CN101604005A - A kind of estimation method of battery dump energy based on combined sampling point Kalman filtering - Google Patents

Abstract

The present invention relates to a kind of estimation method of battery dump energy based on combined sampling point Kalman filtering.Existing method can not satisfy online detection requirements, and precision is very poor.The inventive method at first records the battery terminal voltage y constantly at k by metering circuit _kWith the powered battery current i _k, represent with state equation and observation equation then each state-of-charge constantly of battery to adopt standard sample point Kalman filtering to carry out the estimation of battery dump energy again.The inventive method can be carried out the Fast estimation of battery SOC easily, fast convergence rate, and the estimated accuracy height, and be applicable to the Fast estimation of various battery SOCs.

Description

Battery remaining capacity estimation method based on combined sampling point Kalman filtering

Technical Field

The invention belongs to the technical field of lithium batteries, and particularly relates to a battery remaining capacity estimation method based on combined sampling point Kalman filtering.

Background

Batteries have found widespread use as backup power sources in the fields of communications, power systems, military equipment, and the like. Compared with the traditional fuel automobile, the electric automobile can realize zero emission, so the electric automobile is the main development direction of the automobile in the future. In an electric vehicle, the battery directly serves as an active energy supply component, so that the operating state of the battery is directly related to the driving safety and the operational reliability of the whole vehicle. In order to ensure good performance of the battery pack in the electric automobile and prolong the service life of the battery pack, the running state of the battery needs to be known timely and accurately, and the battery needs to be managed and controlled reasonably and effectively.

Accurate estimation of the State of Charge (SOC) of a battery is the most core technology in a battery energy management system. The SOC of the battery cannot be measured directly by a sensor, and must be estimated by measuring some other physical quantity and using a certain mathematical model and algorithm. The current common battery SOC estimation methods include an open-circuit voltage method, an ampere-hour method and the like. By adopting an open-circuit voltage method, the battery must be kept still for a long time to reach a stable state, and the method is only suitable for SOC estimation of the electric automobile in a parking state and cannot meet the requirement of online detection; the ampere-hour method is easily influenced by the current measurement precision, and the precision is very poor under the condition of high temperature or severe current fluctuation.

Disclosure of Invention

The invention aims to overcome the defects of the prior art, provides a battery residual capacity rapid estimation method based on combined sampling point Kalman filtering, can be suitable for all batteries, and has high estimation precision.

The invention relates to a battery remaining capacity estimation method based on combined sampling point Kalman filtering, which comprises the following specific steps:

step (1) measuring the battery terminal voltage y at the moment k through a measuring circuit_kAnd battery supply current i_k，k＝1，2，3，…。

Step (2) using a State equation and an observation equation to represent the State of Charge (SOC) of the battery at each moment

The state equation is as follows: <math> <mrow> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mi>&Delta;t</mi> </mrow> <msub> <mi>Q</mi> <mi>n</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> </mrow> </math>

the observation equation: $y_k=g(p,z_k,i_k)+v_k=K_0-{\mathrm{Ri}}_k-\frac{K_1}{z_k}-K_2{z}_k+K_3\ln z_k+K_4\ln (1-z_k)+v_k$

z is the state of charge of the battery, i.e. the remaining charge, wherein z is 100% and represents that the battery is in a full state, and z is 0% and represents that the battery is in a depleted state; eta_iThe discharge proportionality coefficient of the battery reflects the influence degree of factors such as discharge rate, temperature, self-discharge, aging and the like on the SOC of the battery; q_nThe rated total electric quantity can be obtained when the battery is discharged at the room temperature of 25 ℃ at the discharge rate of 1/30 times of rated current; r is the internal resistance of the battery; k₀、K₁、K₂、K₃、K₄Is constant, p ═ K₀ R K₁ K₂ K₃ K₄]^TP is a parameter of the battery observation model and is a column vector which is invariant to the batteries of the same type; Δ t is the measurement time interval, u is the processing noise, v is the observation noise, and the subscript k is the measurement time.

Wherein, the discharge proportionality coefficient eta_iThe determination steps are as follows:

(a) discharging fully charged batteries at different rates C_i(0＜C_iC is less than or equal to C, and C is the rated discharge current of the battery) is discharged for N (N is more than 10) times in a constant current way, and the total electric quantity Q of the battery under the corresponding discharge rate is calculated_i，1≤i≤N。

(b) Fitting Q according to the least square method_iAnd C_iWith quadratic relation between, i.e. under the criterion of least mean square errorGo out and simultaneously satisfy $Q_i=a{C}_i^2+b{C}_i+c$ (1. ltoreq. i. ltoreq.N) with optimal coefficients a, b, c.

(c) At a discharge current of i_kTime, corresponding discharge proportionality coefficient eta_iComprises the following steps:

<math> <mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>Q</mi> <mi>n</mi> </msub> <mrow> <msubsup> <mi>ai</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>bi</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>c</mi> </mrow> </mfrac> </mrow> </math>

here, the optimal coefficients a, b, and c only need to be determined once for the same type of battery, and after determination, the optimal coefficients can be directly used as known constants for estimating the remaining capacity of all the batteries of the same type.

The battery model parameter p is determined by adopting a center difference sampling point Kalman filtering algorithm, and the method specifically comprises the following steps:

(d) discharging the fully charged battery at room temperature of 25 ℃ at 1/30 times of rated current at constant current until the electric quantity is exhausted;

(e) the terminal voltage y of the battery at the time s is measured by a measuring circuit at a time interval delta t during the discharging process_sAnd M, wherein s-0 corresponds to the initial discharge time after the battery is fully charged, and s-M corresponds to the end time of the battery charge depletion.

(f) Calculating the residual electric quantity z at the moment s_s，z_s＝1-s/M。

(g) Optionally an initial parameter $p={\hat{p}}_0,$ Setting its square root mean square error matrix as

$S_{p_0}=I_6;$ Wherein I₆6 × 6 identity matrix; selecting a proportionality constant h, wherein h is more than 1; setting variable $R_r=\sqrt{{10}^{-3}}{I}_6;$ Setting weighting coefficients $W_0^{\left(m\right)}=\frac{h^2-7}{h^2},$ $W_i^{\left(m\right)}=\frac{1}{2h^2},$ $W_i^{\left(c1\right)}=\frac{1}{2h},$ $W_i^{\left(c2\right)}=\frac{\sqrt{h^2-1}}{2h^2},$ i＝1，2，…，12。

For s 1, 2.. times, M, successive iterations are performed as follows:

(h) calculating time domain update:

calculating estimated values of model parameters

${\hat{p}}_s^{-}={\hat{p}}_{s-1}$

Calculating an estimate of a square root mean square error matrix of model parameters

$S_{p_s}^{-}=S_{p_{s-1}}+D_{r_{s-1}},$ Wherein, $D_{r_{s-1}}=-\mathrm{diag}\left\{S_{p_{s-1}}\right\}+\sqrt{\mathrm{diag}{\left\{S_{p_{s-1}}\right\}}^2+\mathrm{diag}\left\{R_r\right\}},$ diag {. is a column vector formed by diagonal elements of the correspondence matrix.

(i) Computing

Sequence of sampling points

Is a 6 x 1 column vector of the vector,

is a 6 × 6 matrix, therefore

Is a 6 x 13 matrix.

(j) The measurement updates are calculated as follows:

calculating an observation sequence of sample points

Is a 6 x 13 matrix;

calculating an observation sequence

Is estimated value of

Is composed ofThe ith column;

calculating an observation sequence

Square root mean square error matrix of

Computing a covariance matrix

Calculating the Kalman gain K_s： $K_s=(P_{p_s{d}_s}/S_{{\tilde{d}}_s}^T)/S_{{\tilde{d}}_s};$

Computing parameter updates

${\hat{p}}_s={\hat{p}}_s^{-}+K_s(y_s-{\hat{d}}_s^{-});$

Calculating a temporary variable U: $U=K_s{S}_{{\tilde{d}}_s};$

updating square root mean square error matrix of calculation model parameters $S_{p_s}=\mathrm{cholupdate}\{S_{p_s}^{-},U,-1\};$

Wherein qr {. is } represents solving orthogonal triangle decomposition of the matrix, and returning to obtain an upper triangle matrix; (.)^TTranspose operation for matrix;

representation matrix

Cholesky decomposition of (1).

Through the steps, the final product is obtained through iteration

I.e. the estimated battery model parameters.

For the same type of battery, the parameters only need to be determined once, and the determined parameters can be directly used for estimating the residual capacity of all the batteries of the same type as known constants.

And (3) estimating the residual electric quantity of the battery by adopting standard sampling point Kalman filtering, specifically:

the following initialization procedure is executed:

initial state

And its variance P₀Respectively as follows:

${\hat{z}}_0={\mathrm{SOC}}_0=100\%,$ P₀＝var(z₀)＝10^-2，

processing noise variance R_wAnd observed noise variance R_vRespectively as follows:

R_w＝10^-5，R_v＝10^-2

extended state vector

And its covariance P₀ ^aComprises the following steps:

${\hat{z}}_0^a={\left[\begin{array}{ccc}{z}_0& 0& 0\end{array}\right]}^T,$ $p_0^a=\left[\begin{array}{ccc}{P}_0& 0& 0\\ 0& R_w& 0\\ 0& 0& R_v\end{array}\right]$

the scale parameter γ is:

<math> <mrow> <mi>&gamma;</mi> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> </math>

mean weighting factor w_i ^(m)I-0, 1, 2, 6 and a variance weighting factor w_i ^(c)I is 0, 1, 2, 6 is:

$w_0^{\left(m\right)}=0,$ $w_0^{\left(c\right)}=2,$ $w_i^{\left(m\right)}=w_i^{\left(c\right)}=1/6,$ 1≤i≤6

performing cyclic recursion by adopting a standard sampling point Kalman filtering algorithm:

at the measuring time k equal to 1, 2, 3, …, the battery terminal voltage y measured by the measuring circuit in actual operation is used_kAnd supply current i of the battery_kThe recursion calculation is performed according to the following formulas:

(l) Extended state vector based on time k-1

And its covariance P_k-1 ^aCalculating all the sampling point sequences at the time

(m) time domain updating according to the state equation:

from a sequence of sampling points

Calculating sample point updates from equation of state

Updating sampling points

Weighting and calculating the state estimate

Computing state estimates

Variance of (2)

(n) completing measurement update according to the following formula according to the observation equation:

with updating of the sampling points

Computing measurement updates from observation equations

Updating measurementsWeighting and calculating the measurement estimate

Computing measurement estimates

Variance of (2)

Computing

And

cross covariance of

Calculating the Kalman gain K_k： $K_k=P_{z_k{y}_k}{P}_{{\tilde{y}}_k}^{-1}$

Computing state updates

${\hat{z}}_k=z_k^{-}+K_k(y_k-{\hat{y}}_k^{-})$

Computing state updates

Variance of (2)

$P_{x_k}=P_{x_k}^{-}-K_k{P}_{{\tilde{y}}_k}{K}_k^T$

Recursion of the resulting state update value

I.e. the battery estimated at the current time kThe remaining amount of power. The whole cycle recursion process is completed on line, namely the estimation of the remaining battery capacity at each moment is synchronously completed in the actual working process of the battery.

The method can conveniently and quickly estimate the SOC of the battery, has high convergence speed and high estimation precision, and is suitable for quickly estimating the SOC of various batteries.

According to a first aspect of the present invention, a measurement quantity relied on by a sampling point kalman filtering method for estimating a remaining capacity of a battery is disclosed, which is a terminal voltage of the battery and a supply current of the battery, respectively.

According to a second aspect of the present invention, a state equation and an observation equation of sampling point kalman filtering for estimating a remaining capacity of a battery are disclosed. And the battery model parameters in the observation equation are determined by adopting a center difference sampling point Kalman filtering algorithm.

According to a third aspect of the present invention, an initial value on which a standard sample point kalman filter depends for estimating a battery SOC is disclosed. The method comprises the steps of initial SOC, variance of the initial SOC, variance of processing noise and observation noise, and weight values corresponding to sampling points. The initial SOC and the initial SOC variance are not necessarily accurate, and the initial SOC variance can quickly converge to be close to the true value in the subsequent iteration process of sampling point Kalman filtering.

According to a fourth aspect of the invention, a specific process for battery SOC estimation by applying standard sampling point Kalman filtering is disclosed. The method mainly comprises the following steps: calculating a weighted value of the sampling point after the sampling point is transformed by the state equation to serve as an estimated value of the state, and further calculating the variance of the state estimation through weighting; calculating the weighted value of the sampling point after the sampling point is transformed by the observation equation to serve as the estimated value of observation; calculating a Kalman gain; updates of the computed states and their variances, etc.

Detailed Description

The specific method of the battery residual capacity estimation method based on the combined sampling point Kalman filtering is as follows:

step (1) measuring the battery terminal voltage y at the moment k through a measuring circuit_kAnd battery supply current i_k，k＝1，2，3，…。

And (2) expressing the state of charge of the battery at each moment by using a state equation and an observation equation, wherein the state equation is as follows: <math> <mrow> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mi>&Delta;t</mi> </mrow> <msub> <mi>Q</mi> <mi>n</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> </mrow> </math>

the observation equation: $y_k=g(p,z_k,i_k)+v_k=K_0-{\mathrm{Ri}}_k-\frac{K_1}{z_k}-K_2{z}_k+K_3\ln z_k+K_4\ln (1-z_k)+v_k$

z is the state of charge of the battery, i.e. the remaining charge, wherein z is 100% and represents that the battery is in a full state, and z is 0% and represents that the battery is in a depleted state; eta_iThe discharge proportionality coefficient of the battery reflects the influence degree of factors such as discharge rate, temperature, self-discharge, aging and the like on the SOC of the battery; q_nThe rated total electric quantity can be obtained when the battery is discharged at the room temperature of 25 ℃ at the discharge rate of 1/30 times of rated current; r is the internal resistance of the battery; k₀、K₁、K₂、K₃、K₄Is constant, p ═ K₀ R K₁ K₂ K₃ K₄]^TP is a parameter of the battery observation model and is a column vector which is invariant to the batteries of the same type; Δ t is the measurement time interval, u is the processing noise, v is the observation noise, and the subscript k is the measurement time.

Wherein, the discharge proportionality coefficient eta_iThe determination steps are as follows:

(a) discharging fully charged batteries at different rates C_i(0＜C_iC is less than or equal to C, and C is the rated discharge current of the battery) is discharged for N (N is more than 10) times in a constant current way, and the total electric quantity Q of the battery under the corresponding discharge rate is calculated_i，1≤i≤N。

(b) Fitting Q according to the least square method_iAnd C_iIn relation to a quadratic curve, i.e. calculated under the least mean square error criterion while satisfying $Q_i=a{C}_i^2+b{C}_i+c$ (1. ltoreq. i. ltoreq.N) with optimal coefficients a, b, c.

(c) At a discharge current of i_kTime, corresponding discharge proportionality coefficient eta_iComprises the following steps:

<math> <mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>Q</mi> <mi>n</mi> </msub> <mrow> <msubsup> <mi>ai</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>bi</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>c</mi> </mrow> </mfrac> </mrow> </math>

here, the optimal coefficients a, b, and c only need to be determined once for the same type of battery, and after determination, the optimal coefficients can be directly used as known constants for estimating the remaining capacity of all the batteries of the same type.

The battery model parameter p is determined by adopting a center difference sampling point Kalman filtering algorithm, and the method specifically comprises the following steps:

(d) discharging the fully charged battery at room temperature of 25 ℃ at 1/30 times of rated current at constant current until the electric quantity is exhausted;

(e) the terminal voltage y of the battery at the time s is measured by a measuring circuit at a time interval delta t during the discharging process_sAnd M, wherein s-0 corresponds to the initial discharge time after the battery is fully charged, and s-M corresponds to the end time of the battery charge depletion.

(f) Calculating the residual electric quantity z at the moment s_s，z_s＝1-s/M。

(g) Optionally an initial parameter $p={\hat{p}}_0,$ Setting its square root mean square error matrix as

$S_{p_0}=I_6;$ Wherein I₆6 × 6 identity matrix; choosing a proportionality constanth, h is more than 1; setting variable $R_r=\sqrt{{10}^{-3}}{I}_6;$ Setting weighting coefficients $W_0^{\left(m\right)}=\frac{h^2-7}{h^2},$ $W_i^{\left(m\right)}=\frac{1}{2h^2},$ $W_i^{\left(c1\right)}=\frac{1}{2h},$ $W_i^{\left(c2\right)}=\frac{\sqrt{h^2-1}}{2h^2},$ i＝1，2，…，12。

For s 1, 2.. times, M, successive iterations are performed as follows:

(h) calculating time domain update:

calculating estimated values of model parameters ${\hat{p}}_s^{-}={\hat{p}}_{s-1}$

Calculating an estimate of a square root mean square error matrix of model parameters

$S_{p_s}^{-}=S_{p_{s-1}}+D_{r_{s-1}},$ Wherein, $D_{r_{s-1}}=-\mathrm{diag}\left\{S_{p_{s-1}}\right\}+\sqrt{\mathrm{diag}{\left\{S_{p_{s-1}}\right\}}^2+\mathrm{diag}\left\{R_r\right\}},$ diag {. is a column vector formed by diagonal elements of the correspondence matrix.

(i) ComputingSequence of sampling points

Is a 6 x 1 column vector of the vector,

is a 6 × 6 matrix, therefore

Is a 6 x 13 matrix.

(j) The measurement updates are calculated as follows:

calculating an observation sequence of sample points

Is a 6 x 13 matrix;

calculating an observation sequenceIs estimated value of

Is composed of

The ith column;

calculating an observation sequence

Square root mean square error matrix of

Computing a covariance matrix

Calculating the Kalman gain K_s： <math> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow> <msub> <mi>p</mi> <mi>s</mi> </msub> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </msub> <mo>/</mo> <msubsup> <mi>S</mi> <msub> <mover> <mi>d</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <mi>T</mi> </msubsup> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>S</mi> <msub> <mover> <mi>d</mi> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msub> </msub> <mo>;</mo> </mrow> </math>

Computing parameter updates

${\hat{p}}_s={\hat{p}}_s^{-}+K_s(y_s-{\hat{d}}_s^{-});$

Calculating a temporary variable U: $U=K_s{S}_{{\tilde{d}}_s};$

updating square root mean square error matrix of calculation model parameters

$S_{p_s}=\mathrm{cholupdate}\{S_{p_s}^{-},U,-1\};$

Wherein qr {. is } represents solving orthogonal triangle decomposition of the matrix, and returning to obtain an upper triangle matrix; (.)^TTranspose operation for matrix;

representation matrix

Cholesky decomposition of (1).

Through the steps, the final iteration is obtainedIs/are as follows

I.e. the estimated battery model parameters.

For the same type of battery, the parameters only need to be determined once, and the determined parameters can be directly used for estimating the residual capacity of all the batteries of the same type as known constants.

And (3) estimating the residual electric quantity of the battery by adopting standard sampling point Kalman filtering, specifically:

the following initialization procedure is executed:

initial state

And its variance P₀Respectively as follows:

${\hat{z}}_0={\mathrm{SOC}}_0=100\%,$ P₀＝var(z₀)＝10^-2，

processing noise variance R_wAnd observed noise variance R_vRespectively as follows:

R_w＝10^-5，R_v＝10^-2

extended state vector

And its covariance P₀ ^aComprises the following steps:

${\hat{z}}_0^a={\left[\begin{array}{ccc}{z}_0& 0& 0\end{array}\right]}^T,$ $P_0^a=\left[\begin{array}{ccc}{P}_0& 0& 0\\ 0& R_w& 0\\ 0& 0& R_v\end{array}\right]$

the scale parameter γ is:

<math> <mrow> <mi>&gamma;</mi> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> </math>

mean weighting factor w_i ^(m)I-0, 1, 2, 6 and a variance weighting factor w_i ^(c)I is 0, 1, 2, 6 is:

$w_0^{\left(m\right)}=0,$ $w_0^{\left(c\right)}=2,$ $w_i^{\left(m\right)}=w_i^{\left(c\right)}=1/6,$ 1≤i≤6

performing cyclic recursion by adopting a standard sampling point Kalman filtering algorithm:

at the measuring time k equal to 1, 2, 3, …, the battery terminal voltage y measured by the measuring circuit in actual operation is used_kAnd supply current i of the battery_kThe recursion calculation is performed according to the following formulas:

(l) Extended state vector based on time k-1

And its covariance P_k-1 ^aCalculating all the sampling point sequences at the time

(m) time domain updating according to the state equation:

from a sequence of sampling points

Calculating sample point updates from equation of state

For sampling pointNew

Weighting and calculating the state estimate

Computing state estimates

Variance of (2)

(n) completing measurement update according to the following formula according to the observation equation:

with updating of the sampling points

Computing measurement updates from observation equations

Updating measurements

Weighting and calculating the measurement estimate

Computing measurement estimatesVariance of (2)

Computing

And

cross covariance of

Calculating the Kalman gain K_k： $K_k=P_{z_k{y}_k}{P}_{{\tilde{y}}_k}^{-1}$

Computing state updates ${\hat{z}}_k=z_k^{-}+K_k(y_k-{\hat{y}}_k^{-})$

Computing state updates

Variance of (2)

$P_{x_k}=P_{x_k}^{-}-K_k{P}_{{\tilde{y}}_k}{K}_k^T$

Recursion of the resulting state update value

Namely the estimated battery residual capacity at the current moment k. The whole cycle recursion process is completed on line, namely the estimation of the remaining battery capacity at each moment is synchronously completed in the actual working process of the battery.

Claims (1)

1. A battery residual capacity estimation method based on combined sampling point Kalman filtering is characterized by comprising the following specific steps:

step (1) measuring the battery terminal voltage y at the moment k through a measuring circuit_kAnd battery supply current i_k，k＝1，2，3，…；

Step (2) representing the state of charge of the battery at each moment by using a state equation and an observation equation, wherein

The state equation is as follows: <math> <mrow> <msub> <mi>z</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>z</mi> <mi>k</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mi>&Delta;t</mi> </mrow> <msub> <mi>Q</mi> <mi>n</mi> </msub> </mfrac> <mo>)</mo> </mrow> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>u</mi> <mi>k</mi> </msub> </mrow> </math>

the observation equation: $y_k=g(p,z_k,i_k)+v_k=K_0-{\mathrm{Ri}}_k-\frac{K_1}{z_k}-K_2{z}_k+K_3{\ln z}_k+K_4\ln (1-z_k)+v_k$

z is the state of charge of the battery, i.e. the remaining charge, wherein z is 100% and represents that the battery is in a full state, and z is 0% and represents that the battery is in a depleted state; eta_iIs the discharge proportionality coefficient of the battery; q_nThe rated total electric quantity can be obtained when the battery is discharged at the room temperature of 25 ℃ at the discharge rate of 1/30 times of rated current; r is the internal resistance of the battery; k₀、K₁、K₂、K₃、K₄Is constant, p ═ K₀ R K₁ K₂ K₃ K₄]^TP is a parameter of the battery observation model and is a column vector which is invariant to the batteries of the same type; Δ t is the measurement time interval, u is the processing noise, v is the observation noise, and subscript k is the measurement time; wherein

Discharge proportionality coefficient eta_iThe determination steps are as follows:

(a) discharging fully charged batteries at different rates C_iDischarging at constant current for N times, N is more than 10, and calculating the total electric quantity Q of the battery at corresponding discharge rate_i，1≤i≤N；

(b) Fitting Q according to the least square method_iAnd C_iIn relation to a quadratic curve, i.e. solving under the minimum mean square error criterion while satisfying Q_i＝aC_i ²+bC_i+ c optimal coefficients a, b, c;

(c) at a discharge current of i_kTime, corresponding discharge proportionality coefficient eta_iComprises the following steps:

<math> <mrow> <msub> <mi>&eta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>Q</mi> <mi>n</mi> </msub> <mrow> <msubsup> <mi>ai</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>bi</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>c</mi> </mrow> </mfrac> </mrow> </math>

the battery model parameter p is determined by adopting a center difference sampling point Kalman filtering algorithm, and the specific steps are as follows:

(d) discharging the fully charged battery at room temperature of 25 ℃ at 1/30 times of rated current at constant current until the electric quantity is exhausted;

(e) the terminal voltage y of the battery at the time s is measured by a measuring circuit at a time interval delta t during the discharging process_sM, where s-0 corresponds to the initial discharge time after the battery is fully charged, and s-M corresponds to the end time of battery power exhaustion;

(f) calculating the residual electric quantity z at the moment s_s，z_s＝1-s/M；

(g) Optionally an initial parameter $p={\hat{p}}_0,$ Setting its square root mean square error matrix as

$S_{p_0}=I_6;$ Wherein I₆6 × 6 identity matrix; selecting a proportionality constant h, wherein h is more than 1; setting variable $R_r=\sqrt{{10}^{-3}}{I}_6;$ Setting weighting coefficients $W_0^{\left(m\right)}=\frac{h^2-7}{h^2},$ $W_i^{\left(m\right)}=\frac{1}{{2h}^2},$ $W_i^{\left(c1\right)}=\frac{1}{2h},$ $W_i^{\left(c2\right)}=\frac{\sqrt{h^2-1}}{{2h}^2},$ i＝1，2，…，12；

For s 1, 2.. times, M, successive iterations are performed as follows:

(h) calculating time domain update:

calculating estimated values of model parameters ${\hat{p}}_s^{-}={\hat{p}}_{s-1}$

Calculating an estimate of a square root mean square error matrix of model parameters

$S_{p_s}^{-}=S_{p_{s-1}}+D_{r_{s-1}},$ Wherein, $D_{r_{s-1}}=-\mathrm{diag}\left\{S_{p_{s-1}}\right\}+\sqrt{\mathrm{diag}{\left\{S_{p_{s-1}}\right\}}^2+\mathrm{diag}\left\{R_r\right\}},$ diag {. is a column vector formed by diagonal elements of the corresponding matrix;

(i) computing

Sequence of sampling points

Is a 6 x 1 column vector of the vector,

is a 6 × 6 matrix, therefore

Is a 6 x 13 matrix;

(j) the measurement updates are calculated as follows:

calculating an observation sequence of sample points

Is a 6 x 13 matrix;

calculating an observation sequence

Is estimated value of

Is composed ofThe ith column;

calculating an observation sequence

Square root mean square error matrix of

Computing a covariance matrix

Calculating the Kalman gain K_s： <math> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>P</mi> <mrow> <msub> <mi>p</mi> <mi>s</mi> </msub> <msub> <mi>d</mi> <mi>s</mi> </msub> </mrow> </msub> <mo>/</mo> <msubsup> <mi>S</mi> <msub> <mover> <mi>d</mi> <mo>~</mo> </mover> <mi>s</mi> </msub> <mi>T</mi> </msubsup> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>S</mi> <msub> <mover> <mi>d</mi> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msub> </msub> <mo>;</mo> </mrow> </math>

Computing parameter updates

${\hat{p}}_s={\hat{p}}_s^{-}+K_s(y_s-{\hat{d}}_s^{-});$

Calculating a temporary variable U: <math> <mrow> <mi>U</mi> <mo>=</mo> <msub> <mi>K</mi> <mi>s</mi> </msub> <msub> <mi>S</mi> <msub> <mover> <mi>d</mi> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msub> </msub> <mo>;</mo> </mrow> </math>

updating square root mean square error matrix of calculation model parameters

$S_{p_s}=\mathrm{cholupdate}\{S_{p_s}^{-},U,-1\};$

Wherein qr {. is } represents solving orthogonal triangle decomposition of the matrix, and returning to obtain an upper triangle matrix; (.)^TTranspose operation for matrix;

representation matrix

Cholesky decomposition of (1);

through the steps, the final product is obtained through iteration

The estimated battery model parameters are obtained;

and (3) estimating the residual electric quantity of the battery by adopting standard sampling point Kalman filtering, specifically:

the following initialization procedure is executed:

initial stateAnd its variance P₀Respectively as follows:

${\hat{z}}_0={\mathrm{SOC}}_0=100\%,$ P₀＝var(z₀)＝10^-2，

processing noise variance R_wAnd observed noise variance R_vRespectively as follows:

R_w＝10^-5，R_v＝10^-2

extended state vector

And its covariance P₀ ^aComprises the following steps:

${\hat{z}}_0^a={\left[\begin{array}{ccc}{z}_0& 0& 0\end{array}\right]}^T,$ $P_0^a=\left[\begin{array}{ccc}{P}_0& 0& 0\\ 0& R_w& 0\\ 0& 0& R_v\end{array}\right]$

the scale parameter γ is:

<math> <mrow> <mi>&gamma;</mi> <mo>=</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> </math>

mean weighting factor w_i ^(m)I-0, 1, 2, 6 and a variance weighting factor w_i ^(c)I is 0, 1, 2, 6 is:

$w_0^{\left(m\right)}=0,$ $w_0^{\left(c\right)}=2,$ $w_i^{\left(m\right)}=w_i^{\left(c\right)}=1/6,$ 1≤i≤6

performing cyclic recursion by adopting a standard sampling point Kalman filtering algorithm:

at the measuring time k equal to 1, 2, 3, …, the battery terminal voltage y measured by the measuring circuit in actual operation is used_kAnd supply current i of the battery_kThe recursion calculation is performed according to the following formulas:

(l) Extended state vector based on time k-1

And its covariance P_k-1 ^aCalculating all the sampling point sequences at the time

(m) time domain updating according to the state equation:

from a sequence of sampling pointsCalculating sample point updates from equation of state

Updating sampling points

Weighting and calculating the state estimate

Computing state estimates

Variance of (2)

(n) completing measurement update according to the following formula according to the observation equation:

with updating of the sampling points

Computing measurement updates from observation equations

Updating measurements

Weighting and calculating the measurement estimate

Computing measurement estimates

Variance of (2)

Computing

Andcross covariance of

Calculating the Kalman gain K_k： $K_k=P_{z_k{y}_k}{P}_{{\tilde{y}}_k}^{-1}$

Computing state updates

${\hat{z}}_k=z_k^{-}+K_k(y_k-{\hat{y}}_k^{-})$

Computing state updates

Variance of (2)

$P_{x_k}=P_{x_k}^{-}-K_k{P}_{{\tilde{y}}_k}{K}_k^T$

Recursion of the resulting state update value

Namely the estimated battery residual capacity at the current moment k.