Advanced Algorithm for SOC, Internal Resistance, and SOH Co-Estimation of Lithium-Titanate-Oxide Batteries Using Neural Networks

Abstract

Lithium-titanate-oxide batteries can reduce the long charging time of electric vehicles by offering fast charging capabilities. However, high charging currents require an accurate estimation of battery internal state to prevent early aging of the battery and dangerous situations. An accurate algorithm based on neural networks for the co-estimation of state of charge, internal resistance, and capacity state of health is proposed in this work. The algorithm is trained with synthetic data generated by an electric vehicle simulation platform running seven different standard driving cycles at various settings. The algorithm is then validated using an additional standard driving cycle, achieving, for state of charge, internal resistance, and capacity state of health, a root mean square error lower than 2%, 80 μΩ, and 2.9%, and a mean absolute percentage error lower than 3.4%, 4.4%, and 3.3%, respectively. The results obtained and the comparison with literature works indicate that the co-estimation algorithm proposed is able to estimate the considered quantities with very good accuracy.

1. Introduction

Electric vehicles (EVs) are a game-changer in the transition to sustainable transportation [1,2]. However, batteries of large capacity that require long charging times are needed to achieve adequate driving ranges and reduce drivers’ range anxiety. Lithium-titanate-oxide (LTO) batteries have emerged as a promising alternative to other lithium graphite-based technologies, offering high stability and fast charging capabilities [3]. For example, replacing the graphite of nickel-manganese-cobalt (NMC) cells with lithium-titanate nanocrystals as anode material allows LTO cells to be charged with a higher current than NMC cells without speeding up the aging processes. Therefore, the use of LTO technology may lead to EVs equipped with batteries of smaller capacity that can be charged more frequently at very high current rates, a step that reduces the gap between electric and internal combustion engine vehicles when the energy tank refill is considered [4,5].

The optimal use of high charging current rates requires in-depth knowledge of the internal states of battery cells, i.e, state of charge (SOC) and sstate of health (SOH), to avoid accelerated battery aging and potentially unsafe conditions [6,7]. For example, Epding et al. in ref. [8] and Kalk et al. in ref. [9] demonstrated that a SOC-based multi-stage constant current strategy can significantly reduce the aging of LTO and NMC cells during fast charge, when compared to the classic constant current charging profile that only considers the terminal voltages of the battery cells. Unfortunately, state estimation algorithms for LTO batteries have not yet been widely studied in the literature. However, data-driven algorithms are one of the most promising approaches to address this identification task. In fact, neural network and machine learning approaches enable the development of estimation algorithms that continuously improve accuracy over the operational life of the battery as more data become available [10,11,12,13,14,15]. Unfortunately, data-driven algorithms require a considerable volume of data for training, which is not yet available for LTO batteries. However, many works in the literature demonstrate that synthetic datasets can be an excellent alternative to measurement-based ones for training state estimation algorithms [16,17,18,19]. In fact, the use of simulation platforms to create synthetic datasets allows designers to explore a wide range of operating conditions and easily correlate measurable quantities of the cells with their internal states.

Numerous data-driven algorithms have been introduced in recent years to estimate internal state variables of NMC batteries. For instance, Tao et al. [20] present and compare SOC estimation algorithms based on the gated recurrent unit (GRU) [21], long short-term memory (LSTM) [22], bidirectional GRU (BGRU), and bidirectional LSTM (BLSTM). Moreover, fully connected neural network (FNCC), LSTM, GRU, and temporal convolutional network (TCN) approaches are used to implement algorithms for the internal state estimation of NMC batteries and are compared in ref. [23]. Identifying the best neural network architecture to estimate the battery SOC is very difficult because each architecture optimizes different metrics [20]. In addition, the accuracy of the architecture is directly related to the number of layers and nodes that make up the network. For example, the nodes of a feedforward/fully connected neural network (FNN/FCNN) are the least complex, but the network requires a larger number of nodes to obtain good accuracy in estimating SOC [20]. The temporal convolutional network (TCN) is developed from the connected neural network and generally achieves higher accuracy when processing time-series data [23]. Conversely, the LSTM and GRU networks process data step by step. They keep the state of the system in internal node variables, making them well-suited for streaming or real-time sequences. In fact, LSTM generally achieves good accuracy, and its trade-off between accuracy and complexity is generally better than that of other architectures.

To the best of our knowledge, the literature about the internal state estimation of LTO cells is rather poor. For this reason, we developed and presented in this paper a novel co-estimation algorithm based on neural network and machine learning approaches that can estimate the SOC, internal cell resistance, and SOH of LTO batteries for EV applications. As far as SOH is concerned, we will refer from now on to the capacity state of health (SOHc) that considers aging as a loss of battery capacity. The algorithm is trained and validated using a synthetic dataset generated with a simulation platform that includes a vehicle dynamic model, a battery model, and a test control block. The detailed description of the developed simulation platform is provided in the next section, which also details the synthetic dataset. Section 3 explains the proposed co-estimation algorithm and the training procedure, while Section 4 shows the test phase of the proposed co-estimation algorithm and its accuracy. Finally, the conclusions are drawn in the last section.

2. Simulation Platform and Synthetic Dataset Creation

The block diagram of the proposed simulation platform used to create the synthetic dataset is shown in Figure 1. In particular, the platform consists of three main blocks: the dynamic vehicle model, the battery model, and the test control block. The test control block manages the simulation platform to emulate battery usage in a real electric vehicle application. It alternates charge and discharge phases consisting of a standard charging profile and the electrical power profile calculated by the dynamic vehicle model, respectively. Specifically, the dynamic vehicle model evaluates the electrical power drawn by the vehicle from the battery during travel, based on the vehicle speed profile. Finally, the battery model emulates the electrical and aging behaviors of the battery. The following subsection provides a detailed description of these main blocks.

2.1. Dynamic Vehicle Model

The dynamic vehicle block is based on the model presented in ref. [24] which allows us to evaluate the electric power required from the battery starting from the speed profile and the mechanical properties of the vehicle. In particular, the model calculates the total mechanical force of the vehicle $F_{\mathrm{total}}$ as the sum of the accelerating force $F_{\mathrm{accelerating}}$, the aerodynamic drag force $F_{\mathrm{air}}$, the gravitational force due to the slope of the road $F_{\mathrm{slope}}$, and the rolling force $F_{\mathrm{rolling}}$. These contributions are evaluated as:

$$\left\{\begin{array}{c}{F}_{\mathrm{accelerating}}=M{a}_{\mathrm{veh}}\hfill \\ \\ F_{\mathrm{air}}=0.5\rho S{C}_{\mathrm{x}}{v}_{\mathrm{veh}}^2\hfill \\ \\ F_{\mathrm{slope}}=gMsin\left(\alpha \right)\hfill \\ \\ F_{\mathrm{rolling}}=gM{R}_{\mathrm{rolling}}\hfill \\ \\ F_{\mathrm{total}}=F_{\mathrm{accelerating}}+F_{\mathrm{air}}+F_{\mathrm{slope}}+F_{\mathrm{rolling}}\hfill \end{array}\right.$$

(1)

where M is the kerb weight of the vehicle, $a_{\mathrm{veh}}$ is the acceleration, S is the surface area, $C_{\mathrm{x}}$ is the drag coefficient, $v_{\mathrm{veh}}$ is the speed, and $R_{\mathrm{rolling}}$ is the rolling resistance of the vehicle. Moreover, $\rho$ is the air density, $\alpha$ is the slope of the road, and g is the gravitational acceleration. Finally, the electric power required from the battery $P_{\mathrm{bat}}$ is obtained by multiplying $F_{\mathrm{total}}$ by the efficiency of the system $\eta$. The parameters and mechanical properties of the vehicle are obtained from ref. [25] and are summarized in

Table 1. Dynamic vehicle parameters.

Parameter	Value
Kerb weight M	1525 kg
Surface area S	2.27 m2
Drag coefficient $C_{\mathrm{x}}$	0.29
Rolling resistance $R_{\mathrm{rolling}}$	0.01
Air density $\rho$	1.2 kg/m3
Road slope $\alpha$	0
Gravitational acceleration g	9.82 m/s2
System efficiency $\eta$	0.7

.

2.2. Battery Model

The battery model block takes into account the battery behavior. For the sake of simplicity, all battery cells and their internal states are considered identical. Therefore, the battery model block evaluates the output values of a single cell and applies the results to the other cells. In particular, the 2 RC-group equivalent circuit model (ECM) shown in Figure 2 is used to evaluate the evolution of cell voltage and SOC during battery operation [26,27].

The 2 RC-group ECM is very popular in the literature and is composed of two parts. The left-hand side consists of a dependent current generator controlled by the cell current $I_{\mathrm{cell}}$ and a capacitor, whose value is equal to the nominal cell capacity $C_{\mathrm{cell}}$ (expressed in Coulombs) divided by 1 $\mathrm{V}$. This part of the circuit models the charge stored in the cell, and the voltage value of the capacitor corresponds to the SOC multiplied by 1 $\mathrm{V}$. The right-hand side of the circuit reproduces the cell terminal voltage $V_{\mathrm{cell}}$. It is the sum of the open circuit voltage $V_{\mathrm{oc}}$, the voltage drop on the internal Ohmic resistance $R_0$, and a term that accounts for the relaxation phenomena that occur inside the cell. This dynamic term consists of the two voltages $V_1$ and $V_2$ that represent the effects of double layer capacitance and diffusion, respectively. These effects are modeled by the two $R_1{C}_1$ and $R_2{C}_2$ groups, characterized by very different time constants [28]. The cell time-domain state space is described by the following equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}SOC}{\mathrm{d}t}}=-{\displaystyle \frac{I_{\mathrm{cell}}}{C_{\mathrm{cell}}}}\hfill \\ \\ {\displaystyle \frac{\mathrm{d}{V}_1}{\mathrm{d}t}}=-{\displaystyle \frac{V_1}{R_1{C}_1}}+{\displaystyle \frac{I_{\mathrm{cell}}}{C_1}}\hfill \\ \\ {\displaystyle \frac{\mathrm{d}{V}_2}{\mathrm{d}t}}=-{\displaystyle \frac{V_2}{R_2{C}_2}}+{\displaystyle \frac{I_{\mathrm{cell}}}{C_2}}\hfill \\ \\ V_{\mathrm{cell}}=V_{\mathrm{OC}}-R_0{I}_{\mathrm{cell}}-V_1-V_2\hfill \end{array}\right.$$

(2)

The values of $R_0$, $R_1$, $C_1$, $R_2$, and $C_2$ depend on temperature, SOC, and the sign of the current (charge/discharge). The dependencies are not linear, and the model takes this into account with a multidimensional look-up table whose inputs are the cell temperature, the SOC, and the sign of the current, and whose outputs are the $R_0$, $R_1$, $C_1$, $R_2$, and $C_2$ values.

The battery block is also equipped with a cell degradation model that takes into account both calendar and cycling aging phenomena. Calendar aging occurs even when the battery is not in use and is due to unwanted side reactions [29]. The aging speed depends on storage conditions, such as the temperature and SOC of the cell, and thus the voltage. Conversely, cycling aging is caused by the current that flows in the cell. It produces many physical and chemical mechanisms that age the cell, such as local increase in temperature or electrode potential leading to unwanted side reactions, and mechanical stress occurring during the intercalation and de-intercalation of lithium ions in the electrodes. As a result, cycling aging depends on the operating condition of the cell in terms of current, temperature, SOC, and thus voltage values. As a general rule, cells age faster at higher voltages because of the electrolyte decomposition. Higher currents cause increased stress in the materials during the charging and discharging phases. A higher temperature increases the speed of unwanted side reactions. In this work, the effects of aging have been considered on both internal resistance $R_0$ and cell capacity $C_{\mathrm{cell}}$, according to the models described in refs. [30,31]. These models evaluate the cell degradation as the sum of the three contributions due to cell voltage, current, and temperature values according to the following system of equations:

$$\left\{\begin{array}{c}\\ {\alpha }_{\mathrm{r}}\left(t\right)=\left(b_{\mathrm{r}}{V}_{\mathrm{cell}}\left(t\right)+c_{\mathrm{r}}\right)\left(1+e_{\mathrm{r}}|I_{\mathrm{cell}}\left(t\right)|\right)e^{-{\displaystyle \frac{q{d}_{\mathrm{r}}}{K_{\mathrm{b}}T\left(t\right)}}}\hfill \\ \\ {\alpha }_{\mathrm{c}}\left(t\right)=\left(b_{\mathrm{c}}{V}_{\mathrm{cell}}\left(t\right)+c_{\mathrm{c}}\right)\left(1+e_{\mathrm{c}}\left|I_{\mathrm{cell}}\left(t\right)\right|\right)e^{-{\displaystyle \frac{q{d}_{\mathrm{c}}}{K_{\mathrm{b}}T\left(t\right)}}}\hfill \\ \\ {\displaystyle \frac{R_0}{R_{0_{\mathrm{fresh}}}}}=1+{\int }_0^t{\alpha }_{\mathrm{r}}d\tau \hfill \\ \\ \mathrm{SOHc}={\displaystyle \frac{C_{\mathrm{cell}}}{C_{{\mathrm{cell}}_{\mathrm{fresh}}}}}=1-{\int }_0^t{\alpha }_{\mathrm{c}}d\tau \hfill \\ \end{array}\right.$$

(3)

where ${\alpha }_{{\mathrm{r}}_i}$ and ${\alpha }_{{\mathrm{c}}_i}$ represent the increase in resistance and the reduction in capacity of the cell, respectively, $R_{0_{\mathrm{fresh}}}$ and $C_{{\mathrm{cell}}_{\mathrm{fresh}}}$ are the $R_0$ and $C_{\mathrm{cell}}$ values of the fresh cell, and $V_{\mathrm{cell}}$, $I_{\mathrm{cell}}$, and T are the cell voltage, current, and temperature. A brief description of the other parameters of the aging model and their values is reported in

Table 2. Parameters of the aging model of the cell.

Symbol	Description	Value
q	Electron elementary charge	$1.60217663\times {10}^{-19}$ C
K	Boltzmann constant	$1.380649\times {10}^{-23}$${\mathrm{m}}^2\mathrm{k}\mathrm{g}{\mathrm{s}}^{-2}{\mathrm{K}}^{-1}$
$b_r$	Terminal resistance constant gain for voltage	0.0594 ${\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$
$c_r$	Terminal resistance constant offset for voltage	−0.0713 ${\mathrm{s}}^{-1}$
$d_r$	Terminal resistance temperature-dependent exponential increase	0.4985 $\mathrm{V}$
$e_r$	Terminal resistance constant gain for current	2.5867 ${\mathrm{A}}^{-1}$
$b_c$	Terminal capacity constant gain for voltage	0.0594 ${\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$
$c_c$	Terminal capacity constant offset for voltage	−0.0713 ${\mathrm{s}}^{-1}$
$d_c$	Terminal capacity temperature-dependent exponential increase	0.4985 $\mathrm{V}$
$e_c$	Terminal capacity constant gain for current	0.4533 ${\mathrm{A}}^{-1}$

.

It is important to note that the values used derive from an experimental characterization test campaign and reasonable assumptions. In particular, $d_{\mathrm{c}}$ is set to have a capacity reduction of 2% per year at the constant temperature of 45 ^∘C and 0.1% per year at 0 ^∘C. Meanwhile, $b_{\mathrm{c}}$ and $c_{\mathrm{c}}$ are chosen to obtain a reduction of the cell capacity of 2% with a fully charged cell and of 1% with a fully discharged cell. For simplicity, $b_{\mathrm{r}}$, $c_{\mathrm{r}}$, and $d_{\mathrm{r}}$ are set equal to $b_{\mathrm{c}}$, $c_{\mathrm{c}}$, and $d_{\mathrm{c}}$, respectively. Finally, $e_{\mathrm{r}}$ and $e_{\mathrm{c}}$ are evaluated to obtain a resistance increase of 50% per year and a reduction of the cell capacity of 10% per year, respectively, considering a cycling current of 1 C-rate and an average cell voltage of $2.4$ $\mathrm{V}$, i.e., the voltage of an LTO cell with 50% SOC.

Thus, the values of the aging model parameters obtained are able to model a realistic aging rate of the electric vehicle battery. However, this rate is very low and requires the simulation of hundreds of driving cycles to obtain significant battery degradation. An acceleration factor of 250 is applied to ${\alpha }_{\mathrm{r}}$ and ${\alpha }_{\mathrm{c}}$ to speed up aging processes, obtaining a reasonable trade-off between model accuracy and simulation time to generate the synthetic dataset, and dataset size.

2.3. Test Control Block

The dynamic vehicle block and the battery model block are controlled by the test control block. In particular, the test control block manages the battery block to reproduce vehicle usage, consisting of the repetition of a discharge phase followed by a charging phase. The vehicle usage profile starts with a fully charged fresh battery. The battery is then discharged with the current profile obtained by the ratio between $P_{\mathrm{bat}}$ and the battery voltage generated by the battery model block. The $P_{\mathrm{bat}}$ profile is obtained from the dynamic vehicle model using a standard driving cycle. Standard driving cycles are predefined speed profiles that simulate real-world driving conditions to assess fuel consumption, emissions, and energy efficiency. The driving cycles considered in this work include La92short, J1015, HWFET, ArtMw150, ArtUrban, FTP, ArtRoad, and WLTP. Their characteristics are summarized in

Table 3. Standard driving cycles used to generate the synthetic dataset.

Drive Cycle	Scenario	Distance	Duration	Average Speed	Use
		(km)	(min)	(km/h)
La92short	Urban	11.1	16	41.8	Training
J1015	Urban	6.4	15	25.6	Training
HWFET	Highway	16.8	13	77.5	Training
ArtMw150	Motorway	29.8	18	99.5	Training
ArtUrban	Urban	5.0	17	17.6	Training
FTP	Urban	17.6	31	34.1	Validation
ArtRoad	Rural road	17.2	18	57.4	Validation
WLTP	Mixed	23.3	30	46.5	Test

. It can be noted that each driving cycle covers a very short distance. Therefore, it is repeated until the battery SOC reaches 20%, the point at which the charging phase starts.

The charging phase is based on a constant current–constant voltage (CC-CV) profile, where the constant current is equal to 1 C-rate of the nominal cell capacity $C_{{\mathrm{cell}}_{\mathrm{fresh}}}$ and the constant voltage is equal to the $V_{\mathrm{OC}}$ value at 80 % SOC. The discharge and charge phases are continuously repeated with the same current and voltage values until the battery SOHc, i.e., the ratio between $C_{\mathrm{cell}}$ and $C_{{\mathrm{cell}}_{\mathrm{fresh}}}$, reaches the 60% threshold. The procedure described above was repeated for all the 8 different standard driving cycles at 25 ^∘C and 35 ^∘C to generate the synthetic dataset. Therefore, the dataset contains 16 time series that describe the life of 16 identical batteries used in different conditions.

The 12 time series obtained from the La92short, J1015, HWFET, ArtMw150, and ArtUrban driving cycles were used to train the co-estimation algorithm. The validation phase was carried out using the 4 time series obtained from the FTP and ArtRoad driving cycles. Finally, the 2 time series obtained from the WLTP driving cycle were used for the final test.

3. Co-Estimation Algorithm

The developed co-estimation algorithm aims to identify SOC, $R_0$, and SOHc of an LTO cell used in electric vehicles. The algorithm is divided into three cascaded blocks that estimate SOC, $R_0$, and SOHc, respectively. The block diagram of the algorithm is shown in Figure 3.

The partitioning of the co-estimation algorithm allows us to keep the system complexity low and to execute the estimation functions at different frequencies. For example, the $R_0$ and SOHc variables can be estimated at much lower frequencies than the SOC variable.

It is important to note that network inputs are normalized in the range from 0 to 1 using the following equation:

$$inpu{t}_{\mathrm{N}}={\displaystyle \frac{input-inpu{t}_{\min }}{inpu{t}_{\max }-inpu{t}_{\min }}}$$

(4)

where $inpu{t}_{\mathrm{N}}$ is the normalized input, $input$ is the input value, and $inpu{t}_{\min }$ and $inpu{t}_{\max }$ are the maximum and minimum possible values of the relative input. The values of $inpu{t}_{\min }$ and $inpu{t}_{\max }$ are equal to $1.5$ $\mathrm{V}$ and $2.7$ $\mathrm{V}$, $-250$ $\mathrm{A}$ and 250 $\mathrm{A}$, and −20 ^∘C and 70 ^∘C for cell voltage, current, and temperature, respectively. These values are the most common safety thresholds for the LTO cell safe operating area (SOA) [32].

3.1. SOC Estimation Block

The SOC estimation block is based on an LSTM layer followed by a dropout layer and a fully connected layer. The training of this network was carried out employing the Adam optimization algorithm, which offers good efficiency and adaptability in terms of learning rate. Finally, network outputs are post-processed with a rate limiter to address the intrinsic high-frequency noise generated by the network, such as transient spikes and fluctuations.

The training and validation processes of the proposed SOC estimation network were carried out using the first discharge and charge cycle of the generated synthetic dataset, i.e., the cycle with the fresh battery. In particular, the first discharge and charge cycle of the La92short, J1015, HWFET, ArtMw150, and ArtUrban driving cycles were used in the training, while those of the FTP and ArtRoad cycles were used to validate the network.

Different combinations of the number of internal neurons, learning rate, and window size were explored to identify the best hyperparameters. The results of this analysis are reported in

Table 4. Grid search for SOC estimation network with the validation dataset. The best-performing results are highlighted in bold.

Chunk Size	Learning Rate	Hidden Units	MAE (%)	MAPE (%)	RMSE (%)
100	0.01	20	3.0553	7.7688	4.2216
100	0.01	32	27.4707	80.6835	32.9009
100	0.01	50	6.3870	25.1024	10.0717
100	0.02	20	17.2782	31.8322	19.9990
100	0.02	32	20.4943	33.5117	24.5322
1000	0.01	50	1.5717	4.8987	1.8302
1000	0.02	20	1.6536	5.1177	2.0221
1000	0.02	32	3.9563	7.8092	5.2773
1000	0.02	50	24.5365	44.8146	28.2875

, which shows the root mean square error (RMSE), the mean absolute error (MAE), and the mean absolute percentage error (MAPE) obtained with the hyperparameters explored during the training phase carried out over 20 epochs. RMSE, MAE, and MAPE are evaluated with the following equations

$$\left\{\begin{array}{c}\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{k=1}^N{\left(Y_{\mathrm{predicted}}-Y_{\mathrm{true}}\right)}^2}\hfill \\ \mathrm{MAE}={\displaystyle \frac{1}{N}}\sum _{k=1}^N\left|Y_{\mathrm{predicted}}-Y_{\mathrm{true}}\right|\hfill \\ \mathrm{MAPE}=100·{\displaystyle \frac{1}{N}}\sum _{k=1}^N\left|{\displaystyle \frac{Y_{\mathrm{predicted}}-Y_{\mathrm{true}}}{Y_{\mathrm{predicted}}}}\right|\hfill \end{array}\right.$$

(5)

where N is the number of predicted points, $Y_{\mathrm{predicted}}$ is the value predicted by the neural network, and $Y_{\mathrm{true}}$ is the true value.

The best results were obtained with a chunk size of 1000, a learning rate equal to 0.01, and 50 hidden units. Therefore, the results presented in the next section are evaluated using the network obtained with these hyperparameters and trained with 100 epochs.

3.2. Internal Resistance and SOHc Estimation Blocks

The SOHc and $R_0$ estimation blocks are based on the decision tree (DT) algorithm. The proposed DT models were configured so that each node must have at least 10 observations to be considered for splitting (i.e., minParent parameter set to 10). Additionally, each leaf node must contain at least 4 observations (i.e., minLeaf parameter set to 4). These values provide an effective trade-off between complexity and accuracy, helping to prevent overfitting by ensuring that nodes are split only when there are sufficient data to support them. Both networks were trained with the same dataset composed of the time series obtained from the FPT and ArtRoad drive cycles. It is important to note that all inputs to the networks were generated using the simulation platform, ensuring that the training phase of each network remained independent. The cross-validation approach was used during the network training phase with the training dataset divided into 5 folds. The RMSE and MAE values of the $R_0$ estimation network are equal to 16 μΩ, and 10 μΩ, respectively. Instead, SOHc network training achieved an RMSE of 0.82% and an MAE of 0.57%.

4. Test of the Developed Co-Estimation Algorithm

The developed co-estimation algorithm shown in Figure 3 has been verified using the WLTP drive cycle time series from the database. This part of the database was not used during network training to evaluate the ability of the algorithm developed to correctly estimate battery SOC, $R_0$, and SOHc under new and unforeseen conditions. Moreover, the proposed co-estimation algorithm was also tested using voltage and current inputs that were made as realistic as possible by introducing a measurement error consisting of white noise with normal distribution. The mean values of both distributions were set to 0, while the values of $\sigma$ were set to $2.25$ $\mathrm{m}$$\mathrm{V}$ and $0.32$ $\mathrm{m}$$\mathrm{A}$ for the voltage and current measurement system error distributions, respectively.

Figure 4 compares the estimated SOC values with those coming from the model that are used as reference. The figure shows the estimated SOC value of the first and last discharge/charge cycles with SOHc equal to 100% and 60%, respectively, considering the noise on the current and voltage measurement. As we can observe, the SOC network is capable of accurately estimating the SOC values in both cycles, even though it was trained using the first cycle of the dataset time series with a cell SOHc equal to 100%. It should also be noted that the duration of the discharge/charge cycle is reduced to less than one and a half hours when the cell is aged. This is because the levels of charge and discharge currents always refer to the capacity of the fresh cell. The available charge in aged cells is reduced and, therefore, the investigated SOC interval is covered in less time than in fresh cells. The result highlights the robustness of the developed network, which is capable of working with very different cell conditions compared to those seen during the training phase. In fact, RMSE, MAE, and MAPE are very similar for the first and last cycles going from 1.0%, 0.8%, and 1.7% to 2.9%, 2.1%, and 4.2%, respectively. It is important to note that the accuracy in SOC estimation is significantly improved when measurement noise is not introduced. Under these conditions, the mean RMSE, MAE, and MAPE are 0.29%, 0.22%, and 0.44%, respectively.

The estimated $R_0$ values in the first and last two discharge/charge cycles are compared with the model values in Figure 5. The estimated values follow the trend of the $R_0$ model, and the RMSE is only 7.6 μΩ in the first cycle and 24.4 μΩ in the last.

Finally, Figure 6 compares the estimated values of SOC, $R_0$, and SOHc with the model values during all the simulated cycles. The comparison highlights the capability of the developed co-estimation algorithm to accurately estimate every quantity during the aging of the cell. The RMSE, MAE, and MAPE obtained are shown in

Table 5. RMSE, MAE, and MAPE obtained by the co-estimation algorithm in the WLTP drive cycle at 25 ^∘C and 35 ^∘C.

	Temperature	RMSE	MAE	MAPE
SOC	25 ∘C	2.0%	1.5%	3.4%
	35 ∘C	1.8%	1.3%	3.0%
$R_0$	25 ∘C	80 μΩ	58 μΩ	3.8%
	35 ∘C	71 μΩ	55 μΩ	4.2%
SOHc	25 ∘C	2.9%	2.2%	3.22%
	35 ∘C	2.9%	2.4%	3.3%

. These results are very similar to the RMSE, MAE, and MAPE obtained in the test carried out at a temperature of 35% and reported in the same table.

5. Result Discussion and Comparison

The SOC estimation accuracies obtained by the co-estimation algorithm under different test conditions are compared in

Table 6. Comparison of obtained results with those in the literature.

Ref.	Algorithm	Condition	RMSE (%)	MAE (%)	MAPE (%)
our	LSTM	Fresh cell, 25 ∘C	0.3	0.2	0.4
		Fresh cell, noise, 25 ∘C	1.0	0.8	1.7
		Fresh cell, noise, 35 ∘C	0.9	0.7	1.5
		SOH 60%, noise, 25 ∘C	2.9	2.2	4.2
		SOH 60%, noise, 35 ∘C	2.4	1.9	3.7
[20]	GRU	Fresh cell, 25 ∘C	6.3	4.9	15.8
		SOH 98%, 25 ∘C	6.4	5.0	16.1
	LSTM	Fresh cell, 25 ∘C	5.7	4.4	15.9
		SOH 98%, 25 ∘C	5.7	4.5	16.2
	BGRU	Fresh cell, 25 ∘C	5.4	4.5	14.7
		SOH 98%, 25 ∘C	5.5	4.6	15.0
	BLSTM	Fresh cell, 25 ∘C	4.5	3.6	14.1
		SOH 98%, 25 ∘C	4.5	3.6	14.3
[23]	FCNN	Fresh cell, 25 ∘C	3.1	2.34
		Fresh cell, 40 ∘C	3.0	2.3
	LSTM	Fresh cell, 25 ∘C	0.8	0.7
		Fresh cell, 40 ∘C	0.6	0.5
	GRU	Fresh cell, 25 ∘C	0.6	0.4
		Fresh cell, 40 ∘C	0.5	0.3
	TCN	Fresh cell, 25 ∘C	0.85	0.7
		Fresh cell, 40 ∘C	0.6	0.4
[33]	BERT	Fresh cell	0.81		2.5
[34]	FNN	Fresh cell	2.24
	GRU	Fresh cell	1.13
	LSTM	Fresh cell	1.5
[18]	LSTM	Fresh cell			1.64

to those reported in previous works in the literature.

Four SOC estimation neural networks, based on the GRU, LSTM, BGRU, and BLSTM architectures, were compared in ref. [20]. The authors trained and tested these networks using an experimental dataset obtained from an NMC cell discharged using the NEDC, WLTP, and UDDS driving cycles. The accuracies achieved by the networks are reported for fresh and aged cells (SOH of about 98%). A very similar study was presented in ref. [23], where the performances of FCNN-, LSTM-, GRU-, and TCN-based SOC estimation neural networks were compared. The investigation was carried out using a discharging test profile composed of a random mix of US06, HWFET, and UDD driving cycles, performed on fresh cells at various temperatures. Conversely, a synthetic dataset was used to train and test an LSTM-based SOC estimation neural network in ref. [18]. This study only considered fresh batteries and neglected the effects of voltage and current measurement noise. As observed, the network proposed in this work achieved the highest accuracy in estimating SOC for fresh cells and noiseless measurement conditions. The estimation accuracies slightly decrease when voltage and current measurement noise is introduced and highly aged cells are considered. However, the errors achieved remain comparable to or lower than those found in most of the SOC estimation algorithms reported as references.

6. Conclusions

A co-estimation algorithm based on neural networks and machine learning approaches is presented in this work to accurately estimate the state of charge, internal resistance, and capacity state of health of lithium-titanate-oxide batteries for electric vehicles. The algorithm was trained and tested using a synthetic dataset generated from a simulation platform, including a dynamic vehicle model, a battery model, and a test control block. The results demonstrate that the proposed algorithm achieves high accuracy of estimation under various driving conditions and temperatures. The robustness of the algorithm was validated using the WLTP driving cycle, showing its ability to perform well with conditions unexplored during training and also in the presence of measurement noise. The results highlight the potential of the proposed co-estimation algorithm to enhance battery management systems of lithium-titanate-oxide batteries by providing reliable and accurate state estimations. The research is now directed to the investigation of the following steps:

• Developing a reinforcement learning algorithm to improve the algorithm performance.
• Applying model compression and quantization techniques, which are aimed at reducing the model size and enhancing inference speed.
• Deployment on a microcontroller for practical application to a battery management system.