A Multi-Timescale Method for State of Charge Estimation for Lithium-Ion Batteries in Electric UAVs Based on Battery Model and Data-Driven Fusion

Abstract

This study focuses on the critical problem of precise state of charge (SOC) estimation for electric unmanned aerial vehicle (UAV) battery systems, addressing a fundamental challenge in enhancing energy management reliability and flight safety. The current data-driven methods require big data and high computational complexity, and model-based methods need high-quality model parameters. To address these challenges, a multi-timescale fusion method that integrates battery model and data-driven technologies for SOC estimation in lithium-ion batteries has been developed. Firstly, under the condition of no data or insufficient data, an adaptive extended Kalman filtering with multi-innovation algorithm (MI-AEKF) is introduced to estimate SOC based on the Thévenin model in a fast timescale. Then, a hybrid bidirectional time convolutional network (BiTCN), bidirectional gated recurrent unit (BiGRU), and attention mechanism (BiTCN-BiGRU-Attention) deep learning model using battery model parameters is used to correct SOC error in a relatively slow timescale. The performance of the proposed model is validated under various dynamic profiles of battery. The results show that the the maximum error (ME), mean absolute error (MAE) and the root mean square error (RMSE) for zero data-driving, insufficient data-driving, and sufficient data-driving under various dynamic conditions are below 2.3%, 1.3% and 1.5%, 0.9%, 0.4% and 0.4%, and 0.6%, 0.3% and 0.3%, respectively, which showcases the robustness and remarkable generalization performance of the proposed method. These findings significantly advance energy management strategies for Li-ion battery systems in UAVs, thereby improving operational efficiency and extending flight endurance.

1. Introduction

Electric unmanned aerial vehicles (UAVs), as an efficient and convenient flight platform, have been widely utilized in military reconnaissance, surveillance, and patrol missions, as well as in civil activities such as aerial photography, mapping, environmental monitoring, and power line inspections [1,2]. Due to their environmental friendliness, high efficiency, ease of operation, and high safety, electric UAVs can reduce the risks associated with manual operations and enhance the performance of mission execution. The widespread adoption of lithium-ion batteries in electric UAV systems is primarily attributed to their exceptional power density properties [3]. For all-electric UAV platforms, which rely exclusively on lithium-ion battery systems, the SOC directly determines operational safety and mission success.

Aimed at planning different missions and optimizing flight performance in electric UAVs, an efficient battery management system (BMS) can not only accurately evaluate the UAV’s ability to complete specific tasks and enhance flight safety but also increase flight duration by optimizing energy management strategies and extending the lifespan of lithium-ion batteries [4]. As one of the main parameters of BMS, SOC reflects the remaining capacity within the battery [5]. In fact, achieving accurate SOC estimation remains challenging due to the inherent dynamic and heterogeneous nature of battery systems, as well as the influence of numerous external environmental and operational factors [6]. Therefore, attaining precision and robustness in the estimation of SOC is crucial for the BMS of electric UAVs.

Currently, there exist two widely utilized conventional methods for SOC estimation in electric UAVs. These include the ampere-hour method and the open-circuit voltage (OCV) method [7,8]. However, obtaining accurate SOC using the ampere-hour method is challenging owing to the inaccuracy of the measuring sensors and the deficiency in mechanisms for feedback correction [9,10]. On the other hand, the OCV method faces significant challenges, including the need for a substantial amount of time to achieve equilibrium conditions for accurate OCV measurements when establishing the SOC–OCV correlation [11,12]. As a result, researchers have developed alternative approaches, such as model-based and data-driven methods, to estimate SOC more effectively.

The model-based approach estimates the SOC by constructing a mathematical relationship between measurable parameters and internal battery states, enabling real-time monitoring and self-correcting capabilities through continuous state estimation. In recent years, the widely used battery models include the electrochemical model and the equivalent circuit model (ECM). The electrochemical model offers a more precise and explicit representation of the intricate chemical reaction process occurring within the battery. However, it has difficulty in estimating SOC online due to its complex mathematical model form and high computational burden [13,14]. The ECM abstracts the intricate electrochemical processes within a battery by representing the equilibrium potential through OCV and modeling polarization phenomena using resistor–capacitor networks [15,16]. The simplification makes it easy for online SOC estimation. Numerous filtering techniques and state observers have been extensively utilized for the estimation of SOC, including the extended Kalman filter (EKF) [17,18,19], unscented Kalman filter (UKF) [20,21], the H ∞ filters [22,23], the proportional-integral observers [24,25], and the particle filter observer (PF) [26,27]. However, the precision of the battery model and the performance of the filter or observer strongly affect the SOC estimation. Furthermore, as the battery ages and its parameters change, the accuracy of SOC estimation continues to decline.

In recent years, data-driven approaches for SOC estimation have garnered considerable attention. In contrast with the model-based methods, it does not need to analyze the internal electrochemical mechanisms of the battery, instead relying solely on the relationship between external input signals and corresponding output responses to estimate SOC through data-driven mapping. Numerous researchers have tried to use data-driven methods to estimate SOC and have obtained remarkable results. Early research investigated the application of multi-layer perceptron (MLP) networks for SOC estimation, but these approaches demonstrated limitations in effectively capturing temporal dependencies within continuous time-series data [28]. To solve this problem, some machine learning methods have been introduced, such as support vector machines (SVMs) [29], long short-term memory (LSTM) [30,31], gated recurrent units (GRUs) [32,33], and convolution neural networks (CNNs) [33], as they can deal with sequential data and capture mapping features. Although the aforementioned methods have demonstrated promising performance in the domain of SOC estimation, the feature extraction capabilities of single machine learning (ML) models remain constrained, limiting their ability to generalize across diverse operating conditions encountered in SOC estimation tasks. To enhance the accuracy and robustness of SOC estimation, researchers propose a hybrid ML model. Bian et al. [34] introduced a novel network architecture combining multi-channel CNN with bidirectional LSTM for SOC estimation. It used multiple CNNs to extract locally characteristics from different channels and then input these features into the bidirectional LSTM layer to capture intercorrelated features and thus sequentially estimate the SOC. Hu et al. [35] developed a TCN-LSTM architecture that demonstrates superior estimation capabilities compared to standalone TCN or LSTM models. Sherkatghanad et al. [36] proposed a CNN-Bi-LSTM framework which demonstrated robust performance in achieving accurate SOC estimations under varying temperature conditions. Given that the training of data-driven model requires high-fidelity and large-sample sets, it is difficult to obtain ideal training set data in noisy real-world applications.

The aforementioned studies for estimating SOC based on the data-driven method have a poor interpretability due to the fact that their decision-making process is a black box. Therefore, some scholars have explored the fusion of data-driven and model-based methods to estimate the SOC. Chen et al. [37] and Tang et al. [38] utilized data-driven and fusion-based methods as ECMs to predict the battery’s terminal voltage, which is then integrated as the observation equation within the EKF framework for SOC estimation. However, these methods concentrate on the development of battery model construction and filter optimization, ignoring the error compensation brought by the models. From another perspective, Wang et al. [39] used a back-propagation neural network (BPNN) to compensate for the model’s estimation error of the dual EKF method, further improving the SOC estimation precision. However, the parameter identification for DEKF requires high computational complexity and precision. Hou et al. [40] employed the model’s error from the EKF as the input signals, and the SOC was estimated by the eXtreme gradient boosting model (XGBoost), which is used to compensate the estimation error of the model-based method. However, this method relied on data to correct the model due to the simplicity of the battery model. To reduce data dependence, Oh et al. [41] utilized an electrochemical–thermal model to estimate the SOC under insufficient data, and the model results were used for training neural network models to supplement SOC estimation errors. However, the above studies relied heavily on model data to correct the SOC estimation, and the ML method could not be explored in depth due to the need for the related features to compensate the estimation errors.

Due to the challenges faced by SOC estimation for lithium-ion batteries in electric UAVs in practical applications, such as changes in model parameters, fragmented data collection, and the impact of environmental factors on state estimation, a multi-timescale fusion method that integrates a battery model with time memory and a data-driven model trained on post-run data is proposed in this paper. The method, in different timescales, is based on the model-based method and supplemented by the data-driven model. Under the assumption that the test data are insufficient, or even zero, the model-based method is used to estimate the initial SOC with high accuracy in a fast timescale and has excellent interpretability. Then, a BiTCN-BIGRU-Attention model is adopted to correct the SOC estimation error in a relatively slow timescale. Accurate SOC estimation for lithium-ion batteries is achieved across diverse dynamic operating conditions. The main contributions of this work are summarized as follows:

(1) A multi-timescale fusion-based method is introduced to enhance the precision of SOC estimation for lithium-ion batteries in electric UAVs. The proposed method can quickly and accurately estimate the SOC when cycle data are insufficient and correct the SOC estimation error in a relatively slow timescale without adding additional data.

(2) The proposed framework is grounded in a model-based approach and supplemented by a data-driven model operating in different timescales. The initial SOC estimation is obtained by Thévenin ECM and the multi-innovation AEKF (MI-AEKF) algorithm in a fast timescale, and the SOC error correction is obtained by the BiTCN-BIGRU-Attention model in a slow timescale. Integrating both two parts improves the interpretability of the model and reduces the computational burden.

(3) By using multi-history information, the MI-AEKF algorithm can not only reduce error accumulation but also increase noise adaptability, enabling rapid and accurate estimation of the initial SOC in a fast timescale when cycle data are insufficient.

(4) By fusing BiTCN, BiGRU, and the attention mechanism model, the problems of insufficient long-term dependency learning ability in the single BiTCN model and the computational and memory burden of the single BiGRU model are solved, and its performance is optimized by the attention mechanism.

The remainder of this paper is presented as follows: Section 2 introduces the multi-timescale method framework, the Thévenin equivalent circuit model, the MI-AEKF method, and the BiTCN-BIGRU-Attention model. The battery datasets and estimation settings are described Section 3. In Section 4, the performance of the proposed method is thoroughly analyzed and discussed. Finally, the conclusions are presented in Section 5.

2. Multi-Timescale Method

2.1. Architecture of the Method

The architecture of the multi-timescale method is composed of two components—the fast-timescale initial SOC estimation and the slow=timescale SOC error compensation—as indicated in Figure 1. Firstly, under the condition where the battery data are insufficient or entirely absent, the model-based method is employed for high-frequency and accurate SOC estimation in electric UAVs. The voltage, current, and temperature data obtained from the electric UAV mission flight are entered into the fast-timescale battery model, and the parameters of the Thévenin equivalent circuit model are identified through the data. Next, the initial SOC is rapidly estimated through the MI-AEKF method, referred to as ${\overline{SOC}}_{(t,l)}$. Due to errors, such as in model parameter identification and model variation, the SOC estimation is inaccurate. To compensate for the estimation error, the parameter matrix $\stackrel{\rightharpoonup }{A}=\left[U_{(t)}{I}_{(t)}{T}_{(t)}{U}_{ocv}{U}_{(t)error}SO{C}_{(t)}-SO{C}_{(t-1)}{R}_0{R}_p{C}_p\right]$ derived from the battery model operating cycle under the mission profile is incorporated into the BiTCN-BIGRU-Attention model for training, enabling error correction through the robust non-linear processing capabilities inherent in the data-driven model. In order to reduce the data acquisition and storage of the operating cycle and minimize the training amount and difficulty of the data-driven model, a relatively slow timescale is adopted to compensate the SOC estimation and further improve the SOC estimation accuracy, named $SO{C}_{(t,L)error}$. Therefore, the corrected SOC estimation is expressed as follows:

$$SO{C}_{(t)}={\overline{SOC}}_{(t,l)}+SO{C}_{(t,L)error}$$

(1)

where $SO{C}_{(t)}$ is the SOC estimation of the proposed method, ${\overline{SOC}}_{(t,l)}$ denotes the initial estimation results obtained by the battery model, $SO{C}_{(t,L)error}$ is the corrected error value by the BiTCN-BIGRU-Attention model, and the subscript $l\in \left[1,\dots ,L\right]$ and L are the fast timescale and slow timescale, respectively. When the intermediate compensation value of the slow timescale is needed, it can be obtained by the interpolation function.

2.2. Battery Model

In order to accurately obtain lithium-ion battery information and realize battery state estimation, a Thévenin model is chosen in this study. The model is widely used due to its high accuracy in voltage dynamics, static characteristic, and low complexity. Its circuit structure is shown in Figure 2. Based on Kirchhoff’s circuit theory, the ECM for the lithium-ion battery is formulated using the state and observation equations as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{dSOC}{dt}}=-{\displaystyle \frac{\eta I_b}{Q_b}}\hfill \\ {\displaystyle \frac{d{U}_p}{dt}}=-{\displaystyle \frac{U_p}{R_p{C}_p}}+{\displaystyle \frac{I_b}{C_p}}\hfill \\ U_t=U_{ocv}(SOC)-U_p-R_0{I}_b\hfill \end{array}\right.$$

(2)

where $I_b$ is the working current for the battery, the discharge direction of which is positive; $\eta$ is the coulomb efficiency coefficient; $Q_b$ is the nominal capacity of the battery; $R_p$ and $C_p$ are the polarization resistance and polarization capacitance in the battery ECM, respectively; $U_p$ represents the terminal voltage across the battery polarization capacitor $C_p$ in the ECM; $R_0$ is the ohmic resistance; $U_t$ is the battery terminal voltage; and $U_{ocv}$ is the OCV, demonstrating a strictly monotonic non-linear dependence on the battery’s SOC.

Considering that the battery model parameters change slowly during operation, the state and measurement equations of the battery model are discretized as follows:

$$\left[\begin{array}{c}SO{C}_k\\ U_{p,k}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& e^{-T_s/R_p{C}_p}\end{array}\right]\left[\begin{array}{c}\hfill SO{C}_{k-1}\\ \hfill U_{p,k-1}\end{array}\right]+\left[\begin{array}{c}-\eta T_s/Q_b\\ R_p(1-e^{-T_s/R_p{C}_p})\end{array}\right]I_{b,k-1}$$

(3)

$$U_{t,k}=U_{ocv}(SO{C}_k)-U_{p,k}-R_0{I}_{b,k}$$

(4)

where the subscript k is the sampling interval, and $T_s$ is the sampling time.

2.3. SOC Estimation Based on the MI-AEKF

Because of the non-linear functional relationship between $U_{ocv}$ and SOC, EKF is employed to address the non-linear characteristics of the battery model. This method extends the conventional KF through first-order Taylor expansion of both state and measurement functions, thereby achieving local linearization of the non-linear system. The dicretized state-space equations can be expressed as follows:

$$\left\{\begin{array}{c}{x}_k=f\left(x_{k-1},I_{b,k-1}\right)+w_{k-1}\hfill \\ U_{t,k}=g\left(x_k,I_{b,k}\right)+{\nu }_k\hfill \end{array}\right.$$

(5)

where f and g are the state non-linear and measurement non-linear function, respectively; $x_k={[SO{C}_k{U}_{p,k}]}^T$ is the state variable of the battery model; $w_{k-1}\sim N(0,Q)$ and ${\nu }_k\sim N(0,R)$ represent the covariance matrices of the Gaussian process noise and Gaussian measurement noise, respectively. Q and R represent the process noise covariance matrix and measurement noise covariance matrix, respectively.

The SOC estimation based on EKF iterates according to the following steps:

Step1: State one-step prediction

$${\widehat{x}}_{k/k-1}=f\left({\widehat{x}}_{k-1},I_{b,k-1}\right)+{\omega }_{k-1}$$

(6)

where ${\widehat{x}}_{k/k-1}$ is the state variable predicted at time k; ${\widehat{x}}_{k-1}$ is the corrected state variable.

Step2: Error covariance prediction

$$P_{k/k-1}={\varphi }_{k-1}{P}_{k-1}{\varphi }_{k-1}^T+Q_{k-1}$$

(7)

where $P_{k/k-1}$ represents the predicted error covariance at time step k in the estimation process. $P_{k-1}$ represents the posterior error covariance after correction, and $\varphi$ denotes the Jacobian matrix of the state transition function f.

Step3: Calculate the Kalman gain

$$K_k=P_{k/k-1}{H}_k^T{\left(H_k{P}_{k/k-1}{H}_k^T+R_k\right)}^{-1}$$

(8)

where H represents the Jacobian matrix of the measurement function g in the observation model.

Step4: Status update

$${\widehat{x}}_k={\widehat{x}}_{k/k-1}+K_k\left[U_{t,k}-g\left({\widehat{x}}_{k/k-1},I_{b,k}\right)\right]$$

(9)

Step5: Error covariance update

$$P_k=\left(I-K_k{H}_k\right)P_{k/k-1}$$

(10)

where I is the identity matrix.

For a non-linear SOC estimation, the above traditional EKF uses $k-1$ moment state information to update the state variables of the system. Inaccurate state information will decrease the estimation accuracy of EKF. Therefore, the observed value of the error information is defined as follows:

$$e_k=U_{t,k}-g\left({\widehat{x}}_{k/k-1},I_{b,k}\right)$$

(11)

where $e_k$ is the information error at time k.

However, the use of the information error $e_k$ alone at time k may greatly increase the error in the SOC estimation. In order to make full use of the past effective error information, the single-state error information is extended to a multi-information vector, which can be expressed as follows:

$$E_{j,k}=\left[\begin{array}{c}{e}_k\\ e_{k-1}\\ ⋮\\ e_{k-j+1}\end{array}\right]=\left[\begin{array}{c}{U}_{t,k}-g\left({\widehat{x}}_{k/k-1},I_{b,k}\right)\\ U_{t-1,k-1}-g\left({\widehat{x}}_{k-1/k-2},I_{b,k-1}\right)\\ ⋮\\ U_{t-j+1,k-j+1}-g\left({\widehat{x}}_{k-j+1/k-j},I_{b,k-j+1}\right)\end{array}\right]$$

(12)

where the subscript j is the length of the extending information at time k, and $e_{k-j+1}$ is the information error at time $k-j+1$, which is used to correct the current error. In addition, the Kalman gain at time k is extended as follows:

$$K_{j,k}=\left[K_k{K}_{k-1}\dots K_{k-j+1}\right]$$

(13)

Therefore, the state update of Equation (9) based on multi-innovation can be represented as follows:

$${\widehat{x}}_k={\widehat{x}}_{k/k-1}+C_m{K}_{j,k}{E}_{j,k}$$

(14)

where $C_m={\left[c_k{c}_{k-1}\dots c_{k-j+1}\right]}^{\mathrm{T}}$ is the weight coefficient matrix for multi-information vector. For the state update ${\widehat{x}}_k$ at time k, the influence of error information in the extending information vector $E_{j,k}$ on the state update varies. The closer the error information is to time k, the larger the weight coefficient will be. For example, if the length of the error information is taken as $j=3$, and the weight coefficient matrix is taken as $C_m={\left[0.950.50.1\right]}^{\mathrm{T}}$, then the state update is represented as follows: ${\widehat{x}}_k={\widehat{x}}_{k/k-1}+0.95K_k{e}_k+0.5K_{k-1}{e}_{k-1}+0.1K_{k-2}{e}_{k-2}$.

Inaccurate covariance R affects the filtering result by updating the Kalman gain $K_k$, which, in turn, causes a decrease in SOC estimation accuracy. In order to improve the performance of SOC estimation, an adaptive noise method is used in this paper. According to [42], the adaptive covariance ${\widehat{R}}_k$ is expressed as follows:

$${\widehat{R}}_k=\left\{\begin{array}{c}(1-d_k){\widehat{R}}_{k-1}+d_k{D}_k,eig(D_k)>0\hfill \\ (1-d_k){\widehat{R}}_{k-1}+d_k(e_k{e}_k^T),eig(D_k)\le 0\hfill \end{array}\right.$$

(15)

where $D_k=e_k{e}_k^T-H_k{P}_{k/k-1}{H}_k^T$ is the filter stability prediction matrix, and $eig$ is the eigenvalue of the matrix $d_k=(1-b)/(1-b^{k+1})$, where $b\in \left[0,1\right]$ is the forgetting factor and is suggested as the value of 0.95∼0.99.

To sum up, the MI-AEKF is applied to initially estimate the SOC using Equations (6)–(15), and its implementation process is shown in Algorithm 1.

Algorithm 1 The algorithmic procedure of MI-AEKF for SOC estimation

1: Initialize $k=0$, initialize state vector ${\widehat{x}}_0=E(x_0)$, initialize covariance $P_0=E\left[\left(x-{\widehat{x}}_0\right){\left(x-{\widehat{x}}_0\right)}^T\right]$;   2: Iterative calculation: while K ≤ threshold:   3: State one-step prediction: ${\widehat{x}}_{k/k-1}=f\left({\widehat{x}}_{k-1},I_{b,k-1}\right)+{\omega }_{k-1}$   4: Error covariance prediction: $P_{k/k-1}={\varphi }_{k-1}{P}_{k-1}{\varphi }_{k-1}^T+Q_{k-1}$   5: Calculate the Kalman gain: $K_k=P_{k/k-1}{C}_k^T{\left(H_k{P}_{k/k-1}{H}_k^T+{\widehat{R}}_k\right)}^{-1}$   6: Multi-innovation error and Kalman gain matrix: $$\left\{\begin{array}{c}{E}_{j,k}=\left[\begin{array}{c}{e}_k\hfill \\ e_{k-1}\hfill \\ ⋮\hfill \\ e_{k-j-1}\hfill \end{array}\right]=\left[\begin{array}{c}{U}_{t,k}-g\left({\widehat{x}}_{k/k-1},I_{b,k}\right)\hfill \\ U_{t-1,k-1}-g\left({\widehat{x}}_{k-1/k-2},I_{b,k-1}\right)\hfill \\ ⋮\hfill \\ U_{t-j+1,k-j+1}-g\left({\widehat{x}}_{k-j+1/k-j},I_{b,k-j+1}\right)\hfill \end{array}\right]\hfill \\ K_{j,k}=\left[K_k{K}_{k-1}\dots K_{k-j+1}\right]\hfill \end{array}\right.$$   7: Status update: ${\widehat{x}}_k={\widehat{x}}_{k/k-1}+C_m{K}_{j,k}{E}_{j,k}$   8: Error covariance update: $P_k=\left(I-K_k{H}_k\right)P_{k/k-1}$ $$   9: Adaptive covariance ${\widehat{R}}_k$ update: $${\widehat{R}}_k=\left\{\begin{array}{c}(1-d_k){\widehat{R}}_{k-1}+d_k{D}_k,eig(D_k)>0\hfill \\ (1-d_k){\widehat{R}}_{k-1}+d_k(e_k{e}_k^T),eig(D_k)\le 0\hfill \end{array}\right.$$ 10: Prepare for the next moment: $x_k={\widehat{x}}_k$, $k=k+1$

2.4. Data-Driven Model for SOC Error Correction

The SOC error correction is based on the BiTCN-BiGRU-Attention architecture for feature extraction of the input data. The following describes the specific process of SOC error correction from three aspects: input data, internal network, and output layer. Figure 3 illustrates the overall structure for SOC error correction. Firstly, the input data are from model parameters and EFK’s first estimate of SOC. Next, the dimension transformation of the input features is extracted through the BiTCN layer. Then, the BiTCN layer’s output is learned and weighted through the BiGRU and attention layers. In addition, the flatten and dropout layers are used to prevent overfitting. Finally, the SOC error correction is predicted by the output layer.

2.4.1. Structure of the BiTCN

The one-dimensional convolutional neural network (1D-CNN) architecture, an adaptation of conventional CNN frameworks, has proven particularly effective for feature representation learning in sequential data analysis. As an improved form of traditional 1D-CNN, TCN, proposed by Bai et al. [43], is a convolution model that has significant advantages in sequence modeling tasks. The one-dimensional CNN can perform local feature extraction, while TCNs extend to capture long-range features. Furthermore, TCNs’ architecture integrates three fundamental components—causal convolutions for temporal dependency preservation, dilated convolutions for extended receptive fields, and residual connections for enhanced gradient flow—collectively enabling effective extraction of both local and global temporal patterns. The mathematical formula of the TCNs is as follows:

$$G\left(t\right)=f\left(X(t),X(t-1),\dots ,X\left(t-N+1\right)\right)$$

(16)

where $G\left(t\right)$ is the system output at time step t, $X(t)=[x_1,x_2,\dots ,x_t]$ represents the corresponding input vector at the same temporal instance, and N represents the receptive field size, which determines the range of dependencies.

The incorporation of dilated convolutions in TCNs’ architecture enables efficient modeling of long-range temporal dependencies while maintaining computational efficiency through parameter conservation. This architectural framework has been extended to the bidirectional TCN (BiTCN), which implements parallel processing through dual TCN modules operating in complementary temporal directions—forward and backward sequence analysis—as shown in Figure 4. The BiTCN model, by integrating bidirectional information, deeply extracts the features of the input sequence and enhances the robustness of the model. The mathematical representation of the BiTCN is presented below:

$$G\left(t\right)=C\left(\stackrel{\rightharpoonup }{G\left(t\right)},\stackrel{\leftharpoonup }{G\left(t\right)}\right)$$

(17)

where $\stackrel{\rightharpoonup }{G}$ is the forward direction of the input sequence, while $\stackrel{\leftharpoonup }{G}$ represents the backward direction and $C(.)$ represents the fusion concatenation operation of the encoding information from both forward and backward directions.

2.4.2. Structure of the BiGRU

GRUs have emerged as a prominent architecture in sequential data processing tasks due to their ability to selectively retain and forget information, avoiding the gradient disappearance problem encountered by traditional RNNs. Their structure introduces an update gate and a reset gate. The update gate merges the forget gate and input gate of LSTM, which can simplify the structure and improve the performance. Meanwhile, the reset gate can control the degree of forgetting of historical information. By incorporating gating mechanisms, GRUs effectively learn sequential tasks, and their mathmatical formulation is as follows:

$$\left\{\begin{array}{c}{v}_t=\sigma \left(W_z\ast \left[h_{t-1},x_t\right]\right)\hfill \\ r_t=\sigma \left(W_r\ast \left[h_{t-1},x_t\right]\right)\hfill \\ {\tilde{h}}_t=tanh\left(W_w\ast \left[r_t\ast h_{t-1},x_t\right]\right)\hfill \\ h_t=(1-v_t)\ast h_{t-1}+v_t\ast {\tilde{h}}_t\hfill \end{array}\right.$$

(18)

where $v_t$ is the update gate; $r_t$ is the reset gate; $\sigma (.)$ is the activation function; $W_z$, $W_r$, and $W_w$ represent the weight matrix coefficients; $x_t$ denotes the input information; $h_{t-1}$ is output state of the previous time step; ${\tilde{h}}_t$ is the current hidden state information; and $h_t$ indicates the current output value.

The bidirectional GRU (BiGRU) architecture extends the conventional GRU framework through the integration of dual processing layers—a forward-oriented GRU layer for standard sequence analysis, and a reverse-oriented GRU layer for complementary temporal pattern extraction—as shown in Figure 5. This bidirectional configuration facilitates comprehensive sequence understanding by simultaneously capturing both past and future contextual information. The mathematical equation is expressed as follows:

$$h_t^{bi}=\left[h_t^f,h_t^b\right]$$

(19)

where $h_t^f$ and $h_t^b$ are the forward GRU and the backward GRU, respectively.

2.4.3. Attention Mechanism

Recognizing the inherent limitations of the BiGRU in modeling extended temporal sequences, which may compromise the precision of SOC estimation, an attention mechanism is incorporated subsequently into the BiGRU layer. This architectural enhancement processes the hidden state representations from the BiGRU, thereby improving the model’s predictive accuracy and temporal dependency capture.

The attention mechanism dynamically reallocates weights to input features based on their relative significance, enabling the model to efficiently process extensive hidden layer representations and identify critical information for learning. This adaptive weighting process enhances model performance and generalization capabilities by focusing computational resources on the most relevant input features, with the degree of relevance quantified through attention scores. The attention weights are computed through the softmax activation function, followed by output evaluation using the hyperbolic tangent (tanh) activation function to ensure appropriate value normalization. Therefore, the attention mechanism can be mathematically expressed as follows:

$$\left\{\begin{array}{c}score({\tilde{h}}_t,{h^{\prime }}_s)=〈W,〈{\tilde{h}}_t,{h^{\prime }}_s〉〉\hfill \\ a_t={\displaystyle \frac{exp\left(score({\tilde{h}}_t,{h^{\prime }}_s)\right)}{{\displaystyle \sum _s}\left(score({\tilde{h}}_t,{h^{\prime }}_s)\right)}}\hfill \\ c_t=\sum _s{a}_t{h^{\prime }}_s\hfill \\ n_t=tanh\left(W_c\left[c_t;{\tilde{h}}_t\right]\right)\hfill \end{array}\right.$$

(20)

where ${\tilde{h}}_t$ is the output information of the previous hidden layer, W is the learnable weight, $a_t$ represents the computed attention score for historical information, $c_t$ denotes the weighted combination of multiple attention scores through fusion operations, $W_c$ is the weight coefficient, and $n_t$ is the output.

3. SOC Estimation Discription

3.1. Battery Dataset

To verify the accuracy and feasibility of the proposed method, we adopt the Panasonic NCR 18650PF lithium-ion battery, which is a high-performance lithium-ion battery suitable for electric UAVs. The specific parameters of the battery are listed in

Table 1. The specific parameters of Panasonic NCR 18650PF battery.

Type	Value
Capacity	Min. 2.75 Ah/Typ. 2.9 Ah
Mass	48 g
Energy storage	9.9 Wh
Min/max voltage	2.5 V/4.2 V
Nominal open circuit voltage	3.6 V

. Due to the fact that electric UAVs and electric vehicles (EVs) both involve multi-stage task execution, the typical mission profile of a UAV includes takeoff, cruise, mission execution, and return, while the mission profile of an EV involves driving under various types of road conditions. Both types of mission profile require accurate estimation of the lithium-ion battery SOC. Given that electric vehicles (EVs) have established a series of standard test mission profiles, this paper selects nine mission profiles of EVs to approximate the tasks of electric UAVs. These profiles are the US06 Supplemental FTP Driving Schedule (US06), the Urban Dynamometer Driving Schedule (UDDS), the Highway Fuel Economy Driving Schedule (HWFET), the LA92 Dynamometer Driving Schedule (LA92), Cycle 1, Cycle 2, Cycle 3, Cycle 4, and Neural Network (NN). The specific mission profiles were measured by Dr. Phillip Kollmeyer at the McMaster University [44]. Furthermore, the nine drive cycles were tested under five different testing temperature conditions: $-20$ °C, $-10$ °C, $0$ °C, $10$ °C, and $25$ °C. In this paper, four temperature conditions ($-10$ °C, $0$ °C, $10$ °C) from the dataset were used, while zero, two (Cycle 1–2), four (Cycle 1–4), or five drive cycles (Cycle 1–4 and NN) were utilized for model training, respectively, and four drive cycles (US06, HFWET, UDDS, and LA92) were used for model testing.

3.2. Parameter Identification and Setting

The ECM parameters used in the process of estimating $\overline{SOC}$ or SOC the using model-based method are identified by the recursive least squares with the forgetting factor (FFRLS) algorithm in this paper. The sampled current and voltage are used as the inputs, and the outputs are the $Th{e}^{\prime }venin$ model parameters. The sampling time in the fast timescale is 0.1 s for the Panasonic NCR 18650PF battery. The identification results of FFRLS algorithm and the $Th{e}^{\prime }venin$ model parameters under the $25$ °C Hybrid PulsePower Characteristic (HPPC) dynamic profile are shown in Figure 6 and Figure 7. It can be seen from Figure 6a,b that the errors of terminal voltage using FFRLS are basically within $\pm 0.05$ V, which indicates the feasibility and accuracy of the identification method. As shown in Figure 7a–d, a polynomial fitting is performed on the mapping from the $U_{ocv}$, $R_0$, $R_p$, and $C_p$ to SOC to establish the corresponding relationship for use in the subsequent SOC estimation process.

The experimental setup involves a fast timescale $l=0.1$ s for the MI-AEKF and a slow timescale $L=3$ s for the BiTCN-BiGRU-Attention model. The average method is used to process the parameter matrix $\stackrel{\rightharpoonup }{A}=[U_{(t)}{I}_{(t)}{T}_{(t)}$$U_{ocv}{U}_{(t)error}SO{C}_{(t)}-SO{C}_{(t-1)}{R}_0{R}_p{C}_p]$ from the fast timescale to the slow timescale, where the parameter matrix needs to normalize before being input into the BiTCN-BiGRU-Attention model. For the MI-AEKF, we utilize a multi-information length $j=10$ and the forgetting factor b of 0.99. For the BiTCN-BiGRU-Attention, an Adam optimizer is employed during the model training process with a learning rate (lr) of 1 × ${10}^{-3}$. Training hyperparameters of maximum training epoch and batch size are set to 120 and 64, respectively. In the BiTCN, we employ 64 filters with a kernel size set to 3 for positive and inverse residual units. For the BiGRU, we use 128 positive and inverse residual units. A dropout rate of 0.1 is applied to the outputs of the hybrid model.

3.3. Performance Index

In order to assess the effectiveness of the proposed method, the maximum error (ME), mean absolute error (MAE) and the root mean square error (RMSE) are utilized to quantify the accuracy of the SOC estimation and are mathematically defined as follows:

$$ME=\underset{1\le i\le n}{max}(\left|y_i-{\widehat{y}}_i\right|)$$

(21)

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(22)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(23)

where $y_i$ and ${\widehat{y}}_i$ are the actual and estimated SOC values, respectively; and n is the size of the samples to be evaluated in the dataset.

4. Results and Discussion

4.1. Estimation Results Using MI-AEKF

Figure 8 shows the fast-timescale SOC estimation performance of the proposed MI-AEKF across various dynamic operating conditions. The sampling time of the fast timescale adopted by the MI-AEKF method is 0.1 s. Moreover, to effectively utilize previous data for SOC estimation, the innovation length is selected as 10. The results show that the proposed method achieves an RMSE under 1.5% and an ME within $\pm 2.5\%$. It can be seen that the proposed method can achieve perfect performance under different mission conditions by using the MI-AEKF method without battery cycle data. The multi-innovation error from the AEKF can be fully utilized and ultimately achieve precise SOC estimation.

A comprehensive comparison of three critical performance metrics under various dynamic operating conditions is shown in

Table 2. Comparison analysis of the fast-timescale SOC estimation performance under various dynamic profiles.

Dynamic Profiles	ME (%)	MAE (%)	RMSE (%)
US06	0.95	0.42	0.51
UDDS	1.91	1.10	1.19
LA92	2.27	1.23	1.41
HWFT	1.36	0.68	0.82

. The LA92 profile shows the highest error, with an ME of 2.27%, and an MAE and RMSE of 1.23% and 1.41%, respectively. This can be attributed to the highest number of charge and discharge fluctuations resulting in the largest SOC error accumulation. The same situation happens in the UDDS, which has an estimation error with an ME of 1.91%, an MAE of 1.1%, and an RMSE of 1.19%, respectively. This is due to the MI-EKF method only taking the battery model into account for fast-timescale SOC estimation, while the three performance indicators are still ultimately maintained below 2.5%. In addition, Figure 6 shows that the proposed method’s estimation error increases gradually under different dynamic profiles, notably in the low-SOC region, owing to inaccuracies in the identified model parameters. Therefore, it is necessary to compensate for SOC estimation error when using a data-driven model.

4.2. Comparison of Different Data-Driven Models on SOC Error Correction

To assess the impact of data-driven modeling methods on SOC error correction, a comprehensive comparative analysis is employed for comparative analysis, including the backpropagation (BP) neural network, LSTM, gated recurrent unit (GRUs), relevance vector machine (RVM), TCN, and Transformer-LSTM. The four cycles and 1 NN conditions are used as the training and validation datasets, respectively, and both input and output parameters are consistent with the BiTCN-BiGRU-Attention model in the proposed methodology. Figure 9 illustrates the comparative performance metrics of various models evaluated on the NN validation dataset. Within a small value of SOC correction, the BiTCN-BiGRU-Attention model demonstrates the best performance, followed by the TCN, RVM, GRU, Transformer-LSTM, and LSTM, while the BP performs the worst, with an ME of 9.56%, an MAE of 1.26%, and an RMSE of 2.11%. The results show that the BiTCN-BiGRU-Attention, the TCN, the RVM, and the GRU can accurately compensate SOC estimation error. However, the RVM can better compensate for SOC errors in the case of sparse data, but it is inefficient in handling large-scale data and is actually unable to process large-scale data. The TCN can correct SOC error by capturing local dependencies within the battery through convolutional networks, but it operates within a local receptive field and is not as effective as GRU, RVM, and LSTM in handling long-term dependencies, resulting in an ME of 6.15%. The GRU has high computational efficiency with fewer model parameters, particularly in the prediction of small datasets with SOC correction, and it outperforms LSTM. The results of its RMSE of 1.85% and ME of 4.15% are both better than those of LSTM. However, it is less capable of handling the complex internal dependencies of the battery compared to LSTM, resulting in its MAE value being lower than that of LSTM. The BiTCN-BiGRU-Attention method integrates the local capturing ability of TCN for the internal dependencies of the battery and the time series prediction ability of GRU. Through BiTCN, BiGRU, and Attention, it enhances the capturing ability for the internal relationships of SOC correction and the prediction ability for time series. Its SOC correction ability is the best among the seven above-mentioned models. Therefore, the BiTCN-BiGRU-Attention model is used for SOC correction in this paper.

4.3. SOC Estimation Using Fusion Model

In this section, the proposed multi-timescale fusion model is evaluated alongside several estimation methods. In the comparison models, there are model-based methods such as the AEKF and unscented Kalman filter (UKF), traditional machine learning methods such as RVM and random forest (RF), and popular deep learning algorithms such as LSTM and GRU. The timescales and parameters of all data-driven models are consistent, where the sampling times of fast and slow timescales are 0.1 s and 3 s, respectively. The battery model and parameter identification of the AEKF and UKF are the same as the proposed method in fast-timescale estimation models, and only the SOC estimation algorithm is different. The current, voltage, average current, and average voltage are used as the inputs for machine learning and deep learning methods, and the output is SOC. With the exception of the AEKF and UKF, other comparison models are trained with the insufficient datasets (Cycle 1–2) and sufficient datasets (Cycle 1–4 and NN) at $25$ °C to assess the performance of SOC estimation methods.

Figure 10 shows the SOC estimation results for UDDS and LA92 by different models under insufficient data-driven cycles. UDDS and LA92, with the largest SOC error in the fast timescale (see Figure 8), are selected to evaluate the estimation effect of different models. It can be see from

Table 3. Comparative performance of SOC estimation under UDDS dynamic profiles.

Methods		Insufficient Data-Driven			Sufficient Data-Driven
		ME (%)	MAE (%)	RMSE (%)	ME (%)	MAE (%)	RMSE (%)
Model-based	AEKF	4.42	1.49	1.85	4.42	1.49	1.85
	UKF	4.72	1.23	1.63	4.72	1.23	1.63
Mechine learning	RF	13.13	7.42	7.94	9.73	5.87	6.25
	RVM	17.20	4.76	5.36	2.22	1.28	1.49
Deep learning	LSTM	17.20	4.76	5.36	2.22	1.28	1.49
	GRU	11.68	5.27	5.41	3.41	3.12	3.39
Our proposed		0.71	0.33	0.37	0.55	0.21	0.24

and

Table 4. Comparative performance of SOC estimation under LA92 dynamic profiles.

Methods		Insufficient Data-Driven			Sufficient Data-Driven
		ME (%)	MAE (%)	RMSE (%)	ME (%)	MAE (%)	RMSE (%)
Model-based	AEKF	4.50	1.38	1.75	4.50	1.38	1.75
	UKF	7.50	1.27	1.99	7.50	1.27	1.99
Mechine learning	RF	17.14	4.58	5.16	10.96	3.13	3.38
	RVM	15.18	2.69	3.18	2.83	1.22	1.44
Deep learning	LSTM	8.28	3.58	3.86	3.79	1.88	2.37
	GRU	8.73	3.02	3.24	4.01	1.88	2.23
Our proposed		0.88	0.31	0.38	0.57	0.22	0.25

that the two machine learning models—RF and RVM—under insufficient data-driven conditions have the worst performance, with large error fluctuations, and an ME of more than 13%. Similarly, the performance of the deep learning methods—LSTM and GRU—is poor, and the RMSE and MAE are also at high levels. This may be due to overfitting between the time-history information of the model training and the SOC. Compared to data-driven methods, the model-based methods—AEKF and UKF—have a relatively high estimation accuracy under insufficient data-driven conditions, with an MAE and RMSE below 2%. The proposed fusion method, which has the best performance, can partially compensate SOC estimation error on the basis of fast and accurate ${\overline{SOC}}_{(l)}$ estimation by the MI-AEKF model-based method. The results include an ME of 0.71%, an MAE of 0.33%, and an RMSE of 0.37% under UDDS driving conditions, and an ME of 0.88%, an MAE of 0.31%, and an RMSE of 0.38% under LA92 driving conditions.

The SOC estimation results for UDDS and LA92 under sufficient data-driven cycles are presented in Figure 11. Sufficiently data-driven models estimate closer to the SOC reference value than those under conditions of data insufficiency, showing significant improvement in three performance indexes. Compared with the conventional model-based methods of AEKF and UKF, most data-driven methods—excluding the RF method—show better performance for ME, while the MAE and RMSE have a poor performance. This may be caused by the fact that data-driven methods such as LSTM, GRU, and RVM have more powerful non-linear processing abilities to reduce the ME, while the MAE and RMSE are high due to the limited lithium-ion battery operation data in data-driven training sets and the low-timescale processing of the data. However, the proposed fusion method can obtain a good SOC compensation effect in small sufficiently data-driven sets. This can be attributed to the deep architecture and the long-term dependency learning ability of the proposed SOC estimation error compensation model. The performance of the proposed fusion method in Figure 11 and Table 3 and Table 4 shows higher estimation accuracy, which illustrates its superiority and robustness under UDDS and LA92 driving conditions.

In summary, the estimation error distributions of the proposed model under three data-driven conditions are depicted using box plots in Figure 12. It is clearly shown that the SOC estimation errors of the proposed method, when using only the MI-AEKF model, are mostly higher than the average values by 1.04% and 1.14% under UDDS and LA92 operating conditions, respectively. Moreover, the fluctuations in the SOC estimation error are relatively large. Under insufficiently data-driven conditions, the average error distributions of the UDDS and LA92 when using the fusion model are significantly reduced compared to the case using the MI-AEKF model only, with values of 0.34% and 0.29%, respectively. Although the estimation error is still above the average, the error values are more concentrated and the fluctuations are reduced. Under sufficiently data-driven conditions, the SOC estimation errors are lower than the average values by 0.2% and 0.23% under UDDS and LA92 operating conditions, respectively, which indicates that the majority of the SOC estimates are close to the true SOC values. Furthermore, the errors for the UDDS and LA92 conditions decrease between the MI-AEKF model and the sufficiently data-driven model due to the increased accuracy of the SOC error correction as the data cycle increases. The results show that the average error distributions under model-based conditions, insufficient conditions, and sufficient conditions are below 1.2%, illustrateing the superiority and robustness of the proposed method in SOC estimation across various dynamic operating profiles.

4.4. Performance Different Temperature Conditions

The SOC estimation of lithium-ion batteries can be influenced by external environmental factors, especially temperature. Therefore, accurate SOC estimation becomes critical under different temperature conditions. A comparison of the estimation performance of the proposed fusion method with that of the model-based methods (AEKF and UKF) and the data-driven approaches (RF, RVM, LSTM, and GRU) across different temperature conditions is analyzed in this section. In the fusion model, the SOC estimation error is incorporated into the BiTCN-BiGRU-Attention model for training in the slow timescale, which not only reduces the complexity of the battery model in fast-timescale initial SOC estimation but also achieves accurate error compensation. The model-based methods use the same battery model and parameters as the proposed method to estimate SOC on a fast timescale. The temperature, current, voltage, average current, and average voltage are used as inputs to the data-driven models, and the output is the SOC at different temperatures. Two driving conditions are used to reflect the models’ performance at different ambient temperatures, specifically driving in the UDDS for training at an environment of $-10$ °C, $10$ °C, and $25$ °C, and testing at $0$ °C for the UDDS and US06.

The comparison of estimation performance and error distribution at $0$ °C are presented in Figure 13. Operating under the same UDDS conditions, the data-driven methods have the worst performance in SOC estimation, especially the RF, with an error distribution of more than 10%; this may due to overfitting of the models under insufficiently data-driven conditions at different temperatures. Followed by the data-driven methods, the performance of the model-based methods AEKF and UKF was also poor without increasing the complexity of the model, with an ME of more than 5%. The proposed fusion method has the best performance, with an ME of 0.87%, an MAE of 0.43%, and an RMSE of 0.48%. Under the US06 conditions, the estimation performance is worse than it is under the UDDS conditions due to a lack of error compensation information. However, the proposed fusion method can accurately estimate the SOC, with both the MAE and RMSE being less than 1%. The results show that the proposed method has good robustness and accuracy in SOC estimation at different ambient temperatures.

5. Conclusions

In this study, a multi-timescale fusion method for SOC estimation in electric UAVs is proposed that integrates a battery model and data-driven technologies at different timescales. Firstly, the SOC is quickly estimated by the model-based method MI-AEKF in a fast timescale. In addition, the estimation error is corrected by the hybrid BiTCN-BiGRU-Attention model in a relatively slow timescale. Consequently, it achieves high accuracy and robust estimation for lithium-ion battery SOC under the three conditions of zero data, insufficient data, and sufficient data, respectively.

To validate the efficacy of the proposed methodology, a publicly available dataset with rich mission profiles is used. The results show that the ME, MAE, and RMSE for zero data, insufficient data, and sufficient data under various dynamic conditions, respectively, are as follows: below 2.3%, 1.3%, and 1.5%; below 0.9%, 0.4%, and 0.4%; and below 0.6%, 0.3% and 0.3%. The proposed fusion approach, which combines the ECM and data-driven model, demonstrates superior estimation accuracy in dealing with different cycle data, temperature changes, and complex mission conditions, and has exceptional generalizability.

Compared with other methods, the proposed fusion method not only achieves higher accuracy in SOC estimation, but also provides reliability and practical value for the use of lithium-ion batteries in electric UAVs. However, it should be noted that this study does not analyze the effects of battery life attenuation and the state of health. Therefore, it is necessary for future research to incorporate these factors into this algorithm and realize its application at the hardware level of the BMS for electric UAVs.