Dynamically adaptive method for determining the state of charge of a battery
A method for determining the state of charge (SOC) of a lithium ion battery is provided. The method leverages a circuit model of the lithium ion battery, and generates four battery parameters from which the SOC is derived.
1. An automated method for determining state of charge (SOC) of a battery having an ohmic resistance (R), a double layer voltage (V dl ), an open circuit voltage (V oc ), and a characterizing constant
(
A
=
1
C
dl
)
,
where C dl is a double layer capacitance of the battery, the method comprising:
obtaining, at an electronic and processor-based energy management controller, sampled terminal voltage (V) data for the battery;
obtaining, at the energy management controller, sampled terminal current (I) data for the battery;
calculating, with the energy management controller, a vector quantity φ (k) in accordance with the relationship φ (k)=[dt(k)V(k −1), I(k)−I (k−1)dt(k)I(k−1)dt(k)U(k −1)] T , where k indicates a present time step, T indicates a matrix transpose, and U( )indicates a unit step function;
calculating, with the energy management controller, battery parameters θ 1 , θ 2 , θ 3 , and θ 4 in accordance with the relationship V(k)−V(k−1)=θ T θ (k), where θ=[θ 1 , θ 2 , θ 3 , θ 4 ] T and T indicates a matrix transpose;
generating, with the energy management controller, a present SOC value from the battery parameters θ 1 and θ 4 ; and
saving θ in a memory element, for use as an initial condition for a subsequent iteration of the method; wherein
θ
1
=
-
B
=
-
1
τ
,
where τ is a double layer time constant for the battery;
θ 2 =R;
θ 3 =BR +A; and
θ 4 =BV oc .
2. The method of claim 1 , further comprising obtaining sampled battery temperature data for the battery, wherein generating a present SOC value comprises generating the present SOC value from the battery parameters θ 1 and θ 4 and from the sampled battery temperature data.
3. The method of claim 1 , further comprising:
providing an arbitrary initial terminal voltage value;
providing an arbitrary initial terminal current value; and
setting an initial condition for generation of the present SOC value, the initial condition being based upon the arbitrary initial terminal voltage value and the arbitrary initial terminal current value.
4. The method of claim 1 , further comprising:
comparing the present SOC value to a threshold value; and
if the present SOC value is less than the threshold value, initiating charging of the battery.
5. The method of claim 1 , further comprising calculating a present V oc , value for the battery in accordance with the relationship
V
oc
=
-
θ
4
θ
1
,
wherein generating a present SOC value comprises generating the present SOC value from the present V oc value.
6. The method of claim 1 , wherein calculating battery parameters θ 1 , θ 2 , θ 3 , and θ 4 comprises:
computing a gain vector (G) in accordance with the relationship
G
=
P
ϕ
_
(
k
)
λ
+
ϕ
_
T
(
k
)
P
ϕ
_
(
k
)
,
where P is a correlation matrix for the battery parameters θ 1 , θ 2 , θ 3 , and θ 4 , λ is a forgetting factor, and 0 <λ≦1;
computing an estimation error (α) in accordance with the relationship α=V(k)−V(k−1) θ T φ (k); and
adaptively computing θ in accordance with the relationship θ new =θ old +Gα.
7. The method of claim 6 , further comprising updating P in accordance with the relationship
P
update
=
1
λ
P
old
-
1
λ
G
ϕ
_
T
(
k
)
P
old
.
8. A diagnostic and control system for a vehicle battery, the diagnostic and control system comprising:
a lithium ion battery pack having an ohmic resistance (R), a double layer voltage (V dl ), an open circuit voltage (V oc ), and a characterizing constant
(
A
=
1
C
dl
)
,
where C dl is a double layer capacitance of the lithium ion battery pack;
a power inverter module coupled to the lithium ion battery pack; and
a controller coupled to the power inverter module and to the lithium ion battery pack, the controller being configured to regulate operation of the power inverter module, and the controller comprising a computer-readable medium having stored thereon computer-executable instructions for determining state of charge (SOC) of the lithium ion battery pack, the computer-executable instructions being written to:
obtain sampled terminal voltage data (V) for the lithium ion battery pack;
obtain sampled terminal current (I) data for the lithium ion battery pack;
calculate a vector quantity φ (k) in accordance with the relationship φ (k)=[dt(k)V(k −1), I(k) −I (k −1)dt(k)I(k −1), dt(k)U(k −1)] T , where k indicates a present time step, T indicates a matrix transpose, and U( ) indicates a unit step function;
calculate battery parameters θ 1 , θ 2 , θ 3 ,and θ 4 in accordance with the relationship V(k)−V(k −1)=θ T φ (k), where θ=[θ 1 , θ 2 , θ 3 , θ 4 ] T and T indicates a matrix transpose; and
generate a present SOC value from the battery parameters θ 1 and θ 4 ; wherein
θ 1 =−B=−1/τ, where τ is a double layer time constant for the battery;
θ 2 =R;
θ 3 =BR +A; and
θ 4 =BV oc .
9. The diagnostic and control system of claim 8 , the computer-executable instructions being written to:
obtain sampled battery temperature data for the lithium ion battery pack; and
generate the present SOC value from a present V oc , value and the sampled battery temperature data.
10. The diagnostic and control system of claim 8 , the computer-executable instructions being written to adaptively update the present SOC value by repeating over time the steps of obtaining sampled terminal voltage data, obtaining sampled terminal current data, calculating a vector quantity, calculating battery parameters, and generating a present SOC value.
11. The diagnostic and control system of claim 8 , the computer-executable instructions being written to:
provide an arbitrary initial terminal voltage value;
provide an arbitrary initial terminal current value; and
set an initial condition for generation of the present SOC value, the initial condition being based upon the arbitrary initial terminal voltage value and the arbitrary initial terminal current value.
12. The diagnostic and control system of claim 8 , the computer-executable instructions being written to:
compare the present SOC value to a threshold value; and
if the present SOC value is less than the threshold value, initiate charging of the lithium ion battery pack.