Handbook of Energy Storage for Transmission or ... - W2agz.com

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Number of cells
Vm = Va+VaT/100 T=tolerance in %
Vm < Vc
Va+VaT/100 < Vc
Va < Vc/(1 + T/100)
0
Cell Voltage
Va Vm Vc
Figure 35 Histogram of individual capacitor voltages in a hypothetical series-connected string.
The tolerance, T, used in these derivations can be a “hard” tolerance established by
sorting of cells used in a series string. For example, cells could be selected with
properties in the range of +10% of some nominal value. However, for most applications
with a large number of cells, it is more convenient to express cell voltages using a
continuous, rather than a discrete distribution. A normal distribution defined by a mean
and a standard deviation is appropriate for a controlled manufacturing process. The
relative standard deviation, s, in percentage, is related to the tolerance T by s = T/(sd)
where sd is the number of standard deviations in the tolerance band. The sd value
determines what percentage of the population is included in the tolerance band. Thus, to
include 99% of the population within the normal curve, the sd is 2.58, for 99.9 percent of
the population the value is 3.29.
The relationship between the energy density of a series string and the sd value can be
derived. The energy density of a string of cells is greatly reduced from single-cell values
as the tolerance increases. The large tolerance in cell properties also adversely affects the
manufacturing yield. Figure 36 shows yield versus the number of cells in the string for
two sd values. Thus, cell sorting may be required for reasonable sd values with highvoltage
strings of type II cells.
Therefore, without active or passive cell voltage balance, cell in a series-string must have
extremely small variability to make a reliable, high-voltage, capacitor.
Electrochemical Capacitors 65