Research on State of Power Estimation of Echelon-Use Battery Based on Adaptive Unscented Kalman Filter

Abstract

An echelon-use lithium-ion battery (EULB) refers to a powered lithium-ion battery used in electric vehicles when the battery capacity is attenuated to less than 80% and greater than 20%. Aiming at the degradation of the performance of the EULB and the unclear initial value of the state of energy (SOE), estimations of the state of power (SOP) of an EULB are not accurate. An SOP estimation method based on an adaptive dual unscented Kalman filter (ADUKF) is proposed. First, the second-order resistor-capacitance symmetry equivalent model (SRCSEM) of the EULB is established. Second, an unscented transformation (UT) is introduced and the battery parameters estimated by the ADUKF: (a) the SOE is estimated based on an adaptive unscented Kalman filtering (AUKF) algorithm, that uses the observation noise equation γ_k, R_k and the processes noise equation q_k, Q_k, and (b) the ohmic internal resistance (OIR) and actual capacity (AC) are estimated based on the aforementioned algorithm, which uses the observation noise equation γ_θ,k, R_θ,k and the process noise equation q_θ,k, Q_θ,k. Third, the working voltage and OIR are predicted using optimal estimation, and the SOP of the EULB is estimated. MATLAB simulation results show that EULB symmetry capacity decays to 80%, 60%, 40%, and 20% of rated capacity, the proposed algorithm is adaptive regardless of whether the initial SOE value is consistent with the actual value, and the estimation error of the EULB’s SOP is less than 3.28%, showing high accuracy. The results of this study can provide valuable reference for estimating EULB parameters, and help to understand the usage behavior of retired batteries.

1. Introduction

With an increase in the number of retired lithium-ion batteries, the EULB has attract-ed much attention. An EULB refers to a lithium iron phosphate powered lithium battery used in electric vehicles (EVs) when the capacity is attenuated to greater than 20% and less than 80%, which is used for power backup and energy storage of communication base stations.

In order to achieve effective utilization of lithium-ion batteries throughout their life cycle, many researches are devoted to the echelon-use of retired EVs batteries in the references. Neubauer et al. [1] started the study of battery secondary utilization in 2010. In references [2,3], Tong and Omar verified the feasibility of re-application of echelon-use batteries in photovoltaic charging stations, and completed more than 1000 charge discharge cycles at 1 C and 80% discharge depth. Jiang et al. [4], who estimated the residual capacity of echelon-use batteries by the regression method verified the feasibility of the method with an estimated error within 3%. In reference [5], Schuster et al. studied the influence of battery capacity and impedance parameter aging on state of health (SOH) estimation. Zhang et al. [6] conducted performance tests on retired EV battery modules, coordinating the test accuracy and test time. Therefore, the retired batteries were shown to still have a high ability in discharge, indicating that an EULB energy storage system can be applied to achieve the power requirement of an electrical grid.

Although the retired battery still has a high discharge capacity, it is difficult to ensure the safety of EULB due to performance degradation and capacity attenuation. As a result, large-scale promotion and utilization of echelon-use batteries are difficult. As an important parameter of battery safety control and energy recovery, SOP (state of power) has attracted more and more attention. Common methods to estimate SOP include battery experiment and model method. In references [7,8], the United States Advanced Battery Consortium (USABC) adopted a battery experimental method that has the advantage of strong practicability. In references [9,10], the battery experimental method is a test method adopted by Fang and Yu. Owing to disadvantages of testing being complicated and equipment being required, the use of the estimation method has increased based on the equivalent circuit model. The equivalent circuit model was used to estimate SOP firstly in reference [11]. In references [12,13,14,15], the first-order equivalent circuit model was used to estimate SOP. In reference [15], the first-order model was adopted to estimate state of charge (SOC), SOH, and SOP based on an unscented Kalman filter, but observation and process errors were not considered. The second-order resistor-capacitance equivalent model (SRCSEM) was used to estimate SOP in [16,17,18,19,20,21,22,23,24]. In [23], the adaptive estimation algorithm was adopted, and the estimation accuracy of SOP was 4.8%, which was not high. Reference [24] estimated SOC, SOE (state of energy), SOP, and SOH based on SRCSEM, which achieved high accuracy, but lacked adaptive research. The third-order equivalent model was adopted in [25] to carry out SOP estimation. Owing to the high order of the equivalent model, a large amount of calculation is required. Lin [16,26] proposed a new polarization voltage (NPV) model based on current and time, and the overall error of estimated accuracy of SOP was found to be approximately 5%. Liu [27] estimated SOP based on a fractional model, with a maximum error of 1.34% and high accuracy, but an insufficient adaptive ability.

The traditional SOP estimation method either has poor estimation accuracy or inadequate adaptive ability, and neither of them can be satisfied at the same time. Aiming at the degradation of EULB performance, therefore, an important research direction of battery SOP estimation is to improve the accuracy and adaptive ability of the algorithm.

SOE is an important parameter of the battery management system, which is the ratio of the remaining available energy to the maximum available energy. The performance degradation of EULB reduces the maximum available energy, thus reducing the accuracy of SOE estimation. However, few people have studied this topic in SOE estimation methods at present.

Owing to the above problems of EULB performance, in the present work an adaptive dual unscented Kalman filter (ADUKF) algorithm is first applied based on the SRCSEM. That is, based on the ADUKF algorithm used to estimate the SOE, ohmic internal resistance (OIR), and actual capacity (AC), the optimal estimation is used to predict the working voltage and OIR is used to estimate the EULB’s SOP. Finally, the estimation method of the EULB’s SOP is established. It can not only improve the estimation accuracy of SOP, but also enhance the self-adaptability of SOP estimation. It provides safety guarantees for the promotion and utilization of the EULB.

The objective of this paper is to propose an effective SOP estimation method for EULB and analyze the influence of the degradation of the performance of EULB. There are four original contributions as follows:

(1)

Owing to the degradation of EULB, the SOE initial value is not clear, thus parameter estimation of EULB is discussed.

(2)

Aiming at unclear SOE of EULB, UT is adopted and an adaptive method used, that can be adjusted adaptively to estimate the accuracy based on the observation noise equation γ_k, R_k and the processes noise equation q_k, Q_k.

(3)

To improve the accuracy of SOP estimation, UT and ADUKF are adopted to estimate the parameters of EULB based on the observation noise equation γ_θ,k, R_θ,k and the process noise equation q_θ,k, Q_θ,k aimed at performance attenuation of the EULB and the inaccuracy of AC and OIR.

(4)

It provides a theoretical basis for the effective utilization of the whole life cycle of a lithium-ion battery.

2. SRCSEM of EULB

Owing to the complex nonlinear system of the lithium-ion battery, in order to simulate the characteristics of the battery more accurately, it is necessary to select a higher-order battery equivalent model. The SRCSEM can not only better reflect the static and dynamic characteristics of EULB, but also can reduce the order of the model, which can reduce the calculation and makes it easy to apply to engineering [28,29,30].

As shown in Figure 1, the second-order resistor-capacitance symmetry model is used as the equivalent model of an EULB. ${\tau }_1=R_1{C}_1$ is the long time constant, which is used to simulate the process of rapid change of discharge voltage in the dynamic characteristics of EULB; ${\tau }_2=R_2{C}_2$ is the short time constant, which is used to simulate the process of slow and stable discharge voltage in the dynamic characteristics of EULB. In addition, $U_{oc}$ is the open circuit voltage (OCV) of EULB, $U_L$ is the working voltage of EULB, $i_L$ is the charge/discharge current of EULB, and $R_0$ is the OIR of the EULB.

According to Figure 1, the discrete state equation of the second-order resistor-capacitance symmetry equivalent circuit of the EULB is as follows:

$$\left[\begin{array}{c}{U}_{k+1}^{R_1{C}_1}\\ S_{ek+1}\\ U_{k+1}^{R_2{C}_2}\end{array}\right]=\left[\begin{array}{c}exp\left(\frac{-\Delta t}{{\tau }_1}\right)\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ exp\left(\frac{-\Delta t}{{\tau }_2}\right)\end{array}\right]·\left[\begin{array}{c}{U}_k^{R_1{C}_1}\\ S_{ek}\\ U_k^{R_2{C}_2}\end{array}\right]+\left[\begin{array}{c}{R}_1\left(1-exp\left(\frac{-\Delta t}{{\tau }_1}\right)\right)\\ \frac{-U_k\Delta t}{E}\\ R_2\left(1-exp\left(\frac{-\Delta t}{{\tau }_2}\right)\right)\end{array}\right]·i_k+{\omega }_k$$

(1)

According to Figure 1, the discrete observation equation of the second-order resistor-capacitance equivalent circuit of the EULB is as follows:

$$U_k=\left[\frac{d\left(U_{oc}\left(S_e\right)\right)}{d{S}_e}{|}_{S_e=S_{ek}}\right. -1 \left.-1\right]·\left[\begin{array}{c}{S}_{ek}\\ U_k^{R_1{C}_1}\\ U_k^{R_2{C}_2}\end{array}\right]-i_k{R}_0+{\vartheta }_k$$

(2)

As

$$\begin{gathered} A_k=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ exp\left(\frac{-\Delta t}{{\tau }_1}\right)\\ 0\end{array}\begin{array}{c}0\\ 0\\ exp\left(\frac{-\Delta t}{{\tau }_2}\right)\end{array}\right],B_k=\left[\begin{array}{c}\frac{-\Delta t}{E}\\ R_1\left(1-exp\left(\frac{-\Delta t}{{\tau }_1}\right)\right)\\ R_2\left(1-exp\left(\frac{-\Delta t}{{\tau }_2}\right)\right)\end{array}\right], x_k=\left[\begin{array}{c}{S}_{ek}\\ U_k^{R_1{C}_1}\\ U_k^{R_2{C}_2}\end{array}\right], \\ {C}_k=\left[\frac{d\left(U_{oc}\left(S_e\right)\right)}{d{S}_e}{|}_{S_e=S_{ek}}\right. -1 \left.-1\right],u_k=i_k. \end{gathered}$$

Thus

$$f\left(x_k,u_k\right)=A_k{x}_k+B_k{u}_k$$

(3)

$$g\left(x_k,u_k\right)=C_k{x}_k-R_{0,k}{u}_k$$

(4)

where, $S_{ek}, S_{ek+1}$ are the SOE of the EULB at k, k + 1 time in discrete state; $E$ is the energy of the EULB; $i_k$ is the charge/discharge current at k time in discrete state; $U_k$ is the working voltage of the EULB at k time in discrete state; $\Delta t$ is the sampling period; $U_k^{R_1{C}_1}, U_{k+1}^{R_1{C}_1}$ are the estimated voltage values of $R_1$ at k, k + 1 time in discrete state; $U_k^{R_2{C}_2}, U_{k+1}^{R_2{C}_2}$ are the estimated voltage values of $R_2$ at k, k + 1 time in discrete state; ${\omega }_k, {\vartheta }_k$ are independent system noises, ${\omega }_k$ is a 3-dimensional vector quantity; and $U_{oc}\left(S_e\right)$ is the OCV of the EULB corresponding to the SOE value of the EULB at the k time in discrete state.

3. Model Parameter Identification

3.1. OCV

The OCV $U_{OC}$:

(1)

Charging current is 36 A and cut-off voltage is 3.65 V.

(2)

Discharging current is 36 A and cut-off voltage is 2.5 V. Record the discharge voltage $U_{disarge}$.

(3)

Charging current is 36 A and cut-off voltage is 3.65 V. Record the charge voltage $U_{disarge}$.

(4)

The OCV U_OC is as follows:

$$U_{OC}=\frac{U_{discharge}+U_{charge}}{2}$$

(5)

3.2. OIR

Based on Ohm’s law, the OIR $R_0$ is as follows:

$$R_0=\frac{\Delta U}{\left|i_L\right|}$$

(6)

where $i_L$ is the charge/discharge current of EULB and $\Delta U$ is voltage change [31,32].

3.3. Polarization Resistance

The working voltage $U_L$, charge/discharge current $i_L$ and OCV $U_{OC}$ of the EULB is collected through the charge/discharge test. Model parameter identification based on the least square method is not repeated in this paper because the method is described in detail in references [28,32].

4. SOP Estimation

In this paper, an ADUKF algorithm is applied based on the SRCSEM: the first is to estimate the SOE based on the AUKF algorithm; the second is to estimate the OIR and AC based on the AUKF algorithm [33,34].

4.1. UT

UT is the core technology of unscented Kalman filter (UKF). The idea of UT is to approximate the distribution of a probability density function through a set of carefully selected sample points and the corresponding weights of sample points. A certain number of sigma points are selected from the prior distribution according to a certain strategy, and nonlinear transformation is performed on each sigma point to obtain the corresponding transformed sampling points. In addition, the posterior mean and variance are calculated by weighting these sample points. The sample points obtained by UT are not linearized, nor are higher-order terms lost, thus the accuracy of UT is higher than that of the extended Kalman filter algorithm.

For any nonlinear function $y=\mathrm{g}\left(x\right)$, around the n-dimensional state variable $x$ with mean value of $\overline{x}$ and variance of $P_x$, 2n + 1 sigma points and corresponding weights can be selected through the following UT process.

$$\begin{array}{ll}{\chi }_i=\overline{x}& i=0\\ {\chi }_i=\overline{x}+{\left(\sqrt{\left(n+\lambda \right)P_x}\right)}_i& i=1,\dots ,n\\ {\chi }_i=\overline{x}-{\left(\sqrt{\left(n+\lambda \right)P_x}\right)}_{i-n}\hspace{1em}\hspace{1em}& i=n+1,\dots ,2n\\ {\omega }_0^m={\omega }_0^c=\frac{\lambda }{n + \lambda }& i=0\\ {\omega }_i^m={\omega }_i^c=\frac{\lambda }{2n + \lambda }& i=1,\dots ,2n\end{array}$$

(7)

In the above formula, ${\left(\sqrt{\left(n+\lambda \right)P_x}\right)}_i$ is the $i$ column or row of the square root of $\left(n+\lambda \right)P_x$. $\lambda$ is the scale factor used to represent the range of sampling points around equilibrium value points, and ${\omega }_i^m$ is the weight of corresponding sample points. In this paper, $\lambda =0.1, i=2$.

From ${\chi }_i$, the sample point with nonlinear transformation can be obtained:

$$y_i=f\left({\chi }_i\right)$$

(8)

Then, statistical characteristics of output sampling points can be obtained by weighted approximation:

$$y={{\displaystyle \sum }}_i^{2n}{\omega }_i^m{y}_i$$

(9)

$$P_y={{\displaystyle \sum }}_i^{2n}{\omega }_i^c\left(y_i-y\right){\left(y_i-y\right)}^T$$

(10)

4.2. SOE Estimation Based on ADUKF

According to Equations (3) and (4), the variable of the echelon-use battery system is the SOE. In this paper, the ohmic resistance and actual capacity of the echelon-use battery are added into the state variable due to the serious problems of the increase of ohmic resistance and the decline of the actual capacity of the echelon-use battery. The system state variable has three parameters: SOE, ohm internal resistance, and the actual capacity [35,36].

The state and observation equations are as follows:

$$x_{k+1}=f\left(x_k,u_k,{\theta }_k\right)+q_k$$

(11)

$$y_{k+1}=g\left(x_k,u_k,{\theta }_k\right)+{\gamma }_k$$

(12)

where, ${\theta }_k$ is the state variable ohmic internal resistance and actual capacity,${\theta }_k=\left[R_{0,k},C_k\right]$, to reduce the amount of calculation, whenever ${{\displaystyle \sum }}_{k=1}^n(C_k-C_{k-1})>0.72$, that is, the reduction per $C$ exceeds 0.72 Ah, the values of $E,R_0$ in Equations (1) and (2) are updated, $E=C_n, R_0=R_{0,n}$; $x_k$ is the system state variable; $u_k$ is the input of the system and is the echelon-use battery current; and $y_k$ is the system observation variable and is the working voltage of the echelon-use battery.

In this paper, the error covariance matrices of the zero-mean Gaussian white noise $q_k\wedge {\gamma }_k$ are $Q_k\wedge R_k$.

The algorithm flow is as follows:

• Step 1: Inialize $x$ is as follows:

  $${\widehat{x}}_0=E\left(x_0\right).$$

  (13)

  $${\widehat{P}}_0=E[\left(x_0-{\widehat{x}}_0\right){\left(x_0-{\widehat{x}}_0\right)}^T$$

  (14)
• Step 2: Generating sigma points:

  $${\chi }_i=\left[\overline{x},\overline{x}+{\left(\sqrt{\left(n+\lambda \right)P_x}\right)}_i,\overline{x}-{\left(\sqrt{\left(n+\lambda \right)P_x}\right)}_{i-n}\right]$$

  (15)
• Step 3: Time update of the system state $x$ is as follows

  $${\chi }_{i.k}=f\left({\chi }_{i.k-1}\right)$$

  (16)

  $$x_k={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^m{\chi }_{i.k}+q_k$$

  (17)

  $$P_k={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^c\left[{\chi }_{i.k}-x_k\right]{\left[{\chi }_{i.k}-x_k\right]}^T+Q_k$$

  (18)

  $$y_{i,k}=g\left({\chi }_{i.k-1}\right)$$

  (19)

  $$y_k={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^m{y}_{i.k}+{\gamma }_k$$

  (20)

  $$P_{y,k}={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^c\left[y_{i.k}-y_k\right]{\left[y_{i.k}-y_k\right]}^T+R_k$$

  (21)

  $$P_{xy,k}={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^c\left[{\chi }_{i.k}-x_k\right]{\left[y_{i.k}-y_k\right]}^T$$

  (22)
• Step 4: Status update of the system state $x$ is as follows:
  The Kalman gain is as follows:

  $$K_k=P_{xy,k}{P}_{y,k}^{-1}$$

  (23)
  The optimal estimation of state variables is as follows:

  $${\widehat{x}}_k=x_k+K_k\left[y_k-{\widehat{y}}_k\right]$$

  (24)
  The optimal estimate of the covariance is as follows:

  $${\widehat{P}}_k=P_k-K_k{P}_{y,k}{K}_k^T$$

  (25)
• Step 5: Process noise mean and covariance equation is as follows:

  $$q_k=\left(1-d_k\right)q_{k-1}+d_k\left[{\widehat{x}}_k-{{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^{m}f\left({\chi }_{i.k-1},u_k,{\theta }_k\right)\right]$$

  (26)

  $$Q_k=\left(1-d_k\right)Q_{k-1}+d_k\left[K_k({\widehat{y}}_k-y_k\right){({\widehat{y}}_k-y_k)}^T{K}_k^T+P_k-A_{k-1}{\widehat{P}}_{k-1}{A}_{k-1}^T]$$

  (27)
• Step 6: Observe the noise mean and covariance equation as follows:

  $${\gamma }_k=\left(1-d_k\right){\gamma }_{k-1}+d_k\left[y_k-{{\displaystyle \sum }}_{i=0}^{2n}{\omega }_i^{m}g\left({\chi }_{i.k-1}\right)\right]$$

  (28)

  $$R_k=\left(1-d_k\right)R_{k-1}+d_k\left[({\widehat{y}}_k-y_k\right){({\widehat{y}}_k-y_k)}^T-C_k{P}_k{C}_k^T]$$

  (29)

  where $d_k=\frac{1-b}{1-b^k}$, b is the forgetting factor of x, 0 < b < 1, $k=1,2,\cdots ,n$; $x_k$ and ${\widehat{x}}_k$ are the estimation and optimal estimation of the state variable at k time, respectively; $y_k$ and ${\widehat{y}}_k$ are the estimated value and actual observation value at k time; and $P_k$ and ${\widehat{P}}_k$ are, respectively, the estimation and optimal estimation of the error covariance at k time.

4.3. OIR and AC Estimation Based on AUKF

The state and observation formulas of the system with the newly added state parameters:

$${\theta }_{k+1}={\theta }_k+q_{\theta ,k}$$

(30)

$$D_{k+1}=g\left(x_k,u_k,{\theta }_k\right)+{\gamma }_{\theta ,k}$$

(31)

where $q_{\theta ,k}$ is the noise on the input variable, and it is the zero-mean Gaussian white noise; ${\gamma }_{\theta ,k}$ is the noise on the output variable, and it is the zero-mean Gaussian white noise; the error covariance matrices of $q_{\theta ,k} \mathrm{and} {\gamma }_{\theta ,k}$ are $Q_{\theta ,k} \mathrm{and} R_{\theta ,k}$; and the state variable $\theta$ is estimated based on the AUKF algorithm, and the estimated values of the EULB’s OIR and AC are calculated. In order to improve the accuracy, the error between the actual value and the estimated value of the working voltage is optimized [37,38,39,40,41].

In this paper, the error covariance matrices of the zero-mean Gaussian white noise $q_{\theta ,k}\wedge {\gamma }_{\theta ,k}$ are $Q_{\theta ,k}\wedge R_{\theta ,k}$.

The algorithm flow is as follows:

• Step 1: Initialize $\theta$ as follows:

  $${\widehat{\theta }}_0=E\left({\theta }_0\right)$$

  (32)

  $$P_{\theta ,0}=E\left[\left(x_0-{\widehat{x}}_0\right){\left(x_0-{\widehat{x}}_0\right)}^T\right]$$

  (33)
• Step 2: Generating sigma points:

  $${\chi }_{\theta ,i}=\left[\overline{x},\overline{x}+{\left(\sqrt{\left(n+\lambda \right)P_{\theta }}\right)}_i,\overline{x}-{\left(\sqrt{\left(n+\lambda \right)P_{\theta }}\right)}_{i-n}\right]$$

  (34)
• Step 3: Time update of the system state $\theta$ is as follows:

  $${\chi }_{\theta i.k}=f\left({\chi }_{\theta i.k-1}\right)$$

  (35)

  $${\theta }_k={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_{\theta i}^m{\chi }_{\theta i.k}+q_{\theta ,k}$$

  (36)

  $$P_{\theta ,k}={\widehat{P}}_{\theta ,k-1}+Q_{\theta ,k}$$

  (37)

  $$y_{\theta i,k}=g\left({\chi }_{\theta i.k-1}\right)$$

  (38)

  $$y_{\theta ,k}={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_{\theta i}^m{y}_{\theta i.k}+{\gamma }_{\theta ,k}$$

  (39)

  $$P_{\theta y,k}={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_{\theta i}^c\left[y_{\theta i.k}-y_{\theta ,k}\right]{\left[y_{\theta i.k}-y_{\theta ,k}\right]}^T+R_{\theta ,k}$$

  (40)

  $$P_{\theta xy,k}={{\displaystyle \sum }}_{i=0}^{2n}{\omega }_{\theta i}^c\left[{\chi }_{\theta i.k}-{\theta }_k\right]{\left[y_{\theta i.k}-y_{\theta ,k}\right]}^T$$

  (41)
• Step 4: Status update of the system state $\theta$ is as follows:
  The Kalman gain is as follows:

  $$K_{\theta ,k}=P_{\theta xy,k}{P}_{\theta y,k}^{-1}$$

  (42)
  The optimal estimation of state variables is as follows:

  $${\widehat{\theta }}_k={\theta }_k+K_{\theta ,k}\left[y_k-{\widehat{y}}_k\right]$$

  (43)
  The optimal estimate of the covariance is as follows:

  $${\widehat{P}}_{\theta ,k}=P_{\theta ,k}-K_{\theta ,k}{P}_{\theta y,k}{K}_{\theta k}^T$$

  (44)
• Step 5: Process noise mean and covariance equation is as follows:

  $$q_{\theta ,k}=\left(1-d_{\theta ,k}\right)q_{\theta ,k-1}+d_{\theta ,k}\left[{\widehat{\theta }}_k-{{\displaystyle \sum }}_{i=0}^{2n}{\omega }_{\theta i}^{m}f\left({\chi }_{\theta i.k-1},u_k,{\theta }_k\right)\right]$$

  (45)

  $$Q_{\theta ,k}=\left(1-d_{\theta ,k}\right)Q_{\theta ,k-1}+d_{\theta ,k}\left[K_{\theta ,k}({\widehat{y}}_k-y_k\right){({\widehat{y}}_k-y_k)}^{\mathrm{T}}{K}_{\theta ,k}^{\mathrm{T}}+P_{\theta ,k}-A_{k-1}{\widehat{P}}_{\theta ,k-1}{A}_{k-1}^{\mathrm{T}}]$$

  (46)
• Step 6: Observe the noise mean and covariance equation as follows:

  $${\gamma }_{\theta ,k}=\left(1-d_{\theta ,k}\right){\gamma }_{\theta ,k-1}+d_{\theta ,k}\left[y_{\theta ,k}-{{\displaystyle \sum }}_{i=0}^{2n}{\omega }_{\theta i}^{m}g\left({\chi }_{\theta i.k-1}\right)\right]$$

  (47)

  $$R_{\theta ,k}=\left(1-d_{\theta ,k}\right)R_{\theta ,k-1}+d_{\theta ,k}\left[({\widehat{y}}_k-y_k\right){({\widehat{y}}_k-y_k)}^{\mathrm{T}}-C_k{P}_{\theta ,k}{C}_k^{\mathrm{T}}]$$

  (48)

  where $d_{\theta ,k}=\frac{1-b_{\theta }}{1-b_{\theta }^k}$, $b_{\theta }$ is the forgetting factor of $\theta$, 0 < $b_{\theta }$ < 1, $k=1,2,\cdots ,n$.

4.4. SOP Estimation Based on AUKF

The SOP optimal estimate forecast is updated as follows:

The optimal estimation and prediction working voltage at time k + 1:

$${\widehat{U}}_{k+1}=g\left({\widehat{x}}_k,u_k,{\widehat{\theta }}_k\right)=\left[\frac{d\left(U_{oc}\left(S_e\right)\right)}{d{S}_e}{|}_{S_e=S_{ek}}\right. -1 \left.-1\right]·\left[\begin{array}{c}{S}_{ek}\\ U_k^{R_1{C}_1}\\ U_k^{R_2{C}_2}\end{array}\right]-i_k{R}_{0,k}$$

(49)

where $g\left({\widehat{x}}_{k-1},u_{k-1},{\widehat{\theta }}_{k-1}\right)$ is the estimated value of the observed variable at k time of sampling.

The optimal estimation and prediction SOP at time k + 1:

$$P_{k+1}=\frac{{\widehat{U}}_{k+1}^2}{R_{0,k+1}+R_1+R_2}$$

(50)

The effect of polarization resistance is very small, so it can be ignored. That is:

$$P_{SOP}=\frac{{\widehat{U}}_{k+1}^2}{R_{0,k+1}}$$

(51)

4.5. The Flow Chart of SOP Estimation Based on ADUKF

The flow chart of the SOP estimation of an echelon-use battery is shown in Figure 2.

5. Simulation and Discussion

5.1. SOE Estimation Based on ADUKF

According to Figure 3, an EULB (as shown in

Table 1. Parameters of the EULB.

Items	Parameter	Remarks
capacity	72 Ah	72 A
nominal voltage	3.2 V
working voltage	2.5–3.65 V
charging time	3 h	24 A
charging temperature	0–45 °C
discharging temperature	−20–55 °C

) was selected to conduct the charge/discharge experiment using test equipment (BTS20, 5V/4*300A/WD, Hubei Techpow Electric Co., Ltd., Xiangfan, China) at room temperature. In order to simulate the working conditions, the fully charged EULB was discharged several times. Different discharge currents were used each time for simulation verification and analysis based on MATLAB R2021a (MathWorks, Natick, MA, USA).

In order to verify the adaptive characteristics of the ADUKF algorithm, a test experiment was carried out on a fully charged EULB, starting from SOE of 100% and ending at SOE of 25%. In the paper, selected battery symmetry capacity decays to 80%, 60%, 40%, and 20% separately of rated capacity, and the initial SOE values were changed to 70% and 40% separately, and the adaptive and error curves were observed and analyzed.

In the process of simulation verification, the estimated values of SOE, working voltage, and SOP were calculated based on the ADUKF algorithm, and the actual values were acquired by the test equipment (BTS20).

The formula for error is as follows:

According to Equations (1), (2) and (24), ${\widehat{x}}_k=\left[\begin{array}{c}{\widehat{S}}_{ek}\\ {\widehat{U}}_k^{R_1{C}_1}\\ {\widehat{U}}_k^{R_2{C}_2}\end{array}\right]$, ${\widehat{S}}_{ek}$ is the optimal value of SOE.

The SOE error equation is as follows:

$$SOE error={\widehat{S}}_{ek}-S_{actual}$$

(52)

where, $S_{actual}$ is the value collected by the device.

According to Equation (29), the optimal estimation of the working voltage is ${\widehat{U}}_{k+1}$ at time k + 1.

The working voltage error equation is as follows:

$$Working voltage error={\widehat{U}}_{k+1}-U_{actual}$$

(53)

where $U_{actual}$ is the value acquired by the test equipment.

According to Equation (31), the optimal estimation SOP is $P_{SOP}$ at time k + 1.

The SOP error equation is as follows:

$$SOP error=P_{SOP}-P_{actual}$$

(54)

where, $P_{actual}$ is the value acquired by the test equipment.

5.2. Capacity Decays to 80% and SOE Starts at 70%

The simulation verification curve when capacity decays to 80% and SOE starts at 70% is shown in Figure 4.

SOE estimation curves of EULB are shown in Figure 4a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in

Table 2. Estimation error of the EULB’s parameters.

Estimation Error		SOE	Working Voltage	SOP
capacity decays to 80%	SOE = 70%	−2.29% to 2.34%	4.28%	3.25%
	SOE = 40%	−2.31% to 2.34%	4.28%	3.26%
capacity decays to 60%	SOE = 70%	−2.23% to 2.35%	4.28%	3.26%
	SOE = 40%	−2.25% to 2.35%	4.29%	3.26%
capacity decays to 40%	SOE = 70%	−2.36% to 2.34%	4.32%	3.27%
	SOE = 40%	−2.38% to 2.34%	4.33%	3.28%
capacity decays to 20%	SOE = 70%	−2.37% to 2.34%	4.32%	3.28%
	SOE = 40%	−2.39% to 2.34%	4.33%	3.28%

, the SOE error is −2.29% to 2.34%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of EULB are shown in Figure 4b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1369 V (within 4.28%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of EULB are shown in Figure 4c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.49 W (within 3.25%), and SOP estimation is adaptive through curve comparison.

5.3. Capacity Decays to 80% and SOE Starts at 40%

The simulation verification curve when capacity decays to 80% and SOE starts at 40% is shown in Figure 5.

SOE estimation curves of the EULB are shown in Figure 5a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.31% to 2.34%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of the EULB are shown in Figure 5b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1370 V (within 4.28%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 5c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.50 W (within 3.26%), and SOP estimation is adaptive through curve comparison.

5.4. Capacity Decays to 60% and SOE Starts at 70%

The simulation verification curve when capacity decays to 60% and SOE starts at 70% in Figure 6.

SOE estimation curves of the EULB are shown in Figure 6a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.23% to 2.35%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of EULB are shown in Figure 6b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1371 V (within 4.28%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 6c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.50 W (within 3.26%), and SOP estimation is adaptive through curve comparison.

5.5. Capacity Decays to 60% and SOE Starts at 40%

The simulation verification curve when capacity decays to 60% and SOE starts at 40% in Figure 7.

SOE estimation curves of the EULB are shown in Figure 7a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.25% to 2.35%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of the EULB are shown in Figure 7b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1374 V (within 4.29%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 7c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.51 W (within 3.26%), and SOP estimation is adaptive through curve comparison.

5.6. Capacity Decays to 40% and SOE Starts at 70%

The simulation verification curve when capacity decays to 40% and SOE starts at 70% is shown in Figure 8.

SOE estimation curves of the EULB are shown in Figure 8a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.36% to 2.34%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of the EULB are shown in Figure 8b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1381 V (within 4.32%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 8c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.54 W (within 3.27%), and SOP estimation is adaptive through curve comparison.

5.7. Capacity Decays to 40% and SOE Starts at 40%

The simulation verification curve when capacity decays to 40% and SOE starts at 40% is shown in Figure 9.

SOE estimation curves of the EULB are shown in Figure 9a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.38% to 2.34%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of the EULB are shown in Figure 9b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1384 V (within 4.33%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 9c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.55 W (within 3.28%), and SOP estimation is adaptive through curve comparison.

5.8. Capacity Decays to 20% and SOE Starts at 70%

The simulation verification curve when capacity decays to 20% and SOE starts at 70% is shown in Figure 10.

SOE estimation curves of the EULB are shown in Figure 10a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.37% to 2.34%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of the EULB are shown in Figure 10b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1385 V (within 4.33%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 10c, the bottom graph is the SOP error curve and the top graph is the SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.56 W (within 3.28%), and SOP estimation is adaptive through curve comparison.

5.9. Capacity Decays to 20% and SOE Starts at 40%

The simulation verification curve when capacity decays to 20% and SOE starts at 40% is shown in Figure 11.

SOE estimation curves of the EULB are shown in Figure 11a, the bottom graph is the SOE error curve and the top graph is the SOE adaptive estimation curve. As shown in Table 2, the SOE error is −2.39% to 2.34%, and SOE estimation is adaptive through curve comparison.

The working voltage estimation curves of the EULB are shown in Figure 11b, the bottom graph is the working voltage error curve and the top graph is the working voltage adaptive estimation curve. As shown in Table 2, the working voltage error is 0.1386 V (within 4.33%), and the working voltage estimation is adaptive through curve comparison.

SOP estimation curves of the EULB are shown in Figure 11c, the bottom graph is the SOP error curve and the top graph is SOP adaptive estimation curve. As shown in Table 2, the SOP error is 7.56 W (within 3.28%), and SOP estimation is adaptive through curve comparison.

Simulation results show that the estimation accuracy error of the EULB’s SOE is less than 2.38%, that the EULB working voltage is less than 4.33%, and that the EULB’s SOP is less than 3.28%. The method is adaptive regardless of whether the initial SOE value is consistent with the actual value.

6. Conclusions

In this paper, the SRCSEM of an EULB is established first, and then the SOP estimation method of such a battery is established based on the ADUKF algorithm. Simulation shows that, at the same degradation level, the greater the initial SOE error, the greater the error of SOP estimation, the more serious the degradation level of the battery, and the greater the error of estimation of SOE, operating voltage, and SOP. Simulation results show that the estimation accuracy error of the EULB’s SOP is less than 3.28%. The accuracy is higher than reference [23] and lower than reference [24]; thus, the method has high accuracy. Furthermore, regardless of whether the initial value of the SOE is clear or not, the ADUKF algorithm can be adjusted to an adaptive algorithm.

Therefore, on the basis of ensuring estimation accuracy, the adaptive ability of the SOP estimation method is enhanced.