Fractional-order KiBaM battery model considering nonlinear capacity characteristics and parameter identification method

Abstract

A fractional-order KiBaM battery model considering nonlinear capacity characteristics and a parameter identification method includes a temporary capacity portion and an available capacity portion for describing nonlinear capacity characteristics of a battery, wherein the temporary capacity portion represents the power that can be directly obtained during the discharge, indicating the state of charge (SOC) of the battery; the available capacity portion represents the power that cannot be directly obtained, and such two portions are connected; when the battery is discharged, the load current i flows out from the temporary capacity portion, and a power passing rate coefficient of such capacity portions is obtained; and the nonlinear capacity effect and recovery effect of the battery are denoted by the height ratio of the temporary capacity and available capacity portions in view of the magnitude of the fractional order of battery capacity characteristics.

Claims

1. A fractional-order KiBaM battery model considering nonlinear capacity characteristics, comprising: a temporary capacity portion and an available capacity portion for describing nonlinear capacity characteristics of a battery, wherein the temporary capacity portion represents the power that can be directly obtained during the discharge, indicating the state of charge (SOC) of the battery; the available capacity portion represents the power that cannot be directly obtained, and such two portions are connected by a channel; when the battery is discharged, the load current i flows out from the temporary capacity portion, and a power passing rate coefficient of such capacity portions is obtained at the same time; and a nonlinear capacity effect and recovery effect of the battery are expressed by using a height ratio of the temporary capacity and available capacity portions combined with a magnitude of a fractional order of battery capacity characteristics, wherein the fractional-order KiBaM battery model is applied to the battery for simulating internal characteristics of the power battery.

2. The fractional-order KiBaM battery model considering nonlinear capacity characteristics according to claim 1, wherein the sum of the temporary capacity portion and the available capacity portion is the total capacity of the battery.

3. The fractional-order KiBaM battery model considering nonlinear capacity characteristics according to claim 1, wherein when the battery is completely discharged, the height of the temporary capacity portion is zero.

4. The fractional-order KiBaM battery model considering nonlinear capacity characteristics according to claim 1, wherein the temporary capacity is denoted by y.sub.1 and represents the power that can be directly obtained during the discharge, and its height is denoted by h.sub.1 and represents the SOC of the battery; the available capacity is denoted by y.sub.2 and represents the power that cannot be directly obtained, and its height is denoted by h.sub.2; the sum of y.sub.1 and y.sub.2 is the total capacity of the battery; c represents a distribution ratio of the battery capacity between such two portions, and the following relationship exists: $\{\begin{array}{c}{h}_1\left(t\right)=\frac{y_1\left(t\right)}{c}\\ h_2\left(t\right)=\frac{y_2\left(t\right)}{1-c}\end{array}.$

5. The fractional-order KiBaM battery model considering nonlinear capacity characteristics according to claim 1, wherein the relationships between the temporary capacity y.sub.1 and available capacity y.sub.2 and the h.sub.1 and h.sub.2 representing the SOC of the battery are expressed as: $\begin{array}{cc}\{\begin{array}{c}\frac{{\mathrm{dy}}_1^{\alpha }}{{\mathrm{dt}}^{\alpha }}=-i\left(t\right)+k\left[h_2\left(t\right)-h_1\left(t\right)\right]=-i\left(t\right)+k\left[\frac{y_2\left(t\right)}{1-c}-\frac{y_1\left(t\right)}{c}\right]\\ \frac{{\mathrm{dy}}_2^{\alpha }}{{\mathrm{dt}}^{\alpha }}=-k\left[h_2\left(t\right)-h_1\left(t\right)\right]=-k\left[\frac{y_2\left(t\right)}{1-c}-\frac{y_1\left(t\right)}{c}\right]\end{array}& \left(2\right)\end{array}$ in the formula, the temporary capacity is denoted by y.sub.1 and represents the power that can be directly obtained during the discharge, and its height is denoted by h.sub.1 and represents the SOC of the battery; the available capacity is denoted by y.sub.2 and represents the power that cannot be directly obtained, and its height is denoted by h.sub.2; the sum of y.sub.1 and y.sub.2 is the total capacity of the battery; c represents a distribution ratio of the battery capacity between such two portions; k represents a rate coefficient of flow from the temporary capacity to the available capacity; α represents a magnitude of the fractional order of battery capacity characteristics, and 0<α<1.

6. The fractional-order KiBaM battery model considering nonlinear capacity characteristics according to claim 1, wherein through the established fractional-order KiBaM battery model, the current total remaining capacity y(t), available capacity C.sub.avail (t), unavailable capacity C.sub.unavail (t), and SOC of the battery are obtained to capture the internal characteristics of running time and nonlinear capacity of the power battery.

7. The fractional-order KiBaM battery model considering nonlinear capacity characteristics according to claim 1, wherein the SOC of the power battery is expressed as: $\mathrm{SOC}\left(t\right)=\frac{C_{\mathrm{avail}}\left(t\right)}{C_{\mathrm{init}}}{\mathrm{SOC}}_0-\frac{1}{C_{\mathrm{init}}}\left[\int i_{\mathrm{bat}}\left(t\right)dt+C_{\mathrm{unavail}}\left(t\right)\right]$ wherein the unavailable capacity C.sub.unavail of the battery is expressed as: $\begin{array}{cc}{C}_{\mathrm{unavail}}\left(t\right)=\left(1-c\right){\delta }_h\left(t\right)=\left(1-c\right)\left(2C-\frac{I}{c}\right)t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)& \left(10\right)\end{array}$ the capacity relationship of the power battery is expressed as: C.sub.avail(t)=C.sub.init−∫i.sub.bat(t)dt−C.sub.unavail(t), wherein, c represents a distribution ratio of the battery capacity; C represents a total capacity of the battery; I represents a constant current; δ.sub.h (.sup.t) represents a height difference between the two “wells”; E.sub.α,α (z) is the Mittag-Leffler function; k represents a rate coefficient of flow from the temporary capacity to the available capacity; α represents a magnitude of the fractional order of battery capacity characteristics, 0<α<1, the coefficient $k^{\prime }=\frac{k}{c\left(1-c\right)},$ and SOC.sub.0 is the initial SOC.

8. A parameter identification method using the fractional-order KiBaM battery model according to claim 1, comprising the following steps: step 1: performing a constant current charge and discharge experiment on a power battery to restore the power battery to a fully charged state as an initial state of the battery; step 2: performing a low constant current discharge experiment on the power battery to obtain an initial capacity C.sub.init of the power battery; step 3: fully charging the power battery, performing a high constant current discharge experiment, discharging the power battery to a discharge cutoff voltage within a short time as the discharging current is high, then obtaining a capacity C.sub.1 of the power battery, and calculating a distribution ratio of the battery capacity; step 4: performing constant current discharge tests of two groups of different rates on the power battery to obtain unavailable capacity C.sub.unavail and discharge time t.sub.d data of the battery under such discharge rate, and calculating an identifiable parameter k′ and a magnitude of the order α according to the battery discharge end determination condition; and step 5: obtaining parameters of the fractional-order KiBaM electrochemical model of the tested power battery through the above tests and experiments.

9. The parameter identification method according to claim 8, further comprising step 6: performing a constant current discharge test of other rate on the power battery to obtain unavailable capacity and discharge time data of the battery at such discharge rate; and comparing with the unavailable capacity and discharge time calculated by the model to verify the accuracy of the model.

10. The parameter identification method according to claim 8, wherein the battery discharge end determination condition is: $\begin{array}{cc}\{\begin{array}{c}y\left(t\right)=\left(C-I\right)\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}\\ {\delta }_h\left(t\right)=\left(2C-\frac{I}{c}\right)t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)\end{array}.& \left(8\right)\end{array}$ wherein, δ.sub.h (.sup.t) represents the height difference between the two “wells”; C represents the total capacity of the battery; c represents the distribution ratio of the battery capacity; I represents the constant current; Γ(α) and E.sub.α,α(z) are the Gamma function and Mittag-Leffler function respectively; α represents a magnitude of the fractional order of battery capacity characteristics, 0<α<1, the coefficient $k^{\prime }=\frac{k}{c\left(1-c\right)},$ wherein k represents the rate coefficient of flow from the temporary capacity to the available capacity.

Description

BRIEF DESCRIPTION OF THE DRAWING

(1) The accompanying drawing constituting a part of the present application is used for providing a further understanding of the present application, and the schematic embodiments of the present application and the descriptions thereof are used for interpreting the present application, rather than constituting improper limitations to the present application.

(2) FIG. 1 is a structural schematic diagram of a fractional-order KiBaM electrochemical model of a power battery according to the present invention.

DETAILED DESCRIPTION OF EMBODIMENTS

(3) The present invention will be further illustrated below in conjunction with the accompanying drawing and embodiments.

(4) It should be pointed out that the following detailed descriptions are all exemplary and aim to further illustrate the present application. Unless otherwise specified, all technical and scientific terms used in the descriptions have the same meanings generally understood by those of ordinary skill in the art of the present application.

(5) It should be noted that the terms used herein are merely for describing specific embodiments, but are not intended to limit exemplary embodiments according to the present application. As used herein, unless otherwise explicitly pointed out by the context, the singular form is also intended to include the plural form. In addition, it should also be understood that when the terms “include” and/or “comprise” are used in the description, they indicate features, steps, operations, devices, components and/or their combination.

(6) FIG. 1 shows a fractional-order KiBaM electrochemical model of a power battery according to the present invention. The fractional-order KiBaM battery model includes two portions, which can be regarded as two containers connected by a channel and having certain volume, for example, wells. Such two portions are used to describe nonlinear capacity characteristics of the battery, respectively called “temporary capacity” and “available capacity”; the “temporary capacity” is denoted by y.sub.1 and represents the power that can be directly obtained during the discharge, and its height is denoted by h.sub.1 and represents a state of charge (SOC) of the battery; the “available capacity” is denoted by y.sub.2 and represents the power that cannot be directly obtained, and its height is denoted by h.sub.2; the sum of y.sub.1and y.sub.2 is the total capacity of the battery; c represents a distribution ratio of the battery capacity between the two “wells”, and the following relationship obviously exists:

(7) $\begin{array}{cc}\{\begin{array}{c}{h}_1\left(t\right)=\frac{y_1\left(t\right)}{c}\\ h_2\left(t\right)=\frac{y_2\left(t\right)}{1-c}\end{array}& \left(1\right)\end{array}$

(8) In the fractional-order KiBaM battery model, when the battery is discharged, the load current i flows out from a pipe at the lower right corner of y.sub.1 representing “temporary capacity”, the power of “available capacity” y.sub.2 slowly flows into y.sub.1through k at the same time, the speed of outflow is faster than the speed of flow from y.sub.2 to y.sub.1, y.sub.1 drops faster, and the height difference between y.sub.1 and y.sub.2 increases accordingly; when the battery stops discharging, the power of y.sub.1 rises until y.sub.1 and y.sub.2 have an equal height, which represents the recovery effect of the battery; when the discharge current is higher, the amount of electricity discharged is smaller, which represents the nonlinear capacity effect of the battery;

(9) In the fractional-order KiBaM battery model, the relationships between the “temporary capacity” y.sub.1 and “available capacity” y.sub.2 and the h.sub.1 and h.sub.2 representing the SOC of the battery may be expressed as:

(10) $\begin{array}{cc}\{\begin{array}{c}\frac{{\mathrm{dy}}_1^{\alpha }}{{\mathrm{dt}}^{\alpha }}=-i\left(t\right)+k\left[h_2\left(t\right)-h_1\left(t\right)\right]=-i\left(t\right)+k\left[\frac{y_2\left(t\right)}{1-c}-\frac{y_1\left(t\right)}{c}\right]\\ \frac{{\mathrm{dy}}_2^{\alpha }}{{\mathrm{dt}}^{\alpha }}=-k\left[h_2\left(t\right)-h_1\left(t\right)\right]=-k\left[\frac{y_2\left(t\right)}{1-c}-\frac{y_1\left(t\right)}{c}\right]\end{array}& \left(2\right)\end{array}$

(11) In the formula, the “temporary capacity” is denoted by y.sub.1 and represents the power that can be directly obtained during the discharge, and its height is denoted by h.sub.1 and represents the SOC of the battery; the “available capacity” is denoted by y.sub.2 and represents the power that cannot be directly obtained, and its height is denoted by h.sub.2; the sum of y.sub.1and y.sub.2 is the total capacity of the battery; c represents a distribution ratio of the battery capacity between the two “wells”; k represents a rate coefficient of flow from the “temporary capacity” to the “available capacity”; a represents a magnitude of the fractional order of battery capacity characteristics, and 0<α<1.

(12) The height difference between the two “wells” is defined as δ.sub.h(t), obviously:
δ.sub.h(t)=h.sub.2(t)−h.sub.1(t)  (3)

(13) The unavailable capacity of the battery may be expressed as:
C.sub.unavail(t)=(1−c)δ.sub.h(t)  (4)

(14) It is assumed that the initial capacities y.sub.10 and y.sub.20 of the “temporary capacity” y.sub.1and “available capacity” y.sub.2 of the battery are respectively:

(15) $\begin{array}{cc}\{\begin{array}{c}{y}_{10}=y_1\left(t_0\right)=\mathrm{cC};\\ y_{20}=y_2\left(t_0\right)=\left(1-c\right)C;\\ y_0=y_{10}+y_{20}\end{array}& \left(5\right)\end{array}$

(16) In the formula, C represents the total capacity of the battery. When the battery is discharged by constant current I for the first time, and the discharge time interval is t.sub.0≤t≤t.sub.d, if the initial condition is t.sub.0=0, that is, the height difference is initially zero, after a period of time t.sub.d<t<t.sub.r, formula (2) is subjected to Laplace transformation and inverse

(17) Laplace transformation (the transformation process is omitted) to obtain:

(18) 0$\begin{array}{cc}\hspace{1em}\{\begin{array}{c}{y}_1\left(t\right)=c\left[y_1\left(0\right)+y_2\left(0\right)-I\right]\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}+\left[\left(1-c\right)\left(y_1\left(0\right)-I\right)-c{y}_2\left(0\right)\right]t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)\\ y_2\left(t\right)=\left(1-c\right)\left(y_1\left(0\right)+y_2\left(0\right)-I\right]\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}-\left[\left(1-c\right)\left(y_1\left(0\right)-I\right)-c{y}_2\left(0\right)\right]t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)\end{array}& \left(6\right)\end{array}$

(19) In the formula, Γ(α) and E.sub.α,α(z) are Gamma function and Mittag-Leffler function that are commonly used in fractional calculus calculation; and the coefficient is

(20) $k^{\prime }=\frac{k}{c\left(1-c\right)}.$
Obtained by arrangement is:

(21) $\begin{array}{cc}\{\begin{array}{c}y\left(t\right)=\left[y_1\left(0\right)+y_2\left(0\right)-I\right]\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}\\ {\delta }_h\left(t\right)=\left[\frac{y_1\left(0\right)-I}{c}+\frac{y_2\left(0\right)}{1-c}\right]t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)\end{array}& \left(7\right)\end{array}$

(22) By substituting into the initial condition (2), formula (7) can be simplified into:

(23) $\begin{array}{cc}\{\begin{array}{c}y\left(t\right)=\left(C-I\right)\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}\\ {\delta }_h\left(t\right)=\left(2C-\frac{I}{c}\right)t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)\end{array}& \left(8\right)\end{array}$

(24) From the previous analysis, when the battery is completely discharged, the height h.sub.1 is equal to 0; at this time, the total remaining capacity of the battery is equal to the unavailable capacity:
y(t)=C.sub.unavail(t)=(1−c)δ.sub.h(t)  (9)

(25) The unavailable capacity of the battery C.sub.unavail may be expressed as:

(26) $\begin{array}{cc}{C}_{\mathrm{unavail}}\left(t\right)=\left(1-c\right){\delta }_h\left(t\right)=\left(1-c\right)\left(2C-\frac{I}{c}\right)t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}^{\alpha }\right)& \left(10\right)\end{array}$

(27) The capacity relationship of the power battery may be expressed as:
C.sub.avail(t)=C.sub.init−∫i.sub.bat(t)dt−C.sub.unavail(t)  (11)

(28) In the formula, C.sub.maxC.sub.avail.sub.unavail represent the initial capacity, the available capacity, and the unavailable capacity of the battery, respectively; the unavailable capacity C.sub.unavail represents a nonlinear SOC variable affected by the nonlinear capacity characteristic of the battery; Obviously, the SOC of the power battery may be expressed as:

(29) $\begin{array}{cc}\mathrm{SOC}\left(t\right)=\frac{C_{\mathrm{avail}}\left(t\right)}{C_{\mathrm{init}}}{\mathrm{SOC}}_0-\frac{1}{C_{\mathrm{init}}}\left[\int i_{\mathrm{bat}}\left(t\right)dt+C_{\mathrm{unavail}}\left(t\right)\right]& \left(12\right)\end{array}$

(30) The current total remaining capacity y(t), the available capacity C.sub.avail(t), the unavailable capacity C.sub.unavail(t) and the SOC of the battery may be obtained from the fractional-order KiBaM battery model, so that the internal characteristics of running time and the nonlinear capacity of the power battery can be accurately captured.

(31) An identification method using the above fractional-order KiBaM battery model is provided. From the battery model, the parameters to be identified mainly include an initial capacity y.sub.0 of the battery, a distribution ratio c of the battery capacity between the two “wells”, a rate coefficient k indicating the flow from the “temporary capacity” to the “available capacity”, and a fractional order α of the battery capacity characteristic. The method mainly includes the following steps:

(32) step 1: performing a constant current charge and discharge experiment on a power battery to restore the power battery to a fully charged state as an initial state of the battery;

(33) step 2: performing a low constant current discharge experiment on the power battery to obtain an initial capacity C.sub.init of the power battery; step 3: fully charging the power battery, performing a high constant current discharge experiment, discharging the power battery to a discharge cutoff voltage within a short time as the discharging current is high, then obtaining a capacity C.sub.1 of the power battery; the parameter c of the battery model is equal to

(34) $\frac{C_1}{C_{\mathrm{init}}};$

(35) step 4: performing constant current discharge tests of two groups of different rates on the power battery to obtain unavailable capacity C.sub.unavail and discharge time t.sub.d data of the battery under such discharge rate and the like, and calculating an identifiable parameter k′ and an order a according to formula (8)

(36) $C_{\mathrm{unavail}}\left(t\right)=\left(1-c\right){\delta }_h\left(t_d\right)=\left(1-c\right)\left(2C-\frac{I}{c}\right)t_d^{\alpha -1}{E}_{\alpha ,\alpha }\left(-k^{\prime }{t}_d^{\alpha }\right)$
for determining the battery discharge end;

(37) step 5: obtaining parameters of the fractional-order KiBaM electrochemical model of the tested power battery through the above tests and experiments; and

(38) step 6: performing a constant current discharge test of other rate on the power battery to obtain unavailable capacity and discharge time data of the battery and the like at that discharge rate; and comparing with the unavailable capacity and discharge time calculated by the model to verify the accuracy of the model.

(39) Described above are merely preferred embodiments of the present application, and the present application is not limited thereto. Various modifications and variations may be made to the present application for those skilled in the art. Any modification, equivalent substitution, improvement or the like made within the spirit and principle of the present application shall fall into the protection scope of the present application.

(40) Although the specific embodiments of the present invention are described above in combination with the accompanying drawing, the protection scope of the present invention is not limited thereto. It should be understood by those skilled in the art that various modifications or variations could be made by those skilled in the art based on the technical solution of the present invention without any creative effort, and these modifications or variations shall fall into the protection scope of the present invention.