1. Introduction
Lithium-ion (Li-ion) batteries are widely used in many applications, such as cell phones, electric and hybrid electric vehicles, since they exhibit a higher energy density and have a relatively longer life compared to other batteries [
1]. In these systems, Li-ion batteries must possess a high reliability and pose no safety threats [
2]. However, the thermal behavior can greatly affect the safety, durability, and performance of Li-ion batteries [
3]. For example, fire and explosions caused by thermal runaway were reported [
4]. Thus, reliable battery management systems are essential to mitigate negative effects (e.g., thermal runaway) and avoid catastrophic failures [
5]. As a key component of the battery management system, fault detection and diagnosis play an important role in the management of Li-ion batteries [
6].
Fault detection and diagnosis (FDD) methods generally can be classified into two major groups, i.e., first-principle model-based methods and data driven (or empirical) methods [
7]. For the former, models describing the physical mechanisms of the fault dynamics are oftentimes used, while historical data are typically collected for data driven methods to derive empirical models. Each of these approaches has its own advantage and drawback depending on the specific problems. It is recognized that first-principle model-based methods exhibit a better extrapolation ability, whereas data-driven methods are easier to design [
8]. This work focuses on the use of the first-principle models for FDD, since these models provide a fundamental understanding of the thermal physics of batteries [
9].
Several first-principle thermal models have been previously developed for Li-ion batteries. For example, a three-dimensional thermal finite element model was developed to investigate the cell behavior under abnormal events such as overheating and external short circuits [
10]. This model requires high computational capabilities, and its application is limited to stationary storage [
11]. Compared to the three-dimensional models, the one-dimensional model of Li-ion batteries, developed using the average lumped temperature of the cell, is viable for real-time applications and can enable online battery management [
12]. However, such a model may fail to provide insights into the thermal (fault) dynamics due to its simplicity [
13]. As a trade-off, a two-dimensional thermal model was developed, which can predict the core and the surface temperature of Li-ion battery cells [
3,
13]. Since the two-dimensional model can provide a better understanding of the thermal dynamics of battery cells, while maintaining the computational complexity, it is used in this work for the design of a stochastic FDD scheme.
Measurements of temperatures such as surface and core temperatures are often used for FDD in Li-ion batteries, but there is no direct measurement of the core temperature. To take the core temperature into account, estimation techniques are often required. In the literature, several estimation techniques have been developed. For example, an adaptive observer based on the lumped thermal model [
14] and state observer using partial differential algebraic equations [
15] were proposed to estimate the temperature. Compared to these estimation techniques, the real-time monitoring and diagnosis of faults in batteries are less explored. Although there have been several proposed works related to diagnostic algorithms for internal faults in Li-ion batteries [
3,
16,
17], it is important to note that previously reported FDD work mostly investigated sensor or actuator fault detection problems [
18,
19,
20].
In this work, we propose to estimate the core temperature and use the estimation results to identify and classify two sets of faults. That is, faults that can introduce dynamic changes in core temperatures and faults that can affect the surface temperatures. The FDD scheme in this work can potentially provide more information about the thermal dynamics of batteries and enable an internal thermal fault detection to improve the performance of the Li-ion battery.
For FDD, the available algorithms compare the observed behavior to the corresponding model results, estimated from first-principle models [
21]. When a fault is detectable, the FDD scheme will generate fault signatures, which in turn can be referred to an FDD scheme to identify the root cause of faults using a threshold [
22]. However, the main restrictive factor for the first-principle model-based FDD is the model uncertainty [
23]. The accuracy of the fault detection algorithm can be affected by any uncertainty in the model parameters. Such an uncertainty may result from intrinsic time varying phenomena or originate from the model calibration with noisy measurements [
24]. The uncertainty can be quantitatively approximated by a calibration with experimental data, which include principles such as least squares errors or the Delphi method [
25,
26].
The procedures that firstly quantify the uncertainty and then propagate the uncertainty onto the FDD scheme are typically omitted in previously reported works. This subsequently may lead to a loss of information about the effect of uncertainty on FDD performance. Recently, several techniques, such as the adaptive observer [
27,
28] and the sliding mode observer [
29], were developed for FDD in the presence of uncertainty. However, most of these methods cannot provide information, such as the probability that a fault has occurred. In addition, since the faults in the batteries may happen in a stochastic fashion, the use of fixed thresholds to identify the root cause of faults may not be effective.
There are differences between the actual thermal dynamics of Li-ion batteries and fundamental models derived from physical phenomena. For example, to make models tractable and useful, it is common to make simplifications during the model development, which will introduce a mismatch between the model and the Li-ion battery system of interest. Thus, the first principle model-based FDD scheme should be designed to compensate the mismatch. Specifically, a set of fixed model parameters may not be accurate enough for estimating the core temperature in the presence of a model mismatch. Consequently, any inaccuracy in the temperature estimation may potentially lead to a low fault detection rate. To ensure the accuracy of FDD, it is essential to simultaneously calibrate the model parameters and adjust the FDD scheme. However, this is generally challenging due to the presence of uncertainty such as the measurement noise and an unknown model mismatch.
In this work, we propose to address these aforementioned limitations by developing an FDD scheme for Li-ion batteries described by a two-dimensional first-principle thermal dynamic model, for which both model parameters and faults are of a stochastic nature. Specifically, the faults considered in this work, such as the thermal runaway, are stochastic perturbations superimposed on step changes in the specific thermal dynamic parameter and electric current. The objective is to identify the changes in the mean values of the thermal dynamic parameter and current in the presence of random perturbations, the measurement noise, and a model mismatch. As compared to other existing thermal diagnostic techniques, the main feature of the FDD scheme is the efficient quantification of the effect of stochastic changes in model parameters on fault detection, and the rapid propagation of the stochasticity onto the estimation of temperatures that are required for FDD.
Note that one possible way to propagate uncertainty in model parameters onto temperature estimates is the use of Monte Carlo (MC) simulations [
30]. However, methods such as MC may be computationally demanding, since they often require a larger number of simulations in order to obtain accurate results. It is worth mentioning that although the calibration of an FDD scheme can be performed offline, the online re-calibration of the model in the presence of a model mismatch with MC as shown later in current work is computationally prohibitive. Recently, the uncertainty propagation with generalized Polynomial Chaos (gPC) expansion has been studied in different modelling [
31], optimization [
32], and fault detection problems [
24]. As compared to MC, the advantage of gPC is that it can propagate a complex probability distribution of uncertainty in model parameters onto model predictions rapidly and can analytically approximate the statistical moments of model predictions in a computationally efficient manner [
31]. The improvement in computational time may facilitate its application in the real-time model adjustment for improved FDD.
The FDD algorithm in this work is specifically targeted to identify and diagnose stochastic thermal faults consisting of uncertainty around a set of mean values of thermal properties in the presence of a model mismatch. In summary, the contributions in this work include: (i) The use of an intrusive gPC model for stochastic FDD of Li-ion batteries by approximating the uncertainty in thermal dynamics with gPCs and by propagating the uncertainty directly onto temperatures that can be used for FDD; (ii) the identification and classification of a fault based on the probability information of temperatures other than a single point estimate or threshold; (iii) the formulation of an optimization to account for a model mismatch and adjust the thermal dynamic models by incorporating the discrepancy between model predictions and measurements.
This paper is organized as follows.
Section 2 presents the theoretical background and the principal methodologies in this work, including a two-dimensional thermal dynamic model, the introduction of generalized polynomial chaos (gPC) expansion, and the formulation of the stochastic fault detection and diagnosis (FDD) problem. The methodology for FDD and the formulation of an optimization for model correction to account for the model mismatch is presented in
Section 3. The analysis and discussion of the results are given in
Section 4, followed by conclusions in
Section 5.