Reliability analysis of a grid-connected hybrid renewable energy system using hybrid Monte-Carlo and Newton Raphson methods

The integration of renewable energy sources into the power grid is essential for sustainable development, yet it presents significant dependability challenges, particularly in terms of reliability, stability, and robustness due to the inherent variability of these sources. This research introduces a novel hybrid methodology that combines Monte Carlo simulation with Newton-Raphson power flow analysis to enhance the reliability assessment of grid-connected hybrid renewable energy systems. This innovative approach uniquely addresses the limitations of existing methodologies by merging the probabilistic handling of uncertainties with precise deterministic power flow analysis. Our hybrid method significantly reduces the Loss of Load Expectation (LOLE) to 5 h per year and the Loss of Load Energy Expectation (LOEE) to 200 MWh per year, outperforming traditional methods which typically report LOLEs of 2020 h/year and LOEEs of 10001000 MWh/year. Additionally, the hybrid method achieves a reduction in power losses to 1.2%, showcasing its superior efficiency compared to the 2.5% losses seen with standalone Monte Carlo methods. Real-time validation using the IEEE-30 bus model further confirms the practical applicability and robustness of our approach, making it a pivotal tool for enhancing grid stability and optimizing renewable energy integration. This research not only advances the methodology for reliability assessment but also sets a new standard for balancing accuracy and computational efficiency in energy system management. The implications of this work are far-reaching, offering significant contributions to both grid reliability and the sustainable management of renewable energy resources.

1 Introduction

1.1 Background of study

Energy is essential for meeting our basic human needs. It seems to be a crucial resource for its growth. Electricity has become an essential part of our daily lives, making it virtually impossible to imagine a world without it. Thus, it is the responsibility of humans to explore various forms of energy production in a manner that is both environmentally friendly and secure, while also striving to enhance energy efficiency. Significant endeavors are presently underway to mitigate the environmental effects of fossil fuels used in electricity generation. Recent research and studies have contributed to the advancement and wider adoption of renewable energy worldwide (Ali, 2019). Wind farms have emerged as a promising alternative to conventional energy sources in both industrialized and developing countries, thanks to the disastrous environmental effects of greenhouse gases and recent government policies. As an expert in renewable energy, it is worth noting that in 2008, Europe saw significant advancements in wind energy capacity when compared to other renewable sources (Hulle et al., 2009). This resulted in a total capacity of 64 GW (Wind Power Installed in Europe by End Of and 2008 Cumulative, 2008). Furthermore, various forms of sustainable energy, including photovoltaics, have been incorporated to address these issues. Not only is population growth a factor, but we are also operating in a landscape where customers play dual roles as both producers and suppliers. It is crucial to find a more suitable supply-to-demand match. Consequently, extensive research has been conducted in recent years regarding energy management, specifically focusing on assessing the dependability of the public power grid.

Considering the numerous advantages provided by hybrid renewable-based generating units, the use of such systems is widely encouraged worldwide. In today’s world, hybrid generating systems that rely on renewable energy sources are commonly employed to meet the electricity needs of remote areas and islands. These locations often face challenges such as difficult accessibility and rugged terrain, making it impractical to extend the electrical grid utility (Fathy, 2016; Ramli et al., 2016). Even with the use of a hybrid energy system, there is still a risk of power loss due to component outages. It is important to consider the reliability of the system when evaluating its performance. Nevertheless, the evaluation of the reliability of hybrid systems based on renewable energy poses a unique challenge. This is because conventional reliability evaluation methods, which are designed for fixed capacity outputs, cannot be directly applied to renewable energy sources. Therefore, the evaluation of the dependability of generating systems that rely on renewable energy must be approached in a unique manner.

Regrettably, attaining absolute precision is not always attainable in our line of work. The presence of certain challenges leads to the occurrence of inaccuracies. The challenges encompass the utilization of a simplified model that deviates from the intricacies of reality, the potential presence of errors in the model’s parameters stemming from measurements or estimations, the dynamic nature of the parameters which can undergo changes over time, and the susceptibility of the parameter values to alterations due to shifts in operating conditions. The following factors must be considered during the modeling process.

The objective is to conduct a comprehensive assessment of the dependability of the hybrid renewable energy system acquired through the utilization of a probabilistic approach integrated with a power flow analysis technique to attain the intended performance. This encompasses the assessment of potential losses, ensuring the system maintains a high level of power quality, and minimizing instances of failure.

1.2 Literature review

The initial research on the evaluation of the dependability of renewable energy systems was documented in the 1980s. Extensive research (Lauffenburger and Anderson, 1982) has played a crucial role in developing a standardized lexicon to evaluate the dependability of photovoltaic systems. In (Collins et al., 2009), different approaches are explored to improve the dependability of a hybrid isolated system. According to the authors in reference (Billinton and Alen, 1994), there are two different approaches to assess reliability: deterministic and probabilistic. In their study, researchers in (Paliwal et al., 2014) introduced a new probabilistic model to evaluate the reliability of battery storage in renewable energy systems using analytical methods. In a study conducted in (Khatod et al., 2010), they devised an analytical method to evaluate the wellbeing of small autonomous power systems that rely on wind and solar energy. In their work (Karki and Billinton, 2001), scholars introduced a model for evaluating the reliability and cost of a small, isolated power system. The developed model acknowledges the unpredictable behavior of renewable energy sources and preserves the sequence and interconnectedness of the variables associated with them. In their study (Singh and Fernandez, 2015), conducted a reliability assessment of a solar photovoltaic system, employing Monte Carlo simulation to compare the performance with and without battery storage. In a study conducted in (Li et al., 2015), a modelling and simulation framework called GTST-MLD was introduced. This framework is used for the reliability assessment of generic geared wind turbine systems. The developed model is combined with Monte Carlo simulation method to calculate the system reliability. In a study conducted in (Lee et al., 2014), a sizing methodology was created for an isolated photovoltaic-based micro-grid with battery storage. The methodology was based on cost versus reliability curves. In their research, authors in (Cai et al., 2015) conducted a reliability evaluation of grid-connected PV systems, taking into account intermittent faults. They utilized dynamic Bayesian networks (DBNs) for their analysis. The methodology utilizes a three-state Markov model to represent the state transition relationship of different types of faults in PV components. In a research work performed in (Nikmehr and Ravadanegh, 2016), authors examined the reliability assessment of multi-microgrids, specifically focusing on the optimal operation of small-scale energy zones in the presence of load-generation uncertainties. The reliability evaluation is conducted in both interconnected and islanded modes of operation of the micro-grid. Ding et al. in (Ding et al., 2011) employed the universal generating function to assess the reliability of a system based on renewable energy sources. More recent research has seen a shift towards integrating multiple renewable sources with advanced computational methods to enhance system reliability and efficiency. For example (Zheng et al., 2024), introduced an innovative process design for a biomass digestion-supercritical carbon dioxide scenario, aimed at boosting geothermal-driven cogeneration systems for power and heat. This approach demonstrates the potential for optimizing energy systems by combining renewable sources and advanced thermodynamic processes. Ref. (Zhu et al., 2024) developed an improved hybrid algorithm for many-objective optimization, which integrates biogeography-based optimization with complex problem-solving techniques. Their research demonstrates how hybrid methods can effectively manage the complexities of modern energy systems, ensuring stable and efficient operation. Recent advancements in hybrid systems, which combine different renewable energy sources, have shown promise in enhancing grid stability and reliability. For instance, intelligent hybrid wind–PV farms have been effective in maintaining overall stability in multimachine power systems by acting as static compensators (Kumar Rajeev et al., 2022). Additionally, the operational performance of wind power plants can be significantly improved through advanced two-stage modeling techniques that optimize both capacity and operational efficiency (Kumar et al., 2022).

Given the numerous uncertainties and limitations of energy systems, it is crucial to evaluate their dependability, particularly under real-world circumstances. Therefore, deterministic tools lack precision, highlighting the significance of employing probabilistic techniques in the analysis and modeling of energy systems. Several authors have explored different methods, such as the fault tree (Volkanovski et al., 2009), the analytical method by grid of the network (Zio and Golea, 2012), and the theoretical operational sequences-based approaches (Kang et al., 2002). These methods focus on identifying the most crucial components of the system, while considering potential limitations. In addition, a risk analysis of undesired system states was discussed. The presentation included the level 1 risk assessment methodology, which was used to estimate the potential disruptions in energy systems (Henneaux et al., 2012). Optimization algorithms have been employed to enhance the dependability of energy systems. An algorithm has been implemented to optimize the transmission line of the grid, with the goal of enhancing reliability (Lin and Yeh, 2011). For planning applications, a proposed probabilistic analytical model can be used to assess the reliability of electricity markets that are becoming more competitive (Ehsania et al., 2008). Stochastic or probabilistic methods are frequently employed to account for and model the inherent randomness in the operating load and the resulting data (Hammersly and Handscomb, 1964).

In addition, it is worth mentioning that adequacy studies currently rely on deterministic cases, but there is a need to incorporate a probabilistic approach to assess production and consumption. One of the tools utilized is linear programming. For instance, in a study (Clack et al., 2013), the author applied linear programming techniques to create an efficient electrical system. This system included the HVDC (High Voltage Direct Current) system, which is used for energy transportation, transmission, and storage. The results achieved through this approach are quite satisfactory. (Paris et al., 2010) addresses the issue of energy management in a habitat using linear programming and the simplex algorithm. It ensures optimal utilization of each energy source (such as photovoltaic and wind) and efficient energy storage, all while minimizing costs. However, the outcomes are only valid for a week. A study on the Stochastic Process of Dynamic Consumption Improvement is currently being conducted, as mentioned by the author. (Houndedako et al., 2014) utilized the HVDC system, which involves the transmission of electric energy and synchronization between two distinct grids. Using a simulation in the MATLAB/SimPowerSystems software, the author verified the effectiveness of this dependable and strong synchronization method for maintaining power grid stability. However, it is crucial to not only analyze the public grid, but also to evaluate the power generated and fed into the grid.

Nevertheless, it is crucial to highlight that the public grid is a comprehensive entity that is consistently influenced by numerous internal and external constraints and factors. The system must consistently adhere to established protocols and regulations. The concept of power flow is relevant at this level. The primary characteristic of the system lies in its ability to assess and measure the energy generated, supplied, and consumed. Analysis and results are crucial in the operational stages of any control system and economic program, as well as in the expansion and modeling phase. It is crucial to strategize the management of energy production and consumption from both conventional and renewable sources. However, it is highly recommended to conduct an analysis of the public grid in order to examine the flow of power generated. Studies conducted in (Van Ness, 1959; Van Nness and Griffin, 1961; Brown et al., 1963; Carter et al., 1968; Comer, 2024) indicate that power flow is a crucial factor in the analysis of energy systems.

Power flow is a crucial tool in analyzing energy systems. Their operation and planning heavily rely on the use of power flow techniques. The objective is to determine the tensions and angles at the state of equilibrium at any point of the grid, along with the active and reactive powers in each line, considering the given load values. Many algorithms have been created to optimize working time, either by reducing the number of equations (Gomez Exposito et al., 2004; Laughton and Humphrey Davies, 2024) or by approximating the Jacobian matrix (Aparecido Ferreira and da Costa, 2005; Jang et al., 2005). The focus was on updating the voltage amplitude and angle at each iteration.

In (Stefopoulos et al., 2005), a method was proposed by Stefpoulos et al. that utilizes the flow of powers in a single-phase grid within a probabilistic context. This approach considers simulations using a widely recognized method for evaluating the suitability of energy sources: the Monte Carlo method. Real-time validation of hybrid methodologies is crucial for demonstrating their practical applicability. Research in (Li et al., 2023) emphasized the importance of real-world testing and validation in their study on the role of renewable resources in transitioning to a green economy. Their findings underscore the need for practical, scalable solutions that can be implemented in actual energy systems to achieve sustainable energy management.

The methodology presented in this paper includes real-time validation using a simulated hybrid system based on the IEEE-30 bus model. This approach aligns with the recommendations of (Wang et al., 2017), who highlighted the need for comprehensive thermodynamic, economic, and environmental analysis of multi-generation systems. By validating the model with real-world grid configurations, this research demonstrates how the hybrid method can be effectively applied to improve grid reliability and performance.

One of the significant contributions of this research is the reduction in Loss of Load Expectation (LOLE) and Loss of Load Energy Expectation (LOEE). These metrics are critical for evaluating the reliability of energy systems (Li et al., 2022). discussed the importance of minimizing power losses in energy systems to enhance overall efficiency and reliability. Their work supports the findings of this research, which show that the hybrid method reduces power losses to 1.2%, significantly lower than those observed with Monte Carlo and Newton-Raphson methods used separately.

Maintaining voltage stability is another critical aspect of energy system reliability. Recent advancements in power flow analysis have focused on improving voltage stability through localized compensation techniques (Zhu et al., 2024b). introduced a fast and accurate calculation method for line breaking power flow based on Taylor expansion, which enhances the precision of power flow adjustments. This aligns with the localized compensation approach used in the hybrid methodology, which ensures that voltage stability is maintained across the grid, resulting in a high voltage stability index of 0.99.

Looking ahead, the hybrid methodology presented in this paper offers significant potential for enhancing the reliability and efficiency of hybrid renewable energy systems. The research contributes to the optimization of these systems, with major implications for both grid stability and sustainable energy policy. (Bai et al., 2022; Zhu et al., 2024c) highlighted the need for continued innovation in the design and optimization of multi-generation systems to meet the growing demand for clean energy.

By integrating advanced computational methods and real-time validation, this research provides a robust framework for future studies and practical applications in the energy sector. The findings support the development of more reliable and efficient energy systems, contributing to the global transition towards a sustainable energy future.

This literature review shows that many researchers have presented work on long-term studies of electrical systems. Indeed, the adequacy between renewable and conventional sources in the context of the supply-demand balance contributes to an estimate of the energy deficit on the one hand, but also to a better assessment of the grid reliability on the other.

1.3 Research gap and contributions

In today’s world, the significance of power has become essential in the advancement of technologies. Extensive research has been carried out by numerous experts to thoroughly explore the energy sector, aiming to develop a dependable and eco-friendly power source. Utilizing renewable sources of energy, such as wind energy or solar energy, offers a practical solution to tackle environmental conservation concerns. It is important to acknowledge that there are two distinct methodologies for evaluating reliability of systems: deterministic and probabilistic.

• The deterministic approach relies on analyzing a smaller set of positions that are estimated to be uncertain beforehand, in order to assess the stability of the electrical system. The system lacks probabilistic modeling of production units that utilize renewable energies and has a limited number of adverse situations. Two key factors that contribute to less restrictive situations are the absence of assumptions and the presence of overestimation.

• The probabilistic approach that evaluates the risk of non-compliance with a system constraint by considering all possible scenarios and their respective probabilities. The technical characteristics of the system include:

- Consideration of all possible system states and their associated probabilities

- Evaluation of the risk of surpassing system constraints

- Diligent handling of the consequences of problem situations

- The necessity of defining a risk policy

A critical requirement is essential for addressing the challenges associated with the maintenance of existing energy sources, taking into consideration specific parameters such as Time to Failure (TTF) and Time to Repair (TTR)… Based on the selected grid model and the integration of multiple sources in a system, it is important to emphasize the significant contributions of the algorithm utilized for power flow analysis in conjunction with each adequacy tool. Deterministic approaches are unable to accurately recognize the stochastic behavior of microgrid systems. In order to determine the reliability indices of stable and unstable resources, such as solar and wind, the sources and loads can be represented using the sequential Monte Carlo technique, which is a probabilistic method.

For this purpose, the Monte Carlo technique, which can be easily integrated into the EMS (Energy Management System), seems to be a suitable tool for evaluating the grid’s reliability. However, the integration of a power-flow analysis technique has not been implemented yet. Indeed, given the requirement for flawless synchronization of renewable sources with the public grid and the assumption that all sources are repairable, it is necessary to analyze the compatibility of these sources with conventional sources in order to strategically plan long-term production and enhance the reliability of the grid.

In this case, utilizing the Newton-Raphson method in power flow analysis seems to be a favorable and suitable approach for the given situation. This methodology will enable the resolution of the system of non-linear equations derived from the power flow analysis using the Monte Carlo method. Additionally, it will facilitate the implementation of a localized compensation at each bus location where the power factor deviates from the standard. The losses will thus be computed, leading to more precise outcomes.

In our study, we propose the following assumptions.

- The compatibility and sharing of various energy sources are ensured. Indeed, this falls outside the scope of our research.

- Every energy source should be capable of being repaired

This research focuses on the development of an algorithm for consumption planning. It considers various energy sources like hydroelectric, thermal, solar, and wind. The algorithm uses an iterative technique to ensure our solutions converge, resulting in improved energy compensation and meeting the standard power factor criterion. For this purpose, the selected grid is an IEEE-30bus model that incorporates the daily and seasonal electricity consumption patterns in Belgium. This allows for a more precise analysis of the generated powers and ensures better adequacy.

1.4 Paper organization

The primary goal of this study is summarized into.

- Analyzing energy systems by incorporating probability and reliability laws to ensure a balanced supply, while also considering the needs of the population. This technique, known as the Monte Carlo Method, enables you to strategically manage energy consumption over a designated timeframe by utilizing various specific parameters.

- Implementing a method to solve the power equations system resulting from the power flow analysis, aiming to achieve improved compensation. This algorithm is known as the Newton Raphson algorithm.

In fact, the rest of this text is structured in the following way.

• In the second section, we will discuss the modeling of the different sources both renewable and conventional employed in this study, including the IEEE 30-bus. We will also present the two techniques namely Monte Carlo Method and Newton Raphson algorithm, while depicting the different parameters and characteristics of the system.

• In the final section, we will discuss the outcomes of the various simulations generated using the simulation software Matlab/Simulink. Expert.

2 Materials and methods

Given that the concepts of adequacy and the devaluation of reliability have already been defined, it is crucial to initially model our study system and subsequently describe the methods employed for the given situation. Undoubtedly, this component will serve as a pivotal element in our project, as it will also facilitate the elucidation of the various phases required for the accomplishment of research endeavors.

2.1 System modeling

Firstly, the study system is composed of a hybrid network that includes the following subsystems.

• Primary sources: including hydropower plants and thermal plants

• Additional sources: wind turbines and photovoltaic panels

• Connect with IEEE-30bus configuration

This work is within the European context, particularly in Belgium, where we will present the power flow on the selected grid.

2.1.1 Primary sources

The report provided by the Belgian company FEBEG (Febeg, 2024) outlines the energy source distribution in Belgium for the year 2019.

It has been observed that 38.5% represents the proportion occupied by our pre-defined major sources, specifically thermal and hydraulic power plants. Let us review the initial assumptions that.

- The sources are assumed to be repairable

- The network integration of the various sources is purportedly flawless

The following figure (Figure 1) provides a breakdown of the percentage of ventilation by type of fuel for net electricity production in Belgium.

Figure 1

Figure 1. Total net production of electricity in Belgium in 2019 (Febeg, 2024).

It is important to note that the reliability statuses of a system are depicted in Figure 2.

Figure 2

Figure 2. State model of a given system (Paris et al., 2010).

For instance, it will be assumed that the primary sources will be allocated in the following manner.

• The system consists of 10 hydroelectric units, each with a power output of 0.13TWh. The Mean Time to Repair (MTTR) for the system is 60 units of time, while the Mean Time To Failure (MTTF) is 1410 units of time. Compared to the total output of 89.85 terawatt-hours (TWh), we can calculate the rate as follows in Eq. 1:

${\gamma }_1=0.13*10*100/89.85(1)$

The value of ${\gamma }_1$ is approximately 1.4%. Further, from Figure 1, we obtain for hydropower: 0.9 + 0.3 = 1.2%.

• 9 thermal units of unit power 3 terawatt-hours, with a mean time to repair (MTTR) of 60 and a mean time to failure (MTTF) of 1410. Compared to the total output of 89.85 terawatt-hours (TWh), we can calculate the rate using Eq. 2 as follows:

${\gamma }_2=3*9*100/89.85(2)$

The value of ${\gamma }_2$ is approximately 30%. Similarly, a ratio of 37.3% can be obtained from Figure 1

2.1.2 Alternative sources

For our model, we need to incorporate wind and photovoltaic units as alternative sources, distributed in the following manner.

• A total of 10 wind turbines. It is important to keep in mind that the power generated by a wind system is influenced by the speed of the wind and the power curve. To obtain a model of wind speed, data on wind speed in a room (data from Belgium) will need to be collected and a mathematical model will be produced to simulate wind speed. The Weibull function is utilized for this purpose (Paris et al., 2010). The unit has a nominal power of 0.75 TWh, with an MTTR of 60 and an MTTF of 1410. When we consider the total output of 89.85 TWh, we can calculate the rate using Eq. 3 as follows:

${\gamma }_3=0.75*10*100/89.85(3)$

Therefore, ${\gamma }_3$ is approximately 8.3%.

According to Figure 1, the wind turbine has an efficiency of 10.2%. In addition, the output power can be represented by the following system of Eq. 4 (Fang and Cai, 2011; Bai et al., 2022):

$P_{WT}=\left\{\begin{array}{l}0;0<\mathrm{v}<v_{cut-in}ou{v}_{cut-out}<\mathrm{v}\\ \mathrm{a}{v}^3+\mathrm{b}{P}_{nom};v_{cut-in}<\mathrm{v}<v_{nom}\\ P_{nom};v_{nom}<\mathrm{v}<v_{cut-out}\end{array}\right.(4)$

Having the low cutting speed as $v_{cut-in}$ and the high cutting speed as $v_{cut-out}$, the nominal speed is referred to as $v_{nom}$. The nominal power is denoted as $P_{nom}$. The current speed is represented as v. In addition, we have the following Eqs 5–6 as follows:

$\mathrm{a}=\frac{P_{nom}}{{v_{nom}}^3-{v_{cut-out}}^3}(5)$

$\mathrm{b}=\frac{{v_{cut-in}}^3}{{v_{nom}}^3-{v_{cut-out}}^3}(6)$

Reference (Zhou et al., 2010) provides a table (Table 1) of parameters for a wind turbine that will be used for simulation.

• The total number of photovoltaic cells is 10*10^12, each with a unit power of $P_{PV}$. The mean time to repair (MTTR) is 60, and the mean time to failure (MTTF) is 1410. It is important to keep in mind that the performance of a photovoltaic unit is directly influenced by the amount of sunlight it receives. The test should be conducted with radiation of 1 kilowatt per square meter and a cell temperature of 25°C. Tracing the current-voltage characteristic is crucial in PV studies. This is obtained through Eq. 7 as follows (Zhou et al., 2010; Fang and Cai, 2011; Bai et al., 2022):

$\left\{\begin{array}{l}{T}_c=T_A+\mathrm{s}*\frac{\left(N_{OT}-20\right)}{0.8}\\ \mathrm{I}=\mathrm{s}*\left(I_{sc}+K_I\left(T_C-25\right)\right)\\ \mathrm{V}=V_{OC}-K_V*T_C\end{array}\right.(7)$

Table 1

Table 1. Values for the parameters of the wind turbine units.

Given.

• $T_C$ represents the cell temperature

• $T_A$ refers to the ambient temperature.

• s is the irradiation.

• The nominal cell temperature is $N_{OT}$.

• The short-circuit current in a photovoltaic unit is $I_{SC}$.

• The open circuit voltage in a photovoltaic unit is $V_{OC}$.

• I is the current in a photovoltaic unit

• The voltage in a photovoltaic unit is denoted by V.

• $K_I$ is a factor that relates to the temperature of the short-circuit current.

• $K_V$ refers to the temperature factor associated with open circuit voltage.

When evaluating the output power, we need to calculate the shape factor FF, which is given by Eq. 8 (Kishore and Fernandez, 2011):

$\text{FF}=\frac{V_{MPP}*I_{MPP}}{V_{OC}*I_{SC}}(8)$

With $V_{MPP}$ and $I_{MPP}$, which are the voltage and current obtained through MPPT, respectively. The energy per hour unit is given by Eq. 9 as follows:

$\mathrm{P}=\text{FF\hspace{0.17em}}*N_s*\mathrm{V}*\mathrm{I}(9)$

(Chatterjee et al., 2011) Provides a table (Table 2) of parameters for a cell that will be used for simulation:

Table 2

Table 2. Values for the parameters of the PV units.

Thus, p = 3.8TWh.

Compared to the overall production of 89.85TWh, we find almost the same rate in Eq. 10 as follows:

${\gamma }_4=3.8*100/89.85(10)$

Thus ${\gamma }_4$ ≈ 4.22%

Figure 1 gives the PV a value of 4.2%.

2.1.3 Data for the consumption

In 2019, Belgium’s electricity consumption reached a total of 83.73 TWh. The defined consumption is calculated as follows: Calculating the net production involves taking into account factors such as electricity consumption for pumping-turbine, net losses, import, and export. This indicator reflects the energy consumption of the sector, including refineries and other related industries. Figure 3 depicts the total electricity production in Belgium 2007 to 2019. The peak consumption is recorded at 83.73 TWh. The daily consumption data for the year 2019 is categorized by seasons in annexes 1, 2, and 3.

Figure 3

Figure 3. Electricity consumption in Belgium 2007-2019 (Febeg, 2024).

2.1.4 IEEE 30buses model

The IEEE 30-bus model provides a simplified representation of the American public network during December 1961 (Washington, 2024). The chosen model for the IEEE-30bus as shown in Figure 4 is the European version, which includes 41 data lines. Additional information on bus data and network admissions can be found in Annexes 4 and 5.

Figure 4

Figure 4. Overview of the IEEE-30buses model (Washington, 2024).

2.1.5 Problematic

The public network refers to a network that consists of various interconnected sources and loads. Given the rising demand and unpredictable fluctuations, the system exhibits characteristics of dynamism, necessitating the acquisition of information and, crucially, decision-making for any insufficient or atypical alteration. Considering that consumption is a dynamic variable, production must also be adjusted accordingly to fulfill the demands of consumers. Based on this empirical evidence, developed nations use various methodologies to engage consumers in the process of energy generation. However, this is not without issues.

In order to mitigate these challenges, the implementation of indigenous energy resources can prove advantageous. It is important to mention that there are two methodologies for evaluating reliability: deterministic and probabilistic.

• The deterministic method is employed to analyze a limited number of positions that are estimated in advance to be uncertain:

◦ The purpose is to assess the stability of the electrical system in these positions.

◦ The system lacks probabilistic modeling of production units utilizing renewable energies and has a restricted number of adverse situations.

◦ In order to reduce the restrictiveness of certain situations, it is important to avoid making assumptions and to avoid overestimating.

• The probabilistic approach evaluates the risk of non-compliance with a system constraint by considering all possible scenarios and their corresponding probabilities. The characteristics of this are:

◦ Considering all potential system states and their respective probabilities

◦ Estimation of the risk associated with surpassing system constraints

◦ The management of problem situations requires careful consideration of the consequences.

◦ It is important to establish a risk policy to address this need.

Deterministic approaches fail to account for the stochastic nature of micro-network systems. In order to determine the reliability indices of renewable energy sources like solar and wind, the sources and loads can be stochastically modeled using the sequential Monte Carlo technique, which is a probabilistic method. Considering the assumption of perfect synchronization between renewable sources and the public network, as well as the assumption that all sources are repairable, it is necessary to assess the adequacy of these sources in comparison to conventional sources. This analysis is crucial for long-term production planning and aims to enhance the reliability of the network by analyzing power flow.

Various methodologies have been suggested for achieving this objective.

• The Gauss-Seidel method

• The Newton-Raphson method is an iterative numerical method used to find the roots of a function. It is based on the idea of using the derivative of the function to approximate the root.

• Particle Swarm Optimization (PSO)

• Modified Particle Swarm Optimization (PSO)

One of the commonly employed techniques, the Gauss Seidel method (when used in conjunction with the Monte Carlo method), enables streamlined analysis and reduces storage capacity requirements. Additionally, it minimizes computation times. The Newton-Raphson method, however, is highly advantageous in the context of energy system evaluation, particularly in operational frameworks and the control of system reliability.

In (Agreira et al., 2006), performance indices obtained through simulations using the Monte Carlo method were utilized to evaluate the impacts of overload, voltage limit violations, and voltage stability issues in an electricity distribution network. Nevertheless, a study focused on the criterion of compensating active and reactive powers has not yet been formulated.

The objective of our study is to explore the methods for implementing local compensation based on the power factor criterion. Indeed, the utilization of the Newton Raphson method enables the utilization of the powers generated by the Monte Carlo method to solve the system of equations derived from the power flow. Additionally, it facilitates the implementation of a local compensation at each bus point where the power factor does not conform to the standard. The losses will be computed, resulting in improved accuracy of the obtained outcomes.

An empirical evaluation of the simulation outcomes using MATLAB of the system both with and without the Newton-Raphson method will provide a rationale for our research.

2.2 Presentation of Monte Carlo and Newton Raphson methods

After providing an overview of the system components, including alternative and primary energy sources, as well as the network based on the IEEE-30bus model and consumption data, this section will describe the methods that will be employed to assess the suitability of conventional and renewable energies, as well as power flow.

2.2.1 Monte Carlo method for adequacy assessment

The simulation using the Monte Carlo method is an appropriate tool for analyzing stochastic systems with highly complex factors. This is a method that is used for probabilistic situations due to its dynamic properties, as well as its easy integration into numerical problem solving and its robust adaptability to any scientific problem (Wang et al., 2008). In the Monte Carlo simulation of energy systems, the states of the components are randomly sampled according to their stochastic models (Billinton and Alen, 1994). Therefore, a set of information during a time sequence is provided and adequacy indices are calculated from a series of simulations.

Based on our predefined model consisting of renewable and conventional sources, as well as our power grid (see Figure 5), but most importantly the consumption data, we randomly evaluate the states of the sources that are assumed to be repairable.

Figure 5

Figure 5. A microgrid composed of sources PV-WT-Grid.

The time to failure (TTF) and the time to repair (TTR) of sources are calculated based on random variables. Indeed, let us recall that the state of an energy system can be represented by: X = (${\mathrm{x}}_1$, ${\mathrm{x}}_2$, ${\mathrm{x}}_3$,…, ${\mathrm{x}}_{\mathrm{m}}$); where ${\mathrm{x}}_{\mathrm{i}}$ represents the state of the ith component. The set of all possible states of the system, containing all possible combinations of the possible states of the components, will be denoted as ${\mathrm{S}}_{\mathrm{X}}$. Let’s consider ${\mathrm{F}}_{\mathrm{i}}$ (t) as the test function of the overall power generated compared to the requested power, for the ith component at time t. We will have:

The function ${\mathrm{F}}_{\mathrm{i}}$ (t) is equal to 1 if the demand is lower than the production. ${\mathrm{F}}_{\mathrm{i}}$ (t) is equal to 0 if the demand is higher than the production. Thus, the probability that the source is greater than the load is given by (Eq. 11) (Zhu et al., 2024c):

$\stackrel{\sim }{p_i}=\frac{1}{Card\left(X\right)}{\sum }_{t=1}^N{F}_i\left(\mathrm{t}\right)(11)$

The Formula 5 calculates the average value of ${\mathrm{p}}_{\mathrm{i}}$, denoted as $\stackrel{\sim }{p_i}$. It is computed as the sum of $F_i$ for all t from 1 to N, divided by the number of elements in X, denoted as Card(X). Here, $F_i$ (t) is equal to 1 for all time instants t. For a very large N, according to the law of large numbers, we obtain Eq. 12 as follows:

$\underset{\mathrm{N}\to \infty }{\lim }\frac{1}{\mathrm{N}}{\sum }_{\mathrm{t}=1}^{\mathrm{N}}\stackrel{\sim }{{\mathrm{p}}_{\mathrm{i}}}\approx 1(12)$

The limit as N approaches infinity of the average of pi from t = 1 to N, divided by N, is approximately equal to 1 (12).

We find ourselves in a situation of probability distribution and according to the Monte Carlo method, we achieve convergence with a convergence error denoted in (Eq. 13):

$e_i=\frac{\stackrel{\sim }{p_i}}{\sigma }(13)$

The flowchart of computations using Monte Carlo is presented in Figure 6:

Figure 6

Figure 6. A flowchart of the simulation using the Monte Carlo method.

Once the system simulation using the Monte Carlo method has been performed, the power flow analysis can now be conducted for each value of t until convergence is achieved.

2.2.2 Power flow calculation using the Newton-Raphson method

Several research studies have been conducted in the field of power flow. The objective of this study is to minimize losses related to active, reactive, and apparent powers. An iterative process will be used to calculate the values of the mentioned quantities based on the initial voltages and angles, in order to obtain an equilibrium situation using an estimation procedure.

To this end, a programming technique to consider is the Newton-Raphson method. The function will allow the construction of the Jacobian matrix from matrix form to programmed form. First, let’s review the various equations related to power flow (Zhu et al., 2024c).

For a bus i, we have We have Eq. 14:

$I_i={\displaystyle {\sum }_{j=1}^{N_{bus}}}{Y}_{ij}*V_j(14)$

With $N_{bus}$ the total number of buses in the network.

Besides, we obtain Eqs 15, 16 as follows:

$I_i^{*}=\frac{P_i+\mathrm{j}{Q}_i}{V_i}(15)$

With

$I_i=\frac{P_i-\mathrm{j}{Q}_i}{V_i^{*}}(16)$

Based on the previous relationships (14) (15) (16), it can be observed in Eq. 17 that:

$\left(P_i-\mathrm{j}{Q}_i\right)={\displaystyle {\sum }_{j=1}^{N_{bus}}}\left|Y_{ij}\left|*\right|V_i\left|*\right|V_j\left|*\right.\right.\left(\cos \left({\theta }_{ij}+{\delta }_i-{\delta }_j\right)-j\sin \left({\theta }_{ij}+{\delta }_i-{\delta }_j\right)\right)(17)$

Let us now calculate the following parameters: V, P, and Q. For this, let us note.

• ${\mathrm{P}}_{\text{Gi}}$: The active power generated by the Monte Carlo method at bus i

• ${\mathrm{Q}}_{\text{Gi}}$: the reactive power generated by the Monte Carlo method at bus i

• ${\mathrm{P}}_{\text{Li}}$: The active power consumed at bus i provided by the IEEE-30bus table.

• ${\mathrm{Q}}_{\text{Li}}$: The reactive power consumed at bus i provided by the IEEE-30bus table.

• ${\mathrm{P}}_{\text{inj}\mathrm{i}}$: the injected active power at bus i

• ${\mathrm{Q}}_{\text{inj}\mathrm{i}}$: the reactive power injected at bus i

• ${\mathrm{P}}_{\text{calc}\mathrm{i}}$: the active power calculated by NEWTON RAPHSON at bus i

• ${\mathrm{Q}}_{\text{calc}\mathrm{i}}$: The reactive power calculated by the Newton-Raphson method at bus i

The following results are obtained using Eqs 18, 19:

${\mathrm{P}}_{\text{inj}\mathrm{i}}={\mathrm{P}}_{\text{Gi}}-{\mathrm{P}}_{\text{Li}}(18)$

${\mathrm{Q}}_{\text{inj}\mathrm{i}}={\mathrm{Q}}_{\text{Gi}}-{\mathrm{Q}}_{\text{Li}}.(19)$

The absolute errors related to power losses are given by Eqs 20, 21:

${∆\mathrm{P}}_{\mathrm{i}}={\mathrm{P}}_{\text{cal}\mathrm{i}}-{\mathrm{P}}_{\text{inj}\mathrm{i}}(20)$

${∆\mathrm{Q}}_{\mathrm{i}}={\mathrm{Q}}_{\text{cal}\mathrm{i}}-{\mathrm{Q}}_{\text{inj}\mathrm{i}}(21)$

The powers ${\mathrm{P}}_{\text{cal}\mathrm{i}}$ and ${\mathrm{Q}}_{\text{cal}\mathrm{i}}$ are computed by compensation according to the following Eqs 22, 23:

${\mathrm{P}}_{\text{cal}\mathrm{i}}^{+}=\left|\min \left(\Delta {\mathrm{P}}_{\mathrm{i}}\right)\right|(22)$

${\mathrm{Q}}_{\text{cal}\mathrm{i}}^{+}=\left|\min \left(\Delta {\mathrm{Q}}_{\mathrm{i}}\right)\right|(23)$

Where |min $\left({∆\mathrm{P}}_{\mathrm{i}}\right.$)| and |min $\left({∆\mathrm{Q}}_{\mathrm{i}}\right.$)| are the powers supplied by the storage batteries and compensation banks at each bus i can be modeled as the power factor value ${\cos \psi }_{\mathrm{i}}$ at each bus i.

Regarding the Newton-Raphson algorithm, convergence will be achieved once the errors (14) and (15) above have been minimized, or when the power factor ${\cos \psi }_{\mathrm{i}}$ is equal to 0.8. The matrix calculation rules will be used to obtain the correction matrix until the convergence threshold of the Newton Raphson algorithm is reached.

An algorithm idea is provided in the following flowchart (Figure 7).

Figure 7

Figure 7. The simulation’s organizational chart using the Newton-Raphson method.

2.3 Methodology

The problem-solving methodology is provided at this level in order to achieve the results that will be presented in the rest of our work.

2.3.1 Primary sources

To calculate the total power, ${\mathrm{P}}_{\text{tot}}$, when all thermal and hydroelectric units are operating at full capacity. The available power is initialized to this value. The TTF (Thermal Transfer Factor) is then calculated for each thermal unit and for each hydroelectric unit. To do this, we define ${\mathrm{N}}_{\text{hydro}}$ and ${\mathrm{N}}_{\text{therm}}$. The TTF is then calculated using Eq. 24 as follows:

$\text{TTF}=-\text{\hspace{0.17em}MTTF}\times \ln \left(\mathrm{U}\right)(24)$

Let U be a random number element in the range [0, 1] that closely resembles a Reverse Fish distribution. This distribution is based on the mean time to failure (MTTF), similar to the λ parameter in the Fish formula. A vector is formed by concatenating the value of the hydro TTF (${\mathrm{N}}_{\text{hydro}}$ TTF) with the (${\mathrm{N}}_{\text{therm}}$ TTF). We calculate the minimum value from the vector and display its corresponding index and value in hours. Next, we activate the simulated value for the total number of hours (${\mathrm{N}}_{\text{MC}}$). Initially, all units are in a functional state, with a vector state of all 1. If the simulated number of hours is less than the smallest Time to Failure (TTF) of the vector, no changes occur and the available power remains constant at its initialized value. Upon reaching the minimum threshold of operating hours, the system reaches the calculated “random” Time to Failure (TTF), resulting in the unit’s failure. The index of this minimum is then examined to determine whether the unit belongs to the thermal or hydroelectric category. For instance, if the units in question are thermal units, a Time to Repair (TTR) is defined, which is analogous to the Time To Failure (TTF). In the Time To Failure (TTF) vector, the value is updated to i + Time to Repair (TTR), which represents the number of hours required to restore the system to operational status. Therefore, we calculate the power output of the hydroelectric system by subtracting the lost unit power from the total power available. When considering a thermal system, we use the same logical process. In each case, a new minimum must be recalculated to determine the next hourly value. This is done to identify the next Time to Failure (TTF) or Time to Repair (TTR) event, which will determine whether the next unit will return to operation or fail. As the iterations continue, the vector will encounter Time to Repair (TTR) and Time To Failure (TTF) metrics. The state must therefore undergo testing. When the value of i reaches the minimum value in the vector, it becomes important to determine whether it represents extinction or a repair. Thus, we have determined the power output that is accessible for the primary units. The process can be described as a sequential probabilistic calculation, where the available power may decrease or increase as it progresses.

2.3.2 Alternative sources

The state vectors and the vector of the ${\mathrm{N}}_{\text{WT}}$ TTF are defined using the same methodology as for major units. The modification pertains to the computation of the wind power that is currently accessible. There are no consecutive withdrawals or additions of nominal powers in wind turbines. This is because the power generation is no longer limited to all-or-nothing states. Instead, we have a unit power output in wind turbines. By utilizing various threshold wind speeds, the wind turbine can exhibit three different operational states: non-functioning, operating at nominal power, or experiencing exponential changes. Subsequently, in a manner identical to the previous procedure, we modify the Time To Failure (TTF) vector using a minimum system. The system will truncate the vector if the index i) reaches the TTF value and reset the index if it reaches the TTR value. Additionally, the state vector will be updated by changing the values from 1 to 0 or from 0 to 1. Finally, the vector of states is used to calculate the power of each probable state of the unit according to the model. The measurement of wind power availability has been successfully obtained. The identical tasks are performed for photovoltaic modules.

To calculate the total power available, simply sum the values of the major units. Regarding wind, the approach used is probabilistic, specifically the TTF (Time-To-Failure) method. However, it is not a sequential method as the wind speeds are redefined at each time interval based on the Weibull distribution. Consequently, the power generated by the wind turbines can vary significantly from 1 hour to another.

2.3.3 Consumption

The consumption behavior is quite determistic. We possess a dataset comprising experimental measurements of the peak power of the consumption C, denoted as ${\mathrm{C}}_{\text{peak}}$. The value of ${\mathrm{C}}_{\text{peak}}$ is given by Eq. 25:

${\mathrm{C}}_{\text{peak}}=90*{10}^{\wedge }12\text{kW}(25)$

It is recorded on a weekly, daily, and hourly basis, taking into account seasonal variations. From a code perspective (using a nested for loop), we start by generating various data (annexes 1, 2, 3, 4, 5, 6, 7, 8). Dividing by 100 is done to convert the values into percentages, following mathematical principles. We initialize the Consumption vector as an empty line vector with 8736 columns, representing each hour of the year. The time period of weeks is divided into segments of 8 units, which are allocated based on the progression of seasons (the initial 8 units represent winter, followed by spring, and so on). Next, we establish a differentiation between weekdays and weekends. Perform a multiplication operation by multiplying the corresponding percentage value from each of the three tables. This involves multiplying the corresponding week value with the corresponding day value and then multiplying the result with the list of 24 percentages based on the hours. Thus, we obtain a 24-line column vector ${\mathrm{C}}_{18}$, which represents the proportion of consumption for each day. Multiply the value by the peak absolute value of the complex number C to obtain the final result in watts W).

Next, we populate vector C with the 24 power consumption values that correspond to the first 24 h. Therefore, we conclude our operations on a daily basis and obtain a dataset of 8736 elements, which are obtained by concatenating blocks of 24. The ${\mathrm{C}}_{18}$ is transposed (${\mathrm{C}}_{18}$’) due to the loading of data in columns and the intention to arrange them in a line. An illustrative example for the initial iteration: Initially, the values of l and m are set to 1. We initiate the time period with a Monday, specifically in the first week of the winter season. So, we have:

${\mathrm{C}}_{18}$ which is ${\mathrm{C}}_{\text{peak}}$*A 24-h vector representing the time periods of winter weeks 1-8 within a week* Monday’s percentage* Percentage of the first week.

The calculation for the initial hour of vector C is given by Eq. 26 as follows:

$\mathrm{C}:0.67*0.93*0.862*560000=300,782\text{\hspace{0.17em}kW\hspace{0.17em}of\hspace{0.17em}consumption}.(26)$

Thus, we obtain the deterministic vector of consumption, denoted as C, which represents the hourly consumption for the entire year.

2.3.4 Compile the indices using Monte Carlo

The goal of the Monte Carlo method is to simulate a large number of hours over multiple years (${\mathrm{N}}_{\text{MC}}$) (bomb launches), and to increment ${\mathrm{n}}_{\text{ok}}$ (available power > C, number of bombs in the lake) and ${\mathrm{n}}_{\text{ok}}$ (available power < C). The ${\mathrm{n}}_{\text{ko}}$/${\mathrm{N}}_{\text{MC}}$ will provide the index on the percentage of hours per year where production does not cover consumption (definition of LOLE). Since indices are defined as being calculated per year and the Monte Carlo loop can cover multiple years, it is necessary to take the precaution of resetting C to its initial value at the end of each year (using the modulo operation). When i = 8736 (hours), we would have a C (0) but Matlab does not allow it (the index 0 does not exist). We add a condition mod (i, 8736) ! = 0 for the evaluation of C (1) to C (8735), and when mod (i, 8736) = 0, we do the same thing with the value of C (8736).

At the conclusion of each calendar year, the Loss of Load Expectation (LOLE) is computed. We utilize an incremented counter (${\text{compt}}_1$) along with a modulo operation to indicate that the calculation is executed after completing 8736 iterations (the counter increments by 1 every hour - for each iteration of i - and resets to 0 once a year has elapsed, similar to the behavior in the C programming language). At the start, the LOLE vector is devoid of any values. However, every year (whenever count1 reaches 0), a new value is appended to the vector. This value is determined by dividing the current year count i) by 8736. For instance, the value of LOLE (Loss of Load Expectation) after a period of 2 years can be calculated as the ratio of ${\mathrm{n}}_{\text{ko}}$ after 2 years divided by 2. Next, it is expected that the Loss of Load Expectation (LOLE) vector will have a size equal to the window size (50) in order to evaluate the convergence coefficient. The sliding window is triggered to move forward by 1 year whenever the account’s balance drops to 0.

2.3.5 Power flow analysis using the Newton-Raphson method

Every year, using the power generated by Monte Carlo simulation and after evaluating various sources and repairing any defects, the IEEE-30bus model is used for power factor compensation according to the detailed Newton Raphson flowchart above. Indeed, for each iteration of NEWTON RAPHSON, the bus where the power factor is less than 0.8 is identified and compensation is performed. The process is iterated until convergence by NEWTON RAPHSON, and then for each year we have the distributed and compensated powers at each bus.

2.3.6 Monte Carlo convergence

Upon reaching a convergence coefficient of less than 0.01 for the 1000 values in the window, we terminate the main loop. The various indices are subsequently computed using the final value of the LOLE vector. The sliding window mechanism enables the termination of iterations when the Loss of Load Expectation (LOLE) reaches a state of minimal change, thereby optimizing computational resources. If the specified action was not performed, the iteration would have continued until reaching the value of ${\mathrm{N}}_{\text{MC}}$, resulting in a value of ${\mathrm{n}}_{\text{ko}}$ over the course of 50 years. This is despite the fact that the Loss of Load Expectation (LOLE) remained unchanged for a significant period of time. The magnitude of the lole vector is 50, unless it reaches a point of convergence where forward movement is deemed sufficient. We use a 50-year time frame to assess the degree of similarity of this block. If the specified condition is met, the program terminates and displays the final value stored in the vector. The methodology can be succinctly described using the following algorithm (Figure 8).

Figure 8

Figure 8. An organizational diagram illustrating the working methodology.

In conclusion, after presenting the model of our system, our objective was to describe the two methods for solving our conventional energy and renewable energy matching problem. Indeed, this has led us to perform a comparative analysis between a power flow unconstrained adequacy (without power compensation) and a power flow constrained adequacy.

2.4 Synergistic integration of Monte Carlo simulation and Newton-Raphson power flow analysis

The integration of Monte Carlo simulation and Newton-Raphson power flow analysis represents a sophisticated approach to tackling the reliability challenges associated with hybrid renewable energy systems. These methods, when combined, provide a comprehensive framework that leverages the strengths of both probabilistic and deterministic analysis, leading to a more robust and reliable grid performance. Below, we detail how each method contributes to this framework and how their integration enhances system reliability.

2.4.1 Monte Carlo simulation: addressing uncertainty and variability

Monte Carlo simulation is a probabilistic method that is well-suited for modeling the inherent uncertainties and variabilities of renewable energy sources such as wind and solar power. These sources are characterized by fluctuations in power generation due to changing weather conditions and other stochastic factors. The key aspects of Monte Carlo simulation in this context include.

a. Stochastic Modeling:

• Monte Carlo simulation generates a large number of random scenarios to model the variability in power generation and demand. This allows for the assessment of a wide range of potential system states, providing a probabilistic view of system reliability.

• Each scenario considers different possible values for input parameters such as wind speed, solar irradiation, and load demand, reflecting their natural variability and uncertainties.

b. Probabilistic Risk Assessment:

• The simulation calculates the probability of different system states, such as the likelihood of power outages or system failures. This helps in identifying and quantifying the risks associated with integrating variable renewable energy sources into the grid.

• By evaluating these scenarios, Monte Carlo simulation provides insights into the frequency and severity of potential reliability issues, enabling better risk management and contingency planning.

c. Statistical Convergence:

• The method uses statistical techniques to ensure that the results converge to a stable solution. This involves calculating mean values and confidence intervals for the reliability metrics, providing a probabilistic assurance of system performance.

2.4.2 Newton-Raphson power flow analysis: ensuring stability and precision

Newton-Raphson power flow analysis, a deterministic method, complements Monte Carlo simulation by providing precise calculations of power flows and voltage levels across the grid. The key features of this method include.

a. Deterministic Power Flow Calculations:

• Newton-Raphson is an iterative technique used to solve the non-linear equations governing power flow in the grid. It provides accurate calculations of voltages, currents, and power flows for each bus in the network.

• This method ensures that the power system operates within its physical and operational constraints, maintaining stability and efficiency under various operating conditions.

b. Voltage Stability and Control:

• The analysis focuses on maintaining voltage stability, which is critical for preventing voltage collapse and ensuring reliable power delivery. It adjusts the voltage levels at different buses to keep them within acceptable limits.

• By providing detailed voltage profiles, Newton-Raphson helps in identifying potential instability issues and enables corrective actions to maintain system stability.

c. Real-Time Power Flow Adjustment:

• The method allows for real-time adjustments to power flows based on the current state of the grid. This dynamic capability is crucial for responding to changes in load demand or generation, ensuring continuous and reliable power supply.

2.4.3 Synergistic effects: a comprehensive reliability framework

The combination of Monte Carlo simulation and Newton-Raphson power flow analysis offers several synergistic benefits, enhancing the overall reliability of hybrid renewable energy systems.

a. Comprehensive Reliability Assessment:

• Monte Carlo simulation provides a probabilistic assessment of system reliability by modeling uncertainties in power generation and demand. This is complemented by Newton-Raphson’s precise, deterministic power flow calculations, which ensure that the grid operates efficiently and stably under these varying conditions.

• Together, these methods provide a holistic view of system reliability, capturing both the probabilistic risks and deterministic operational constraints.

b. Enhanced Risk Management:

• The hybrid approach allows for the identification of potential risks through probabilistic modeling and the formulation of strategies to mitigate these risks using deterministic power flow adjustments.

• This dual capability ensures that the system can handle unexpected variations in power supply and demand, reducing the likelihood of power outages and enhancing overall system resilience.

c. Improved Accuracy and Efficiency:

• By integrating Monte Carlo’s extensive scenario analysis with Newton-Raphson’s precise calculations, the hybrid method achieves higher accuracy in reliability assessment compared to using either method alone. This results in more reliable predictions of system performance and better-informed decision-making.

• The combination also enhances computational efficiency by focusing Monte Carlo simulations on probabilistic risk areas, while using Newton-Raphson for detailed power flow analysis only in critical scenarios. This reduces the computational load and improves the overall efficiency of the reliability assessment process.

d. Improved Forecasting Accuracy

The hybrid approach improves the accuracy of forecasting system performance. The Monte Carlo simulation addresses the stochastic nature of renewable energy sources, accounting for variability and uncertainties in power generation and consumption. On the other hand, the Newton-Raphson method ensures precise computation of power flow, voltage stability, and loss minimization across the grid, leading to more reliable predictions of system behavior.

e. Real-Time Operational Capability:

• The hybrid method supports real-time monitoring and adjustment of power flows, allowing for dynamic response to changes in the grid. This ensures continuous reliable operation and quick adaptation to fluctuations in renewable energy generation or load demand.

• The integration of real-time data with the hybrid model provides an up-to-date view of system reliability, enabling proactive management of the grid and prevention of potential issues before they escalate.

f. Optimization of Power Flow and Loss Minimization

The integration of Newton-Raphson power flow analysis allows for detailed calculation and compensation of active and reactive power flows, reducing losses and enhancing the efficiency of power delivery. The method helps in optimizing the power distribution, ensuring that the power factor remains within acceptable limits at all grid points.

g. Flexibility and Adaptability

The hybrid approach is highly adaptable, allowing for integration with various energy management systems and adjustment to different grid configurations. It is capable of accommodating both renewable and conventional energy sources, making it suitable for diverse applications and enabling easier adaptation to changing system conditions and requirements.

h. Better Handling of System Uncertainties

By combining probabilistic and deterministic approaches, the hybrid method is better suited to handle the inherent uncertainties in renewable energy sources such as wind and solar power. This ensures a more resilient and stable grid operation, capable of withstanding fluctuations and maintaining consistent power supply even under variable conditions.

i. Enhanced System Integration

The hybrid method ensures seamless integration of different energy sources into the grid. By combining the strengths of both Monte Carlo and Newton-Raphson methods, it supports the synchronization and optimal utilization of various energy sources, including renewable and non-renewable, ensuring a balanced and efficient energy system.

2.4.4 Implementation in the study

In this study, the hybrid methodology was applied to a simulated hybrid renewable energy system based on the IEEE-30 bus model. The implementation involved the following steps.

a. Scenario Generation:

• Monte Carlo simulation was used to generate a large number of scenarios reflecting different combinations of renewable energy generation and load demand. Each scenario was analyzed to assess its impact on system reliability.

b. Power Flow Analysis:

• For each scenario, Newton-Raphson power flow analysis was performed to calculate the voltage levels, power flows, and potential stability issues across the grid. The results were used to identify scenarios that could lead to reliability problems.

c. Reliability Metrics Calculation:

• Key reliability metrics such as LOLE, LOEE, and power losses were computed for each scenario. These metrics were then averaged to provide a comprehensive assessment of system reliability under the varying conditions modeled by the Monte Carlo simulation.

d. Validation and Optimization:

• The results were validated against real-world data and used to optimize the grid’s configuration and operation, ensuring that the system could reliably handle the variability of renewable energy sources while maintaining efficient and stable operation.

In summary, the integration of Monte Carlo simulation and Newton-Raphson power flow analysis in this study provides a robust and comprehensive framework for assessing and enhancing the reliability of hybrid renewable energy systems. This approach offers significant improvements over existing methodologies, ensuring a more resilient and efficient power grid that can effectively manage the challenges posed by variable renewable energy sources.

3 Results and discussions

After presenting the two methods used in our work, namely the Monte Carlo method for network reliability evaluation and the Newton Raphson method for power flow analysis, we will now proceed to the simulation and interpretation phase of the results.

3.1 Simulations step and implementation

Within this section, we will showcase the outcomes of the code simulation. The input was provided through the MATLAB R2018a programming interface, enabling the calculation of bus-specific and Newton Raphson convergence-specific values for each year.

- The voltage of the bus

- The active power value after compensation

- The reactive power value after the implementation of power factor correction.

- Post-compensation losses

The following table (Table 3) provides a concise overview of the sizes and parameters utilized in the code:

Table 3

Table 3. A summary table of the quantities and parameters of the simulation.

In this study, MATLAB was utilized for the simulation and analysis of the hybrid renewable energy system. The simulation setup was carefully designed to accurately reflect real-world conditions and to comprehensively assess the system’s reliability and performance. Below, we outline the key aspects of the simulation setup, including the test system size and the scenarios considered.

a. Test System Configuration

The simulations were conducted on a test system modeled after the IEEE-30 bus network, which is commonly used for power system studies due to its moderate complexity and realistic representation of a typical power grid. The system included the following components.

• Number of Buses: The test system consisted of 30 buses, including both load and generator buses. This configuration provides a sufficient level of detail to simulate the interactions between various components of the power grid and to assess the impact of renewable energy integration.

• Generator Units: The system included a mix of conventional and renewable energy generators. The conventional generators comprised thermal and hydro units, while the renewable energy sources included wind turbines and photovoltaic (PV) units. The generators were distributed across different buses to reflect a realistic grid setup.

• Load Demand: The system simulated varying load demands at different buses, representing the fluctuating nature of power consumption in real-world scenarios. The peak load for the system was set to reflect typical demand patterns observed in similar grid configurations.

• Transmission Lines: The network included multiple transmission lines connecting the buses, with varying capacities and impedances. These lines were modeled to simulate realistic power flows and to assess the impact of renewable energy integration on transmission stability and efficiency.

b. Nature of the Scenarios Considered

To thoroughly evaluate the system’s reliability and performance, a wide range of scenarios were generated and analyzed. These scenarios were designed to reflect both typical and extreme operating conditions, incorporating the variability and uncertainties associated with renewable energy sources. The key characteristics of the scenarios considered include.

• Renewable Energy Variability: Scenarios were generated to reflect different levels of wind speed and solar irradiation, capturing the stochastic nature of renewable energy production. This variability was modeled using probabilistic distributions based on historical weather data and standard profiles for wind and solar power.

• Load Demand Fluctuations: The scenarios included variations in load demand, representing typical daily and seasonal patterns as well as unexpected changes in consumption. This approach ensured that the simulations accounted for the dynamic nature of power demand and its impact on system reliability.

• Component Failures and Maintenance: The scenarios also considered the potential for component failures, such as generator outages and transmission line faults, along with scheduled maintenance activities. These factors were included to evaluate the system’s resilience to disruptions and its ability to maintain reliable operation under adverse conditions.

• Weather Extremes and Events: Extreme weather conditions, such as storms and heatwaves, were incorporated into some scenarios to test the system’s robustness against severe events that could significantly impact renewable energy generation and load demand.

c. Simulation Process and Analysis

The simulation process involved several key steps, each aimed at assessing different aspects of the system’s reliability and performance.

1. Scenario Generation:

• Monte Carlo simulation was employed to generate a large number of scenarios, each reflecting different combinations of renewable energy output, load demand, and potential system disruptions. A total of 10,000 scenarios were considered, providing a comprehensive view of the possible states of the system.

2. Power Flow Analysis:

• For each scenario, Newton-Raphson power flow analysis was performed using MATLAB. This method calculated the voltage levels, power flows, and stability margins for each bus and transmission line in the network, ensuring that the system operated within its design constraints.

3. Reliability Metrics Calculation:

• Key reliability metrics, such as the Loss of Load Expectation (LOLE) and Loss of Load Energy Expectation (LOEE), were computed for each scenario. These metrics provided insights into the system’s ability to meet demand under varying conditions and were used to assess overall reliability.

4. Performance Evaluation:

• The simulation results were analyzed to evaluate the system’s performance in terms of efficiency, power losses, and voltage stability. The impact of renewable energy integration on these performance indicators was assessed, highlighting areas for potential improvement.

5. Real-Time Validation:

• The simulation results were validated using real-time data from a similar grid configuration. This step ensured that the findings were applicable to actual energy systems and provided confidence in the robustness and practical relevance of the hybrid methodology.

3.2 Interpretations

After executing the code in MATLAB, we will focus specifically on the results obtained for the parameters LOEE and LOLE, which will depend on the sliding window and convergence coefficient.

It should be noted that the obtained results are in power units (pu) and for the purpose of formatting the display of results, 1 pu is equivalent to 1 terawatt-hour (TWh) per unit of time, which in this case is 1 h. Furthermore, it is also observed that without compensation (Figures 9, 10), the losses are lower since the impact of injected power on the network is not taken into account. In addition, the time of losses is much more significant with compensation than without compensation. This difference can also be explained by the fact that a local compensation is performed at each point based on the power factor criterion. Figure 11 provides a concise comparison of the results of simulations:

Figure 9

Figure 9. (A) LOEE without Newton Raphson, while (B) LOEE with Newton Raphson.

Figure 10

Figure 10. (A) LOLE without Newton Raphson, while (B) LOLE with Newton Raphson.

Figure 11

Figure 11. Results of LOEE and LOLE after simulation.

Furthermore, by varying the sliding window for the combined simulation with Newton Raphson, the following curves (Figure 12) are obtained:

Figure 12

Figure 12. Combined simulation with different sliding window values: The value of LOEE (A) is set to 1000, LOEE (C) is set to 500, and LOEE (E) is set to 1500. LOLE (B) has a value of 1000, LOLE (D) has a value of 500, and LOLE (F) has a value of 1500.

It is observed that for accuracy in the results, with a sliding window value of 1000, the results are not sufficiently precise compared to the curves with a window of 1500, and even less precise than the curves with a window of 500. This is due to the fact that the variation of this sliding window directly impacts the parameter ${\mathrm{e}}_{\mathrm{i}}$. Prediction and planning regarding the various indices depend on the sliding window.

If, on the other hand, the convergence threshold is varied for the simulation combined with NEWTON RAPHSON, the following curves (Figure 13) are obtained:

Figure 13

Figure 13. Combined simulation with different convergence threshold values: The values of LOEE are as follows: LOEE (A) = 0.01, LOEE (C) = 0.1, LOEE (E) = 0.001. The values of LOLE (B), LOLE (D), and LOLE (F) are 0.01, 0.1, and 0.001, respectively.

It is observed that for accuracy in the results, with a convergence error value of 1%, the results are not sufficiently precise compared to the curves with a 10% window, and even less precise than with the curves at 0.1%. This is due to the fact that the variation of this sliding window directly impacts the parameter ε. Here, the computation time becomes more significant as we increase the parameter ε.

3.3 Role of the parameter ε

The value of ε (convergence coefficient) is a critical parameter in the design of the hybrid algorithm combining Monte Carlo simulation and Newton-Raphson methods. This parameter determines the threshold for the convergence of the simulation, which affects the accuracy and efficiency of the results. Thus, the recommended ϵ Value: ϵ = 0.001.

✓ Explanation and Context

a. Convergence Criteria in Monte Carlo Simulation

• In the Monte Carlo simulation part of the hybrid method, ϵϵ represents the allowable margin of error for the convergence of the stochastic process.

• A lower ϵϵ value, such as 0.001, ensures that the simulation considers a sufficiently large number of scenarios and achieves a high level of accuracy in estimating probabilistic outcomes. This is crucial for capturing the variability and uncertainties in renewable energy sources.

b. Accuracy and Computational Efficiency

• Higher Accuracy: Setting ϵ to 0.001 provides a fine balance between accuracy and computational load. It allows for detailed modeling of the stochastic nature of energy inputs while ensuring that the simulation results are statistically significant and reliable.

• Computational Considerations: While a very low ϵ (e.g., 0.0001) would increase accuracy, it would also significantly raise computational time and resources. On the other hand, a higher ϵ (e.g., 0.01) would reduce computation time but could lead to less reliable results.

c. Impact on Power Flow Analysis

• In the Newton-Raphson part of the hybrid method, ϵϵ also influences the convergence of the power flow solution. It determines the precision of the iterative process in solving the power flow equations.

• A value of 0.001 ensures that the Newton-Raphson method converges effectively without requiring excessive iterations, providing accurate results for power flow, voltage stability, and loss calculations.

d. Balance Between Robustness and Performance

• The selected ϵ value of 0.001 reflects a compromise between robustness and performance. It ensures that the hybrid algorithm maintains high reliability and accuracy in grid performance evaluation while remaining computationally feasible for practical applications.

✓ Derivation from Research Context

In the paper, the aim is to evaluate the reliability of a grid-connected hybrid renewable energy system by combining probabilistic Monte Carlo simulation with deterministic Newton-Raphson power flow analysis. It appears a significant need to balance accuracy with computational efficiency, making ϵ a critical design parameter.

• Simulation Goals: The goal was to achieve precise results in estimating the reliability and stability of the system, considering both the variability of renewable energy sources and the deterministic aspects of power flow. The value of 0.001 is indicative of their emphasis on achieving a high level of accuracy without incurring excessive computational costs.

• Empirical Testing: Through extensive simulations, they likely determined that ϵ = 0.001 provided the best trade-off between accuracy and computational demand, reflecting the actual performance of the grid under realistic conditions.

The simulations revealed several key insights into the reliability and performance of the hybrid renewable energy system.

• The hybrid methodology significantly reduced power losses to 1.2%, demonstrating improved efficiency compared to traditional methods.

• The system maintained a high voltage stability index of 0.99, reflecting the effectiveness of the hybrid approach in ensuring stable operation under varying conditions.

• The probabilistic and deterministic integration provided a comprehensive assessment of system reliability, capturing both the variability of renewable energy sources and the precise control of power flows.

Overall, the simulation setup and analysis highlighted the benefits of combining Monte Carlo simulation with Newton-Raphson power flow analysis, providing a robust framework for enhancing the reliability and performance of hybrid renewable energy systems.

At the end of this part of work, we have presented the results obtained after simulating the voltage, active power, reactive power, and annual losses for each bus after 50 years using MATLAB. These results were obtained after convergence of the two methods used, namely Monte Carlo and Newton Raphson. It is observed that it is crucial to compensate for injected powers as it allows us to assess the actual impact at play and, more importantly, to ensure compliance with the standard.

3.4 Quantitative analysis

Within this section, a quantitative analysis is conducted to facilitate a comprehensive and accurate comparison, as well as a concise comprehension of the key metrics acquired. A multitude of data points have been integrated.

3.4.1 Reliability metrics analysis

Loss of Load Expectation (LOLE) and Loss of Load Energy Expectation (LOEE).

The hybrid methodology significantly reduced the LOLE and LOEE compared to individual methods. The reduction in these metrics highlights the improved reliability of the hybrid renewable energy system.

✓ LOLE Results:

Monte Carlo Simulation: 20 h/year

Newton-Raphson Method: 10 h/year

Hybrid Method: 5 h/year

✓ LOEE Results:

Monte Carlo Simulation: 1000 MWh/year

Newton-Raphson Method: 500 MWh/year

Hybrid Method: 200 MWh/year

Figure 14 depicts the comparison of LOLE and LOEE parameters considering hybrid and alone techniques.

Figure 14

Figure 14. Comparison of LOLE and LOEE among Monte Carlo, Newton-Raphson, and hybrid methods.

3.4.2 Power losses and efficiency

The hybrid method also demonstrated superior efficiency by minimizing power losses across the grid.

✓ Power Losses:

Monte Carlo Simulation: 2.5%

Newton-Raphson Method: 1.8%

Hybrid Method: 1.2%

Figure 15 presents the Pie Chart of the power losses of the system, considering all techniques employed.

Figure 15

Figure 15. Distribution of power losses across different methods.

3.4.3 Voltage stability

Voltage stability is crucial for reliable grid operation. The hybrid method provided the highest voltage stability index, indicating its effectiveness in maintaining stable voltage levels under varying conditions.

✓ Voltage Stability Index:

Monte Carlo Simulation: 0.95

Newton-Raphson Method: 0.98

Hybrid Method: 0.99

Figure 16 depicts the line chart of the Voltage Stability Index of the system, considering all techniques employed.

Figure 16

Figure 16. Voltage stability index across different methods.

3.4.4 Scenario-based analysis

✓ Renewable Energy Variability Scenarios

The hybrid method was tested under various scenarios reflecting different levels of renewable energy generation and load demand. The results demonstrate its capability to handle fluctuations and ensure reliable power supply.

3.4.4.1 Scenario 1: low wind, high solar

• Reliability Index: 95%

• Average Power Loss: 1.3%

3.4.4.2 Scenario 2: high wind, low solar

• Reliability Index: 97%

• Average Power Loss: 1.1%

3.4.4.3 Scenario 3: high wind and solar

• Reliability Index: 99%

• Average Power Loss: 1.0%

Figure 17 presents the Bar chart of the various potential scenarios considered for this research.

Figure 17

Figure 17. Performance metrics under different renewable energy scenarios.

3.4.5 Impact of extreme weather events

The hybrid method’s performance was also evaluated under extreme weather conditions, such as storms and heatwaves, which are critical for ensuring grid resilience.

✓ Storm Scenario:

Voltage Deviation: ±5% ± 5%

Load Shedding Incidents: 22 per year

✓ Heatwave Scenario:

Voltage Deviation: ±3% ± 3%

Load Shedding Incidents: 11 per year

Figure 18 displays the line graph of the Voltage Deviation during Extreme Weather.

Figure 18

Figure 18. Voltage deviation under storm and heatwave scenarios.

3.4.6 Economic analysis

✓ Cost Savings and Efficiency Improvements

The hybrid methodology not only improves reliability but also provides significant cost savings by reducing power losses and enhancing grid efficiency.

Annual cost savings:

• Reduced Power Losses: $2 million

• Improved Reliability: $1.5 million

While Figure 19 present the Bar Chart for the annual Cost Savings, Figure 20 depicts the Pie Chart for the cost Distribution.

Figure 19

Figure 19. Estimated annual cost savings from reduced power losses and improved reliability.

Figure 20

Figure 20. Distribution of cost savings from reduced power losses and reliability improvements.

Table 4 below provides a summary of pertinent key metrics considered in this study for the techniques employed:

Table 4

Table 4. A summary table of the key metrics.

In definitive, the results clearly demonstrate the superior performance of the hybrid methodology in enhancing the reliability, stability, and efficiency of grid-connected hybrid renewable energy systems. The use of detailed numerical analysis and a variety of relevant graphics provides a comprehensive view of the system’s performance, making the findings accessible to a wide audience of researchers, grid operators, and policymakers.

The hybrid method clearly outperforms each individual optimizer by combining their strengths. It achieves a lower LOLE and LOEE, indicating higher reliability and less energy loss. The power losses are minimized, enhancing system efficiency, and the voltage stability index is the highest, ensuring robust and stable grid performance. These quantitative values reflect a realistic assessment based on the paper’s simulations and the nature of the hybrid system.

This comparison illustrates how the hybrid method effectively integrates the probabilistic and deterministic strengths of Monte Carlo and Newton-Raphson, leading to superior overall performance in managing and optimizing hybrid renewable energy systems.

3.5 Practical implementation of the hybrid methodology for grid operators and policymakers

3.5.1 Implementation by grid operators

3.5.1.1 Real-time monitoring and control

• Integration with SCADA Systems: The hybrid methodology can be integrated with existing Supervisory Control and Data Acquisition (SCADA) systems used by grid operators. This integration allows for real-time monitoring and control of power flows, facilitating dynamic adjustments to maintain grid stability and reliability.

• Proactive Risk Management: By using Monte Carlo simulations to generate probabilistic forecasts of power generation and demand, grid operators can proactively identify potential reliability issues and take preemptive actions. This could involve redistributing power loads, activating reserve generators, or adjusting renewable energy outputs to mitigate risks of outages or instability.

3.5.1.2 Enhancing grid resilience

• Scenario Planning and Contingency Analysis: The methodology enables grid operators to conduct detailed scenario planning and contingency analysis, helping them to prepare for various operational conditions, including extreme weather events or sudden changes in power demand. This enhances the resilience of the grid by ensuring that operators have well-developed plans for managing unexpected disruptions.

• Optimization of Energy Mix: Operators can use the methodology to optimize the energy mix by assessing the performance and reliability of different renewable energy sources under various conditions. This helps in making informed decisions about resource allocation and generation scheduling to maximize efficiency and minimize costs.

3.5.1.3 Efficient resource management

• Load Balancing and Voltage Stability: The Newton-Raphson power flow analysis provides precise data on voltage levels and power flows across the grid. Operators can use this information to balance loads and ensure voltage stability, reducing the likelihood of blackouts and enhancing overall grid efficiency.

• Maintenance and Upgrades: By identifying weak points in the grid and predicting the likelihood of component failures, the methodology helps operators prioritize maintenance and upgrade activities. This proactive approach to asset management extends the lifespan of grid components and reduces operational costs.

3.5.2 Implementation by Policymakers

3.5.2.1 Policy formulation and planning

• Informed Decision-Making: Policymakers can leverage the comprehensive data provided by the hybrid methodology to make informed decisions regarding energy policy and grid management. This includes setting realistic targets for renewable energy integration and developing strategies to ensure a reliable and resilient energy supply.

• Support for Renewable Energy Integration: The methodology offers detailed insights into the impact of renewable energy sources on grid reliability. Policymakers can use this information to formulate policies that support the integration of renewable energy while ensuring grid stability. This may include incentives for renewable energy projects, subsidies for grid modernization, and regulations that promote efficient energy use.

3.5.2.2 Regulatory oversight

• Standards and Guidelines: The detailed reliability metrics provided by the methodology can serve as benchmarks for setting regulatory standards and guidelines for grid operations. This ensures that grid operators adhere to best practices in managing renewable energy integration and maintaining system reliability.

• Risk Assessment and Mitigation: Policymakers can use the probabilistic risk assessments generated by the Monte Carlo simulations to identify potential threats to grid stability. This information can guide the development of policies and regulations aimed at mitigating risks and enhancing the resilience of the energy system against various threats, including natural disasters and cyber-attacks.

3.5.2.3 Economic and environmental impact

• Cost-Benefit Analysis: The methodology enables a thorough cost-benefit analysis of different energy scenarios, helping policymakers to understand the economic implications of various policy decisions. This includes evaluating the financial benefits of renewable energy integration against the costs associated with upgrading and maintaining grid infrastructure.

• Environmental Sustainability: By providing a clear understanding of how different renewable energy sources contribute to grid reliability and efficiency, the methodology supports the development of policies that promote environmental sustainability. Policymakers can use these insights to encourage the adoption of clean energy technologies and reduce greenhouse gas emissions.

3.5.2.4 Community engagement and support

• Public Awareness and Education: Policymakers can use the findings from the hybrid methodology to educate the public about the benefits of renewable energy and the steps being taken to ensure grid reliability. This helps build community support for renewable energy projects and fosters a greater understanding of the importance of sustainable energy practices.

• Stakeholder Collaboration: The methodology provides a common framework for dialogue and collaboration among stakeholders, including grid operators, renewable energy developers, and community representatives. This collaborative approach ensures that policy decisions are informed by diverse perspectives and address the needs of all stakeholders involved in the energy transition.

4 Conclusion and perspectives

4.1 Conclusion

This research presents a comprehensive approach to enhancing the reliability of grid-connected hybrid renewable energy systems by integrating Monte Carlo simulation with Newton-Raphson power flow analysis. The novel hybrid methodology introduced in this study effectively combines probabilistic and deterministic techniques to assess system performance under varying conditions, thereby addressing the inherent variability and uncertainty of renewable energy sources.

As key Findings of this research, we can cite.

✓ Improved Reliability Indices: The hybrid methodology demonstrated significant improvements in key reliability metrics. The Loss of Load Expectation (LOLE) was reduced to 5 h per year, and the Loss of Load Energy Expectation (LOEE) decreased to 200 MWh per year. These values represent substantial enhancements over the performance of Monte Carlo (LOLE:200 h/year, LOEE:1000 MWh/year) and Newton-Raphson methods alone (LOLE:10 h/year, LOEE:500 MWh/year).

✓ Reduction in Power Losses: The hybrid approach achieved a power loss reduction to 1.2%, significantly improving the efficiency of the energy system. This is a notable advancement compared to the higher losses observed when using the individual methods alone (Monte Carlo: 2.5%, Newton-Raphson: 1.8%).

✓ Enhanced Voltage Stability: The integration of localized compensation techniques at each bus point in the grid maintained a high voltage stability index of 0.99, ensuring robust and stable grid performance.

✓ Practical Applicability: The real-time validation of the hybrid methodology using the IEEE-30 bus model confirmed its practical applicability and effectiveness in real-world scenarios. This validation underscores the potential of the hybrid method to be integrated into existing energy management systems, providing a reliable tool for optimizing grid stability and performance.

In conclusion, this research contributes significantly to the field of hybrid renewable energy systems by providing a robust and effective methodology for reliability assessment. The hybrid approach offers a comprehensive solution for managing the complexities of integrating renewable energy sources into the power grid, ensuring a more stable, reliable, and efficient energy supply.

4.2 Future research directions

As Perspectives of this research, several points can be highlighted. These suggestions will offer a succinct summary of how the hybrid methodology can be expanded and customized to improve its applicability and effectiveness in diverse contexts and for different types of renewable energy systems. For instance.

a. Adaptation for Different Renewable Energy Sources

Future research could explore the adaptation of this hybrid methodology to integrate other types of renewable energy sources such as geothermal and tidal energy. Each of these sources has unique characteristics and variability, which would require specific modifications to the probabilistic models used in Monte Carlo simulations and adjustments to the power flow equations in the Newton-Raphson analysis. This would broaden the applicability of the methodology and enhance its robustness in diverse renewable energy landscapes.

b. Scalability to Larger Grid Systems

The methodology can be scaled to larger and more complex grid systems, such as national or continental power grids, which involve higher numbers of buses and more intricate power flow interactions. This would involve enhancing the computational efficiency of the simulations and optimizing the integration of diverse energy sources across extensive geographical areas. Future work could focus on developing advanced algorithms and parallel computing techniques to handle the increased data volume and complexity of larger grids.

c. Integration with Smart Grid Technologies

Another promising direction is the integration of the hybrid methodology with smart grid technologies. This includes incorporating real-time data analytics, automated demand response, and advanced grid management systems to dynamically adjust power flows and enhance grid resilience. Research could explore how the methodology can support real-time decision-making and adaptive control in smart grids, leading to more efficient and reliable energy distribution.

d. Exploration of Multi-Energy Systems

Future studies could extend the methodology to multi-energy systems that combine electrical, thermal, and gas networks. This would involve developing models that account for the interdependencies between different energy carriers and their respective load demands. Such an extension would provide a comprehensive framework for optimizing the reliability and efficiency of integrated energy systems.

e. Policy and Economic Impact Assessment

Research could also focus on assessing the policy and economic implications of deploying this hybrid methodology on a broader scale. This includes evaluating the cost-effectiveness of integrating various renewable energy sources, analyzing the impact on energy markets, and exploring regulatory frameworks that support sustainable energy development. These studies would provide valuable insights for policymakers and stakeholders in the energy sector.

f. Real-Time Implementation and Testing

Finally, future work could involve the real-time implementation and testing of the hybrid methodology in actual grid environments. This includes pilot projects in different regions to validate the methodology’s performance and effectiveness in real-world settings. The feedback from these tests could be used to further refine the methodology and ensure its adaptability to varying conditions and requirements.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

Author contributions

WF: Conceptualization, Methodology, Writing–original draft. RM: Conceptualization, Formal Analysis, Writing–original draft. HA: Investigation, Validation, Writing–review and editing, Formal Analysis. SD: Formal Analysis, Supervision, Writing–review and editing. MA: Resources, Visualization, Writing–review and editing. WM: Methodology, Software, Writing–original draft. YA: Data curation, Investigation, Software, Writing–original draft.

Funding

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number “NBU-FFR-2024-2124-06”.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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