JP6755162B2 - Estimator - Google Patents

Description

The present invention relates to an estimation device that estimates the charge rate of a battery using a battery equivalent circuit model.

Conventionally, a device for estimating the internal state and parameters of a battery is known. For example, the battery parameter estimation device described in Patent Document 1 estimates the hysteresis voltage of the battery equivalent circuit model using the Hysteresis model of Plett described in Non-Patent Document 1. The estimation device estimates the open circuit voltage (OCV) and charge rate (SOC) of the battery based on the estimated hysteresis voltage.

JP-A-2016-090322

G. L. Plett: "Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs Part 2. Modeling and identification", Journal of Power Sources 134 (2004) 262-276

However, when trying to estimate the hysteresis voltage using the above-mentioned hysteresis model, the estimated value may in principle obtain an arbitrary value in the range of −∞ to + ∞. On the other hand, in an actual hysteresis phenomenon, it is usual to obtain an arbitrary value within a constrained predetermined range. That is, the conventional estimation device does not consider the restrictions caused by such an actual hysteresis phenomenon.

An object of the present invention made in view of such a viewpoint is to provide an estimation device that enables more accurate estimation in consideration of restrictions caused by an actual hysteresis phenomenon.

により定義される制約の範囲内において、

という傾きを有し、

により定義される制約の範囲外において、

という傾きを有する。 In order to solve the above problems, in the estimation device for estimating the charge rate of the battery using the equivalent circuit model of the battery according to the embodiment of the present invention,
Variables _x hysteresis voltage is expressed by a function of _{h (t) v h (t} ) , using the maximum value m of the hysteresis due to charging and discharging of the battery, and a variable _x h (t),

Within the constraints defined by

With the inclination

Outside the constraints defined by

It has an inclination of.

According to the present invention, it is possible to provide an estimation device that enables more accurate estimation in consideration of the constraints caused by the actual hysteresis phenomenon.

It is a figure which shows the functional block of the estimation apparatus which concerns on this embodiment connected to a battery. It is a measurement result of SOC-OCV characteristics of a lithium iron phosphate battery. It is a circuit diagram which shows the equivalent circuit model of a battery. It is a figure which shows the relationship between the hysteresis voltage and a variable x _h according to the formula (27). It is a figure which shows the estimation result of the hysteresis voltage at the time of estimating by UKF with respect to the conventional Hysteresis model by Plett which does not consider the constraint. It is a figure which shows the estimation result of the hysteresis voltage by UKF when the constraint is taken into consideration in the estimation apparatus which concerns on this embodiment. It is a schematic diagram which showed the relationship between an equivalent circuit model and a Kalman filter. It is a figure which showed the current value and voltage observation value input to a Kalman filter. It is a figure which shows the estimation result of the hysteresis voltage at the time of inputting the double hysteresis voltage as an observation value into UKF in the estimation apparatus which concerns on this embodiment. It is a figure which shows the relationship between the hysteresis voltage and a variable x _h according to the formula (46).

Hereinafter, an embodiment of the present invention will be described with reference to the accompanying drawings. FIG. 1 is a diagram showing a functional block of the estimation device 10 according to the present embodiment connected to the battery 1.

The estimation device 10 for the battery 1 of the present embodiment is used for a vehicle such as an electric vehicle or a hybrid electric vehicle. Such a vehicle is equipped with an electric motor for driving the vehicle, a battery, a controller thereof, and the like. Then, electric power is supplied (discharged) to the electric motor, braking energy is regenerated from the electric motor during braking, and electric power is recovered (charged) from the ground charging facility to the battery. The state inside the battery changes due to the input / output of the charge / discharge current to and from the battery. By monitoring the internal state while estimating it with the estimation device 10, necessary information such as the remaining battery level is collected.

As shown in FIG. 1, the estimation device 10 of the battery 1 includes a voltage sensor (terminal voltage detection unit) 2, a current sensor (charge / discharge current detection unit) 3, an estimation unit 4, a charge amount calculation unit 5, and the like. A charge rate calculation unit 6 and a soundness calculation unit 7 are provided. The estimation unit 4, the charge amount calculation unit 5, the charge rate calculation unit 6, and the soundness calculation unit 7 are composed of, for example, an in-vehicle microcomputer.

The battery 1 is, for example, a rechargeable battery (secondary battery). The battery 1 will be described as being a lithium ion battery in the present embodiment, but the battery 1 is not limited thereto. The battery 1 may be another type of battery.

The terminal voltage detection unit 2 is, for example, a voltage sensor and detects the terminal voltage value v of the battery 1. The terminal voltage detection unit 2 outputs the detected terminal voltage value v to the estimation unit 4.

The charge / discharge current detection unit 3 is, for example, a current sensor and detects the charge / discharge current value i of the battery 1. The charge / discharge current detection unit 3 outputs the detected charge / discharge current value i to the estimation unit 4.

The estimation unit 4 includes an equivalent circuit model 41 of the battery 1 and a Kalman filter 42. The estimation unit 4 can estimate (calculate) the parameter value of the equivalent circuit model 41, the open circuit voltage OCV (Open Circuit Voltage) of the battery 1, and the internal state amount of the battery 1 by using the Kalman filter 42. In the present embodiment, the estimation unit 4 simultaneously estimates the parameter value and the internal state quantity based on the terminal voltage v from the terminal voltage detection unit 2 and the charge / discharge current i from the charge / discharge current detection unit 3. The estimation unit 4 calculates the open circuit voltage OCV based on the estimated parameter value. Details of the estimation and calculation processing performed by the estimation unit 4 will be described later. Further, the estimation unit 4 outputs the calculated open circuit voltage OCV to the charge rate calculation unit 6 and the soundness calculation unit 7.

The equivalent circuit model 41 is a Foster-type RC ladder circuit represented by an approximation based on the sum of infinite series, in which a parallel circuit of a resistor and a capacitor is connected, or an approximation by continuous fraction expansion in which resistors connected in series are grounded by a capacitor. It is composed of a Cowell type RC ladder circuit represented by. The resistor and capacitor are parameters of the equivalent circuit model 41.

The Kalman filter 42 compares the outputs of both the target system model (equivalent circuit model 41 in this embodiment) and the actual system when the same input signal is input. If there is an error in the outputs of both, the Kalman filter 42 multiplies this error by the Kalman gain and feeds it back to the above model. As a result, the Kalman filter 42 modifies the model so that the error between the two is minimized. The Kalman filter 42 estimates the parameters of the model by repeating these steps.

The charge amount calculation unit 5 acquires the charge / discharge current value i of the battery 1 detected by the charge / discharge current detection unit 3. The charge amount calculation unit 5 obtains the charge amount output from the battery 1 and the charge amount input to the battery 1 by sequentially accumulating these values. The charge amount calculation unit 5 calculates the charge amount Q of the current battery 1 by subtracting the input / output charge amount from the residual charge amount stored before the sequential integration calculation. This charge amount Q is output to the soundness calculation unit 7.

Since the relationship between the open circuit voltage OCV and the charge rate SOC (State of Charge) is not easily affected by the temperature and deterioration of the battery 1, the charge rate calculation unit 6 obtains correlation data obtained by obtaining these relationships in advance through experiments or the like. Remember. The charge rate calculation unit 6 may store, for example, the SOC-OCV characteristics as a table. The charge rate calculation unit 6 estimates the charge rate SOC at the corresponding time point from the open circuit voltage OCV estimated by the estimation unit 4 based on the corresponding table. The estimated charge rate SOC is used for battery management of battery 1.

The soundness calculation unit 7 has a characteristic table showing the correlation between the charge amount Q and the open circuit voltage OCV for each soundness SOH (State of Health) divided by a predetermined width. Details of the characteristic table are disclosed in, for example, Japanese Patent Application Laid-Open No. 2012-57956 filed by the applicant. The open circuit voltage OCV estimated by the estimation unit 4 and the charge amount Q calculated by the charge amount calculation unit 5 are input to the soundness calculation unit 7. The soundness calculation unit 7 calculates which of the soundness SOH ranges in the above-mentioned characteristic table the input values fall into. The soundness calculation unit 7 outputs the applicable soundness SOH.

Next, the processing of the estimation unit 4 will be described.

In the present embodiment, the estimation unit 4 simultaneously estimates the internal state quantity of the battery 1 and the parameter value using the Kalman filter 42 in the equivalent circuit model 41. It is preferable that the internal state amount of the battery 1 includes the charge rate SOC of the battery 1 and the parameter value includes at least one of the diffusion capacity C _d or the diffusion resistance R _d described later. In the present embodiment, the Kalman filter 42 is, for example, an unscented Kalman filter (UKF), but is not limited thereto. UKF approximates the probability distribution using weighted sample points called sigma points and calculates each weighted transition. Specifically, UKF calculates the mean and variance after the transition for each sigma point and adds them according to the weights. As a result, UKF can approximate the probability distribution after the transition so that it is closer to the true value and the amount of calculation does not increase too much. In addition, UKF does not approximate the system, but approximates the probability distribution with sigma points, so there are no restrictions on the non-linearity of the system.

In an actual battery, a hysteresis phenomenon showing different SOC-OCV characteristics after charging and discharging may occur. In this case, the SOC-OCV characteristics of the battery cannot be accurately represented. The hysteresis phenomenon occurs depending on the material of the electrode, and the effect of the hysteresis phenomenon is particularly large when lithium phosphate is used.

FIG. 2 shows the measurement results of the SOC-OCV characteristics of the lithium iron phosphate battery. Focusing on FIG. 2A, it can be seen that there is a difference in OCV between the characteristics during charging and the characteristics during discharging. Further, in FIG. 2 (b) in which the portion surrounded by the broken line in FIG. 2 (a) is enlarged, it can be seen that the hysteresis characteristic is exhibited even if the SOC is discharged at the time of 68%. More specifically, even in a small loop when discharging is performed until the SOC decreases from 68% to 38% and then charging is performed until the SOC increases from 38% to 68%, the charging time There is a difference in OCV between the characteristics and the characteristics at the time of discharge. Such a small loop is formed inside the entire loop shown in FIG. 2 (a). That is, assuming that the maximum value of the OCV difference between the characteristics during charging and the characteristics during discharging caused by hysteresis is 2 × m, the hysteresis voltage v _h is restricted by the following equation (1).

Plett's hysteresis model is known as one of the models expressing such a hysteresis phenomenon. Estimating apparatus 10 of the present embodiment, as the equivalent circuit shown in FIG. 3, employs the equivalent circuit model 41 incorporating the hysteresis model Plett constituted by a parallel connected variable resistor R _h and the variable capacitor C _h ..

FIG. 3 is a circuit diagram showing an equivalent circuit model 41 of the battery 1. FIG. 3A is a circuit diagram showing the entire equivalent circuit model 41 of the battery 1. FIG. 3B is a circuit diagram showing an equivalent circuit model of the Warburg impedance Z _w constituting the equivalent circuit model 41.

The equivalent circuit model 41 of the battery 1 will be described with reference to FIG. As shown in FIG. 3, an equivalent circuit model 41 of the battery 1 has an open voltage OCV, and a variable resistor _{R h} and the variable capacitor _{C h} connected in parallel, the internal resistance _{R 0,} the Warburg impedance _{Z w,} Suppose an open circuit in which is connected in series.

とすると、連続時間系の状態空間表現は、

となる。ただし、

である。また、

である。この状態空間表現で、ＦＣＣはバッテリ１の満充電容量、Ｒ_０はバッテリ１の直達抵抗である。また，Ｃ_ｄ及びＲ_ｄはバッテリ１の拡散現象を表すパラメータであり、それぞれ拡散容量及び拡散抵抗である。Ｃ_ｈ及びＲ_ｈはバッテリ１のヒステリシス現象を表すパラメータであり、それぞれヒステリシス容量及びヒステリシス抵抗である。パラメータγはヒステリシスの速さを表すパラメータである。さらに、ｆ_ＯＣＶ（ＳＯＣ（ｔ））は、充電側及び放電側のＳＯＣ-ＯＣＶ特性の平均を表す非線形な関数（図１のＯＣＶ_ｍとＳＯＣとの関係に相当）である。 In the model shown in FIG. 3, the input is the current u = i, the output is the terminal voltage y = v, and the state variable is

Then, the state-space representation of the continuous time system is

Will be. However,

Is. Also,

Is. In this state-space representation, FCC is the full charge capacity of the battery 1 and R ₀ is the direct resistance of the battery 1. Further, C _d and R _d are parameters representing the diffusion phenomenon of the battery 1, and are diffusion capacity and diffusion resistance, respectively. _Ch and R _h are parameters representing the hysteresis phenomenon of the battery 1, and are the hysteresis capacitance and the hysteresis resistance, respectively. The parameter γ is a parameter representing the speed of hysteresis. Further, f _OCV (SOC (t)) is a non-linear function (corresponding to the relationship between OCV _m and SOC in FIG. 1) representing the average of SOC-OCV characteristics on the charging side and the discharging side.

と選ぶと、拡大系状態ベクトルは、

となる。このとき、拡大系は、

となる。ただし、

である。ここで、パラメータベクトルを、式（１２）のように定義したが、ヒステリシスの最大値ｍは、上述のとおり、充放電のＳＯＣ-ＯＣＶ特性の幅（の２分の１）に相当する量であるので、予め計測しておくことができる。そのため、この値については既知として、推定するパラメータから除外してもよい。すなわち、

であってもよい。これにより、推定するパラメータが少なくなるので、他のパラメータの推定精度が向上する。以下では、説明の簡略化のために、式（１８）のパラメータベクトルを用いる。 Here, the estimation unit 4 constitutes an expansion system in which the parameters to be estimated in the equivalent circuit model 41 of the battery 1 are added as state variables, and simultaneously estimates the internal state amount of the battery 1 and the parameter values. Select the direct resistance R ₀ , the diffusion resistance R _d , and the diffusion capacitance C _d as the parameters to be estimated, and set the parameter vector.

If you select, the expanded state vector is

Will be. At this time, the expansion system

Will be. However,

Is. Here, the parameter vector is defined as in the equation (12), but the maximum value m of the hysteresis is an amount corresponding to (1/2) the width of the SOC-OCV characteristic of charge / discharge as described above. Since there is, it can be measured in advance. Therefore, this value may be known and excluded from the estimated parameters. That is,

It may be. As a result, the number of parameters to be estimated is reduced, so that the estimation accuracy of other parameters is improved. In the following, the parameter vector of equation (18) is used for simplification of the description.

となる。これにより、拡大系状態ベクトルは、

となる。このとき、拡大系は、

となる。ただし、

である。 If the above expansion system is used as it is, the difference in the order of the parameters to be estimated is large, and the accuracy of the numerical calculation by the computer deteriorates. Therefore, the accuracy of numerical calculation is improved by logarithmizing each parameter and estimating the exponential part. Logarithmic parameter vector

Will be. As a result, the expanded state vector becomes

Will be. At this time, the expansion system

Will be. However,

Is.

より具体的には、式（１）により定義される制約の範囲内において、ヒステリシス電圧ｖ_ｈは、変数ｘ_ｈの変化に対して傾きが略１となるような関数によって置換される。一方で、制約の範囲外では、ヒステリシス電圧ｖ_ｈは、変数ｘ_ｈの変化に対して傾きが１よりも大きいような関数によって置換される。 Here, the estimation device 10 according to the present embodiment performs the following processing in order to consider the restriction due to hysteresis represented by the above-mentioned equation (1). That is, the estimation unit 4 substitutes the hysteresis voltage v _h as follows, and estimates the variable x _h (t) instead of the hysteresis voltage v _h .

More specifically, within the range of the constraint defined by Eq. (1), the hysteresis voltage v _h is replaced by a function such that the slope becomes approximately 1 with respect to the change of the variable x _h . On the other hand, outside the constraints, the hysteresis voltage v _h is replaced by a function such that the slope is greater than 1 with respect to changes in the variable x _h .

図４は、式（２７）に従った、ヒステリシス電圧ｖ_ｈと変数ｘ_ｈとの関係を示す図である。図４に示すとおり、ヒステリシス電圧ｖ_ｈは、式（１）により定義される制約の範囲内においては、傾き略１で変化する。一方で、ヒステリシス電圧ｖ_ｈは、制約の範囲外では、変数ｘ_ｈの絶対値が大きくなるにつれて、その傾きを増大させる。以下では、式（２７）におけるｘ_ｈ（ｔ）／ｍを改めてｘ_ｈ（ｔ）と置き直す。すなわち、以下では、変数ｘ_ｈ（ｔ）の軸をｍ分の１倍にして、各式の導出を行う。 One differentiable function that satisfies both of the above conditions is:

FIG. 4 is a diagram showing the relationship between the hysteresis voltage v _h and the variable x _h according to the equation (27). As shown in FIG. 4, the hysteresis voltage v _h changes with a slope of approximately 1 within the range of the constraint defined by the equation (1). On the other hand, the hysteresis voltage v _h increases its slope as the absolute value of the variable x _h increases outside the constraint range. In the following, x _h (t) / m in the equation (27) is replaced with x _h (t). That is, in the following, each equation is derived by multiplying the axis of the variable x _h (t) by 1 / m.

となる。式（２８）のヒステリシス電圧ｖ_ｈを式（２７）で置き換えると、

が得られる。ただし、式（２９）の導出過程において、

と仮定した。すなわち、ヒステリシスの最大値ｍは、時間変化しないものと仮定する。 Here, when extracting the equation for hysteresis voltage v _h from Equation (23),

Will be. Replacing hysteresis voltage _{v h} of formula (28) in equation (27),

Is obtained. However, in the process of deriving equation (29),

Was assumed. That is, it is assumed that the maximum value m of hysteresis does not change with time.

とすると、拡大系状態ベクトルは、

と表せる。このとき、拡大系は、

となる。ただし、

である。 The estimation unit 4 puts the above result into a logarithmic expansion system. More specifically, it is as follows. That is, first, the state vector and the parameter vector

Then, the expanded state vector is

Can be expressed as. At this time, the expansion system

Will be. However,

Is.

The expansion system that is logarithmic as described above and that considers the constraint of hysteresis is a state-space representation of a continuous system. Therefore, the estimation unit 4 finally discretizes using the Runge-Kutta method, the Euler method, or the like. In the present embodiment, as an example, the estimation unit 4 discretizes by the Runge-Kutta method. As a result, the estimation unit 4 can reduce the discretization error as compared with the Euler method which is usually often used.

となる。ただし、

である。また、

である。 When the sampling period is T _S seconds,

Will be. However,

Is. Also,

Is.

Subsequently, the estimation result by UKF in consideration of the constraint obtained by the processing of the estimation unit 4 will be described.

Figure 5 does not consider the constraint is a diagram showing the estimation result of the hysteresis voltage v _h in the case of performing estimation by UKF against hysteretic model conventional Plett. 6, the estimation device 10 according to the present embodiment and showing the estimation result of the hysteresis voltage v _h by UKF in consideration of the constraints. FIG. 6A is a diagram comparing the output value by Plett's hysteresis model and the estimated value by UKF. 6 (b) is a diagram showing the estimation error of the hysteresis voltage v _h.

For comparison the beginning, with reference to FIG. 5, without considering constraints, will be described estimation result of hysteresis voltage v _h in the case of using a hysteresis model by conventional Plett. In FIG. 5, the output value by the conventional Plett's hysteresis model is shown by a broken line, and the estimated value by UKF is shown by a solid line. As shown in FIG. 5, when the constraint is not taken into consideration as described above, the output value by the conventional Plett's hysteresis model and the estimated value by UKF are significantly different in a part of the time domain. In particular, in the two dashed lines in FIG. 5, the output value and the estimated value do not match. Further, in the two dashed lines enclosing parts, the estimated value by the UKF hysteresis voltage _{v h} is also greatly exceed constraints (about -10 mV).

Meanwhile, with reference to FIG. 6, the estimation result of the hysteresis voltage v _h by UKF in the case of considering a constraint is described. In FIG. 6A, the output value by the hysteresis model of Plett is shown by a broken line, and the estimated value by UKF is shown by a solid line, as in FIG. In the estimation device 10 according to the present embodiment in consideration of the restrictions, as shown in FIG. 6A, the output value by the hysteresis model of Plett and the estimation value by UKF are in good agreement. In fact, as shown in FIG. 6 (b), the estimated error of the hysteresis voltage v _h is within the range of a few mV. Furthermore, the estimated value by the UKF hysteresis voltage _{v h} is falls constraint in all the time domain (about ± 10 mV).

Then, consider a case where input to the UKF twice the hysteresis voltage v _h as observed values. FIG. 7 is a schematic diagram showing the relationship between the equivalent circuit model 41 and the Kalman filter 42. FIG. 8 is a diagram showing a current value and a voltage observed value input to the Kalman filter 42. FIG. 9 is a diagram showing an estimation result of the hysteresis voltage v _h when the double hysteresis voltage v _h is input to UKF as an observed value in the estimation device 10 according to the present embodiment. FIG. 9A is a diagram comparing the output value by Plett's hysteresis model and the estimated value by UKF. 9 (b) is a diagram showing the estimation error of the hysteresis voltage v _h.

As shown in FIG. 7, the estimation unit 4 branches the output of the true voltage value from the equivalent circuit model 41, that is, the hysteresis model of Plett, and doubles the true voltage value (α = 2) at the branched output. .. The estimation unit 4 inputs the doubled true voltage value to the Kalman filter 42 as a voltage observation value.

At this time, as shown in FIG. 8, the hysteresis voltage v _h, observations are that indicated by the solid line, in all the time domain, the multiple of the true value indicated by a broken line. That is, the observed values exceed the constrained range (about ± 10 mV) in some time domains.

On the other hand, focusing on FIG. 9, even when the observed voltage value exceeds the range of the constraint, the output value by the hysteresis model of Plett, that is, the true value and the estimated value by UKF constituting the Kalman filter 42 are often found. Match. In fact, as shown in FIG. 9 (b), the estimated error of the hysteresis voltage _{v h} is approximately zero. Furthermore, the estimated value by the UKF hysteresis voltage v _h, although beyond the scope of some limitations in the dashed line enclosed portion immediately fits within the constraints, falls within constraints (about ± 10 mV) at substantially all time domain.

The estimation device 10 as described above enables more accurate estimation in consideration of the restrictions caused by the actual hysteresis phenomenon. That is, estimator 10 can estimate the hysteresis voltage v _h that fall within the constraints actual range of the hysteresis phenomenon. In particular, estimator 10, by replacing the hysteresis voltage v _h by the equation (27), as shown in FIGS. 6 and 9, can be reduced estimation error. Further, the estimation unit 10, as shown in FIG. 9, even when the voltage observed value exceeds the range of constraints, it is possible to accurately estimate within the constraints of the hysteresis voltage v _h. This is considered to be due to the following reasons. That is, in the equation (27), when the variable x _h becomes + m or more or −m or less, the absolute value of the hysteresis voltage v _h increases sharply. On the other hand, in an actual phenomenon, the hysteresis voltage v _h takes a value of −m or more and + m or less. Therefore, the discrepancy between the estimated value and the measured value increases as compared with the case where the replacement is not performed by the equation (27). When the divergence increases, the Kalman filter 42 adjusts the Kalman gain so as to reduce it and immediately feeds back. As a result, the estimation device 10 can keep the estimated value within a range that does not deviate significantly from the range of the constraint.

It will be apparent to those skilled in the art that the present invention can be realized in certain forms other than those described above without departing from its spirit or its essential features. Therefore, the above description is exemplary and is not limited thereto. The scope of the invention is defined by the appended claims, not by the earlier description. Some of all changes that are within their equality shall be included therein.

ただし、ｋは１よりも大きな値の係数である。 For example, the estimation apparatus 10 according to the present embodiment, a hysteresis voltage v _h is replaced by equation (27), but is not limited thereto. For example, the estimation device 10 may replace the hysteresis voltage v _h with the following equation.

However, k is a coefficient having a value larger than 1.

FIG. 10 is a diagram showing the relationship between the hysteresis voltage v _h and the variable x _h according to the equation (46). As shown in FIG. 10, the hysteresis voltage v _h changes with a slope of 1 within the range of the constraint defined by the equation (1) (−m ≦ x _h ≦ m). On the other hand, the hysteresis voltage v _h changes with a slope k outside the constraint range. Thus, estimating apparatus 10, since replacing the hysteresis voltage v _h a simple linear function, can be simplified equation in the estimation process.

As described above, the estimation device 10 may replace the hysteresis voltage v _h with any function as long as it is a function satisfying the conditions of the equations (25) and (26).

1 Battery 2 Voltage sensor (terminal voltage detector)
3 Current sensor (charge / discharge current detector)
4 Estimating unit 41 Equivalent circuit model 42 Kalman filter 5 Charge amount calculation unit 6 Charge rate calculation unit 7 Soundness calculation unit 10 Estimator

Claims (3)

An estimation device that estimates the charge rate of the battery using the equivalent circuit model of the battery.
Variables _x hysteresis voltage is expressed by a function of _{h (t) v h (t} ) , using the maximum value m of the hysteresis due to charging and discharging of the battery, and a variable _x h (t),

Within the constraints defined by

With the inclination

Outside the constraints defined by

Has an inclination,
Estimator. The hysteresis voltage v _h (t) and the variable x _h (t) are

Satisfy the relationship,
The estimation device according to claim 1. The hysteresis voltage v _h (t) and the variable x _h (t) have k as a coefficient having a value larger than 1.

Satisfy the relationship,
The estimation device according to claim 1.

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