Joint Estimation of SOC and SOH for Single-Flow Zinc–Nickel Batteries

Abstract

The single-flow zinc–nickel battery (ZNB) is a new type of flow battery with a simple structure, large-scale energy storage, and low cost, and thus has attracted much attention in the battery field recently. The state of charge (SOC) and state of health (SOH) are key indicators of the battery, and their inaccurate estimation can damage the battery. However, little has been done so far to study how to jointly estimate SOC and SOH for the ZNB. In this paper, the method of adaptive IDUKF is proposed. A second-order equivalent circuit model is applied to improve the accuracy. At the same time, the double unscented Kalman filter (DUKF), which is optimized by the improved Harris hawk optimization (IHHO) algorithm, is used to estimate the SOC and parameters online. Then, the capacity update model is introduced to simulate the change in SOH. Finally, the proposed method is applied to a 16 Ah ZNB, and the experimental results confirm the validity of the proposed method.

1. Introduction

The redox flow battery (RFB) is widely applied in many fields, such as large-scale energy storage systems and electric vehicles [1], the reasons being its simple structure, large-scale energy storage, and long cycle life. It is promising that the all-vanadium redox flow battery (VRB) has become the latest grid-tied energy storage system [2]. Nevertheless, with the development of VRB, the high price of the ion exchange membrane is unignorable.

In order to deal with the problem of high cost, the single-flow zinc–nickel battery (ZNB) without a membrane has been proposed by Cheng et al. in 2007 [3]. Meanwhile, the electrolyte is stored in a single reservoir and only a single circulation pump is used for one-sided circulation, which eliminates the problem of cross-contamination of the solution and also improves the energy efficiency of the battery to a certain extent. However, the capacity of a conventional ZNB will decay rapidly because of the zinc deposition that happens in a zinc anode [4]. Therefore, regular capacity monitoring is necessary to determine the maintenance time [5]. A well-conditioned battery management system (BMS) can optimize the applied current and reduce the effect of polarization. In particular, battery modelling and states estimation are key indicators of BMS. The common modeling methods currently used in the battery field are the electrochemical models, equivalent circuit model (ECM) [6], and neural network model. ECM is used in this work because of real-time performance and accuracy [7]. Moreover, state of charge (SOC) and state of health (SOH) are key indicators of the state estimation. So far, the methods of SOC estimation are Coulomb counting (CC), the open circuit voltage method (OCC), the AC impedance method (ACI), the data-driven method (DD), and the ECM-based SOC observer. Among them, the CC error will accumulate over time with an incorrect initial value, and then the OCC looks up the UOC-SOC table to estimate the SOC but cannot be applied to real-time estimation. Next, similar to OCC, the online performance of ACI is terrible, and DD need no specific mathematical model but intensively relies on the quality and quantity of the sample. Finally, the ECM-based SOC observer is widely used in real-time applications [8,9], which applies a suitable filter to the estimation of the internal states. The details are as follows: recursive least squares filter (RLS), particle filter (PF), Kalman filter (KF), and various improved methods [10,11,12,13,14,15]. Furthermore, the methods of SOH estimation are as follows: the method of equivalent DC resistance (EDCR), which estimates SOH by establishing the relation of SOH and DC resistance [16]; and then the SOH estimation based on the group method of data handling, as proposed in [17], which achieves the estimated targets by using a neural network. Chen, Z et al. studied SOC and SOH estimation of Li-ion Batteries, which is based on adaptive square-root unscented Kalman filters [18]. However, little has been done so far to study how to jointly estimate SOC and SOH for the ZNB. In this paper, a second-order ECM is used to simulate the dynamic behavior of the ZNB, and the improved Harris hawk optimization (IHHO) algorithm is applied to the offline identification of the RC parameters. Meanwhile, the double unscented Kalman filter (DUKF) optimized by IHHO is applied to the online estimation of SOC and the parameters. Finally, the capacity update model is introduced to simulate the change in SOH, which improves the accuracy of SOC estimation, and then the joint estimation of SOC and SOH is successfully completed.

The rest of this paper is organized as follows. In Section 2, the 2-RC equivalent circuit is applied to the battery model, and the RC parameters are identified precisely with the help of the above model. The experimental settings of the CCPC tests are also displayed in Section 2. The improvement steps of the IHHO are introduced in Section 3, and the comparison results with other typical algorithms are presented in Section 3. The proposed scheme and experimental results are displayed and analyzed in Section 4. The experimental settings of the VCCC tests are also introduced in Section 4. The conclusion is given in Section 5.

2. Battery State Space Model

2.1. Battery Modeling

In order to accurately simulate the dynamic operating characteristics of the battery, it is necessary to build an accurate battery equivalent circuit model. A second-order RC equivalent circuit model (ECM) has been widely used so far, having a simple structure, high accuracy, and low computational complexity. It also facilitates battery characterization and parameter identification [19], and a second-order ECM is thus used in this work. The circuit form of the model is shown in Figure 1. The ohmic resistance R0 stands for the resistant losses of the electrodes and electrolytes. The two RC networks present the terminal voltage dynamic response [20].

The equivalent circuit model of the cell is described by the following equation:

$$\left\{\begin{array}{l}I=C_1\frac{d{U}_1}{dt}+\frac{U_1}{R_1}\\ I=C_2\frac{d{U}_2}{dt}+\frac{U_2}{R_2}\\ U_{cell}=U_{oc}-U_1-U_2-R_0\cdot I\end{array}\right.$$

(1)

where U_oc is the open-circuit voltage; U_cell is the battery voltage; I is the battery current; U₁ is the voltage across the R₁C₁ network; U₂ is the voltage across the R₂C₂ network; R₁, C₁, R₂, and C₂ are the activation polarization resistance, activation polarization capacitance, concentration polarization resistance, and concentration polarization capacitance, respectively; and R₀ is the internal resistance.

A widely accepted SOC recursion model is given by [21]:

$$SOC(t)=SOC(t_0)-{\displaystyle {\int }_{t_0}^t\frac{\eta \cdot I(t)}{C_n}}dt$$

(2)

where η is the Coulomb efficiency factor and C_n is the calibrated capacity of the cell.

Combining Equations (1) and (2), the state–space model of the cell can be obtained as follows:

$$\left\{\begin{array}{ll}& \left[\begin{array}{l}\stackrel{\wedge }{U_1}(k+1)\\ \begin{array}{c}\stackrel{\wedge }{U_2}(k+1)\hfill \\ \stackrel{\wedge }{SOC}(k+1)\hfill \end{array}\end{array}\right]=\left[\begin{array}{cc}{e}^{-T_s/{\tau }_1}\hfill & 0\\ \begin{array}{c}0\hfill \\ 0\hfill \end{array}\hfill & \begin{array}{c}{e}^{-T{}_s/{\tau }_2}\hfill \\ 0\hfill \end{array}\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]·\left[\begin{array}{c}{U}_1\left(k\right)\hfill \\ \begin{array}{c}{U}_2\left(k\right)\hfill \\ SOC\left(k\right)\hfill \end{array}\hfill \end{array}\right]+\left[\begin{array}{c}{R}_1(1-e^{-T_s/{\tau }_1})\hfill \\ \begin{array}{c}{R}_2(1-e^{-T_s/{\tau }_2})\hfill \\ {}^{-\eta }/{}_{C_n}\hfill \end{array}\hfill \end{array}\right]·I\left(k\right)+\left[\begin{array}{c}{w}_1\left(k\right)\\ \begin{array}{c}{w}_2\left(k\right)\\ w_3\left(k\right)\hfill \end{array}\end{array}\right]\\ & \stackrel{\wedge }{U_{cell}}\left(k\right)=U_{oc}\left[\stackrel{\wedge }{SOC}\left(k\right)\right]-\stackrel{\wedge }{U_1}\left(k\right)-\stackrel{\wedge }{U_2}\left(k\right)-R_0·I\left(k\right)+v\left(k\right)\hfill \end{array}\right.$$

(3)

where T_s is the sampling time; ${\tau }_1=R_1{C}_1,{\tau }_2=R_2{C}_2$; $I(k)$ is the input; $\stackrel{\wedge }{U_{cell}}(k)$ is the output; $U_1(k),U_2(k),SOC(k)$ are the state; $w_i(k) (\mathrm{i}=1,2,3)$ is the state noise; and $v(k)$ is the measurement noise.

2.2. Model Parameter Recognition

The experimental object is a third-generation zinc–nickel single-flow battery with a capacity of 16 Ah. The thickness of the porous positive electrode and negative electrode are 0.64 mm and 0.1 mm, respectively. The electrolyte is paired with a 20 mol/L KOH + 2 mol/L ZnO + 0.5 mol/L LiOH solution, and the volume of the electrolyte added in the electrolyte tank is about 10 L. The battery charging and discharging equipment is the CT-3004-5V200A-NTFA battery test platform produced by Newware Electronics Co., Ltd., Shenzhen, China. The experimental platform is displayed in Figure 2.

Parameters to be identified in this model are R₀, R₁, C₁, R₂, C₂, and U_oc. A constant current pulse condition is used in this study. The battery is charged with 1 C (16 A) to achieve the capacity of 16 Ah, and then a 0.75 C (12 A) constant discharging current pulse is loaded to the battery over every 5% SOC drop interval. A relaxation period of 20 min is set between two discharging intervals to allow for the equilibrium. Certainly, when the terminal voltage drops to the lower cut-off value 1.2 V, the battery stack is treated as fully discharged. The specific pulse process is shown in Figure 3. The blue line represents the voltage, and the red line stands for the current.

The U_oc-SOC look-up table can be obtained by the pulse data above, and determined as a sixth-order polynomial expression. The fitted curve is shown in Figure 4.

$$U_{oc}=h(SOC)={\displaystyle \sum _{i=0}^{n=6}{p}_{i}SO{C}_i}$$

(4)

where $p_i$ are the polynomial coefficients to fit the nonlinear correlation.

The ohmic internal resistance of the battery can be calculated from the data of voltage dip at the moment of power-on or voltage surge at the moment of power-off, corresponding to the voltage between points A and B and C and D. The process B to C represents the zero state response and the DE segment represents the zero input response. The identification process is shown in Figure 5.

$$\left\{\begin{array}{l}\Delta U_1=U_A-U_B,\Delta U_2=U_D-U_C\\ R_0=\frac{(\Delta U_1+\Delta U_2)}{2I}\end{array}\right.$$

(5)

where $\Delta U_1$ stands for voltage dip, $\Delta U_2$ denotes voltage surge, and $I$ is the terminal current.

The swarm intelligence optimization algorithm can optimize the parameters to minimize the objective function. The improved Harris hawk optimization (IHHO) algorithm is used in this paper, and the details are described in Section 3. $R_1$, $C_1$, $R_2$, $C_2$ can be obtained by Equation (6), combined with Equation (3). Certainly, the smaller the value of ff, the more accurate the RC parameters. The offline identification results are shown in Figure 6.

$$ff={\displaystyle \sum _{k=1}^L{(\stackrel{\wedge }{U_{cell}}(k)-U_{cell}(k))}^2}$$

(6)

where $U_{cell}(k)$ is the voltage measurement value; $\stackrel{\wedge }{U_{cell}}(k)$ is the estimation derived from Equation (3); and ff is the objective function value.

As we can see from the chart above, $R_0$, $R_1$, $R_2$ gradually decrease as the discharge process proceeds, and $C_1$, $C_2$ show an overall increasing trend. The parameter changes can respond well to the polarization process.

2.3. Model Validation

The initial RC parameters are set to the value of SOC = 1, and the voltage prediction curve is shown in Figure 7. The error curve is shown in Figure 8.

The error gradually becomes larger as the discharge process proceeds, which reaches the maximum at the end of discharge process. It is evident that the error keeps within 0.08, except for the last two stages. The error of the end originates from the violent electrochemical reaction at the late stage of discharge process, so the method of online parameters estimation is used to reduce error in this paper. The details are described in Section 4.

3. Improved Harris Hawk Optimization Algorithm

3.1. Standard HHO

HHO is a heuristic algorithm proposed by Heidari [22] in 2019, which originates from the chase and escape behavior between Harris hawks and their prey (rabbits). The HHO is widely used because of its relatively simple model and strong search capability. The basic optimization search principle is as follows.

(1)

Exploration phase. The Harris hawk is in a waiting mode during this phase; it is based on two strategies to find prey in random places, and iterates with probability $q$. The expression is shown in Equation (7).

$$X_{t+1}=\left\{\begin{array}{l}{X}_{rand}-r_1|X_{rand,t}-2r_2{X}_t|,q>=0.5\\ (X_{rabbit,t}-X_{m,t})-r_3(lb+r_4(ub-lb)),q<0.5\end{array}\right.$$

(7)

where $X_t$, $X_{t+1}$ stands for the position at the t and t + 1th iteration, respectively; $X_{rabbit,t}$ stands for the position of prey at tth; $q$, $r_1$, $r_2$, $r_3$, $r_4$ are random numbers in the interval (0,1); and $lb$, $ub$ are the lower and upper borders of space. $X_{rand,t}$ is the random position, and $X_{m,t}$ is mean position at tth, which is described as follows.

$$X_{m,t}=\frac{1}{N}{\displaystyle \sum _{i=1}^N{X}_{i,t}}$$

(8)

(2)

Transition from exploration to exploitation. The HHO algorithm can transfer from exploration to exploitation through the energy equation of the prey. The model is described as follows.

$$E=2E_0(1-\frac{t}{T})$$

(9)

where $E$ indicates the escaping energy of the prey; $E_0$ is the initial value of energy, which is modeled as Equation (10); and T and t are the maximum number of iterations and current value.

(3)

Exploitation phase. Depending on the probability of escape of its prey, the Harris hawk performs four types of attacks. The following r is a random number indicating the probability of escape.

Case 1: Soft besiege

When r $\ge$ 0.5, $E$ $\ge$ 0.5, Case 1 is executed.

$$X_{t+1}=\Delta X_t-E|J{X}_{rabbit,t}-X_t|$$

(10)

$$\Delta X_t=X_{rabbit,t}-X_t$$

(11)

Case 2: Hard besiege

When r $\ge$ 0.5, $E$ $<$ 0.5, Case 2 is executed.

$$X_{t+1}=X_{rabbit,t}-E|\Delta X_t|$$

(12)

Case 3: Soft besiege with progressive rapid dives

When r $<$ 0.5, $E$ $\ge$ 0.5, Case 3 is executed.

$$\begin{array}{l}{X}_{t+1}=\\ \left\{\begin{array}{l}Y:X_{rabbit,t}-E|J{X}_{rabbit,t}-X_t|, if F(Y)<F(X_t)\\ Z:Y+S\times LF(D), if F(Z)<F(X_t)\end{array}\right.\end{array}$$

(13)

Case 4: Hard besiege with progressive rapid dives

When r $<$ 0.5, $E$ $<$ 0.5, Case 4 is executed.

$$\begin{array}{l}{X}_{t+1}=\\ \left\{\begin{array}{l}Y:X_{rabbit,t}-E|J{X}_{rabbit,t}-X_{m,t}|,if F(Y)<F(X_t)\\ Z:Y+S\times LF(D), if F(Z)<F(X_t)\end{array}\right.\end{array}$$

(14)

where $X_{rabbit,t}$ is the local optimum; LF is the levy flight function; $J=2(1-r_5)$, representing the random jump strength of the rabbit; and $r_5$ is a random value.

3.2. Improved HHO

(1)

Stochastic contraction exponential function. The magnitude of prey energy (E) plays an important role in balancing exploration and exploitation. Mathematicians simulate the interaction between predator and prey [23], and then draw conclusion that the stochastic contraction exponential function is better suited to express the energy change during prey escape. The energy equation is given by (15).

$$E=2E_0(2rand\times \exp (-(\frac{\pi }{2}\times \frac{t}{T}\left)\right))$$

(15)

(2)

Adaptive weights. In order to improve the local capability of the algorithm, it is necessary to update the neighborhood of the prey position to find a better solution. The expressions are as follows.

$$w=\sin (\frac{\pi \times t}{2T}+\pi )+1$$

(16)

$$\stackrel{\wedge }{X_{rabbit}}=w\times X_{rabbit}$$

(17)

where w is the weight and $\stackrel{\wedge }{X_{rabbit}}$ is the neighborhood of the prey position.

(3)

Polynomial variation. To address the problem that HHO tends to fall into the local optimum, the method of polynomial variation is used in this work. The expressions are as follows.

$$\stackrel{\wedge }{X_{best,j}}=X_{best,j}+\alpha \cdot (u_j-l_j)$$

(18)

$$\alpha =\left\{\begin{array}{l}(2r+(1-2r)(1-{\alpha }_1{)}^{\beta +1}{)}^{\frac{1}{\beta +1}}-1, if \mathrm{r}<0.5\\ 1-(2(1-r)+2(r-0.5)(1-{\alpha }_2)\beta +1)\frac{1}{\beta +1},if \mathrm{r}\ge 0.5\end{array}\right.$$

(19)

$$\left\{\begin{array}{l}{\alpha }_1=(X_{best,j}-l_j)/(u_j-l_j)\\ {\alpha }_2=(u_j-X_{best,j})/(u_j-l_j)\end{array}\right.$$

(20)

where $r$ is a random inside (0, 1); $\beta$ is a distribution exponent; ${\alpha }_{}$ is a variational operator; $u_j$ and $l_j$ are the upper and lower limits of the position; $j=1,\dots ,n$, where n is the dimension; $X_{best,j}$ is the historical optimal position; and $\stackrel{\wedge }{X_{best,j}}$ is the value after variation.

3.3. Testing

Several test functions are used to test the IHHO’s ability to find the best solution for complex problems, and the Shifted functions are suitable for the problems in this paper. The expressions are shown in

Table 1. Test functions.

Function Name	Expressions	Range	Optimal Value
Shifted Sphere	$f_1(x)={\displaystyle \sum _{i=1}^n{(x_i-o_i)}^2}$	(−100, 100)	0
Shifted Schwefel 2.21	$f_2(x)=\max \left\{\right|x_i-o_i|,1\le i\le n\}$	(−10, 10)	0

.

The experimental setup is as follows: N (population size) = 50; maximum number of evaluation (T) = 20,000; $o_i$ ($i=1,\dots ,n$) are random values within the search range, which stand for the offset; and number of tests is 30. For a fair comparison, other algorithms, including the sine cosine algorithm (SCA), genetic algorithm (GA), and standard Harris hawk optimization (HHO) algorithm, were set to the same values for the same parameters. The four algorithms were run 30 times independently for each test function. Then, the mean (Ave) and standard deviation (Std) of the optimal function values were recorded to evaluate the performance of the algorithms. Ave reflects the solution accuracy of the algorithm and Std denotes the stability of the solution. The test results are shown in

Table 2. Comparison of the test function optimization results.

Function	N	Indication	SCA	GA	HHO	IHHO
$f_1$	4	Ave	8.5476	2.2238	7.0177 × 10−3	0
		Std	2.8957	1.5334	6.3068 × 10−3	0
$f_2$	4	Ave	2.3606 × 10−1	1.0444 × 101	4.5452 × 10−2	9.474 × 10−16
		Std	7.4158 × 10−2	2.2259 × 102	2.2962 × 10−2	1.211 × 10−15
$f_1$	18	Ave	5.39305 × 103	5.52848 × 102	1.93612 × 101	4.165 × 10−6
		Std	1.2103409 × 103	9.70237 × 101	1.68195 × 101	3.762 × 10−6
$f_2$	18	Ave	4.5491	1.0576	1.3325	3.852 × 10−2
		Std	6.2022 × 10−1	1.4971 × 10−1	5.8435 × 10−1	4.074 × 10−2

.

From Table 2, we can see that the optimal value of the solution of IHHO has the smallest result and the smallest standard deviation. Therefore, in terms of convergence accuracy and stability, the IHHO algorithm is significantly better than the other algorithms.

4. Joint Estimation of SOC and SOH

4.1. Multi-Timescale DUKF Estimation

The UKF algorithm is a nonlinear filtering method proposed by Julier, which uses the Kalman linear filtering framework and the Unscented Transform (UT) to handle the mean and covariance of the nonlinear transfer problem [24]. DUKF uses two UKFs to perform SOC estimation and the online RC parameters update of the process, respectively. Meanwhile, the updated values of the former influence the RC parameters, and the estimated values of the latter influence the former, too.

The battery system state–space model is described in (21):

$$\left\{\begin{array}{l}{x}_{k,l+1}=f(x_{k,l},{\theta }_l,I_{k,l})+w_{k,l},{\theta }_{l+1}={\theta }_l+{\rho }_l\\ y_{k,l}=g(x_{k,l},{\theta }_l,I_{k,l})+{\nu }_{k,l}\end{array}\right.$$

(21)

where $\theta$ = [R₀, R₁, C₁, R₂, C₂]^T; k is the microscopic scale; and l is the macroscopic scale.

The steps of the DUKF algorithm are as follows:

(1)

Initialize the state variables and error covariance matrix:

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

(22)

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

(23)

where $P_{x,0}$, and $P_{\theta ,0}$ describe the initial values of the state error covariance for the SOC estimation model and the parameter estimation model, respectively.

(2)

Generate Sigma points:

$$\left\{\begin{array}{l}{x}_{0,k-1}=x_{k-1}\\ x_{i,k-1}=x_{k-1}+(\sqrt{(n+\lambda )P_{k-1}})i,i=1,2,\dots ,n\\ x_{i,k-1}=x_{k-1}-(\sqrt{(n+\lambda )P_{k-1}})i,i=n+1,n+2,\dots 2n\end{array}\right.$$

(24)

where n stands for number of state variables, $\lambda$ is a coefficient, and the expression is as follows:

$$\lambda ={\alpha }_{}^2(n+h)-n$$

(25)

where $\alpha$ describes the extent to which the Sigma point deviation from the state value, h is a coefficient to guarantee variance matrix is semi-positive definite. $\alpha$ = 1 × 10⁻³, h = 0.

(3)

Determination of the weighting factors:

$$\left\{\begin{array}{l}{{\displaystyle w}}_0^m=\frac{\lambda }{\lambda +n}\\ {{\displaystyle w}}_0^c=\frac{\lambda }{\lambda +n}+(1+\beta -{\alpha }_{}^2)\\ {{\displaystyle w}}_i^m={{\displaystyle w}}_i^c=\frac{\lambda }{2(n+\lambda )},i=1,2,\dots ,2n\end{array}\right.$$

(26)

where $w_{}^m$ and $w_{}^c$ are the weights, and $\beta$ = 2.

(4)

Time refresh:

$$\left\{\begin{array}{l}{x}_{i,k|k-1}=f(x_{i,k-1},{\theta }_{l-1},I_k),i=0,1,\dots ,2n\\ x_{k|k-1}={\displaystyle \sum _{i=0}^{2n}{{\displaystyle w}}_i^m}f(x_{i,k-1},{\theta }_{l-1},I_k)+q_k,i=0,1,\dots ,2n\end{array}\right.$$

(27)

$$P_{k|k-1}={\displaystyle \sum _{i=0}^{2n}{{\displaystyle w}}_i^c}(x_{i,k|k-1}-x_{k|k-1}){(x_{i,k|k-1}-x_{k|k-1})}_{}^T+Q_k$$

(28)

where $q_k$ is the mean of process noise, and $Q_k$ is the covariance of the process noise.

(5)

Measurement update:

$$\left\{\begin{array}{l}{y}_{i,k|k-1}=g(x_{i,k-1},{\theta }_{l-1},I_k)\\ y_{k|k-1}={\displaystyle \sum _{i=0}^{2n}{{\displaystyle w}}_i^{m}g(x_k,{\theta }_{l-1},I_k)+r_k}\end{array}\right.$$

(29)

$$\left\{\begin{array}{l}{P}_{xy,k}={\displaystyle \sum _{i=0}^{2n}{{\displaystyle w}}_i^c}(x_{i,k|k-1}-x_{k|k-1})(y_{i,k|k-1}-y_{k|k-1})T\\ P_{yy,k}={\displaystyle \sum _{i=0}^{2n}{{\displaystyle w}}_i^c}(y_{i,k|k-1}-y_{k|k-1})(y_{i,k|k-1}-y_{k|k-1})T\\ L_k=P_{xy,k}{P}_{yy,k}^{-1}\end{array}\right.$$

(30)

$$\left\{\begin{array}{l}{x}_k=x_{k|k-1}+L_k(y_k-y_{k|k-1})\\ P_k=P_{k|k-1}-L_k{P}_{yy,k}{L}_k{}_{}^T\end{array}\right.$$

(31)

If the SOC estimation of (31) is brought into Equation (21), then the RC parameters can be refreshed by replacing $x_k$ with ${\theta }_k$ and repeating the steps of (24) to (31). It is worth noting that the RC parameters are refreshed while l = 5 s, and at the same time, l is reset to 0. The voltage prediction curve of DUKF is shown in Figure 9. Combined with Figure 8, The comparisons of the error curves are displayed in Figure 10.

It is not difficult to find the voltage prediction error of DUKF being less than the offline UKF. The DUKF calibrates error in a very short time, and thus the robustness of the DUKF is better than UKF. The result demonstrates the superiority of the online DUKF.

4.2. Improved DUKF

In Section 3, the IHHO is proposed, and now is used to find the optimal initial parameters for DUKF, such that the $Q_1$, $Q_2$, $P_{0,1}$, $P_{0,2}$, $R_1$, $R_2$ matrix, and a total of 18 parameters are determined. The parameters are set up as follows: $Q_1$ = diag (x(1), x(2), x(3)), $Q_2$ = diag (x(4), x(5), x(6)), $P_{0,1}$ = diag (x(7), x(8), x(9), x(10), x(11)), $P_{0,2}$ = diag (x(12), x(13), x(14), x(15), x(16)), $R_1$ = x(17), $R_2$ = x(18), where diag denotes the diagonal matrix.

$$\stackrel{\wedge }{SOC}=FF(Q_1,Q_2,P_{0,1},P_{0,2},R_1,R_2)$$

(32)

$$fit={\displaystyle \sum _{k=1}^L{(\stackrel{\wedge }{SOC}(k)-SOC(k))}^2}$$

(33)

where $FF$ stands for the function of the DUKF, whose output is a series of estimated $\stackrel{\wedge }{SOC}(k)$; $SOC(k)$ is reference value; and $fit$ is the fitness of the improved DUKF. The part results of optimized parameters are as follows: $R_1$ = 1.01718439826945 × 10⁻⁵ and $R_2$ = 0.322416048219495. It is difficult for empirical methods to achieve the optimal parameters; however, the proposed method in this paper easily can find the optimal parameters to minimize $fit$.

4.3. The Capacity Update Model

The default battery capacity was constant in the previous study. However, battery capacity is actually variable with the start of the charging or discharging process. Therefore, it is necessary to introduce the capacity update model shown in (34), and the SOC estimation is more accurate with the help of the corrected capacity, which has an influence on SOH. The estimation result of capacity is shown in Figure 11, and the expression of SOH is indicated in (35). What is more, SOH can be expressed by Equation (36). The comparison of the $R_0$ estimations and references are shown in Figure 12, and then the predicted results of the SOH are shown in Figure 13.

$$\left\{\begin{array}{l}{C}_{now,k+1}=C_{now,k}+{\beta }_k\\ d_k=SO{C}_k-SO{C}_{k-1}+\frac{\eta I_{k-1}T}{C_{now,k-1}}+{\gamma }_k\end{array}\right.$$

(34)

$$SOH=\frac{C_{now}}{C_n}$$

(35)

$$SOH=\frac{2R_{new}-R_{now}}{R_{new}}$$

(36)

where ${\beta }_k$ and ${\gamma }_k$ are the process noise and measurement noise, respectively. $C_{now}$ is the current maximum capacity, $C_n$ is the rated capacity, and the goal of $d_k$ is zreo. $R_{now}$ is the current value of ohm internal resistance, and $R_{new}$ is the value of new ZNB.

From Figure 12, we can see that estimated values of $R_0$(EST1, EST2) can track the reference values well, where EST1 and EST2 are the estimated values at the wrong and true initial $R_0$. It is not hard to find the proposed method can estimate the reference $R_0$ accurately, even if the initial value is wrong. In addition, Figure 11 and Figure 13 present the slow reductions in capacity and SOH, conforming to the actual aging characteristics during the use of the battery.

4.4. Results Validation and Analysis

Several test conditions are carried out in this work, such as constant current pulse condition (CCPC) and variable current continuity condition (VCCC). Comparisons of the UKF, DUKF, and IDUKF algorithms are as follows. The estimation errors of IDUKF, DUKF, and UKF are bounded within 0.00719, 0.014, and 0.02663 in the CCPC tests, and IDUKF converges the fastest, followed by DUKF and then UKF. The mean squared errors (MSE) of them are 1.1827 × 10⁻⁵, 1.3925 × 10⁻⁴, and 2.4772 × 10⁻⁴, respectively, and the mean absolute errors (MAE) are 0.0027, 0.0063, and 0.0098, respectively, which denotes that the error of IDUKF is far less than other methods. The results are displayed in Figure 14 and Figure 15.

VCCC tests are applied to validate the effectiveness of the proposed method in complex working conditions, and the details of the tests are described in Figure 16. The blue line represents the voltage, and the red line stands for the current.

The predicted SOC curves and error curves are shown in Figure 17 and Figure 18. At this moment, it is not difficult to find the predicted curve as the UKF method deviates from the reference value. Furthermore, the DUKF method can track the reference curve with an error of 0.012; however, the error reaches 0.0266 at the end. Meanwhile, the SOC estimation error of the IDUKF is bounded within 0.0059. The MSE of IDUKF, DUKF, and UKF are 4.3893 × 10⁻⁵, 2.3957× 10⁻⁴, and 0.0022 respectively, and the MAE are 0.0059, 0.0130, and 0.0432, respectively. The results again confirm the effectiveness and robustness of the proposed method.

5. Conclusions

Accurate estimations of SOC and SOH are important in practical applications, which can avoid overcharging or overdischarging. However, little has been done to jointly estimate SOC and SOH for the ZNB. In this paper, the proposed IDUKF combines DUKF with IHHO, improving the estimation accuracy. In the CCPC tests, the MSE of the IDUKF decreases by about 91.5 percent and 95.2 percent, respectively, compared to the DUKF and UKF; also, the MAE drops by about 57.2 percent and 72.4 percent compared to the DUKF and UKF. In the VCCC tests, the MSE of the IDUKF decreases by 81.7 percent and 98 percent relative to the DUKF and UKF, and the MAE reduces by 54.6 percent and 86.3 percent compared to the DUKF and UKF. As described above, the IDUKF is more accurate in terms of SOC estimation, compared with the traditional method. Meanwhile, the proposed method also takes aging into account, which avoids the inaccuracy of a single SOC estimation, and then the joint estimation is successfully completed.