1 Introduction

Global pollution and climate change are major threats to human survival. In an effort to reduce carbon emissions, several developed countries have announced plans to phase out diesel and gasoline cars in favor of electric vehicles (EVs) in the near future [8]. Electric vehicles (EVs) are still struggling to become a popular and efficient mode of transportation because of their limited range and short lifecycles of their energy storage devices [18]. To power a wide range of essential products, batteries are the most often used green energy storage technology [17]. There are a variety of batteries in the market. Most often used rechargeable batteries are lithium-ion, lead-acid, and metal-nickel hydride batteries [9]. Lithium-ion batteries (LIBs) have several advantageous properties, including low chemical fuel consumption, low carbon emission, low memory, high power density, high voltage, low self-discharge rate, rapid changes, and a long service life [5].

The state of charge (SoC), state of health (SoH), and state of temperature (SoT) properties of the LIBs are crucial components of the BMS. Battery state of charge is often described as the ratio of residual capacity to total capacity [12, 15]. While an accurate SoC estimation model would be helpful, it is challenging to develop due to internal and external factors. Internal challenges include LIB’s materials, self-discharge, aging, thermal runaway, and hysteresis, external challenges include changes in ambient temperature and charging technique [4]. LIBs are notoriously difficult to understand because of the complexity of the problems they cause. Neither SoC nor SoH are readily available for on-site measurement, and instead must be determined by means of complex algorithms and measurable quantities like as current, voltage, and temperature [10]. More research in field of SoC, SoH, and SoT models is required to improve EV performance and extend battery lifetimes. Using Coulomb-counting, an open circuit voltage, an impedance analysis, a Kalman filter, and Artificial neural network (ANN) are all viable options for estimating SoC [21]. The Coulomb-counting approach represents model-free estimation, derived from the ampere-hour method to determine SoC but its accuracy decreases with changing temperature [14]. OCV is the simplest approach for estimating SoC. The OCV voltage is a linear function of SoC. By measuring OCV, we can simply anticipate the battery’s SoC [11]. Impedance analysis method is sensitive to noise and requires exact measurement of different parameters [19]. Kalman filter can measure SoC without need of information about initial SoC, meanwhile, its filtering method greatly reduces sampling noise [13]. The ANN-based model is developed using realistic battery test results, Artificial intelligence (AI) methods are more effective and accurate than traditional methods [6].

The primary objective and contribution of this paper is to devise a proficient algorithm for estimating the SoC, SoH, and SoT of lithium-ion batteries. This study presents an analysis of the driving cycle calculation and the subsequent validation of different parameters such as the root mean square error (RMSE) and mean absolute error (MAE). Additionally, there has been a reduction in the percentage error during state of charge (SoC) estimation.

2 Problem statement

Lithium-ion batteries are extensively utilized throughout a range of applications, encompassing electric vehicles, consumer gadgets, and renewable energy systems. The precise determination of the SoC, SoH, and SoT of a battery is of utmost importance in order to optimize battery performance, guarantee safe operation, and prolong the overall lifespan of the battery. The SoC of a battery indicates its present level of charge, whereas the state of health (SoH) describes the overall condition or degree of degeneration of the battery. The monitoring of temperature is a fundamental component of battery management systems, and the maintenance of batteries within an appropriate temperature range is crucial for ensuring both safety and performance.

The issue under consideration pertains to the estimation error that is inherent in the methodologies employed for estimating SoC, SoH, and SoT of lithium-ion batteries. Despite the considerable progress made in battery management systems and estimate approaches, the issue of inaccuracy in estimating SoC, SoH, and SoT remains unresolved. The presence of these inaccuracies may result in compromised performance, diminished battery longevity, and potential safety risks. The aim of this study is to implement an enhanced techniques for estimating SoC, SoH, and SoT in lithium-ion batteries, with the goal of minimizing estimation inaccuracies.

3 Proposed methodology

In order to estimate SoC, SoH, and SoT, a data-driven technique that utilizes hybrid optimum Deep Learning strategies has been proposed. Initially, the information from the DST datasets have been taken, which include a wide range of features and Li-ion battery is used in EV driving cycles. Data normalization is employed for increase the learning rate and pre-processing of data increase the accuracy. To make an accurate prediction, Simulated-Annealing-based Golden Eagle optimization [1, 16] is offered as feature extraction and selection approach. The data is also segmented to create a distinct training set and testing set. Then, the new optimized hybrid EP-based R110-BLSTM have been employed for training the model. After that the loss functions had been approximated using a combination of Cross entropy and Focal function [3, 16]. Finally for training the model the new optimized hybrid EP-based R110-BLSTM is used [2]. After validating the trained model with the testing set, SoC, SoH, and SoT values have been computed. Finally, the efficacy of the suggested method is determined by analyzing the performance measures. The flowchart of proposed algorithm is shown in Fig. 1.

Fig.1
figure 1

Flowchart of proposed algorithm

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To build a network model, a solid foundation of preprocessed data is required. Numerical differences occur between the acquired battery data indicators for voltage, current, SoT, SoH, and SoC, which will slow down training and reduce the model’s accuracy and precision. Accordingly, normalizing the data samples is necessary to reduce the influence of scale between the indicators and preserve the model’s accuracy and generalizability. Further, the collection of normalized data is broken up into segments. This normalized dataset (\(x^{*}\)) can be represented by using Eq. (1):

$$ x^{*} = \frac{{x - x_{\_} }}{{x_{ + } - x_{\_} }} $$
(1)

EP based R110-BLSTM is used to train the dataset and then losses and error are to be calculated for parameter evaluation. There are a variety of factors, including voltage (\(v_{{\text{i}}}\)), current (\(i_{i}\)), temperature (\(t_{i}\)), and the rated capacity (\(c_{i}\)), that may be used as input data (\(x_{i}\)) for this model. Equation (2) is used to represent the input sequence:

$$ x_{i} = \left\{ {v_{i} ,i_{i} ,c_{i} ,t_{i} } \right\} $$
(2)

In developing the proposed algorithm, some constraints are there. In order to safeguard the battery, it is imperative to restrict the SoC within the designated range of SoCmin and SoCmax. Also, the voltage (\(v_{{\text{i}}}\)) and current (\(i_{i}\)) must be within the limit. One of the primary limitations in battery technology is the limits on storage capacity Ci. Some of the constraints were taken into account are shown in Eq. (3):

$$ \left. {\begin{array}{*{20}l} {{\text{SoC}}_{\min } \le {\text{SoC}} \le {\text{SoC}}_{\max } } \hfill \\ {V_{i,\min } \le V_{i} \le V_{i,\max } } \hfill \\ {i_{i,\min } \le i_{i} \le i_{i,\max } } \hfill \\ {C_{i,\min } \le C_{i} \le C_{i,\max } } \hfill \\ \end{array} } \right\} $$
(3)

Next, we couple the BiLSTM’s forward and backward implied level outputs and feed them into ResNet 110’s fully connected layer. The output state of the assumed layer is used as input by the fully linked layer. The residual mapping is the value added to the input to approximate the block’s final function. Dimensionality reduction is performed using the sigmoid activation function, and the ultimate result is estimates as shown in Eqs. (4)–(6)

$$ {\text{SoC}}_{i} = \ell W_{k} + \alpha $$
(4)
$$ {\text{SoH}}_{i} = \ell W_{k} + \alpha $$
(5)
$$ {\text{SoT}}_{i} = \ell W_{k} + \alpha $$
(6)

The output of the fully connected layer is indicated by \(\ell\), the weight matrix is written as \(W_{k}\), and the output of the regression layer’s variance is marked by \(\alpha\).

For calculation of correntropy induced loss along with focal loss, Eq. (7) is being used:

$$ {\text{Loss}}_{cf} \left( {\hat{x}_{i} ,x_{i} } \right) = \left\{ {\begin{array}{*{20}l} {f_{l} \left( {\hat{x}_{i} ,x_{i} } \right),} \hfill & {0 < T \le k} \hfill \\ {c_{l} \left( {\hat{x}_{i} ,x_{i} } \right)} \hfill & {k < T < K} \hfill \\ \end{array} } \right. $$
(7)

where total epochs are denoted by \(K\), whereas the current epoch and the threshold value are denoted by \(T\) and \(k\), respectively. In addition, both the learning parameters and the backpropagation parameters are computed. In the event that the precondition is fulfilled, the training phase is finished, and accuracy and loss in the proposed model’s ability to detect SoC, SoH, and SoT are evaluated with the use of test data. Mean absolute error (MAE), root mean square error (RMSE), mean square error (MSE), and coefficient of determination (R2) are some of the metrics used to evaluate the model’s accuracy in estimating the effect SoC, SoH, and SoT on lithium batteries by using the following Eqs. (8)–(11):

$$ E_{{{\text{rms}}}} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {\hat{x}_{i} - x_{i} } \right)}^{2} } $$
(8)
$$ E_{{{\text{ms}}}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {\hat{x}_{i} - x_{i} } \right)}^{2} $$
(9)
$$ E_{{{\text{ma}}}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {\hat{x}_{i} - x_{i} } \right|} $$
(10)
$$ R^{2} = \sum\limits_{i = 1}^{n} {\left( {x_{i} - \hat{x}_{i} } \right)\left( {x_{i} - \hat{x}_{i} } \right)} /\sqrt {\sum\limits_{i = 1}^{n} {\left( {x_{i} - \hat{x}_{i} } \right)^{2} \sum\nolimits_{i = 1}^{n} {\left( {x_{i} - \hat{x}_{i} } \right)} } } $$
(11)

4 Result and discussion

All of the simulated CPU tests are performed in MATLAB 2018b. This information is then used to develop an estimate for SoC, SoH, and SoT using a hybrid optimum method. The discharge tests employ a variety of DST driving cycle input data sets. In this study, a model is proposed and then trained and assessed using discharge cycles at different temperatures.

Figure 2 represents the input dataset for DST drive cycle. The DST drive cycles begin with a collection of data. The data includes measurements of both current and voltage. The features of the dataset are retrieved based on the fitness of the SAGEO algorithm. In this context, extraneous data has been excluded, and only pertinent variables like as current, voltage, temperature, capacity, and others have been identified and retained.

Fig. 2
figure 2

Input dataset for DST drive cycle

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Based on the results obtained, it can be concluded that the proposed EP-based R110-BLSTM model demonstrates a high level of accuracy in estimating the SoC of the battery under different constant temperature conditions, as depicted in Fig. 3. The internal temperature of lithium-ion batteries undergoes continuous fluctuations during operation, and this temperature variation significantly affects the battery, especially in relation to battery SoC estimation. SoC estimation is a critical factor that must be taken into account irrespective of the model employed. Therefore, it is imperative to evaluate the efficacy of the proposed methodology in accurately estimating values within a setting characterized by varying temperatures. For each operational discharge, we have selected data sets corresponding to three specific temperatures: 0 °C, 10 °C, and 25 °C. The statistics clearly demonstrate that the estimated SoC value closely aligns with the real value during the driving cycles. This observation provides evidence of the proposed model’s robustness and its ability to minimize estimation errors.

Fig. 3
figure 3

Findings of SoC for DST drive cycle over a range of temperatures

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The discrepancy between the estimated SoC and the actual SoC is denoted by the SoC error, which necessitates diligent monitoring. The figure labeled as Fig. 4 illustrates the SoC error obtained at various temperatures. The observed discrepancy ranges from − 0.06 to 0.06% at 0 °C, − 0.06 to 0.065% at 10 °C, and − 0.06 to 0.065% at 25 °C throughout DST cycles.

Fig. 4
figure 4

SoC errors for DST at varying temperature

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Figure 5 illustrates the comparison between the estimated and real SoH values in relation to the number of cycles at various temperatures, including 0 °C, 10 °C, and 25 °C, for the DST drive cycle. The findings indicate that the proposed methodology has demonstrated effective performance, resulting in enhanced accuracy. A total of 500 cycles were taken into account for the execution of the SoH analysis. In the DST cycle at 0 °C, the observed and predicted values are found to be nearly identical. At 10 °C, the predicted value deviates slightly from the observed value due to an increase in the cycle. Furthermore, at 25 °C, there is some variation between the predicted value and the observed value, which can be attributed to the rise in temperature. Nevertheless, the suggested system exhibits a significantly low error value for the SoH.

Fig. 5
figure 5

SoH with respect to different temperature for DST

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Figure 6 depicts the SoT for the DST cycle at various temperatures, 0 °C, 10 °C, and 25 °C. The temperature of target and predicted performance of 0 °C, 10 °C, and 25 °C for DST cycle is mostly same with each other and this shows the effective performance of the proposed SoT estimation. The estimation results obtained at the three different temperatures are deemed to be both realistic and devoid of any errors. Moreover, the estimated SoT derived from DST cycle, exhibits a high level of accuracy when compared to the actual SoT. When the impact of noise is taken into account, the SoT estimate exhibits some small oscillations but the error is still under acceptable limit.

Fig. 6
figure 6

Comparing SoT for DST at a range of temperatures

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The validation of the root mean square error (RMSE) and loss function is conducted for the DST cycle, as depicted in Fig. 7. The outcome indicates that the validation root mean square error (RMSE) is 0.023 on the final iteration. The last epoch of the DST training cycle is 200, and the total training time is 23 min and 15 s. The frequency of validation is 30 iterations. In this context, the piecewise learning rate is implemented and its numerical value is recorded as 0.01. The observation demonstrates that the suggested system has enhanced functionality in estimating the SoC, SoH, and SoT for the DST cycle. This improvement is evidenced by the effective performance, as indicated by the reduced root mean square error (RMSE) and loss.

Fig. 7
figure 7

DST drive cycle loss function and training distribution

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5 Performance analysis

In this research, a significant reduction in RMSE, MAE, and MSE values with an increase in R2 for DST driving cycles have been observed. These results showed that the suggested SAGEO and EP based R110-BLSTM technique performed very well in estimating SoC, SoH, and SoT for LIBs.

The novelty of this work is its ability to estimate SoC, SoH, and SoT combinedly. To develop an algorithm which can estimate the SoC, SoH, and SoT combinedly is very difficult and challenging task. The proposed algorithm can estimate the SoC, SoH, SoT combinedly. Also, the predicted SoC, SoH, and SoT is coinciding with the targeted one. So, the error while estimation of SoC, SoH, and SoT is very less. The findings indicate that the root mean square error (RMSE) for validation is 0.023 during the last iteration. The final MAE%, RMSE%, MSE%, and R2 is 0.3024, 0.252, 0.2352, and 0.9900, respectively, at 25 °C. The results demonstrate a significant drop in the values of RMSE, MAE, and MSE, indicating an improvement in R2. The approach that has been suggested demonstrates enhanced resilience and accuracy in estimating SoC, SoH, and SoT. The results demonstrate the noteworthy effectiveness of the proposed SAGEO and EP based R110-BLSTM method in estimating the SoC, SoH, and SoT of lithium-ion batteries (LIBs). The performance metrics for the DST EV cycle are presented in Table 1.

Table 1 Comparative analysis with other methodology
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6 Conclusion

The primary contribution of this work is the development of an intelligent SoC, SoH, and SoT estimation model based on SAGEO and EP-based R110-BLSTM. Under a wide range of EV driving cycles, temperatures, and monitoring disturbances, our model enhances performance in terms of resilience, accuracy, and estimate speed. The model’s durability is validated by using two dynamic temperature characterizations (DST) of electric vehicle (EV) driving profiles at different temperatures. The predicted SoC is almost coinciding with the targeted SoC and SoC error is less than 0.06%. With a DST cycle RMSE is less than 0.252% and MAE is less than 0.3024%. So, the estimated SoC is accurate and error is very less near about zero. Also, the predicted SoH and SoT is almost coinciding with the targeted SoH and SoT. So, the proposed algorithm is successfully estimating the SoH and SoT more accurately. Hence, SoC, SoH, and SoT accuracy have dramatically increased. Future research will also evaluate how the proposed model fares in comparison with other advanced machine learning approaches.