1 Introduction

In recent decades, the massive implementation of Artificial Intelligence (AI) strategies into electrical energy field provided effective solutions for several obstacles associated with the efficient operation of electrical networks (Mellit and Kalogirou 2021; Veisi et al. 2022). Economic dispatching, reliability guarantee, quality improvement and infrastructure strengthen of energy networks are the main domains of the integration of AI methodologies into the electrical issues (Abdalla et al. 2021). In these domains, the main roles and challenges of AI strategies considering the energy systems can be highlighted in Table 1 regarding the previous scientific contributions.

Table 1 Effective contributions and future challenges of AI strategies within energy systems
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Considering the major covered aspects in Table 1, the AI strategies represent the magic wand for supporting the simulation studies to improve the reliable quality of energy systems particularly the operation of RESs.

Since the emergence of AI techniques on the scientific research arena, researchers are still competing in an open challenge to develop the different AI branches for obtaining optimal solutions of several case studies. Regarding energy issues, the AI branches are categorized based on the applied-operational methodology into three major types; Artificial Neural Networks (ANNs), Adaptive Neural Fuzzy-Logic Networks (ANFLNs) and computerized optimization algorithms (Abdalla et al. 2021). ANN is a flexible-domain methodology that is inspired from superior operational features and merits of the human brain (Jain et al. 1996; Ghadami et al. 2021; Bienvenido-Huertas et al. 2020). The detailed overview of ANN features, models, applications, and obstacles are expressed in the following sections. ANFLN is a well-known dynamic methodology flourished in the middle of the nineteenth century as a powerful controller to treat with complicated processes that are characterized by little bit mathematical and physical definitions (Lee 1990). As a powerful digital-computerized method, ANFLN is an effective methodology to be massively implemented in electrical issues such as modeling and simplification of electrical networks as will be expressed in the following sections (Fattahi et al. 2016). The prominent boarding of AI methods reached its climax point with the integration of computerized-optimization approaches (Ibrahim et al. 2022). This hybrid combination represents an iconic alternative for analytical methods in solving complex non-linear problems with open and bounded constraints as will be mentioned later (Yang 2020).

Comprehensively, Table 2 expands in highlighting some examples of previous contributions to show the roles of AI branches in the applied studies within energy systems considering the merits and demerits of each contribution.

Table 2 Samples of previous contributions showing effect of AI branches within energy systems
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There are more and more research examples that are strongly similar in terms of idea as well as motivations whereas different in the proposed methodology to achieve this idea (Garud et al. 2021; Mas’ud et al. 1060; Farah et al. 2020).

Regarding the large expansion of the infrastructure of the energy systems as one of the most important polarizing platforms of many electrical issues and phenomena, the degree of complexity in construction and operation has increased affecting directly the quality of the whole network. These obstacles can be summarized into the following points (Ahmad et al. 2020):

  • Complicated modeling of network stages leads to massive increase in operations.

  • The increase in malfunctions due to intensified magnetic and electrical fields.

  • The difficulty to monitor and control the several operational processes.

  • The remarkable increase in the overall running cost.

  • The difficulty of conducting empirical studies that are related to different stages of the energy networks in case of extreme modes of High Voltage (HV), Extra High Voltage (EHV) and Ultra-High Voltage (UHV) transmission networks.

  • Operational obstacles of simulation studies using traditional numerical approaches.

  • The tuning errors from the operation of electrical or electronic devices and topologies embedded in the electrical networks for quality control.

Due to the massive bifurcation of conducted studies for these obstacles and the corresponding optimal solutions, it would be difficult to broach such a large number of studies in only one research (Ibrahim et al. 2022). So, there will be a focus shot on the role of numerical simulation modeling and their computational methods in solving the obstacles of energy systems of several domains.

In the past, analytical approaches and their corresponding computational techniques provided effective solutions for several obstacles of electrical networks and issues. As a simple method in processing and evaluation objectives, the functions of analytical methods and their computational techniques can be expressed as follows (Girdinio et al. 1988):

  • Constructing a well-defined embodied model to imitate simple electrical issues.

  • Expressing the constructed models into a meaningful mathematical representation in a simplified form based on the nature of the proposed electrical model or issue.

  • Solving the provided mathematical models using the available conventional analytical techniques to validate the effectiveness of the proposed electrical model or issue.

In recent decades, researchers knocked the door of numerical methods to take the lead at the expense of the analytical methods in solving the obstacles of electrical issues. This preference is due to the difficulty in obtaining analytical solutions for complex mathematical models using physical approaches and computerizing the graphical mapping for electrical parameters obtained from analogue modeling techniques. Regarding the need of powerful computing analysis, numerical techniques practically become the most effective choice to compute the complicated operations of different parameters of electrical domains (Elayyan and Abderrazzaq 2005). Table 3 reviews the major previous contributions of numerical methodologies within energy systems.

Table 3 Samples of contributions showing effect of numerical methods within energy systems
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There are more research examples highlighting the roles of numerical methods within energy systems that are strongly similar in terms of idea and motivated contributions (Chang 2011).

Simulation model is considered as the magic tool to describe the features of electrical issues or phenomena in order to avoid the complexity of assembling experimental components, the highly cost of conducting empirical models with expected dangers (Talaat et al. 2019). These simulation models contribute alongside to conducted experimental studies in enriching the discussion of electrical scientific topics and provide a possible domain to simulate the complex practical electrical issues. As a natural result, these unique features of simulation models paved the way to the appearance of numerical software methods to provide accurate, safe and precise evaluation of different parameters within the integrated energy generation system (Qi et al. 2018).

Numerical analysis is considered as a fertile field for computational methodologies to obtain approximate solutions for complex mathematical problems. Also, it enables the simulation studies to be more effective than the traditional methods in extracting the best solutions in case of complicated electrical issues. Until the fifties of the twentieth century, Numerical methods depended only on manual calculations with a complete absence of the technological analysis. With the development of information revolution and technological advances in the second half of twentieth century, the implementation of the numerical analysis is enriched by computing machine techniques making the analysis fast and economic. In this research, the light will be focused on the classification of numerical methods for the simulation modeling of electrical energy domains. This classification is based on two common factors; the type of solved mathematical equations and the applied constraints. Numerical methods are strongly concerned with the mathematical models of ordinary Partial Differential Equations (PDEs) besides the Integral Equations (IEs) (Ciarlet et al. 1990).

In this context, numerical methods for simulation modeling of electrical energy domains are divided into two major categories. One of them is an open domain method with no limiting boundaries and is applied to solve mathematical models of IEs. The other one of constrained domain that is applied to solve proposed mathematical models of PDEs (Rabah et al. 2017). Based on the nature of the proposed issues to be numerically simulated within the integrated energy generation systems, it is very important to choose the convenient numerical method to be utilized in each study considering the computational requirements of simple representation of complex boundaries in addition to accurate and free-susceptible computation.

The firm relation between AI methodologies and the quality control of electrical networks were rooted many years ago thanks to the massive obstacles that were overcome efficiently. The numerical simulation methods form with AI methodologies a powerful tool to obtain the optimal approximate solutions for the most complicated problems in the electrical energy networks either in the presence of any type of constraints or not. In this research, a comprehensive study is provided to show the effective integration of AI approaches in the electrical energy domain with different simulated study issues to overcome the operational problems of different applied numerical methods for efficient quality enhancement (Lee et al. 2022).

In this research, it is worth to be mentioned that the concept of integrated energy systems is related to three common approaches. The first one is to combine different energy generation sources of uneven operational features in one hybrid model. The second one is to combine different controlling strategies with the proposed model of one energy generation system to improve its operation. The third one is to combine several improvement strategies with different energy sources to enhance the reliability of generation. The complexity of integrated energy systems is attributed to high cost of conducting an experimental integration of large-scaled system components, the long processing time of the utilized controlling strategies, the difficulty to simulate the complicated mathematical models considering the applied boundaries and the transformation of digital controlling actions to analogue ones considering the proposed system. Regarding the complicated configuration of integrated energy systems, the provided practical examples in this paper highlight the difference between the two topologies of conventional integrated energy systems and the complex integrated ones.

The main structural framework of this research is divided into four levels as shown in Fig. 1. The fundamental level focuses on reviewing innovatively the operational and mathematical features of the numerical simulation methods within energy domains. The next level discusses the effective role of integrating the AI strategies with the numerical simulation methods to overcome and optimize their obstacles within energy issues. The third level reviews the practical implementation of the combination of AI approaches with numerical simulation methods in several energy issues to show the effectiveness of this hybrid integration highlighting the experimental and simulation studies of energy systems. The final level extracts the concluded points from the provided studies to make innovative discussions highlighting the importance of these studies. All discussions will be organized to pave the way for the upcoming researches to effectively continue the studies in the future.

Fig. 1
figure 1

Structural framework of the proposed studies in this research

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2 Impact of the numerical methodologies within the complicated electrical domains

2.1 Operational features of numerical approaches over analytical ones

Although the simple operation and fast response of the conventional analytical approaches regarding simple structured electrical systems, they are susceptible to several operational errors collapsing their role in coping with the complicated electrical systems. The operational obstacles within complicated electrical domains are expressed in Fig. 2 (Han and Liu 2017).

Fig. 2
figure 2

Major categories of impediments within complicated large-scale electrical systems

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Due to the mentioned obstacles in Fig. 2, the numerical approaches are expressed as an iconic alternatives for analyzing the multi-objective processes within the complicated electrical systems (Sevgi 2003). The difficulty in evaluating the values of electromagnetic fields is the most effective reason for the rise of numerical methods on scientific research domain regarding the large-scaled electrical systems. For highlighting the computational impact of numerical methods over traditional ones, Table 4 reviews the major discriminations among them (Sevgi 2003).

Table 4 Analytical methods vs. numerical methods for the simulation of electrical domains
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Schematically, the operational methodology of each category may support the comparison to be more comprehensive. Figure 3 illustrates significantly the process of computation for each category regarding the defined aspects in Table 4 (Sevgi 2003).

Fig. 3
figure 3

Computational process flowchart; a analytical method, b numerical method (Sevgi 2003)

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Regarding Fig. 3, there are some explanations to be comprehensively discussed as follows:

  • The operation of the numerical approaches is subjected to two control signals that increase the computational accuracy of these methods over analytical ones regarding the large-sized electrical systems.

  • The mapping criterion of the computational domain selection of the numerical method is based on the mathematical representation of the proposed simulation model of the defined system parameters. Manually or using a programmed software, the defined equations of the proposed simulation model either PDEs or IEs and methodologies either iterative or mesh in addition to the electromagnetic domain are to be matched to the suitable computational domain in the side of the utilized numerical method.

  • The utilized platform of software program should be a powerful tool to apply the desired operational or controlling tasks regarding the defined system scale.

2.2 Computational domains of numerical approaches

In this context, a comprehensive study for each type of computational domains of numerical approaches is expressed considering the mathematical representation of each domain:

2.2.1 Maxwell’s equations

Maxwell’s equations are the base structure of the modeling and computation processes of the electromagnetic fields. Maxwell’s theorem is mainly subjected to the Curl and Divergence theorems as follows (Jin 2011; Baek et al. 2015):

$$\xrightarrow{{{\text{Divergence Laws}}}} \;\left\{ {\begin{array}{*{20}l} {\nabla \cdot {\text{B}} = \nabla \cdot \mu {\text{H}} = 0\;(Magnetic\;Concept)} \hfill \\ {\nabla \cdot {\text{D}} = \nabla \cdot \varepsilon {\text{E}} = \rho _{v} \;(Electrical\;Concept)} \hfill \\ \end{array} } \right.$$
(1)
$$\xrightarrow{\text{Curl Laws}} \left\{ {\begin{array}{*{20}{l}} {\nabla \times {\text {H}} = \frac{{\partial {\text{D}}}}{{\partial {\text{t}}}} + {\text {J}}(Magnetic\,Concept)}\\ {\nabla \times {\text {E}} = - \frac{{\partial {\text{B}}}}{{\partial {\text{t}}}}\,({{Electrical}}\;{{Concept}})} \end{array}} \right.$$
(2)

For conducting medium with movable charges, the previous equations are supported with some formulas to express the features of these moving charges as follows (Jin 2011; Baek et al. 2015; Guru and Hiziroglu 2009; Hildebrand 2012; Stakgold and Holst 2011):

$$\xrightarrow{{\text{Charge}}\;{\text{Conservation}}\;{\text{and}}\;{\text{Current}}\;{\text{Continuity}}\;{\text{Laws}}} \left\{ {\begin{array}{*{20}{l}} {\nabla \cdot {\text {J}} + \frac{{\partial {\rho _v}}}{{\partial {\text{t}}}} = 0}\\ {{\text {J}} = \sigma {\text {E}}}\\ {{\text {E}} = {v_d} \times {\text {B}}} \end{array}} \right.$$
(3)

For dielectric medium, the Maxwell’s equations are to be formulated to fit the features of the polarization and magnetization concepts as follows:

$$\xrightarrow{{\text{Divergence}}\;{\text{Laws}}} \left\{ {\begin{array}{*{20}{l}} {\nabla \cdot {\text {B}} = \nabla \cdot \mu {\text{H}} = 0(Magnetic\;Concept)}\\ {\nabla \cdot {\text {E}} = \frac{1}{\varepsilon }\left[ {{\rho _v} - \left( {\nabla \cdot {\text{P}}} \right)} \right]\,(Electrical\;Concept)} \end{array}} \right.$$
(4)
$$\xrightarrow{{\text{Curl}}\;{\text{Laws}}} \left\{ {\begin{array}{*{20}{l}} {\nabla \times {\text {B}} = \mu \left[ {\left( {\varepsilon \frac{{\partial {\text{D}}}}{{\partial {\text{t}}}}} \right) + {\text{J}} + \frac{{\partial {\text{P}}}}{{\partial {\text{t}}}} + \left( {\nabla \times {\text{M}}} \right)} \right]}&{(Magnetic\;Concept)}\\ {\nabla \times {\text {E}} = - \frac{{\partial {\text{B}}}}{{\partial {\text{t}}}}}&{(Electrical\;Concept)} \end{array}} \right.$$
(5)

2.2.2 Green’s theorem

This theorem provides an iconic solution for electrical issues with complex IEs converting them into simple linear equations. This theorem depends mainly on the position; surface and physical concept of bounded medium of the simulated electrical issue (Guru and Hiziroglu 2009). Based on Divergence theorem, the general form of Green’s function is expressed as follows:

$${\int _{{\text{Region}}\;\varphi }}\left( {\nabla \cdot {\text{Z}}} \right){\text{d}}\varphi = {\oint _{{\text{Surface}}\;\vartheta \;{\text{of}}\;{\text{Region}}\;\varphi }}{\text{Z}} \cdot {\text{d}}\vartheta$$
(6)

This general form is derived in such manners based on physical structural definition into:

2.2.2.1 Charge simulation concept

Green’s function is used to simplify the complex integral form of evaluating the electric potential at different positions of the fictitious simulation charges on the surface of a given bounded region as follows (Zhou 2012):

$$\lambda ( {\text {r}}) = \frac{1}{{4\pi \varepsilon }}{\int _{{\text{Region}}\;\varphi }}\frac{{{\rho _v}( {\text{r}^\prime} )}}{{\left| {{\text {r}} - {{\text {r}}^{\prime}}} \right|}}d\varphi + \frac{1}{{4\pi }}{\oint _{{\text{Surface}}\;\vartheta }}\left[ {\frac{1}{{\left| {{\text {r}} - {{\text {r}}^{\prime}}} \right|}}\frac{{\partial \lambda }}{{\partial n}} - \lambda \frac{\partial }{{\partial n}}\left( {\frac{1}{{\left| {{\text {r}} - {{\text{ r}}^{\prime}}} \right|}}} \right)} \right]{\text{d}}\vartheta$$
(7)
2.2.2.2 Mathematical forms of different concepts of boundary conditions

Furthermore, Gree’s function simplifies the complex non-homogenous integral forms of different boundary conditions regarding the Poisson’s boundary condition as follows (Zhou 2012):

Laplacian boundary condition

$$A( {\text {r}}) = {\int _{{\mathop{\text {Re}}\nolimits} {\text{gion}}\;\;\varphi }}{\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}} ){\text {f}}( {{{\text {r}}^{\prime}}} ){\text {d}}\varphi + {\oint _{{\text{Surface\;}}\vartheta }}\left[ { - A({\text {r}} )\frac{{\partial {\text{ K}}( {{\text {r}},{{\text {r}}^{\prime}}})}}{{\partial n}} + {\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}})\frac{{\partial A}}{{\partial n}}} \right]{\text {d}}\vartheta$$
(8)

Dirichlet boundary condition

$$A( {\text {r}}) = {\int _{{\mathop{\text {R}e}\nolimits} {\text{gion}}\;\varphi }}{\text{K}}( {{\text {r}},{{\text {r}}^{\prime}}} ){\text {f}}( {{{\text {r}}^{\prime}}} ){\text {d}}\varphi - {\oint _{{\text{Surface}}\;\vartheta }}A( {\text {r}})\frac{{\partial {\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}})}}{{\partial n}}{\text {d}}\vartheta$$
(9)

Neumann boundary condition

$$A( {\text {r}}) = {\int _{{\mathop{\text {R}}\nolimits} {\text{egion}}\;\varphi }}{\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}} ){\text {f}}( {{{\text {r}}^{\prime}}} ){\text {d}}\varphi + {\oint _{{\text{Surface}}\;\vartheta }}{\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}})\frac{{\partial A}}{{\partial n}}{\text {d}}\vartheta$$
(10)

Boundary condition of robin problems

$$A( {\text {r}}) = {\int _{{\text{Region}}\;\varphi }}{\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}})f( {{{\text {r}}^{\prime}}}){\text {d}}\varphi + {\oint _{{\text{Surface}}\;\vartheta }}\frac{{{f_2}( {\text {r}})}}{{{f_1}}}\frac{{\partial {\text {K}}( {{\text {r}},{{\text {r}}^{\prime}}})}}{{\partial n}}{\text {d}}\vartheta$$
(11)

Boundary condition of Poisson’s equation

$$K( {{\text{ r}},{{\text {r}}^{\prime}}}) = \frac{1}{{4\pi \varepsilon }}\;\frac{1}{{\left| {{\text {r}} - {{\text {r}}^{\prime}}} \right|}}$$
(12)

2.2.3 Huygen’s theorem

Huygen’s theorem is related to the modeling of wave equations. This theorem depends on the iterative methodology. The mathematical representation of this theorem is divided into three forms based on the complexity of simulated model as follows (Guru and Hiziroglu 2009; Zhenxiu 2003):

$$\xrightarrow{\text{First Form}}\; \left\{ {\begin{array}{*{20}{l}} {\Psi ( {{\text{r}^\prime}} ) = {\oint _{Surface\;\vartheta }}\hat {\text {n}} \cdot \left[ {G( {{\text {r}},{\text{r}^\prime}} )\nabla \Psi ( {\text {r}}) - \Psi ( {\text {r}} )\nabla {\text{G}}( {{\text {r}},{\text{r}^\prime}})} \right]{\text {d}}\vartheta }&{{\text{if}}}&{\;{\text{r}^\prime} \in {\cal V}}\\ {{\oint _{Surface\;\vartheta }}\hat {\text {n}} \cdot \left[ {{\text {G}}( {{\text {r}},{\text{r}^\prime}})\nabla \Psi ( {\text {r}}) - \Psi ( {\text {r}})\nabla {\text {G}}( {{\text {r}},{\text{r}^\prime}})} \right]{\text {d}}\vartheta = 0\;\;}&{{\text{if}}}&{{\text{r}^\prime} \notin V} \end{array}} \right.$$
(13)
$$\xrightarrow{{\text{Second}}\;{\text{Form}}} \begin{array}{*{20}{l}} {\Psi ({\text{r}^\prime}) = - {\oint _{{\text{Surface}}\;\vartheta }}\left[ {\Psi (r)\hat {\text {n}} \cdot \nabla {\text{G}}({\text{r}},{\text{r}^\prime})} \right]{\text{d}}\vartheta }&{\;{\text{r}^\prime} \in {\cal V}} \end{array}$$
(14)
$$\xrightarrow{{\text{Third}}\;{\text{Form}}} \begin{array}{*{20}{l}} {\Psi ({\text{r}^\prime}) = - {\oint _{{\text{Surface}}\;\vartheta }}\left[ {{\text{G}}( {\text {r}}, {\text{r}^\prime} ){{\hat {\text {n}}}} \cdot \nabla \Psi ({\text{r}})} \right]{\text{d}}\vartheta }&{\;{\text{r}^\prime} \in {\cal V}} \end{array}$$
(15)

2.2.4 Variational theorems

These theorems are based on dividing the solution domain into small regional elements to facilitate the computation process. The desired solution of the whole complicated system can be obtained by assembling all equations of the divided sub-sectional elements. The methodology of these theorems is based on the following equations (Zienkiewicz et al. 2005):

$$\xrightarrow{{\text{General}}\;{\text{Form}}} \;\zeta = \int_{{\text{Region}}\;\varphi } {\xi \left( {{\text{c}},\frac{{{\text{dc}}}}{{{\text{dx}}}}, \ldots } \right){\text{dx}} + {{\left. {\varrho \left( {{\text{c}},\frac{{{\text{dc}}}}{{{\text{dx}}}}, \ldots } \right)} \right|}_\varpi }}$$
(16)
$$\xrightarrow{{\text{General}}\;{\text{Form}}\;{\text{with}}\;{\text{Higher}} - {\text{Order}}\;{\text{Elements}}} \;{{\text{c}}^{\text{e}}} \approx {{{{\hat {\text c}}}}^{\text{e}}} = \sum\nolimits_{{\text{b}} = 1}^{\text{m}} {{N_b}} {{\tilde c}_b}$$
(17)
$$\begin{array}{l}\xrightarrow{{\text{Derived}}\;{\text{Equation}}\;{\text{of}}\;{\text{One}}\;{\text{Regional}}\;{\text{lement}}\;{\text{Regarding}}\;{\text{Higher}} - {\text{Orders}}} \zeta = - 0.5{{{{\tilde {\text {c}}}}}^{\text{T}}}a\tilde c\\ \end{array}$$
(18)

2.2.5 Parabolic equations

These equations are based on the solution of PDEs of wave propagation of electromagnetic fields. The general form of this equation in Cartesian coordinates is expressed as (Gao et al. 2019):

$$\frac{{\partial {\text{S}}( {{\text{x,z}}} )}}{{\partial {\text{x}}}} = \frac{{{\text{j}}\Pi }}{2}\;\left[ {\frac{1}{{{\Pi ^2}}}\frac{{{\partial ^2}}}{{\partial {{\text{z}}^2}}} + {\chi ^2}( {{\text{x,z}}}) - 1} \right]{\text{S}}( {{\text{x,z}}})$$
(19)

3 Numerical simulation modeling of complex integrated energy systems

3.1 Operational methodology

Modeling of complex electrical issues is one of the most pioneering trends in the research arena. The modeling simulation is scientifically related to the mathematical embodiment and imitated implementation of any realistic problem. Imitating the dynamic performance of electrical problems requires an effective computerized-numerical methodology to apply the computational simulation (Sevgi 2003; Papas 2014). Schematically, the operational methodology of numerical simulation method is classified into six stages as shown in Fig. 4 (Han and Liu 2017; Sevgi 2003).

Fig. 4
figure 4

Methodology flowchart of numerical simulation modeling of a complicated problem

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Regarding Fig. 4, all the operational stages are programmed in a coded library within a computational platform of a software program such as the embedded library of the Finite Element Method (FEM) in the platform of COMSOL Multiphysics program. Based on this programmable code, data actions or questions in the methodology flowchart are exposed to saved scenarios and operational steps that control the orientation of answer according to the matching criteria with the embedded studies regarding the control actions. Furthermore, the control signals are subjected to the error tolerance within difference counters representing a judgmental tool for accepting the results or controlling the data for optimal solution.

3.2 Types of numerical simulation software programs

There are several categories to classify the numerical simulation software approaches in the domains of electrical energy based on different aspects as follows:

3.2.1 Sensitivity analysis category

This category is based on simulating the proposed model regarding a reasonable tolerance of computational errors that is defined as sensitivity index (Han and Liu 2017). The methodology of this category is computationally based on the precise evaluation of the sensitivity index that arranges the parameters of the simulated problem regarding their importance. The derivation process of the sensitivity index is expressed as (Han and Liu 2017; Sevgi 2003):

$$\xrightarrow{{\text{Sampling}}\;{\text{Step}}} \;\Omega ( {\text{x}}) = {\Omega _{\text{o}}} + \sum\nolimits_i {{\Omega _i}( {{x_i}})} + \sum\nolimits_{\begin{array}{*{20}{c}} {1 \le \;i < \;j}\\ {i < j < k\;} \end{array}} {{\Omega _{ij}}( {{x_i},{x_j}}) + {\text{HOE}}}$$
(20)
$$\xrightarrow{{\text{Evaluation}}\;\;{\text{of}}\;{\text{TVR}}} \;{\text{TVR}} = \int_0^1 {{\Omega ^2}( {\text{x}}){\text{dx}} - {\Omega ^2}_{\text{o}}}$$
(21)
$$\xrightarrow{{\text{Evaluation}}\;{\text{of}}\;{\text{PVR}}} \;{\text{PVR}} = \int_0^1 {{\Omega ^2}_{{i_1},{i_2}, \ldots {i_q}\;}\left( {{x_{{i_1}}},{x_{{i_2}}}, \ldots {x_{{i_q}}}} \right){\text{d}}{x_{{i_1}}}{\text{d}}{x_{{i_2}}} \ldots {\text{d}}{x_{{i_q}}}}$$
(22)
$$\xrightarrow{{\text{Estimation}}\;{\text{of}}\;{\text{Sensitivity}}\;{\text{Index}}\;{\text{for}}\;{\text{Sampled}}\;{\text{HOE}}} {\theta ^I}_{HOE} = {\text{PVR}}/{\text{TVR}}$$
(23)
$$\xrightarrow{{\text{Estimation}}\;{\text{of}}\;{\text{Sensitivity}}\;{\text{Index}}\;{\text{for}}\;{\text{One}}\;{\text{Sampled}}\;{\text{Item}}} {\theta ^I}( {{x_i}}) = 1 - {\theta ^I}_{HOE} - {\theta ^I}_i$$
(24)

The equations are validated in their current formulation for the stationary centralized energy frameworks at stable modes and simple operations. Considering the transient stability and complicated controlling operations of energy systems in case of hybrid integrations, centralized domains are not sufficient for expressing the dynamic behaviour of such complicated systems and the computational equations of sensitivity analysis will be subjected to the mathematical modeling of complex energy system and its constraints. Significantly, the concepts of transient stability assessment and decentralized frameworks of energy systems strongly affect the reformulation of the methodology of computing the sensitivity index. The decentralized frameworks of energy systems are related to the complexity of solving PDEs considering the dynamic behaviour of momentum and kinetic energy. The dynamic decentralized equations of sensitivity analysis can be expressed as follows (Zareian Jahromi et al. 2020):

(25)
(26)

Then, the sensitivity index is obtained considering the defined mathematical domain of the proposed energy system and their corresponding parameters as well as constraints. Estimation process of sensitivity analysis in decentralized frameworks is related to the comparison of the calculated kinetic energy to the referenced critical one.

This category is sub-divided into two major approaches: Local Sensitivity Approach (LSA) and Global Sensitivity Approach (GSA). An innovative comparison for explaining the basic features of the two approaches is expressed in Table 5 (Han and Liu 2017; Sevgi 2003).

Table 5 Comparison between the different approaches of sensitivity analysis category
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Comprehensively, the classified methods of LSA and GSA are expressed in Fig. 5.

Fig. 5
figure 5

Classification chart of sensitivity analysis methods

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According to the chart shown in Fig. 5, there are six computational methods to be applied for the studies of LSA. The features of these methods are explained in Table 6. Also, the chart shows different applied computational categories for the studies of GSA. Furthermore, this diagram highlights the effectively common sensitivity methods to be utilized in the electrical energy domains with noting that there are many other sensitivity techniques to be used for different studies or integrated models combine these methods with other mentioned ones. To provide a comprehensive study for the GSA categories and their corresponding methods, Table 7 is provided expressing the operational analysis of each category (Han and Liu 2017; Sevgi 2003; Campolongo et al. 2007; Castillo et al. 2006; Sobol 2001; Zhang and Heitjan 2006).

Table 6 Comparison between the different methods of LSA
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Table 7 Comparison between the different categories of GSA
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To improve the accuracy of LSA, the different categories of GSA are emerged on scientific area. Significantly, GSA is the natural development of LSA for the emergence of complex problems. With a simple analysis for LSA, researchers usually tend to use the methods of GSA due to iconic features of GSA over LSA for complex problems (Han and Liu 2017; Sevgi 2003; Campolongo et al. 2007; Castillo et al. 2006; Sobol 2001; Zhang and Heitjan 2006).

It is worth to be mentioned that the similarities among LSA methods are summarized into the accurate computation in providing a solution of a unique-characterized model with simple objectives. In spite of the operational simplicity of LSA methods, there is a limited analysis for processing complex objectives with multiple variations regarding the simulation design factors. Significantly, the unique property of each LSA method is based on the ability of processing higher derivatives without disturbances and acceptable sensitivity index.

Regarding Fig. 5, there are four major categories for the operational methods of GSA as an iconic alternative for processing multi-objective models precisely with less number of simulations and complex boundaries. All GSA categories are based on the computation of complex models regarding the operational disturbances. The difficulties of these categories are graded from time consuming to the failure of convergence with operational disturbances within dependent elemental systems. Considering these obstacles, the classified methods of each category are recommended for sampling the domain of complicated models with uneven degree in computational time, sampling capability and accurate operation with less tuning errors. Although the merits of sensitivity analysis methods, its reliability decreases sharply due to the massive disturbances in output response for irregular issues (Han and Liu 2017).

3.2.2 Regularization category

This category can be generally applied for one of the following points to increase accuracy of numerical simulation and parameterization (Engl et al. 1996; Sun et al. 2014):

  • There are ill-defined or non-subsistent parameters within the simulated model.

  • The numerical simulation process is unstable for non-linear problems.

  • There is a noise due to the massive disturbances in the output response.

  • The existence of multi-variations that leads to discontinuous linearity in response.

The mathematical definition of this category is based on the following formula (Han and Liu 2017):

$${\ell ^\delta } = {{\text{I}}^{ - 1}}{{\text{L}}^\delta } = {{\text{O}}\dot D}({\Im ^{ - 1}}){{\text{W}}^{\text{T}}}[{{\text{W}}\dot D}( \Im){{\text{O}}^{\text{T}}}\ell + {\text{noi}}] = \ell + \sum\limits_{i = 1}^p {{\Im _i}^{ - 1}({{\text{w}}_{\text{i}}}^{\text{T}} \cdot {\text{noi}}){{\text{o}}_{\text{i}}}}$$
(27)

The most effective factors on the operational response of this category are noise and error coefficient singular matrix that affect the computational effectiveness. Comprehensively, the different methods of regularization category are listed in Fig. 6 (Han and Liu 2017; Sun et al. 2014).

Fig. 6
figure 6

Comprehensive classifications of regularization methods

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Significantly, the main distinguishable factors between these categories are based on reducing computational errors that are related to ill-defined parameters, noise and derived error coefficient singular matrix. Furthermore, these computational errors are minimized using derived objective functions from the general form in Eq. (25) regarding the application of stability constraints. These objectives and corresponding constraints are expressed as (Han and Liu 2017):

$$\xrightarrow{{\text{VPM}}\;{\text{Category}}} \;{\ell ^{\beta ,\;\delta }} = {[{{\cal I}^T}{\cal I} + \beta I]^{ - 1}}{{\cal I}^T}{{\cal L}^\delta }\;\;\;\xrightarrow{{\text{under}}\;{\text{constrints}}} \;\left\| {{{\cal L}^\delta } - {\cal I}\ell } \right\|\;\left\| \ell \right\|\;$$
(28)
$$\xrightarrow{{\text{SAM}}\;{\text{Category}}} \;{\ell ^{\beta ,\delta }} = \mathop \sum \limits_{{\Im ^2} \ge \beta } {\Im _i}^{ - 1}\left[ {{w_i}^T{{\cal L}^\delta }} \right]{o_i}\;\;\;\xrightarrow{{\text{under}}\;{\text{constrints}}} \;\;\left\| {\;1 - \frac{4}{\pi }{{\tan }^{ - 1}}\left( {{e^{ - \frac{\Im }{\beta }}}} \right)} \right\|$$
(29)
$$\xrightarrow{{\text{IM}}\;{\text{Category}}} \;{{\cal L}^\delta } = \Im \ell + {\text{noi}}\xrightarrow{{\text{under}}\;{\text{constrints}}} \left\| {\;{\ell ^*}} \right\|\;$$
(30)

Mainly, the effective factor to discriminate between the methods of each methodology is the arithmetic programming manner for simulating the objective function and boundaries of the proposed model (Tarantola and Valette 1982). Using computerized codes, the basic operational methodology of the regularization method is subjected to embedded criteria in a programmed platform to control the data and signals, see Fig. 7.

Fig. 7
figure 7

The basic operational methodology of the regularization method

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Despite the enough points about regularization methods that are described, it is necessary to show that the major classification methods of IM category are considered the most common ones due to their various merits, obstacles and applications over the other methodologies as shown in Table 8. The sub-classified methods that are branched from the major ones of IM categories are subjected to the same assumption of ignoring the discrimination criterion of the programming process in the electrical engineering studies (Han and Liu 2017; Engl et al. 1996; Sun et al. 2014; Tarantola and Valette 1982).

Table 8 Comparison between the major classified-methods of IM of regularization category
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3.2.3 Inverse computational approaches

This category is the most effective one to completely define the parameters of the proposed problem, cover comprehensively the overall constraints of the parameterized model as well as the simulation software and increase computational stability of numerical response. The bifurcation of this category in the domains of electrical energy are considered as the most massive one at all among the other different categories of numerical simulation software programs as shown in Fig. 8 (Aster et al. 2018).

Fig. 8
figure 8

Classification of inverse computational methods in electrical energy domains

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3.2.3.1 Gradient iteration methods

These methods are one of the earliest computational approaches to simply obtain the inverse arithmetic solutions for the complicated problems based on the following merits (Aster et al. 2018):

  • Optimizing the objective function of the simulated model based only on the positions and directions of the search steps without any complicated controlling processes.

  • Converging the computational response whatever the analyzed complicated problems.

  • The massive variety of programmable software programs related to the computation of these methods such as International Mathematics and Statistics Library (IMSL).

  • Applying the iterative methodologies to avoid large simulated blocks, ensure the stability of the computation and increase the accuracy regarding tuning errors.

Significantly, the iterative methodologies of these techniques depend on the Homotopy Algorithm (HA) for processing the objective function of the simulated complicated model. HA is a tracing and scrutinizing methodology for the response path of the computational analysis. This algorithm involves four major operational stages as shown in Fig. 9 (Liu and Han 2003). The initial points of numerical computation are obtained in the initialization stage. Choosing the optimal step length and tracing direction are processed in the forecasting stage based on Euler’s approach. Testing the effectiveness of the optimally forecasted data is embedded in correction stage based on Newton-adoption approach. Checking the convergence process and validity of the results are expressed in validation stage, see Fig. 10 (Liu and Han 2003).

Fig. 9
figure 9

Sequential operational stages of HA

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Fig. 10
figure 10

The operational methodology flowchart of HA performance

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To innovatively describe the computational methodology and stages of HA, the mathematical modeling of HA must be expressed based on the following formulas (Han and Liu 2017):

$$\xrightarrow{\begin{array}{*{20}{c}} {{\text{The}}\;{\text{General}}\;{\text{Non - Linear}}\;{\text{Expression}}\;{\text{of}}\;{\text{the}}\;{\text{Mathematical}}}\\ {{\text{Objective}}\;{\text{Function}}\;{\text{to}}\;{\text{Be}}\;{\text{Processed}}} \end{array}} \;( \tau) = h( \tau) - {{\cal L}^\delta }$$
(31)
$$\xrightarrow{\begin{array}{*{20}{c}} {{\text{The}}\;{\text{Applied}}\;{\text{Mathematical}}\;{\text{Modeling}}\;{\text{Function}}\;{\text{of}}\;{\text{HA}}}\\ {{\text{for}}\;{\text{The}}\;{\text{Model}}\;{\text{Objective}}\;{\text{Function}}} \end{array}} \;\varepsilon ( {\tau ,u}) = u\left[ {( \tau )} \right] + ( {1 - u})\left[ {R( \tau)} \right]$$
(32)

Deepening more into the properties of HA, the following innovatively expressed points will show the performance and applications of this algorithm:

  • This algorithm is one of the most iconic methodologies for the computation of simple non-linear mathematical modeling formulas.

  • It is suitable for the iterative computation due to its comprehensive convergence with a global set of converged-computations for any applied initial arithmetic point.

  • This algorithm experiences a poor performance for sophisticated non-linear models due to the difficulty in computing just complicated derived-mathematical equations

Regarding the embedded data as well as programmed strategies within a software program, the applied questions and actions are accredited in the suitable orientation.

Traditional iterative methods Regarding Fig. 8, the traditional iterative methods are one of the most common numerical simulation methods with a several bifurcated classes (Aster et al. 2018). The merits of all classes are almost the same involving the convergence capability for non-linear, ill-defined and large-scale simulated problems. The other advantages are based on fast response and stagnation to get the optimal solutions with higher accuracy. Furthermore, the obstacles of the classes are mainly related to large number of iterations, time waste, slow convergence and simulation failure regarding the applied constraints for large-scale parameters. The recommendations of these methods are associated with simple structural models and simulated grids (Aster et al. 2018)

Regarding the domain of energy systems and the electrical nature of solved problems inspired from the practical power grids as shown in Fig. 8, the traditional iterative methods are categorized into six major classes addressed by the most common pioneering method of each class. The features of these classes are to be innovatively expressed in Table 9 (Han and Liu 2017; Aster et al. 2018; Liu and Han 2003).

Table 9 Comparison between the traditional iterative classes in electrical engineering domain
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Integral simulation methods Considering Fig. 8, the integral simulation methods are classified into two common major types. These types provide effective solutions for the complex obstacles of electromagnetic fields. The discrimination between CSM and BEM is expressed innovatively in Table 10 regarding the role of iterative methodology (Han and Liu 2017; Wu 2019; Talaat et al. 2020).

Table 10 Comparison among the iteratively integral simulation methods within electrical domain
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Based on programmed software code, the applied actions through structural methodologies as mentioned in Table 10 are answered in a suitable orientation regarding validation criteria.

Intelligent evolutionary methods In spite of the merits of gradient iteration methods, there are several obstacles as follows (Han and Liu 2017):

  • The operational incompetence in case of large-parameterized complicated problems due to the exaggerated needed number of iterations and low processing response.

  • The convergence shortages and slow operation for evaluating non-linear problems.

  • The difficulty in determining the initial guess point and the optimum step size for the integral simulation methods negatively affects the accurate computation.

To overcome these obstacles, intelligent evolutionary methods are provided as a magic tool to enhance the overall efficiency of inverse computational category. The merits, obstacles and recommendations of these methods can be listed as shown in Fig. 11.

Fig. 11
figure 11

Major properties of intelligent evolutionary methods within electrical domains

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ANNs and ANFLNs Considering Fig. 11, a comprehensive overview for ANNs and ANFLNs operational features are provided in Table 11 (Talaat et al. 2022; Han and Liu 2017). Significantly, ANNs are smart functionally systems at which their operational nature inspired from the transmission manner of electrical signals between the nerve cells. On the other side, ANFLNs are logical control systems that mimic the human thinking to dominate the electrical signals among nerve cells. ANNs are utilized in medical and energy domains such as voice recognition and medical diagnosis whereas ANFLNs are used in controllable energy systems such as robotics and automation.

Table 11 Overview on ANNs and ANFLNs regarding the electrical domains
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Based on the embedded data in a programmed software code, the applied questions and actions through the structural methodologies that are mentioned in Table 4 are answered in a suitable orientation regarding the validation criteria.

Differential simulation methods Considering Fig. 8, these methods are categorized into three effective approaches. These approaches inspired their operational performance either practically for SAA regarding the melting and cooling behaviour of a body or statistically for MCM regarding the estimated risks within contingency analysis. Furthermore, FEM is based on mesh generation concept for simplifying the large-scale systems. With higher accuracy, these approaches deal with non-uniform shaped models; multi-sampled and random complex models especially for electromagnetic issues. Regarding operational factors, these approaches depend in an uneven manner on the complexity of simulated models, applied constraints and operational stability.

Computational optimization algorithms Considering Fig. 8, these algorithms are smart computerized tools imitating natural behavior of practical embodiments such as natural selection process for GA, searching process for the position of food supply for ACA and sudden orientation for PSO. The searching domain of these algorithms is mainly meta-heuristic with a probabilistic mathematical behavior. The merits of these algorithms are based in an uneven degree on the fast processing response and less number of iterations for large scale objective functions. The deviated disturbances in the optimized results and time waste for complex multi-objectives are the main obstacles for these algorithms. The computational operators vary definitely for each algorithm. For GA, the crossover as well as mutation processes are the main operators whereas clustering and searching speed processes are the main ones regarding the optimal studies (Nguyen and Jung 2021).

Significantly, the hybrid methods provide an effective solution for complicated energy problems. These hybrid combination methods involve multiple aspects such as:

  • Optimization algorithms assisted by ANNs to increase accuracy of global searching.

  • Gradient iteration methods assisted with ANFLN to increase the convergence.

  • Bayesian methodologies assisted with the evolutionary optimization algorithms.

4 Numerical simulation methods with ANNs for integrated energy issues

4.1 Sensitivity approaches with ANNs; illustrative and prospective practical examples

The integration of ANNs with the operational analysis of sensitivity simulation approaches generally provides the following merits:

  • Precisely estimating the sensitivity indices with less operational disturbances.

  • Enabling the sensitivity approaches to forecast the influence of the coefficient factors on the operation of different electrical systems.

  • Optimizing the number of effective factors within any electrical system and arranging the coefficient factors based on their impact degree.

  • Enhancing the reliability and efficiency of different electrical issues or systems.

4.1.1 Integrated strategy of sensitivity analysis with ANNs within energy system

4.1.1.1 Improving and optimizing the operation of photo–voltaic (PV) modules

In this practical experimental study, the proposed framework is to integrate the sensitivity analysis with ANNs in one hybrid controlling model to enhance the generation features of the Photo-Voltaic (PV) energy system. As one of the most effective energy generation systems to practically exploit the solar effect, PV networks require more enhancements to achieve a sufficient accuracy to be a vital source in the energy generation market (Bazilian et al. 2013; Tzuc et al. 2019). Considering this conducted study, the obstacles of PV systems can be expressed as follows (Tzuc et al. 2019):

  • At the highest level of energy conversion, the generated electricity from PV modules do not exceed 15% from the total incident solar radiation lines resulting in the need of large number of PV modules to obtain the desired amount of electricity.

  • The massive amount of heat dissipation affects the operation of PV modules by minimizing the operational voltage and power of the output generated electricity.

  • The longtime of exposure to solar radiation increases the temperature of PV modules above the acceptable limit that may lead to the operational damage of these modules.

In this study, a computational forecasting-optimization software is constructed to precisely predict the temperature level of PV modules. This software arranges optimally the impact degrees of estimated temperatures to improve the overall generation efficiency of PV energy system. The hybrid integration of GSA with ANNs is provided in this study as a powerful computerized software to enhance the experimental operation of PV modules regarding Fig. 12 (Tzuc et al. 2019). The effectiveness of the controlling results of this integrated strategy is validated considering the Elementary Effect Test (EET). Regarding EET, the ambient temperature and solar radiations are the most effective factors on the operation of PV modules (Tzuc et al. 2019).

Fig. 12
figure 12

Methodology of integrated GSA with ANNs for improving operation of PV module

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Considering the performed studies, the achieved targets can be expressed as follows:

  • Validating the effectiveness of EET to detect the most influential parameters on the PV module based on the sensitivity index of each parameter.

  • Confirming arithmetically and graphically that the increase in the studied random samples of GSA decreases deviations in the performance of PV module parameters.

  • Prospectively, integrating different methods of GSA with ANNs may affect response of the forecasting and optimization processes to provide advanced future studies.

4.1.1.2 Improving the operational feasibility and reliability of wind power generation

In this practical simulation study, the proposed framework is to integrate the hybrid model of sensitivity analysis and ANNs with NPV strategy to enhance the generation features of the wind energy system. To improve the reliability and profitability of wind energy systems, it is necessary to identify the impact of coefficients that affect the operation of these systems. Regarding this study, the proposed integrated controlling strategy is provided to detect the most influential parameters on the operation of wind energy systems (Rotela Junior et al. 2281).

The economic feasibility for constructing an efficient wind energy system is a major obstacle for the reliability of these systems in addition to the sudden changes in their parameterized variables and factors. The recent studies provide NPV method as a powerful methodology to be integrated with the hybrid model of sensitivity analysis and ANNs to determine the impact of financial and operational parameters of wind energy system (Rotela Junior et al. 2281). The proposed model improves the reliability of wind system regarding the following achieved targets (Rotela Junior et al. 2281):

  • The most influential factors on the operation of wind energy system are tariff of generated energy, wind velocity and the total investment size.

  • NPV indicator detects that the most effective factor is the wind velocity.

  • The usage of ANNs is numerically classifying the impact levels of the influential factors to validate the performance of NPV approach as shown in Fig. 13.

Fig. 13
figure 13

The results of proposed model for improving the operation of wind energy system

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4.1.1.3 Impact of ANN to control the generated energy of different integrated sources

In this practical experimental study, the proposed framework is to integrate ANN topology to validate the experimental capability of Field Programmable Gate Array (FPGA) technology as an innovative digital methodology to efficiently integrate PV, wave and Fuel Cells (FCs) energy generation systems in one hybrid model. This study emphasized the ability of FPGA technology in obtaining reasonable results of voltage, current and power profiles of proposed integrated energy system. The maximum percentage deviations between the experimental study with FPGA and controlling study of ANN with FPGA for the energy generation profiles of the integrated sources are expressed in Table 12. As vital operator of generation process in this study, the voltage behaviour of both experimental and controlling scenarios is shown in Fig. 14. For comprehensive illustration, current and power profiles of the integrated sources of both experimental and controlling scenarios are also shown in Fig. 14 (Talaat et al. 2022).

Table 12 Generation analysis of integrated energy sources for practical and controlling scenarios
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Fig. 14
figure 14

Generation analysis of integrated system considering practical and controlling scenarios; a voltage profile, b current profile, c power profile

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4.2 Integration of regularization methods with ANNs for analyzing complicated fields

4.2.1 General overview

There is an extreme rarity for the conducted scientific researches on the role of ANNs to improve the operational performance of regularization methods. This scarcity is due to the deep overlap of regularization methods with inverse computational considering the Bayesian Regularization Approach (BRA). Correspondingly, the integration process of ANN with the regularization methods is implicitly hidden in the studies combining numerical simulation methods together. The roles of ANNs with regularization methods are defined as follows:

  • Improving the operational capability of regularization methods for precisely imitating, estimating, and optimizing the spatially scattered electromagnetic fields.

  • Increasing computation accuracy with fast response for complex energy systems.

  • Enabling the regularization methods to reach the optimum response with low number of iteration and reasonable convergence time.

4.2.2 Practical implementation

In this practical simulation study, the proposed framework is to integrate ANN topology with Tikhonov Regularization Approach (TRA) to numerically simulate the scattering behaviour of antenna signals due to the operational disturbances. This integrated controlling strategy of ANN with TRA provides a proper investigation of the directional disturbances of spatially scattered electromagnetic fields on the carriers of antenna signals (Sandhu et al. 2021). This strategy paves the way for normalizing optimally these scattered fields to enhance the reliability and stability of the transmission process (Sandhu et al. 2021). Furthermore, the proposed integrated controlling model improves the operation of GPOA for sampling effectively the complicated field signals by minimizing the computational error and enhancing convergence stagnation. Considering the proposed strategy, the following points can be deduced (Sandhu et al. 2021):

  • The usage of ANN enables Tikhonov method to detect precisely number of samples of the linearized-optimized fields in order to construct easily the sampling matrix.

  • The integration of ANN provides a magic tool for reaching the optimal distribution of scattered fields by ordering each sample according to its importance level.

  • Furthermore, ANN boosts the operational performance of Tikhonov regularization method and GPOA to achieve convergence with a smaller number of generations

  • Prospectively, there is a future priority to such strategies to break into scientific arena.

Schematically, the effect of integrated controlling strategy of ANN with TRA is expressed in reducing the fitness value of GPOA considering four training samples as shown in Fig. 15.

Fig. 15
figure 15

Convergence of GPOA considering the TRA model trained by ANN samples

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4.3 Inverse computational methods assisted by ANNs; illustrative practical examples

The vital impact of ANNs clearly appears in improving the analysis of inverse computational methods due to the massive variety of the practical applications within electrical domains. Regarding the features of the inverse computational methods, the following points summarize the powerful effects of ANNs for enhancing the simulation process of these methods:

  • Testing the effectiveness of inverse computational methods to assess their efficiency.

  • Obtaining the optimal coefficients that minimize the operational errors and control the accuracy of inverse computational methods.

  • Improving the computational analysis of hybrid integrated energy systems.

4.3.1 Integrated controlling strategies with energy of different integrated sources

In this practical experimental study, the proposed framework is to integrate ANN topology with MFO to optimally forecasting the controlling scenarios of FPGA as an innovative digital step to enhance the generation process of the hybrid integrated model of PV, wave and FCs. This study emphasized the ability of the integrated controlling strategy of ANN with MFO to enhance the tuning errors of ANN topology considering the training samples and stagnation. Also, this study validates the effectiveness inverse computational methods and optimization algorithms in relaxing optimally the energy generation distribution profiles. Furthermore, the strategy of ANN with MFO enhances the experimental studies considering the controlling scenarios of FPGA method. Comprehensively, the controlling strategy of ANN with Particle Swarm Optimization (PSO) is innovatively provided to show effectiveness of optimization algorithms in such studies. Voltage behaviour of both experimental and controlling scenarios is shown in Fig. 16. For comprehensive illustration, current and power profiles of integrated sources of both experimental and controlling scenarios are also shown in Fig. 16 (Talaat et al. 2022).

Fig. 16
figure 16

Generation analysis of integrated system considering practical and controlling scenarios; a current profile, b power profile

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4.3.2 Integrated controlling strategies within energy systems

In the upcoming practical studies, the proposed frameworks that are expressed in Table 13 show the integrated controlling strategies combining ANN with the defined type of inverse computational methods to simulate and validate the operation of the proposed energy system.

Table 13 Practical energy systems with the hybrid integration of ANNs with inverse methods
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5 Numerical simulation methods with ANFLNs for integrated energy issues

5.1 General overview

There is a severe scarcity in practical experimental and simulation studies that are related to energy systems utilizing the ANFLNs for improving the operation of different numerical simulation methods. The reasons for this rarity can be traced back to the following points:

  • The arithmetic dominance of ordinary ANN and FLCS methodologies for improving the computation of simulated problems due to their simplicity and higher accuracy.

  • The permanent dependence of ANFLN on the optimization algorithms or ANN for training its parameters that increases the complexity degree of the computation.

  • Despite the iconic property of global searching, ANFLN suffers from the failure of convergence regarding complex simulated problems due to low speed of processing.

5.2 Practical implementations

In the upcoming practical studies, the proposed frameworks that are expressed in Table 14 show integrated controlling strategies combining ANFLN with the defined type of numerical methods to simulate and validate the operation of the proposed energy system.

Table 14 Practical energy systems integrating ANFLNs with numerical simulation methods
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In the prospective studies, the following points may be useful to increase the computational dependence on ANFLN methodology for improving the numerical simulation analysis:

  • Improving optimally the mathematical operators of ANFLN to cope with complicated electrical issues as well as obtaining the desired output response.

  • The accurate analysis of the role of ANN and the different optimization algorithms to be integrated with ANFLN to test the effectiveness of this approach.

6 Numerical simulation methods with optimization ones for integrated systems

6.1 General overview

As the most common AI type to be used in improving the simulation analysis of different complicated electrical issues, the computerized optimization algorithms provide powerful roles in enhancing the operational performance of numerical simulation methods as follows:

  • Increasing the simulation accuracy by lowering the tuning errors.

  • Estimating the optimal coefficients affecting the performance of simulation methods.

  • Reaching the optimum shape, size, cost, and any other performance parameters for enhancing the operational features and reliability of simulated electrical issues.

6.2 Practical implementations of integrating CSM and FEM with proposed optimizers

In the upcoming practical studies, the proposed frameworks that are expressed in Table 15 show integrated controlling strategies combining proposed optimizers with CSM or FEM to simulate and validate the operation of proposed energy system. It is worth to be mentioned that experimental and simulation studies involving CSM or FEM depend mainly on obtaining the optimum computational parameters considering the topologies of AI.

Table 15 Practical energy systems integrating CSM or FEM with optimization algorithms
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6.3 Practical examples of integrating hybrid simulation methods with optimizers

In the upcoming practical studies, the proposed frameworks that are expressed in Table 16 show integrated controlling strategies combining proposed optimizers with any defined type of numerical simulation methods except CSM and FEM.

Table 16 Practical energy systems integrating hybrid simulation methods with optimizers
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7 Conclusions

The iconic computational analysis of AI strategies is clearly investigated in this research showing their effective capabilities for overcoming the operational obstacles of the numerical simulation methodologies regarding the energy issues. The conducted studies in this paper show the effectiveness of AI strategies as smart solutions for improving the efficiency of numerical simulation methods based on the intelligent controllability of the hybrid models integrating the AI operators with proposed numerical methods for simulating practical energy systems. Significantly, the supreme dominance of numerical methods remains the optimal solution for simulating complicated-structural energy systems. Considering the massive bifurcation of computational domains of numerical simulation methods, there are multiple studies that are reviewed in this paper showing the operational features of each category detecting the desired choice for applied case study. Furthermore, the operational hybrid models integrating numerical simulation methods with AI strategies provide a powerful way for increasing the reliability and quality of energy systems whatever the complexity degree regarding the provided practical examples. Regarding the reviewed practical examples, the unique merits of optimized numerical simulation methods are validated paving the way for several studies to highlight the significant effectiveness of this hybrid integration within different sciences in the upcoming studies. In this paper, the renewable energy systems are provided with different studies as effective practical systems for accrediting the optimal operation of the hybrid integrated simulation methods with AI strategies.