† Corresponding author. E-mail:
This paper focuses on the modeling of LiFePO4 battery open circuit voltage (OCV) hysteresis. There exists obvious hysteresis in LiFePO4 battery OCV, which makes it complicated to model the LiFePO4 battery. In this paper, the recursive discrete Preisach model (RDPM) is applied to the modeling of LiFePO4 battery OCV hysteresis. The theory of RDPM is illustrated in detail and the RDPM on LiFePO4 battery OCV hysteresis modeling is verified in experiment. The robust of RDPM under different working conditions are also demonstrated in simulation and experiment. The simulation and experimental results show that the proposed method can significantly improve the accuracy of LiFePO4 battery OCV hysteresis modeling when the battery OCV characteristic changes, which conduces to the online state estimation of LiFePO4 battery.
In recent years, electric vehicles (EVs) have been more and more popular due to the advantages of zero pollution and high energy efficiency.[1–3] Li–ion batteries are considered to be the most promising energy storage system to meet the specific energy and power demands of EVs, which has high energy density, good cycle-life performance, and low self-discharge rate.[4–10] LiFePO4 battery is one of the most promising solutions for EV energy storage system with the benefits of excellent thermal stability and low cost. As a crucial characteristic relationship, the state of charge (SOC)–OCV relationship plays an important role in EV battery manage system (BMS). However, there exhibits a pronounced hysteresis between the SOC–OCV relationship of LiFePO4 battery.[11] The existence of hysteresis means that the SOC–OCV curve is not a one-to-one mapping but the OCV value varies at the same SOC point between charge and discharge. This special characteristic makes it difficult to accurately model the SOC–OCV relationship of LiFePO4 battery.
There have been extensive studies dealing with battery hysteresis modeling.[12–25] The Preisach model (PM) is one of the most popular methods for battery hysteresis modeling. The Preisach model was proposed by Preisach[21] in 1935 and has been used for hysteresis modeling in various kinds of systems such as magnetics, piezoelectric actuators, batteries, etc. In Ref. [22], the discrete Preisach model was used to describe the OCV hysteresis for NiMH battery. In Ref. [23], the same approach was applied to the OCV hysteresis modeling of LiFePO4 battery. In Refs. [22] and [23], the discrete Preisach model worked effectively both for NiMH battery and LiFePO4 battery. In Ref. [24], the Everett function was adopted to discretize the Preisach model, and the first-order reversal branch test was used to identify the density function, which simplifies the measurement and reduces the computation resources. In Ref. [25], an adaptive discrete Preisach model is used to model the hysteresis of LiFePO4 battery.
It is noticeable that the battery parameters are subjected to change due to different working and aging conditions, which means that the SOC–OCV model works in a certain condition may not work well in another condition. This phenomenon makes it necessary to build a model that can be adapted to the change of the hysteresis characteristic of LiFePO4 battery. However, this problem has rarely been discussed in existing researches. As a main contribution of this paper, an SOC–OCV modeling method that can be adapted to the change of battery OCV characteristic is proposed based on the recursive discrete Preisach model. The rest of this paper is organized as follows. In Section
The classical Preisach model is defined as an infinite collection of ideal relay operators as shown in Fig.
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![](cpb_26_12_127503/cpb_26_12_127503_ieqn1.gif)
And the Preisach model is defined as
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In practical cases, the threshold of α and β is limited by the value of u(t), so the integration domain can be restricted to a triangular domain, which is defined as the Preisach triangle. As illustrated in Fig.
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The characteristic of the Preisach operator enables the Preisach model to memorize the historical paths of the input. The history of the input can be represented in the Preisach triangle by a staircase line L(t), which is composed of horizontal and vertical line(s). Each segment corresponds to a rising or falling part of the input history. For the increasing input, the memory curve is horizontal and moves up towards the local maximum value along the vertical axis; for the decreasing input, the memory curve becomes vertical and moves towards the local minimum value along axis β. With the rising and falling of the input values, new extrema are produced consistently and old extrema are replaced by the new ones if the input exceeds the old extrema.[26]
As shown in Fig.
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To calculate the output of the Preisach function, the weight function
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![](cpb_26_12_127503/cpb_26_12_127503_ieqn19.gif)
![](cpb_26_12_127503/cpb_26_12_127503_ieqn20.gif)
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As mentioned in the Introduction, the characteristic of the battery changes under different working and aging conditions. To solve this problem, an online SOC–OCV modeling algorithm should be adopted. Inspired by the work of Ruderman M et al.,[27] a recursive discrete Preisach model is adopted in this paper. The main difference between CPM and RDPM is that in RDPM the Preisach density function is identified online based on the output increment error As shown in Fig.
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From Eq. (
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It is evident that the error increase relates to the difference between
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![](cpb_26_12_127503/cpb_26_12_127503_ieqn30.gif)
The experimental setup consists of (i) LiFePO4 test cells, (ii) chroma 72001 battery test system, (iii) a host computer, and (iv) a thermal chamber. Two LiFePO4 test cells are used and the details are given in Table
![]() | Table 1.
Details of cells used. . |
Pulse current tests (PCTs) are carried out to measure the OCVs of the battery at different SOCs. As shown in Fig.
To identify the PDF of the battery, the charging first order reversal (FOR) branches are tested and the method of Everett function is used. In a charging FOR test, the cell is first charged to 100% SOC and then discharge to the full discharge state. Then the cell is charged to 100% SOC again and discharge to 10% SOC. The charge-discharge loop repeats until the discharge point reaches 90% SOC. The detailed procedure of the charging FOR test is shown in Fig.
To verify the PDF identified with the charging FOR branch test, another hysteresis input beyond the training data of the FOR branch is designed, which is named loop H2. To examine the performance of the tested PDF in the whole SOC range, loop H2 is designed to be comprised of three minor loops that spread at the high, medium and low location of the whole SOC range. Figure
![](cpb_26_12_127503/cpb_26_12_127503_ieqn31.gif)
To model the SOC–OCV relationship of LiFePO4 battery online with RDPM, the battery SOC is chosen as an input variable and OCV is chosen as the output variable. A very important issue for RDPM is to determine the value of N (the Preisach plane is divided by
The PDFs identified in the experiments with the charging FOR branches are shown in Figs.
Model verification experiments are carried out for both cell A and cell B to verify the performance of DPM under the extended input beyond the training data. Loop H2 is used as the verification hysteresis input and the OCV relative error is used to judge the accuracy of the estimated PDF. The OCV relative error is computed from
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![](cpb_26_12_127503/cpb_26_12_127503_ieqn33.gif)
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To test the robust of DPM when the PDF of the cell changes, the PDF of cell A is used to estimate the OCV value of cell B with loop H2 used as the input to simulate the change of the cell hysteresis characteristic. The estimated OCV curve and the measured OCV curve are shown in Fig.
![]() | Fig. 12. (color online) DPM modeling results when cell PDF changes. (a) Comparison result of SOC–OCV curve; (b) OCV estimation error scatter diagram. |
![]() | Table 2.
Comparisons between RMS and peak value of the OCV relative errors estimated with DPM and RDPM. . |
The online identification of the PDF is carried out by using the loop H3 as an input and the PDF of Cell A is used as a reference PDF to calculate the reference OCV in the RDPM process. The initial value of the estimated PDF is set to be zero. The Preisach distributions taken for example at the recursive steps of n = 50, 100, 150, and 1000 in the estimation process are shown in Fig.
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![]() | Fig. 13. (color online) Estimated PDF at the recursive steps of (a) n = 50; (b) n = 100; (c) n = 150; (d) n = 1000. |
The resulting cumulative absolute error in the RDPM process is shown in Fig.
![]() | Fig. 15. (color online) Comparison between estimated OCV and reference OCV in the estimation process as well as the logarithm of the OCV estimation error. |
To further verify the performance of RDPM, the PDF acquired from the RDPM process is used to model the SOC–OCV relationship under the input of loop H2 and the modeling result is compared with the result in Subsection
The influence of initial PDF on the performance of RDPM is also studied. In this condition, cell B is used as a reference cell and the initial PDF is set to be the PDF of cell A and zero. Loop H3 is still used as the hysteresis input. The comparison result of the PDF cumulative absolute error is shown in Fig.
![]() | Fig. 17. (color online) Variations of PDF cumulative absolute error with recursive step number for different initial PDF values. The insert shows the linear amplification. |
![](cpb_26_12_127503/cpb_26_12_127503_ieqn35.gif)
In actual usages, the value of OCV cannot be measured directly and is usually estimated with battery models. However, as there exist errors in model parameters, the estimated OCV usually deviates from the true OCV. To testify the reliability of the proposed RDPM algorithm under the existence of reference OCV uncertainty, random errors with 5-mV and 10-mV magnitude (the mean values are both zero) are added to the reference OCV in the estimation process. Figure
![]() | Fig. 19. (color online) Variations of estimated PDF cumulative absolute error for different reference OCV uncertainties. |
This paper focuses on the modeling of LiFePO4 battery OCV hysteresis. The RDPM algorithm which is originally used in the magnetic material hysteresis characterization is used for the hysteresis modeling of LiFePO4 battery. The experimental result shows that the proposed method can significantly improve the accuracy of LiFePO4 battery OCV hysteresis modeling when the battery OCV characteristic changes. Besides, the influence of initial estimation PDF and reference OCV uncertainties are also discussed. The simulation and experimental results show that the proposed method has a stable working performance under different initial estimation PDFs and reference OCV uncertainties. By means of this research, battery parameter and state estimation technologies such as battery ECM and Kalman filter can further be combined with to realize a more accurate state estimation of LiFePO4 battery. The proposed method shows a promising performance that will benefit the state estimation of LiFePO4 battery.