1. Introduction
With the energy crisis and the deterioration of the ecological environment, electric vehicles (EVs) and smart grids have received widespread attention around the world. Among the existing energy storage devices, lithium-ion (Li-ion) batteries have been widely utilized for their distinctive advantages, such as high energy density, long service life, low self-discharging rate, and no memory effect [
1,
2,
3]. In order to ensure the safety and reliability and achieve the best working performance of the energy storage system with a proper management strategy, accurate state monitoring is needed for the battery management system (BMS). Battery state is a complicated concept that usually includes multitudes of branches, such as state of charge (SOC), state of health (SOH), state of power (SOP), etc. One of the key variables that must be estimated in the BMS is the SOC, which is the ratio of the battery’s current remaining capacity to its maximum value [
4,
5,
6].
However, it is inaccessible to accomplish the direct SOC measurement by BMS. Usually, some relevant measurable variables, such as current, voltage, and temperature, are needed for battery SOC estimation. The commonly used SOC estimation methods include the ampere-hour integration method [
7,
8], the open circuit voltage (OCV) method [
9,
10], and the model-based method. The ampere-hour integration method has obvious drawbacks of the measurement error accumulation and the uncertainty of the initial SOC, whereas it is the simplest method. Besides, the ampere-hour integration method only uses the current to estimate SOC without the correction and information from terminal voltage. The OCV method can recompense the abovementioned shortcoming utilizing the determinate relationship between SOC and OCV. Nevertheless, the SOC estimation cannot be achieved online by the OCV method because battery OCV measurement requires a long relaxation period to reach its internal equilibrium.
The model-based methods commonly deploy the equivalent circuit model (ECM) or the electrochemical model (EM) to reach the goal of SOC estimation. Based on battery model, filtering algorithm, such as extended Kalman filter (EKF) [
11], unscented Kalman filter (UKF) [
12], and particle filter (PF) [
13], can be applied to achieve SOC estimation. Meanwhile, a battery model can be combined with an observer, such as Luenberger observer (LO) [
14], sliding-mode observer (SMO) [
15], or nonlinear observer (NO) [
16], to reach the same goal. ECM [
17,
18], including the Rint model, Thevenin model [
19,
20], first-order RC model [
21], partnership for new generation of vehicles (PNGV) model, etc., use resistors, inductors, and capacitors to simulate the external characteristics of a Li-ion battery, and have been widely applied to estimate SOC in practice thanks to their simple structure, high calculation efficiency, and few parameters to be identified. However, there remain some drawbacks for ECM, e.g., parameters of the ECM are not directly related to the electrochemical reactions inside the Li-ion battery. In this way, the parameters in ECM lose their specific physical meaning in reality, which further contributes to the incapability to reflect the internal states, such as the overpotential and the lithium-ions concentration, and also leading to limitations in terms of modeling accuracy. The SOC estimation errors can be divided into dynamic transient error and steady state error. The dynamic transient error can be eliminated if the gain of observer is selected to be a big value. However, the steady state error is independent of the observer gain and cannot be neglected. So, one consensus for the model-based SOC estimation method is that the estimation algorithm cannot well handle the battery modeling errors, which restricts the usage of ECMs for advanced BMS applications [
22,
23]. Meanwhile, the EM can provide more internal details on the battery, which could support an accurate battery model, as well as the charging strategy and power prediction of the Li-ion battery. Thus, researchers focus on using EM to obtain a superior model-based SOC estimation method.
The original EM is a pseudo-two-dimensional porous electrode (P2D) model proposed by Doyle, Fuller, and Newman [
24,
25]. The P2D model describes the battery internal reactions according to five subsections: porous positive electrode, porous negative electrode, separator, and the current collector in both positive and negative area. The P2D model is composed of a series of coupled partial differential equations (PDEs) that reflect the internal electrochemical reactions of the Li-ion battery, including the diffusion equation in solid and electrolyte phase, ohm equation in solid and electrolyte phase, charge conservation equation, and the Butler–Volmer equation. Although it can improve the modelling accuracy and widens the application range of the battery model to some extent, the computational burden of EM is aggravated owing to the difficulty of solving the coupled PDEs [
26]. An immense amount of research has emerged to reduce the complexity of EM. Mathematical approaches, such as finite element method (FEM) [
27], method of linear (MOL) [
28], finite difference method (FDM) [
29], finite volume method (FVM) [
30], and Padé approximation [
31], have been widely studied. Meanwhile, B. Haran et al. [
30] propose the concept of the single particle model (SPM) [
32] using single spherical particles to represent the positive and negative electrodes. It is assumed the current is uniformly distributed in the electrode and the electrolyte dynamics are ignored. In this way, the mathematical calculation of the EM is simplified, and the SPM is widely used for the online estimation of SOC [
33,
34,
35]. Unfortunately, the simplification of electrolytes may decrease the calculation burden, and at the same time cause some unwanted errors which limit the application of the SPM in low C-rates circumstances. It is noted that SPM will lose fidelity once the C-rates is above C/2. Therefore, one challenge here is to properly choose a suitable EM for Li-ion battery SOC estimation.
Since the EM has a good interpretability and can precisely describe the state variations inside the battery, it has become a trend in the future development of the model-based SOC estimation method. The computational accuracy of EM depends remarkably on the accuracy of the parameters, so how to obtain parameters for such an intricate model is a crucial yet challenging task [
36]. Some parameters can be acquired by directly disassembling the commercial battery through laboratorial tests, such as geometric parameters and SOC-OCV curves in the electrodes. However, those microscopic parameters, such as diffusion coefficients and kinetic reaction rates, cannot be measured experimentally and need to be identified by specific algorithms. Li-ion battery parameter identification is generally performed using the nonlinear least squares (NLS) method [
37,
38], genetic algorithm (GA) [
39,
40], and particle swarm optimization (PSO) method [
41,
42]. Due to the simplicity of the components and the fewness of parameters in ECM, the NLS method can well complete the identifying task. However, the parameters required to reflect the internal electrochemical reaction mechanisms are extensively increasing in EM. Meanwhile, EM has become highly nonlinear and due to a lack of analytical solutions, more advanced algorithms for parameter identification are in an urgent demand. Hence, the second challenge using EM for SOC estimation is the demand for a parameter adjustment strategy, which should be a trade-off of the accuracy and complexity. Although GA and PSO can well identify the parameters in EM, a large number of microscopic parameters will increase the computing burden.
This paper attempts to solve the abovementioned issues by utilizing a single particle model with electrolyte dynamics (SPME) based SOC online estimation framework using a sensitive parameter adjustment strategy and PF. In this thread, a proper selection procedure is designed to choose only the representative ones from SPME for parameter adjustment. There are three main contributions in this paper:
- (1)
A simplification of the P2D model, SPME, is combined with PF for Li-ion battery SOC estimation, which involves a trade-off of both the modeling accuracy and simplicity.
- (2)
The elementary effect test (EET) is used for the parameter sensitivity analysis (SA) of EM to improve the calculational efficiency, and only highly sensitive parameters are identified by PSO.
- (3)
The performance of the proposed method is validated on three different driving cycles and compared with the EKF and the SPME without PF.
The remainder of this paper is organized as follows.
Section 2 introduces the SPM with electrolyte dynamics and its simplification procedure.
Section 3 describes representative parameter selection and the adjustment strategy with SA, EET, and PSO.
Section 4 illustrates the state-space equation about the lithium-ions concentration in the solid phase, and then the SOC online estimator based on the PF and the SPME. The experimental results are exhibited in
Section 5, and the conclusion is provided in
Section 6.
3. Global Sensitivity Analysis and Parameter Identification
As previously described, the EM usually comprises a group of microscopic parameters related to battery design. Abovementioned parameters, such as solid phase lithium ions concentration
, exchange current density
, moles of lithium ions
, and electrolyte activity coefficient
, are simplified in SPME, that is, assumed to be constant at any positions in electrodes and invariable along the x-coordinate. In order to achieve the goal of accurate SOC estimation and voltage prediction, more than twenty parameters still need to be identified in SPME. The parameters of the EM can be divided into the following three categories, as shown in
Table 2. One is the geometric parameters which can be obtained through the battery disassembling test, the thickness of electrodes and separator
,
,
, and the area of electrodes S belong to this kind of parameters. Another is the performance parameters, including diffusion coefficients, solid phase conductivity
, electrolyte ionic conductivity
, volume fraction
, etc., which can be obtained from the relevant literature or parameter identification method. The third category is those fixed parameters, such as the molar gas constant R, Faraday’s constant F, reaction transfer coefficients
, etc. It should be noted that the performance parameters are difficult to obtain directly from experimentation, while those parameters greatly affect the accuracy of the EM. Considering the computational burden of the BMS, the sensitivity of the parameters is analyzed and further identified by a special designed algorithm in this work.
3.1. Global Sensitivity Analysis
SA is used to verify the influence of the specific parameters on the output of the model. Through sensitivity analysis, the significant parameters of the EM can be selected from numerous variables to reduce the labor of parameter identification. One way to qualify the SA is changing the parameters within a specific range to evaluate the variation of the EM’s output. This paper uses the EET to analyze the parameters’ sensitivity [
46]. Assuming that the state-space equation of the system is:
where z is the state vector, y is the output vector, t represents the time, u is the system input, and p is the system parameters. Assuming each parameter
(i = 1,2,…,n) is independent in the system and varies in n-dimensional space, the elementary effect (EE) of j-th trajectory, i.e., the effect of i-th parameter on the output, can be defined as:
Here, we set two metrics to evaluate the parameters: the average
and the standard deviation
of EE.
A high value of means that the parameter has a significant effect on the output, in which indicates the parameter is sensitive. Similarly, a high means that the parameter interacts with other parameters, or that the effect of this parameter is nonlinear. Then, these interactions should be taken into consideration for parameter identification. After understanding the sensitive parameters, PSO can be used to identify these parameters. The principle and calculating steps are described as follows.
3.2. Parameter Identification Based on PSO
PSO is originally inspired by the behavior of birds foraging in nature. Through the cooperation and information sharing between individuals of the swarm, PSO can achieve the goal of finding the optimal solution. This algorithm treats each individual as a particle without weight and volume, flying with a certain speed in the n-dimensional searching space. The flying speed is synchronously affected by the flying experience of the individual and swarm. The flowchart of PSO is illustrated in
Figure 2 and detailed steps are as follows:
(1) Initialization: Randomly initialize the particle swarm, the position of i-th particle in n-dimensional space is expressed as , the velocity can be described as .
(2) Best particle selection: Record individual suitability, individual optimal suitability, and global optimal suitability; the particle with individual optimal suitability is the personal best particle . Similarly, the particle with global optimal suitability is defined as the global best particle .
(3) Update: Update the speed and position of particle flight.
where w is the inertial coefficient,
,
are the learning factors representing the influence of the best particles. If the suitability is more optimal, then update the best particle; otherwise, continue the algorithm.
(4) Recursion: Determine whether the termination criterion is met. The termination criterion could be the value of suitability or the cycle number.
6. Conclusions
This paper developed SPME using PF to achieve the online accurate estimation of SOC under different conditions. Considering the accuracy and fidelity of EM, this paper uses SPME to find a compromise involving both accuracy and low computational burden. After completing the battery model, the elementary test was applied for the SA of parameters to reduce the calculational burden, and eight parameters were identified using PSO algorithm. The solid diffusion equations were simplified and transformed into state-space equations to adjust the PF and achieve the SOC estimation.
The battery tests were applied under NEDC, FTP, and UDDS conditions. The experimental results show that the method introduced in this paper has high accuracy. Among three different working conditions, the proposed method has maximum MAE = 0.0076, RMSE = 0.0086, while another model-based method has maximum MAE = 0.0641, RMSE = 0.0766. Therefore, the proposed method demonstrates remarkable robustness and reliability for different battery working conditions.
The proposed method is only experimentally validated on a regular lithium-ion batteries, namely the ICR18650-26J battery manufactured by SAMSUNG. With the development of batteries, solid-state and anode-free batteries are now widely used in BMS. With some adjustment to the model structure, the proposed method can be applied to these kinds of batteries. The adjustment of model structure also can be the future trend and our next research goal.