AI-Driven Digital Twin Model for Reliable Lithium-Ion Battery Discharge Capacity Predictions

Te present study proposes a novel method for predicting the discharge capabilities of lithium-ion (Li-ion) batteries using a digital twin model in practice. By combining cutting-edge machine learning techniques, such as AdaBoost and long short-term memory (LSTM) network, with a semiempirical mathematical structure, the digital twin (DT)—a virtual representation that mimics the behavior of actual batteries in real time is constructed. Various metaheuristic optimization methods, such as antlion, grey wolf optimization (GWO), and improved grey wolf optimization (IGWO), are used to adjust hyperparameters in order to optimize the models. As indicators of performance, mean absolute error (MAE) and root-mean-square error (RMSE) are applied to the models after they have undergone extensive training and ten-fold cross-validation. Te models are rigorously trained and cross-validated using the NASA battery aging dataset, a widely accepted benchmark dataset for battery research. Te IGWO-AdaBoost digital twin model emerges as the standout performer, achieving exceptional accuracy in predicting the discharge capacity. Tis model demonstrates the lowest mean absolute error (MAE) of 0.01, showcasing its superior precision in estimating discharge capabilities. Additionally, the root mean square error (RMSE) for the IGWO-AdaBoost DTmodel is also the lowest at 0.01. Te fndings of this study ofer insightful information about the potential utilization of the digital twin model to accurately predict the discharge capacity of batteries.


Introduction
In the modern world, greenhouse gas emissions and their efects on global warming continue to be major concerns.To address this challenge, nations worldwide have implemented robust measures aimed at reducing waste emissions and promoting a sustainable future [1][2][3].Energy storage devices, frequently utilizing batteries, are essential elements in addressing the need for more sustainable energy sources.Energy storage systems not only mitigate reliance on nonrenewable energy sources but also enhance the stability and cost-efectiveness of renewable energy [4].Batteries fnd extensive utilization across a wide spectrum of applications throughout the domain of renewable energy, apart from their primary roles in energy storage and transmission.Additionally, they are employed to energize of-grid infrastructure, such as telecommunications and meteorological stations [5,6].Te degradation of battery performance is a result of the repeated charging and discharging cycles [7].As per the fndings of studies [8], the malfunctioning of an aged battery has the potential to result in considerable loss of human life and fnancial harm.Te evaluation of the health status of a lithium-ion battery, commonly referred to as the state of health (SOH), is of utmost importance in the reduction of system risks and the minimization of maintenance costs [9,10].Vigilantly monitoring the degradation of a battery is essential to maximizing its energy delivery, preventing premature failures, and improving its dependability and lifetime.Within a battery management system (BMS), this oversight involves a meticulous evaluation of the battery's state of health (SOH), an accurate prediction of its state of charge (SOC), and a precise estimation of its remaining useful life (RUL).A considerable investigation has been carried out to assess SOH in various studies.Capacity directly measures the quantitative assessment of a battery's current state in relation to its ideal state, known as SOH.Battery modeling is required to establish correlations between various battery operating characteristics, such as temperature, life cycle, and charging and discharging voltage.Te electrochemical model proposed by Goebel et al. [11] employs electrochemical impedance spectrometry to assess the internal impedance of the battery.Te model establishes a negative linear correlation between the battery's impedance and capacity to determine its overall capacity.Te battery aging model proposed by Daigle and Kulkarni [12], which is based on electrochemical principles, was subjected to a series of randomized discharge patterns for testing purposes.
In the past twenty years, machine learning (ML) has been a popular method for creating accurate prediction models.Numerous engineering felds have benefted from its successful applications, including the prediction of bearing degradation [13], RUL estimation [14], and the prediction of fractures in welded connections [15].ML is a data analysis technique that automates the creation of analytical models.It focuses on the idea that machines are capable of learning from information, identifying trends, and making judgments or predictions with little assistance from humans.Computational methodologies and AI techniques have signifcantly advanced the domain of battery systems.Patil et al. [16] conducted a study to determine the RUL of Li-ion batteries from the features extracted by discharge cycles and support vector machine.A model based on a feed-forward neural network to track estimated lithium-ion battery RUL is proposed in [17,18].Testing and numerical evaluations are conducted in order to verify the accuracy and validity of the proposed method.Te fndings indicate that the proposed methodologies exhibit a high degree of accuracy when applied in practical scenarios.Wang et al. [19] suggested a better robust multitime scale singular fltering-Gaussian process regression-long short-term memory modeling method for fguring out the remaining life-cycle capacity.In another study, Deng et al. [20] proposed a battery capacity prognostic method based on charging data and data-driven algorithms.Gaussian process regression and a sequence-tosequence model were employed to predict future capacity trajectories.Interesting research was conducted to identify the degradation patterns from discharge capacity curves of batteries using a transfer learning approach.Data of 124 cells from public sets were used for verifcation, and the LSTM model gives the best prediction accuracy with less errors [21].Jiao et al. [22] conducted a study that examines the use of a regularized extreme learning machine trained using the spectral Fletcher-Reeves algorithm to achieve accurate and resilient state of charge (SOC) estimation, providing accurate predictions about the capacity of batteries.Liu et al. [23] introduced an improved sparrow search-optimized LSTM network to estimate the RUL of the battery.Comparative analysis with other ML algorithms suggests that an optimized LSTM model is more accurate and robust.Recently, DT has gained worldwide attention due to its utility in providing solutions for various applications.Tis virtual counterpart provides ongoing monitoring, captures complex patterns with ML approaches, and promotes adaptation in the context of predictions.Incorporating ML into the digital twin framework [24] improves the models' ability to manage complicated patterns and nonlinear interactions.Additionally, the digital twin facilitates predictive maintenance by employing machine learning and historical data to identify potential issues before they become signifcant.Recently, several authors utilized and applied DT in various domains.Ramos et al. [25] conducted a comprehensive study to establish a smart water grid (SWG) with DT live monitoring of system components and to improve system efciency.In another study, the role of DT in improving urban water system efciency and achieving sustainable development goals was explored after utilizing a pressure-reduction strategy by developing a digital twin model [26].
Monitoring the dynamic properties of batteries in real time is challenging due to their complex and nonlinear behavior.Traditional empirical models are insufcient for accurately identifying degradation patterns and faults in Li-ion batteries, as they overlook unknown factors and lack adaptability.By leveraging the algorithms, DT is capable of determining the intricate patterns and dependencies in the battery's performance data, accounting for previously unconsidered factors.Te present work proposes constructing an AI-assisted data-driven DT model that seamlessly integrates various metaheuristic optimization algorithms and deep learning models with a semiempirical battery capacity estimation model that can accurately estimate the discharge capacity of a lithium-ion battery.Tese integrated models proposed by the authors are further analyzed using the data acquired from diferent batteries, resulting in the formulation of proposed DT models.
Te study summarizes the key objectives and contributions as follows: (a) Te main goal of the research is to build a datadriven, AI-assisted DT model that accurately estimates lithium-ion battery discharge capacity by utilizing a semiempirical model and a combination of deep learning and metaheuristic optimization algorithms.
(b) Tis study makes a signifcant contribution by combining advanced machine learning techniques with traditional empirical models.Te methodology resolves the issues that current models have with understanding how batteries behave in a complex Figure 1 shows a fowchart that illustrates the methodical process of creating an optimal DT model for accurately predicting the discharge capacity of lithium-ion batteries.Te remaining sections of the research are organized as follows.Section 2 provides a concise explanation of machine learning and metaheuristic algorithms.Section 3 provides a description of the dataset for Li-ion batteries, enabling the utilization of the suggested approach.In Section 4, the authors describe the semiempirical model in a general sense, and in Section 5, they discuss the optimized digital twin model.Section 6 summarizes and examines the results in more depth, while Section 7 highlights the conclusions.

Machine Learning and Metaheuristic Optimization
ML is a branch of AI that studies training computers to accomplish intelligent tasks [27].Researchers accomplish this by searching for connections, patterns, or correlations among the variables in a dataset.Te process usually begins with a mathematical algorithm or model.After training on a set of datasets, standard metrics measure the model's accuracy.Te trained model may categorize new data (called "classifcation") or predict a number value (called "regression") [28].Choosing the best hyperparameter values within a specifc search space reduces machine learning prediction errors.Battery state-of-health is nonlinear; therefore, machine learning can improve prediction accuracy and real-time management decision-making.Tis method speeds up calculations and boosts system efciency, making it a strong tool for real-time battery health and performance optimization [29].Te ML and optimization algorithms used in this work are briefy explained as follows.
2.1.AdaBoost.Te iterative AdaBoost methodology converts multiple low-performing classifers into an individual, more resilient classifer by incrementally changing the weight of each instance in the data distribution.It builds this algorithm by incrementally changing the weight of each instance in the data distribution.Te weight is determined based on the characterization reliability of the associated instance and the general accuracy of predictions from the last iteration.Te instances and the changed weights are further trained for the respective iterations.Te weight scores of poor classifers are progressively aggregated through successive steps to produce the fnal robust classifer.AdaBoost will allow the classifer to concentrate on the examples that need to be accurately classifed by ignoring the unneeded instances [30].Similarly, the algorithm can be adapted to predict continuous variables, such as in regression [31].Within the AdaBoost algorithm, every incident i as well as its associated labels y ∈ Y will receive an initial weight denoted as w 1 i,y .Te function w 1 representing initial weight is based on the density D (i) defned in Algorithm 1. Te weights are normalized in Step 1 to obtain a density ρ t .Subsequently, the weak learner employs the information obtained in Step 1 to determine a hypothesis ht: X ⟶ Y that aims to minimize a loss function defned as ε t , as defned in Step 3. Te weights are revised in accordance with the diminution calculated using p t i,y dy| as described in [32].An algorithmic representation of the working methodology of AdaBoost is shown in Algorithm 1.

Long Short-Term Memory (LSTM).
LSTMs, a subset of RNN networks, are widely used in modern ML/AI tasks for their ability to efectively process sequential data.It has been demonstrated that LSTMs perform better than typical RNNs in time series prediction tasks by their capacity to handle the problem of short-term memory loss.Tis is accomplished with the use of cells and gates that regulate the fow of data throughout the network.LSTM gates play a critical role in determining which information to retain and discard, while the cell maintains essential data processing information [33].A network based on LSTM consists of an input layer, one or more hidden layers, and an output layer.Te fundamental characteristic of LSTM networks resides in the presence of memory cells within the hidden layer(s).Te memory cells are equipped with three gates, namely, the forget gate (ft), the input gate (it), and the output gate (ot), computed utilizing (1)-(3), respectively.Tese gates are responsible for maintaining and regulating the cell states "t."At every timestep when the network is provided with an input X t and the output h t−1 , the gates serve their purpose, such as defning which information is removed or added and which information is used as output from the cell state by the forget, input, and output, respectively [34].Te input node memory cell is computed using (4), while the memory cell's internal state is calculated using (5).Te hidden state is calculated using an activation function as indicated in ( 6).An algorithmic representation of the working methodology of LSTM is shown in Algorithm 2.

International Journal of Intelligent Systems
Li-ion

Data
Figure 1: Digital twin blueprint for the estimation of discharge capacity of lithium-ion battery. 4 International Journal of Intelligent Systems Te term "Vanilla LSTM" refers to a type of neural network architecture that has a separate hidden layer made up of LSTM units and then a normal forward-thinking output layer.Stacked LSTM is a modifed version that integrates several concealed LSTM layers, which consist of an extensive number of memory cells.

Antlion Optimization (ALO).
ALO is an optimization method based on antlion hunting behavior, aiming to fnd the best hyperparameters for an objective function.Antlions randomly place themselves in the search domain and hunt for prey, which are candidate solutions with the highest ftness value in their vicinity.Antlions use a random walk tactic to approach their prey, and the mathematical implementation of ALO is described in [35,36].Te ants' movement is infuenced by the antlion, and vectors C and d determine their random walks.
A roulette wheel operator relying on ftness values emulates the antlion's capacity to construct traps.It is presumed that the antlion with the highest ftness rating has captured the ants.Te diameter that defnes the random movements of the ants reduces adaptively to simulate confned ants slipping toward an antlion.Equations ( 7)-( 10) describe the methods.
Here, t represents the most recent iteration, throughout which c t is the smallest and d t is the largest variable at the beginning of t th repetition, and I represents the ratio of the current repetitions to the highest number of repetitions multiplied by w th to the 10th power, when w is an unchanged value used to modify the level of exploitation accuracy.
Equation (11) represents updating procedure during which an antlion revises the location to the most recent of the preyed upon ant (least solution) in order to get close to the most optimum solution.
Input: Set of N instances from (x1, y1) . .., to (x N , y N ) with identifers yi belongs to Y (between 0 and 1) standalone algorithms such as decision trees (Weak Learn) preset iteration counter (T) Initialize the weight vector: (2) Call Weak Learn, where ht is the hypothesis got back.
(3) Loss calculation ε t is greater than ffty percent, then reset T to previous iteration (t − 1).( 4 ALGORITHM 1: AdaBoost regression. ( where Antlion t j represents location of j th antlion, and Ant t i represents the location of i th ant at t th iteration.Te best antlion is maintained as the elite with every ant randomly walking around the elite and other antlions by roulette wheel using the following equation: where r t a and r t e are random walks in vicinity of the selected antlion and elite antlion.An algorithmic representation of the working methodology of ALO is shown in Algorithm 3.

Grey Wolf Optimization (GWO).
GWO is a populationbased algorithm based on metaheuristics presented by Mirjalili et al. [37].Te algorithm starts with an initial set of potential solutions, each representing a unique combination of hyperparameters.Pack hierarchy and the predatory behavior of grey wolves are utilized to iteratively update the population.Te search is conducted by alpha, beta, and delta wolves, representing the optimal, second-, and third-best solutions in the current population [38,39].
Te mathematical structure of GWO is as follows.
Te algorithm initiates by dissociating the positions of grey wolves as they engage in the pursuit of prey.During the pursuit, the AM component regulates the degree of deviation exhibited by a search agent from its target, thereby controlling the level of randomness.Likewise, CM employs stochastic weights to explore the search space for potential prey.Similarly, CM initiates arbitrary weight values in order to search for prey (ft values) in the domain.CM also depicts the efect of a predator's approaching prey.r 1

→
and r 2 → vectors both have values between 0 and 1.Furthermore, vector a → undergoes a linear decrease from a value of 2 to 0.
Te top three wolves in the hierarchy labelled as alpha, beta, and delta evaluate as well as infuence the positions of remaining pack of wolves.D M

�→
represents the encircling movement of the grey wolf which can be expressed as equation (14).Te following equations theoretically express the anticipated boundary: Here, t indicates the latest iteration cycle, X P → (t) is the location of the prey, and X → (t) represents the location vector of the grey wolf.
Te analogous of hunting expressed theoretically in mathematical terms is indicated in equations ( 16)- (19), where the alpha wolf guides for hunting after encircling, followed by the delta and beta wolves. where

�→
, and X 3 �→ can be calculated as follows: Grey wolves hunt their target by attacking it when it stops moving.Te alpha, beta, and delta wolves' locations revise the search agents' locations during the hunting phase.Tis process is summarized in Algorithm 4.

Improved Grey Wolf Optimization (IGWO).
IGWO is an improved version of GWO that aims to enhance tuning performance.Nadimi-Shahraki [40] proposed IGWO in 2020.Te improvement in IGWO is its dynamic search mechanism, which adapts to the optimization process by adjusting the exploration factor.Te algorithm dynamically adjusts this factor based on its performance, controlling the degree of exploration and exploitation in the search process.Te algorithm proceeds as follows.
To initialize the algorithm, N number of wolves are arbitrarily distributed in the search space with range [l i , u j ], represented as follows: where X i (t) represents the i th wolf in the t th iteration, and the population is expressed by an array with N rows and D columns.
IGWO includes an independent movement strategy inspired by wolves' occasional independent hunting behavior, referred to as a dimension of learning-based hunting.Tis allows the wolf to learn from its adjacent wolves, thereby enhancing its qualifcations for the prospective role of X i (t).Te DLH selection method involves the utilization of equation (23) to calculate the value of each dimension that ranges in the most recent location of the wolf X i (t).Tis calculation is based on the knowledge acquired from neighbouring wolves and a randomly chosen wolf from the group of wolves.DLH strategy creates new candidate wolf X i−DLH (t + 1) for the new position of X i (t) apart from X i−GWO (t + 1), which is obtained from the canonical GWO.
Te neighbor's N i (t) is created utilizing equation (22), where D i is the Euclidean distance between X i (t) and X j (t).

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International Journal of Intelligent Systems 2.5.1.Selecting and Updating Phase.Te process of determining the more exceptional candidate involves a comparison of the ftness values derived from the GWO and DLH strategies, as depicted in the following equation: Te processes mentioned involved in the functioning of IGWO can be compiled as Algorithm 5.

Experimentation and Proposed Framework
Te authors showcased the proposed methodology by utilizing the publicly released battery aging dataset from NASA [41].At the Prognostic Center of Excellence (PCoE) of NASA Ames, researchers conducted the study using a customized testing apparatus on commercially available lithium-ion batteries [42].Te batteries utilized by NASA were of the LiNi 0.8 Co 0.15 Al 0.05 O 2 type and had a capacity of 2 Ah.Furthermore, constant current-constant voltage (CC-CV) charging and constant current (CC) discharging profles were employed to assess the batteries' aging characteristics, as depicted in Figure 2. Te charging rate was set at 0.75 C, and the discharging rate was set at 1 C. Te investigation aimed to produce accelerated degradation on these batteries by operating the various batteries through consecutive charge and discharge cycles.At the conclusion of each discharge cycle, measurements were conducted of battery capacity as well as temperature, current, and voltage at both the load and the battery.

Empirical Model for Capacity Estimation
A semiempirical model has been utilized to estimate the battery capacity for batteries B0005, B0006, B0007, and B0018.Xu et al. [43] introduced a semiempirical technique to calculate the SOC of a battery by combining fundamental theories of battery degradation.Te model accounts for the nonlinear nature of the battery's fade capacity by considering various factors that afect the deterioration of a Li-ion battery, especially solid electrolyte interphase (SEI) flm formation [44].Te empirical model utilizes the initial rated capacity (C), temperature (T), time taken for a discharge cycle (t i ), and cycle index (i) to calculate the expected discharge capacity for a given lithium-ion battery.Te model suggests lifespan of a Li-ion battery can be accurately predicted using a semiempirical formula, as outlined in the following equation: Equation ( 25) calculates the battery life L of a fresh battery.For a used battery, the existing battery life must be calculated.In such conditions, equation ( 25) can be modifed as follows: where f d is the nonlinear degradation function of the battery expressed as an expression of time, measured temperature, and repetitive cycles of operation that can be calculated as follows: Here, C o indicates rated initial capacity while C is estimated capacity of battery.
Te study derived the features for predicting the discharge capabilities of lithium-ion batteries from the digital twin model, which serves as a virtual representation mirroring the physical battery's behavior and characteristics in real time.Te empirical component of the proposed model is based on a robust semiempirical mathematical structure meticulously designed to encapsulate the degradation in the battery's ampere-hour (Ah) capacity.Tis is achieved through the systematic integration of key battery attributes, including battery temperature during discharge, the number of discharge cycles from its initial state, and the temporal duration of the current discharge cycle, as refected in Table 3.Researchers extract these crucial features from the Liion battery aging dataset corresponding to each discharge cycle.Subsequently, the semiempirical model employs these features to compute the battery's discharge capacity, as explicitly outlined in equations ( 27) and (28).Te combination of empirical and ML techniques ensures that both known and previously unknown factors are considered, contributing to the reliability of the selected features.Te initial step in the proposed framework involves segregating the discharge cycle data from the broader Li-ion battery dataset.Tis process also includes the normalization of features, ensuring they are standardized for subsequent analysis.Te feature listed in Table 3 serves as inputs for building a semiempirical model to calculate the discharge capacity of batteries.A variable delta is introduced to assess the accuracy of the model, representing the diference between the calculated capacity and the actual capacity obtained from the dataset.Te subsequent step involves training the models to predict the value of delta using the calculated capacity as the target variable.Te training phase incorporates optimized hyperparameters obtained through the application of ALO, GWO, and IGWO algorithms.Tese metaheuristic optimization techniques aid in fne-tuning the models and improving their predictive capabilities.During the testing phase, the predicted delta obtained from the trained models is added back to the calculated capacity.Additionally, tenfold cross-validation (10-CV) is employed to aid the training and testing of algorithms as well as establish robustness in the results.Tis enables the framework to accurately estimate the discharge capacity of Li-ion batteries.By incorporating the predicted delta, the framework compensates for any discrepancies between the calculated capacity and the actual capacity, resulting in more precise and reliable predictions.It is anticipated that in International Journal of Intelligent Systems future scenarios where the direct determination of the battery's discharge capacity is not feasible, the semiempirical model will serve as a crucial tool, facilitating accurate determination in these contexts.Te semiempirical model will serve as a crucial tool in future scenarios where direct determination of the battery's discharge capacity is not feasible, facilitating accurate determination in these contexts.It is believed that this approach ofers a comprehensive and efective method for predicting the discharge capabilities of lithium-ion batteries in real-time.

Optimized Digital Twin Model
A digital twin refers to a virtual version of a physical process or system that leverages data from sensors and other sources to simulate its behavior and performance.To obtain a precise estimate of the lithium-ion battery's ability to hold a charge, the model must be as close as possible to the real thing.In machine learning modeling, the utilization of digital twins has the potential to enhance the precision and efectiveness of predictive models.It is possible to train and test machine learning models in a simulated environment by using realtime sensor data in conjunction with other sources to create a virtual model of a physical system or process.Similarly, the application of a digital twin can be leveraged within the domain of anticipatory maintenance to track the behavior and performance of equipment, such as turbines or engines, and predict when it needs to be fxed before it breaks down.
In the current work, authors integrate the semiempirical model with the experimental Li-ion battery dataset to create a DT model.Te diference, delta (Δ), is calculated as ∆ � C − C act , where C is the calculated battery capacity procured from the semiempirical model and C act is the actual capacity.Te ML models are given the value of C as the input feature and the target is to predict the corresponding value of Δ.In ML models, hyperparameter tuning is extremely important and needed to signifcantly reduce prediction errors.Te necessity of metaheuristic optimization techniques for hyperparameter tuning arises because the hyperparameter space is often huge, and the exhaustive search for the ideal hyperparameters may be computationally costly and time-consuming.Metaheuristic optimization methods can compress the search space greatly and identify efective solutions quicker, making them a powerful tool for hyperparameter tuning.Tables 4-6 highlight the default hyperparameters and the optimized parameters obtained after applying antlion, GWO, and IGWO to three ML models.Te RMSE values of the machine learning model are selected as an objective function for the optimization algorithms, intending to minimize them.Each optimization algorithm is made to iterate 100 times for the hyperparameter search space, and the hyperparameter combination that yields the least RMSE value is subsequently selected.Te hyperparameters obtained from antlion, GWO, and IGWO are incorporated individually in all ML models, and the optimal combination of ML and the optimization algorithm is selected to construct the DT to estimate the discharge capacity.

Results and Discussion
Tis section conducts a thorough investigation of the efcacy of a novel approach that uses digital twin technology to predict Li-ion battery discharge capacity.Te dataset is frst split into training and testing portions, with 70% of the data assigned for DT model training and the remaining 30% put aside for accuracy testing.A 10-fold cross-validation (10-CV) technique is used, which is a common practice in ML and statistical analysis for assessing model performance, to assure unbiased predictions.Tis method divides the dataset into ten equal subsets or "folds."Every iteration rotates over all folds, using one for validation and the remaining nine for training.Before undergoing validation testing, the model goes through nine training cycles.Each fold acts as the validation set once, and this cycle is repeated ten times.As explained in Section 5, for every battery cycle, the diference (Δ) between the actual capacity measured from the dataset and the estimated discharge capacity, derived using empirical equation (28), is computed.By allowing the machine learning models to forecast the variance of the calculated discharge capacity from the actual data, this deviation aids in the integration of the DT model.).Tis showed that it was better at predicting the future and was more stable.Models with the default DT confguration had the highest values of MAE and RMSE, indicating a major contribution of hyperparameter altering to improved model performance.Models optimized using IGWO consistently displayed lower error metrics than the Vanilla LSTM and Stacked LSTM models, reinforcing the significance of metaheuristic optimization techniques in fnetuning ML models.Te results of the sensitivity analysis show that the AdaBoost model performs better than other models, especially when boosted with IGWO.AdaBoost's capacity to integrate several weak learners and adapt to prediction errors is probably the cause of this.Furthermore, it appears that the IGWO optimization method is useful for exploring the hyperparameter space and discovering confgurations that reduce prediction errors.Te efectiveness of various ML models applied to the NASA AMES battery dataset is compared in Table 7, with an emphasis on how well models predict the discharge capacity of Li-ion batteries.Tere are six diferent studies, with recent publications varying from 2021 to 2023, and each study incorporated a diferent methodology.Te table presents an overview of how predictive modeling techniques have advanced over time, with the most current models, particularly the proposed work, displaying exceptionally precise predictions.Te proposed work achieved equal RMSE and MAE values of 0.01, the lowest among the studies, highlighting its signifcant efcacy.International Journal of Intelligent Systems Te practical impact of research results for the battery industry includes creating methodologies for the development of more robust and efcient battery systems, which will lower costs and improve the economic feasibility of energy storage solutions.Te increased discharge capacity prediction accuracy of Li-ion batteries can help manufacturers optimize battery design and performance, which could result in a decrease in greenhouse gas emissions and a shift to cleaner energy sources.Tis is especially important when considering environmental sustainability and climate change.However, it is important to acknowledge certain constraints in the proposed methodology.Several issues in   International Journal of Intelligent Systems battery chemistry and physics.Te dependability of assumptions on the efcacy of metaheuristic optimization techniques, such as antlion, GWO, and IGWO, could be improved by investigating alternative methodologies.
Additionally, the predictive models' exceptional accuracy is based on the dataset used, and their applicability to other datasets or diverse battery scenarios requires further investigation.International Journal of Intelligent Systems

Conclusion
DT technology builds data-driven ML models that estimate lithium-ion battery discharge capacity in this research.Te models have been developed through the integration of ML techniques, specifcally AdaBoost and LSTM, with empirical battery capacity estimation models.Te DT models are created by applying 10-CV approaches in conjunction with normal training.Te authors have employed the application of metaheuristic optimization techniques to optimize the DT models' hyperparameters.Te authors outline the principal discoveries as follows: ( (3) For RMSE, a crucial metric for model accuracy, the IGWO-AdaBoost DT model again excels, registering a minimal RMSE only 0.01.Tis is in stark contrast to the default-Vanilla LSTM DT model, which exhibits a notably higher maximum RMSE of 0.19.
(4) When it comes to predicting the discharge capacity, the AdaBoost DT model consistently demonstrates remarkably low prediction errors, far surpassing the performance of both the Vanilla LSTM DT and Stacked LSTM DT models.(5) Te use of the IGWO technique for hyperparameter tuning in the DT models has been shown to slightly enhance performance.Tis improvement is evident when compared against models using GWO, ALO, and default hyperparameter settings.
Te deployment of DT-assisted, optimized ML models for Li-ion battery prediction promises to vastly expand the scope of possibilities in the coming years, potentially transforming the landscape of the energy storage industry.Tese cutting-edge techniques combine the accuracy of DT with the fexibility of ML to improve battery performance prediction, optimize battery design, and expedite maintenance and operations procedures.In order to solve the present problems with battery technology and provide more reliable, efective, and long-lasting energy storage solutions, this integration is essential.Researchers expect considerable improvements in performance, cost reduction, and overall efciency as enterprises increasingly adopt Li-ion battery technology.Te insights gained from this research are particularly salient for sectors, such as automotive, where the shift to electric vehicles necessitates reliable and efcient battery systems.In the context of renewable energy, better battery technology can also greatly increase the stability and storage of solar and wind energies, which help to lessen the intermittent problems that come with these renewable energy sources.Moreover, this study's fndings about the improved predictability and efciency of Li-ion batteries play a signifcant role in encouraging the wider adoption of a variety of sustainable technologies.Tis advances the goal of a more sustainable, low-carbon future by lowering carbon emissions and supporting global environmental objectives.
Te authors also emphasize the signifcance of investigating the actual implementation of the generated models in real-world situations.Larger-scale feld testing and validation with multiple battery types and operating conditions may ofer insightful information about how well the models function in various scenarios.Exploring this particular research direction would not only exclusively strengthen the validity of the digital twin-assisted models that are being proposed but also provide pragmatic recommendations for their implementation in a wide range of battery technology applications.

Figure 2 :
Figure 2: Voltage/current observed during CC-CV charge and CC discharge of Li-ion batteries as specifed in Table1.
Figures 3(a)-3(c) indicate a sample of the prediction results with grey wolf optimized for all three ML models: AdaBoost, Vanilla LSTM, and Stacked LSTM.Te estimation results seem to have varied considerably after looking at the fgure, but careful observations reveal that the numerical prediction errors are in the permissible range.Figures 4(a)-4(c) show the mean absolute error (MAE) obtained in training and 10-CV in three ML models with and without hyperparameter tuning with metaheuristic optimization algorithms.MAE quantifes the average absolute disparity between the observed and predicted values.Figure 4(a) shows that the IGWO-AdaBoost DT model has the least MAE, followed by the GWO-AdaBoost DT, the ALO-AdaBoost DT, and the default-AdaBoost DT model after training and a ten-fold on Liion batteries.Here, the default condition refers to the DT model without hyperparameter optimization.Te IGWO-Vanilla LSTM DT and IGWO-Stacked LSTM DT models from all Li-ion batteries achieved the lowest MAE when conducting training and 10-CV, as demonstrated in Figures 4(b) and 4(c).A lower MAE value indicates that the

Figures 5 (
a)-5(c) illustrate the efcacy of decision tree models in terms of RMSE values.RMSE represents the square root of the average squared diference between the actual values and predicted values.Te highest RMSE was observed from the default-AdaBoost DT model as compared to other models (Figure 5(a)), while training as well as 10-CV were conducted on Li-ion batteries.Similarly, the lowest RMSE values are observed with the IGWO-AdaBoost DT model in all the cases.Figures 5(b) and 5(c) show the RMSE values when Vanilla LSTM and Stacked LSTM DT models are considered.Here, again, maximum RMSE values are observed with default models as compared to the optimized DT models.While considering Vanilla and Stacked LSTM DT models in combination with optimized models, the IGWO-Vanilla LSTM DT model and IGWO-Stacked LSTM DTexhibit the lowest RMSE consistently with all batteries, as observed.Terefore, the IGWO-AdaBoost DT model exhibits the least RMSE as compared to the IGWO-Stacked LSTM DT model, followed by the IGWO-Vanilla LSTM DT model.Figures 6(a)-6(c) indicate the robustness of ML algorithms to predict capacity under various conditions.It is observed that AdaBoost's prediction capability is much better as compared to the other two ML algorithms.Using the NASA AMES dataset, ML models were analyzed for their sensitivity in predicting Li-ion battery discharge capacity.Te objective is to evaluate the relative impact of model confgurations and optimization strategies on each model's predictive ability, as demonstrated by two important metrics: RMSE and MAE.Te IGWO-AdaBoost DT model had the lowest MAE and RMSE values (0.01) for the battery datasets (B0005, B0006, B0007, and B0018

Figure 6 :
Figure 6: Digital twin predicted discharge capacity ftted against observed and empirically calculated capacity for AdaBoost DT, Vanilla LSTM, and Stacked LSTM model, respectively, in ten-fold CV.

( 2 )
1) Te IGWO-AdaBoost DT model demonstrates outstanding predictive accuracy in estimating the discharge capacity of Li-ion batteries.It outperforms both the IGWO-Vanilla LSTM DT model and the IGWO-Stacked LSTM DT model, as evidenced by key performance metrics such as MAE and RMSE.In terms of MAE, the IGWO-AdaBoost DT model achieves impressive precision, recording a minimal MAE of just 0.01.Tis is signifcantly more accurate compared to the default-Stacked LSTM DT model, which shows a higher maximum MAE of 0.20.
Te establishment of ML models aided by DT for estimating Li-ion battery discharge capacity represents a signifcant advancement in achieving global energy and environmental and sustainability goals.Numerous applications in industries, such as boosting renewable energy storage systems or increasing battery efciency in electric cars, could beneft from this research.Tis work advances the global search for sustainable energy solutions by tackling the difculties associated with predicting and optimizing battery performance.It is in line with the United Nations Sustainable Development goals, especially those that deal with clean and afordable energy, industry innovation, and responsible consumption.
Table 1 indicates the steps Initialize GW packs.Initialize the numeric value of the variables a, A, and C Compute the population's ftness score X α , X β , and X δ represents the three best obtained values in ftness While (t ≤ t max ) For each wolf: Revised location of every Greywolf End for Update values of a, A, and C variables Compute ftness values of all wolves Update the best wolves X α , X β , and X δ Move to next iteration End While Return value of the best wolf X α ALGORITHM 4: Grey wolf optimization (GWO) algorithm.Input: ftness function, domain of search, quantity of ants and antlions, preset max iterations (T) Output: Te fttest antlion and its value (1) initiate arbitrary collection of ants and antlions.(2) Calculate the ftness of all ants and antlions.(3) identify fttest antlion and call it Elite.(4) set the current iteration to a variable "t." (5) while (t ≤ T) For each ant i (i) Select an antlion employing the roulette wheel.(ii) Move ants towards antlion.(iii) Initiate random walks in the vicinity of the antlion and normalize it to contain within domain of search.end (6) Calculate ftness of the ants.(7) fttest ant if found is replaced by the antlion (Catching Prey); (8) revise the elite if any antlion becomes ftter than the elite.Input: numeric values of N, D, maximum iterations Output: Te Global Optimum Initialize arbitrarily generate the starting points of a N wolves in search domain For iterations between 2 to maximum iterations Find X α , X β , X δ International Journal of Intelligent Systems involved in the lithium-ion battery aging experiment.Te batteries were operated in an environmental chamber to maintain and control the ambient characteristics of the investigation.Te batteries were charged at 1.5 A as they reached 4.2 V, followed by discharge at 2 A current until the voltages of the batteries fell to their respective cut-of voltages.Te cut-of voltages were set under the original equipment manufacturers (OEMs) recommended voltage threshold of 2.7 V to induce profound aging efects and thereby accelerate the fade in battery capacity.Table2 in Table1.

Table 1 :
Description of the experimental cycle.

Table 2 Table 2 :
Selected batteries and their operational parameters.

Table 3 :
Features corresponding to discharge profle of the Li-ion battery used to build the digital twin model.