Instrumentation & Measurement Magazine 25-7 - 16

```its impedance increases over time. When the impedance of the
battery increases, the output power from it decreases; this phenomenon
is known as the PF.
PF is formally defined as [12], [13]:
PF




1
Pk
P 0
Pk v


100%
where the available power at time k is defined as:


 


Vs v


Rk0
R0(k) is the ohmic resistance at time k, V0
(2)
(3)
(s) is the OCV of the
battery at a certain reference SOC s, and v is the measured voltage
across battery terminals. Here, the time index k indicates the
elapse of life-cycle event, such as time and charge-discharge cycle;
it is assumed that k = 0 denotes the very initial cycle, e.g., brand
new battery. The above definition of PF is only in terms of the
series Ohmic resistance R0
Estimating Resistance and Capacity
In practice, the exact values of the resistance R0
. In reality, the battery behavior is represented
using higher-order electrical equivalent circuit models [1].
(k) and capacity
Q(k) of the battery at time k are not known-they need to
be estimated. Hence, the PF can CF equations need to be rewritten
as:
PF
CF




ˆ
1
1
Pk
P 0
ˆ






Qk
Q 0


100%
100%
(5)
(6)
where R0(k) and Q(k) are the estimated values of the resistance
and capacity, respectively, at time k. That is, to compute its final
outputs (SOC, SOH, TTS and RUL), a BFG needs to estimate
the capacity and the impedance of the battery. The different approaches
to estimate these parameters are the subject of very
active ongoing research [1], [2].
There is always an uncertainty in an estimated parameter;
the associated uncertainty of an estimated parameter is usually
denoted by the variance of the estimator:
2 
2
 
Rk 00
0   E R k ER k
2
Qk
   
ˆˆ
 E Qk E Qk

   


ˆˆ

where Rk denotes the variance of estimating R0
2
0 
2
(8)
On the other hand, in applications where running-time is
(k), and 
2
Qk
denotes the variance of estimating Q(k). Usually, as the battery
ages, the uncertainty in estimating these two parameters also
increases.
Two major indicators of the SOH are PF and CF that are
computed in terms of the estimated resistance and capacity
of the battery, respectively. Fig. 3a shows a generic plot of
the PF (i.e., the increase of resistance R0
over time) in a battery.
Possible estimated values of R0(k) by a hypothetical BFG
is shown along the true value at discrete time instances k. The
16
more critical than the peak output power, the SOH could be
defined as:
SOH CF
(10)
In applications where both power and capacity are critical,
SOH could be defined through various design criteria; for instance,
a stricter SOH measure is:
SOH 100 max PF, CF
IEEE Instrumentation & Measurement Magazine
  
(11)
October 2022
(7)
Fig. 3. State of health (SOH) of a battery. (a) Battery impedance over time. (b)
Battery capacity over time.
uncertainty of R0(k) estimation is indicated in dashed lines
with ellipses. As previously mentioned, the uncertainty of estimation
is indicated to be increasing with battery age.
Similar to the R0
(k) estimation in Fig. 3a, the true value of
capacity and its possible estimated values by a BFG are shown
in Fig. 3b. Similar to before, the uncertainty of the estimation is
shown to increase over time.
Having defined PF and CF, the SOH of the battery is a
means to unify them and to express the " state of health " using
a single quantity. In applications where the output power
is more critical, the SOH could be defined as:
SOH PF
(9)
```

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