoltaic system intended for installation in regions of Cyprus. After employing the mathematical models presented in the previous chapter for the various components and their interconnections, the next step is to input data into these models to assess the functionality, reliability, and efficiency of the photovoltaic system being designed. The input data include temperature, solar irradiance, and the daily electrical load that the system is required to support.

5.1. Input Data

5.1.1. Temperature

All temperature data were obtained from the NSRDB: National Solar Radiation Database. The selected locations for the construction and installation of the photovoltaic system are three cities in Cyprus: Limassol, Larnaca, and Nicosia. Cyprus was chosen due to its high levels of sunshine almost year round and its elevated temperatures. The following figures, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7, display the monthly temperature distribution for each month of 2019, as well as the periods during which maximum and minimum temperatures were recorded throughout the year.

It is observed that the highest temperatures in all three cities occur in the eighth month of the year, August. Specifically, in Nicosia, the peak temperature was recorded on 11 August at 1:30 p.m., reaching 36.1 °C. In Limassol, the maximum was also on 11 August at 11:30 a.m., with a temperature of 34.1 °C, while in Larnaca, it occurred on 12 August at 12:30 p.m., reaching 35.3 °C. Conversely, the minimum temperatures were recorded in all three cases on 9 January at 5:30 a.m., with values of 7.7 °C, 9.0 °C, and 8.8 °C, respectively.

5.1.2. Solar Irradiance

The solar irradiance data for each of the selected cities were also obtained from the same website for the year 2019. These data are displayed in the following charts, Figure 8, Figure 9 and Figure 10, providing insights into the distribution and intensity of solar irradiance across the year for each location. These charts help in assessing the potential solar energy yield for the photovoltaic system based on seasonal variations in solar exposure.

The unit of measurement for solar irradiance is watts per square meter (W/m²). For the three cities in question, higher values are naturally observed during the summer months—June, July, and August. Specifically, the maximum solar irradiance for each city is as follows: in Nicosia, the peak solar irradiance was recorded on 1 July at 11:30 a.m., with a value of 1024 W/m². In Limassol, the maximum value occurred on the same date and time, reaching 1016 W/m². In Larnaca, the highest value was recorded on 2 July at 11:30 a.m., also with a value of 1016 W/m². The minimum solar irradiance for all three locations is zero during the nighttime hours.

5.1.3. Daily Load Profile

For the implementation of the standalone photovoltaic system, three load scenarios will be studied. These scenarios, illustrated in Figure 11, Figure 12 and Figure 13, assume average daily loads of 10 kWh, 15 kWh, and 20 kWh, respectively. Three hypothetical hourly profiles, illustrating the distribution of the required load throughout the day for each of these cases, will be used and repeated consistently throughout the year. These profiles represent the expected daily energy demand patterns, allowing for analysis and system design tailored to each load requirement.

6. Results

To obtain optimal solutions for the two objective functions—Loss of Load Probability (LLP) and life cycle cost (LCC)—three optimization algorithms were utilized: Non-dominated Sorting Genetic Algorithm II (NSGA-II), Particle Swarm Optimization (PSO), and Differential Evolution for Multi-objective Optimization (DEMO). The Pareto fronts generated by all three algorithms provided insights into the best solutions, and the results allowed for a comparative analysis to determine which of the three methods ultimately offered the best outcomes. Finally, using four multi-criteria decision-making methods—AHP-TOPSIS, ELECTRE, PROMETHEE, and VIKOR—the optimal solution was identified, along with the best values for the decision variables Npm (number of photovoltaic modules) and Cbat (battery capacity), as well as for LLP and LCC.

The following tables,

Table 1. Technical specifications of the components used in the construction of the standalone photovoltaic system.

Component	Specifications	Value
PV Panel (JA Solar JAM72S10-405/PR)	Maximum power at STC (Pmax)	405 W
	Voltage at maximum power point (Vmp)	41.4 V
	Current at maximum power point (Imp)	9.77 A
	Open-circuit voltage (Voc)	49.1 V
	Short-circuit current (Isc)	10.32 A
	Efficiency at STC (npv)	20.33%
	Module Area (Apv)	2.02 m2
Battery (Trojan T-150 Renewable Energy Battery)	Nominal Voltage (V)	6 V
	Nominal Capacity (Ah)	150 Ah
	Charge/Discharge Efficiency (nb)	85–90%
	Maximum State of Charge (SOCmax)	100%
	Minimum State of Charge (SOCmin)	20%
Inverter (SMA Sunny Boy 5.0)	Efficiency (ninv)	97%
	Nominal Voltage (VAC)	230 V
	DC Voltage (VDC)	175–500 V
Charge Controller (Victron Energy SmartSolar MPPT 150/35)	Type	MPPT
	Efficiency	98%
	Maximum PV Open-Circuit Voltage (V)	150 V
	Maximum Charge Current (A)	35 A
Load	AC Voltage (VAC)	230 V
	DC Voltage (VDC)	12 V
	Daily Average Load Energy	10–20 kWh

and

Table 2. Unit costs for the components and services of the system.

Component	Specifications	Value
PV Module	120 Wp	160/module
Battery	225 Ah, 12 V	200/unit
Inverter	5 kW	1.200
Charge Controller	MPPT	400
Installation	Residential Setup	1.000–2.000
PV Maintenance	Wp/year	0.05/Wp
Battery Maintenance	Wh/year	0.003/Wh
Replacement Cost	% battery cost/year	25% of battery cost
Circuit Breaker	Standard unit	30
Miscellaneous	Wiring, mounting, etc.	200–400

, present the components, types, specifications, and corresponding costs. The selection of these specific components was based on the current market, prevailing prices, and available technologies, with the primary objective of meeting the requirements of the problem.

6.1. Algorithm Measurements and Results

After extensive testing and experimentation with the NSGA-II and DEMO algorithms, during which the most appropriate values for the mutation factor, crossover rate, maximum generations, and population size needed to be determined, the combination yielding the best results was identified as:

CR = 0.7,  F = 0.05, population_size = 120, max_generations = 35

The best results were defined as those where the objective function LLP exhibited the smallest possible values, both minimum and maximum, with greater weight given to LLP compared to LCC.

On the other hand, for the Particle Swarm Optimization algorithm, the number of iterations and the number of particles were determined empirically, based on optimization problems involving two objective functions. The final values were:

Num_particles = 30, max_iterations = 100

For the three different load scenarios, the three cities in Cyprus, and the three algorithms applied to produce results, numerous diagrams were generated. However, the following observations emerged as key: Across all cities, load scenarios, and algorithms, the NSGA-II algorithm demonstrated greater diversity, covering the entire range of values between the maximum and minimum for both objective functions. Additionally, in nearly all cases, NSGA-II achieved both the smallest and largest values for LLP and LCC. However, it was also observed that all three algorithms yielded very similar results, with some optimal solutions appearing on the Pareto fronts being common across multiple cases.

Regarding the different load scenarios, the following figures for each city with the corresponding algorithm, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22, for calculating the optimal final solutions are presented below. The three load cases are collectively displayed in a single diagram for each city, highlighting the variations in the objective function values resulting from changes in the load.

For Limassol:

Figure 14. 3 DEMO cases.

Figure 14. 3 DEMO cases.

Figure 15. 3 NSGA-II cases.

Figure 15. 3 NSGA-II cases.

Figure 16. 3 PSO cases.

Figure 16. 3 PSO cases.

For Larnaca:

Figure 17. 3 DEMO cases.

Figure 17. 3 DEMO cases.

Figure 18. 3 NSGA-II cases.

Figure 18. 3 NSGA-II cases.

Figure 19. 3 PSO cases.

Figure 19. 3 PSO cases.

For Nicosia:

Figure 20. 3 DEMO cases.

Figure 20. 3 DEMO cases.

Figure 21. 3 NSGA-II cases.

Figure 21. 3 NSGA-II cases.

Figure 22. 3 PSO cases.

Figure 22. 3 PSO cases.

The comparison of the three Pareto fronts reveals that as the load increases, the Pareto front shifts to the right and upwards, indicating an increase in the values of the two objective functions. This suggests that higher loads lead to a greater probability of load loss (LLP) and higher life cycle cost (LCC). This occurs because larger loads place increased demand on the system’s components, requiring more substantial energy storage and generation capacity. As a result, both the likelihood of supply shortages and the overall system investment tend to rise.

Furthermore, the values for LLP and LCC show greater dispersion for the 20 kWh load, indicating that larger loads exhibit greater variability in the objective functions. Conversely, smaller loads (10 kWh and 15 kWh) demonstrate less dispersion in their values.

This pattern highlights the trade-offs involved in managing larger loads within a standalone photovoltaic system, emphasizing the need for careful optimization to balance reliability and cost.

6.2. MCDM’s Measurements and Results

After identifying the optimal solutions for each load scenario, city, and algorithm, the multi-criteria decision-making methods were employed to select the best among the optimal solutions, based on the specific characteristics of each case. Specifically, each method identified the top five optimal solutions according to its respective calculation approach.

In all the applied multi-criteria decision-making methods, identical weights were assigned to the two objective functions (LLP and LCC). Two weighting scenarios were implemented:

Scenario 1: A weight of 0.8 was assigned to LLP and 0.2 to LCC, emphasizing the importance of meeting energy needs over system costs. Scenario 2: Equal weights of 0.5 were assigned to both LLP and LCC, treating the reliability of energy supply and system cost as equally significant.

The selection of these weighting scenarios aimed to illustrate the differences in the optimal solutions depending on whether greater emphasis is placed on meeting energy demands or when both objectives are considered equally important.

It is observed that the optimal solutions for each case are quite close to one another. No method demonstrates significant advantages or disadvantages compared to the others, indicating overall competitive performance. Below are 12 figures, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28, illustrating the cases for Nicosia at 10 kWh and 20 kWh, representing the minimum and maximum loads, respectively. These diagrams compare the optimal solutions obtained for each algorithm from each method under the weighting scenario of 0.8 for LLP and 0.2 for LCC.

Corresponding diagrams for the case of equal weights assigned to the two objective functions, 0.5 and 0.5, for the city of Nicosia under the minimum and maximum load scenarios are presented below:

Figure 26. LLP and LCC for different methods for Nicosia 10 kWh (NSGA-II).

Figure 26. LLP and LCC for different methods for Nicosia 10 kWh (NSGA-II).

Figure 27. LLP and LCC for different methods for Nicosia 10 kWh (DEMO).

Figure 27. LLP and LCC for different methods for Nicosia 10 kWh (DEMO).

Figure 28. LLP and LCC for different methods for Nicosia 10 kWh (PSO).

Figure 28. LLP and LCC for different methods for Nicosia 10 kWh (PSO).

7. Discussion and Comparison

According to the above diagrams, as well as the remaining results for the cities and load scenarios, the following conclusions can be drawn: Under the AHP-TOPSIS method, the five best solutions for all load scenarios, cities, and algorithms exhibit relatively low LLP values, while the LCC values remain at acceptable levels. This is primarily due to the structure of these two methods: AHP allows for the evaluation of criteria through pairwise comparisons and the assignment of weights to them, while TOPSIS uses the same weights to calculate the distance of each solution from the ideal one. This approach enables a balance of performance across both criteria, maintaining both LLP and LCC at low levels in both weighting scenarios. Simultaneously, the NSGA-II algorithm appears to outperform others in certain cases concerning the proximity to the ideal solution, as indicated by a higher closeness coefficient.

In the VIKOR method, the optimal solutions exhibit higher LLP values and lower LCC values, even in the first weighting scenario, where greater emphasis is placed on the LLP objective function. This result stems from the fact that the VIKOR method focuses on optimizing solutions through compromise, emphasizing both collective utility and individual regret. This means that the solutions may achieve a lower LCC but might not be as successful in reducing LLP due to the method’s design, which prioritizes the criterion deemed to have a greater impact on the final ranking of solutions.

Finally, in the PROMETHEE method, the lowest LLP values are achieved compared to the two previous methods, while the LCC also remains at competitive levels. Like AHP-TOPSIS, this method relies on pairwise comparisons of alternative solutions and the use of preference functions, allowing for a detailed analysis of criterion preferences. As a result, it leads to solutions that achieve lower LLP while maintaining LCC at satisfactory levels.

This variation in results across the methods can be attributed to the underlying decision-making logic and prioritization strategies embedded within each algorithm. While AHP-TOPSIS and PROMETHEE are structured to systematically incorporate the decision maker’s preferences—ensuring a more consistent trade-off between LLP and LCC—VIKOR emphasizes compromise solutions that may not always align with strict minimization of critical objectives like LLP. Particularly in VIKOR, the balance between group utility and individual regret often results in solutions that favor cost effectiveness (LCC) at the expense of reliability (LLP), especially when the weights do not heavily penalize load loss.

Moreover, the differences observed suggest that methods emphasizing relative distance from an ideal solution (such as TOPSIS and PROMETHEE) are more effective at maintaining overall system performance within acceptable bounds. In contrast, methods based on compromise ranking may yield solutions that are mathematically balanced but practically less desirable in applications where reliability is paramount, such as autonomous photovoltaic systems. This highlights the importance of selecting a decision-making method that aligns with the system’s operational priorities and constraints.

The framework and methods discussed here offer considerable flexibility in adapting to various geographic regions and energy demands. For instance, the algorithms and multi-criteria methods employed could be applied to regions with different climate conditions, solar radiation levels, and energy profiles. In regions with higher solar energy potential or more variable weather patterns, adjustments to the models could be made to reflect these differences, potentially enhancing the framework’s utility in optimizing photovoltaic systems for diverse environments.

Moreover, the decision-making methods like AHP-TOPSIS, VIKOR, and PROMETHEE could be tailored to specific regional needs by adjusting the weighting criteria, such as prioritizing environmental factors or focusing on cost effectiveness, depending on local priorities. The balance between LLP and LCC could also shift depending on the energy demand and local infrastructure—areas with higher energy consumption might place a greater emphasis on minimizing cost, while others with more frequent grid failures or energy insecurity might prioritize reliability over cost savings.

Additionally, as energy demand patterns can vary significantly across different regions (e.g., seasonal variations in energy use), the proposed framework could be further adapted to model such fluctuations. For instance, some regions may have high energy demand during hot seasons (requiring more energy for cooling), while others may have peak loads during colder periods (for heating). Tailoring the model to account for these demands could result in more accurate and applicable solutions, making the proposed framework more versatile.

8. Conclusions

Based on the analysis of the three multi-criteria methods (AHP-TOPSIS, VIKOR, and PROMETHEE) and three optimization algorithms (NSGA-II, DEMO, and PSO) for 10, 15, and 20 kWh loads in Nicosia, Larnaca, and Limassol, the following conclusions were drawn:

NSGA-II Algorithm: It showed excellent performance in terms of LLP and LCC, providing stable results across all load levels due to its ability to evenly distribute the Pareto front, offering a range of optimal solutions. DEMO Algorithm: While delivering some of the best solutions, particularly for AHP-TOPSIS, it exhibited instability due to large variations in objective function values, making it less reliable for this optimization. PSO Algorithm: Demonstrated satisfactory performance with competitive results, especially in minimizing LCC, though its performance varied depending on the method applied, with PROMETHEE offering the best results.

Multi-Criteria Methods:

AHP-TOPSIS: Provided consistent, accurate results with high closeness coefficients, particularly in scenarios where LLP was prioritized. VIKOR: Focused more on LLP, even under equal weightings, resulting in higher LLP values. PROMETHEE: Stable, but not as consistent in identifying optimal solutions as AHP-TOPSIS. Weight Scenario Impact: The differences between the 0.8–0.2 and 0.5–0.5 weight scenarios were minimal, but the 0.8–0.2 scenario produced more reliable solutions, avoiding high LLP values.

Optimal Combination: NSGA-II and AHP-TOPSIS under the 0.8–0.2 weight scenario were identified as the optimal combination for all cities and load conditions, while the top-ranked configurations for each city and load were the following (

Table 3. Top ranked configurations for each city and load.

Nicosia 10 kWh Load

Top-ranked configuration: LLP: 0.0455, LCC: 59,670.7675, closeness coefficient: 0.8948

15 kWh Load

Top-ranked configuration: LLP: 0.065, LCC: 71,571.8243, closeness coefficient: 0.8636

20 kWh Load

Top-ranked configuration: LLP: 0.0897, LCC: 83,985.7914, closeness coefficient: 0.8564

Larnaca

10 kWh Load

Top-ranked configuration: LLP: 0.0469, LCC: 58,173.6822, closeness coefficient: 0.8948

15 kWh Load

Top-ranked configuration: LLP: 0.0688, LCC: 69,240.2017, closeness coefficient: 0.8694

20 kWh Load

Limassol

10 kWh Load

Top-ranked configuration: LLP: 0.0589, LCC: 56,943.5076, closeness coefficient: 0.8869

15 kWh Load

Top-ranked configuration: LLP: 0.073, LCC: 70,080.4614, closeness coefficient: 0.8731

20 kWh Load

Top-ranked configuration: LLP: 0.0919, LCC: 84,444.3978, closeness coefficient: 0.8481

):

Based on the above solutions and the decision variables corresponding to each of them, the optimal number of parallel photovoltaic modules and the optimal battery capacity to be used for the construction of the photovoltaic system are determined. Specifically:

For Nicosia and 10 kWh, the selected decision variable values are: Npm = 73, Cbat = 16,753 Wh.

For Nicosia and 15 kWh, the selected decision variable values are: Npm = 84, Cbat = 23,029 Wh.

For Nicosia and 20 kWh, the selected decision variable values are: Npm = 102, Cbat = 28,331 Wh.

For Larnaca and10 kWh, the selected decision variable values are: Npm = 66, Cbat = 17,149 Wh.

For Larnaca and 15 kWh, the selected decision variable values are: Npm = 82, Cbat = 21,868 Wh.

For Larnaca and 20 kWh, the selected decision variable values are: Npm = 97, Cbat = 29,206 Wh.

For Limassol and 10 kWh, the selected decision variable values are: Npm = 67, Cbat = 15,925 Wh.

For Limassol and 15 kWh, the selected decision variable values are: Npm = 80, Cbat = 22,827 Wh.

For Limassol and 20 kWh, the selected decision variable values are: Npm = 103, Cbat = 28,399 Wh.

During this study, several limitations influenced the accuracy of the results. The reliance on climate data from specific periods and locations could affect the validity of the findings, as even data from reputable sources may contain inaccuracies. Additionally, simplifications and assumptions made in the simulation models, especially when calculating the objective function (LLP), were necessary due to the absence of a predefined method for considering all relevant parameters. The economic estimates and technological specifications used for the LCC objective function were based on current costs, which may change over time due to material price fluctuations, installation costs, and technological advancements. Furthermore, the algorithms (NSGA-II, DEMO, PSO) have their own limitations, and their sensitivity to initial conditions and the need for specific parameterization could influence their performance and reliability.

The multi-criteria evaluation method, relying on subjective judgments and weightings assigned to the criteria, also introduces variability. In this study, two scenarios were implemented with different weightings, but further testing with different preferences could lead to different outcomes. Recognizing these limitations is essential for interpreting the results and making informed decisions.

Looking ahead, future improvements could include testing additional optimization algorithms and multi-criteria methods to compare their effectiveness. Conducting tests with smaller data sets may help assess the models’ reliability in scenarios with less information. The integration of smart control systems for energy management could also enhance photovoltaic system efficiency. Moreover, incorporating more objective functions, both energy and economic, could provide additional parameters for optimization and yield better solutions. Finally, investigating hybrid systems that combine photovoltaic power with other renewable energy sources, such as wind or geothermal, could offer increased flexibility and reliability, improving energy security.

Limitations and Future Improvements

The results obtained in this study were influenced by several limitations that could affect their accuracy. The reliance on climate data from specific periods and locations may affect the generalizability of the findings, as even reputable data sources may have inaccuracies. Additionally, simplifications in the simulation models, especially when calculating the LLP objective function, were necessary due to the lack of a predefined method for considering all relevant parameters. The economic estimates used for LCC were based on current cost assumptions, but fluctuations in material prices, installation costs, and technological advancements over time may impact these estimates.

The algorithms used in this study (NSGA-II, DEMO, PSO) also have their own inherent limitations, such as sensitivity to initial conditions and the need for specific parameterization, which can influence their reliability and performance. Furthermore, the multi-criteria evaluation method relied on subjective judgments for the weightings of criteria, introducing variability in the results. While two weighting scenarios were tested, exploring additional scenarios could yield different outcomes.

Looking ahead, further research could focus on testing additional optimization algorithms and multi-criteria methods to assess their comparative effectiveness. Smaller data sets could be tested to evaluate the models’ performance in scenarios with less information. Integrating smart control systems for energy management may enhance the overall efficiency of photovoltaic systems. Additionally, expanding the analysis to include more objective functions—such as energy generation and economic considerations—could yield more comprehensive optimization results. Finally, exploring hybrid systems combining photovoltaic energy with other renewable sources, like wind or geothermal, could increase system flexibility and reliability, offering a more robust solution to meet varying energy demands.