RECHARGEABLE BATTERY PARAMETER ESTIMATION APPARATUS AND RECHARGEABLE BATTERY PARAMETER ESTIMATION METHOD

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a rechargeable battery parameter estimation apparatus and a rechargeable battery parameter estimation method for estimating an offset error of measurements of a current flowing through a rechargeable battery, and estimating a capacity error, which is a difference of a typical full charge capacity (FCC) of the battery measured in advance from an actual FCC of the battery at a time when the battery is in use.

2. Description of the Related Art

In order to efficiently use a rechargeable battery such as a lithium-ion battery, a nickel metal-hydride battery, and a lead-acid battery, it is indispensable to highly precisely estimate a state of charge (SoC) of the battery. However, the SoC of the battery is not a physical quantity that is directly measurable, and hence the SoC has to be estimated from a current flowing through the battery, a voltage between terminals of the battery, measurements of the battery temperature, and histories of these physical quantities. As the most basic methods for estimating the SoC of the battery, the open circuit voltage method and the Coulomb-counting method are known.

The open circuit voltage method is a method in which a rechargeable battery is leaved out offline, and the voltage between terminals of the battery which sufficiently reaches the thermodynamic equilibrium, namely, an open circuit voltage (OCV), is measured, then the SoC of the battery is calculated on the basis of a dependence of the OCV on the SoC, which is determined in advance. Therefore, the battery has to be suspended for a long period during the SoC estimation by means of the open circuit voltage method.

On the other hand, the Coulomb-counting method is a method in which the current flowing through the battery is integrated, and the integrated current is devided by an FCC of the battery, to thereby calculate a change in the SoC from the integration start time. The Coulomb-counting method is capable of estimating the SoC even while the battery is online, by considering a state in which the SoC is apparent, that is, a state in which the battery is fully charged or fully discharged as a reference. However, the Coulomb-counting method has such a problem that an estimation accuracy of the SoC is greatly degraded by a small off set error of the current measurements, which is inevitable in widely used Hall effect sensors.

As a related-art apparatus using the Coulomb-counting method, there is disclosed a battery charging rate estimation apparatus, in which a rechargeable battery is assumed as a linear system based on an equivalent circuit model of the battery, the current flowing through the battery is assumed as an input of the linear system, and the voltage between the terminals of the battery is assumed as an output of the linear system; a Kalman filter is then constructed on the basis of an extended system in which the offset error of the current measurements is appended to the state of the linear system, to thereby precisely estimate the SoC of the battery and the offset error of the current measurements (for example, refer to Japanese Patent Application Laid-open No. 2012-57964).

As a related-art apparatus combining the open circuit voltage method and the Coulomb-counting method, there is disclosed a battery capacity calculation apparatus in which a current flowing through a rechargeable battery is integrated only in a period in which an absolute value of the current flowing through the battery is equal to or more than a certain value and the influence of the offset error of current measurements is thus relatively small, and comparing changes in the SoC with that in the integrated current in this period, to thereby estimate the FCC of the battery (for example, refer to Japanese Patent Application Laid-open No. 2012-58028).

According to Japanese Patent Application Laid-open No. 2012-58028, the SoC is estimated from the OCV, and the OCV is acquired by means of recursive calculation based on an equivalent circuit of the battery. Characteristic parameters of the equivalent circuit are adjusted by an adaptive mechanism from transitions of the current flowing through the battery and the voltage between terminals of the battery. As a result, the influence of the offset error of the current measurements is affirmed to be decreased by limiting the current integration period to the period in which the absolute value of the current is equal to or more than the certain value.

However, the related arts have the following problems.

According to Japanese Patent Application Laid-open No. 2012-57964, the typical FCC of the battery and the characteristic parameters of the equivalent circuit are considered to be known, and hence there is such a problem that when the typical FCC of the battery and the characteristic parameters of the equivalent circuit have errors, those errors propagate to the estimated value of the SoC and that of the offset error of the current measurements.

On the other hand, according to Japanese Patent Application Laid-open No. 2012-58028, the current flowing through the battery, which includes an offset error, is referred to by the adaptive mechanism for adjusting the characteristic parameters of the equivalent circuit. The characteristic parameters of the equivalent circuit thus has an error caused by the offset error. This error propagates to the estimated value of the OCV, which is calculated in a manner that depends on the characteristic parameters of the equivalent circuit, further to the estimated value of the SoC, and finally to the estimated value of the FCC of the battery.

Further, for example, even if the FCC of the battery is estimated by using the technology disclosed in Japanese Patent Application Laid-open No. 2012-58028, and the offset error of the current measurements is estimated by using the estimated value of the FCC by means of the method disclosed in Japanese Patent Application Laid-open No. 2012-57964, the estimated value of the offset error is superimposed with an error caused by the actual offset error of the current measurements, and hence the correct value of the offset error cannot be obtained.

As described above, in the related-art apparatuses, there is such a problem that the FCC of a rechargeable battery cannot be correctly estimated as long as the measurements of the current flowing through the battery have an offset error, because the error caused by the offset error propagates the estimated value of the FCC of the battery.

SUMMARY OF THE INVENTION

The present invention has been made in view of the above-mentioned problem, and therefore has an object to provide a rechargeable battery parameter estimation apparatus and a rechargeable battery parameter estimation method that are capable of highly precisely estimating an offset error of measurements of a current flowing through a rechargeable battery and a capacity error, which is a difference of a typical full charge capacity (FCC) of the battery measured in advance from an actual FCC of the battery at a time when the battery is in use even if the current measurements have an offset error.

According to one embodiment of the present invention, there is provided a rechargeable battery parameter estimation apparatus, including: a voltage measurement part for measuring a voltage between terminals of the battery; an SoC estimation part for estimating an estimated value of the SoC of the battery in a manner that depends on a typical FCC of the battery measured in advance and the voltage between the terminals of the battery, the typical FCC has a capacity error; a current measurement part for measuring a current flowing through the battery, the measurements of the current has an offset error; a first coefficient calculation part for calculating a first coefficient, which is a partial derivative of the estimated value of the SoC with respect to the offset error of the current measurements; a second coefficient calculation part for calculating a second coefficient, which is a partial derivative of the estimated value of the SoC with respect to the capacity error, which is a difference of the typical FCC measured in advance from an actual FCC of the battery at the battery in use; and an error estimation part for estimating the offset error of the current measurements and the capacity error of the FCC from derivative information including the first coefficient and the second coefficient, the current flowing through the battery, and the estimated value of the SoC.

According to one embodiment of the present invention, there is provided a rechargeable battery parameter estimation method, including: measuring a voltage between terminals of the battery; estimating an estimated value of the SoC of the battery in a manner that depends on a typical FCC of the battery measured in advance and the voltage between the terminals of the battery, the typical FCC has a capacity error; measuring a current flowing through the battery, the measurements of the current has an offset error; calculating a first coefficient, which is a partial derivative of the estimated value of the SoC with respect to the offset error of the current measurements; calculating a second coefficient, which is a partial derivative of the estimated value of the SoC with respect to the capacity error, which is a difference of the typical FCC measured in advance from an actual FCC of the battery at the battery in use; and estimating the offset error of the current measurements and the capacity error of the FCC from derivative information including the first coefficient and the second coefficient, the current flowing through the battery, and the estimated value of the SoC.

According to the one embodiment of the present invention, assuming either one or both of the current measurements and the typical FCC have errors, the errors are estimated from the estimated value of the SoC, which is estimated from the measurements of the voltage between the terminals of the battery. As a result, it is possible to provide the rechargeable battery parameter estimation apparatus and the rechargeable battery parameter estimation method that are capable of highly precisely estimating the offset error of the current measurements and the capacity error of the FCC, even if the measurements of the current have an offset error.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus according to a first embodiment of the present invention.

FIG. 2 illustrates an equivalent circuit of a rechargeable battery according to the first embodiment of the present invention.

FIG. 3 is an overall configuration diagram of an SoC estimation part according to the first embodiment of the present invention.

FIG. 4 is an overall configuration diagram of a first coefficient calculation part according to the first embodiment of the present invention.

FIG. 5 is an overall configuration diagram of a second coefficient calculation part according to the first embodiment of the present invention.

FIG. 6 is an overall configuration diagram of an error estimation part according to the first embodiment of the present invention.

FIG. 7 is a block diagram of an explanatory variable calculation part according to the first embodiment of the present invention.

FIG. 8 is a block diagram of an observed variable calculation part according to the first embodiment of the present invention.

FIG. 9 is a flowchart of a rechargeable battery parameter estimation method according to the first embodiment of the present invention.

FIG. 10 is a flowchart of a processing executed by the SoC estimation part in a cycle of periodic operations according to the first embodiment of the present invention.

FIG. 11 is a flowchart of a processing executed by the first coefficient calculation part in a cycle of periodic operations according to the first embodiment of the present invention.

FIG. 12 is a flowchart of a processing executed by the second coefficient calculation part in a cycle of periodic operations according to the first embodiment of the present invention.

FIG. 13 is a flowchart of a processing executed by the error estimation part in a cycle of periodic operations according to the first embodiment of the present invention.

FIG. 14 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus according to a second embodiment of the present invention.

FIG. 15 is a flowchart of a rechargeable battery parameter estimation method according to the second embodiment of the present invention.

FIG. 16 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus according to a third embodiment of the present invention.

FIG. 17 is an overall configuration diagram of an error estimation part according to the third embodiment of the present invention.

FIG. 18 is a block diagram of an explanatory variable calculation part according to the third embodiment of the present invention.

FIG. 19 is a block diagram of an observed variable calculation part according to the third embodiment of the present invention.

FIG. 20 is a flowchart of a rechargeable battery parameter estimation method according to the third embodiment of the present invention.

FIG. 21 is a flowchart of a processing executed by the error estimation part in a cycle of periodic operations according to the third embodiment of the present invention.

FIG. 22 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus according to a fourth embodiment of the present invention.

FIG. 23 is an overall configuration diagram of an error estimation part according to the fourth embodiment of the present invention.

FIG. 24 is a flowchart of a rechargeable battery parameter estimation method according to the fourth embodiment of the present invention.

FIG. 25 is a flowchart of a processing executed by the error estimation part in a cycle of periodic operations according to the fourth embodiment of the present invention.

FIG. 26 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus according to a fifth embodiment of the present invention.

FIG. 27 is an overall configuration diagram of an error estimation part according to the fifth embodiment of the present invention.

FIG. 28 is a flowchart of a rechargeable battery parameter estimation method according to the fifth embodiment of the present invention.

FIG. 29 is a flowchart of a processing executed by the error estimation part in a cycle of periodic operations according to the fifth embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A description is now given of a rechargeable battery parameter estimation apparatus and a rechargeable battery parameter estimation method according to preferred embodiments of the present invention referring to the accompanying drawings. It should be noted that throughout the drawings, like or corresponding components are denoted by like reference numerals to describe those components.

First Embodiment

FIG. 1 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus 101 according to a first embodiment of the present invention. A current measurement part 102 measures a current 104 flowing through a rechargeable battery 103. Moreover, a voltage measurement part 105 measures a voltage 106 between terminals of a positive electrode and a negative electrode of the battery 103. On this occasion, the measurements of the current 104 measured by the current measurement part 102 has an offset error 116.

A SoC estimation part 107 estimates an estimated value of the SoC 108 of the battery 103 in a manner that depends on a typical full charge capacity (FCC) of the battery 103 measured in advance and the voltage 106 measured by the voltage measurement part 105 on the basis of an equivalent circuit of the battery 103. It should be noted that, as illustrated in FIG. 1, the SoC estimation part 107 may further refer to the current 104 measured by the current measurement part 102. On this occasion, the typical FCC of the battery 103 has a capacity error 117, which is a difference of the typical FCC of the battery 103 measured in advance from an actual FCC of the battery 103 at a time when the battery 103 is in use.

A first coefficient calculation part 109 calculates, while referring to the internal calculations of the SoC estimation part 107, a first coefficient 110, which is a partial derivative of the estimated value of the SoC 108 with respect to the offset error 116 of the current 104 measured by the current measurement part 102, on the basis of the equivalent circuit of the battery 103.

A second coefficient calculation part 111 calculates, while referring to the internal calculations of the SoC estimation part 107, a second coefficient 112, which is a partial derivative of the estimated value of the SoC 108 with respect to the capacity error 117 of the FCC of the battery 103 measured in advance, on the basis of the equivalent circuit of the battery 103.

A current integration part 113 integrates the current 104 measured by the current measurement part 102, thereby calculating a current integration value 114.

An error estimation part 115 estimates the offset error 116 and the capacity error 117, from the first coefficient 110 calculated by the first coefficient calculation part 109, the second coefficient 112 calculated by the second coefficient calculation part 111, the current integration value 114 calculated by the current integration part 113, and the estimated value of the SoC 108 estimated by the SoC estimation part 107, then outputs the offset error 116 and the capacity error 117.

FIG. 2 illustrates the equivalent circuit of the battery 103 according to the first embodiment of the present invention. A voltage source 201 having a voltage E [V] illustrated in FIG. 2 represents an open circuit voltage (OCV) of the battery 103. The OCV is a function of the SoC, and is defined for each battery 103. A parallel circuit 204 constructed by an electrical resistor 202 having a resistance R_d[Ω] and a capacitor 203 having a capacitance C_d[F] represents a relaxation process of the battery 103. An electrical resistor 205 having a resistance R₀[Ω] represents a battery internal resistance other than the relaxation process.

On this occasion, let q_b[C] denote an electrical quantity, which is an amount of electricity accumulated in the battery 103, F_cc[C] denote the typical FCC of the battery 103, and s [%] denote the SoC of the battery 103, the electrical quantity q_b[C] is expressed by:

q_b=F_cc^S (1)

Because of the Kirchhoff's current law relating to the equivalent circuit and the Ohm's law relating to the electrical resistances, the voltage 106 of the battery 103 denoted by V [V] is determined by the following differential equation system:

$\begin{matrix} \dot{x} = F_{c} x + G_{c} I, V = h (x) + R_{0} I, & (2) \\ where \\ F_{c} := [\begin{matrix} - \frac{1}{C_{d} R_{d}} \\ 0 \end{matrix}], G_{c} := [\begin{matrix} 1 \\ 1 \end{matrix}] . & (3) \end{matrix}$

Moreover, h(x) and x are defined by:

$\begin{matrix} h (x) := E (q_{b} / F_{cc}) + \frac{1}{C_{d}} q_{d}, x := {[\begin{matrix} q_{d} & q_{b} \end{matrix}]}^{T} . & (4) \end{matrix}$

Moreover, I denotes the current 104 flowing through the battery 103, and C_d, R_d, and R₀are assumed as characteristic parameters of the equivalent circuit of the battery 103 illustrated in FIG. 2.

Let t_s[second] be a sampling period of the current measurement part 102 and the voltage measurement part 105, the differential equation system (2) is discretized to the following difference equation system:

$\begin{matrix} x_{k + 1} = {Fx}_{k} + {G I}_{k}, V_{k} = h (x_{k}) + R_{0} I_{k}, & (5) \\ where \\ t_{k} := t_{Sk}, x_{k} := x (t_{k}), I_{k} := I (t_{k}), V_{k} := V (t_{k}), and & (6) \\ F := [\begin{matrix} e^{- \frac{t_{s}}{R_{d} C_{d}}} \\ 1 \end{matrix}], G := [\begin{matrix} R_{d} C_{d} (1 - e^{- \frac{t_{s}}{C_{d} R_{d}}}) \\ t_{s} \end{matrix}] . & (7) \end{matrix}$

FIG. 3 is an overall configuration diagram of the SoC estimation part 107 according to the first embodiment of the present invention. In the SoC estimation part 107 according to the first embodiment, an extended Kalman filter is used to estimate the estimated value of the SoC 108 of the battery 103 on the basis of the equivalent circuit illustrated in FIG. 2, namely, the difference equation system (5).

The SoC estimation part 107 periodically operates for each of the measurements carried out by the current measurement part 102 and the voltage measurement part 105. In the operation for a certain cycle, a state prediction part 301 calculates a predicted state estimate x̂_kfrom a current denoted by I_k−l, which is the current 104 of the battery 103 measured by the current measurement part 102 and stored in the current storage part 302 one cycle before, and an updated state estimate x_k−1, which is calculated by a state estimation part 310 and stored in a state storage part 303 one cycle before, by following:

{circumflex over (x)}
_k
=Fx
_k−1
+GI
_k−1. (8)

A measurement residual calculation part 304 calculates a measurement residual e_kfrom the predicted state estimate x^̂_kcalculated by the state prediction part 301, a current denoted by I_k, which is the current 104 of the battery 103 measured by the current measurement part 102 in a cycle, and a voltage denoted by V_k, which is the voltage 106 of the battery 103 measured by the voltage measurement part 105 by following:

e
_k
:=V
_k
−h({circumflex over (x)}_k)−R₀I_k. (9)

A covariance prediction part 305 calculates a predicted estimate covariance P̂_kfrom an updated estimate covariance P_k−1calculated by a covariance estimation part 311 and stored in a covariance storage part 306 one cycle before by following:

{circumflex over (P)}
_k
=FP
_k−1
F
^T
+Q. (10)

It should be noted that Q is calculated in advance by:

Q:=Φ
₁₂Φ₂₂⁻¹, (11)

where Φ₁₂and Φ₂₂are matrices defined by:

$\begin{matrix} [\begin{matrix} Φ_{11} & Φ_{12} \\ Φ_{21} & Φ_{22} \end{matrix}] := \exp ([\begin{matrix} F_{c} & σ_{I}^{2} G_{c} G_{c}^{T} \\ 0 & - F_{c}^{T} \end{matrix}] t_{s}) . & (12) \end{matrix}$

Moreover, σ_I²is a predetermined positive real constant representing a variance of an observation error of the current 104.

An observation matrix calculation part 307 calculates an observation matrix H_kfrom the predicted state estimate x^̂_kcalculated by the state prediction part 301 by following:

$\begin{matrix} H_{k} := \frac{\partial h ({\hat{x}}_{k})}{\partial x} . & (13) \end{matrix}$

An residual covariance calculation part 308 calculates an residual covariance S_kfrom the observation matrix H_kcalculated by the observation matrix calculation part 307 and the predicted estimate covariance P̂_kcalculated by the covariance prediction part 305 by following:

S
_k=σ_V²+H_k{circumflex over (P)}_kH_k^T. (14)

where σ_V²is a predetermined positive real constant representing a variance of an observation error of the voltage 106.

A Kalman gain calculation part 309 calculates an optimal Kalman gain K_kfrom the predicted estimate covariance P̂_kcalculated by the covariance prediction part 305, the observation matrix H_kcalculated by the observation matrix calculation part 307, and the residual covariance S_kcalculated by the residual covariance calculation part 308 by following:

K
_k
={circumflex over (P)}
_k
H
_k
^T
S
_k
⁻¹. (15)

The state estimation part 310 calculates an updated state estimate x_kfrom the predicted state estimate x̂_kcalculated by the state prediction part 301, the optimal Kalman gain K_kcalculated by the Kalman gain calculation part 309, and the measurement residual e_kcalculated by the measurement residual calculation part 304 by following:

x
_k
={circumflex over (x)}
_k
+K
_k
e
_k, (16)

and stores a result in the state storage part 303.

The covariance estimation part 311 calculates an updated estimate covariance P_kfrom the optimal Kalman gain K_kcalculated by the Kalman gain calculation part 309, the observation matrix H_kcalculated by the observation matrix calculation part 307, and the predicted estimate covariance P̂_kcalculated by the covariance prediction part 305 by following:

P
_k
={circumflex over (P)}
_k
−K
_k
H
_k
{circumflex over (P)}
_k, (17)

and stores the updated estimate covariance P_kin the covariance storage part 306.

An SoC calculation part 312 calculates an estimated value of the SoC 108 by dividing an amount of charge q_b,k, which is the second element of the updated state estimate x_kcalculated by the state estimation part 310 by the typical FCC F_ccof the battery 103 measured in advance.

The SoC estimation part 107 refers to the measured value of the current 104 in order to estimate the SoC 108. If the measured value of the current 104 is strictly equal to the current actually flowing through the battery 103, and the characteristics of the battery 103 strictly matches the equivalent circuit illustrated in FIG. 2, the SoC estimation part 107 provides the best SoC estimation in terms of the mean square error.

However, if the measured value of the current 104 has an offset error of a value ΔI [A], the estimated value of the SoC 108 has a bias error caused by the offset error of the current measurements. The first coefficient calculation part 109 of the first embodiment approximately quantifies the bias error of the estimated value of the SoC 108 caused by the offset error of the current measurements.

FIG. 4 is an overall configuration diagram of the first coefficient calculation part 109 according to the first embodiment of the present invention. A predicted state derivative calculation part 401 calculates a partial derivative of the predicted state estimate with respect to a parameter p₁denoted by dx̂_k/dp₁from a partial derivative of the updated state estimate with respect to the parameter p₁, which is denoted by dx_k−1/dp₁, calculated by an estimated state derivative calculation part 404 and stored in an estimated state derivative storage part 402 one cycle before by following:

$\begin{matrix} \frac{\partial {\hat{x}}_{k}}{\partial p_{1}} = F \frac{\partial x_{k - 1}}{\partial p_{1}} + {GI}_{typ} . & (18) \end{matrix}$

On this occasion, the offset error ΔI is expressed by ΔI=I_typ×p₁, where I_typis a constant introduced to ensure the absolute value of the parameter p₁sufficiently smaller than 1.

An measurement residual derivative calculation part 403 calculates a partial derivative of the measurement residual with respect to the parameter p₁denoted by de_k/dp₁from the partial derivative of the predicted state estimate with respect to the parameter p₁calculated by the predicted state derivative calculation part 401 by following:

$\begin{matrix} \frac{\partial e_{k}}{\partial p_{1}} = - H_{k} \frac{\partial {\hat{x}}_{k}}{\partial p_{1}} - R_{0} I_{typ} . & (19) \end{matrix}$

It should be noted that the observation matrix H_kis calculated from the predicted state estimate x̂_k, which is calculated by the state prediction part 301 by Equation (13).

The estimated state derivative calculation part 404 calculates a partial derivative of the updated state estimate with respect to the parameter p₁denoted by dx_k/dp₁from the partial derivative of the predicted state estimate with respect to the parameter p1, which is denoted by dx̂_k/dp₁and calculated by the predicted state derivative calculation part 401, the optimal Kalman gain K_kcalculated by the Kalman gain calculation part 309, and the partial derivative of the measurement residual with respect to the parameter p₁, which is denoted by de_k/dp₁and calculated by the measurement residual derivative calculation part 403 by following:

$\begin{matrix} \frac{\partial x_{k}}{\partial p_{1}} = \frac{\partial {\hat{x}}_{k}}{\partial p_{1}} + K_{k} \frac{\partial e_{k}}{\partial p_{1}} . & (20) \end{matrix}$

An SoC derivative calculation part 405 calculates the first coefficient 110 by dividing a second element of the partial derivative of the updated state estimate with respect to the parameter p₁, which is denoted by dq_b,k/dp₁and calculated by the estimated state derivative calculation part 404 by the typical FCC F_ccof the battery 103 measured in advance.

The SoC estimation part 107 uses the typical FCC F_ccof the battery 103 in order to calculate the estimated value of the SoC 108. If the typical FCC F_ccused by the SoC estimation part 107 matches the actual FCC of the battery 103, and the measurements of the current do not have an offset error, and other characteristics of the battery 103 match the equivalent circuit illustrated in FIG. 2, the SoC estimation part 107 illustrated in FIG. 3 provides the best SoC estimation in terms of the mean square error.

However, if the typical FCC F_ccused by the SoC estimation part 107 has a capacity error, the estimated value of the SoC 108 has a bias error. The second coefficient calculation part 111 approximately quantifies the bias error in the estimated value of the SoC 108 caused by the capacity error of the typical FCC F_cc.

FIG. 5 is an overall configuration diagram of the second coefficient calculation part 111 according to the first embodiment of the present invention. In the following, the actual FCC of the battery 103 is denoted by F_cc, and the typical FCC used by the SoC estimation part 107 is denoted by F˜_cc. On this occasion, it is assumed that F˜_ccis differ from F_ccas:

{tilde over (F)}
_cc
=F
_cc(1+p₂). (21)

On this occasion, it is assumed that the difference is sufficiently smaller than F_cc, in other words, the absolute value of the parameter p₂is sufficiently smaller than 1.

A predicted state derivative calculation part 501 calculates a partial derivative of the predicted state estimate with respect to a parameter p₂, which is denoted by dx̂_k/dp₂, from a partial derivative of the updated state estimate with respect to the parameter p₂, which is denoted by dx_k−1/dp₂, calculated by an estimated state derivative calculation part 509 and stored in an estimated state derivative storage part 502 one cycle before by following:

$\begin{matrix} \frac{\partial {\hat{x}}_{k}}{\partial p_{2}} = F \frac{\partial x_{k - 1}}{\partial p_{2}} . & (22) \end{matrix}$

An observation matrix derivative calculation part 503 calculates a partial derivative of the observation matrix with respect to the parameter p₂, which is denoted by dH_k/dp₂, from the predicted state estimate x̂_kcalculated by the state prediction part 301.

An measurement residual derivative calculation part 504 calculates a partial derivative of the measurement residual with respect to the parameter p₂, which is denoted by de_k/dp₂, from the partial derivative of the observation matrix with respect to the parameter p₂, which is denoted by dH_k/dp₂and calculated by the observation matrix derivative calculation part 503, the predicted state estimate x̂_kcalculated by the state prediction part 301, the observation matrix H_kcalculated by the observation matrix calculation part 307, and the partial derivative of the predicted state estimate with respect to the parameter p₂, which is denoted by dx̂_k/dp₂and calculated by the predicted state derivative calculation part 501 by following:

$\begin{matrix} \frac{\partial e_{k}}{\partial p_{2}} = - \frac{\partial H_{k}}{\partial p_{2}} {\hat{x}}_{k} - H_{k} \frac{\partial {\hat{x}}_{k}}{\partial p_{2}} . & (23) \end{matrix}$

A predicted covariance derivative calculation part 505 calculates a partial derivative of the predicted estimate covariance with respect to the parameter p₂, which is denoted by dP̂_k/dp₂, from a partial derivative of the updated estimate covariance with respect to the parameter p₂, which is denoted by dP_k−1/dp₂, calculated by a estimated covariance derivative calculation part 510 and stored in an estimated covariance derivative storage part 506 one cycle before by following:

$\begin{matrix} \frac{\partial {\hat{P}}_{k}}{\partial p_{2}} = F \frac{\partial P_{k - 1}}{\partial p_{2}} F^{T} . & (24) \end{matrix}$

An residual covariance derivative calculation part 507 calculates a partial derivative of the residual covariance with respect to the parameter p₂, which is denoted by dS_k/dp₂, from the partial derivative of the observation matrix with respect to the parameter p₂, which is denoted by dH_k/dp₂and calculated by the observation matrix derivative calculation part 503, the predicted estimate covariance P̂_kcalculated by the covariance prediction part 305, the observation matrix H_kcalculated by the observation matrix calculation part 307, and the partial derivative of the predicted estimate covariance with respect to the parameter p₂, which is denoted by dP̂_k/dp₂and calculated by the predicted covariance derivative calculation part 505 by following:

$\begin{matrix} \frac{\partial S_{k}}{\partial p_{2}} = \frac{\partial H_{k}}{\partial p_{2}} {\hat{P}}_{k} H_{k}^{T} + H_{k} \frac{\partial {\hat{P}}_{k}}{\partial p_{2}} H_{k}^{T} + H_{k} {\hat{P}}_{k} \frac{\partial H_{k}^{T}}{\partial p_{2}} . & (25) \end{matrix}$

A Kalman gain derivative calculation part 508 calculates a partial derivative of the Kalman gain with respect to the parameter p₂, which is denoted by dK_k/dp₂, from the partial derivative of the predicted estimate covariance with respect to the parameter p₂, which is denoted by dP̂_k/dp₂and calculated by the predicted covariance derivative calculation part 505, the partial derivative of the observation matrix with respect to the parameter p₂, which is denoted by dH_k/dp₂and calculated by the observation matrix derivative calculation part 503, the residual covariance S_kcalculated by the residual covariance calculation part 308, the predicted estimate covariance P̂_kcalculated by the covariance prediction part 305, the partial derivative of the observation matrix with respect to the parameter p₂, which is denoted by dH_k/dp₂and calculated by the observation matrix derivative calculation part 503, and the partial derivative of the residual covariance with respect to the parameter p₂, which is denoted by dS_k/dp₂and calculated by the residual covariance derivative calculation part 507 by following:

$\begin{matrix} \frac{\partial K_{k}}{\partial p_{2}} = \frac{\partial {\hat{P}}_{k}}{\partial p_{2}} H_{k}^{T} S_{k}^{- 1} + {\hat{P}}_{k} \frac{\partial H_{k}^{T}}{\partial p_{2}} S_{k}^{- 1} - {\hat{P}}_{k} H_{k}^{T} S_{k}^{- 1} \frac{\partial S_{k}}{\partial p_{2}} S_{k}^{- 1} . & (26) \end{matrix}$

The estimated state derivative calculation part 509 calculates a partial derivative of the updated state estimate with respect to the parameter p₂, which is denoted by dx_k/dp₂, from the partial derivative of the updated state estimate with respect to the parameter p₂, which is denoted by dx̂_k/dp₂and calculated by the predicted state derivative calculation part 501, the partial derivative of the optimal Kalman gain with respect to the parameter p₂, which is denoted by dK_k/dp₂and calculated by the Kalman gain derivative calculation part 508, the measurement residual e_kcalculated by the measurement residual calculation part 304, the optimal Kalman gain K_kcalculated by the Kalman gain calculation part 309, and the partial derivative of the measurement residual with respect to the parameter p₂, which is denoted by de_k/dp₂and calculated by the measurement residual derivative calculation part 504 by following:

$\begin{matrix} \frac{\partial x_{k}}{\partial p_{2}} = \frac{\partial {\hat{x}}_{k}}{\partial p_{2}} + \frac{\partial K_{k}}{\partial p_{2}} e_{k} + K_{k} \frac{\partial e_{k}}{\partial p_{2}}, & (27) \end{matrix}$

and stores the resulting partial derivative of the updated state estimate with respect to the parameter p₂, which is denoted by dx_k/dp₂, in the estimated state derivative storage part 502.

The estimated covariance derivative calculation part 510 calculates a partial derivative of the updated estimate covariance with respect to the parameter p₂, which is denoted by dP_k/dp₂, from the partial derivative of the predicted estimate covariance with respect to the parameter p₂, which is denoted by dP̂_k/dp₂and calculated by the predicted covariance derivative calculation part 505, the partial derivative of the optimal Kalman gain with respect to the parameter p₂, which is denoted by dK_k/dp₂and calculated by the Kalman gain derivative calculation part 508, the observation matrix H_kcalculated by the observation matrix calculation part 307, the predicted estimate covariance P̂_kcalculated by the covariance prediction part 305, the optimal Kalman gain K_kcalculated by the Kalman gain calculation part 309, and the partial derivative of the observation matrix with respect to the parameter p₂, which is denoted by dH_k/dp₂and calculated by the observation matrix derivative calculation part 503 by following:

$\begin{matrix} \frac{\partial P_{k}}{\partial p_{2}} = \frac{\partial {\hat{P}}_{k}}{\partial p_{2}} - \frac{\partial K_{k}}{\partial p_{2}} H {\hat{P}}_{k} + K_{k} \frac{\partial H_{k}}{\partial p_{2}} {\hat{P}}_{k} + K_{k} H_{k} \frac{\partial {\hat{P}}_{k}}{\partial p_{2}}, & (28) \end{matrix}$

and stores the resulting updated estimate covariance with respect to the parameter p₂, which is denoted by dP_k/dp₂, in the estimated covariance derivative storage part 506.

The SoC derivative calculation part 511 calculates a second coefficient, which is denoted by ds_k/dp₂, from the typical FCC of the battery 103 measured in advance, a partial derivative of the amount of charge, which is denoted by dq_b,k/dp₂, obtained as the second element of the partial derivative of the updated state estimate with respect to the parameter p₂and calculated by the estimated state derivative calculation part 509, and the amount charge q_b,k, which is obtained as the second element of the estimated state x_kestimated by the state estimation part 310 by following:

$\begin{matrix} \frac{\partial s_{k}}{\partial p_{2}} = \frac{1}{F_{cc}} (\frac{\partial q_{b, k}}{\partial p_{2}} - q_{b, k}) . & (29) \end{matrix}$

The current integration part 113 calculates an integration value of the current 104 from the initial time by following:

q
_cc,k
=q
_cc,k−1
+t
_s
I
_k−1. (30)

Incidentally, if the SoC of the battery 103 is correctly estimated, the measurements of the current do not have an offset error, and the typical FCC is equal to the actual FCC of the battery 103, the following equation holds:

q
_cc,k
=F
_cc
s
_k
−q
₀, (31)

where q₀denotes an initial charged capacity.

However, actually, the measurements of the current have an offset error, the typical FCC differs from the actual FCC, and further, the estimated value of the SoC 108 has a bias error caused by the offset error of the current measurements and the capacity error of the FCC. Thus, Equation (31) is corrected as:

$\begin{matrix} q_{cc, k} = \frac{{\tilde{F}}_{cc}}{1 + p_{2}} ({\tilde{s}}_{k} - \frac{\partial s_{k}}{\partial p_{1}} p_{1} - \frac{\partial s_{k}}{\partial p_{2}} p_{2}) + t_{k} I_{typ} p_{1} - q_{0} . & (32) \end{matrix}$

By omitting the higher order terms with respect to the parameters p₁and p₂, Equation (32) is transformed into:

$\begin{matrix} q_{cc, k} - {\tilde{F}}_{cc} {\tilde{s}}_{k} = (t_{k} I_{typ} - {\tilde{F}}_{cc} \frac{\partial s_{k}}{\partial p_{1}}) p_{1} - {\tilde{F}}_{cc} ({\tilde{s}}_{k} + \frac{\partial s_{k}}{\partial p_{2}}) p_{2} - q_{0} . & (33) \end{matrix}$

On this occasion, the left hand side of Equation (33) is a constant at each cycle, and the right hand side of Equation (33) is a linear expression in terms of the parameters p₁, p₂, and the initial charged capacity q₀at each cycle. Thus, the optimal estimation of the parameters in terms of the mean square error, which is denoted by p̂_k, is obtained with a recursive least squares filter which minimizes the following objective function:

$\begin{matrix} J ({\hat{p}}_{k}) := {λ^{k - 1} (y_{l} - {\hat{p}}_{k}^{T} u_{l})}^{2}, & (34) \end{matrix}$

where λ is a forgetting factor which is a predetermined real constant more than 0 and equal to or less than 1.

In the first embodiment, the vector u_kreferred to as an explanatory variable is defined by the estimated value of the SoC s_k, the first coefficient ds_k/dp₁, and the second coefficient ds_k/dp₂as follows:

$\begin{matrix} u_{k} := [\begin{matrix} t_{k} I_{typ} - {\tilde{F}}_{cc} \frac{\partial s_{k}}{\partial p_{1}} \\ - {\tilde{F}}_{cc} (s_{k} + \frac{\partial s_{k}}{\partial p_{2}}) \\ - 1 \end{matrix}] . & (35) \end{matrix}$

Moreover, in the first embodiment, the scalar y_kreferred to as an observed variable is defined by:

y
_k
:=q
_cc,k
{tilde over (F)}
_cc
s
_k. (36)

Moreover, in the first embodiment, the vector p̂_kreferred to as an estimated parameter vector is composed of an estimated value of the parameter p₁as the first element, an estimated value of the parameter p₂as the second element, and an estimated value of the initial charge capacity q₀as the third element. The gist of the present invention is to use this fact, to thereby estimate the parameters p₁and p₂, by which the offset error and the capacity error of the battery are able to be corrected.

FIG. 6 is an overall configuration diagram of the error estimation part 115 according to the first embodiment of the present invention. An explanatory variable calculation part 601 calculates the explanatory variable u_kby Equation (35) from the estimated value of the SoC 108 estimated by the SoC estimation part 107, the first coefficient 110 calculated by the first coefficient calculation part 109, and the second coefficient 112 calculated by the second coefficient calculation part 111.

An observed variable calculation part 602 calculates the observed variable y_kby Equation (36) from the estimated value of the SoC 108, and the current integration value 114 calculated by the current integration part 113.

An RLS gain calculation part 603 calculates an RLS gain g_kfrom the explanatory variable u_kcalculated by the explanatory variable calculation part 601, and an inverse variance Y_k−1, which is calculated by an inverse variance calculation part 605 and stored in an inverse variance storage part 604 one cycle before by following:

g
_k
=Y
_k−1
u
_k(λ+u_k^TY_k−1u_k)⁻¹. (37)

The inverse variance calculation part 605 calculates the inverse variance Y_kfrom the inverse variance Y_k−1, which is calculated by the inverse variance calculation part 605 and stored in the inverse variance storage part 604 one cycle before, the RLS gain g_kcalculated by the RLS gain calculation part 603, and the explanatory variable u_kcalculated by the explanatory variable calculation part 601 by following:

Y
_k=λ⁻¹(I_d−g_ku_k^T)Y_k−1, (38)

where I_ddenotes a unit matrix.

A parameter estimation part 606 calculates the estimated parameter vector p̂_kfrom the explanatory variable u_kcalculated by the explanatory variable calculation part 601, the observed variable y_kcalculated by the observed variable calculation part 602, the RLS gain g_kcalculated by the RLS gain calculation part 603, and a estimated parameter vector p̂_k−1stored in a parameter storage part 607 one cycle before by following:

{circumflex over (p)}
_k
={circumflex over (p)}
_k−1
+g
_k(y_k−u_k^T{circumflex over (p)}_k−1), (39)

then stores the estimated parameter vector p̂_kin the parameter storage part 607.

The offset error ΔI of the current measurements is obtained by ΔI=I_typ×p₁, where the parameter p₁is the first element of the estimated parameter vector p̂_k, and the actual FCC of the battery is obtained by F_cc=F˜_cc(1−p₂), where p₂is the second element of the estimated parameter vector p^̂_k.

An offset calculation part 608 and a capacity calculation part 609 carry out this calculation, and respectively estimate the offset error 116 of the current measurements and the capacity error 117 of the battery.

FIG. 7 is a block diagram of the explanatory variable calculation part 601 according to the first embodiment of the present invention. A time point t_kis calculated by adding a sampling period t_sto a time point t_k−1for one cycle before.

The first element of the explanatory variable u_kis calculating by acquiring a difference between a product of the time point t_kand the constant I_typand a product of the first coefficient 110 calculated by the first coefficient calculation part 109 and the typical FCC F_cc. The second element of the explanatory variable u_kis calculated by acquiring a sum by adding the second coefficient 112 calculated by the second coefficient calculation part 111 to the estimated value of the SoC 108 estimated by the SoC estimation part 107, acquiring a product by multiplying the sum by the typical FCC F_cc, and negating the product. The third element of the explanatory variable u_kis set a constant of −1.

FIG. 8 is a block diagram of the observed variable calculation part 602 according to the first embodiment of the present invention. The observed variable y_kis calculated by acquiring a difference between the current integration value 114 calculated by the current integration part 113 and a product acquired by multiplying the typical FCC F_ccby the estimated value of the SoC s_kestimated by the SoC estimation part 107.

FIG. 9 is a flowchart of a rechargeable battery parameter estimation method according to the first embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the rechargeable battery parameter estimation apparatus 101 illustrated in FIG. 1. It should be noted that an execution sequence of the rechargeable battery parameter estimation method according to the first embodiment is not limited to the execution sequence illustrated in FIG. 9. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 10 is a flowchart of a processing executed by the SoC estimation part 107 in a cycle of periodic operations according to the first embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the SoC estimation part 107 illustrated in FIG. 3. It should be noted that an execution sequence by the SoC estimation part 107 according to the first embodiment is not limited to the execution sequence illustrated in FIG. 10. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 11 is a flowchart of a processing executed by the first coefficient calculation part 109 in a cycle of periodic operations according to the first embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the first coefficient calculation part 109 illustrated in FIG. 4. It should be noted that an execution sequence by the first coefficient calculation part 109 according to the first embodiment is not limited to the execution sequence illustrated in FIG. 11. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 12 is a flowchart of a processing executed by the second coefficient calculation part 111 in a cycle of periodic operations according to the first embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the second coefficient calculation part 111 illustrated in FIG. 5. It should be noted that an execution sequence by the second coefficient calculation part 111 according to the first embodiment is not limited to the execution sequence illustrated in FIG. 12. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 13 is a flowchart of a processing executed by the error estimation part 115 in a cycle of periodic operations according to the first embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the error estimation part 115 illustrated in FIG. 6. It should be noted that an execution sequence by the error estimation part 115 according to the first embodiment is not limited to the execution sequence illustrated in FIG. 13. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

Moreover, a detailed calculation method according to the first embodiment is not limited that described in the first embodiment. For example, Equation (38) is an equation for recursively calculating the inverse variance Y_k, which is an inverse of the weighted covariance matrix R_kof the explanatory variable u_k, but the inverse variance Y_kmay be calculated by means of usual matrix inversion, which increases a calculation complexity, by following:

$\begin{matrix} Y_{k} = {(\sum_{l = 0}^{k} λ^{k - l} u_{k} u_{k}^{T})}^{- 1} . & (40) \end{matrix}$

Similarly, the estimated parameter vector p̂_kcan be calculated from a vector r_kdefined by:

$\begin{matrix} r_{k} := \sum_{l = 0}^{k} λ^{k - l} y_{k} u_{k}, & (41) \end{matrix}$

as follows:

{circumflex over (p)}_k=Y_kτ_k. (42)

Alternatively, in the first embodiment, the forgetting factor λ is assumed as a constant, but may be changed in a manner that depends on k.

Further, in the first embodiment, the SoC estimation part 107 uses a Kalman filter to obtain the estimated value of the SoC 108 from the current 104 flowing through the battery 103 and the voltage 106 between the terminals of the battery 103, but the SoC estimation part 107 according to the first embodiment is not limited to the implementation using the Kalman filter. As long as the partial derivatives of the estimated value of the SoC 108 with respect to the offset error of the current measurements and the capacity error of the FCC can be calculated, the present invention is applicable.

The rechargeable battery parameter estimation apparatus 101 according to the first embodiment may be realized, for example, as a microcontroller with a memory and an input/output function. In this case, each part other than the current measurement part 102 and the voltage measurement part 105 is realized as software embedded in the microcontroller.

As described above, according to the first embodiment, assuming either one or both of current measurements of a rechargeable battery and the typical FCC of the battery measured in advance have errors, the errors are estimated from an estimated value of the SoC, which is estimated from measurements of a voltage between terminals of the battery on the basis of an equivalent circuit of the battery. As a result, it is possible to provide a rechargeable battery parameter estimation apparatus capable of highly precisely estimating the offset error of the current measurements and the capacity error of the FCC, even if the measurements of the current have an offset error.

As a result, more efficient energy management can be provided, which contributes to energy saving. Moreover, the highly precise estimation of rechargeable battery parameters enables a continuous degradation assessment of rechargeable batteries, which contributes lifetime management of rechargeable batteries, a reduction in a maintenance cost of a electrical power storage system, and the like.

Second Embodiment

FIG. 14 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus 101 according to a second embodiment of the present invention. The rechargeable battery parameter estimation apparatus 101 according to the second embodiment illustrated in FIG. 14 is different from that illustrated in FIG. 1 of the first embodiment in such a point that the offset error 116 of the current measurements and the capacity error 117 of the FCC are estimated while errors in the characteristic parameters R_d, C_d, and R₀of the equivalent circuit of the battery 103 illustrated in FIG. 2 are additionally considered.

Specifically, typical parameters R_d, C_d, and R₀measured in advance for the use in the SoC estimation part 107 are considered to have errors compared with those of the actual battery at the battery in use, and are parameterized as:

$\begin{matrix} {\tilde{R}}_{d} {\tilde{C}}_{d} = R_{d} C_{d} (1 + p_{3}), \frac{1}{{\tilde{C}}_{d}} = \frac{1}{C_{d}} (1 + p_{4}), {\tilde{R}}_{0} = R_{0} (1 + p_{5}) . & (43) \end{matrix}$

Then, a third coefficient, a fourth coefficient, and a fifth coefficient, which are change rates of the estimated value of the SoC s_kfor the parameters p₃, p₄, and p_s, are respectively calculated as in the first embodiment by a third coefficient calculation part 1401, a fourth coefficient calculation part 1402, and a fifth coefficient calculation part 1403 respectively corresponding to the parameters p₃, p₄, and p_s.

The error estimation part 1404 uses, in addition to Equation (31), the following relationships:

$\begin{matrix} q_{d, k} = e^{- \frac{t_{s}}{R_{d} C_{d}}} q_{d, k - 1} + R_{d} C_{d} (1 - e^{- \frac{t_{s}}{R_{d} C_{d}}}) I_{k - 1}, & (44) \\ V_{k} = E (s_{k}) + \frac{1}{C_{d}} q_{d, k} + R_{0} I_{k} . & (45) \end{matrix}$

In addition to derivative information including the first coefficient and the second coefficient according to the first embodiment, Equation (32) is extended by adding the third coefficient, the fourth coefficient, and the fifth coefficient as follows:

$\begin{matrix} q_{cc, k} = \frac{{\tilde{F}}_{cc}}{1 + p_{2}} ({\tilde{s}}_{k} - \sum_{j = 1}^{5} \frac{\partial s_{k}}{\partial p_{j}} p_{j}) + t_{k} I_{typ} p_{1} - q_{0} . & (46) \end{matrix}$

Moreover, Equations (44) and (45) can be similarly transformed into:

$\begin{matrix} {\tilde{q}}_{d, k} - \sum_{j = 1}^{5} \frac{\partial q_{d, k}}{\partial p_{j}} p_{j} = ({\tilde{F}}_{11} - \frac{\partial F_{11}}{\partial p_{3}} p_{3}) ({\tilde{q}}_{d, k - 1} - \sum_{j = 1}^{5} \frac{\partial q_{d, k - 1}}{\partial p_{j}} p_{j}) + ({\tilde{G}}_{1} - \frac{\partial G_{1}}{\partial p_{3}} p_{3}) (I_{k - 1} - I_{typ} p_{1}), & (47) \\ V_{k} = E ({\tilde{s}}_{k}) - \frac{\partial E}{\partial s_{k}} \sum_{j = 0}^{5} \frac{\partial s_{k}}{\partial p_{j}} p_{j} + \frac{1 - p_{4}}{{\tilde{C}}_{d}} ({\tilde{q}}_{d, k} - \frac{\partial q_{d, k}}{\partial p_{j}} p_{j}) + {\tilde{R}}_{0} (1 - p_{5}) (I_{k} - I_{typ} p_{1}), where & (48) \\ {\tilde{F}}_{11} : = e^{- \frac{t_{s}}{{\tilde{R}}_{d} {\tilde{C}}_{d}}}, & (49) \\ {\tilde{G}}_{1} := {\tilde{R}}_{d} {\tilde{G}}_{d} (1 - e^{- \frac{t_{s}}{{\tilde{R}}_{d} {\tilde{C}}_{d}}}), & (50) \end{matrix}$

respectively.

By omitting the high order terms with respect to the parameters, Equations (46), (47), and (48) are transformed into linear relationships in terms of a parameter p_j.

The error estimation part 1404 according to the second embodiment minimizes the following objective function J in place of Equation (34):

$\begin{matrix} J ({\hat{p}}_{k}) : = \frac{1}{2} \sum_{l = 0}^{k} {λ^{k - l} (y_{l} - u_{l}^{T} {\hat{p}}_{k})}^{T} Σ^{- 1} (y_{l} - u_{l}^{T} {\hat{p}}_{k}), & (51) \end{matrix}$

where Σ is a predetermined positive definite symmetric weighting matrix. Moreover, an observed variable y₁is defined as follows:

$\begin{matrix} y_{l} := [\begin{matrix} y_{1, l} \\ y_{2, l} \\ y_{3, l} \end{matrix}] := [\begin{matrix} {\tilde{q}}_{cc, k} - {\tilde{F}}_{cc} {\tilde{s}}_{k} \\ {\tilde{q}}_{d, k} - {\tilde{F}}_{11} {\tilde{q}}_{d, k - 1} - {\tilde{G}}_{1} I_{k - 1} \\ V_{k} - E ({\tilde{s}}_{k}) - \frac{1}{{\tilde{C}}_{d}} {\tilde{q}}_{d, k} - {\tilde{R}}_{0} I_{k} \end{matrix}] . & (52) \end{matrix}$

Moreover, the explanatory variable u_kis a matrix [u_1,ku_2,ku_3,k] where

$\begin{matrix} u_{1, k} : = [\begin{matrix} - \frac{\partial q_{b, k}}{\partial p_{1}} + I_{typ} t_{k} \\ - \frac{\partial q_{b, k}}{\partial p_{2}} \\ - \frac{\partial q_{b, k}}{\partial p_{3}} \\ - \frac{\partial q_{b, k}}{\partial p_{4}} \\ - \frac{\partial q_{b, k}}{\partial p_{5}} \\ - 1 \end{matrix}] & (53) \\ u_{2, k} : = [\begin{matrix} \frac{\partial q_{d, k}}{\partial p_{1}} - {\tilde{F}}_{11} \frac{\partial q_{d, k - 1}}{\partial p_{1}} - {\tilde{G}}_{1} I_{typ} \\ \frac{\partial q_{d, k}}{\partial p_{2}} - {\tilde{F}}_{11} \frac{\partial q_{d, k - 1}}{\partial p_{2}} \\ \frac{\partial q_{d, k}}{\partial p_{3}} - {\tilde{F}}_{11} \frac{\partial q_{d, k - 1}}{\partial p_{3}} - \frac{\partial F_{11}}{\partial p_{3}} q_{d, k - 1} - \frac{\partial G_{1}}{\partial p_{3}} I_{k - 1} \\ \frac{\partial q_{d, k}}{\partial p_{4}} - {\tilde{F}}_{11} \frac{\partial q_{d, k - 1}}{\partial p_{4}} \\ \frac{\partial q_{d, k}}{\partial p_{5}} - {\tilde{F}}_{11} \frac{\partial q_{d, k - 1}}{\partial p_{5}} \\ 0 \end{matrix}] & (54) \\ u_{3, k} := [\begin{matrix} - \frac{1}{{\tilde{F}}_{cc}} \frac{\partial E}{\partial s_{k}} \frac{\partial q_{b, k}}{p_{1}} - \frac{1}{{\tilde{C}}_{d}} \frac{\partial q_{d, k}}{\partial p_{1}} = {\tilde{R}}_{0} I_{typ} \\ - \frac{1}{{\tilde{F}}_{cc}} \frac{\partial E}{\partial s_{k}} (\frac{\partial q_{b, k}}{p_{2}} - {\tilde{q}}_{b, k}) - \frac{1}{{\tilde{C}}_{d}} \frac{\partial q_{d, k}}{\partial p_{2}} = \\ - \frac{1}{{\tilde{F}}_{cc}} \frac{\partial E}{\partial s_{k}} \frac{\partial q_{b, k}}{p_{3}} - \frac{1}{{\tilde{C}}_{d}} \frac{\partial q_{d, k}}{\partial p_{3}} \\ - \frac{1}{{\tilde{F}}_{cc}} \frac{\partial E}{\partial s_{k}} \frac{\partial q_{b, k}}{p_{4}} - \frac{1}{{\tilde{C}}_{d}} (\frac{\partial q_{d, k}}{\partial p_{4}} + {\tilde{q}}_{d, k}) \\ - \frac{1}{{\tilde{F}}_{cc}} \frac{\partial E}{\partial s_{k}} \frac{\partial q_{b, k}}{p_{5}} - \frac{1}{{\tilde{C}}_{d}} \frac{\partial q_{d, k}}{p_{5}} {\tilde{R}}_{0} I_{k} \\ 0 \end{matrix}] . & (55) \end{matrix}$

In place of Equation (37), let RLS gain g_kbe defined as:

g
_k
=Y
_k−1
u
_k(λΣ+u_k^TY_k−1u_k)⁻¹, (56)

in combination with Equations (38) and (39), the estimated parameter vector p̂_kcan be estimated completely in the same way as in the first embodiment.

FIG. 15 is a flowchart of a rechargeable battery parameter estimation method according to the second embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the rechargeable battery parameter estimation apparatus 101 illustrated in FIG. 14. It should be noted that an execution sequence by the rechargeable battery parameter estimation apparatus 101 according to the second embodiment is not limited to the execution sequence illustrated in FIG. 15. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

As described above, according to the second embodiment, the offset error 116 of the current measurements and the capacity error 117 of the FCC can be estimated while the errors in the characteristic parameters R_d, C_d, and R₀of the equivalent circuit of the battery 103 illustrated in FIG. 2 are considered.

It should be noted that the rechargeable battery parameter estimation apparatus 101 according to the second embodiment is not limited to the configuration for correcting the characteristic parameters of the equivalent circuit illustrated in FIG. 2. For example, considering a circuit includes plural CR circuits or a diffusion impedance and the like as the equivalent circuit, and a configuration for correcting parameters included therein may be employed.

Moreover, a temperature dependency of parameters such as the resistances and the diffusion coefficient may be expressed by the Arrhenius equation or the like, and a frequency factor, an activation energy, and the like included therein may be corrected as parameters. A configuration for estimating only a part of the characteristic parameters R_d, C_d, and R₀of the equivalent circuit may be employed.

Third Embodiment

In the first embodiment, the estimated parameter vector p̂_kis estimated from the explanatory variable u_kand the observed variable y_kdefined by Equations (35) and (36). On this occasion, the first element of the explanatory variable u_kincludes a term proportional to t_k, and the term unboundedly increases as the time elapses. A calculation process including an amount unboundedly increasing in this way is not appropriate for a software implementation in embedded systems.

A third embodiment of the present invention serves to solve this problem, and effects similar to those of the first and second embodiments are provided without directly treating the unboundedly increasing amount by considering the observed variable y_kand the explanatory variable u_kin an affine space. A description is now given mainly of the difference of a rechargeable battery parameter estimation apparatus 101 according to the third embodiment from the rechargeable battery parameter estimation apparatus 101 according to the first embodiment.

First, weighted time averages of the observed variable y_kand the explanatory variable u_kare defined by:

$\begin{matrix} 〈 y_{k} 〉 := \frac{1}{S_{k}} \sum_{\underset{^{'}}{l = 0}}^{k} λ^{k - 1} y_{l} : 〈 u_{k} 〉 : = \frac{1}{S_{k}} \sum_{l = 0}^{k} λ^{k - 1} u_{l}, & (57) \end{matrix}$

where S_kis a sum of the first k terms of a geometric sequence having the scale factor of 1 and having a common ratio of λ. On this occasion, a vector space to which a pair (y_k,u_k) belongs is considered as an affine space, and a local coordinate (z₁,v₁) at a time point t_kis defined as z₁:=y₁−y_kand v₁:=u₁−u_k. Then, weighted time averages of the observed variable y_kand the explanatory variable u_krepresented in the local coordinate are calculated by:

$\begin{matrix} 〈 z_{k} 〉 = λ \frac{S_{k - 1}}{S_{k}} [〈 z_{k - 1} 〉 - (y_{k} - y_{k - 1})], & (58) \\ 〈 v_{k} 〉 = λ \frac{S_{k - 1}}{S_{k}} [〈 v_{k - 1} 〉 - (u_{k} - u_{k - 1})] & (59) \end{matrix}$

Both differences y_k−y_k−1and u_k−u_k−1are bounded, and λS_k−1/S_kis less than 1, and hence z_kand v_kare always bounded.

On this occasion, the estimated parameter vector p̂_kcan be calculated as in the error estimation part 115 according to the first embodiment by replacing the observed variable y_kwith <z_k>, and replacing the explanatory variable u_kwith <v_k>. It should be noted that, in the third embodiment, the parameters are considered in the affine space, and q₀cannot thus be estimated. Thus, the explanatory variable u_kforms a two-dimensional vector by stripping the last element of −1, and the estimated parameter vector p̂_kalso forms a two-dimensional vector.

FIG. 16 is an overall configuration diagram of a rechargeable battery parameter estimation apparatus 101 according to the third embodiment of the present invention. The rechargeable battery parameter estimation apparatus 101 according to the third embodiment has the same configuration as that of the rechargeable battery parameter estimation apparatus 101 according to the first embodiment except for such a point that the current integration part 113 is no longer necessary, and the current 104 measured by the current measurement part 102 is directly input to an error estimation part 1601.

FIG. 17 is an overall configuration diagram of the error estimation part 1601 according to the third embodiment of the present invention. The error estimation part 1601 according to the third embodiment has the same configuration as that of the error estimation part 115 of the first embodiment except for such a point that the current 104 is input to an observed variable calculation part 1702 in place of the current integrated value 114.

FIG. 18 is a block diagram of an explanatory variable calculation part 1701 according to the third embodiment of the present invention. A first element of the explanatory variable u_k−u_k−1is calculated by acquiring a difference by subtracting the first coefficient 110 of previous sampling from the first coefficient 110 calculated by the first coefficient calculation part 109, and subtracting a product acquired by multiplying the difference by the typical FCC F_ccfrom a product acquired by multiplying the sampling period t_sby a constant I_typ.

A second element of the explanatory variable u_k−u_k−1is calculated by acquiring a sum of the estimated value of the SoC 108 calculated by the SoC estimation part 107 and the second coefficient 112 calculated by the second coefficient calculation part 111, acquiring a difference of the calculated sum from the value at the previous sampling, further acquiring a product by multiplying the difference by the typical FCC F_cc, and negating the product.

On the other hand, λ_kis calculated by multiplying λ_k−1calculated at the previous sampling by the predetermined forgetting factor λ. The geometric series S_k, which is a sum of the first k terms of the geometric sequence having a scale factor of 1 and the common ratio λ, is calculated by adding λ_kto the geometric series S_k−1calculated at the previous sampling. λS_k−1/S_kis calculated by acquiring a ratio of S_k−1to S_k, and multiplying the ratio by the factor λ. The explanatory variable <v_k> is calculated by acquiring a difference between <v_k−1> at the previous sampling and the explanatory variable u_k−u_k−1, and further multiplying the difference by λS_k−1/S_k.

FIG. 19 is a block diagram of the observed variable calculation part 1702 according to the third embodiment of the present invention. The observed variable y_k−y_k−1is calculated by acquiring a product by multiplying the estimated value of the SoC 108 calculated by the SoC estimation part 107 by the typical FCC F_cc, acquiring a difference of the product from that at the previous sampling, and subtracting the difference from a product acquired by multiplying the current 104 by the sampling period t_s. The observed variable <z_k> is calculated by acquiring a difference between <Z_k−1> at the previous sampling and the observed variable y_k−y_k−1, and further multiplying the difference by λS_k−1/S_k.

For combining the third embodiment with the second embodiment of the present invention, it is only required that the symbol y_kis reinterpreted as a vector, and the symbol u_kis reinterpreted as a matrix. Moreover, in the calculation of the explanatory variable and the observed variable, the terms u_k−u_k−1, and the like have to be calculated as defined except the terms relating to the time point t_kand a current integration amount q_cc,k.

FIG. 20 is a flowchart of a rechargeable battery parameter estimation method according to the third embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the rechargeable battery parameter estimation apparatus 101 illustrated in FIG. 16. It should be noted that an execution sequence of the rechargeable battery parameter estimation apparatus 101 according to the third embodiment is not limited to the execution sequence illustrated in FIG. 20. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 21 is a flowchart of a processing executed by the error estimation part 1601 in a cycle of periodic operations according to the third embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the error estimation part 1601 illustrated in FIG. 17. It should be noted that an execution sequence by the error estimation part 1601 according to the third embodiment is not limited to the execution sequence illustrated in FIG. 21. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

As described above, according to the third embodiment, without directly treating the amount unboundedly increasing as the time elapses, the same effects as those of the first embodiment can be achieved.

Fourth Embodiment

In the first embodiment, the recursive least squares filter (RLS filter) is used to estimate the offset error 116 of the current measurement and the capacity 117 error of the FCC. This method provides the optimal estimation in terms of the mean square error only if a random error is superimposed only on the observed variable y_k, and a random error on the explanatory variable u_kis negligibly small.

However, it is known that if the explanatory variable u_khas a large random error, the estimated value is biased. A fourth embodiment of the present invention solves this problem by using the recursive total least squares filter (RTLS filter) to estimate the offset error 116 of the current measurements and the capacity error 117 of the FCC.

In the fourth embodiment, the rechargeable battery parameter estimation apparatus 101 is designed to find the estimated parameter vector p̂_kwhich minimizes the following objective function J defined by:

$\begin{matrix} J ({\hat{p}}_{k}) := \frac{1}{2} \sum_{l = 0}^{k} λ^{k - l} δ_{l}^{2} + ε_{l}^{T} W^{- 1} ε_{l} & (60) \end{matrix}$

in place of Equation (51), where δ_kand ε_ksatisfy the following equation for an arbitrary k:

y
_k−δ_k={circumflex over (p)}_k^T(u_k−ε_k), (61)

where u_kis the explanatory variable and y_kis the observed variable. Let y_k:=y₁−δ₁and û₁:=u₁−ε₁, a relationship

ŷ₁={circumflex over (p)}_k^Tû₁ (62)

holds from Equation (61). Using this equation, Equation (60) is able to be transformed into:

$\begin{matrix} J ({\hat{p}}_{k}) = \frac{1}{2} \sum_{l = 0}^{k} {λ^{k - l} ([\begin{matrix} y_{l} \\ u_{l} \end{matrix}] - [\begin{matrix} {\hat{p}}_{k}^{T} \\ I \end{matrix}] {\hat{u}}_{l})}^{T} [\begin{matrix} 1 \\ W^{- 1} \end{matrix}] ([\begin{matrix} y_{l} \\ u_{l} \end{matrix}] - [\begin{matrix} {\hat{p}}_{k}^{T} \\ I \end{matrix}] {\hat{u}}_{l}) . & (63) \end{matrix}$

The vector û₁which minimizes Equation (63) is obtained as a solution of the following equation:

$\begin{matrix} - [\begin{matrix} {\hat{p}}_{k} & I \end{matrix}] [\begin{matrix} 1 \\ W^{- 1} \end{matrix}] ([\begin{matrix} y_{l} \\ u_{l} \end{matrix}] - [\begin{matrix} {\hat{p}}_{k}^{T} \\ I \end{matrix}] {\hat{u}}_{l}) = 0. & (64) \end{matrix}$

Now, let

$\begin{matrix} A_{k} := [\begin{matrix} {\hat{p}}_{k}^{T} \\ I \end{matrix}], Λ := [\begin{matrix} 1 \\ W^{- 1} \end{matrix}], ξ_{l} := [\begin{matrix} y_{l} \\ u_{l} \end{matrix}], & (65) \end{matrix}$

the solution of Equation (64) is written as:

û
_l:=(A_k^TΛA_k)⁻¹A_k^TΛξ_l. (66)

Substituting the solution to Equation (63), Equation (63) is transformed into:

$\begin{matrix} J ({\hat{p}}_{k}) = \frac{1}{2} \sum_{l = 0}^{k} λ^{k - l} ξ_{l}^{T} [Λ - Λ {A_{k} (A_{k}^{T} Λ A_{k})}^{- 1} A_{k}^{T} Λ] ξ_{l} . & (67) \end{matrix}$

Further, let Λ^1/2denote the Cholesky decomposition of the matrix Λ, B_k:=Λ^1/2A_kand ζ₁:=Λ^1/2ξ₁, the following equation holds:

$\begin{matrix} J ({\hat{p}}_{k}) = \frac{1}{2} \sum_{l = 0}^{k} λ^{k - l} ζ_{l}^{T} [I - {B_{k} (B_{k}^{T} B_{k})}^{- 1} B_{k}^{T}] ζ_{l} . & (68) \end{matrix}$

On this occasion, a non-zero vector b_kexists such that B_k^Tb_k=0, because the null space of the matrix A_k^Tis 1-dimensional. This b_kis unique except for scalar multiplies. Because of the proparties of a projection, the vector b_ksatisfies the following relationship:

$\begin{matrix} I = {B_{k} (B_{k}^{T} B_{k})}^{- 1} B_{k}^{T} = \frac{b_{k} b_{k}^{T}}{b_{k}^{T} b_{k}} . & (69) \end{matrix}$

Consequently, Equation (68) is transformed into follows:

$\begin{matrix} \begin{matrix} J ({\hat{p}}_{k}) = \frac{1}{2} \sum_{l = 0}^{k} λ^{k - l} ζ_{l}^{T} \frac{b_{k} b_{k}^{T}}{b_{k}^{T} b_{k}} ζ_{l} \\ = \frac{1}{2} \sum_{l = 0}^{k} λ^{k - l} \frac{b_{k}^{T} \tilde{ζ} ζ^{T} b_{k}}{b_{k}^{T} b_{k}} \\ = \frac{1}{2 b_{k}^{T} b_{k}} b_{k}^{T} {\tilde{R}}_{k} b_{k}, where {\tilde{R}}_{k} := \sum_{l = 0}^{k} λ^{k - l} ζ ζ^{T} . \end{matrix} & (70) \end{matrix}$

Remark B_k^Tb_k=0, namely, A_k^TΛ^1/2b_k=0, and thus, let a_k:=Λ^1/2b_k, Equation (70) is further transformed into:

$\begin{matrix} J ({\hat{p}}_{k}) = \frac{1}{2} \frac{a_{k}^{T} R_{k} a_{k}}{a_{k}^{T} Λ a_{k}}, where R_{k} := \sum_{l = 0}^{k} λ^{k - l} {ξξ}^{T} . & (71) \end{matrix}$

Therefore, a_kis obtained as an eigenvector corresponding to the minimum eigenvalue which is a solution of a generalized eigenvalue problem relating to the pair (R_k, Λ). Remark A_k^Ta_k=0 again, and let a_0,kdenote the first element of the vector a_kand â_kdenote the other elements, p̂_ka_0,k=â_k. Thus, the estimated parameter vector p̂_kis expressed by â_k/a_0,k.

As a method of finding the eigenvector, a publicly known method may be used. For example, the Householder transformation, the Givens rotation method, and the Arnoldi's method are known. On this occasion, the covariance matrix R_kis a positive definite symmetric matrix, thus the Lanczos algorithm can be used. Further, it is required to find only the eigenvector corresponding to the minimum eigenvalue, and hence the inverse iteration and the Rayleigh quotient iteration are also applicable.

A description is now mainly given of the difference of a rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment from the rechargeable battery parameter estimation apparatus 101 according to the first embodiment.

FIG. 22 is an overall configuration diagram of the rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment of the present invention. The rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment has the same configuration as that of the rechargeable battery parameter estimation apparatus 101 according to the first embodiment except for an error estimation part 2201.

FIG. 23 is an overall configuration diagram of the error estimation part 2201 according to the fourth embodiment of the present invention. The explanatory variable calculation part 601 and the observed variable calculation part 602 are the same as those of the first embodiment.

A covariance calculation part 2301 calculates a covariance matrix R_kfrom a covariance matrix R_k−1at the previous sampling stored in a covariance storage part 2304, the explanatory variable u_kcalculated by the explanatory variable calculation part 601, and the observed variable y_kcalculated by the observed variable calculation part 602 by following a recursive equation:

R
_k
=λR
_k−1+ξ_kξ_k^T, (72)

and stores a result in the covariance storage part 2304.

An eigenvector calculation part 2302 solves the generalized eigenvalue problem of Equation (71) based on the covariance matrix R_kcalculated by the covariance calculation part 2301 and a predetermined weighting matrix Λ, and acquires the eigenvector a_kcorresponding to the minimum eigenvalue.

An parameter calculation part 2303 calculates the estimated parameter vector p̂_kby dividing the second or subsequent elements of a_kby the first element thereof. The offset calculation part 608 and the capacity calculation part 609 are the same as those of the first embodiment.

FIG. 24 is a flowchart of the rechargeable battery parameter estimation method. Each step of the flowchart corresponds to a processing executed by the corresponding part of the rechargeable battery parameter estimation apparatus 101 illustrated in FIG. 22. It should be noted that an execution sequence of the rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment is not limited to the execution sequence illustrated in FIG. 24. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 25 is a flowchart of a processing executed by the error estimation part 2201 according to the fourth embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the error estimation part 2201 illustrated in FIG. 23. It should be noted that an execution sequence of the error estimation part 2201 according to the fourth embodiment is not limited to the execution sequence illustrated in FIG. 25. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

The fourth embodiment can be combined with the second embodiment. In order to achieve this, let Σ denote a positive definite symmetric weighting matrix, it is only required that the covariance R_kis calculated as:

$\begin{matrix} R_{k} = λ R_{k - 1} + [\begin{matrix} y_{1, k} & y_{2, k} & y_{3, k} \\ u_{1, k} & u_{2, k} & u_{3, k} \end{matrix}] {Σ^{- 1} [\begin{matrix} y_{1, k} & y_{2, k} & y_{3, k} \\ u_{1, k} & u_{2, k} & u_{3, k} \end{matrix}]}^{T} . & (73) \end{matrix}$

The fourth embodiment can be combined with the third embodiment. This is apparent by considering that a difference between the first embodiment and the third embodiment is limited to the explanatory variable calculation part 601 and the observed variable calculation part 602. It should be understood that, as described in the third embodiment, the third embodiment combined with the second embodiment can further be combined with the fourth embodiment.

As described above, according to the fourth embodiment, even if measurements of the current 104 and the voltage 106 have large random errors, the same effects as those of the first embodiment can be provided.

Fifth Embodiment

The rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment solves the generalized eigenvalue problem in the eigenvector calculation part 2302, and this is generally a processing having a large calculation complexity. Thus, in a fifth embodiment of the present invention, a configuration is disclosed where a method disclosed in D.-Z. Feng, X.-D. Zhang, D.-X. Chang and W. X. Zheng, “A fast recursive total least squares algorithm for adaptive FIR filtering,” IEEE Transaction on Signal Processing, vol. 52, No. 10, 2004 is applied to reduce the calculation complexity by using the fact that the covariance matrix R_kchanges only in a gradual manner.

First, the estimated parameter vector p̂_kis expressed as â_k/a_0,k, and the estimated parameter vector p̂_khas a freedom of scaling, and thus p̂_kcan be parameterized as [−1 a_k^T]^T.

Then the following update scheme:

{circumflex over (p)}
_k
={circumflex over (p)}
_k−1+θ_kw_k (74)

is employed where θ_kis a scalar, w_kis a time series such that a series (w_k, w_k−1, . . . ) efficiently spans an image of A_k.

Although the explanatory variable u_kis used as w_kaccording to D.-Z. Feng, X.-D. Zhang, D.-X. Chang and W. X. Zheng, “A fast recursive total least squares algorithm for adaptive FIR filtering,” IEEE Transaction on Signal Processing, vol. 52, No. 10, 2004, a random non-zero vector is employed in the fourth embodiment in the present invention. As another option, for example, generation and use of a directional vector orthogonal to previous n−1 w_ks is conceivable.

As the scalar θ_kincluded in Equation (74), the scalar θ_kthat minimizes the objective function J represented as Equation (71) is calculated. By substituting Equation (74) to Equation (71) is rewritten as follows:

$\begin{matrix} J ({\hat{p}}_{k - 1} + θ_{k} w_{k}) = \frac{N_{k}}{D_{k}} := \frac{1}{2} \frac{[\begin{matrix} - 1 & {\hat{p}}_{k - 1}^{T} + θ_{k} w_{k}^{T} \end{matrix}] [\begin{matrix} d_{k} & c_{k}^{T} \\ c_{k} & Q_{k} \end{matrix}] [\begin{matrix} - 1 \\ {\hat{p}}_{k - 1} + θ_{k} w_{k} \end{matrix}]}{[\begin{matrix} - 1 & {\hat{p}}_{k - 1}^{T} + θ_{k} w_{k}^{T} \end{matrix}] [\begin{matrix} 1 \\ W^{- 1} \end{matrix}] [\begin{matrix} - 1 \\ {\hat{p}}_{k - 1} + θ_{k} w_{k} \end{matrix}]}, & (75) \\ where \\ [\begin{matrix} d_{k} & c_{k}^{T} \\ c_{k} & Q_{k} \end{matrix}] = R_{k} . & (76) \end{matrix}$

A scalar θ_k, which minimizes Equation (75), is given as a solution of the following equation:

$\begin{matrix} \frac{\partial}{\partial θ_{k}} J ({\hat{p}}_{k - 1} + θ_{k} w_{k}) = \frac{\frac{\partial N_{k}}{\partial θ_{k}} D_{k} - N_{k} \frac{\partial D_{k}}{\partial θ_{k}}}{D_{k}^{2}} = 0. & (77) \end{matrix}$

Because of the denominator D_k²is always strictly positive, and thus it is sufficient that a θ_kwhich make a numerator equal to zero. On this occasion, the numerator N_kand the denominator D_kare expanded into:

$\begin{matrix} N_{k} = w_{k}^{T} Q_{k} w_{k} θ_{k}^{2} + 2 (w_{k}^{T} Q_{k} {\hat{p}}_{k - 1} - c_{k}^{T} w_{k}) θ_{k} + ({\hat{p}}_{k - 1}^{T} Q_{k} {\hat{p}}_{k - 1} - 2 c_{k}^{T} {\hat{p}}_{k - 1} + d_{k}), & (78) \\ D_{k} = w_{k}^{T} W w_{k} θ_{k}^{2} + 2 w_{k}^{T} W {\hat{p}}_{k - 1} θ_{k} + (1 + {\hat{p}}_{k - 1}^{T} W {\hat{p}}_{k - 1}) . & (79) \end{matrix}$

For simplicity, the numerator N_kis rewritten as N_2,kθ_k²+2N_1,kθ_k+N_0,k, and the denominator D_kis written as D_2,kθ_k²+2D_1,kθ_k+D_0,k, derivatives of those terms with respect to the scalar θ_kare expressed dN_k/dθ_k=2N_2,kθ_k+2N_1,k, dD_k/dθ_k=2D_2,kθ_k+2D_1,k, respectively. Thus, the numerator of Equation (77) is written as:

$\begin{matrix} \frac{\partial N_{k}}{\partial θ_{k}} D_{k} - N_{k} \frac{\partial D_{k}}{\partial θ_{k}} = 2 (N_{2, k} D_{1, k} - N_{1, k} D_{2, k}) θ_{k}^{2} + 2 (N_{2, k} D_{0, k} - N_{0, k} D_{2, k}) θ_{k} + 2 (N_{1, k} D_{0, k} - N_{0, k} D_{1, k}) . & (80) \end{matrix}$

For simplicity, Equation (80) is rewritten as 2α_kθ_k²+2β_kθ_k+2γ_kθ_k. This is a quadratic expression, which possibly has two distinct real roots:

$\begin{matrix} θ_{k}^{-} = \frac{- β_{k} - \sqrt{β_{k}^{2} - 4 α_{k} γ_{k}}}{2 α_{k}}, θ_{k}^{+} = \frac{- β_{k} + \sqrt{β_{k}^{2} - 4 α_{k} γ_{k}}}{2 α_{k}} . & (81) \end{matrix}$

According to first derivative tests with respect to the objective function J shown in Tables 1 and 2, the root θ⁺_kis apparently the solution that minimizes J. It should be noted that C is a constant N_2,k/D_2,kin the tables.

TABLE 1

First derivative test of J for α_k> 0

θ_k
−∞

θ⁻_k

θ⁺_k

∞

dJ/dθ_k
0
POSITIVE
0
NEGATIVE
0
POSITIVE
0

J
C
INCREASE

DECREASE

INCREASE
C

TABLE 2

First derivative test of J for α_k< 0

θ_k
−∞

θ⁺_k

θ⁻_k

∞

dJ/dθ_k
0
NEGATIVE
0
POSITIVE
0
NEGATIVE
0

J
C
DECREASE

INCREASE

DECREASE
C

According to the first derivative tests, the quadratic expression α_kθ_k²+β_kθ_k+γ_kθ_kalways has two distinct real roots unless α_kis 0. However, if α_kis accidentally equal to 0 or close to 0, the root finding is numerically unstable. In this case, a zero is employed as the root θ_k.

A description is now given mainly of the difference of a rechargeable battery parameter estimation apparatus 101 according to the fifth embodiment from the rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment.

FIG. 26 is an overall configuration diagram of the rechargeable battery parameter estimation apparatus 101 according to the fifth embodiment of the present invention. The rechargeable battery parameter estimation apparatus 101 according to the fifth embodiment has the same configuration as that of the rechargeable battery parameter estimation apparatus 101 according to the fourth embodiment except for an error estimation part 2601.

FIG. 27 is an overall configuration diagram of the error estimation part 2601 according to the fifth embodiment of the present invention. The explanatory variable calculation part 601 and the observed variable calculation part 602 are the same as those of the first embodiment. The variance calculation part 2301 and the variance storage part 2304 are the same as those of the fourth embodiment.

A coefficient calculation part 2701 calculates the coefficients α_k, β_k, and γ_kof the quadratic expression by Equations (78), (79), and (80) from the explanatory variable u_kcalculated by the explanatory variable calculation part 601, the covariance matrix R_kcalculated by the covariance calculation part 2301, and the parameter estimated value p̂_k−1at the previous sampling stored in the parameter storage part 2704.

A quadratic expression solving part 2702 acquires the real root θ⁺_kof the quadratic expression having the coefficients calculated by the coefficient calculation part 2701 by the second equation of Equation (81).

A parameter calculation part 2703 calculates the estimated parameter vector p̂_kby Equation (74) from the coefficients of the quadratic expression calculated by the coefficient calculation part 2701 and the real root θ⁺_kof the quadratic expression found by the quadratic expression solving part 2702, and stores the estimated parameter vector p̂_kin a parameter storage part 2704. It should be noted that if the coefficient of the second order term of the quadratic expression is almost 0, p̂_kis set to p̂_k−1. The offset calculation part 608 and the capacity calculation part 609 are the same as those of the first embodiment.

FIG. 28 is a flowchart of a rechargeable battery parameter estimation method according to the fifth embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the rechargeable battery parameter estimation apparatus 101 illustrated in FIG. 26. It should be noted that an execution sequence of the rechargeable battery parameter estimation apparatus 101 according to the fifth embodiment is not limited to the execution sequence illustrated in FIG. 28. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

FIG. 29 is a flowchart of a processing executed by the error estimation part 2601 in a cycle of periodic operations according to the fifth embodiment of the present invention. Each step of the flowchart corresponds to a processing executed by the corresponding part of the error estimation part 2601 illustrated in FIG. 27. It should be noted that an execution sequence of the error estimation part 2601 according to the fifth embodiment is not limited to the execution sequence illustrated in FIG. 29. The execution sequence of the steps can be shuffled as long as the dependencies among the steps are not violated.

The fifth embodiment can be combined with the second embodiment.

As described above, according to the fifth embodiment, the estimated parameter vector p̂_kis calculated without solving the generalized eigenvalue problem, and the same effects as those of the fourth embodiment can be provided. As a result, the calculation complexity can be reduced, and the fifth embodiment is appropriate for an software implementation in embedded systems.

Claims

1. A rechargeable battery parameter estimation apparatus, comprising: a voltage measurement part for measuring a voltage between terminals of the battery;an SoC estimation part for estimating an estimated value of the SoC of the battery in a manner that depends on a typical full charge capacity of the battery measured in advance and the voltage between the terminals of the battery, the typical FCC has a capacity error;a current measurement part for measuring a current flowing through the battery, the measurements of the current has an offset error;a first coefficient calculation part for calculating a first coefficient, which is a partial derivative of the estimated value of the SoC with respect to the offset error of the current measurements;a second coefficient calculation part for calculating a second coefficient, which is a partial derivative of the estimated value of the SoC with respect to the capacity error, which is a difference of the typical full charge capacity measured in advance from an actual full charge capacity of the battery at the battery in use; andan error estimation part for estimating the offset error of the current measurements and the capacity error of the full charge capacity from derivative information including the first coefficient and the second coefficient, the current flowing through the battery, and the estimated value of the SoC.
2. A rechargeable battery parameter estimation apparatus according to claim 1, further comprising a current integration part for calculating a current integration value by integrating the current measured by the current measurement part, wherein the error estimation part estimates the offset error and the capacity error from the derivative information, the current integration value, and the estimated value of the SoC.
3. A rechargeable battery parameter estimation apparatus according to claim 1, wherein the error estimation part estimates the offset error and the capacity error from a difference between a current calculated value and a previous calculated value of the derivative information, a difference between a current calculated value and a previous calculated value of the estimated value of the SoC, and the current flowing through the battery.
4. A rechargeable battery parameter estimation apparatus according to claim 1, wherein: the SoC estimation part estimates the estimated value of the SoC of the battery from the typical full charge capacity, the voltage between terminals of the battery, and a characteristic parameter of an equivalent circuit of the battery measured in advance;the rechargeable battery parameter estimation apparatus further comprises a third coefficient calculation part for calculating a third coefficient, which is a partial derivative of the estimated value of the SoC with respect to a difference between a typical value and an actual value of the characteristic parameter of the equivalent circuit of the battery; andthe derivative information referred to by the error estimation part includes the first coefficient, the second coefficient, and the third coefficient.
5. A rechargeable battery parameter estimation apparatus according to claim 2, wherein: the SoC estimation part estimates the estimated value of the SoC of the battery from the typical full charge capacity, the voltage between terminals of the battery, and a characteristic parameter of an equivalent circuit of the battery measured in advance;the rechargeable battery parameter estimation apparatus further comprises a third coefficient calculation part for calculating a third coefficient, which is a partial derivative of the estimated value of the SoC with respect to a difference between a typical value and an actual value of the characteristic parameter of the equivalent circuit of the battery; andthe derivative information referred to by the error estimation part includes the first coefficient, the second coefficient, and the third coefficient.
6. A rechargeable battery parameter estimation apparatus according to claim 3, wherein: the SoC estimation part estimates the estimated value of the SoC of the battery from the typical full charge capacity, the voltage between terminals of the battery, and a characteristic parameter of an equivalent circuit of the battery measured in advance;the rechargeable battery parameter estimation apparatus further comprises a third coefficient calculation part for calculating a third coefficient, which is a partial derivative of the estimated value of the SoC with respect to a difference between a typical value and an actual value of the characteristic parameter of the equivalent circuit of the battery; andthe derivative information referred to by the error estimation part includes the first coefficient, the second coefficient, and the third coefficient.
7. A rechargeable battery parameter estimation apparatus according to claim 1, wherein the SoC estimation part uses the Kalman filter to estimate the estimated value of the SoC of the battery.
8. A rechargeable battery parameter estimation apparatus according to claim 2, wherein the SoC estimation part uses the Kalman filter to estimate the estimated value of the SoC of the battery.
9. A rechargeable battery parameter estimation apparatus according to claim 3, wherein the SoC estimation part uses the Kalman filter to estimate the estimated value of the SoC of the battery.
10. A rechargeable battery parameter estimation apparatus according to claim 1, wherein the error estimation part comprises: an explanatory variable calculation part for calculating an explanatory variable from: one of the derivative information and a difference between a current calculated value and a previous calculated value of the derivative information; and one of the estimated value of the SoC and a difference between a current calculated value and a previous calculated value of the estimated value of the SoC;an observed variable calculation part for calculating an observed variable from: one of the derivative information and the difference between the current calculated value and the previous calculated value of the derivative information; and one of the estimated value of the SoC and the difference between the current calculated value and the previous calculated value of the estimated value of the SoC;a parameter estimation part for calculating an estimated parameter vector, which is an estimated value of a parameter relating to the offset error and the capacity error, based on the explanatory variable and the observed variable;an offset calculation part for correcting the offset error by using the estimated parameter vector; anda capacity calculation part for estimating the capacity error by using the estimated parameter vector.
11. A rechargeable battery parameter estimation apparatus according to claim 2, wherein the error estimation part comprises: an explanatory variable calculation part for calculating an explanatory variable from: one of the derivative information and a difference between a current calculated value and a previous calculated value of the derivative information; and one of the estimated value of the SoC and a difference between a current calculated value and a previous calculated value of the estimated value of the SoC;an observed variable calculation part for calculating an observed variable from: one of the derivative information and the difference between the current calculated value and the previous calculated value of the derivative information; and one of the estimated value of the SoC and the difference between the current calculated value and the previous calculated value of the estimated value of the SoC;a parameter estimation part for calculating an estimated parameter vector, which is an estimated value of a parameter relating to the offset error and the capacity error, based on the explanatory variable and the observed variable;an offset calculation part for correcting the offset error by using the estimated parameter vector; anda capacity calculation part for estimating the capacity error by using the estimated parameter vector.
12. A rechargeable battery parameter estimation apparatus according to claim 3, wherein the error estimation part comprises: an explanatory variable calculation part for calculating an explanatory variable from: one of the derivative information and a difference between a current calculated value and a previous calculated value of the derivative information; and one of the estimated value of the SoC and a difference between a current calculated value and a previous calculated value of the estimated value of the SoC;an observed variable calculation part for calculating an observed variable from: one of the derivative information and the difference between the current calculated value and the previous calculated value of the derivative information; and one of the estimated value of the SoC and the difference between the current calculated value and the previous calculated value of the estimated value of the SoC;a parameter estimation part for calculating an estimated parameter vector, which is an estimated value of a parameter relating to the offset error and the capacity error, based on the explanatory variable and the observed variable;an offset calculation part for correcting the offset error by using the estimated parameter vector; anda capacity calculation part for estimating the capacity error by using the estimated parameter vector.
13. A rechargeable battery parameter estimation apparatus according to claim 10, wherein: the error estimation part further comprises: an RLS gain calculation part for calculating an RLS gain from the explanatory variable and an inverse variance in a previous cycle; andan inverse variance calculation part for calculating the inverse variance from the inverse variance in the previous cycle, the RLS gain, and the explanatory variable; andthe parameter estimation part calculates the estimated parameter vector from the explanatory variable, the observed variable, the RLS gain, and the estimated parameter vector in the previous cycle.
14. A rechargeable battery parameter estimation apparatus according to claim 11, wherein: the error estimation part further comprises: an RLS gain calculation part for calculating an RLS gain from the explanatory variable and an inverse variance in a previous cycle; andan inverse variance calculation part for calculating the inverse variance from the inverse variance in the previous cycle, the RLS gain, and the explanatory variable; andthe parameter estimation part calculates the estimated parameter vector from the explanatory variable, the observed variable, the RLS gain, and the estimated parameter vector in the previous cycle.
15. A rechargeable battery parameter estimation apparatus according to claim 12, wherein: the error estimation part further comprises: an RLS gain calculation part for calculating an RLS gain from the explanatory variable and an inverse variance in a previous cycle; andan inverse variance calculation part for calculating the inverse variance from the inverse variance in the previous cycle, the RLS gain, and the explanatory variable; andthe parameter estimation part calculates the estimated parameter vector from the explanatory variable, the observed variable, the RLS gain, and the estimated parameter vector in the previous cycle.
16. A rechargeable battery parameter estimation apparatus according to claim 10, wherein: the error estimation part further comprises: a covariance calculation part for calculating a covariance matrix from the explanatory variable, the observed variable, and the covariance matrix in a previous cycle; andan eigenvector calculation part for calculating an eigenvector corresponding to a minimum eigenvalue of the covariance matrix by solving an eigenvalue problem relating to the covariance matrix; andthe parameter estimation part calculates the estimated parameter vector from the eigenvector.
17. A rechargeable battery parameter estimation apparatus according to claim 11, wherein: the error estimation part further comprises: a covariance calculation part for calculating a covariance matrix from the explanatory variable, the observed variable, and the covariance matrix in a previous cycle; andan eigenvector calculation part for calculating an eigenvector corresponding to a minimum eigenvalue of the covariance matrix by solving an eigenvalue problem relating to the covariance matrix; andthe parameter estimation part calculates the estimated parameter vector from the eigenvector.
18. A rechargeable battery parameter estimation apparatus according to claim 12, wherein: the error estimation part further comprises: a covariance calculation part for calculating a covariance matrix from the explanatory variable, the observed variable, and the covariance matrix in a previous cycle; andan eigenvector calculation part for calculating an eigenvector corresponding to a minimum eigenvalue of the covariance matrix by solving an eigenvalue problem relating to the covariance matrix; andthe parameter estimation part calculates the estimated parameter vector from the eigenvector.
19. A rechargeable battery parameter estimation apparatus according to claim 10, wherein: the error estimation part further comprises: a covariance calculation part for calculating a covariance matrix from the explanatory variable, the observed variable, and the covariance matrix in a previous cycle;a coefficient calculation part for calculating coefficients of a quadratic expression for acquiring the estimated parameter vector from the explanatory variable, the covariance matrix, and the estimated parameter vector in the previous cycle; anda quadratic expression solving part for acquiring a real root of a quadratic expression having the coefficients of the quadratic expression; andthe parameter estimation part calculates the estimated parameter vector from the coefficients of the quadratic expression and the real root of the quadratic expression.
20. A rechargeable battery parameter estimation apparatus according to claim 11, wherein: the error estimation part further comprises: a covariance calculation part for calculating a covariance matrix from the explanatory variable, the observed variable, and the covariance matrix in a previous cycle;a coefficient calculation part for calculating coefficients of a quadratic expression for acquiring the estimated parameter vector from the explanatory variable, the covariance matrix, and the estimated parameter vector in the previous cycle; anda quadratic expression solving part for acquiring a real root of a quadratic expression having the coefficients of the quadratic expression; andthe parameter estimation part calculates the estimated parameter vector from the coefficients of the quadratic expression and the real root of the quadratic expression.
21. A rechargeable battery parameter estimation apparatus according to claim 12, wherein: the error estimation part further comprises: a covariance calculation part for calculating a covariance matrix from the explanatory variable, the observed variable, and the covariance matrix in a previous cycle;a coefficient calculation part for calculating coefficients of a quadratic expression for acquiring the estimated parameter vector from the explanatory variable, the covariance matrix, and the estimated parameter vector in the previous cycle; anda quadratic expression solving part for acquiring a real root of a quadratic expression having the coefficients of the quadratic expression; andthe parameter estimation part calculates the estimated parameter vector from the coefficients of the quadratic expression and the real root of the quadratic expression.
22. A rechargeable battery parameter estimation method, comprising: measuring a voltage between terminals of the battery;estimating an estimated value of the SoC of the battery in a manner that depends on a typical full charge capacity of the battery measured in advance and the voltage between the terminals of the battery, the typical FCC has a capacity error;measuring a current flowing through the battery, the measurements of the current has an offset error;calculating a first coefficient, which is a partial derivative of the estimated value of the SoC with respect to the offset error of the current measurements;calculating a second coefficient, which is a partial derivative of the estimated value of the SoC with respect to the capacity error, which is a difference of the typical full charge capacity measured in advance from an actual full charge capacity of the battery at the battery in use; andestimating the offset error of the current measurements and the capacity error of the FCC from derivative information including the first coefficient and the second coefficient, the current flowing through the battery, and the estimated value of the SoC.

RECHARGEABLE BATTERY PARAMETER ESTIMATION APPARATUS AND RECHARGEABLE BATTERY PARAMETER ESTIMATION METHOD

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