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An Estimation Algorithm of Extended Kalman Filter based on improved Thevenin Model for the management of Lithium Battery System

We proposed a new estimation algorithm of extended Kalman filter (EKF) based on improved Thevenin model; Experiments were carried out to verify the validity with seven 4Ah lithium cobalt acid batteries in series. The experimental results showed that when using the algorithm, the estimation error of SOC is in the scope of error allowed, and the requirement of online SOC estimation can be satisfied.


Introduction
State Of Charge (SOC) is an important parameter to characterize the state of residual battery power. Accurately estimating SOC can effectively avoid battery overcharge or overdischarge and prolong the service life of the battery.SOC estimation is an essential function of the battery management system and also the key to reliable operation of cells [1].
SOC is the ratio of the remaining capacity of the battery to the role of its fully charged state. It can not be measured directly, and we can only estimate it by measuring physical quantities such as current and voltage of batteries.
Because the noise in practical application polluted the data of voltage and current measured by sensors, the traditional estimation methods of SOC, such as time integral method and open-circuit voltage method, are not ideal for reducing noise, so there are some shortcomings such as more error and difficulty in real-time estimation of SOC of batteries.
Kalman filtering algorithm [2] is a powerful tool to deal with noise. Because Classical Kalman is only suitable for the narrow field while Lithium battery SOC is a non-linear system, we use an extended Kalman algorithm to estimate the SOC of the battery.
By building a lithium battery test platform, we tested the algorithm by experiments and verified the validity of the algorithm. The results show that the state estimation satisfied the requirement of the online SOC estimation with faster speed and less error.

Research on Lithium Battery State Model and Parameter Identification
The accurate battery model is significant for SOC estimation of batteries. At present, the commonly used battery models include electrochemical model [3], artificial neural network model [4], an equivalent circuit model and so on.
The electrochemical model is generally suitable for the development and improvement of lithium batteries, but not for the research of the battery management system. Artificial Neural Network (ANN) model needs a lot of experimental data, which is too dependent on the historical data of batteries and is also not suitable for this study. Typical circuit models include Rint model, Thevenin model[ [5], PNGV model [6], Masimo Ceraolo model and so on. The comparison analysis shows that the Thevenin model can represent not only the sudden change of battery terminal voltage, but also the gradual shift in battery terminal voltage, and the calculation amount is appropriate. So this paper uses the Thevenin model.
We show the battery Thevenin model in Figure 1. In this model, the lithium battery is equivalent to a model consisting of electromotive force E, ohmic resistance R0, polarization resistance R1, polarization capacitance C, battery current I and terminal voltage V. The model have the electrical relations as follows:

Research on SOC Estimation Method for Lithium Batteries
There are many methods for estimating SOC of lithium batteries. The typical methods are the current integration method, open-circuit voltage method, discharge experiment method, neural network method, Kalman filter method [8] and so on. We estimated the current integration method according to the definition of SOC. The description of SOC is shown in Formula (2): Generally, we take the full state of the battery as t0, and the nominal capacitance of the cell as Q0. is a parameter that compensates for the effects of battery discharge rate, temperature and battery health on SOC. Because there are errors in obtaining the charging and discharging current of lithium batteries under the working condition, there are also more errors in SOC estimation. Discharge experiment method is quite reliable, but it consumes a lot of workforces and material resources and can not be estimated online. It is generally used in experimental research. Neural network method needs many training data, and it relies heavily on historical data and has a large amount of calculation. The measurement accuracy seriously affects the estimation accuracy. The application of the Kalman filter algorithm to estimate SOC of lithium battery has the advantages of high efficiency and good real-time performance. In this paper, we take the polarization voltage UC and SOC of lithium batteries as the system state variables, and the current I is considered as the system input. Then the state equation and observation equation are constructed. Then the comparison is discretized and linearized. Finally, the SOC is estimated by the Kalman algorithm.

Research on Extended Kalman Filtering Algorithms
Kalman filtering algorithm is an algorithm that calculates the current state of the system by using the linear state equation and input and output data of the system. Because Kalman filtering algorithm estimates in time domain entirely and does not transform between time domain and frequency domain, so the computation is small and the real-time estimation effect is quite good.
The classical Kalman filtering algorithm is suitable for linear systems, while the lithium battery SOC system is nonlinear. Extended Kalman filtering algorithm is an improvement of the traditional Kalman algorithm. It is an algorithm that expands the non-linear space equation by Taylor series and obtains the approximate linear space equation by discarding the second order and above terms. Then the Kalman filtering algorithm is applied to the linear space equation to estimate the current space state. It is suitable for discrete nonlinear systems.
The space of discrete nonlinear systems can be expressed as: The first part of the formula (3) represents the equation of state, and the second part describes the observation equation. To apply the Kalman filter, we expanded the non-linear functions f (*) and H (*) by the first order Taylor series and the results are shown in formula (4).
In Formula (4), when: Then formula (4) can be linearized as follows: We use the basic Kalman filter equation for the linearized model (5), then get the recursive process of extended Kalman filter.
In formula (6), P represents the mean square error, and K represents the Kalman gain. I am an n×m matrix. Q and R are the variances of ω and υ, and generally, do not change with the system.

Experimental verification
In this paper, the commonly used lithium cobalt oxide battery was used for experimental verification. The rated capacitance of the cell is 4Ah, and rated voltage is 3.7V working voltage range is 3.7~4.15V. We simulated the working state under actual working conditions. This paper adopts the lithium battery in a series of 7 for experimental verification.

The relationship between E-SOC curve
We left the cell under different SOC for 40 minutes, and measured the open circuit voltage, while the open circuit voltage is equal to the lithium battery E. Then we carried out the 7-order polynomial fitting of e-soc data. The obtained E-SOC curve is shown in figure 3, and the relationship of E-SOC is shown in equation (7) Fig. 3 The E-SOC curve of lithium cobalt oxide battery

Experiments for Model parameter identification
In this paper, Hybrid Pulse Power Characteristic (HPPC) is applied to determine the parameters of the model, and in the experiment, nine sampling points were set to collect the relationship between each parameter and SOC. These sampling points collected data of SOC from 100% to 10 %, once for every 10% change of SOC, and the obtained data are shown in table 1.

EKF estimation of SOC experiments for lithium batteries
We verified the EKF SOC algorithm by using the battery pack with a constant current of 0.5c, combined with equations (1), (2), (6) and (7)  In figure 4, the red curve is the real SOC, and the black curve is the extended Kalman estimated SOC curve.  figure 5 is the difference between the extended Kalman estimation of SOC and the actual value. It is found that the error of SOC estimation is less than 4%, which is within the allowable range.

Conclusion
Aiming at the SOC online estimation problem of the lithium battery pack, we conduct the Thevenin model equivalence for the lithium battery pack and propose an estimation method based on the extended Kalman filter algorithm. A series of 7 4Ah lithium cobalt battery pack was used to simulate the lithium battery pack for experimental verification. The results show that the estimation error is less than 4%, and within the allowable range, it can meet the requirement of online estimation of SOC of the lithium battery.