Method and system for identifying third-order model parameters of lithium battery based on likelihood function
A method and a system for identifying third-order model parameters of a lithium battery based on a likelihood function are provided, which relates to a method for estimating battery model parameters of a lithium battery under different temperatures, different system-on-chips (SOCs), and charge-discharge currents. The method includes the following steps. A third-order battery model of the lithium battery is established. A battery model output voltage U d and a total battery current I under different temperatures, different SOCs, and charge-discharge currents are collected. The likelihood function is adopted to construct an identification model, and the collected data is substituted into the identification model to calculate the battery model parameters. Identified parameters are substituted into the third-order battery model to obtain a battery terminal voltage to be compared with a measured terminal voltage. The operation method of the disclosure is simple and effective, and can accurately estimate internal resistance parameters of the lithium battery.
1. A method for identifying third-order model parameters of a lithium battery based on a likelihood function, characterized by comprising the following steps executing by a processor:
Step (1) of establishing a third-order battery circuit of the lithium battery;
Step (2) of collecting an output voltage and a total battery current of the third-order battery circuit under different temperatures, different system-on-chips (SOCs), and charge-discharge currents;
Step (3) of adopting the likelihood function to construct an identification model; and
Step (4) of substituting the collected output voltage and total battery current of the third-order battery circuit into the identification model to calculate the third-order battery circuit parameters;
wherein the third-order battery circuit comprises a battery (OCV), an ohmic resistor (R ac ), a charge transfer resistor (R ct ), a charge transfer capacitor (C ct ), a first diffused resistor (R wb1 ), a first diffused capacitor (C wb1 ), a second diffused resistor (R wb2 ), and a second diffused capacitor (C wb2 ), wherein
a first terminal of the battery (OCV) is connected to a first terminal of the ohmic resistor (R ac ); a second terminal of the ohmic resistor (R ac )is connected to a first terminal of the charge transfer resistor (R ct ) and a first terminal of the charge transfer capacitor (C ct ); after being connected, a second terminal of the charge transfer resistor (R ct ) and a second terminal of the charge transfer capacitor (C ct ) are connected to a first terminal of the first diffused resistor (R wb1 ) and a first terminal of the first diffused capacitor (C wb1 ); after being connected, a second terminal of the first diffused resistor (R wb ) and a second terminal of the first diffused capacitor (C wb1 ) are connected to a first terminal of the second diffused resistor (R wb2 ) and a first terminal of the second diffused capacitor (C wb2 ); and after being connected, a second terminal of the second diffused resistor (R wb2 ) and a second terminal of the second diffused capacitor (C wb2 ) are connected to a second terminal of the battery (OCV).
2. The method according to claim 1 , characterized in that Step (1) comprises:
a discretized battery state equation of the third-order battery circuit is:
{
U
c
t
(
k
+
1
)
=
U
c
t
(
k
)
*
exp
(
-
T
/
τ
c
t
)
+
I
(
k
)
R
c
t
*
(
1
-
exp
(
-
T
/
τ
c
t
)
)
U
w
b
1
(
k
+
1
)
=
U
w
b
1
(
k
)
*
exp
(
-
T
/
τ
w
b
1
)
+
I
(
k
)
R
w
b
1
*
(
1
-
exp
(
-
T
/
τ
w
b
1
)
)
U
w
b
2
(
k
+
1
)
=
U
w
b
2
(
k
)
*
exp
(
-
T
/
τ
w
b
2
)
+
I
(
k
)
R
w
b
2
*
(
1
-
exp
(
-
T
/
τ
w
b
2
)
)
S
O
C
(
k
+
1
)
=
S
O
C
(
k
)
-
T
*
I
(
k
)
C
where T is a sampling interval, k is a sampling time, I represents a total battery current, U ct represents a voltage of R ct C ct network, τ ct represents a time constant of R ct C ct network, U wb1 represents a voltage of R wb1 C wb1 network, τ wb1 represents a time constant of R wb1 C wb1 network, U wb2 represents a voltage of R wb2 C wb2 network, τ wb2 represents a time constant of R wb2 C wb2 network, SOC represents a battery state of charge, C represents a battery capacity, and R wb1 =3R wb2 and τ wb1 =τ wb2 ; and
a discretized predicted battery module terminal voltage of the third-order battery circuit is:
y k+1 =U ocv −U ct ( k )− U wb1 ( k )− U wb2 ( k )− R ac I ( k )
where y k+1 represents the predicted battery module terminal voltage, U ocv , represents a battery open circuit voltage, and R ac represents an ohmic internal resistance.
3. The method according to claim 2 , characterized in that Step (3) comprises:
Step (3.1) of performing an inverse Z-transformation after performing a Z-transformation on the battery state equation and the predicted battery module terminal voltage, so as to obtain:
U d ( k+ 3)=θ 1 U d ( k+ 2)+θ 2 U d ( k+ 1)+θ 3 U d ( k )+θ 4 I ( k+ 3)+θ 5 I ( k+ 2)+θ 6 I ( k+ 1)+θ 7 I ( k ) and U d ( k )=( y k −U OCV ); and
Step (3.2) of obtaining the identification model from y k =θ T φ(k), where y k is the predicted battery module terminal voltage, where: θ=[θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 ],
φ(k)=U d (k)=[U d (k−1) U d (k−2) U d (k−3) I(k) I(k−1) I(k−2) I(k−3)], θ 1 =b 1 +b 3 +b 5 , θ 2 =−(b 1 b 3 +b 1 b 5 +b 3 b 5 ), θ 3 =b 1 b 3 b 5 , θ 4 =−R ac , θ 5 =R ac (b 1 +b 3 +b 5 )−(b 2 +b 4 +b 6 ), θ 6 =(b 3 +b 5 )b 2 +(b 1 +b 5 )b 4 +(b 1 +b 3 )b 6 −R ac (b 1 b 3 +b 1 b 5 +b 3 b 5 ), θ 7 =b 1 b 3 b 5 R ac −b 3 b 5 b 2 −b 1 b 4 b 5 −b 1 b 3 b 6 , b 1 =exp(−T/τ ct ), b 2 =R ct *(1−exp(−T/τ ct )), b 3 =exp(−T/τ wb1 ), b 4 =R wb1 *(1−exp(−T/τ wb1 )), and b 5 =exp(−T/τ wb2 ).
4. The method according to claim 3 , characterized in that Step (4) comprises:
Step (4.1) of taking a logarithm of a distribution function of the battery module terminal voltage to obtain:
ln
L
(
θ
)
=
-
1
2
σ
2
∑
k
=
1
n
(
y
(
k
)
-
θ
T
φ
(
k
)
)
2
-
n
ln
σ
2
π
,
where n represents a total sample number of the collected output voltage and total battery current of the battery model, σ represents a variance, where y( k ) is an actual measured battery module terminal voltage, and φ(k)=U d (k)=(y k −U OCV );
Step (4.2) of setting a gradient operator as
∇
θ
=
[
∂
∂
θ
1
∂
∂
θ
2
∂
∂
θ
3
∂
∂
θ
4
∂
∂
θ
5
∂
∂
θ
6
∂
∂
θ
7
]
,
substituting the collected output voltage and total battery current of the third-order battery circuit into an equation in Step (4.1), and
∇
θ
ln
L
(
θ
)
=
-
1
σ
2
∑
k
=
1
n
(
y
(
k
)
-
θ
T
φ
(
k
)
)
2
(
-
φ
(
k
)
)
=
0
to calculate each element value of a matrix θ=[θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 ]; and
Step (4.3) of inferring a magnitude of a third-order battery circuit parameter value according to a relationship between the third-order battery circuit parameter value and θ.
5. A system for identifying third-order model parameters of a lithium battery based on a likelihood function, characterized by comprising:
a battery model construction module, configured to establish a third-order battery circuit of a lithium battery;
a data collection module, configured to collect an output voltage and a total battery current of the third-order battery circuit under different temperatures, different SOCs, and charge-discharge currents;
an identification model construction module, configured to adopt the likelihood function to construct an identification model; and
a parameter determination module, configured to substitute the collected output voltage and total battery current of the third-order battery circuit into the identification model to calculate the third-order battery circuit parameters;
wherein the third-order battery circuit comprises a battery (OCV), an ohmic resistor (R ac ), a charge transfer resistor (R ct ), a charge transfer capacitor (C ct ), a first diffused resistor (R wb1 ), a first diffused capacitor (C wb1 ), a second diffused resistor (R wb2 ), and a second diffused capacitor (C wb2 ), wherein
a first terminal of the battery (OCV) is connected to a first terminal of the ohmic resistor (R ac ) a second terminal of the ohmic resistor (R ac )is connected to a first terminal of the charge transfer resistor (R ct ) and a first terminal of the charge transfer capacitor (C ct ); after being connected, a second terminal of the charge transfer resistor (R ct ) and a second terminal of the charge transfer capacitor (C ct ) are connected to a first terminal of the first diffused resistor (R wb1 ) and a first terminal of the first diffused capacitor (C wb1 ); after being connected, a second terminal of the first diffused resistor (R wb1 ) and a second terminal of the first diffused capacitor (C wb1 )are connected to a first terminal of the second diffused resistor (R wb2 ) and a first terminal of the second diffused capacitor (C wb2 ); and after being connected, a second terminal of the second diffused resistor (R wb2 ) and a second terminal of the second diffused capacitor (C wb2 ) are connected to a second terminal of the battery (OCV).
6. The system according to claim 5 , characterized in that the battery model construction module comprises:
a battery state equation establishment module, configured to establish a discretized battery state equation of the third-order battery circuit:
{
U
c
t
(
k
+
1
)
=
U
c
t
(
k
)
*
exp
(
-
T
/
τ
c
t
)
+
I
(
k
)
R
c
t
*
(
1
-
exp
(
-
T
/
τ
c
t
)
)
U
w
b
1
(
k
+
1
)
=
U
w
b
1
(
k
)
*
exp
(
-
T
/
τ
w
b
1
)
+
I
(
k
)
R
w
b
1
*
(
1
-
exp
(
-
T
/
τ
w
b
1
)
)
U
w
b
2
(
k
+
1
)
=
U
w
b
2
(
k
)
*
exp
(
-
T
/
τ
w
b
2
)
+
I
(
k
)
R
w
b
2
*
(
1
-
exp
(
-
T
/
τ
w
b
2
)
)
S
O
C
(
k
+
1
)
=
S
O
C
(
k
)
-
T
*
I
(
k
)
C
where T is a sampling interval, k is a sampling time, I represents a total battery current, U ct represents a voltage of R ct C ct network, τ ct represents a time constant of R ct C ct network, U wb1 represents a voltage of R wb1 C wb1 network, τ wb1 represents a time constant of R wb1 C wb1 network, U wb2 represents a voltage of R wb2 C wb2 network, τ wb2 represents a time constant of R wb2 C wb2 network, SOC represents a battery state of charge, C represents a battery capacity, and R wb1 =3R wb2 and τ wb1 =3τ wb2 ; and
a battery module terminal voltage acquisition module, configured to acquire a discretized predicted battery model terminal voltage of the third-order battery circuit:
y k+1 =U ocv −U ct ( k )− U wb1 ( k )− U wb2 ( k )− R ac I ( k )
where y k+1 , represents the predicted battery module terminal voltage, U ocv , represents a battery open circuit voltage, and R ac represents an ohmic internal resistance.
7. The system according to claim 6 , characterized in that the identification model construction module comprises:
a transformation module, configured to perform an inverse Z-transformation after performing a Z-transformation on the battery state equation and the predicted battery module terminal voltage, so as to obtain:
U d ( k+ 3)=θ 1 U d ( k+ 2)+θ 2 U d ( k+ 1)+θ 3 U d ( k )+θ 4 I ( k+ 3)+θ 5 I ( k+ 2)+θ 6 I ( k+ 1)+θ 7 I ( k ) and U d ( k )=( y k −U OCV ); and
an identification model construction submodule, configured to obtain the identification model from y k =θ T φ(k), where y k is the predicted battery module terminal voltage, where θ=[θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 ],
φ(k)=U d (k)=[U d (k−1) U d (k−2) U d (k−3) I(k) I(k−1) I(k−2) I(k−3)], θ 1 =b 1 +b 3 +b 5 , θ 2 =−(b 1 b 3 +b 1 b 5 +b 3 b 5 ), θ 3 =b 1 b 3 b 5 , θ 4 =−R ac , θ 5 =R ac (b 1 +b 3 +b 5 )−(b 2 +b 4 +b 6 ), θ 6 =(b 3 +b 5 )b 2 +(b 1 +b 5 )b 4 +(b 1 +b 3 )b 6 −R ac (b 1 b 3 +b 1 b 5 +b 3 b 5 ), θ 7 =b 1 b 3 b 5 R ac −b 3 b 5 b 2 −b 1 b 4 b 5 −b 1 b 3 b 6 , b 1 =exp(−T/τ ct ), b 2 =R ct *(1−exp(−T/τ ct )), b 3 =exp(−T/τ wb1 ), b 4 =R wb1 *(1−exp(−T/τ wb1 )), and b 5 =exp(−T/τ wb2 ).
8. The system according to claim 7 , characterized in that the parameter determination module comprises:
a first calculation module, configured to take a logarithm of a distribution function of the battery module terminal voltage, so as to obtain:
ln
L
(
θ
)
=
-
1
2
σ
2
∑
k
=
1
n
(
y
(
k
)
-
θ
T
φ
(
k
)
)
2
-
n
ln
σ
2
π
,
where n represents a total sample number of the collected output voltage and total battery current of the battery model, σ represents a variance, where y( k ) is an actual measured battery module terminal voltage, and φ(k)=U d (k)=(y k −U OCV );
a second calculation module, configured to set a gradient operator as
∇
θ
=
[
∂
∂
θ
1
∂
∂
θ
2
∂
∂
θ
3
∂
∂
θ
4
∂
∂
θ
5
∂
∂
θ
6
∂
∂
θ
7
]
,
substitute the collected output voltage and total battery current of the third-order battery circuit into an equation of the first calculation module, and
∇
θ
ln
L
(
θ
)
=
-
1
σ
2
∑
k
=
1
n
(
y
(
k
)
-
θ
T
φ
(
k
)
)
2
(
-
φ
(
k
)
)
=
0
to calculate each element value of a matrix θ=[θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 θ 7 ]; and
a parameter determination submodule, configured to infer a magnitude of a third-order battery circuit parameter value according to a relationship between the third-order battery circuit parameter value and θ.