Download SOC(state of charge)

State-of-Charge
Algorithm for Batteries
Juexiao Su
Zhuo Jia
Outline
1. Brief Introduction of State-of-Charge
2. A Universal State-of-Charge Algorithm for
Batteries
3. A Behavioral Algorithm for State of Charge
Estimation
4.Discussion and Conclution
What is State-of-Charge
• State of Charge (SOC): a measure of energy available in an
electrochemical battery
• Energy management in Hybrid Electric Vehicles (HEV) by
controlling the electric motor power and gasoline engine
power to:
• Optimize power performance,
• Reduce the emissions,
• Improve the fuel economy.
How to find SOC
• SOC is an internal chemical state of the battery, it is not
possible to measure SOC directly.
• We have to develop algorithms that estimate SOC using the
available physical measurements such as terminal voltage
and terminal current.
Two types of SOC estimation
• Coulomb-Counting Based Estimation
• SoC is an integration function of time.
1 t
SOCc (t ) = SOCc (0) -  I (t ) dt ,
Q 0
• However, error will be accumulated over time.
• Voltage-Based Estimation
• Bijection between SoC and Open-Circuit Voltage (OCV)
• Then how to obtain OCV from the terminal voltage and current?
Existing work on Open-Circuit-Voltage estimation
Simplified circuit models applied to reduced the complexity
Parameters need to be tuned for different battery types and
individual battery cells
to be
decided
predetermined
Outline
1. Brief Introduction of State-of-Charge
2. A Universal State-of-Charge Algorithm for
Batteries
3. A Behavioral Algorithm for State of Charge
Estimation
4.Discussion
Proposed Approach
• Problem of Existing Work
• Models are developed for specific types of batteries
• Characteristics of Proposed Approach
• Using linear system analysis but without a circuit model
• Low complexity for real-time battery management
• The Only Assumption Used in Proposed Approach
• Within the short observing time window, a battery is
treated as a time-invariant linear system and the SOC
and accordingly the OCV is treated as constants.
+
V
-
Linear
System
+
V
-
Framework of Linear-Time-Invariant system
For single input and single output system, the relationship
between input and output can be represented by a deferential
equation d nc(t ) d n1c(t )
d m r (t ) d m1r (t )
dt
n

dt
n 1
 ......  c(t ) 
dt
n

dt
n
 .....r (t )( m  n)
r(t)
c(t)
Laplace Transformation
(S  S
n
n 1
 ....  1)  c(s )  [S
Zero-state response
(s ) 
C (s )
R(s )
n 1
d n 1c(t )
'
m
m 1


....

c
(
0
)]

(
S

S
 ....  1)  R(s )
t

0
n 1
dt
zero-input response
r(t )  (t )  c(t )
r(t )  h(t )  C (t )
C (t )  C (0)  r(t )  h(t )
Initial time windows
Assumption: Within the short observing time window, a battery is
treated as a time-invariant linear system
r(t) models the current
entering the battery
c(t) models the
voltage across the
battery terminal
Battery
v(t )  vzi (t )  vzs (t )
 vzi (t )  i(t )  h(t )
[v(t )  v zi (t )] * f (t )  [i (t )  h(t )]  f (t )
v f (t )  OCV  u f (t )  [i (t )  h(t )]  f (t )
vzi (t )  OCV  u(t ), 0  t  tw
v f (t )  OCV  u f (t )   (t )  h(t )
 1, t  0
u (t )  
 0, t  0
v f (t )  OCV  u f (t )  h(t )
f (t )  i(t )   (t ), 0  t  tw
 , t  0
 0, t  0
 (t )  



 (t )dt  1
u f (t )  f (t )  u (t )
lim
t 
lim h(t )  lim sH ( s)  0
t 
v f (t )
u f (t )
s 0
 OCV
Algorithm for compute f(t)
The process to calculate f(t) is very likely to the
process of solving the inverse matrix
Next time windows
vnext (t )  v(t  te ) inext (t )  i(t  te ), 0  t  tw
vzi (t )  OCV  u(t ) vhistory (t )
h(t )  v f (t ) - OCV  u f (t )
v f (t )  OCV  u f (t )  h(t ), 0  t  t w
te
 (t )  vnext (t )   i( )h(t  te   )d , 0  t  tw
vnext
t
f next (t )  inext (t )   (t ), 0  t  tw
lim
t 
vfnext (t )
u fnext (t )
 (t ) u fnext (t )  f next (t )  u (t )
vfnext (t )  f (t )  vnext
te : end time of previous window
vzi (t ) : zero-input response
 OCVnext
Special cases
Case I:
uf (t) also converges to zero as t approaches infinity.
I.e., uf(t) = 0 for t > 0.
Then, the terminal current is constant and the battery becomes a
pure resistance network.
OCV = V (t )  I (t ) Reff
lim
t 
v f (t )
 OCV
u f (t )
Case II:
The first sample of terminal current in the window is close to 0.
Then move the window to the next sample as the starting point.
The extreme case is that the sampled current is keeps 0
battery in open-circuit state.
OCV = V (t )
Experiment result
• The extracted SoC fits well with the simulated data
(labeled as simulated) for different current profiles.
Simulated
Ours
100
80
20
0
0
-20
60
Current (A/m2)
40
SOC %
60
Current (A/m2)
SOC %
80
0
Simulated
Ours
100
40
-40
1000
500
1010
Time (s)
20
1020
1000
Time (s)
1500
0
2000
0
Simulated
Ours
0
1000
Time (s)
1000
Time (s)
2000
1500
2000
Simulated
Ours
100
80
20
0
-18
60
-20
40
-22
20
0
500
1000
Time (s)
2000
1000
Time (s)
(c) Constant Load
1500
2000
0
Current (A/m2)
40
SOC %
60
Current (A/m2)
SOC %
-28
(b) Constant Power
80
0
-26
500
(a) Periodical Discharge
100
-24
0
0
-20
-40
0
500
1000
Time (s)
1000
Time (s)
2000
1500
(d) Piecewise Discharge
2000
Universality
• Error within 4% for different materials for active positive
material / electrolyte / negative positive material of
batteries (Labeled).
• For each type of battery
• Only a discharge from fully-charged to empty-charged is
conducted to build up the bijection between OCV and SoC.
• No other tuning is needed.
10%
Graphite/LiPF 6/CoO2
SOC error
8%
Tungsten oxide/Perchlorate/CoO2
Graphite/30% KOH in H2O/V2O5
6%
4%
2%
0%
0
500
1000
Time (s)
1500
2000
Robustness
100%
OCV error
SOC error
• The algorithm converges quickly to the correct
SoC despite an upset on SoC.
50%
0%
0.1
0.2
0.3
Time (s)
0.4
0.5
0.1
0.2
0.3
Time (s)
0.4
0.5
20%
10%
0%
Outline
1. Brief Introduction of State-of-Charge
2. A Universal State-of-Charge Algorithm for
Batteries
3. A Behavioral Algorithm for State of Charge
Estimation
4.Discussion and Conclusion
Behavioral Framework
LTI system
r(t) models the current
entering the battery
Battery
c(t) models the
voltage across the
battery terminal
• We want a specific free response (zero input response) of the
system.
• We want to compute this free response,
• directly from input and output data without constructing any
model, and by making the least assumptions on the system
as possible.
• Behavioral framework (J.Willems, 1986) fits perfectly well to
this problem!
The data IS my model.
LTI system representations
r(t) models the current
entering the battery
Battery
c(t) models the
voltage across the
battery terminal
d n c(t ) d n 1c(t )
d m r (t ) d m1r (t )

 ......  c(t ) 

 .....r (t )( m  n)
n
n 1
n
n
dt
dt
dt
dt
Where u(t) is input, y(t) is
output
Some Definitions(Willem,1986)
• Definition: A discrete dynamical system is 3-tuple
,
with
the time axis, the signal space, and
the
behavior. The behavior is the set of all legitimate functions,
according to the system .
– When the behavior is restricted to an interval [1,T], we denote the
behavior as
.
• Definition: A .
• A system
vector space and
• A system
where
is called a trajectory.
is a linear when the signal space
is a linear subspace of
.
is time-invariant if for any
r(t) models the
current entering
the battery
Battery
is a
c(t) models
the voltage
across the
battery
terminal
Hankel Matrix
• The L-deep block Hankel matrix of a trajectory
denoted by
and defined as follows:
is
Fundamental Lemma(Willems et al., 2005)
Let:
•
be a trajectory of length T of the linear time
invariant system
,
• the system be controllable,
• and the input sequence u is persistently exciting of order n+L,
i.e.
is full rank.
• Then,
Fundamental Lemma(Willems et al., 2005
• Obvious direction: colspan
:
• Each column of
is in B. (shift invariance)
• Any linear combination of columns of
remember: B is a vector space!)
• Interesting direction: colspan
Any trajectory of length L of
combination of columns of
is also in B (linearity,
:
can be written as a linear
.
Which is free response?
• Let w0 = (0; y0d ) be a free response of the system of length L.
• Then
by the fundamental lemma
:
A Behavioral Algorithm
• Solve
• Then find
Notes
• The battery is not an LTI system.
• We are actually constructing the free response, y0 that best explains the
observed data among all the LTI systems within a certain model class
determined by the length of the “past”.
Does it work?
• Input (Terminal Current): Real current data obtained from an electrical
golf cart
• Output (Terminal Voltage): Simulated terminal voltage data using Dualfoil
Simulator
• Sampling period: 6 seconds, T=500, Length of the free response
computed=80.
Does it work?
• l2 error of Xiao et al: 0.505274
• l2 error of the Behavioral Algorithm:0.066759
Sometimes it doesn’t work.
• Since the current is changing slowly,
becomes
"almost" rank defficient. Hence, the persistency of excitation
assumption is "almost" violated and the problem becomes ill
conditioned.
• In this specific case, at a certain instant there is always only
one singular value that is really large and the others are 6
order of magnitude smaller than the maximum singular value.
Real Data
• Sampling Period=0.008 seconds, Length of the free response
computed=50,T=20.
• The data is provided by Hughes Research Laboratory.
Outline
1. Brief Introduction of State-of-Charge
2. A Universal State-of-Charge Algorithm for
Batteries
3. A Behavioral Algorithm for State of Charge
Estimation
4.Discussion
Discussion
r(t) models the current
entering the battery
Build certain
model
Battery
c(t) models the
voltage across the
battery terminal
System identification
Data-driven method
Reference
[1]Bingjun Xiao, Yiyu Shi, and Lei He. A universal state-of-charge algorithm for batteries. In
DAC, pages 687–692, 2010.
[2]Jan W. Polderman and Jan C. Willems. Introduction to mathematical systems theory: a
behavioral approach. Texts in applied mathematics, 26. Springer, 1998.
[3] Ayca Balkan, Min Gao, Paulo Tabuada, Lei He. A Behavioral Algorithm for State of Charge
Estimation