Systems and methods for battery performance monitoring and management using discrete-time state-space overpotential battery models
View Patent ↗Disclosed are systems, methods, and other implementations, including a method for monitoring and managing battery performance that includes deriving a representation of diffusion overpotential behavior for a lithium-ion battery according to a discrete-time state-space approximation of a convolution-defined diffusion (CDD) model for the lithium-ion battery, and determining behavior of the lithium-ion battery according to the discrete-time state-space approximation of the convolution-defined diffusion (CDD) model for the lithium-ion battery.
1 . A method for monitoring and managing battery performance comprising:
deriving a battery health model for a lithium-ion battery based on overpotential behavior of the lithium-ion battery caused by diffusion dynamics in the lithium-ion battery, wherein the overpotential behavior caused by diffusion dynamics is modeled according to a discrete-time state-space approximation of a convolution-defined diffusion (CDD) model for the lithium-ion battery representing diffusion states in the lithium-ion battery; and
determining behavior of the lithium-ion battery, including battery degradation, according to the discrete-time state-space approximation of the convolution-defined diffusion (CDD) model for the lithium-ion battery representing the diffusion states in the lithium-ion battery.
2 . The method of claim 1 , wherein determining the behavior of the lithium-ion battery comprises:
applying input current to the lithium-ion battery;
capturing voltage response of the lithium-ion battery resulting from applying the input current; and
determining the overpotential behavior of the lithium-ion battery caused by diffusion dynamics based on analysis of the voltage response according to the discrete-time state-space approximation of the convolution-defined diffusion (CDD) model.
3 . The method of claim 1 , wherein deriving the representation of the overpotential behavior caused by diffusion dynamics comprises:
deriving the discrete-time state-space approximation based on a recursive formulation using diffusion state data computed by the model extending back to a pre-determined number of instances defining a finite time horizon.
4 . The method of claim 1 , wherein deriving the representation of the overpotential behavior caused by diffusion dynamics comprises:
deriving a diffusion related constant, A D , at steady state for the lithium-ion battery, with the lithium-ion battery comprising a nickel-manganese cobalt (NMC) cell, according to:
A
D
=
2
β
v
M
S
F
D
π
where 1/β is the maximum stoichiometric added lithium, v M is the molar volume of the NMC cell, S is the active surface area, F is Faraday's constant, and D is the Lithium-ion diffusion coefficient.
5 . The method of claim 1 , wherein deriving the representation of the overpotential behavior caused by diffusion dynamics comprises:
determining the CDD model representing the overpotential behavior caused by diffusion dynamics in the lithium-ion battery as the product of a diffusion related constant A D , and a convolution of a unit impulse response, g z (t), with a time-dependent diffusion state amplitude, ζ, for the lithium-ion battery according to:
V
D
(
t
)
=
A
D
ζ
(
t
)
*
g
z
(
t
)
,
wherein the time-dependent diffusion state amplitude, ζ, is determined based on a step change ΔI and a gradient of the open-circuit voltage (OCV) curve for the lithium-ion battery, given by:
ζ
n
(
t
)
=
Δ
I
(
t
n
)
∇
V
OC
t
-
t
n
,
and wherein g z (t) is defined as g z (t)=√{square root over (t)}−√{square root over (t−Δt)}.
6 . The method of claim 5 , further comprising:
deriving the discrete-time state-space approximation according to a recursive relationship represented as:
x
v
(
t
k
+
1
)
=
A
v
x
v
(
t
k
)
+
B
v
u
v
(
t
k
)
V
D
(
t
k
)
=
C
v
x
v
(
t
k
)
where x v (t k ) is a diffusion state vector comprising diffusion state samples, u v (t k ) is an input current vector applied to the lithium-ion battery, C v is a vector with values depending on a diffusion related constant A D , A v is a matrix with values depending on a relative sampling time for state samples of the discrete-time state-space model, and B v is a row vector with an entry that depends on an OCV gradient and differential current applied to the discrete-time state-space model.
7 . The method of claim 6 , wherein deriving the discrete-time state-space approximation according to the recursive relationship comprises:
computing the diffusion overpotential V D based on a current diffusion state sample and M preceding diffusion state samples, where M is a pre-determined tunable value defining a computational horizon.
8 . The method of claim 6 , wherein computing the diffusion overpotential V D comprises:
weighing the diffusion states samples arranged in x v (t k ) with diminishing weights, specified by the matrix A v , to decrease the contribution of earlier computed diffusion states.
9 . The method of claim 8 , wherein the non-zero diminishing weights specified in the matrix A v are computed according to
a
m
=
1
+
1
m
,
where m=1, 2, . . . , M, where M represents the number of samples in a computational horizon for computing the diffusion overpotential V D .
10 . The method of claim 1 , further comprising:
deriving based on the discrete-time state-space approximation one or more battery performance metrics and/or battery degradation data.
11 . The method of claim 10 , further comprising:
determining based on the battery degradation data one or more of: state of health (SoH) of the lithium-ion battery, or state of charge (SoC) for the lithium-ion battery.
12 . The method of claim 1 , wherein deriving the battery health model based on overpotential behavior of the lithium-ion battery caused by diffusion dynamics comprises:
deriving the overpotential behavior caused by diffusion dynamics according to the discrete-time state-space CDD module representing the diffusion states in the lithium-ion battery using a diffusion constant, A D, determined based on characteristics of the lithium-ion battery and representative of diffusion behavior for a discrete number of current step changes, combined with a circuit model for the lithium-ion battery comprising a resistor in series with N pairs of resistors and capacitors in parallel, with N>2.
13 . A battery performance monitoring and management system comprising:
one or more memory storage devices to store data and executable instructions; and
a processor-based controller, coupled to the one or more memory storage devices, configured to:
derive a battery health model for a lithium-ion battery based on overpotential behavior of the lithium-ion battery caused by diffusion dynamics in the lithium-ion battery, wherein the overpotential behavior caused by diffusion dynamics is modeled according to a discrete-time state-space approximation of a convolution-defined diffusion (CDD) model for the lithium-ion battery representing diffusion states in the lithium-ion battery; and
determine behavior of the lithium-ion battery, including battery degradation, according to the discrete-time state-space approximation of the convolution-defined diffusion (CDD) model for the lithium-ion battery representing the diffusion states in the lithium-ion battery.
14 . The system of claim 13 , wherein the processor-based controller configured to determine the behavior of the lithium-ion battery is configured to:
apply input current to the lithium-ion battery;
capture voltage response of the lithium-ion battery resulting from applying the input current; and
determine the overpotential behavior of the lithium-ion battery caused by diffusion dynamics based on analysis of the voltage response according to the discrete-time state-space approximation of the convolution-defined diffusion (CDD) model.
15 . The system of claim 13 , wherein the processor-based controller configured to derive the representation of the overpotential behavior caused by diffusion dynamics is configured to:
derive the discrete-time state-space approximation based on a recursive formulation using diffusion state data computed by the model extending back to a pre-determined number of instances defining a finite time horizon.
16 . The system of claim 13 , wherein the processor-based controller configured to derive the representation of the overpotential behavior caused by diffusion dynamics is configured to:
derive a diffusion related constant, Ap, at steady state for the lithium-ion battery, with the lithium-ion battery comprising a nickel-manganese cobalt (NMC) cell, according to:
A
D
=
2
β
v
M
S
F
D
π
where 1/β is the maximum stoichiometric added lithium, v M is the molar volume of the NMC cell, S is the active surface area, F is Faraday's constant, and D is the Lithium-ion diffusion coefficient.
17 . The system of claim 13 , wherein the processor-based controller configured to derive the representation of the overpotential behavior caused by diffusion dynamics comprises:
determine the CDD model representing the diffusion overpotential behavior of the lithium-ion battery as the product of a diffusion related constant A D , and a convolution of a unit impulse response, g z (t), with a time-dependent diffusion state amplitude, ξ, for the lithium-ion battery according to:
V
D
(
t
)
=
A
D
ζ
(
t
)
*
g
z
(
t
)
,
wherein the time-dependent diffusion state amplitude, ξ, is determined based on a step change ΔI and a gradient of the open-circuit voltage (OCV) curve for the lithium-ion battery, given by:
ζ
n
(
t
)
=
Δ
I
(
t
n
)
∇
V
OC
t
-
t
n
,
and wherein g z (t) is defined as g z (t)=√{square root over (t)}−√{square root over (t−Δt)}.
18 . The system of claim 17 , wherein the processor-based controller is further configured to:
derive the discrete-time state-space approximation according to a recursive relationship represented as:
x
v
(
t
k
+
1
)
=
A
v
x
v
(
t
k
)
+
B
v
u
v
(
t
k
)
V
D
(
t
k
)
=
C
v
x
v
(
t
k
)
where x v (t k ) is a diffusion state vector comprising diffusion state samples, u v (t k ) is an input current vector applied to the lithium-ion battery, C v is a vector with values depending on a diffusion related constant A D , A v is a matrix with values depending on a relative sampling time for state samples of the discrete-time state-space model, and B v is a row vector with an entry that depends on an OCV gradient and differential current applied to the discrete-time state-space model.
19 . The system of claim 18 , wherein the processor-based controller configured to derive the discrete-time state-space approximation according to the recursive relationship is configured to:
compute the diffusion overpotential V D based on a current diffusion state sample and M preceding diffusion state samples, where M is a pre-determined tunable value defining a computational horizon.
20 . The system of claim 18 , wherein the processor-based controller configured to compute the diffusion overpotential V D is configured to:
weigh the diffusion states samples arranged in x v (t k ) with diminishing weights, specified by the matrix A v , to decrease the contribution of earlier computed diffusion states;
wherein the non-zero diminishing weights specified in the matrix A v are computed according to
a
m
=
1
+
1
m
,
where m=1, 2, . . . , M, where M represents the number of samples in a computational horizon for computing the diffusion overpotential V D .
21 . A non-transitory computer readable media comprising computer instructions executable on a processor-based device to:
derive a battery health model for a lithium-ion battery based on overpotential behavior of the lithium-ion battery caused by diffusion dynamics in the lithium-ion battery, wherein the overpotential behavior caused by diffusion dynamics is modeled according to a discrete-time state-space approximation of a convolution-defined diffusion (CDD) model for the lithium-ion battery representing diffusion states in the lithium-ion battery; and
determine behavior of the lithium-ion battery, including battery degradation, according to the discrete-time state-space approximation of the convolution-defined diffusion (CDD) model for the lithium-ion battery representing the diffusion states in the lithium-ion battery.