Li-Ion Battery Anode State of Charge Estimation and Degradation Monitoring Using Battery Casing via Unknown Input Observer

Abstract

The anode state of charge (SOC) and degradation information pertaining to lithium-ion batteries (LIBs) is crucial for understanding battery degradation over time. This information about each cell in a battery pack can help prolong the battery pack’s life cycle. Because of the limited observability, estimating the anode state and capacity fade is difficult. This task is even more challenging for the cells in a battery pack, as the current through the individual cell is not constant when cells are connected in parallel. Considering these challenges, this paper presents a novel method to set up three-electrode cells by using the battery’s casing as a reference electrode for building a three-electrode battery pack. This work is a continuation of the authors’ previous research. An unknown input observer (UIO) is employed to estimate the anode SOC of an individual battery in the battery pack. To ensure the stability of a defined Lyapunov function, the UIO parameter matrices are expressed as a linear matrix inequality (LMI). The anode SOC of a lithium nickel manganese cobalt oxide (NMC) battery is estimated by using the standard graphite potential (SGP) and state of lithiation (SOL) characteristic curve. The anode capacity is then calculated by using the total charge transferred in a charging cycle and the estimated SOC of the anode. The degradation of the battery is then evaluated by comparing the capacity fading of the anode to the total charge carried to the cell. The proposed method can estimate the anode SOC and capacity fade of an individual battery in a battery pack, which can monitor the degradation of the individual batteries and the battery pack in real time. By using the proposed method, we can identify the over-degraded batteries in the pack for remaining useful life analysis on the battery.

1. Introduction

1.1. Background

Lithium-ion batteries are now the most popular alternative energy source for the automotive industry. LIBs are in high demand as an energy storage medium, mainly due to their high energy density, long service life, slow self-discharge rate, and zero harmful gas emission. In particular, the increasing demand for lithium-ion batteries in electric vehicles (EVs) made researchers interested in improving battery technology further to enhance its performance and reliability. This development vastly depends on the battery’s internal state information, such as the state of charge, degradation patterns, and degradation status of the electrodes [1].

Among the most critical battery internal parameters, the energy-indicating parameter SOC provides remaining capacity information in the battery. Accurate SOC estimation in the battery-management system is critical for reliability and for prolonging battery life. Various SOC estimating methods have been developed considering different approaches [2,3,4]. The ampere-hour (Ah) or coulomb counting estimating method [2,3,5,6,7] is the most conventional among the estimation methods used based on calculating the transferred charge to the cell by integrating the charge and discharge current over time. This low-cost computational method can estimate SOC in real time. However, the significant drawbacks of this technique are measurement error accumulation during operation [5], and the requirement of a known initial SOC in the algorithm. The open-circuit voltage (OCV) measurement method [2,7,8] is another widely used SOC estimation technique. However, with the OCV-SOC characteristic relationship [9] OCV-based methods can predict SOC more accurately if the OCV-SOC curve is not very flat. However, a long relaxation time for voltage stabilization [3,5] makes it an offline estimation technique. Neural network (NN)-based techniques [7,8] have gained popularity in the SOC estimation due to their ability to estimate parameters without knowing system behavior [7]. Nevertheless, considering the amount of training data required for accurate prediction, it falls into the costliest SOC estimation method. Moreover, the complexity of developing an NN-based estimation method is another drawback of this estimation technique. Mathematical model-based estimation techniques [4,10,11,12] gained popularity in EV applications due to their estimation reliability when employed in EV real-time applications. These methods are developed by using a battery model with a mathematical model to estimate SOC [11]. The performance and reliability of this method can be further improved upon by employing an observer or adaptive filter [2].

The degradation of lithium-ion batteries is an irreversible and complex phenomenon and an essential parameter to monitor in the BMS. These degradation mechanisms are mainly a result of various side reactions [13]. Solid electrolyte interface (SEI) formation [13,14], and lithium plating are the most significant and concerning anode side reactions. A passivation layer [14] of SEI is formed due to electrolyte decomposition on the electrode surface. Lithium plating is another significant anode side reaction, mainly caused due to fast-charging or low-temperature charging. In lithium plating [13,15] a layer of metallic lithium is formed on an anode surface. These side reactions consume cyclable lithium-ions and electrolytes, resulting in capacity fade [13,14]. Notable cathode side reactions are transition metal dissolution and structure disordering [13]. At a high charging voltage [16], cathode-active material dissolves and deposits on the anode [13,16,17]. Other causes of degradation demonstrated in Figure 1 includes binder decomposition [13], graphite exfoliation [13,16], electrode particle cracking [13], and current collectors’ electrical connection loss [13]. These degradation mechanisms result in the capacity fade of the electrodes and cell. Therefore, monitoring the electrode level capacity fade is advantageous for BMS.

1.2. Current Status

There have been numerous methodologies developed for SOC and state of health (SOH) estimation based on different estimation techniques such as adaptive filter, machine learning, or observer-based [2,7,18] techniques. Chen et al. [19] proposed an uncentered Kalman filter-based SOC estimation method. A non-parametric lithium-ion battery model based on the Gaussian process regression is proposed. The Gaussian process regression is utilized to develop the observation models by learning experimental data. Then to improve the estimation accuracy, an unscented Kalman filter algorithm is utilized. The proposed method is an offline estimation method. Lai et al. [20] developed an equivalent circuit model by considering partial electrochemical properties. Battery parameters are then identified by using a particle swarm optimization method. Then an extended Kalman filter is employed to estimate SOC. They have achieved a less than 1% estimation error for the proposed estimation method. The proposed method adopted multiple algorithms to develop the estimator, thus introducing application complexity. Dai et al. [21] proposed an online cell SOC estimation method in a battery pack. An equivalent circuit model is developed for the battery, and then the battery pack’s average SOC is estimated. Then the performance divergences between the “averaged cell” and each cell are incorporated to estimate SOC for all cells by using an extended Kalman filter. Guo et al. [22] proposed an SOH estimation method by employing an equivalent circuit model to characterize the CC part of the charging curve and derived a transformation function and a time-based parameter. Equivalent circuit model parameters are evaluated by using a nonlinear least-squares method (LSM). Li et al. [17] developed a single particle-based degradation model which includes SEI formation coupled with crack propagation and a model of lithium-ion loss. The proposed SOH estimation methods heavily depend on degradation models to propagate LIB degradation, which may not be the same as the actual battery degradation. He et al. [23] proposed an artificial neural network (ANN)-based SOC estimation method. An ANN is utilized to develop a battery model for SOC estimation with an unscented Kalman filter to reduce estimation error. Wei et al. [24] employed an improved dynamic neural network to estimate the SOC through an open-loop and a closed-loop nonlinear auto-regressive model. This method utilized multiple models and a dynamic NN which introduces more complexity. To reduce the complexity of employing model-based SOC estimation methods for battery packs, Deng et al. [25] proposed a data-driven SOC estimation method for battery packs based on Gaussian process regression (GPR). First, features are extracted through correlation analysis and principal component analysis to obtain an input dataset with a high correlation with SOC. Second, the wights of the features are optimized by utilizing the squared exponential kernel function and automatic relevance determination. To improve the estimation accuracy, the authors used an autoregressive GPR model along with the regular GPR model. The estimation error achieved in this study is lower than 3.9%. Most existing SOC and SOH estimation methods focus on model-based cell level SOC/SOH estimations, wherein electrode-level SOC is important to get battery internal status information, which helps analyze overall battery degradation.

Tian et al. [26] developed a technique for estimating electrode-level parameters by using a convolutional neural network. Without feature extraction, the proposed technique can estimate electrode capacities and start SOCs from segments of the charging curve, enabling quickly aging diagnostics of electrodes. Additionally, these calculated characteristics are utilized to reconstruct OCV-Q (charge amount) curves, with an accuracy greater than 99% for estimating battery capacity. S. Lee et al. [27] proposed voltage fitting and differential voltage analysis methods to estimate the electrode state of health by using half-cell potential. The proposed method isolates the voltage contribution of individual electrodes from the cell voltage by utilizing peak information in the differential voltage curve. The proposed estimation techniques are developed chiefly based on parameter estimation models without measuring actual electrode parameters. These studies also consider single cells only for electrode-level estimation, which might not be practical in real-time applications wherein multiple cells are used to meet energy requirements.

Battery degradation is an intricate process, and to understand such a mechanism, electrode-level monitoring is crucial. Electrode-level monitoring involves electrode parameter identification. Electrode potential is one of the most critical electrode parameters acquired by utilizing a reference electrode. To get the electrode potential, electrode voltage is measured with respect to the reference electrode [28]. The most conventional way to make a three-electrode cell is by inserting external metal [15,29,30] inside the cell by either cutting the positive terminal or cutting a patch (pouch cell). These external metals are carefully chosen depending on the behavior of the cell and the metal characteristics [31]. Metal lithium rods or other lithium-plated metals are the most commonly used reference electrode materials [32,33,34,35]. Lithium is not very suitable for this application as it is a very reactive material [35,36] that readily reacts with electrolytes, resulting in electrolyte loss in the battery. These metal-insertion methods are performed in the laboratory for testing purposes only under certain precautions to avoid hazards. Thus, the practical application of these methods is still not feasible where multiple cells are required. Our proposed three-electrode cell setup does not require additional material. Moreover, the proposed process requires minimal modification to the cell, which can be avoided by manufacturing the battery casing with a spacer between the reference and negative terminal. Thus, a three-electrode battery pack setup can also investigate the possible practical application.

1.3. Research Gaps and Contributions

To the best of our knowledge, this paper presents a three-electrode-based battery pack setup for the first time by using the battery casing as the reference electrode. The benefit of this method is that it does not require additional materials to be added to the cell and can help monitor individual battery status in the battery pack. It does not require additional sensors to employ in the BMS other than measuring the current through each cell. Further, a UIO model is developed to estimate anode SOC by using the SGP-SOL characteristic curve. An unknown input observer is used because the floating anode voltage is an unknown input to the standard battery system, which shifts up and down over time. The anode storage capacity of a commercially available 18,650 NMC lithium-ion battery is then calculated by using the estimated anode SOC and total charge carried to the cell. The anode SOC and capacity of an individual battery in a battery pack can now be estimated and monitored even for variable current.

The main contributions are as follows.

A novel approach is proposed to set up a three-electrode battery pack by using a battery casing as the reference electrode. Using the casing as a reference for the three-electrode battery pack would be beneficial for estimating individual battery electrode parameters, such as electrode capacity fade and SOC.

A model-based anode state of the charge estimation method is presented for an individual battery in the battery pack.

The anode state of the charge is estimated for variable charge and discharge current.

The capacity fade of the anode can be calculated and monitored.

2. Methodology

2.1. Experimental Setup

The tested battery pack used in this study is made of commercially available Sanyo UR18650AA lithium-ion 18,650 cells. The single cell consists of a lithium nickel manganese cobalt oxide (NMC) cathode and a graphite anode. The nominal capacity of the single cell is 2.25 Ampere-hour (Ah), and the charge cut-off voltage is 4.2 V. The cells were discharged to their standard discharge cut-off voltage, 2.5 V, before the experimental setup. Three of these 18,650 cells were connected in parallel to set up the battery pack. Thus, the nominal capacity of the battery pack is 6.75 Ah, where the charge and discharge cut-off voltage remain the same as a single cell. The experiment was conducted at room temperature. Specifications of the battery are shown in

Table 1. Specifications of the battery.

Specification	Battery
Cell type	UR18650AA. Sanyo
Electrode material	Li(Ni0.8Co0.1Mn0.1)O2/graphite
Nominal capacity (C)	2.25 Ah
Charge cut-off voltage	4.2 V
Discharge cut-off voltage	2.5 V
Charge and discharge cut-off current	0.02 C
Standard charging current	0.7 C
Testing Ttemperature	25 °C

.

Three Sanyo UR18650AA cells are used for the three-electrode battery pack. To set up a three-electrode battery, the cell was first tabbed in three positions: positive and negative terminals and the shell, with nickel tabs. Then the cell′s shell was carved while being held vertically to separate it from the negative electrode. An adhesive non-reactive silicone glue was utilized to seal the gap once the reference case was detached from the negative electrode. The anode and cathode voltage were perceived as 2.7 V and 0.8 V, respectively, as to the reference electrode shortly after the detachment. Newly formed working three-electrode cells then rested overnight to desiccate the silicon glue. Figure 2 shows a three-electrode cell setup.

Prepared three-electrode cells are then connected in parallel for battery pack setup. The battery pack negative and positive terminal is then connected to a channel of the cycler through cable. The reference electrode and respective anode of each cell in the pack are connected to a channel via cable to record the voltage difference between the negative terminal and that cell’s reference. Three channels of the cycler are used for recording the voltage difference of the anode and reference for three cells. A current sensor is connected between each two cell’s cathode connection. Furthermore, the current sensors are connected to an Arduino nano for recording the current input through each cell. The battery pack was cycled by using the standard constant current and constant voltage (CC-CV) charge and discharge protocol. The pack was charged and discharged with 0.7 C (4725 mA) current, where the charge and discharge cut-off voltages were set to 4.2 V and 2.5 V, respectively. The charge and discharge cut-off current were set to 1/50 C (135 mA). The anode voltage of each cell in the battery pack is recorded with respect to the respective reference electrode of the cell. The experimental setup of the battery pack and electrical wiring is shown in Figure 3a,b.

We noticed that the recorded anode voltage progressively increases and decreases during cycling. We call this voltage shift anode floating voltage. This voltage shift of $L{i}_x{C}_6$ with respect to the battery casing can be described with the Nernst equation [37] as stated below,

$$E=E^0-\frac{RT}{nF}lnC,$$

(1)

where $E$, $E^0$ represents reduction and standard potential, respectively. $R$ and $F$ are gas and Faraday constant, $T$ is the ambient temperature in Kelvin, $n$ is the number of electrons transferred in the reaction, and $C$ denotes the reaction quotient, which is the ratio of products and reactants available during a reaction at a given time. If $T$ is constant, then $E$ merely depends on the metal ion concentration $C$ in the redox reaction. Thus, we can conclude that the anode voltage shift regarding the metal casing is due to the low casing material ion concentration in the solution. This study uses this floating voltage as the unknown input to the system as the voltage up and downshifting pattern is unknown. The anode voltage with respect to the reference casing is shown in Figure 4.

2.2. Observer Design

To design an unknown input observer for anode SOC estimation, we considered a linear system, and the mathematical model of the system is as follows,

$$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)+Dv\left(t\right)$$

(2)

$$y\left(t\right)=Cx\left(t\right),$$

(3)

where $x\left(t\right)$, $u\left(t\right)$, and $v\left(t\right)$ are the state vector, the input vector, the unknown input vector, and $y\left(t\right)$ is the output vector. $A$, $B$, $C$, and $D$ are dynamic parameters of the considered system consisting of appropriate dimensions. Rank ($C\ast D)=$ rank ($D$) is required for the system to be observable where (C,A) is an observable pair.

The structure of the UIO for the abovementioned (2) linear system can be described as follows [38]. The structure of such UIO is shown in Figure 5 [39].

$$\dot{z}\left(t\right)=Nz\left(t\right)+Ly\left(t\right)+Gu\left(t\right)$$

(4)

$$\widehat{x}\left(t\right)=z\left(t\right)-Ey\left(t\right)$$

(5)

where $z\left(t\right)$ is the UIO state vector and $\widehat{x}\left(t\right)$ is the estimated state vector. UIO parameter matrices are defined as $N$, $L$, $G$ and $E$ with appropriate dimensions. For the estimated state vector, $\widehat{x}\left(t\right)$ convergence to $x\left(t\right),$ the UIO parameters are determined by formulating into LMI.

UIO state estimation error can be noted as follows [38],

$$e\left(t\right)=\widehat{x}\left(t\right)-x\left(t\right)=z\left(t\right)-Mx\left(t\right),$$

(6)

with

$$M=I+EC,$$

(7)

where $e$ and $I$ are the state estimation error and the identity matrix, respectively.

Hence, state estimation error dynamics can be given as stated below [38]:

$$\dot{e}\left(t\right)=\dot{z}\left(t\right)-M\dot{x}\left(t\right).$$

(8)

Substituting $\dot{x}\left(t\right)$ and $\dot{z}\left(t\right)$ from Equations (2) and (4) into Equation (8) yields

$$\dot{e}\left(t\right)=Nz\left(t\right)+Ly\left(t\right)+Gu\left(t\right)-M\left(Ax\left(t\right)+Bu\left(t\right)+Dv\left(t\right)\right).$$

(9)

With $e\left(t\right)=z\left(t\right)-Mx\left(t\right)$ Equation (9) reads as

$$\begin{gathered} \dot{e}\left(t\right)=Ne+\left(NM+LC-MA\right)x\left(t\right) \\ +\left(G-MB\right)u\left(t\right) \\ -MDv\left(t\right). \end{gathered}$$

(10)

To ensure that the system is independent from the unknown input and converges, the following conditions must be satisfied [38],

$$NM+LC-MA=0$$

(11)

$$G-MB=0 \mathrm{or} G=MB$$

(12)

$$MD=0.$$

(13)

If the above conditions are satisfied, the Equation (10) yields

$$\dot{e}\left(t\right)=Ne\left(t\right).$$

(14)

From Equation (14), if $N$ is stable, the system asymptotically converges.

For $K=L+NE$ Equation (11) can be rewritten as [40]

$$N=MA-KC.$$

(15)

Thus, $L$ can be solved by using Equation (15),

$$L=K\left(I+CE\right)-MAE,$$

(16)

where $I$ is an identity matrix with the same dimension as $C\ast E$.

For the considered system $C\ast D=1,$ which is considered a full rank. Thus, from Equation (13),

$$E=-D{\left(CD\right)}^{+}+Y\left(I-\left(CD\right){\left(CD\right)}^{+}\right),$$

(17)

where, ${\left(CD\right)}^{+}={\left({\left(CD\right)}^T\left(CD\right)\right)}^{-1}{\left(CD\right)}^T$ and $Y$ is an arbitrary matrix [38].

For notational convenience, Equation (17) is rewritten as

$$E=U+YV,$$

(18)

where, $U=-D{\left(CD\right)}^{+}$ and $V=I-\left(CD\right){\left(CD\right)}^{+}$.

A Lyapunov function $V\left(t\right)=e{\left(t\right)}^{T}pe\left(t\right)$ with $P>0$ can be considered to formulate the UIO matrices into LMI to determine the matrices such that $e$ is asymptotically stable and the system converges [38,41].

The dynamics of the Lyapunov function can be described as [38]

$$\dot{V}\left(t\right)=e{\left(t\right)}^T\left(N^{T}P+PN\right)e\left(t\right),$$

(19)

where $N^{T}P+PN<0$ has to be satisfied for the system to converge.

With the general solution of $E$ from Equation (18), LMI [41] for the UIO matrices yields

$$\begin{gathered} {\left(\left(I+UC\right)A\right)}^{T}P+P\left(I+UC\right)A \\ +{\left(VCA\right)}^T{Y}^T+Y\left(VCA\right) \\ -C^T{K}^T-KC<0. \end{gathered}$$

(20)

By solving the LMI described in (20), $Y, K,$ and $P>0$ matrices are determined, which is used to determine the UIO parameter matrices, $M, N, L, G,$ and $E$ by using Equations (7), (12), (15), (16) and (18), respectively.

2.3. Thevenin Equivalent Circuit Model

A Thevenin equivalent circuit model is utilized for the battery half-cell model. For simplicity of the model, the resistor–capacitor (RC) is used to describe the polarization and diffusion effect. The utilized equivalent battery half-cell model [42] is shown in Figure 6. In the model $V_t$ is terminal voltage, and standard graphite potential is used as OCV, $I$ is the current carried through the battery, and battery internal resistance is represented by the ohmic resistor $R_a$, $V_b$ is the voltage of the RC branch, $R_b$ and $C_b$ are the parallel connected RC branch that describes the polarization and diffusion effect, and $V_f$ is the unknown input to the system due to the floating issue mentioned earlier. The electrical behavior of this equivalent battery half-cell model can be described by the following equations and the relative parameters.

The RC branch voltage is

$${\dot{V}}_b=-\frac{V_b}{R_b{C}_b}+\frac{I}{C_b}.$$

(21)

The terminal voltage $V_t$ is defined as

$$V_t=V_p+V_b+I{R}_a.$$

(22)

2.4. Model Parameter Identification

The first order RC model parameters, $R_a$, $R_b$, $C_b$, are tuned to obtain the minimum error between the estimated anode voltage and measured anode potential. The initial guesses for the parameters are obtained from [43]. Then the parameters are tuned based on a trial and error until a minimum error value is obtained. For computational simplicity, the parameters are considered to be constant. The parameter identification results are shown in

Table 2. Parameter identification results of the utilized model.

Parameters	Symbol	Value
Cell capacity	$Q_{charge}$	2.22 Ah
Ohmic resistance	$R_a$	0.012 Ω
Polarization resistance	$R_b$	0.031 Ω
Polarization capacitance	$C_b$	1900 F

.

2.5. Anode SOC and Capacity Estimation

The anode SOC is estimated based on the considered linear system described in Equation (2). System matrices A, B, C, and D can be evaluated from the equivalent battery model described above.

$$A=\left[\begin{array}{cc}0& 0\\ 0& -\frac{1}{R_b{C}_b}\end{array}\right], B=\left[\begin{array}{c}\frac{1}{Q_n}\\ \frac{1}{C_b}\end{array}\right], D=\left[\begin{array}{c}0\\ 1\end{array}\right], C=\left[\begin{array}{cc}\frac{OCV\left(SOC\right)}{SOC}& 1\end{array}\right],$$

where, $Q_n$ is the estimated anode capacity, $\mathrm{OCV}\left(\mathrm{SOC}\right)$ is the open-circuit voltage (standard graphite potential) as a function of SOC (state of lithiation here). As this study focuses on anode SOC estimation, the standard graphite potential versus the state of lithiation curve is considered as an OCV-SOC characteristic curve for the anode SOC estimation. This characteristic curve is evaluated by using an 11th-order polynomial function as follows:

$$P\left(x\right)=P_1{x}^n+P_2{x}^{n-1}+\dots +P_{n}x+P_{n+1},$$

(23)

where, $P_1,P_2\dots P_{n+1}$ are coefficients with constant values.

With the system matrices, the linear system (2) can be rewritten as

$$\dot{x}\left(t\right)=\left[\begin{array}{cc}0& 0\\ 0& -\frac{1}{R_b{C}_b}\end{array}\right]x\left(t\right)+\left[\begin{array}{c}\frac{1}{Q_n}\\ \frac{1}{C_b}\end{array}\right]u\left(t\right)+\left[\begin{array}{c}0\\ 1\end{array}\right]v\left(t\right),$$

(24)

and the system output vector,

$$y\left(t\right)=\left[\begin{array}{cc}\frac{OCV\left(SOC\right)}{SOC}& 1\end{array}\right]x\left(t\right),$$

(25)

where system input, $u\left(t\right)=I\left(t\right)$, $y\left(t\right)=V_t\left(t\right)-I\left(t\right)R_a,$ the unknown input vector, $v\left(t\right)=V_f\left(t\right)$, and the state space vector $x\left(t\right)$ has the following representation.

$$x\left(t\right)=\left[\begin{array}{c}SOC\\ V_b\end{array}\right]$$

The dynamics of the SOC estimation rate can be evaluated from (24) as follows:

$$\dot{SOC}=\frac{1}{Q_n}\int I\left(t\right)dt.$$

(26)

The Algorithm 1 to estimate SOC is shown below.

Algorithm 1 SOC Estimation

1: Initialize state-space vector 2: Evaluate SGP-SOL characteristic curve 3: Define system matrices, A, B, C, D 4: Solve LMI (19) to compute Y, K, and p > 0 5: Compute UIO matrices, M, N, L, G, and E 6: Estimate state 7: Calculate and update anode capacity

Anode capacity is directly related to the amount of charge carried to the cell during a charging cycle. Thus, in order to calculate anode capacity, the amount of charge transferred to a single cell during a charging cycle is calculated first. Though we have adopted the CC-CV charging protocol for the battery pack cycling, the current following through, the individual battery is no longer constant in the CC part of the cycling due to parallel cell connection. Thus, the amount of charge carried to the cell during a charging cycle is calculated by integrating current over the charging time as stated in Equation (27):

$$Q_{charge}={{\displaystyle \int }}_{t_0}^{t_{end}}I\left(t\right)dt,$$

(27)

where, $Q_{charge}$ is the total charge transferred to the individual cell during a charging cycle, $t_0$ and $t_{end}$ represent the starting and ending times of the charging cycle, $I\left(t\right)$ is the charging current as a function of time.

Anode capacity is calculated by dividing the total charge transferred to the cell by the difference between the estimated end of charge SOC and the initial SOC. In this study, the SOC of the anode is represented as the anode state of lithiation. Anode storage capacity is calculated by using Equation (28), as shown below:

$$Q_{anode}=\frac{Q_{charge}}{SO{C}_{end}-SO{C}_0 },$$

(28)

where, $Q_{anode}$ is the anode storage capacity. $SO{C}_0$ and $SO{C}_{end}$ represent the estimated initial and end of charging cycle SOC of the anode.

3. Results and Discussion

Anode SOC estimation result is evaluated in this section. We calculated cell capacity over cycles by using the coulomb counting method and compared it with the anode capacity. Anode SOC and capacity fade are then discussed, followed by a detailed discussion on the battery degradation based on the capacity fade.

3.1. Cell Capacity Fade

The individual cell capacity in the battery pack is calculated by using Equation (27). Figure 7 shows the CCCV current profile of the battery pack along with the voltage graph. A 0.7 C current was applied for both charging and discharging the battery pack. Although we have adopted a CCCV charging profile for this study, due to parallel cell connection, the measured current through individual cells does not have a CC part in the charging cycle. An individual cell charging cycle is shown in Figure 8. We see that the current profile has the shape of a CCCV charging profile, but a lot of fluctuation is observed for both the CC and CV parts of charging. This variable current was induced by the current sensor. This variable current is directly used as input to the system for anode SOC estimation. Figure 8 shows the variable charging profile of the individual cell in the battery pack, cell voltage, and capacity graph of the cell. Over the 68 charging cycles, the cell capacity decreases by 0.06 Ah, which is 2.7% of the initial cell capacity.

Three-Electrode Cell vs. Fresh Cell Capacity Fade

We compared the capacity fade of the fresh cell to the proposed three-electrode cell to demonstrate the effect of the proposed three-electrode cell setup on battery performance.

Figure 9 shows that the three-electrode cell capacity remains almost the same as the fresh cell. The fresh cell capacity degrades by 0.03% over the 20 cycles, whereas the three-electrode cell capacity degrades by 0.032%. Based on the capacity comparison, the proposed three-electrode cell setup does not significantly affect cell performance.

3.2. Anode Capacity Fade and Anode SOC

The anode SOC is estimated by using the UIO by observing the state estimation vector $\widehat{x}\left(t\right)$. This study shows that the estimated full charge anode SOC gradually decreases from 72.85% to 65.40%, and the initial guess for anode SOC in the state-space vector is set to 20%; moreover an initial guess for the anode capacity is made to be the cell initial capacity 2.22 Ah. the estimated anode SOC is shown in Figure 10b. From the anode SOC graph, we see that the initial SOC starts with a high value and then gradually stabilizes to a consistent upper value. The reason for this initial high SOC is the initial anode capacity guess for estimation. After each cycle, the anode capacity for that cycle is calculated by using Equation (28), and the system matrix, B is updated with the estimated anode capacity. The anode capacity is usually higher than the cell capacity; thus, the SOC estimation rate described in Equation (26) decreases after the estimated anode capacity update. The anode SOC can also be considered as a stoichiometric range or state of lithiation of the anode. We observed that the stoichiometric range of the anode decreases over the charging cycle due to cyclable lithium loss in the cell. The capacity of the anode is calculated by dividing the total charge carried to the cell in a charging cycle by the percentage difference of the estimated state of lithiation of the anode after each cycle, as stated in Equation (28). The calculated anode capacity over 68 cycles is shown in Figure 10a. From the figure, we see the initial anode capacity is much lower than the next cycle capacity. This is because the initial capacity guess for estimation is the cell capacity which is much smaller than the anode capacity and the initial SOC end-of-charge value is very high compared to the other cycles. Thus, from Equation (28) the difference between the end of charge SOC and the start of charge SOC is a higher number. Dividing the initial guess for anode capacity by this higher number produces a smaller anode capacity estimation value. This is why the initial anode capacity starts with a smaller number as shown in Figure 10a. This study shows that the anode capacity decreases by 0.062 Ah over 68 cycles which is 2.15% of the mean anode capacity of 2.88 Ah. We observed that anode capacity fade is less than the overall cell capacity fade by 1.27%. We also noticed that the anode capacity is higher than the cell capacity because there are not enough lithium ions to fill the total capacity of the anode [1].

3.3. Degradation Discussion

Based on the capacity fade discussion of the anode and cell, we can see the overall degradation pattern of the cell, which states that anode capacity is much higher than the cell capacity, and over time, the anode degrades slower than the cell. This could be due to the much higher anode storage capacity compared to the cell capacity. A comparison between the anode and cell capacity fade is shown in Figure 11 below. Figure 10 shows that anode capacity decreases gradually while the stoichiometric range (or SOC) also decreases. This is because the cell charge carrying capacity decreases faster, possibly due to various cell degradation mechanisms and cyclable lithium-ion loss. Figure 11 shows that cell storage capacity is always less than the anode storage capacity because there are not enough lithium ions in the cell to fill the anode’s remaining capacity [1]. This study presents an anode SOC estimation method by utilizing the variable current and the anode floating voltage as an unknown input to the system. A three-electrode battery pack setup is also presented that helps monitor an individual battery’s internal parameters.

3.4. Comparison to the Existing Research

To evaluate the accuracy of the designed observer, we have compared the estimated anode and cell capacity to the calculated capacity and the estimated anode SOC to the anode SOC presented in [1]. From Figure 12, we see that the estimated anode SOC is similar to the one presented in [1] with an estimation error of 5.33%. The estimated mean anode capacity for the single-cell varies by 0.21 Ah from the calculated value of the reference [1]. This could be due to the initial SOC estimated by the herein proposed method, which is close to zero at the beginning of each cycle. However, this value is not always zero, as presented in [1]. Figure 13 shows the anode capacity comparison between the estimated value and the calculated value from [1].

The capacity comparison in

Table 3. Capacity comparison to the existing works.

Initial Capacity	Estimated (Ah)	Calculated (Ah)
Cell type	UR18650AA. Sanyo	UR18650AA. Sanyo
Cell	2.22	2.22
Anode	2.88	3.09

shows that the cell capacity is the same as the reference [1] whereas the anode capacity varies by 0.21 Ah with an estimation error of 6.7%.

4. Conclusions

This paper presents a novel method to set up a three-electrode battery pack by using a battery casing as the reference electrode. To the best of our knowledge, a three-electrode battery pack has never been presented in studies. Furthermore, an unknown input observer-based estimation method is presented to estimate the anode SOC of an individual battery in the battery pack for variable current input. An anode capacity calculation method is described by using the estimated SOC to analyze battery degradation. This innovative approach uses the battery casing as a reference electrode to provide the battery’s internal state in real time. Thus, manufacturing a battery pack for practical application is feasible. This study shows that the cell degrades faster than the anode and also shows that the initial anode capacity is 30% higher than the cell′s initial capacity. Overall cell degradation can now be described from the capacity fade comparison between the anode and the cell.