
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12777-6
10.1016/j.heliyon.2024.e36746
e36746
Research Article
Are hybrid approaches combining EKF-UKF and artificial neural networks key to unlocking the full potential of sustainable energy technologies and reducing environmental footprint?
Liang WeiFang a
Maesoumi Mohsen mohsen.maesoumi@iau.ac.ir
b⁎
Basem Ali c
Jasim Dheyaa J. d
Sultan Abbas J. e
Al-Rubaye Ameer H. f
Zhang Jingyu g
a School of Computer Science and Engineering, Hunan University of Information Technology, Changsha, 410151, China
b Department of Electrical and Computer Engineering, Jahrom Branch, Islamic Azad University, Jahrom, Iran
c Faculty of Engineering, Warith Al-Anbiyaa University, Karbala, 56001, Iraq
d Department of Petroleum Engineering, Al-Amarah University College, Maysan, Iraq
e Department of Chemical Engineering, University of Technology- Iraq, Baghdad, Iraq
f Department of Petroleum Engineering, Al-Kitab University, Altun Kupri, Iraq
g School of Computer & Communication Engineering, Changsha University of Science & Technology, Changsha, 410004, China
⁎ Corresponding author. mohsen.maesoumi@iau.ac.ir
04 9 2024
30 9 2024
04 9 2024
10 18 e3674611 5 2024
10 7 2024
21 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The integration of traditional state estimation techniques like the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) with modern artificial neural networks (ANNs) presents a promising avenue for advancing state estimation in sustainable energy systems. This study explores the potential of hybridizing EKF-UKF with ANNs to optimize renewable energy integration and mitigate environmental impact. Through comprehensive experimentation and analysis, significant improvements in state estimation accuracy and sustainability metrics are revealed. The results indicate a substantial 8.02 % reduction in estimation error compared to standalone EKF and UKF methods, highlighting the enhanced predictive capabilities of the hybrid approach. Moreover, the integration of ANNs facilitated a 12.52 % increase in renewable energy utilization efficiency, leading to a notable 5.14 % decrease in carbon emissions. These compelling outcomes underscore the critical role of hybrid approaches in maximizing the efficiency of sustainable energy technologies while simultaneously reducing environmental footprint. By harnessing the synergies between traditional filtering techniques and machine learning algorithms, hybrid EKF-UKF with ANNs emerges as a key enabler in accelerating the transition towards a more sustainable and resilient energy landscape.

Keywords

Hybrid state estimation
EKF
UKF
ANNs
Renewable energy integration
Carbon emissions reduction
Sustainable energy systems
==== Body
pmc1 Introduction

Acronyms	
EKF	Extended Kalman Filter	
UKF	Unscented Kalman Filter	
ANNs	Artificial Neural Networks	
LSTM	Long Short-Term Memory	
SoC	State of Charge	
MOFs	Metal-Organic Frameworks	
CCS	Carbon Capture and Storage	
GIoT	Global Internet of Things	
Unscented Transform	UT	
PV	Photovoltaic	
RMSE	Root Mean Square Error	
MAE	Mean Absolute Error	

1.1 Motivation

The transition towards sustainable energy systems has become imperative in the face of escalating environmental concerns and the pressing need to mitigate climate change [1]. Central to this transition is the optimization of state estimation methodologies [2], which play a pivotal role in ensuring the efficient operation and integration of renewable energy sources [3]. Traditional techniques such as the EKF and UKF have long been employed for state estimation, offering valuable insights into system dynamics [4]. However, the inherent limitations of these methods, particularly in handling nonlinearities and uncertainties prevalent in renewable energy systems [5], have spurred the exploration of novel approaches [6].

In recent years, the integration of ANNs with traditional filtering techniques has emerged as a promising avenue for enhancing state estimation accuracy and robustness [7]. By leveraging the nonlinear modeling capabilities of ANNs, researchers have sought to augment the capabilities of EKF and UKF, thereby addressing the complex challenges posed by renewable energy [8] integration and environmental sustainability [9].

Motivated by the imperative to accelerate the transition towards a greener and more resilient energy landscape, this study investigates the potential of hybridizing EKF-UKF with ANNs for state estimation in sustainable energy systems. The aim is to elucidate the efficacy of this hybrid approach in optimizing renewable energy integration, reducing carbon emissions, and advancing environmental sustainability. By thorough experimentation and examination, this study seeks to uncover the transformative impact of hybrid state estimation methodologies on the efficiency, reliability, and sustainability of energy systems.

By bridging the gap between traditional filtering techniques and modern machine learning algorithms, this research endeavors to contribute to the ongoing discourse on sustainable energy transition and climate change mitigation. Through innovation and collaboration, this research aspires to pave the way for a future where renewable energy sources are seamlessly integrated, emissions are drastically reduced, and environmental stewardship is prioritized.

1.2 Literature review

The pursuit of sustainable energy systems has garnered significant attention in recent years, driven by mounting environmental concerns and the imperative to mitigate climate change [10]. Central to the development and operation of such systems is the accurate estimation of system states, which enables effective decision-making [11] and optimization of energy resources [12]. Traditional state estimation techniques, including the EKF and UKF, have long served as cornerstones in this domain, offering valuable insights into system dynamics [13]. However, these methods are often challenged by the nonlinear and uncertain nature of renewable energy systems [14], necessitating the exploration of innovative approaches [15].

In response to these challenges, researchers have increasingly turned to the integration of ANNs [16] with traditional filtering techniques to enhance state estimation capabilities [17]. ANNs, with their ability to model complex nonlinear relationships, offer a promising solution for addressing the shortcomings of traditional methods and improving estimation accuracy in sustainable energy systems. Several studies have demonstrated the efficacy of this hybrid approach across various domains, ranging from power systems to renewable energy integration [18].

For instance [19], proposed a hybrid EKF-ANN framework for wind speed forecasting, achieving superior prediction accuracy compared to standalone EKF and UKF methods. Similarly [20], employed a combination of UKF and Long Short-Term Memory (LSTM) networks for photovoltaic power forecasting [21], demonstrating significant improvements in forecast accuracy and reliability. These studies underscore the potential of hybrid state estimation methodologies in optimizing renewable energy integration [22]and enhancing system performance.

Furthermore, the importance of accurate state estimation in supporting sustainable energy transition and climate change mitigation cannot be overstated [23]. With the increasing adoption of renewable energy sources [24] such as wind, solar, and hydroelectric power, there is a growing need for robust estimation techniques [25] that can effectively manage variability and uncertainty [26]. By leveraging the complementary strengths of traditional filtering techniques and machine learning algorithms, hybrid state estimation approaches offer a promising avenue for advancing the sustainability and resilience of energy systems [27].

In light of these developments, this study aims to contribute to the existing body of literature by investigating the effectiveness of hybridizing EKF-UKF with ANNs for state estimation in sustainable energy systems. By systematic experimentation and review, this research seeks to elucidate the potential of the hybrid approach in optimizing renewable energy integration, reducing carbon emissions, and advancing environmental sustainability. By synthesizing insights from previous research and exploring novel methodologies, this study endeavors to provide valuable contributions to the ongoing discourse on sustainable energy transition and climate change mitigation.

In article [4], hybrid state estimation methodologies combining Kalman filters and neural networks are evaluated for their effectiveness in accurately estimating system states and navigating complexities in nonlinear systems. This summary is essential for researchers and practitioners seeking to optimize renewable energy integration and advance sustainability in energy systems. Article [28] focuses on hybrid methodologies for estimating the state of charge (SoC) in lithium-ion batteries. It highlights these methods' reliability in predicting SoC under various conditions, leveraging neural networks and Kalman filters to improve accuracy and robustness. The summary underscores their potential to enhance battery management systems and prolong battery lifespan. Article [29] evaluates a novel hybrid neural network model for SoC estimation in lithium-ion batteries under low-temperature conditions. It demonstrates the model's effectiveness in predicting SoC in challenging thermal environments, outperforming traditional methods. This summary emphasizes its role in bolstering battery reliability and efficiency in cold climates.

A novel SOC estimation approach for E-cell batteries is comprehensively assessed in Ref. [30]. Through meticulous analysis, the method proves effective and reliable in predicting SOC accurately. By integrating a backpropagation neural network with the EKF algorithm, this hybrid approach surpasses conventional methods, improving battery management system accuracy, optimizing performance, and extending lifespan. An evaluation of a novel SoC estimation approach for lithium-ion batteries is presented in Ref. [31]. Through rigorous experimentation, the method demonstrates effectiveness and robustness in accurately predicting SoC. By combining an adaptive extended Kalman filter with an ANN, this hybrid method outperforms traditional techniques, enhancing battery management system accuracy, optimizing performance, and extending lifespan. A comprehensive evaluation of a hybrid approach for runoff estimation is provided in Ref. [32]. Through meticulous analysis, the method shows effectiveness and robustness in accurately predicting runoff. By integrating ANNs with the EKF, this hybrid approach exhibits superior performance compared to conventional methods, offering easy-to-implement models that improve accuracy and reliability in runoff estimation.

A performance summary evaluates an augmented forecasting model for renewable energy consumption in Ref. [33]. Through extensive experimentation, the model proves effective in accurately predicting short-term renewable energy consumption. By integrating a modified grey model with a Kalman filter, the approach outperforms traditional methods, providing reliable forecasts crucial for efficient energy management in renewable systems. Synopsis of findings assesses the integration of Kalman filters with artificial intelligence techniques in Ref. [34]. Through a comprehensive review, it highlights hybrid approaches' effectiveness in state estimation and prediction tasks. Combining KF with ANNs, fuzzy logic, and genetic algorithms enhances accuracy and robustness, addressing nonlinearities. A performance summary evaluates a dynamic data reconciliation approach for nonlinear process systems in Ref. [35]. Through rigorous analysis, the approach reconciles process data effectively by integrating Elman neural networks with extended Kalman filters.

The paper offers a comprehensive overview of the current landscape of AI applications in the sustainable energy industry in Ref. [36]. Through meticulous analysis and synthesis of existing literature and industry practices, the study sheds light on the efficacy, challenges, and opportunities associated with the integration of AI in sustainable energy applications. By examining case studies, technological advancements, and emerging trends, the summary provides valuable insights into the performance and potential of AI techniques such as machine learning, deep learning, and reinforcement learning in enhancing energy efficiency, optimizing renewable energy integration, and addressing environmental concerns. The reference provides a thorough evaluation of a novel approach for predicting the thermophysical properties of graphene oxide and MXene hybrid nanofluids in Ref. [37]. Via diligent experimentation and study, the article elucidates the effectiveness and reliability of the proposed model in accurately predicting the thermophysical profile of nanofluids, which is crucial for sustainable energy applications. By employing a Bayesian-optimized neural network with K-cross-fold validation, the model demonstrates superior performance compared to traditional methods. The paper provides a comprehensive evaluation of a novel approach for predicting CO2 storage capacity in metal-organic frameworks (MOFs) using neural networks in Ref. [38]. Through meticulous experimentation and analysis, it elucidates the effectiveness and reliability of the proposed model in accurately predicting CO2 storage capacity, a critical aspect of carbon capture and storage (CCS) technologies. By employing neural networks, the model demonstrates superior performance compared to traditional methods, offering valuable insights into the potential of machine learning techniques for enhancing CCS processes.

The study offers a comprehensive evaluation of a novel approach for integrating eco-environmental impact assessment with eco-tourism using deep neural network algorithms within the context of the Global Internet of Things (GIoT) environment in Ref. [39]. Through meticulous experimentation and analysis, the assessment overview elucidates the effectiveness and reliability of the proposed model in accurately predicting and analyzing eco-environmental impacts, as well as optimizing eco-tourism activities. By leveraging deep neural network algorithms, the model demonstrates superior performance compared to traditional methods, providing valuable insights into the potential of artificial intelligence for sustainable tourism management and environmental conservation efforts. The evaluation report provides a comprehensive ]overview of sustainable investment strategies aimed at reducing environmental footprint in Ref. [40]. Through meticulous analysis and synthesis of existing literature and industry practices, the article elucidates the effectiveness, challenges, and opportunities associated with sustainable investment approaches in mitigating environmental impact. By examining case studies, regulatory frameworks, and emerging trends, the summary offers valuable insights into the performance and potential of sustainable investment practices in fostering environmental stewardship and promoting sustainable development.

1.3 Research gaps and contributions

Traditional techniques such as the EKF and UKF have been widely employed, yet they often encounter challenges in accurately capturing the nonlinear dynamics and uncertainties inherent in renewable energy integration. Moreover, while ANNs have shown promise in addressing these challenges, their integration with traditional filtering methods remains relatively unexplored in the context of sustainable energy systems.

This gap in the literature underscores the need for novel approaches that leverage the complementary strengths of both traditional filtering techniques and modern machine learning algorithms. In this context, this article makes several notable contributions to the field:1. Novel Hybrid Methodology: A novel hybrid methodology is proposed that combines EKF-UKF with ANNs for state estimation in sustainable energy systems. This approach capitalizes on the strengths of traditional filtering techniques and machine learning algorithms to improve estimation accuracy and robustness.

2. Enhanced Sustainability and Efficiency: By leveraging the capabilities of hybrid state estimation techniques, the methodology aims to significantly enhance the sustainability and efficiency of energy systems. Through accurate estimation of system states, optimized integration of renewable energy sources, reduction of carbon emissions, and overall improvement in environmental sustainability are facilitated.

1.4 Organization

In the second section of this article, Problem Formulation, the intricacies of implementing various hybrid estimation algorithms, including the Hybrid Extended Kalman Filter Algorithm, Integrated Hybrid EKF and Artificial Neural Network, Hybrid Unscented Kalman Filter Algorithm, and Hybrid Unscented Kalman Filter combined with ANN, are meticulously discussed step by step. Moving on to the third part, Results and Discussion, the focus is on presenting the outcomes of simulations and analyses, particularly in a case study titled “Optimizing Hybrid Renewable Energy Systems: A Case Study in Sustainable Energy Management and Environmental Footprint Reduction." Here, the results are thoroughly documented, and comparisons are drawn to underscore the efficacy of each hybrid estimation algorithm, with tables aiding in succinctly presenting the findings. Finally, the fourth part concludes the article by summarizing the key findings and insights derived from the research, emphasizing the significance of hybrid estimation algorithms in optimizing renewable energy systems and reducing environmental footprint, while also hinting at potential avenues for future research and applications in sustainable energy management and environmental conservation.

2 Problem Formulation

The accurate estimation of system states is paramount for the effective operation and management of sustainable energy systems, particularly in the context of renewable energy integration. However, traditional state estimation techniques such as the EKF and UKF face challenges in accurately capturing the nonlinear dynamics and uncertainties inherent in renewable energy sources. Furthermore, while ANNs offer a promising solution for addressing these challenges, their integration with traditional filtering methods remains relatively unexplored in the context of sustainable energy systems. This disparity underscores the need for innovative approaches that leverage the complementary strengths of both traditional filtering techniques and modern machine learning algorithms. In this section, the problem statement is formulated, and the proposed methodology for addressing these challenges is outlined.

The flowchart in Fig. 1 serves to visually depict the sequential steps and interactions involved in hybrid approach integrating EKF-UKF and ANN for state estimation in sustainable energy systems. By including this schematic representation, aim to enhance the clarity and comprehensibility of proposed methodology, facilitating a deeper understanding of how these advanced techniques synergistically contribute to improved accuracy and efficiency in energy management.Fig. 1 The flowchart of the proposed filter design.

Fig. 1

2.1 Hybrid extended Kalman Filter algorithm

The Hybrid Extended Kalman Filter algorithm is a state estimation technique that combines the principles of the traditional EKF with additional mechanisms, typically drawn from other methodologies such as neural networks [41], particle filters, or other machine learning techniques. The EKF algorithm is an extension of the Kalman filter, specifically designed to handle nonlinear systems by linearizing them at each time step. However, in complex nonlinear systems, the linearization assumptions may not hold, leading to suboptimal performance. To address this limitation, the Hybrid EKF algorithm incorporates additional techniques to improve estimation accuracy and robustness [42].

One common approach in the Hybrid EKF algorithm involves integrating neural networks [43] or machine learning models to enhance the prediction step of the EKF. These models learn the system dynamics from historical data and can capture nonlinearities more accurately than traditional linearization methods. By incorporating the predictions from the neural network [44] into the EKF framework, the Hybrid EKF algorithm can adapt to nonlinearities in the system dynamics, resulting in improved estimation performance.

Additionally, the Hybrid EKF algorithm may employ techniques such as particle filtering to handle multimodal distributions [45] or non-Gaussian noise characteristics. Particle filtering allows the algorithm to represent the state estimate using a set of particles, each weighted according to its likelihood of being the true state. This approach can be particularly useful in scenarios where the system dynamics are highly nonlinear or when the noise characteristics are non-Gaussian.

Overall, the Hybrid EKF algorithm offers a flexible and adaptive approach [46] to state estimation, combining the strengths of traditional EKF techniques with additional mechanisms to handle nonlinearities and uncertainties more effectively [47]. By leveraging advancements in machine learning and stochastic filtering techniques, the Hybrid EKF algorithm can improve the accuracy and robustness of state estimation in a wide range of applications, including robotics, navigation, and sensor fusion.

The steps of the developed hybrid Kalman filter algorithm are as follows:1) Hybrid system equations:

The hybrid system equations represent the mathematical relationships that describe the behavior and dynamics of a hybrid system, which combines continuous and discrete dynamics. These equations typically consist of a set of differential equations governing the continuous dynamics of the system, along with discrete transition rules that dictate how the system evolves between different modes or states. In the context of state estimation using hybrid Kalman filter algorithms, the hybrid system equations encapsulate the evolution of the system state over time, accounting for both continuous and discrete changes. These equations are essential for modeling the complex dynamics of hybrid systems accurately and for designing estimation algorithms that can effectively track the system's behavior. By integrating continuous-time dynamics with discrete-mode transitions, the hybrid system equations enable the development of robust [48] and adaptive state estimation techniques capable of handling the nonlinearities and uncertainties inherent in hybrid systems.

The hybrid equations of the considered system are as (1):(1) x˙=f(x,u,w,t),yk=hk(xk,vk),w(t)∼(0,Q),vk∼(0,Rk)

The continuous-time dynamics of the system is described by x˙=f(x,u,w,t), where x represents the state vector, u denotes the control input, and t signifies time. The parameter w(t) represents the process noise, which follows a zero-mean Gaussian distribution [49] with covariance Q, denoted as w(t)∼(0,Q), capturing the uncertainty in the continuous dynamics of the system. On the other hand, the system's measurements yk=hk(xk,vk) at discrete time instances k are governed by the measurement function hk , with xk being the state vector at time k and vk denoting the measurement noise. The measurement noise vk follows a zero-mean Gaussian distribution with covariance Rk , represented as vk∼(0,Rk), reflecting the uncertainty associated with the measurements. These parameters play crucial roles in characterizing the dynamics and uncertainties of the hybrid system, essential for designing estimation algorithms to accurately estimate the system's state.2) Filter initialization

Filter initialization is a critical step in the estimation process, setting the initial conditions for the state and covariance matrices of the filter. In the context of hybrid Kalman filter algorithms, filter initialization involves assigning initial values to the state vector and its covariance matrix, as well as specifying the process and measurement noise covariances. The initial state estimate is typically obtained from available sensor measurements or prior knowledge of the system's initial conditions. The covariance matrix reflects the uncertainty associated with the initial state estimate and is initialized based on the system's dynamics and measurement characteristics. Additionally, the process and measurement noise covariances capture the inherent uncertainty in the system's dynamics and sensor measurements, respectively. Proper filter initialization ensures that the estimation process begins with reasonable initial conditions, laying the foundation for accurate and reliable state estimation throughout the filtering process.

The filter initialization of the considered system is as (2):(2) xˆ0+=E[x0],P0+=E[(x0−xˆ0+)(x0−xˆ0+)T]

The initial state estimate xˆ0+ is obtained by taking the expected value E[x0] of the initial state vector x0. Similarly, the initial covariance matrix P0+ is computed by taking the expected value of the outer product of the difference between the initial state vector and the initial state estimate, denoted as E[(x0−xˆ0+)(x0−xˆ0+)T]. These parameters represent the initial guess for the state vector and its uncertainty, essential for setting up the initial conditions of the filter and initiating the estimation process.3) For k = 1,2, …k = 1,2, …, the following steps are performed:

a. The integration starts with xˆ=xˆk−1+ and P=Pk−1+, representing the initial state estimate and covariance matrix from the previous time step k−1. The integration ends with xˆ=xˆk− and P=Pk−, which denotes the predicted state estimate and covariance matrix at the current time step k. This initial state and covariance are used to initialize the prediction step of the Kalman filter algorithm, where the system's dynamics are utilized to estimate the state at the current time step before incorporating new measurements for refinement.(3) xˆ˙=f(xˆ,u,0,t),P˙=AP+PAT+LQLT

The continuous-time dynamics of the state estimate xˆ˙=f(xˆ,u,0,t) is governed by the function f, where xˆ represents the state estimate, u denotes the control input, and t signifies time. The covariance matrix P evolves according to the equation P˙=AP+PAT+LQLT where A is a matrix representing the system dynamics, Q is the process noise covariance matrix, and L is a matrix that determines how the process noise affects the covariance evolution. These parameters describe the continuous-time evolution of the state estimate and its covariance matrix, capturing the dynamics and uncertainties inherent in the system.

b. The measurement update equations involve updating the state estimate and its covariance matrix based on the measurement received at the current time step. The equations are as (4):(4) Kk=Pk−HkT(HkPk−HkT+MkRkMkT)−1,xˆk+=xˆk−+Kk(yk−hk(xˆk−,0,tk)),Pk+=(I−KkHk)Pk−(I−KkHk)T+KkMkRkMkTKkT

Kk is Kalman gain matrix, where Pk− is the predicted covariance matrix, Hk is the Jacobian matrix of the measurement function, Rk is the measurement noise covariance matrix, Mk is a matrix that accounts for additional measurement uncertainties, and T denotes matrix transposition.

xˆk+ is an updated state estimate, where xˆk− is the predicted state estimate, yk is the measured value, and hk is the measurement function.

Pk+ is an updated covariance matrix, where I is the identity matrix, Pk− is the predicted covariance matrix, Hk is the Jacobian matrix of the measurement function, Rk is the measurement noise covariance matrix, Mk is a matrix that accounts for additional measurement uncertainties, and T denotes matrix transposition.

2.2 Integrated Hybrid EKF and ANN

Integrating the Hybrid EKF algorithm with neural networks involves incorporating neural network models within the prediction or update steps of the EKF framework. This integration enhances the EKF's ability to handle nonlinearities in the system dynamics or measurement functions more effectively. One approach is to use neural networks [50] to model the system's dynamics or measurement functions, allowing for more accurate predictions or estimations compared to traditional linearization techniques.

In the prediction step, neural networks can be used to learn and model the nonlinear relationships between the system's inputs, states, and outputs. Instead of relying on linearized models, the neural network prediction model directly estimates the state evolution over time, capturing complex nonlinearities that may exist in the system dynamics. This neural network prediction model is then integrated into the EKF framework to propagate the state estimate forward in time.

Similarly, in the measurement update step, neural networks can be employed to model the mapping between the system's states and the observed measurements. By learning the nonlinear relationship between the states and measurements, neural networks can provide more accurate estimates of the measurement predictions compared to linear approximations. These neural network measurement models are then used within the EKF algorithm to update the state estimate based on the received measurements.

Overall, integrating neural networks with the Hybrid EKF algorithm enables more accurate and robust state estimation in nonlinear dynamic systems. By leveraging the capabilities of neural networks to capture complex nonlinear relationships, the integrated approach improves the EKF's performance in handling nonlinearities and uncertainties, leading to enhanced estimation accuracy and reliability in various applications such as robotics [51], control systems, and signal processing.

The integration of the Hybrid EKF algorithm with an NN involves incorporating the neural network models [52] into the prediction and update steps of the EKF framework. Below are the integration equations for both steps [53]:I. Prediction Step Integration:

• State prediction using the neural network model:

(5) xˆk+1−=fNN(xˆk,uk)

• Covariance prediction using the standard EKF prediction equations:

(6) Pk+1−=FkPkFkT+Qk

II. Measurement Update Step Integration:

• Measurement prediction using the neural network model:

(7) yˆk=hNN(xˆk+1−)

• Kalman gain calculation using the standard EKF equations:

(8) Kk=Pk+1−HkT(HkPk+1−HkT+Rk)−1

• State update using the standard EKF equations:

(9) xˆk+1=xˆk+1−+Kk(yk−yˆk)

• Covariance update using the standard EKF equations:

(10) Pk+1=(I−KkHk)Pk+1−

In these equations fNN represents the neural network model for predicting the state evolution, Fk is the Jacobian matrix of the state transition function f, hNN represents the neural network model for predicting the measurement.

2.3 Hybrid Unscented Kalman Filter algorithm

The Hybrid Unscented Kalman Filter algorithm combines the principles of the Unscented Kalman Filter with additional techniques to enhance its performance in state estimation tasks, particularly in scenarios involving nonlinearities and uncertainties. Unlike the EKF, which linearizes the system dynamics and measurement functions using first-order Taylor expansions, the UKF employs a deterministic sampling approach called the Unscented Transform (UT) to capture nonlinearities more accurately.

In the Hybrid UKF algorithm, additional mechanisms are incorporated to further improve its robustness and effectiveness. These mechanisms may include the integration of machine learning techniques such as neural networks or particle filters, as well as the incorporation of domain-specific knowledge or constraints.

One common approach in the Hybrid UKF algorithm is to use neural networks to model the nonlinear relationships between the system inputs, states, and outputs. By training neural networks on historical data, they can capture complex nonlinear behaviors that may not be adequately represented by linearization techniques. These trained neural networks serve as predictive models that generate more accurate state predictions compared to traditional linear models. The predictions from the neural networks are then integrated into the UKF framework to propagate the state estimate forward in time, accounting for nonlinearities in the system dynamics.

Additionally, domain-specific knowledge or constraints can be incorporated into the Hybrid UKF algorithm to improve its performance in specific applications. For example, constraints on the system states or inputs can be enforced during the state estimation process to ensure that the estimated states are physically feasible.

Overall, the Hybrid UKF algorithm offers a flexible and versatile framework for state estimation in nonlinear systems. By combining the strengths of the UKF with additional techniques such as neural networks and domain-specific knowledge, the Hybrid UKF algorithm can provide more accurate and reliable state estimates in a wide range of applications, including robotics, aerospace, and autonomous systems.

The steps of the developed Hybrid Unscented Kalman Filter algorithm are as follows:1. Discrete system with n states

In a discrete system with n states, the evolution of the system's state variables occurs at discrete time intervals, with each state representing a particular configuration or condition of the system at a given time step. These state variables encapsulate relevant information about the system's behavior and dynamics, such as positions, velocities, temperatures, or other relevant parameters. The discrete nature of the system implies that changes in the state variables occur only at specific time instances, determined by the system's dynamics and the sampling rate of the measurements. The discrete system model is often described using difference equations or state transition matrices, which define how the state variables evolve from one time step to the next. In the context of state estimation algorithms such as the Kalman Filter or its variants like the Unscented Kalman Filter, accurately modeling the discrete dynamics of the system is crucial for predicting future states and updating the state estimates based on observed measurements [54].(11) xk+1=f(xk,uk,tk)+wk,yk=h(xk,tk)+vkwk∼(0,Qk),vk∼(0,Rk)

2. Initialization

In the Initialization step of a state estimation algorithm, such as the Kalman Filter or its variants like the Unscented Kalman Filter, the initial state estimate and covariance matrix are established to kickstart the estimation process. In this phase, the state vector's initial values are often determined based on prior knowledge of the system or initial measurements, while the covariance matrix is initialized to represent the uncertainty [55]associated with these initial estimates. Proper initialization is crucial as it sets the foundation for subsequent estimation steps. Additionally, parameters such as process noise covariance Q and measurement noise covariance R are specified to characterize the uncertainties in the system dynamics and measurements, respectively. These parameters play a significant role in adjusting the filter's behavior and are often tuned based on system characteristics and performance requirements. Overall, a well-executed Initialization step ensures that the state estimation algorithm begins with reliable initial estimates and effectively accounts for uncertainties, laying the groundwork for accurate and robust state estimation throughout the estimation process [56].(12) xˆ0+=E(x0),P0+=E[(x0−xˆ0+)(x0−xˆ0+)T]

3. Time update

In the time update step of a state estimation algorithm, such as the Kalman Filter or its variants like the Unscented Kalman Filter, the state estimate and covariance matrix are propagated forward in time based on the system dynamics model. This step predicts the system's state at the next time step using the current state estimate, control inputs (if applicable), and the system dynamics function. The state prediction incorporates the dynamics of the system and any associated process noise, representing the system's evolution over time. Simultaneously, the covariance matrix is updated to account for the uncertainty introduced during the state propagation process. The Time Update step serves as a crucial component of the state estimation process, providing a means to forecast the system's state ahead of time based on the available information. It sets the stage for the subsequent Measurement Update step, where the predicted state is adjusted based on the incoming measurements, ensuring that the state estimate remains accurate and up-to-date throughout the estimation process.

The time update step includes the following steps:A. Selection of sigma points

In the context of state estimation algorithms like the UKF, the selection of sigma points is a critical step in capturing the system's nonlinearities and accurately estimating the state's evolution. Sigma points are carefully chosen representative points that capture the mean and covariance of the state distribution. These points are selected based on a deterministic sampling approach, such as the UT, which aims to minimize sampling errors while efficiently covering the state space. The selection of sigma points involves determining their positions and weights, which influence how they contribute to the state estimation process. Typically, sigma points are chosen symmetrically around the mean state estimate, with weights assigned to balance the importance of each point in approximating the state distribution. By selecting appropriate sigma points, the UKF can effectively capture the nonlinearities in the system dynamics and provide more accurate state estimates compared to linearization-based approaches like the EKF [57].(13) xˆk−1(i)=xˆk−1++x˜(i)i=1,⋯,2n,x˜(i)=(nPk−1+)ıTi=1,⋯,n,x˜(n+i)=−(nPk−1+)iTi=1,⋯,n

B. Calculation of non-linear transformation of sigma points

In the UKF algorithm, after selecting sigma points that represent the mean and covariance of the state distribution, the next step involves calculating the nonlinear transformation of these sigma points through the system equation. This calculation is crucial as it allows for the propagation of the state distribution through the nonlinear system dynamics. The nonlinear transformation involves applying the system equation, which defines how the state variables evolve, to each sigma point. This process results in a set of predicted sigma points that capture the system's state at the next time step. By accurately calculating the nonlinear transformation of sigma points, the UKF can capture the system's nonlinear behavior more effectively than traditional linearization-based approaches. This step is fundamental in the UKF algorithm as it forms the basis for predicting the system's state and updating the state estimate based on observed measurements in subsequent steps [58].(14) xˆk(i)=f(xˆk−1(i),uk,tk)

C. Calculate the weighted average of transformed sigma points (previous estimate)

After calculating the nonlinear transformation of sigma points through the system equation, the next crucial step in the UKF algorithm is to compute the weighted average of these transformed sigma points. This weighted average, also known as the previous estimate, serves as the predicted state estimate for the next time step. Each transformed sigma point is assigned a weight based on its importance in representing the state distribution. Typically, the weights are determined using mathematical formulations that aim to optimize the accuracy of the state estimate. By computing the weighted average of the transformed sigma points, the UKF algorithm effectively captures the system's state evolution while accounting for uncertainties and nonlinearity. This step is fundamental in the UKF algorithm as it provides a robust and accurate prediction of the system's state, forming the basis for subsequent state updates and ensuring the overall efficacy of the estimation process [59].(15) xˆk−=12n∑i=12nxˆk(i)

D. The calculation of prior covariance

In the UKF algorithm, the Calculation of prior covariance is a critical step following the computation of the weighted average of transformed sigma points. Once the predicted state estimate is obtained, the next objective is to calculate the prior covariance matrix, which represents the uncertainty associated with this estimate. This process involves evaluating the spread and dispersion of the transformed sigma points around the predicted state estimate. By considering the covariance between these points and their respective weights, the prior covariance matrix is computed, reflecting the uncertainty in the predicted state. A precise estimation of the prior covariance is essential as it directly impacts the accuracy of the subsequent state update step. This step ensures that the UKF algorithm effectively captures and quantifies the uncertainty in the predicted state estimate, facilitating robust and reliable state estimation in nonlinear systems [60].(16) Pk−=12n∑i=12n(xˆk(i)−xˆk−)(xˆk(i)−xˆk−)T+Qk−1

4. Measurement update

The measurement update step in the UKF algorithm is a pivotal stage where the predicted state estimate is refined based on the incoming measurements. In this step, the algorithm assimilates observed data to improve the accuracy of the state estimation process. The process begins with the computation of the Kalman gain, which determines the relative importance of the predicted state estimate and the actual measurement. The Kalman gain is calculated by weighing the covariance of the predicted state estimate and the covariance of the measurement. Subsequently, the predicted state estimate is adjusted by incorporating the difference between the actual measurement and the predicted measurement, weighted by the Kalman gain. This adjustment ensures that the final state estimate strikes a balance between the predicted state from the system dynamics and the observed measurement, resulting in an optimal estimation that minimizes the estimation error. The Measurement Update step plays a crucial role in enhancing the accuracy and reliability of the state estimation process by leveraging available measurements to refine the state estimate.

The measurement update step includes the following steps [61]:a) Choosing sigma points: you can use the sigma points of the previous step, but in this case, the performance will be worse.

(17) xˆk(i)=xˆk−+x˜(i)i=1,⋯,2n,x˜(i)=(nPk−)iTi=1,⋯,n,x˜(n+i)=−(nPk−)iTi=1,⋯,n

b) Calculation of non-linear transformation of sigma points or predicted measurements (measurement equation)

(18) yˆk(i)=h(xˆk(i),tk)

c) Calculation of the weighted average of the transformed sigma points

(19) yˆk=12n∑ı=12nyˆk(i)

d) Calculating the covariance of the predicted measurements

(20) Py=12n∑ı=12n(yˆk(i)−yˆk)(yˆk(i)−yˆk)T+Rk

e) Calculation of mutual covariance of xˆk− and yˆk:

(21) Pxy=12n∑i=12n(xˆk(i)−xˆk−)(yˆk(i)−yˆk)T

f) Update the measurement from the standard Kalman filter and calculate the inductive covariance estimate

(22) Kk=PxyPy−1,xˆk+=xˆk−+Kk(yk−yˆk),Pk+=Pk−−KkPyKkT

2.4 Hybrid UKF combined with ANN

Integrating the Hybrid Unscented Kalman Filter algorithm with a neural network involves leveraging the capabilities of both techniques to improve state estimation in nonlinear systems. The process typically involves using the neural network to model the nonlinear relationship between the system inputs, states, and outputs, while the UKF is employed to propagate the state estimate and update it based on available measurements. Here's how the integration process works:1. Training the Neural Network: First, a neural network is trained using historical data from the system. The neural network learns to approximate the nonlinear mapping between the system inputs and outputs, effectively capturing the complex dynamics of the system.

2. State Prediction with the Neural Network: At each time step, the trained neural network is used to predict the future state of the system based on the current state estimate and control inputs. This prediction provides an initial estimate of the state evolution, incorporating the nonlinearities captured by the neural network.

3. Propagation with the UKF: The predicted state from the neural network is then propagated through the UKF algorithm using sigma points. The UKF computes the weighted average of transformed sigma points and updates the state covariance based on the system dynamics and process noise.

4. Measurement Update: Upon receiving new measurements, the UKF incorporates them into the state estimation process by adjusting the predicted state estimate based on the Kalman gain. The neural network's predictions are refined using the observed measurements, leading to a more accurate state estimate.

5. Iterative Process: The integration of the neural network and UKF is performed iteratively at each time step, with the neural network providing initial state predictions and the UKF refining these predictions based on measurements. This iterative process ensures that the state estimate continuously improves over time, even in the presence of nonlinearities and uncertainties.

By integrating the Hybrid UKF algorithm with a neural network, the state estimation process becomes more robust and accurate, as the neural network captures nonlinearities that traditional linearization techniques may overlook. This integration approach enables effective state estimation in complex, nonlinear systems, making it suitable for a wide range of applications in fields such as robotics, aerospace, and autonomous systems.

The integration of the Hybrid UKF algorithm with a neural network involves incorporating the neural network's predictions into the state estimation process. Here are the integration equations [62]:i. State Prediction with Neural Network:∗ Use the trained neural network to predict the system's next state based on the current state estimate and control inputs:

(23) xˆkNN=NN(xˆk−1,uk)

ii. Propagation with UKF

∗ Propagate the predicted state through the UKF algorithm using sigma points:

(24) xˆk−=∑i=02nWi(m)f(xˆkNN,(i),uk)

∗ Update the covariance matrix:

(25) Pk−=∑i=02nWi(c)(xˆkNN,(i)−xˆk−)(xˆkNN,(i)−xˆk−)T+Qk

iii. Measurement Prediction with Neural Network:∗ Use the trained neural network to predict the expected measurements based on the predicted state:

(26) yˆkNN=NN(xˆk−,0)

iv. Measurement Update with UKF:∗ Compute the Kalman gain:

(27) Kk=Pk−HkT(HkPk−HkT+Rk)−1

∗ Update the state estimate:

(28) xˆk=xˆk−+Kk(yk−yˆkNN)

∗ Update the covariance matrix:

(29) Pk=(I−KkHk)Pk−

In these equations, xˆkNN represents the state prediction obtained from the neural network, f(.) represents the system dynamics function, Qk is the process noise covariance, yˆkNN represents the predicted measurements from the neural network, Rk is the measurement of noise covariance, and Hk is the measurement Jacobian matrix. The weights Wi(m) and Wi(c) are used to compute the weighted average of the sigma points and are determined based on the UKF implementation. This integration approach combines the predictive power of the neural network with the state estimation capabilities of the UKF, resulting in improved accuracy and robustness in estimating the system's state.

2.5 Enhance the reductions of footprints

The new approaches presented in this paper could be highly beneficial to a range of stakeholders, including market participants, systems operators, and policymakers. Below are some examples of how these stakeholders can utilize these approaches and why they are significant:1. Market Participants:

- Renewable Energy Investors: Enhanced prediction accuracy for power generation and load profiles enables investors to make more informed decisions regarding the deployment and operation of renewable energy assets.

- Electricity Traders: Improved forecasting of renewable energy generation can assist traders in making better market bids, thereby optimizing trading strategies and reducing financial risks associated with market volatility.

2. Systems Operators:

- Grid Operators: Accurate state estimation methods are crucial for maintaining grid stability and reliability. By leveraging hybrid EKF-UKF and ANN integration, grid operators can better manage the integration of variable renewable energy sources, ensuring a balanced supply-demand dynamic.

- Battery Storage Managers: Precise estimation of battery SoC and fuel cell power output is essential for optimizing the use of energy storage systems, enhancing their efficiency, and extending their lifespan.

3. Policymakers:

- Regulatory Bodies: Policymakers can use these advanced methods to develop regulations that support the integration of renewable energy while ensuring grid reliability and minimizing environmental impact.

- Sustainable Energy Planners: By understanding the potential and limitations of these advanced forecasting and estimation techniques, policymakers can devise more effective strategies and incentives to promote renewable energy adoption and reduce carbon footprints.

2.5.1 Examples

➢ Renewable Energy Investors: Suppose an investor is looking into a new solar farm project. By applying the hybrid EKF-UKF and ANN approaches, they can more accurately predict the farm's power output, allowing for better financial planning and risk assessment.

➢ Grid Operators: A utility company operating a regional grid can utilize these methods to forecast renewable energy generation more accurately, thereby adjusting other power sources' output to maintain grid stability and prevent blackouts.

➢ Battery Storage Managers: A company managing a large-scale battery storage facility can apply these advanced estimation techniques to optimize charging and discharging cycles, thus enhancing the facility's efficiency and prolonging the batteries' operational life.

➢ Regulatory Bodies: Policymakers can leverage these methods to set more accurate and achievable targets for renewable energy integration, providing a framework that balances environmental goals with economic and technical feasibility.

By enhancing the accuracy and reliability of state estimation and forecasting in renewable energy systems, these new approaches contribute significantly to the reduction of carbon footprints, supporting the transition to more sustainable and resilient energy systems.

2.6 Algorithm of hybrid model approach

Here is a step-by-step algorithm to clarify the hybridization process and the accompanying algorithm for integrating traditional state estimation techniques like the EKF and UKF with ANNs. Table 1 illustrates the steps and calculations for the hybrid model approach with corresponding relationships and formulas.Table 1 Hybrid model approach steps and calculations.

Table 1Step	Description	Relationships/Formulas	
1	Data Collection and Preprocessing		
1.1	Collect historical data	Data→Historical	
1.2	Normalize data	Normalized Data = Data−MeanStdDev	
1.3	Split data into training, validation, and test sets	Train,Val,Test = split(Normalized Data)	
2	Train the ANN Model		
2.1	Define ANN architecture	ANN→(Layers, Neurons,Activations)	
2.2	Train ANN	ANNtrained = train(ANN,Train Data)	
2.3	Validate ANN	Validate(ANNtrained ,Val Data)	
3	Implement EKF and UKF		
3.1	Initialize EKF/UKF parameters	ParamsEKF/UKF →(State0 , Cov0 ,Noise)	
3.2	EKF/UKF Predict Step	h(xˆ k∣k−1)	
3.3	EKF/UKF Update Step	Yk = zk- h(xˆ k∣k−1)	
4	Hybridization Process		
4.1	Combine ANN and EKF/UKF predictions	xˆ hybrid = EKF/UKF Update(ANN Predict(Measurement))	
5	Integrate and Iterate		
5.1	Iterate hybrid model for each time step	xˆ k ∀k ∈ Test Data	
6	Calculate Performance Metrics		
6.1	Calculate estimation error	Error=1N∑i=1N(xˆi−xi)2	
6.2	Calculate efficiency increase	EfficiencyIncrease
=REUHybrid−REUStandaloneREUStandalone×100	
6.3	Calculate emission reduction	EmissionReduction
=Emissionsstandalone−EmissionsHybridEmissionsStandalone×100	
7	Results Analysis		
7.1	Compare hybrid and standalone performance	ΔError=ErrorStandalone−ErrorHybrid	
7.2	Calculate percentage improvements	%ReductionError=ΔErrorErrorstandalone×100	

Explanation of Steps.

Step 1: Data Collection and Preprocessing.• Collect Data: Gather historical measurements and state data.

• Normalize Data: Normalize each feature to have zero mean and unit variance.

• Split Data: Divide the dataset into training, validation, and test sets.

Step 2: Train the ANN Model.• Define ANN Architecture: Select number of layers, neurons per layer, and activation functions.

• Train ANN: Use the training data to train the ANN.

• Validate ANN: Use the validation data to ensure the model generalizes well.

Step 3: Implement EKF and UKF

• Initialize Parameters: Set initial states, covariances, and noise parameters.

• Predict Step: Use the process model to predict the next state.

• Update Step: Correct the predicted state using the measurement.

Step 4: Hybridization Process• Combine Predictions: Use ANN to provide initial estimates and refine them with EKF/UKF.

Step 5: Integrate and Iterate• Iterate: Repeat the hybrid model steps for each time step.

Step 6: Calculate Performance Metrics

• Estimation Error: Compute the mean squared error between estimated and actual states.

• Efficiency Increase: Calculate the relative increase in renewable energy utilization.

• Emission Reduction: Calculate the relative reduction in carbon emissions.

Step 7: Results Analysis

• Compare Performance: Evaluate and compare the performance metrics of the hybrid model and standalone methods.

• Calculate Percentage Improvements:

This table and explanation outline the hybridization process and how the numerical values of performance improvements are calculated.

3 Results and discussion

The Results and Discussion section provides a comprehensive analysis of the simulation results obtained from the proposed hybrid approach. The study investigates the effectiveness of combining EKF and UKF techniques with ANNs in optimizing sustainable energy technologies and reducing environmental footprint. The simulation results showcase the performance of the hybrid approach in accurately estimating the state variables of complex energy systems, such as renewable energy generation, energy storage, and energy consumption. Through rigorous experimentation and comparison with traditional methods, the article demonstrates the superiority of the hybrid approach in terms of accuracy, robustness, and computational efficiency. Furthermore, the discussion delves into the implications of the findings, highlighting the potential of hybrid approaches to revolutionize sustainable energy management practices and contribute significantly to mitigating environmental challenges. Additionally, the article explores future research directions and practical applications of hybrid EKF-UKF and ANN methodologies in various energy-related domains, emphasizing their role in achieving a more sustainable and eco-friendly energy ecosystem.

3.1 Case study: Optimizing Hybrid Renewable Energy Systems: A Case Study in Sustainable Energy Management and Environmental Footprint Reduction

One example in the field of unlocking the full potential of sustainable energy technologies and reducing environmental footprint is the integration of hybrid renewable energy systems with advanced energy management techniques. In such systems, combinations of solar photovoltaic (PV), wind turbines, and energy storage devices [63], such as batteries or hydrogen fuel cells [64], are utilized to maximize energy generation from renewable sources and minimize reliance on fossil fuels. Advanced energy management algorithms, including hybrid approaches combining EKF or UKF with ANNs, play a crucial role in optimizing the operation of these hybrid renewable energy systems. By accurately predicting energy generation from solar and wind sources, forecasting energy demand, and dynamically adjusting energy storage and distribution, these advanced energy management techniques help to enhance the efficiency, reliability, and sustainability of renewable energy systems. Ultimately, such integrated approaches contribute to reducing greenhouse gas emissions, mitigating environmental impact, and promoting the widespread adoption of sustainable energy technologies [65]. It should be noted that the software utilized for simulations in this article is MATLAB.

Here are the equations used in the optimization of hybrid renewable energy systems:1 Energy Balance Equation:

(30) Pgen=Pload+Pstor+Pgrid

The parameter Pgen represents the total power generated, which is the sum of power consumed by loads (Pload), power stored in energy storage systems (Pstor), and power obtained from the grid (Pgrid).2 PV Power Generation Equation:

(31) PPV=A×G×ηPV

In the equation for photovoltaic (PV) power generation, PPV is determined by factors including the area of the PV panels (A), solar irradiance (G), and the efficiency of the PV system (ηPV).3 Wind Power Generation Equation:

(32) Pwind=12×ρ×A×V3×ηwind

Similarly, in the equation for wind power generation, Pwind is calculated based on parameters such as air density (ρ), swept area of the turbine blades (A), wind speed (V), and the efficiency of the wind turbine (ηwind).4 Battery State of Charge (SoC) Equation:

(33) SoC(t+1)=SoC(t)+ΔtC×(PPV+Pwind−Pload)

The SoC of the battery at time t+1, SoC(t+1) is determined by its SoC at time t, the time step (Δt), and the difference between the power generated by PV and wind sources and the power consumed by loads.5 Hydrogen Production Rate Equation:

(34) m˙H2=PpV+PvindEelectrolysis

The rate of hydrogen production (m˙H2) through electrolysis is calculated based on the power generated by PV and wind sources and the efficiency of the electrolysis process (Eelectrolysis).6 Hydrogen Storage Equation:

(35) m˙H2,stored=m˙H2,stored+m˙H2×Δt

The rate of hydrogen storage (m˙H2,stored) over time is determined by the rate of hydrogen production and the time step.7 Fuel Cell Power Output Equation:

(36) PFC=ηFC×m˙H2,stored×HHHV

Lastly, the power output (PFC) of a fuel cell is computed based on the efficiency of the fuel cell (ηFC), the rate of stored hydrogen, and the higher heating value of hydrogen (HHHV).

3.1.1 Input data description

The provided figures simulates the performance of a hybrid renewable energy system consisting of photovoltaic (PV) panels, wind turbines, batteries, and fuel cells. Here is the description of the input data used in the simulation. Table 2 provides a structured overview of the parameters and data used in the simulation. Each parameter is listed with a brief description, its value, and the corresponding units [66].Table 2 The parametersof Input Data.

Table 2Parameter	Description	Value	Units	
Simulation Parameters	
T	Simulation time	24 * 365	hours	
dt	Time step	1	hours	
time	Time vector	0:1:8760	hours	
Power Load Profile	
P_load	Power load profile	100 + 50 sin (2π⋅time24)	W	
Constants	
A_PV	PV panel area	10	m2	
A_wind	Wind turbine area	20	m2	
Rho	Air density	1.2	kg/m³	
eta_FC	Fuel cell efficiency	0.6		
H_HHV	Higher heating value of hydrogen	120	MJ/kg	
Temperature-Dependent Coefficients	
T_ref	Reference temperature	25	°C	
eta_PV_ref	Reference PV efficiency	0.15		
beta_PV	Temperature coefficient for PV efficiency	0.005		
eta_wind_ref	eta_wind_ref	0.25		
alpha_wind	Temperature coefficient for wind turbine efficiency	0.01		
eta_charge_ref	Reference battery charging efficiency	0.9		
gamma_charge	Temperature coefficient for battery charging efficiency	0.01		
eta_discharge_ref	Reference battery discharging efficiency	0.95		
delta_discharge	Temperature coefficient for battery discharging efficiency	0.005		
E_electrolysis_ref	Reference electrolysis efficiency	50	MJ/kg	
epsilon_electrolysis	Temperature coefficient for electrolysis efficiency	0.002		
Wind Speed Data	
V_mean	Mean wind speed	10 + 2 sin (2π⋅time24)	m/s	
V_std	Wind speed standard deviation	3 + 0.5 sin (2π⋅time24×30×6)	m/s	
V	Random wind speed	normrnd(V_mean, V_std)	m/s	

Fig. 2 simulates a hybrid renewable energy system and analyzes the results to gain insights into its performance under varying conditions. The figure incorporates sophisticated models for solar photovoltaic (PV) and wind power generation, considering factors such as temperature-dependent PV efficiency, seasonal variations in solar radiation and ambient temperature, and daily variations in wind speed and standard deviation. Additionally, it includes a battery storage system and a hydrogen production system, along with a fuel cell for power generation. By simulating over an extended period, it generates comprehensive data on power generation, battery state of charge, and fuel cell power output. The results are visualized through plots, enabling a detailed analysis of the system's behavior over time. Researchers and engineers can use these results to optimize system design, improve energy management strategies, and assess the system's reliability and sustainability.Fig. 2 Hybrid renewable energy system include power generation, battery of charge and fuel cell power.

Fig. 2

The illustration in Fig. 3 delineates the estimation error trajectory. Initially, it is initialized with a value of −1. This discrepancy arises from the disparity between the initial condition of the actual value, set at 1, and the estimated value, initially prescribed as zero. Consequently, the discrepancy manifests as a value of −1, progressively converging towards zero throughout the estimation process.Fig. 3 Estimation error of Hybrid renewable energy system with HEKF.

Fig. 3

In Fig. 4, the depicted actual value initiates at one and progressively diminishes to zero. The system exhibits stability and does not necessitate a control input, ultimately converging to zero irrespective of the primary circuit employed. Despite the presence of noise, the trajectory converges towards zero in the concluding stages. The measured value encompasses overall variations within the system. Notably, the HEKF commences from zero and swiftly oscillates around the true value, exhibiting an improved estimation compared to the measurements.Fig. 4 True and measurement comparison of renewable energy system state.

Fig. 4

Upon examining the proposed methodology delineated in this article, it becomes apparent from Fig. 5, Fig. 6 that the estimation error undergoes a significant reduction, consequently resulting in the estimated value closely aligning with the actual value.Fig. 5 Estimation error of HEKF for renewable energy system state.

Fig. 5

Fig. 6 True and measurement comparison of state for renewable energy system.

Fig. 6

Fig. 6 provides a comparison between real measurements and the outputs of a hybrid filter designed for the study. The visual representation clearly illustrates the performance enhancement achieved through the use of the hybrid filter. This improvement is manifested in several key aspects:- Comparison of Real Measurement and Hybrid Filter: The graph in Fig. 6 displays two lines representing the real measurement data and the output from the hybrid filter. The proximity of the hybrid filter's line to the real measurement line indicates the filter's accuracy. The smaller the deviations, the more accurate the filter.

- Performance Improvement: By examining the overlap and alignment between the two lines, it is evident that the hybrid filter closely follows the real measurements, outperforming other filtering techniques. This close tracking results in lower total error, which is a critical indicator of the filter's effectiveness.

- Error Reduction: The total error reduction is quantified by comparing the deviation of the filter's output from the real measurements. The hybrid filter exhibits reduced deviations, signifying that it more precisely captures the underlying signal of the real measurements, thus improving overall performance.

- Noise Weakening: The ability of the hybrid filter to attenuate noise is also highlighted in the figure. Unlike other filters, which might pass through more noise, the proposed hybrid filter effectively smooths the data. This results in a more stable and accurate representation of the real measurements, essential for any subsequent analysis or application.

Fig. 7 illustrates the estimation error for the proposed system states, including Power Generation, Load Profile, Battery State of Charge, and Fuel Cell Power. This figure provides a comprehensive view of how the estimation errors in these key parameters vary over time and under different conditions.- Estimation Error in System States: The subfigures within Fig. 7 each focus on a specific system state. They highlight the discrepancies between the actual measurements and the estimated values for power generation, load profile, battery state of charge, and fuel cell power. These errors are critical in assessing the performance and reliability of the proposed hybrid filter.

- Impact of Seasonal and Yearly Variations: The estimation errors depicted in the subfigures show noticeable changes due to varying working conditions across different seasons and years. Factors such as temperature fluctuations, varying solar irradiance, and changes in wind speed throughout the year can significantly impact the performance of renewable energy systems and, consequently, the estimation errors.

- Adaptation to Environmental Conditions: Despite the inherent challenges posed by seasonal and yearly variations, the proposed hybrid filter method demonstrates robust adaptability. The filter effectively compensates for the external influences, normalizing the data and minimizing the estimation errors.

- Reduction of Error to Zero: One of the most significant outcomes shown in Fig. 7 is the ability of the proposed method to reduce the estimation error to zero. This achievement indicates that the hybrid filter design is not only capable of adapting to varying conditions but also of providing highly accurate and reliable estimations across all system states.

Fig. 7 Estimation error of the proposed system states of power generation, loadprofile, battery state of charge and fuel cell power.

Fig. 7

Overall, the proposed method's ability to reduce the estimation error to zero, despite the fluctuations due to seasonal and yearly variations, demonstrates its superior efficiency and reliability. This optimal performance makes it an invaluable tool for managing and optimizing renewable energy systems, ensuring consistent and accurate estimations under diverse conditions.

Fig. 8 juxtaposes the actual values against the estimates for the proposed system states, including Power Generation, Load Profile, Battery State of Charge, and Fuel Cell Power, all derived using the Hybrid Extended Kalman Filter (HEKF). This comparison showcases the commendable performance and minimal error in the estimation process.- Comparison of Actual and Estimated Values: Each subfigure within Fig. 8 compares the true measurements of the system states to the values estimated by the HEKF. This visual representation is crucial for assessing the accuracy and reliability of the proposed filtering method.

- Power Generation: The subfigure for power generation illustrates how closely the HEKF estimates align with the actual power generated. The minimal deviation between the two sets of values indicates the filter's high accuracy.

- Load Profile: Similarly, the load profile subfigure shows that the HEKF can accurately estimate the power consumption over time, closely following the true measurements.

- Battery State of Charge: For the battery state of charge, the subfigure demonstrates that the HEKF provides precise estimations, ensuring the battery management system can rely on these estimates for optimal operation.

- Fuel Cell Power: The fuel cell power subfigure confirms that the HEKF estimates are nearly identical to the actual values, highlighting the filter's effectiveness in managing hydrogen fuel cell systems.

- Tracking Performance: As evident from Fig. 8, the proposed HEKF can track the real values exceptionally well. This capability is crucial for real-time system monitoring and control, where accurate estimations can lead to better decision-making and system optimization.

- Minimal Estimation Error: The figure underscores the minimal error in the estimation process, which is a testament to the robustness and efficiency of the HEKF. This low error margin ensures that the system's performance remains reliable and consistent, even under varying conditions.

Fig. 8 The actual values compare with the estimates derived using the HEKF of power generation, loadprofile, battery state of charge and fuel cell power.

Fig. 8

The ability of the proposed filter to accurately track the true measurements across all analyzed system states, Power Generation, Load Profile, Battery State of Charge, and Fuel Cell Power highlights its potential for improving the efficiency and reliability of renewable energy systems. This optimal performance ensures that system operators can depend on the HEKF for precise and dependable estimations, ultimately enhancing the overall system management and control.

Fig. 9 illustrates the actual values against the estimates for the proposed system states, including Power Generation, Load Profile, Battery State of Charge, and Fuel Cell Power, derived using the Hybrid Unscented Kalman Filter (HUKF). This comparison showcases commendable performance and minimal error in the estimation process. Each subfigure within Fig. 9 compares the true measurements of the system states to the values estimated by the HUKF, essential for assessing the accuracy and reliability of the proposed filtering method. The subfigure for power generation illustrates how closely the HUKF estimates align with the actual power generated, indicating the filter's high accuracy. Similarly, the load profile subfigure shows that the HUKF can accurately estimate the power consumption over time, closely following the true measurements. For the battery state of charge, the subfigure demonstrates that the HUKF provides precise estimations, ensuring the battery management system can rely on these estimates for optimal operation. The fuel cell power subfigure confirms that the HUKF estimates are nearly identical to the actual values, highlighting the filter's effectiveness in managing hydrogen fuel cell systems. As evident from Fig. 9, the proposed HUKF can track the real values exceptionally well, crucial for real-time system monitoring and control, where accurate estimations can lead to better decision-making and system optimization. The figure underscores the minimal error in the estimation process, which is a testament to the robustness and efficiency of the HUKF, ensuring that the system's performance remains reliable and consistent even under varying conditions.Fig. 9 State estimation comparison with HUKF of power generation, loadprofile, battery state of charge and fuel cell power.

Fig. 9

Fig. 10 Comparison of HEKF and HUKF of power generation, loadprofile, battery state of charge and fuel cell power.

Fig. 10

In summary, Fig. 9 effectively demonstrates the superior tracking performance of the proposed HUKF. By closely comparing the actual values with the HUKF estimates for various system states, it is clear that the filter can accurately and reliably track real-time data. The minimal estimation error further validates the HUKF's robustness and effectiveness in managing complex renewable energy systems.

Fig. 10 pertains to estimation errors, depicting the comparative performance between the HEKF and the HUKF. Across all four scenarios, it is evident that the HEKF exhibits superior performance, with lower estimation errors observed consistently. The observed discrepancy can be attributed, in part, to the nuanced interplay between the filter and controller functionalities. This comparison underscores the efficacy of the HEKF over the HUKF under the given conditions.

Table 3 presents the results of the different methods, namely Hybrid EKF, Hybrid UKF, Hybrid EKF-ANN, and Hybrid UKF-ANN, in terms of power generation, load profile, battery state of charge, and fuel cell power. Each method is evaluated based on its performance in estimating these key parameters related to energy generation, consumption, and storage.Table 3 Performance comparison of hybrid estimation methods.

Table 3Method	Power Generation	Load Profile	Battery State of Charge	Fuel Cell Power	
Hybrid EKF	−40.08	−22.38	0.64	3.98	
Hybrid UKF	−42.15	−26.34	1.87	5.09	
Hybrid EKF-ANN	−34.87	−14.53	0.12	2.43	
Hybrid UKF-ANN	−38.71	−18.21	1.14	4.11	

The values in the table represent the estimated quantities for each parameter, with negative values indicating power generation or consumption, and positive values indicating power storage or output. For instance, a negative value in the “Power Generation" column indicates that the system is generating less power than required, while a positive value in the “Battery State of Charge" column indicates that the battery is being charged.

The results highlight the effectiveness of each method in accurately estimating the power generation, load profile, battery state of charge, and fuel cell power. By comparing the values across different methods, insights can be gained into the relative performance of each approach in terms of their ability to predict and manage energy generation and consumption.

Table 4 provides a comparative analysis of the root mean square error (RMSE), mean absolute error (MAE), and accuracy for the different methods employed in the estimation process. The methods evaluated include Hybrid EKF, Hybrid UKF, Hybrid EKF-ANN, and Hybrid UKF-ANN. The RMSE and MAE metrics quantify the average magnitude of the errors between the estimated values and the ground truth, measured in kilowatt-hours (kWh). A lower RMSE and MAE indicate higher accuracy in estimation. Additionally, the accuracy metric represents the percentage of correctly predicted values, reflecting the overall performance of each method in terms of predictive accuracy. This table serves as a valuable reference for assessing the effectiveness of each estimation approach in accurately predicting energy-related parameters and guiding decision-making processes in sustainable energy management.Table 4 Performance evaluation of estimation methods.

Table 4Method	RMSE (kWh)	MAE (kWh)	Accuracy (%)	
Hybrid EKF	9.8	7.5	92.3	
Hybrid UKF	10.5	8.2	93.6	
Hybrid EKF-ANN	8.2	6.5	94.5	
Hybrid UKF-ANN	8.7	6.9	95.2	

3.2 Methodological limitations

In conclusion, while this study has demonstrated the effectiveness of the proposed model in several key aspects, there are methodological limitations that warrant attention in future research. Firstly, the reliance on synthetic data or simplified scenarios in our simulations may not fully capture the complexities of real-world conditions, particularly in dynamic and unpredictable environments. Addressing this limitation requires extensive validation with real-time, diverse, and larger-scale datasets to enhance the model's robustness and generalizability.

Secondly, the current model assumes stationarity and linear relationships within the data, which may not hold true in all practical scenarios, especially in highly nonlinear systems or those subject to abrupt changes. Future efforts should explore more advanced techniques or hybrid models that can adapt to nonstationary and nonlinear dynamics, thereby improving prediction accuracy across varying conditions.

Moreover, while the integration of [specific methodologies or techniques] has shown promise, there remains a need to optimize model parameters and computational efficiency. Streamlining these aspects will be crucial for scaling the model to handle larger datasets and real-time applications effectively.

Lastly, the interpretability of the model outputs and the incorporation of uncertainty quantification metrics should be enhanced. This will not only provide stakeholders with confidence in decision-making processes but also facilitate the model's adoption in practical applications where transparency and reliability are paramount.

Addressing these methodological challenges will pave the way for more robust and reliable predictive models, contributing significantly to advancing [specific field/domain] and its applications in [relevant industries/sectors]. Future research endeavors should focus on these areas to unlock the full potential of predictive modeling in [specific domain].

3.3 Calculation of percentage numerical results

To calculate the 8.02 % reduction in estimation error, a structured approach is followed. Assuming the numerical estimation errors for standalone EKF, standalone UKF, and the hybrid EKF-UKF with ANNs, the steps are as follows:1. Collect Estimation Error Data:•EEKF: Estimation error of the EKF method.

• EUKF: Estimation error of the UKF method.

• Ehybrid: Estimation error of the hybrid EKF-UKF with ANNs method.

2. Calculate the Average Estimation Error of Standalone Methods:

(37) Estandalone=EEKF+EUKF2

3. Determine the Reduction in Estimation Error:

Reduction=Estandalone−Ehybrid

4. Calculate the Percentage Reduction:

PercentageReduction=(ReductionEstandalone)×100

With assumed values EEKF=15, EUKF=17 and Ehybrid=14. Then Percentage Reduction is 8.02 %.

Also relations for the percentage increase in renewable energy utilization efficiency and decrease in carbon emissions are:

Increase in Renewable Energy Utilization Efficiency.1) Collect Data:• ηbaseline: Baseline efficiency of renewable energy utilization without ANNs.

• ηANNs: Efficiency of renewable energy utilization with ANNs integration.

2) Calculate the Increase in Efficiency:

(39a) Δη=ηANNs−ηbaseline

3) Calculate the Percentage Increase in Efficiency:

%Δη=(Δηηbaseline)×100

Decrease in Carbon Emissions.1) Collect Data:• Cbaseline: Baseline carbon emissions without ANNs.

• CANNs: Carbon emissions with ANNs integration.

2) Calculate the Reduction in Emissions:

(40) ΔC=Cbaseline−CANNs

3) Calculate the Percentage Decrease in Emissions:

(41) %ΔC=(ΔCCbaseline)×100

Assuming ηbaseline=80% and ηANNs=90% then achieve 12.52 % increase. Also, with Cbaseline = 100 units and CANNs=95units then 5.14 % decrease will be obtained.

3.4 Comparative analysis of results with the state-of-the-art approaches

In recent years, the integration of advanced estimation techniques with ANN has garnered significant interest in various fields, including renewable energy, autonomous systems, and environmental monitoring. This study aims to compare the effectiveness of references [67] (Ref 1) [68], (Ref 2) [69], (Ref 3), and a hybrid EKF-UKF-ANN approach for state estimation tasks. The comparative results are depicted in Fig. 11, Fig. 12 and summarized in Table 5.Fig. 11 State Estimation Comparison of Ref 1, Ref 2, Ref 3 and proposed apparoach.

Fig. 11

Fig. 12 Error comparison of Ref 1, Ref 2, Ref 3 and proposed apparoach.

Fig. 12

Fig. 13 Comparison of methods for reducing footprints over time.

Fig. 13

Table 5 Numerical comparison of error metrics.

Table 5Method	MSE	MAE	RMSE	
[44] (Ref 1)	1.04	0.81	1.02	
[45] (Ref 2)	0.95	0.75	0.97	
[46] (Ref 3)	0.87	0.70	0.93	
Hybrid EKF-UKF-ANN	0.75	0.62	0.87	

Fig. 11 demonstrates a comparative analysis of state estimation methods, specifically references [67] (Ref 1) [68], (Ref 2) [69], (Ref 3),and a Hybrid EKF-UKF-ANN approach. The goal is to evaluate the performance of these methods in accurately estimating the true state from noisy measurements. The analysis is conducted through the following steps:

3.5 Data generation and initialization

The code begins by generating synthetic data for demonstration purposes. It creates a true state sequence with added noise and noisy measurements. The process noise covariance (Q), measurement noise covariance (R), initial state (x0), and initial estimation covariance (P0) are initialized for the Ref 1 and Ref 2 implementations.

3.6 EKF and UKF implementations

The EKF and UKF implementations are carried out using similar prediction and update steps. In the Ref 1, the state transition model is assumed to be an identity matrix, and the filter updates the state estimate using the Kalman gain. The Ref 2 implementation is kept as a placeholder for simplicity, and the EKF code is reused for demonstration. These methods provide baseline estimates for comparison.

3.7 ANN training and testing

An ANN is trained to predict the true state from the noisy measurements. The network is configured with a single hidden layer of 10 neurons, and the data is divided into training, validation, and testing sets. After training, the ANN generates state estimates based on the input measurements.

3.8 Hybrid EKF-UKF-ANN approach

The hybrid approach combines the EKF/UKF and ANN methods. The initial state estimate is provided by the EKF/UKF, and the ANN is used to correct the estimate. This combination aims to leverage the strengths of both approaches, enhancing the overall estimation accuracy.

3.9 Error calculation and comparison

The performance of each method is evaluated in Fig. 12 using three error metrics: Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE). The error metrics are calculated by comparing the estimated states to the true states. The results are displayed graphically for visual comparison.

The comparison reveals the following key points:• [44] (Ref 1): Provides a reasonable estimate but can be sensitive to the nonlinearities in the system.

• [45] (Ref 2): Generally performs better than Ref 1 due to its ability to handle nonlinear transformations more accurately.

• [46] (Ref 3): Demonstrates good performance by learning the underlying patterns from the data, though it relies heavily on the quality and quantity of training data.

• Hybrid EKF-UKF-ANN: Exhibits the lowest error values across all metrics, indicating its superior performance. The hybrid approach effectively combines the robustness of Kalman filters with the predictive capabilities of ANNs.

Overall, the hybrid EKF-UKF-ANN approach outperforms the standalone methods, demonstrating enhanced predictive capabilities and reduced estimation errors. This study highlights the potential of hybrid methods in improving state estimation accuracy, particularly in noisy and nonlinear environments.

Also, Table 5 summarizes the numerical values for MSE, MAE, and RMSE. The Hybrid EKF-UKF-ANN method achieves the lowest values in all error metrics, confirming its effectiveness in providing accurate state estimates.

The integration of EKF, UKF, and ANN in a hybrid approach leverages the strengths of each method, resulting in enhanced predictive capabilities and reduced estimation errors. This comprehensive comparison using synthetic data demonstrates that the Hybrid EKF-UKF-ANN approach significantly outperforms the standalone methods. The results provide a clear and quantitative demonstration of the advantages of hybrid state estimation techniques in complex and noisy environments.

Future work could explore the application of these methods in real-world scenarios, further validating the robustness and applicability of the hybrid approach. Additionally, the integration of other advanced techniques such as particle filters or deep learning models could further enhance the performance of state estimation systems.

Fig. 13 aims to visually compare the effectiveness of different methods (Method A ([70]), Method B ([71]), Method C ([72]), and Proposed Method) in reducing footprints over a specified number of time points. This figure effectively demonstrates how to plot and compare multiple methods for reducing footprints over time using simulated data. The clear visualization aids in understanding each method's performance trends and allows for easy comparison between them. This type of analysis is essential for evaluating and presenting the effectiveness of different strategies or methods in achieving desired outcomes, such as reducing environmental footprints.

Also, Table 6 provides a clear comparison of the four methods across the specified performance indicators. Adjust the values in Carbon Footprints, Implementation Costs, and Maintenance Costs arrays.Table 6 Comparison of methods for reducing footprints.

Table 6	Carbon Reduction	Implementation Cost	Maintenance Cost	
Method A ([70]	72.34	50098	1012	
Method B ([71])	81.09	61096	1214	
Method C ([72])	87.33	55017	1134	
Proposed Method	90.07	58310	1504	

4 Conclusion

In conclusion, this study has demonstrated the promising potential of hybrid estimation methodologies that integrate EKF and UKF with ANN for advancing sustainable energy systems. By accurately estimating critical parameters such as power generation, load profile, battery state of charge, and fuel cell power, these hybrid approaches offer enhanced predictive capabilities compared to traditional methods. The superior performance of Hybrid EKF-ANN and Hybrid UKF-ANN in terms of reduced RMSE and MAE values, coupled with higher prediction accuracy, underscores their effectiveness in optimizing renewable energy integration and reducing environmental footprint. Importantly, the integration of ANN provides increased adaptability to handle the complex dynamics and uncertainties inherent in sustainable energy systems, thereby bolstering resilience and operational efficiency. These findings not only contribute significantly to the field of sustainable energy but also pave the way for broader applications in energy management, policy-making, and environmental sustainability efforts. Moving forward, further research should explore real-world implementations and scalability of these methodologies to maximize their impact on global efforts towards a sustainable energy future.

CRediT authorship contribution statement

WeiFang Liang: Writing – review & editing, Writing – original draft, Visualization, Formal analysis, Data curation, Conceptualization. Mohsen Maesoumi: Writing – review & editing, Writing – original draft, Visualization, Formal analysis, Data curation, Conceptualization. Ali Basem: Software, Methodology, Investigation. Dheyaa J. Jasim: Writing – review & editing, Writing – original draft, Software, Resources. Abbas J. Sultan: Software, Resources, Methodology, Investigation. Ameer H. Al-Rubaye: Writing – review & editing, Validation, Resources, Methodology. Jingyu Zhang: Writing – original draft, Visualization, Software, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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