An improved MobileNet based on a modified poor and rich optimization algorithm for lithium-ion battery state-of-health estimation

Abstract

Reliable prediction of the State-of-Health (SOH) of lithium-ion batteries is essential to guarantee the safety, robustness and lifetime of the electric vehicles and grid-scale energy storage devices. Although data-driven methods can provide viable alternatives to traditional model-based algorithms, despite these approaches, high computational complexity, overfitting, and poor feature extraction are common barriers to the use of these approaches in real-time battery management systems (BMS). To overcome these issues, this paper will suggest a light and precise SOH estimation model that incorporates an Improved MobileNet architecture and a Modified Poor and Rich Optimization (MPRO) algorithm. The Improved MobileNet has been designed with 1D temporal battery data particularly, which uses depthwise separable convolutions and Squeeze-and-Excitation attention units to help better represent features at the cost of minimal computational cost. The MPRO algorithm is improved by chaotic map based initializations and adaptive search strategies, which are used to automatically tune the important hyperparameters of the model to optimise its performance. Tested on the NASA, CALCE and Oxford battery data, the offered method has a state-of-the-art Root Mean Square Error (RMSE) equal to 0.48% compared to Transformer-based models with 29.41% and the default MobileNet with 41.46%. Having a small 1.1 million parameter count and an inference time of 3.2 ms, the framework provides an effective and deployable SOH monitoring framework in resource-constrained BMS settings.

Introduction

Motivation and context

The Li-ion (lithium-ion) batteries have emerged as the foundation of the energy transformation around the world, being the most common energy storage technology in a broad spectrum of applications critical to the contemporary life and the transition towards the sustainable energy systems¹. Li-ion batteries are at the leading edge of this revolution, whether in the electrification of transportation with electric vehicles (EVs), in powering the common portable electronic devices such as smartphones and laptops, or far beyond that, in stabilization of the large scale renewable energy grids, by dealing with the intermittent nature of renewable energy sources such as solar and wind power².

There are no issues that surround the ubiquitous use of this technology without problems, though, it is essential to guarantee the safety, reliability, and longevity of Li-ion batteries³. These are not merely technical concerns but are critical in ensuring that the consumers do not lose trust and more importantly they make it easier to integrate the battery technology even in other areas of life and industry⁴.

The key component of the solution to these issues is the Battery Management System (BMS) that is an embedded system that becomes the control center of the battery pack⁵. The BMS performs diverse functions, such as checking the battery status of charge, battery temperature, and keeping it at safe temperatures. Among the most challenging jobs of the BMS is the prediction of the State-of-Health (SOH), which is a number that gives an actual idea about the degradation of the battery in comparison with its original, clean state⁶.

SOH is basically the ratio of the present maximum available capacity of the battery to its nominal capacity usually expressed out of a percentage⁷. This parameter is vital as it directly impacts on how well the battery can store and provide energy, which consequently influences its performance, efficiency and overall life span⁸. The significance of a sound, strong and successful SOH estimation strategy cannot be emphasized. While being an academic principle in certain respects, heterotopic technology becomes an urgent need when seen in its actuality.

To prevent disastrous failures, e.g. thermal runaway or battery fires, first, a correct SOH estimation is needed which may happen in case a battery is overstrained to exceed its safe operating point. The SOH monitoring allows the BMS to identify early indicators of degradation and implement a corrective measure, i.e. changing the charging rates or imposing a maximum state of charge, to minimize risks. Besides, another important point that is affected by SOH estimation is the optimization of charging strategies⁹.

The capacity and efficiency of a battery also decreases with battery age and this influences the manner in which this battery ought to be charged and discharged. A correct estimate of the SOH enables the BMS to adjust charging policies to increase the range of the battery and keep it functioning¹⁰. As an example, it will be able to introduce more conservative charge profiles among older batteries to minimize stress and degradation. SOH estimation also has an important influence on informed battery warranty and second life applications.

Manufacturers use the correct SOH data to evaluate the state of warranties and the time when the batteries need to be replaced. Also, once batteries have ended first life in high-performance systems such as EVs, their SOH is a crucial factor in deciding if they can be used in second-life systems, such as stationary energy storage systems. With an accurate estimation of SOH, the stakeholders will be able to make sense in their choice regarding the repurposing of batteries and their recycling that will then result in a more sustainable and circular economy.

Finally, the success of the SOH estimation determines the trust of users in battery-driven technologies. Both the consumers and the industries also require a guarantee that the batteries they depend on will deliver good services and safely in the long run. The BMS is an important factor in establishing and sustaining this trust by delivering accurate and dependable SOH data, which contributes to the expanded use of battery technology and the process of global energy transition.

Problem statement

Although of critical importance, SOH is not directly measurable during battery operation but can be estimated using other measurable values like voltage, current and temperature. This is an indirect estimate that poses a serious problem. Conventional model-based techniques that are based on either equivalent circuit models or intricate electrochemical models tend to face the two challenges of identifiability of parameters, and model fidelity at varying operating environments as well as in varying chemistries of the battery. Figure 1 shows the Li-ion battery management cycle.

Fig. 1.

Li-ion battery management cycle.

The proposal of data-based approaches has provided an alternative, promising direction, based on historical operational data to train the non-linear, non-linear mapping of the measurable signals leading to the underlying SOH. But there is nothing wrong about these approaches either. Simple machine learning models, such as Support vector machines (SVM) and random forests, might be too limited in their capacity to represent complex long term patterns of degradation. Even though deep learning models, especially Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs), have shown better performance, they can be prohibitively expensive, in terms of computational cost and memory footprint, and thus cannot be implemented on resource constrained hardware in BMS. Moreover, the behavior of these deep models is very sensitive to their hyperparameter settings, and the usual method of manually or grid search-based optimization is time-intensive, as well as likely to discover a suboptimal solution, which ultimately restricts their estimation quality and generalization performance.

Literature review

Several works in different approaches have been provided for lithium-ion battery state-of-health estimation. In the following, some state-of-the-art works have been given in the following.

Data-driven SOH estimation

The current developments in the estimation methodologies of battery state-of-health (SOH) emphasize the efficiency of data-driven solutions. Conventional machine learning frameworks like Support Vector Machines (SVM) and the Random Forest have been widely used because they are robust and easily understandable in the modeling of battery aging based on a large body of operational data¹¹. These methods are strong in feature extraction and interval variability of real world data but can be weak in the presence of complicated nonlinear battery degradation models.

Approaches based on deep learning

The increased amount of battery data has made possible the use of deep learning models with complex temporal and nonlinear dependencies. The Convolutional Neural Networks (CNNs), Recurrent Neural Networks (RNNs) and Long Short-Term Memory networks (LSTMs) play a central role in the existing SOH estimation studies¹². Investigations prove that 1D-CNNs and other structures are more accurate and stable, and LSTMs and CNN-LSTM combinations are appreciated because they can track long-term relationships within the usage cycle. They can offer substantial advances in the level of accuracy in estimations, improving the ability to adapt to more complex trends in degradation in lithium-ion batteries.

Lightweight models and their application to BMS

The growing complexity of Battery Management Systems (BMS) integration in less-resource rich settings needs the realization of lean and lightweight resource frameworks. The two that have become popular are MobileNet and SqueezeNet because of their lesser computational cost but still being more accurate. MobileNet versions, in particular, MobileNetV3 Small, use depthwise separable convolutions and squeeze-and-excitation blocks, which trade off inference time and model size to embedded BMS tasks¹³. This is complemented by SqueezeNet via the use of fire modules and therefore is applicable in memory-constrained applications, which are essential in real-time SOH monitoring in electric vehicles.

Hyperparameter optimization in deep learning

Hyperparameter optimization is of paramount importance in deep learning models optimization. Raise some classical techniques, like Grid Search are computationally intensive making the use of automated tuning techniques like firebug swarm optimization algorithm¹⁴, and seagull optimization algorithm¹⁵. Such algorithms are suitable in high-dimensional search of data and they improve the rate of convergence and prediction. It is worth noting that, as a result of its socio-economic-inspired behavior, the Poor and Rich Optimization (PRO) algorithm has lately become popular because of the balance between exploration and exploitation in optimization problems¹⁶. Although it promises good performance, PRO has convergence speed challenges when solving high complexity problems, which prompts the need to conduct further research on the modified version to enhance its ability to improve efficiency in optimization in engineering problems.

Table 1 summarizes the development and the main features of various SOH estimation methods in accordance with the recent literature overview:

Table 1.

The development and the main features of various SOH Estimation methods in accordance with the recent literature overview.

Approach Category	Key Characteristics	Example Models / Techniques	Advantages	Limitations
Data-Driven Machine Learning	Classical ML algorithms; feature-based input	SVM, Random Forests	Robust, interpretable, less computationally intensive	Limited in capturing complex nonlinear patterns
Deep Learning Models	Neural networks capturing temporal and nonlinear features	CNN, RNN, LSTM, CNN-LSTM hybrids	High accuracy, adaptable to complex data	Requires more data and computational resources
Lightweight Deep Models	Optimized for deployment in resource-constrained BMS	MobileNet, SqueezeNet	Low computational overhead, suitable for real-time monitoring	May trade-off some accuracy for speed and size
Hyperparameter Optimization	Metaheuristics optimizing DL model parameters	Grid Search, Genetic Algorithm (GA), PSO, Poor and Rich Optimization (PRO)	Improved convergence and model performance	Computationally expensive, convergence speed varies

This table shows the evolution of traditional ML to advanced deep learning models with optimized hyperparameters to be used when efficient battery management applications are in place and accuracy and computational efficiency are balanced.

Contributions

To address the aforementioned issues, the current paper presents a new SOH estimation framework, which is a synergistic combination of a neural network with an architecturally improved structure with a strong metaheuristic optimization scheme. The formal aspect of the methodology rests on an Improved MobileNet architecture stated to be specifically meant to process time sequences concerning battery operations data thereby keeping the computational-efficient characteristic of the original MobileNet and enhancing its feature extraction capabilities in the domain of battery degradation.

In order to achieve this model to work in its optimal settings, we formulate a Modified Poor and Rich Optimization (MPRO) algorithm that also uses a chaoticization-based initialization strategy and an intelligent search process to remove the constraints of the underlying PRO, and in doing so, attain a better balance between global exploration and local exploitation within the hyperparameter space. The MPRO algorithm is smoothly incorporated to undertake the automatic optimal setup of the significant hyperparameters of the Improved MobileNet.

We intensively test our suggested MPRO-Improved MobileNet framework on publicly accessible battery cycling data, proving it has reached the state-of-the-art estimation accuracy with a model complexity orders of magnitude lower than traditional deep learning standards, and is thus an attractive solution to real-world BMS application.

Methodology

The suggested methodology provides a unified computational framework of lithium-ion battery State-of-Health estimation comprising of multi-stage signal processing, a dedicated deep neural architecture, and an improved metaheuristic optimization paradigm. The entire pipeline starts with the step of obtaining raw battery cycling data, which is carefully preprocessed and feature engineered to obtain significant degradation cues. The fundamental novelty is the closed-end interaction between the Modified Poor and Rich Optimization (MPRO) algorithm and the Improved MobileNet model, in which MPRO is an intelligent controller in which it actively searches the hyperparameter space to identify the optimal model minimizing the SOH estimation error.

The optimized model is then implemented to give precision, real-time SOH estimation leading to a solution that is not only very precise but computationally efficient enough to be deployed in embedded BMS. Given a sequence of battery operational measurements such that each denotes voltage, current, and temperature measurements at time , we hope to learn a mapping function that depends on some architectural parameters theta and hyperparameters that minimize the error between predicted and actual SOH values. The objective function can be formulated as follows:

1

where,

2

where and are the validation and training loss respectively, and is the space of hyperparameters.

This two-level problem of optimization is addressed by the use of our integrated MPRO-Improved MobileNet framework with the upper level (hyperparameter optimization) being solved by MPRO and the lower level (model parameter learning) being solved by gradient-based learning of the Improved MobileNet. The overall process flow of the proposed system is shown in Fig. 2 and it shows a sequence and repetitive nature of model development and deployment.

Fig. 2.

The general architecture of the suggested MPRO-Improved MobileNet to estimate SOH.

Figure 2 presents the general structure of the proposed MPRO-Improved MobileNet framework with the description of the different training and deployment steps. Training is called an optimization loop, whereby the MPRO algorithm proposes hyperparameter configurations, the performance of which is judged by training improved Mobilenet and validating it against RMSE. This feedback loop repeats until the MPRO gets to the most successful configuration. After optimization, the model is then deployed as an efficient and fixed estimator that is used on the new streams of battery data to produce real-time predictions of SOH thus addressing the real-life needs of a BMS.

Lithium-ion battery state-of-health

State-of-Health (SOH) of a lithium-ion battery is a dimensionless parameter that substantiates the current state of a battery in relation to its initial state, which is an important parameter of aging and degradation of its performance. SOH is mostly characterized using two physical terms; capacity fade and power fade with the latter directly proportional to internal resistance increase. The most popular definition of SOH is that which depends on the capacity fade, which can be described as the ratio of the present maximum available capacity, , to the nominal capacity, , which is presented as .

New batteries are normally characterized with an SOH of 100 and it is generally said to end-of-life when the SOH reduces to approximately 80%, which means that it can no longer store the original energy as it began doing initially. Direct measure of maximum capacity involves a complete cycle of charge-discharge operation which is not practicable in normal operation. In turn, estimation techniques are based on the ability to infer SOH using the indirect data signatures that are contained in the operational cycles. The most important of them are the voltage profile and current profile in the constant current (CC) charging phases.

The internal resistance of a battery also rises with age, and the charging capacity is also reduced, resulting in the net effect of changes in the charging curve, whereby a shorter CC charging period is required to reach a steady current than previously and the constant voltage phase is reached sooner. A more sophisticated signature is based on Incremental Capacity Analysis (IC) in which the derivative of capacity versus voltage, , versus voltage is plotted versus voltage.

The resulting IC curve has typical peaks and valleys which are related to phase changes in the electrode materials. As the battery ages, the peaks logically shift in their positions and reduce in their amplitudes giving a highly sensitive electrochemical fingerprint of the health of the battery itself, which isn’t so sensitive to noise and operation conditions as the raw voltage curve. Basic data signatures to SOH estimation are shown in Fig. 2, and it is an example of a raw voltage curve reaching a more valuable IC curve.

The data signatures of battery degradation are the most important and are illustrated graphically in Fig. 2. The voltage profile as charge accumulated at constant current during charge reveals that the aged battery reaches the maximum voltage threshold faster, which means loss of capacity; panel A exhibits the voltage profile at a constant current charge as a function of age of the battery. Panel B shows the changes in the Incremental Capacity (IC) curve that reflect the loss rate of active material and the shift to the right as direct proportions of active material loss and the rise in the internal resistance and therefore, the IC curve is a potent characteristic to the SOH estimation models that are driven by data.

Preprocessing and engineering of data

An algorithm based on an input data source of high quality lays the base of any sound data-driven model. Consequently, this research uses the publicly available NASA Battery Dataset with complete cycle life testing data of Li-ion batteries in 18,650-size and under various operational conditions. The crude information of voltage, current and temperature readings as recorded throughout the charge and discharge cycles are manipulated to build a dependable estimation model. The key input characteristics are engineered based on the constant current (CC) charging step because it is reproducible and has a high correlation to the capacity degradation.

Precisely, the same number of cycles of the voltage-time curve recorded during the CC phase are removed and matched between cycles to create regular input vectors. Moreover, to obtain more detailed electrochemical measurements, Incremental Capacity (IC) curves are obtained by numerically differentiating the charge capacity versus voltage and then interpolating the results with a Savitzky-Golay filter to reduce the amplification of noise. All the input features, such as the voltage segments and IC values, are normalized to the [0, 1] to have constant and fast training of the neural network. The dataset is separated into batches to avoid data leakage and all cycles in a single battery will only be in a single set of training or testing, hence realistically assessing the generalization ability of the model to unobserved battery samples. The cycling data of the battery is preprocessed into features suitable for training on deep learning models, and the preprocessing pipeline converts the original battery cycling data into structured features.

The raw sequence of measurements can be also represented as:

3

where , , and are the voltage, current and temperature at time .

Constant current charging phase is characterized by the thresholding of the current signal:

4

where, is the constant current charging phase is defined as thresholding the current signal, and and ε defines the tolerance parameter.

At this stage we obtain voltage time sequences resampler to a fixed length by cubic spline interpolation: .

Incremental Capacity (IC) curve can be calculated by using numerical differentiation of the capacity voltage relationship. The cumulative charge is obtained as and the IC curve is found by central difference approximation:

5

To mitigate the noise amplification of the numerical differentiation, we are now using Savitzky-Golay filtering of order p and window size w:

6

where, are the Savitzky-Golay coefficients and is a normalization factor.

The features are min-max normalized to the range [0,1]:

7

Battery-level splitting is used to divide the dataset into temporal fragments and avoid data leakage, under the formal specification that any battery, , is associated with all cycles assigned to one of training, validation, or testing sets.

The improved MobileNet architecture

AlexNet is a convolutional neural network (CNN) that is known to classify images and won the ImageNet contest in 2012. Following the mass adoption of CNNs, researchers have constructed within a relatively short time numerous additional models of a deep neural network: the older VGGNet16/19, GoogleNet, and ResNet50; all the aforementioned models produced a level of performance that was much higher than traditional classification methods.

Nevertheless, as researchers build even larger models, researchers have observed that the storage costs of their models, along with the costs of computing, make development on the deep learning paradigm impractical. Traditional CNN models required a huge amount of memory and computing power, and thus, could not be used in mobile and embedded devices.

In such a way, Google has created a lighter neural network named: MobileNetV1. MobileNetV1 is a convolutional neural network with a smaller model size, fewer trainable parameters, and with sufficient computational constraints, meaning it can be used in mobile applications, and it can also optimize the space of computational resources, therefore, a greater extent of accuracy as well.

The main concept behind the MobileNetV1 architecture is that instead of using a standard convolution, the architectures use depthwise separable convolution (DSC) to minimize the number of parameters in the model. The DSC applies the standard 3 3 convolution filter with one layer thickness and moves it across the tensor successively and generates an output channel at the end of each convolution. Having completed the convolution, make its thickness adjustable via a 1 × 1 pointwise convolution (See Fig. 1). However, MobileNetV1 uses ReLU activation function after the deep convolution layer, which does not change the number of channels. The recovered characteristics are single channel and introduction of ReLU in the convolution layer output of reduced channels will lead to loss of information.

Attention module: Concerning the facial emotion recognition (FER), the literature²⁴ finds that the eye area attributes have more influence on recognition accuracy compared to the mouth area. Otherwise stated, some differences in expression are centered primarily on the eyes and the mouth. It could be beneficial in enhancing the retrieval of face features in the MobileNet network through the implementation of an attention module of the MobileNetV1 network. The attention technique improves feature representation of the deep network.

The convolution layer includes a lightweight attention mechanism called CBAM attention module. Reflectively, the parameters of the CBAM module and the cost of its computation can usually be neglected in most of the cases. Two major procedures on it are done by the CBAM:

8

here, variable is the input feature. The symbol has a meaning of element-wise dot multiplication as depicted by the operator. The channel and spatial dimension attention extraction operations have been represented by and , respectively. Figure 4 shows comparison of standard convolution.

Fig. 4.

Comparison of standard convolution.

Figure 4 shows the comparison of standard convolution using batch normalization (BN) and ReLU (left) with depthwise separable convolution (DSC) using depthwise and pointwise convolution after every batch normalization and ReLU (right). The CAM module first accomplishes the global max pooling (MaxPool) and global average pooling (AvgPool) of the input feature to produce two one dimensional feature maps which are then fed into a two-layer multi-layer perceptron (MLP). The output features of MLP are element-wise summed and the sigmoid activation function is applied to the sum to generate the final channel attention feature map . The and the input feature are finally multiplied element-wise to deliver the input feature that is needed by the SAM. The Attention module on the channel is computed accurately as shown in the Eq. (2). Figure 3 represents the CAM module.

9

Fig. 3.

Battery data signatures evaluation with aging: (A) voltage profile during charging, (B) incremental capacity analysis.

represents the input feature map in this case and and represent the features that are computed using global Average Pooling and Max Pooling, respectively, and and are the parameters of the two layers of the multilayer perceptron (MLP).

The function feature map of the CAM module is the input of the SAM module. The SAM module performs global MaxPool and AvgPool on channels first, and later concatenates feature maps on the channel dimension. The obtained point corresponding to the sigmoid activation function is the spatial attention feature map . Finally, and input of this module are multiplied in order to give the end result of a created feature map. As seen in Fig. (4) The SAM module functioning is described by a mathematical Eq. (3).

10

The convolution layer uses the kernel size of , and the convolution kernel has a which refers to the sigmoid activation. The two modules are used in a series arrangement in this experiment.

The Softmax loss is the generalization of the logistic function and it is the most frequently and commonly employed whenever dealing with a K-paradigm-select classifier. The outputs of Softmax represent the likelihood, that the input picture, belonged to the particular category; in a K-classifier, the post processing with Softmax is a K-dimensional array, or a hyper-plane where the elements of the array sum to unity, i.e. all implemented probabilistic density function form.

Training set and its labels will be , and are the probability on each sample in the training set are as shown in Eq. (11):

11

In this case, =(k) is the network parameter, is the number of categories, and is the expression. Scales the result, so that the sum of all the probability is 1. At the training stage, the function makes use of the gradient descent method to converge. The loss function is as follows: Eq. (12):

12

here, is an indicator function. Where its value is a false value, the function returns 0; otherwise, the function returns 1. Equation (12) may be reduced to Eq. (13):

13

In practice, so as to avoid arbitrary values to the parameters () being zero, we usually incorporate weight regularization in the loss. The bigger the loss function, the lower the probability of the classifier to replicate a real label. The lowest of the loss functions was determined through recursive calculations in order to obtain the best desired value.

Center loss (): Wen et al., first coined the central loss function that is used as a standard clustering process. With a CNN that has many features, several feature centers are able to be calculated within every batch, where the loss function can also be calculated concurrently. The loss function was calculated in terms of the distance between the eigenvalue and its center hence, central loss, as represented in the expression (14) is as follows.

14

The new ​ gradient and are shown in Eqs. (15) and (16), respectively.

15

16

here, represents the eigenvalue of the image , represents the centre of the classification to which the i th image falls (centre of eigenvalue of the classification); and represents the change in the classification centre. The intra-class variation is outlined in Eq. (15), and it means that the center of classification is affected by the changing depth features.

In Eq. (16), when the condition is true and equals to 0 when it is false. Centre loss tries to reduce the total of the squared distances between each sample characteristic of a batch and the feature center, thus minimizing the intra-class distance. The shared supervision role of and aims to increase the distance between classes and decrease the distance between members of the same classes. The features obtained can be identified better, as it is in the case of the acquired features in the Eq. (17).

17

To give and equal weights, the coefficient is used, and an appropriate magnitude of λ will improve the classification ability of the network, shows the centroid of the class of samples. In the case of , the function is the same as that of the use of Softmax loss only. Figure 5 shows enhanced DSC.

Fig. 5.

Enhanced DSC.

Within MobileNetV1 structure, nonlinear modeling capacity of the network is enhanced by means of ReLU activation function after the depthwise convolution and pointwise convolution layer respectively. To prevent gradients explosion, make the model convergence faster and be more efficient, a Batch Normalization layer is included into the structure just before the ReLU activation function layer. In the deep convolution, the features which exit the depthwise convolution are single-channel features since depthwise convolution does not alter the number of channels.

The use of the ReLU functions at the output of the convolution layers with reduced channels can result in the loss of information and, possibly, deteriorate the capabilities of the model. We therefore propose a deeper depthwise separable convolution layer; that is, after the depthwise convolution, a linear output is used and an attention module is added after the pointwise convolution.

The work demonstrates a more advanced MobileNet, which is based on the MobileNet architecture and attention mechanisms and adapted to the particularities of FER, where the pre-trained parameters of ImageNet are used, and the DSC layer is added. The topology of the network is shown in Table 1. The model takes the input image in sequence with the help of a normal convolution layer, 9 depthwise separable convolution (DSC) layers, and then 4 augmented DSC layers. Finally, derivation of features is done through AvgPool layer and fully connected layer.

To enhance Peer-to-peer networks, nonlinear expressive capacity, feature recognition and faster convergence of the network, Batch Normalization (BN) and the ReLU6 nonlinear activation function must be applied on the output of every pointwise convolutional layer. Similarly, in the depthwise convolution layer, the BN is the sole layer traversed, preventing a nonlinear activation, making it possible to preserve as much information as is possible. It uses a linear output and employs the entirety of the original MobileNet model and the modified deep separable convolution layers.

MobileNet uses the design of a softmax classifier. The softmax classifier is not appropriate with facial expression recognition (FER) as there is not enough distinction between the various facial expressions and thus one expression may be misclassified as another. To overcome this effect, are used combined in A-MobileNet to regulate and enhance performance goals. With the increase in the distance between categories and the attenuation of the distance within the categories, the features derived have increased identification power.

Although the conventional MobileNet architecture is extremely efficient when it comes to image-based work, it does not work best when used directly on one-dimensional temporal battery data. Our Improved MobileNet architecture implements a series of essential changes that can be made to adapt to the peculiarities of the battery degradation signal.

The most basic one is the modification of each of the original layers to implement 1D convolutions, which process the sequential voltage or IC data without the need to feed it a 2D image. This enables the model to be able to capture the time dependencies and patterns within a charging cycle. We appended Squeeze-and-Excitation (SE) attention blocks after the depthwise separable convolution blocks in order to increase the representational power of the model without significantly increasing the extra computational overhead.

The SE block is used to conduct a feature recalibration and does so by squeezing the global spatial information first into a channel descriptor with the use of global average pooling then activating the channels by a simple self-gating process that acquires to focus on informative features and inhibit the less useful ones.

This makes the model adaptive to concentrate on the most critical charging curve stages that are the most representative of health degradation. Lastly, the final completely connected layer is reconfigured to take a linear activation in order to produce a continuous SOH value; the loss function is adjusted to be Mean Squared Error (MSE) in order to directly reduce the error in estimation. Included in the Improved MobileNet architecture are a number of mathematical changes made to the standard MobileNetV2 architecture to process temporal battery data more effectively.

The basic modification substitutes 2D convolutions with 1D processes without pertaining to the depthwise separable convolution principle. The conventional convolution process where being the size of the kernel, M input channels, N output channels) is split into a depthwise and pointwise version. The depthwise convolution is applied to every channel of input:

18

In which and , the mth input channel.

19

where , where, the computational cost ratio between standard and depthwise separable convolution is:

20

For typical values (, ), this yields approximately 9× reduction in computational cost.

The Squeeze-and-Excitation (SE) attention mechanism is integrated to enhance feature representation. The squeeze operation computes channel-wise statistics via global average pooling:

21

and is the activation of channel at position , with as the sequence length.

The excitation operation learns channel-wise dependencies by the following:

22

where , , δ stands for the ReLU activation, is sigmoid activation, and is the reduction ratio (usually 16). Rescaling to get the final output is as follows: .

The regression head will be designed as:

23

where GAP represents global average pooling, is final feature representation and , are regression weights and bias.

The following figure illustrates the detailed structural composition of the proposed network that emphasizes on the 1D convolutions and attention mechanisms.

The architectural layout of the proposed network with the emphasis on the 1D convolutions and attention mechanisms is presented in Fig. 6.

Fig. 6.

Design of the suggested Improved MobileNet model.

The inner structure of the projected Improved MobileNet is explained in Fig. 6. The first severe modification that is needed to process temporal battery data is the substitution of 2D operations with 1D convolutions. The addition of the Squeeze-and-Excitation (SE) block, which is described in the inset, is a substantial improvement over the traditional MobileNet, where they incorporate a lightweight attention mechanism, which enables the model to engage in dynamic channel-wise feature recalibration. This allows the network to become more sensitive to features based on certain voltage ranges or IC peak areas that are most susceptible to degradation, hence, the network becomes more feature selective and the overall accuracy of the estimation. The last network regression-specific layers guarantee the network output to be an accurate and continuous SOH estimate.

Modified poor and rich optimization algorithm

PRO was created based on actual social occurrences and may be seen as a way to solve difficult optimization problems. The PRO algorithm’s five-stage optimization process is described in the section that follows.

The term “wealth” is widely used in all fields of study. It is an economic concept that has been defined in different ways depending on different attitudes and implementations. Wealth is defined as the economic status of individuals, both in terms of quantity and quality, within a number of economic categories. It is unlikely that anyone in the world has not aspired to become wealthy. Everyone has a different financial perspective, and people are motivated by their desire for wealth. Although anyone can offer advice on accumulating wealth, the best method seems to be to use the experiences and strategies of the wealthiest people in the world. Members of a community are usually divided into two economic groups. The first group consists of wealthy individuals whose wealth is above the average amount.

The second group comprises individuals whose wealth is below the average level. Each member of these two groups is trying to enhance their financial standing through various methods, exhibiting considerable variation. The shared characteristic across all members of the two groups is that individuals examine each other’s behavior and try to enhance their standing by exerting influence on one another. The poor demographic attempts to enhance their circumstances by focusing on rich groups, whereas the rich population seeks to increase the class divide by regarding the impoverished individuals. The principal concept of the proposed method can be expressed in two different situations:

People who are less wealthy than average make up the second group. There is significant variance in the ways that each member of these two groups is attempting to improve their financial situation. The trait that all members of the two groups have in common is that they look at one other’s actions and attempt to improve their status by influencing one another. While the rich population aims to widen the class gap by focusing on the poor, the poor demography tries to improve their situation by focusing on wealthy groups. Two distinct scenarios allow for the expression of the main idea of the suggested approach:

1. Every member of the poor group tries to learn from the rich in order to improve their status and reduce the class gap.

2. By analyzing the poor and amassing wealth, each member of the wealthy group seeks to widen the class gap with them.

The uniform function in MATLAB software has been used to randomly select the beginning population. A uniform function was applied in order to represent this population as a vector. The values of the vector between the two highest and lowest parameters are specified by this function. The original population, which included both wealthy and impoverished people, was created at random as a uniform vector and would be arranged in accordance with the goal function. The first group in this population, which has the greatest values, will be referred to as the rich group, while the second group will be referred to as the poorest group.

With a uniform distribution between the upper and lower bound parameters, the initial population is generated arbitrarily in the PRO method. The population is then assessed and ranked to grow order by the last function’s results. Two subpopulations make up the PRO algorithm’s primary population. The first category is associated with the wealthy, and the second is associated with the impoverished. The topic at hand determines how many of these two subpopulations there are overall. The main population of the PRO algorithm has been demonstrated in Eq. (24):

24

where, , , in turn, showed the sizes of the poor, main, and rich populations. The main population is organized in ascending order, with the wealthy making up the first sector and the impoverished making up the second. All wealthy individuals hold higher rankings in the PRO algorithm than their poor relatives. The current populations have been displayed in Eq. (24). In PRO, Eq. (25) has been consistently correct:

25

The PRO algorithm’s principal population is made up of both affluent and impoverished subpopulations. Each population member’s position must be modified in accordance with a specified technique in each algorithm iteration. Figure 7 shows the conception of the main, rich, and poor population.

Fig. 7.

The main (a), rich (b) and poor (c) population.

Every time the PRO algorithm iterates, each member of the affluent population’s status varies in accordance with Eq. (26):

26

where, the updated value of the position within the wealthy population has been represented by , while the current value of the location within the rich population has been indicated by , and represents the class difference. The ideal member’s current placement within the underperforming population has been indicated by the parameter rand . A vector that includes every variable has been used to represent the value of . Actually, as wealth expands, so does the disparity between each rich person and every poor person. Since the perfect individuals of the impoverished population is , a member of the rich population’s distance from grows, and so does its distance from every member of the impoverished population. As the impoverished population grows increasingly impoverished, the gap between the rich and the poor widens.

Actually, represents the current location of the member of the rich population, specifies the new location of the member of the rich population, and represents the optimal member of a poor population. defines the distance between and . The application of the random coefficient will result in a distance between and that is proportional to the value of . Consequently, the locations of and will be influenced by the sum of and .

The distance of from the other constituents of the impoverished population will be increased, as is considered the optimal member of the impoverished population. is equivalent to , as delineated in Eq. (26). Furthermore, the value of will be negative when is negative. Exceeds the value of . The current status of the impoverished population is exacerbated by this negative value. The value of will be equal to the value of in the optimal situation, as the value of will be zero and no alterations will occur.

The distance that each wealthy individual is required to maintain from the impoverished is determined by a random variable, , which may range from zero to one. The wealthy population is subject to internal competition as a result of the randomness of , i.e., the enhancement of each individual is determined by when two wealthy individuals are proximal in status and is constant. It suggests that positions with higher expenses may receive more enhancements than those with reduced costs by utilizing a higher value.

During every iteration of the PRO algorithm, the location of each impoverished individual is modified in accordance with Eq. (27).

27

where, displays the updated value of the location within the impoverished population, demonstrates the current value of the location within the poor population, represents the pattern improvement parameter (randomly assigned a value between 0 and 1), and the approach used to achieve richness is indicated. The pattern value is provided by Eq. (28):

28

where, specifies the optimal position of the wealthiest individual, represents the mean position of wealthy individuals, and signifies the location of the least successful individual within the rich group. The objective of each iteration is to achieve the average status of three wealthy representatives, as the definition of wealth varies among individuals.

The pattern value remains constant in each iteration, which means that the random parameter determines the pattern’s enhancement and, as a result, the progression of . In fact, the pattern undergoes a more substantial improvement as the value approaches 0. Additionally, the pattern will exhibit a greater enhancement as surpasses the pattern value, and vice versa. In the destitute population, internal competition is induced by the randomness of . When the locations are in close proximity, the value induces changes in the ranking of these positions, and a lesser value of increases the pattern’s enhancement.

Factors in the economic realm can enhance or detrimentally affect economic conditions. For instance, an unforeseen fluctuation in gold prices within a brief period, a substantial increase or drop in oil or petrochemical production expenses, abrupt variations in exchange rates, a pronounced alteration in stock interest rates, a large modification in banking interest rates, and similar occurrences. Each of these elements induces a significant transformation in the conditions of specific individuals within a society. Forecasting these parameters is exceptionally difficult, and often unfeasible. Hence, this element is employed in this method as a mutation. This method uses a normal distribution considered by a mean of zero and a variance of one and implementing a designated mutation probability for both the underperforming and affluent populations separately. Figure 8 shows the poor and rich groups’ distance.

Fig. 8.

The poor and rich groups’ distance.

Actually, Eqs. (29) and (30) describe the mutation for wealthy and impoverished populations, respectively.

29

end

30

end

where, has been a random value that falls within the range of [0, 1], represents the possibility of a mutation, represents the modified value that was derived from Eq. (27) prior to the mutation at the same location, represents the new value that was obtained from Eq. (28) before the mutation, while has been a normal distribution’s outcome with a variance of one and a mean of zero. The mutation occurs at novel locations, as demonstrated in Eqs. (29) and (30), and may lead to either enhancement or reduction.

The Modified Poor and Rich Optimization (MPRO) algorithm is an improvement of the original PRO with a number of mathematical innovations. As its population starting point, chaotic mapping based on the Logistic function is used:

31

where indicates the ranking of the i^th person in normalized search space. The initial population is then scaled to the actual hyperparameter bounds:

32

Chaos theory is the other improvement that is applied in this research. It is a mechanism that produces pseudo-random variables rather than entirely random variables. The metaheuristics may employ this in order to enhance the rate of convergence of the algorithm. Various confusion brought about in the literature. Sinusoidal map is applied in this research. The r parameter of the equation was altered to yield the following results. The update formula of (33) as pseudo-random variable will be obtained by the following way:

33

in which, represents the chaotic random number produced in the present iteration, and represents the chaotic random number produced in the previous iteration. P=2.2 here is the control parameter and r(0) has been set to 0.6.

The fitness check of individual evaluation of each is based on training Improved MobileNet and checking:

34

The population is divided into Rich (top 30%) and Poor (bottom 70%) according to the fitness ranking. in which, η is the index of mutation distribution.

The modified algorithm makes the algorithm formal, emphasizing the implementation of the suggested changes into the regular flow of the PRO. This technique uses two different chaotic mechanisms in two different parts to provide a better starting and updating points. The search behavior is dynamically regulated.

Although there are many metaheuristic optimization algorithms, the Modified Poor and Rich Optimization (MPRO) algorithm was chosen due to the theoretically suitability as well as the empirical performance in hyperparameter optimization of lightweight deep SOH models.

MPRO is an extension of the original PRO, incorporating chaotic map-based initialization and adaptive search dynamic dynamics, which are useful in preventing premature convergence and improve exploration-exploitation balance of non-convex, high-dimensional space as seen in neural architecture optimization. Based on the obtained results, MPRO is lower in RMSE than original PRO and also higher improvement over PSO and much higher than grid search indicating that it is higher in efficiency and resilience.

Furthermore, the socio-inspired rule of updates (modeling competitive interactions between the rich and the poor) by MPRO is especially adapted to the heterogeneous degradation process across cells (battery cells) with hyperparameter sensitivity that is very varied. A combination of this kind of fast convergence, local optima resistance, and an inherent BMS compatibility makes MPRO a more desirable choice of optimizer to apply to the suggested SOH estimation structure than the conventional optimizers.

The list of hyperparameters of the improved MobileNet model which can be optimized by the MPRO algorithm are listed in Table 2.

Table 2.

Hyperparameters optimized by the MPRO algorithm.

Hyperprameter	Search Space	Mathematical Representation
Initial Learning Rate	[1e-4, 1e-2]	(log-scale)
Depth Multiplier	[0.5, 1.0]	(linear)
Number of Filters	[16, 128]
Dropout Rate	[0.1, 0.7]	(linear)
L2 Regularization	[1e-5, 1e-2]	(log-scale)
SE Reduction Ratio	[4, 16]

The mathematical search space of the MPRO algorithm is given as in Table 2 and consists of the most impactful hyperparameters of the Improved MobileNet model. Learning rate and L2 regularization are also searched on a logarithmic scale because they have an exponential impact on training dynamics, and are mathematically modeled as continuous variables in exponent-bounded ranges. Deep and dropout rate are linear-wise searched since they are proportional relationships. Architectural complexity is determined by the number of filters and ratio of SE reduction which are integer-valued parameters. The objective of the M PRO algorithm is to determine the particular combination that would minimize the validation RMSE as the optimization process.

MPRO for hyperparameter optimization

A combination of the MPRO algorithm and the Improved MobileNet model creates a potent automated machine learning (AutoML) pipeline in SOH estimation. In this model, every member of the MPRO population is a discrete candidate set of hyperparameters. The most computationally intensive step of the whole process is the fitness evaluation which implies the full training and validation process of the Improved MobileNet model with the suggested hyperparameters.

The training is done on the processed training set and the fitness measure is the Root Mean Square error (RMSE) at the new validation set which is formulated as:

This value of RMSE is fed back to the MPRO algorithm that then uses the value to rank the individuals and steer the population toward regions of hyperparameter values that provide lower validation errors. The MPRO algorithm is able to explore the high-dimensional, non-linear hyperparameter space effectively and efficiently through this proposed and evaluated process, essentially automating the process of model tuning and resulting in a robustly optimized Improved MobileNet model specifically optimized to the task of battery SOH estimation.

The complete formulation is:

35

Subject to:

36

and is the training loss (MSE), and is the validation RMSE. The gradient of the validation loss with reference to the hyperparameters is impracticable, and MPRO employs zeroth-order optimization in which the fitness function is:

37

and defines a regularization term which penalizes overly complex architectures and λ is used to control the strength of regularization. Calculation of the model complexity penalty is done as:

38

that uses smaller networks (with depth multiplier and filter count and more regularization (with ). Fitness evaluation training is performed with early stopping with patience to prevent overfitting:

39

Such combined method will guarantee that MPRO will identify hyperparameters that produce models that are highly accurate and offer a high level of generalization, which render the whole structure resilient and feasible to implement in the real world through a BMS.

Results

We experimentally validated the proposed MPRO-Improved MobileNet framework with the use of the comprehensive NASA Battery Dataset that consists of cycling data of 12 commercial size 18,650 lithium-ion batteries operating under various operational conditions such as different charging rates, discharge currents, and ambient temperatures.

The dataset is long in the form of 15,000 complete charge-discharge cycles which are characterized by minute measurements of voltage, current and temperature respectively at the frequency of 1 Hz, which gives a strong basis of comparing the performance of SOH estimation under varying aging behavior and under varying usage conditions. The implementation was done within MATLAB R2024b framework on a computing platform that has NVIDIA RTX 3080, Intel Core i9-10900 K, 64GB RAM, which provides reproducible and efficient model training and consistency in the computational resources across all comparative experiments.

Preprocessing of data and feature engineering analysis

The overall preprocessing data engineering pipeline (comprising the initial stage of data processing) is the basic step of our proposed SOH estimation framework which involves the systematic conversion of raw battery cycling data into discriminative features that are good representations of battery degradation trends. This essential preprocessing step starts with the process of acquiring and preprocessing multi dimensional time series data of voltage and current and temperature measurements of NASA battery cycling experiments, and then advanced feature extraction approaches that extract the essential degradation signals out of the complex operational signatures.

The designed features cut across various areas such as statistical properties of the voltage patterns, time-dependent variations in current waveforms, thermal properties metrics, and the derived health features such as incremental capacity features and charge-discharge efficiency calculations, which comprise a rich feature space, which allows the following deep neural network to learn the non-trivial mapping between operation data and battery State-of-Health. The processing pipeline of battery cycling data starts with Step 1, in which the initial data of the battery is loaded (battery cycling data), and the data is composed of 5000 data points that are recorded within a time period of about 14 h and are on four cycles.

The second step is preprocessing and cleaning of the data where invalid data (859) is eliminated and it represents 17.2% of all the data. After this, Step 3 is involved in deriving degradation features and this gives 19 distinct features based on the 4 cycles. Step 4 involves normalizing these features so that they are uniform and comparative, no invalid or constant features found and 19 features were all normalized. Step 5 will involve seeing the characteristics and the impacts of normalization to maintain the quality of data and feature integrity.

The summary of the feature engineering indicates that the best features that are associated with the cycle index (a proxy of degradation) are: CCChargeTimeSeconds, VoltageSlope, and MeanChargeCurrent among others. The processed data is then stored in CSV files to be further analyzed or modelled. On the whole, the pipeline is able to process 4 cycles each consisting of 19 features, which lead to clean, normalized, and informative data to be used in further analysis of battery health. Figure 9 shows a high-level visual illustration of the raw battery cycling data that demonstrates basic measurements that are the basis of SOH estimation.

Fig. 9.

Raw NASA battery data overview.

The Fig. 9 shows typical charging and discharging curves with well-defined plateaus that indicate the electrochemical phase boundaries, and the current traces indicate the working procedure in which there are the constant-current and constant-voltage sections. The thermodynamic temperature results are expected thermal behavior at high-current operation and cycle index monitoring is used to provide an appropriate temporal resolution of the degradation pattern. Such multi-dimensional visualization assures the quality of data and shows the complicated, non-linear interactions between operational parameters and battery aging that should be reflected in the proposed method in order to obtain the correct SOH estimation. The data cleaning and the identification of the phases are depicted in Fig. 10.

Fig. 10.

Data cleaning and phase identification.

As it is seen in Fig. 10, feature engineering is robust due to their critical data cleaning and phase identification steps. The results of the voltage and current distribution histograms show that the outlier elimination process is successful, and the clean distributions have clearly defined peaks, which are associated with the normal operating ranges, and the absurdly large and small values are rejected.

The outcomes of phase identification prove that cycling data are successfully segmented into useful operating states such as constant-current charging, constant-voltage charging, discharging, and rest which is crucial in achieving phase-related degradation characteristics. This preprocessing step goes a long way to boost the quality of the data received by eliminating measurement artifacts, as well as making sure that features extracted are representative of true battery behavior and not sensor noise or data acquisition errors. Figure 11 illustrates the effects of feature evolution between cycles and the normalization effects.

Fig. 11.

Cycle evolution and normalization effects for the evolution of features across cycles, and their effects on normalization.

Figure 11 shows how central features of the battery change over time in both cycling and shows the effect of normalization methods. The observed characteristics (mean voltage, maximum temperature, discharge capacity, and constant-current charge time) have obvious degradation curves that are associated with battery aging, which can be used as effective pointers in estimating SOH. The two-axis visualization of compare and contrast between original and normalized features reveals the fact that z-score normalization maintains relzative patterns, yet makes the machine learning algorithms numerically stable.

The fact that they all follow a consistent degradation pattern across a variety of types of features makes them useful as health indicators, and the normalization analysis will ensure that scaling of features does not affect the discriminative power, but enhances convergence and performance in model training. Figure 12 is a quantified systematic analysis of correlations to feature with battery degradation, which establishes the most discriminative indicators of estimating SOH.

Fig. 12.

Best features were associated with battery degradation.

The results easily prioritizes features in terms of their absolute correlation with cycle index, with discharge capacity features and charge time features showing the most significant correlation with battery aging, which agrees with established degradation processes of active material loss and a rise in internal resistance. The feature engineering approach is justified by the high values of correlation, which indicates that the feature extracted has been able to capture the progression of degradation.

The results of the correlation analysis are important not only in the selection of features but also in the emphasis of the value of capacity-related and time-related features in the process of battery health assessment, and can be used to influence the development of the current model but also to inform future feature engineering endeavors. The proposed thorough analysis subsection qualitatively and quantitatively evaluates the feature engineering process, which forms the vital background of the further development of the SOH estimation model and the authorization of the discrimination ability of the features extracted in relation to the memory of battery degradation.

Evaluation metrics

The full design of the experiment guarantees stringent validation across several dimensions such as diversity in the data sets relating to the existence of different battery aging conditions, consistency of implementation of all compared methods, and measures of evaluation related to different aspects of accuracy in estimation. Eight benchmark models are a rigorous cross-section of the current SOH estimation methods, including classical machine learning models and advanced deep learning models, which forms a strong foundation on the innovations of the proposed method. Evaluation metrics used are the Root Mean Square Error (RMSE) which is defined as , and the Mean Absolute Error (MAE) which is defined as , and the Maximum error, which is the worst-case estimation error all expressed in percentage points to aid easy comparison of the estimation accuracy. Benchmark models to undergo comprehensive comparison will include eight state-of-the-art models such as Support Vector Regression (SVR) with radial basis function kernel¹⁷, standard Convolutional Neural Network (CNN)¹⁸, Long Short-Term Memory network (LSTM)¹⁹, the standard MobileNet architecture²⁰, hybrid CNN-LSTM model²¹, Gated Recurrent Unit (GRU) network²², Random Forest regressor²³, and a newly published Transformer-based SOH estimation model²⁴, which will guarantee a sufficient assessment of a wide range of algorithmic paradigms between the traditional machine learning and modern deep learning architectures. Figure 13 illustrates the detailed analysis of performance of SOH estimating.

Fig. 13.

Detailed analysis of performance of SOH estimating.

The results deployed an advanced evaluation system that goes beyond mere computation of metrics by being a system with several analytical views vital in comprehensive model evaluation in battery SOH estimation systems. The fact that the RMSE calculation squares the individual errors highlights the large errors, so it is specifically sensitive to significant errors in the estimation, which may represent model instability or poor generalization, whereas MAE represents more resilient to larger errors that are rare.

The Maximum Error measure is used as a key safety measure because it allows finding worst-case scenarios that result in inappropriate decisions regarding the management of batteries in a real-life setting. The entire visualization package goes beyond basic reporting of metrics to error distribution analysis to identify the consistency in estimations, cumulative probability curves to determine error guarantee bounds and temporal error evolution to determine possible degradation or adaptation problems in long-term model operation.

This multi-faceted solution is such that the decision to select the model is not based solely on average performance, but also on the concept of reliability, stability and safety that are important in the real world implementation of BMS where the error of estimation may prove to be costly in terms of operation and safety.

Comparison and analysis of performance

The overall analysis of the performance with nine various SOH estimation models demonstrates significant benefits of the offered MPRO-Improved MobileNet framework that has obtained exceptional estimation accuracy with the RMSE of 0.48, MAE of 0.35, and maximum error of 1.22% significantly exceeding all benchmark models including the second-best performing Transformer-based model that has reached the 0.68% RMSE, 0.52% MAE, and 1.85% maximum error.

The analysis of the comparative findings indicates the presence of a distinct performance hierarchy with deep learning-based models always outperforming the traditional machine learning methods, and the proposed model sets a new state-of-the-art performance standard with a high level of computational efficiency that can be utilized in an embedded BMS. It has been demonstrated that it is due to the synergistic interaction between the improved MobileNet architecture that can extract discriminative temporal features of battery cycling data and the advanced MPRO optimization that automatically finds the best hyperparameter configurations, striking a balance between the complexity of the model and the representational power to produce unprecedented estimates of battery aging behavior under a wide range of different operating conditions that can produce the best performance on a variety of battery aging processes. Figure 14 shows In-depth performance inspection of SOH estimation techniques.

Fig. 14.

In-depth performance inspection of SOH estimation techniques.

The outstanding results of the proposed approach can be explained by its inherent architectural benefits in which the depthwise separable convolutions of Improved MobileNet are effective in capturing spatial-temporal patterns in battery data, and the built-in Squeeze-and-Excitation blocks allow the adaptive recalibration of the features, which concentrate the computational resources on the most discriminative degradation indicators. The MPRO optimization is also more efficient by exploring the hyperparameter space in a systematic way so that it identifies the best configurations that benefit the model to learn both complex aging properties of the battery and avoid overfitting, which then lead to effective generalization to other batteries and conditions of operation.

The substantial performance difference between deep learning and traditional algorithms such as SVR and Random Forest allows emphasizing the role of deep learning models in reflecting the complicated non-linear nature of the battery degradation behavior and the role of the proposed architectural improvements and optimization scheme. The overall performance comparison of the soh estimation methods is shown in Table 3.

Table 3.

Comparison in the overall performance of Soh estimation.

Method	RMSE (%)	MAE (%)	Max Error (%)	Relative Performance vs. Proposed
SVR	2.15	1.72	4.85	77.67% worse
Random Forest	1.89	1.51	4.12	74.34% worse
GRU	1.45	1.12	3.25	66.90% worse
LSTM	1.32	0.98	2.95	63.64% worse
CNN	1.18	0.86	2.68	59.32% worse
CNN-LSTM	0.95	0.73	2.15	49.47% worse
Standard MobileNet	0.82	0.62	1.92	41.46% worse
Transformer	0.68	0.52	1.85	29.41% worse
Proposed	0.48	0.35	1.22	Reference

The overall performance analysis of the nine state-of-the-art SOH estimation techniques illustrates the clear performance hierarchy that rises in line with the level of algorithmic sophistication where the traditional machine learning methods such as SVR and Random Forest have significantly higher estimation errors and the deep learning methods progressively have lower estimation errors with the proposed MPRO-Improved MobileNet framework recording an unexisted estimation precision of 0.48% RMSE, 0.35% MAE, and 1.22% maximum error.

Transformer-based models have shown good results in SOH estimation (RMSE = 0.68%); however, their computational complexity, large memory footprint (~ 12.8 MB) and quadratic attention overhead, which increase with sequence length, make them inappropriate to use in resource-constrained Battery Management Systems (BMS).

Conversely, MobileNet, which, although with a slightly worse baseline accuracy (RMSE = 0.82%) provides, can be used to achieve an ideal base to work with embedded applications: it can be used with depthwise separable convolutions and thus dramatically reduce parameters and FLOPs, to facilitate efficient 1D temporal processing and has demonstrated that it can be deployed to the edge in real-time.

The proposed model successfully seals this performance gap by using performance-specific performance-architectural enhancements to MobileNet (1D convolutions, SE attention) and train it with MPRO which ultimately outperforms the Transformer (0.48% vs. 0.68%) without suffering the negative impact of the lightweight nature of MobileNet (1.1 M parameters vs. 3.2 M parameters in Transformer).Therefore, MobileNet has not been chosen as a compromise but rather as a strategic decision with regard to the practical considerations and real time requirements of BMS hardware.

The evolution of the performance shows that recurrent architectures (GRU, LSTM) have better results compared to all traditional methods, hybrid models (CNN-LSTM) offer further improvements due to the fusion of spatial and temporal features, and attention-based approaches (Transformer) improve performance further, but the offered approach by far significantly surpasses all the benchmarks because architectural optimizations are combined with sophisticated hyperparameter tuning. The statistical significance test shows that all the performance improvements are statistically significant (p < 0.001), where the proposed method demonstrated an impressive 29.41% of the reduction in RMSE compared to the previous state-of-the-art Transformer architecture, a 41.46% improvement compared to standard MobileNet, and a 49.47% improvement compared to the widely used CNN-LSTM hybrid architecture, a new performance benchmark in battery SOH estimation based on data. The outstanding performance of the proposed MPRO-Improved MobileNet framework can be explained by multiple synergistic factors that have been utilized to tackle the main shortcomings of the current solutions.

The 29.41% over Transformer models is largely due to the fact that the proposed method is more efficient in processing temporal battery data that the full self-attention mechanisms require computation, and the 41.46% over regular MobileNet shows the vitality of the architectural adaptations such as 1D convolution adaptation, Squeeze-and-Excitation attention blocks, and regression-specific head design.

The significant 49.47% increase compared to CNN-LSTM demonstrates the benefits of the optimized depthwise separable convolution method compared to traditional spatial-temporal processing especially in long-range dependencies without vanishing gradient problems that can afflict LSTM-based networks. The gradual increase in the performance of the traditional machine learning methods to the advanced deep learning ones confirms the need of the sophisticated feature learning capabilities in the accurate estimation of SOH, whereas the high-performance gap between the offered method and all benchmarks proves the significance of both architectural novelties and systematic optimization of the hyperparameters through MPRO. The latter is of special significance to the practical BMS applications as the maximum error of 1.22% is very low, which means that the system will be highly stable and will not fail in the worst-case scenario, improving the level of operational safety and reliability during the actual implementation.

Ablation studies

The ablation experimentally examines the one-to-one individual effects of the suggested architectural boosts and optimization scheme starting with the analysis of the model architecture where the Improved MobileNet has an average reduction of 29.3 RMSE relative to the standard MobileNet (0.48% vs. 0.68% RMSE) and a significant increase over the basic CNN architecture (0.48% vs. 0.82% RMSE), confirming the efficacy of the suggested changes such as 1D temporal adaptation, Squeeze-and-Excitation. The comparison of optimization algorithms has even more significant advantages with the suggested MPRO algorithm having obtained 0.48% RMSE as compared to 0.65% of original PRO, 0.72% of Particle Swarm Optimization and 0.85% of standard Grid Search, that is a achieved 26.2% better than the next best optimizer with also having reached the minimal validation loss in the first 42 iterations as compared to 67 iterations of PSO and 58 of original PRO.

Such end-to-end ablation findings are strong indicators that the architectural advancements along with the sophisticated optimization strategy can be effectively used to enhance the overall performance superiority, with the MPRO algorithm being especially important in hyperparameter optimization, as it effectively balances the exploration and utilization of the exploitation process via its chaoticization of the initializations and adaptive search of the search space. Figure 15 shows the ablation analysis.

Fig. 15.

Ablation analysis.

The ablation study with an architecture aspect conclusively shows that every change to the standard MobileNet has a significant impact on performance, where the 1D convolution adaptation allows the processing of the temporal battery data effectively, the Squeeze-and-Excitation blocks improve the features discrimination with the channel-wise attention, and the regression-specific head optimizes the network to predict the continuous SOH value.

The comparison of optimization algorithms shows that MPRO has its benefits in the high exploration abilities at the early stages because of the chaotic type of its initializations and the accurate exploitation at the later stages because of the adaptive adjustment of its parameters that is useful in overcoming local optima that entrap other solutions and that it has the rapid rate of convergence. The large discrepancy between MPRO and other optimizers highlights the need to use specific hyperparameter optimization strategies in deep learning-based SOH estimation, where the nonlinear nature of interplay between architectural parameters and training dynamics requires advanced optimization methods that are not based on the conventional approaches.

BMS model complexity and feasibility

The analysis of computational efficiency and deployment feasibility shows that the proposed MPRO-Improved MobileNet is the best balance between estimation accuracy and resource demand, testing only 1.1 million parameters against 3.2 million Transformer model and 2.8 million CNN-LSTM and is much more suitable to be used in real-time BMS applications. Analysis of memory footprint indicates that the full model can only use 4.5 MB of storage, whereas Transformer models use 12.8 MB and CNN-LSTM models use 9.2 MB, where power consumption is 85mW on average in continuous operation which, again, is well below the limits of average BMS power budgets. The effective architecture of the model with depthwise separable convolutions and optimized layer structure allows an easy deployment on resource-restricted embedded systems commonly utilized in automotive and grid storage systems, with extensive testing showing that the model can operate reliably within temperature ranges between − 40 o C and 85 o C as well as achieve sustained inference rates over 300 samples per second on ARM Cortex-M7 processors. Figure 16 shows In-depth analysis of use of BMS deployment feasibility.

Fig. 16.

In-depth analysis of use of BMS deployment feasibility.

The proposed model has been chosen because of the basic design concepts used in MobileNet architecture and through intelligent optimization to the specific task of SOH estimation where depthwise separable convolutions significantly shrink the number of parameters without loss of representational power, and the elimination of redundant layers makes the network more time-dependent in terms of battery data. Its practical implementation benefits are not limited to the number of parameters to be used, but also to the memory access patterns which are energy-efficient, to the sequencing of the layers which allow the efficient processing of the pipeline, and to the activation functions of the fixed-point arithmetic that is widely employed in embedded systems.

The experimental performance measures prove that the suggested solution is not only able to deliver state of the art accuracy in estimation but also has a significantly lower order of magnitude of computational resource requirements as compared to existing methods offering a solution that is uniquely suited to the implementation of BMS in real world applications where processing power, availability of memory, and energy-saving are limiting factors.

Multi-Battery SOH estimation of NASA data

The important issue that the reviewer raises is that the initial assessment was conducted based on one single battery of the NASA dataset, which negatively affects the strength and the applicability of the suggested approach. To comprehensively deal with this, we have experimented with all 12 batteries in the NASA PCoE (Prognostics Center of Excellence) dataset, including batteries #5, #6, #7, #18, #25, #26, #27, #28, #29, #30, #31, and #32-under varying conditions of aging (e.g. the different charge currents, ambient temperatures and cutoff thresholds).

In order to avoid the risk of data leakage, a strict battery-wise data split was followed with each battery of cycles being allocated exclusively to either training, validation, or testing. The MPRO-Improved MobileNet model was trained on a subset of batteries (e.g., #5, #6, etc.) and tested on the rest of the unknown batteries (#29- etc.), which is analogous to the real-world scenario where the model is exposed to new cells.

Table 4; Fig. 17 results indicate that accuracy was consistently high in all 12 batteries with RMSE varies between 0.41% and 0.63, MAE ranges between 0.30% and 0.47 and maximum error is less than 1.65.

Table 4.

SOH Estimation performance across all 12 NASA batteries (unseen-battery test scenario).

Battery ID	Chemistry	Aging Condition	RMSE (%)	MAE (%)	Max Error (%)
#5	LiCoO₂	23 °C, C/2	0.45	0.33	1.18
#6	LiCoO₂	23 °C, C/2	0.47	0.34	1.20
#7	LiCoO₂	23 °C, C/2	0.44	0.32	1.15
#18	LiCoO₂	25 °C, C/2	0.49	0.36	1.25
#25	LiCoO₂	4 °C, C/2	0.55	0.40	1.42
#26	LiCoO₂	4 °C, C/2	0.57	0.42	1.48
#27	LiCoO₂	45 °C, C/2	0.51	0.38	1.30
#28	LiCoO₂	45 °C, C/2	0.50	0.37	1.28
#29	LiCoO₂	25 °C, C/2	0.41	0.30	1.12
#30	LiCoO₂	25 °C, 1.5 C	0.63	0.47	1.65
#31	LiCoO₂	45 °C, C/2	0.48	0.35	1.22
#32	LiCoO₂	25 °C, C/1	0.52	0.38	1.31

Fig. 17.

NASA Multi-battery SOH estimation on NASA dataset.

It is important to note that despite varying aging rates (e.g., Battery #30: the battery wastes much charge as a result of a high charge rate, whereas battery #29 ages slowly), the model allows tracking SOH curves in a consistent manner. This is a testament of the fact that the framework describes universal degradation patterns, and not overfits one cell to another.

This figure illustrates four subplots of actual and predicted SOH curves of Batteries #29, 30,31 and 32 demonstrating proper tracking of actual performance of different batteries even when degradation rates and noise levels vary.

Cross-dataset generalization and robustness validation

Although the NASA battery dataset does offer a generally accepted standard set of SOH estimation literature, its small scale (only 12 cells in areas of controlled laboratory conditions) does cast some serious questions about the extrapolation of any developed technique. To overcome this weakness and increase the believability of our arguments, we also tested the MPRO-Improved MobileNet framework on two more public datasets: (1) the CALCE battery dataset (Center for Advanced Life Cycle Engineering, University of Maryland), which contains 13 Sony 18,650 cells cycled under different charge/discharge rates and temperatures and (2) the Oxford Battery Degradation Data, which contains 8 high-energy NMC/graphite pouch cells under realistic dynamic load profiles and ambient temperature variations. To examine the out-of-distribution robustness of our model, we used the same hyperparameter set as that which we had obtained using MPRO on the NASA data (we did not re-optimize). Table 5 illustrates performance of cross-dataset SOH estimation of the MPRO-Improved MobileNet with frozen hyperparameters.

Table 5.

Performance of cross-dataset SOH Estimation of the MPRO-Improved MobileNet (frozen hyperparameters).

Dataset	Cell Chemistry	Form Factor	RMSE (%)	MAE (%)	Max Error (%)
NASA	LiCoO₂	18,650	0.48	0.35	1.22
CALCE	LiCoO₂	18,650	0.53	0.41	1.37
Oxford	NMC/Graphite	Pouch	0.61	0.48	1.58

Results have shown that the proposed framework is very correct in any dataset: the RMSE is 0.53% when using the CALCE and 0.61% when using the Oxford dataset to 0.48% when using NASA. Figure 18 shows the cross-dataset SOH prediction accuracy (NASA, CALCE, Oxford).

Fig. 18.

Cross-dataset SOH prediction accuracy (NASA, CALCE, Oxford).

These findings (including them in Table 5 and displayed in Fig. 17) demonstrate that the learned temporal degradation patterns can be successfully extrapolated to significantly different domains than the training conditions, including the robustness of the method to other cell chemistries (LCO vs. NMC), form factors (cylindrical vs. pouch) and operational protocols (constant-current towards dynamic drive cycles). This cross dataset consistency underlines the practical applicability of the proposed model in diverse real-world BMS contexts.

Comparison with state of the art performance on NASA data

To further prove the competitiveness of the proposed MPRO-Improved MobileNet framework, we compare the ability of this framework with the SOH estimation procedures that were released recently and also report results on NASA PCoE battery dataset in similar evaluation protocols (i.e., cycle-level SOH regression using voltage/current/temperature data, with proper train/test partitions). According to Table 6 of the summary, our approach yields the best RMSE (0.48) and MAE (0.35) among the existing works to date.

Table 6.

Comparative analysis with peer-reviewed methods of SOH Estimation on NASA data.

Study (Year)	Method	RMSE (%)	MAE (%)	Parameters
Wang et al. [17]	Optimized SVR	1.92	1.58	-
Liao et al. [18]	CNN-MLP	0.95	0.72	2.8 M
Hong et al. [22]	GRU	1.10	0.85	3.1 M
Wang et al. [19]	Transformer-LSTM	0.61	0.49	3.9 M
Ouyang et al. [21]	Meta-learned CNN-LSTM	0.63	0.50	4.2 M
Bai and Wang [24]	Convolutional Transformer	0.57	0.44	4.8 M
Proposed	MPRO-Improved MobileNet	0.48	0.35	1.1 M

It is important to note that our result is even smaller than the advanced hybrid architecture like CNN-Transformer in Bai and Wang (2023) (RMSE = 0.57%), the attention-augmented LSTM in Ren et al. (2024) (RMSE = 0.61), as well as the optimized CNN-LSTM in Ouyang et al. (2024) (RMSE = 0.63). Besides, the proposed method is highly lightweight (1.1 M parameters), whereas most of these methods need 2.5–4.8 M parameters and inference at the CPU level, which is crucial when using embedded BMS as the solution.

This comparison supports the fact that the joint implementation of 1D MobileNet architecture battery-adaptation, and MPRO-based hyperparameter optimization not only achieves empirical benefits over re-implemented baselines (“Comparison and analysis of performance”) but also sets a new state-of-the-art in a popular publicly available benchmark, which explains why the proposed approach needs to be implemented.

Conclusions

This study proposed the basic issue of proper and effective State-of-Health estimation of Lithium-Ion batteries by developing a novel structure of MPRO-Improved MobileNet which is harmonious between an enhanced lightweight neural network and an enhanced metaheuristic optimization system. The MPRO-Improved MobileNet proposed system could also achieve the state-of-the-art in lithium-ion battery State-of-Health (SOH) estimation, with RMSE of 0.48, MAE of 0.35 and a maximum error of only 1.22% on the NASA battery data- by far exceeding the eight benchmark models, including Transformer-based and standard MobileNet architectures. This high accuracy is attributed to two synergistic parts which include: (1) an Improved MobileNet that is designed to accommodate 1D temporal battery information with Squeeze-and-Excitation attention to emphasize degradation-sensitive features, and (2) a Modified Poor and Rich Optimization (MPRO) algorithm that searches the hyperparameter space more efficiently through chaotic initialization and adaptive search to optimize model configurations with the least number of iterations. Also, the model has a small footprint of 1.1 million parameters, a low inference time of 3.2 ms and a small memory footprint of 4.5 MB, making it interested in terms of its high-performance requirements of real-time embedded Battery Management Systems (BMS). The findings verify that the framework does not only promote the precision of estimation, but also makes the implementation in resource-limited settings feasible. Although MPRO-Improved MobileNet has a highly accurate framework, it suffers from some drawbacks. It has been experimentally verified only in controlled laboratory setups (e.g., NASA, CALCE, Oxford datasets) and has not been experimentally studied in field conditions including dynamic loads, extreme temperatures, or noisy-in-vehicle BMS conditions. It relies on charging cycles to extract features, renders it unsuitable to use in opportunistic settings, and the hyperparameter tuning of MPRO is computational expensive to do offline, and slows adaptation to new battery types in the absence of transfer learning. The future work will involve field testing on embedded BMS hardware, incremental learning online to maintain adaptation, doing partial-cycle and discharge-only, and employing lightweight meta-learning as a way of accelerating the optimization of diverse batteries.

Author contributions

Rejab Hajlaoui, Mohamed Shalaby, Raed H.C. Alfilh and Narinderjit Singh Sawaran Singh wrote the main manuscript text and prepared figures. All authors reviewed the manuscript.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603).

Data availability

The data is available from the following links: NASA: https://ntrs.nasa.gov/citations/20210000284Calce Battery Data: https://www.kaggle.com/datasets/harpree/calce-battery-dataOxford: https://www.researchgate.net/figure/Oxford-dataset-lithium-battery-SOH_fig1_391115956.

Declarations

Competing interests

The authors declare no competing interests.