Modeling of Lithium-Ion Battery for Energy Storage System Simulation

then explained in terms of the battery SOD in a n th order
polynomial. Also, from Fig. 1, the equilibrium potential
E(t,T,i,l) is also seen as a function of V(i,T,t,l) and current i(t).
These relationships are expressed as (1) and (2).
2) Secondly, the discharge rate and temperature corresponding
to the reference curve are treated as the reference discharge
rate and temperature.
Therefore in view of the above, expressions for E, V(i,T,t,l),
SOD and Rint are
Eit [ ( ), T( t), tl , ] = vit [ ( ), T( t), tl , ] + R i( t)
(1)
n
�
int r
v[ i( t), T( t), t, l] = c SOD [ i( t), T( t), t, l]
(2)
k
k = 0
k
t
1
SOD[(), i t T (), t t, l] = � i() t dt
(3)
Q 0
R = R + R
(4)
int 1 2
Where ck is the coefficient of the k th -order term in the
polynomial representation of the reference curve and Qr is the
battery capacity referred to the cutoff voltage for the reference
curve. For k = 0, E = c0 is the open-circuit voltage at the
beginning of discharge at the reference temperature of the
reference curve.
B. Description of the Internal Resistance
Normally, the internal resistance Rint will increase with the
state of discharge. In this model, Rint has two components R1
and R2. R1 is defined as the internal resistance of the lithiumion
battery at SOD=0. It depends on the discharge condition,
i.e. temperature, current level and lifecycle. R2 is the increase
in Rint as SOD increases. R2 can also be affected by the
temperature. However, it is proposed that an n th -order
polynomial is used instead to describe the relationship
between R2 and SOD. A correction factor �(T) will then be
used later to compensate for variation of R2 with T.
Based on the above, R can be defined as a function of
1
discharge current, temperature and lifecycle, as follows,
R = f( i( t), T( t), l)
(5)
1
Take derivative on both sides of (5),
δ f δ f δ f
∂ R = ∂ i+ ∂ T + ∂l
(6)
1
δi δT δl
R1 can be calculated by dividing the initial voltage drop
(shown in Fig. 2) by the discharge current i(t) at SOD=0. From
T
1 n
the experimental data, [ ∂R ∂R
]
1 1
1 1 1
�∂i ∂T ∂l
�
�
�
�
� n
�∂i �
n
∂T �
� can be easily calculated.
�
n
∂l
�
�
Expressed in matrix form,
1 1
�∂R � �∂i � � �
� = �
� � �
n
n
�∂R �
� � i 1 � �∂ 1
∂T �
n
∂T 1
∂l
�
��δ
f
�
���
δi n
∂l
�
�
δ f
δT δ f �
δl
��
1 T
� and
(7)
17
V
16.5
16
15.5
15
0 0.1 0.2
SOD
T=25� I=2 A
Fig. 2 Determination of the voltage drop at SOD=0 for
different discharge condition
Then using least-square method to obtain the values
δ f δ f δ f
of , ,
δiδT δ l
. Then R1 can be obtained as
δ f δ f δ f
R = ( i− i ) + ( T − T ) + ( l − l ) + R (8)
1 ref ref ref 1_ref
δi δT δl
where, iref, Tref, lref are the discharge rate, temperature and
lifecycle of the reference curve. R1_ref is the internal resistance
of the reference curve at SOD=0.
Returning to R2, in general, R2 can be defined as a function
of SOD and temperature, as follows:
R = g( T( t), SOD)
(9)
2
Firstly choose another temperature discharge curve which
is of the same discharge rate (the reference discharge rate).
Then define i*R2_ref as the voltage drop (the difference between
this curve and reference curve) at the same SOD, as for
example shown in Fig. 3. The selection of the SOD point can
be arbitrarily because as shown subsequently, it does not cause
significant difference to the final simulation result. Normally,
selection of the SOD point at the middle of these curves yields
higher overall accuracy. An n th -order polynomial can be used
to fit to that relationship between R2_ref and SOD. The same
order polynomial as that with the potential E is recommended,
again to yield higher accuracy. A correction term �(T) is used
to compensate for the variation of R2 at different discharge
condition. This is illustrated as follows.
The method to determine R2 is illustrated in Fig. 3, where
experimental data from the ULTRALIFE UBBL10 lithium-ion
battery is used. The reference curve and another curve at -20�
are chosen to calculate R2_ref, which is shown in Fig. 4. Then,
n
k
R = � r * SOD [ i( t), t]
(10)
2_ref
k
k = 0
Fig. 3. Determination of the R2 for discharge condition at
different temperatures.
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