Interval estimation for state-of-charge and temperature in battery packs with heterogeneous cells
An interval observer based on an equivalent circuit-thermal model for lithium-ion batteries is presented. State of charge-temperature-dependent parameters are considered as unknown but bounded uncertainties in a single cell model. A parallel and a series arrangement of five cells are used for observer design, where cell heterogeneity is accounted for through the uncertainty bounding functions.
1. A method for operating batteries belonging in a battery pack comprising a plurality of cells of batteries, each of the cells being represented by an R-C pair comprising a resistor with resistance R 2 and a capacitor with capacitance C, the R-C pair being in series with a resistor with resistance R 1 , the method comprising:
estimating a state of charge (SOC) of each of the cells of the batteries belonging in the battery pack based on a coupled electrical-thermal model, wherein
the SOC of each of the cells is derived from the non-linear differential equation system:
dξ/dt=A 0 ξ+δA (θ)ξ+ b (ξ,θ)
y=Hξ+v (θ, t )
where
ξ
=
(
ξ
1
ξ
2
)
=
(
OCV
(
x
1
)
+
x
2
-
τ
k
,
0
x
2
)
A
0
=
[
0
1
0
-
τ
k
,
0
]
,
δA
(
θ
)
=
[
0
δτ
k
/
τ
k
,
0
0
-
δτ
k
]
,
b
(
ξ
,
θ
)
=
(
(
1
/
Q
k
)
φ
(
ξ
2
/
τ
k
,
0
)
+
1
/
C
k
-
τ
k
,
0
/
C
k
)
I
,
H
=
[
1
,
0
]
,
φ(≤)=dOCV/dx(OCV −1 (·)), v(θ,t)=δh(θ)u, where
u=I k (t)εR is a system input, where I k (t) is a current flowing in the k-th cell, yεR is a system output, xεR n is a state vector, having a state x 1 =z k and a state x 2 =V c,k , where z k (t) represents the SOC for the k-th cell, and V c,k (t) represents a voltage across the R-C pair in the k-th cell, t represents a time variable, OCV is an open circuit voltage across the k-th cell, τ k =1/R 2,k C k =τ k,0 +δτ k where τ k,0 is a deterministic scalar and δτ k represents an uncertain component, {dot over (Q)} k (t) is an internal heat generation from resistive dissipation, C k is the capacitance of the capacitor in the R-C pair in the k-th cell, I is an identity matrix, δh(θ)=R 1,k , R 1,k is R 1 in the k-th cell, R 2,k is R 2 in the k-th cell, where
unknown quantities Q k , C k , δ τk and R 1,k have values belonging within intervals [ Q k , Q k − ], [ C k , C k − ], [ δ τk , δ τk − ], and [ R 1,k , R 1,k − ], respectively, with set upper and lower limits;
estimating a surface temperature T s and a core temperature T c for each of the cells, wherein
the T s and the T c of each of the cells is derived from the non-linear differential equation system:
dx ( t )/ dt=Ax ( t )+ Bu ( t )+ D{dot over (Q)} ( t )
y ( t )= Cx ( t )
where
x
=
(
T
c
T
s
)
,
A
=
[
-
1
/
R
c
C
c
1
/
R
c
C
c
1
/
R
c
C
c
-
(
1
R
u
C
s
+
1
R
c
C
s
)
]
,
B
=
(
0
1
/
R
u
C
s
)
,
D
=
(
1
/
C
s
0
)
,
C
=
[
0
,
1
]
,
u
=
T
f
where R c , R u , C c , and C s represent heat conduction resistance between core and surface, convection resistance between ambient and surface, core heat capacity, and surface heat capacity, respectively, and T f is a coolant flow temperature at a cell; and
adjusting charging of the batteries based on the estimated SOC, T s , and T c of each of the cells.
2. The method according to claim 1 , wherein
the battery pack comprises a string of a plurality of cells connected in series.
3. The method according to claim 2 , wherein
the battery pack comprises a plurality of strings of cells connected in parallel.
4. The method according to claim 1 , wherein
the battery pack comprises 20 strings of cells connected in parallel, each of the strings comprising 400 blocks of cells connected in series, each of the blocks comprising 5 cells connected in parallel.