Electric vehicle

Batteries, Flywheels and Supercapacitors
75
If we combine this equation with the normal equation for power we obtain
P = V × I = (E + IR) × I = EI + RI
2
The sensible, normal efficient operation, solution to this quadratic equation is
I =
−E +
√
E
2
+ 4RP
2
R
(3.28)
The value of R, the internal resistance of the cell, will normally be different when charging as opposed to discharging. To use a value twice the size of the discharge value is a good
first approximation.
When running the simulation, we must remember that the power P is positive and that
Equation (3.28) gives the current into the battery. So when incorporating regenerative braking into battery simulation, care must betaken to use the right equation for the current, and that Equation (3.23) must be modified so that the charge removed from the battery is reduced . Also, it is important to remove the Peukert correction, as when charging a battery large currents do not have proportionately more effect than small ones.
Equation (3.23) thus becomes
CR
n+1
= CR
n
−
δt × I
3600
Ah
(3.29)
We meet this equation again in Section 8.4.2, where we simulate the range and performance of EVs with and without regenerative braking.
3.12.5 Calculating the Peukert Coefficient
These equations and simulations are very important and will be used again when we model the performance of EVs in Chapter 8. There the powers and currents will not be constant, as they were above, but exactly the same equations are used.
However, all this begs the question How do wend out what the Peukert coefficient is It is very rarely given on a battery specification sheet, but fortunately there is nearly always sufficient information to calculate the value. All that is required is the battery capacity at two different discharge times. For example, the nominally 42 Ah (10 h rating)
of Figure 3.2 also has a capacity of 33.6 Ah at ah rating.
The method of finding the Peukert coefficient from 2 Ah ratings is as follows. The two different ratings give two different rated currents
I
1
=
C
1
T
1
and
I
2
=
C
2
T
2
(3.30)
We then have two equations for the Peukert capacity, as in Equation (3.12):
C
p
= I
k
1
× T
1
and
C
p
= I
k
2
× T
2
(3.31)
However, since the Peukert coefficient is constant, the right hand sides of both parts of
Equation (3.31) are equal, and thus
I
k
1
T
1
= I
k
2
T
2

76
Electric Vehicle Technology Explained, Second Edition and so

I
1
I
2

k
=
T
2
T
1
Taking logs, and rearranging, this gives
k =
(log T
2
− log T
1
)
(log I
1
− log I
2
)
(3.32)
This equation allows us to calculate the Peukert coefficient k for a battery, provided we have two values for the capacity at two different discharge times T . Taking the example of our 42 Ah nominal battery, Equation (3.30) becomes
I
1
=
C
1
T
1
=
42 10
= 4.2 A and I
2
=
C
2
T
2
=
33
.6 1
= 33.6 A
Putting these values into Equation (3.32) gives
k =
log 1
− log log 4
.2 − log 33.6
= 1.107
Such calculations can be done with any battery, provided some quantitative indication is given as to how the capacity changes with rate of discharge. If a large number of measurements of capacity at different discharge times are available, then it is best to plot a graph of log(T ) against log(I ). Clearly, from Equation (3.32), the gradient of the best-fit line of this graph is the Peukert coefficient.
As a general rule, the lower the Peukert coefficient, the better the battery. All battery types behave in a similar way and are quite well modelled using this method. The Peukert coefficient tends to be rather higher for lead acid batteries than for other types.
3.12.6 Approximate Battery Sizing
The modelling techniques described above, when used with the models for vehicles described in Chapter 8, should be used to give an indication of the performance that will be obtained from a vehicle with a certain type of battery. However, it is possible,
and sometimes useful, to get a very approximate guide to battery range and/or size using the approach outlined below.
A designer maybe either creating anew vehicle or alternatively adapting an existing vehicle to an electric car. The energy consumption of an existing vehicle will probably be known, in which case the energy used per kilometre can be multiplied by the range and divided by the specific energy of the battery to give an approximate battery mass.
If the vehicle is anew design the energy requirements maybe obtained by comparing it with a vehicle of similar design. Should the similar vehicle have an IC engine, the energy consumption can be derived from the fuel consumption and the engine/gearbox efficiency.
This method is fairly crude, but nonetheless may give a reasonable answer which can be analysed later.

Batteries, Flywheels and Supercapacitors
77
For example, the vehicle maybe compared with a diesel engine car with a fuel consumption of 18 km l mpg. The specific energy of diesel fuel is approximately kWh kg
−1
and the conversion efficiency of the engine and transmission is approximately, resulting in 4 kWh of energy per litre of fuel stored delivered at the wheels.
In order to travel 180 km the vehicle will consume 10 l of fuel, approximately 11 kg allowing for fuel density. This fuel has an energy value of 440 kWh, and the energy delivered to the wheels will be 44 kWh (44 000 Wh) allowing for the 10% efficiency.
This can be divided by the electric motor and transmission efficiency, typically about 0.7
(70%), to give the energy needed from the battery, that is 62.8 kWh or 62 800 Wh.
Hence if a lead acid battery is used (specific energy 35 Wh kg) the battery mass will be 1257 kg, if an NiMH battery (specific energy 60 Wh kg) is used the battery mass will be 733 kg, if a sodium nickel chloride battery is used of a specific energy of 86 Wh kg
−1
then the battery mass will be 511 kg, and if a zinc–air battery of 230 Wh kg
−1
is used then a battery mass of 191 kg is needed.
Care must betaken when using specific energy figures, particularly, for example when a battery such as a lead acid battery is being discharged rapidly when the specific energy actually obtained will be considerably lower than the nominal 35 Wh kg. However, this technique is useful, and in the case quoted above gives a fairly good indication of which batteries would be ideal, which would suffice and which would be ridiculously heavy.
The technique would also give a ballpark figure’ of battery mass for more advanced analysis, using the modelling techniques introduced above, and much further developed in Chapter 8.

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