271
Coupling with Batteries and Fuel Cells
of the supercapacitor V c will vary based on its operation, and likewise the
terminal voltage V uc varies with the input or output current i c .
A general optimization method is presented in relation to the system
and component parameters. Because both the battery and supercapacitor
are composed of several cells connected in a bank (series and parallel), a
per-cell basis is used for each component variable. Modeling each battery
cell as a constant voltage with constant internal series impedance and the
supercapacitor cell as a capacitor with a constant internal impedance using
a linear model is preferred here for initial analysis. Other models can be
used to account for more accurate changes in these parameters upon finalizing a design.
6.6.5.1 Sizing and Costs
Sizing a battery and supercapacitor components through series and parallel strings to assess the total number of cells required is done to control
the voltage at which the battery begins to contribute (i.e., V b to V d , where
V d is the diode voltage drop). This involves simple calculations in which
the subscripts s and p denote cells connected serially and in parallel. It
is important to note that for a battery, the B p (battery in parallel) only
contributes to the size of the limiting current. Thus, for mass and cost
controls, the B s (battery in series) must be optimized. The following equation describes B p [16]:
⎛
⎞
P
V i
low bcell
B p =
ceil
⎜
⎝
⎟
⎠
(6.2)
where the cell function returns the highest rounded integer, V low is the low
limit of the load inverter voltage, i bcell is the maximum discharge current per
battery cell, and P is maximum ESS positive power. Optimizing the B s variable is then done using
V b = V cellb B s
(6.3)
R B s
cellb
R
(6.4)
=
b
B p
where V cellb is nominal battery voltage and R cellb is battery internal impedance at DC. The supercapacitor sizing requires the necessary number of
series-connected cells to match the maximum high voltage of the system
V high by the equation [16]:
Coupling with Batteries and Fuel Cells
of the supercapacitor V c will vary based on its operation, and likewise the
terminal voltage V uc varies with the input or output current i c .
A general optimization method is presented in relation to the system
and component parameters. Because both the battery and supercapacitor
are composed of several cells connected in a bank (series and parallel), a
per-cell basis is used for each component variable. Modeling each battery
cell as a constant voltage with constant internal series impedance and the
supercapacitor cell as a capacitor with a constant internal impedance using
a linear model is preferred here for initial analysis. Other models can be
used to account for more accurate changes in these parameters upon finalizing a design.
6.6.5.1 Sizing and Costs
Sizing a battery and supercapacitor components through series and parallel strings to assess the total number of cells required is done to control
the voltage at which the battery begins to contribute (i.e., V b to V d , where
V d is the diode voltage drop). This involves simple calculations in which
the subscripts s and p denote cells connected serially and in parallel. It
is important to note that for a battery, the B p (battery in parallel) only
contributes to the size of the limiting current. Thus, for mass and cost
controls, the B s (battery in series) must be optimized. The following equation describes B p [16]:
⎛
⎞
P
V i
low bcell
B p =
ceil
⎜
⎝
⎟
⎠
(6.2)
where the cell function returns the highest rounded integer, V low is the low
limit of the load inverter voltage, i bcell is the maximum discharge current per
battery cell, and P is maximum ESS positive power. Optimizing the B s variable is then done using
V b = V cellb B s
(6.3)
R B s
cellb
R
(6.4)
=
b
B p
where V cellb is nominal battery voltage and R cellb is battery internal impedance at DC. The supercapacitor sizing requires the necessary number of
series-connected cells to match the maximum high voltage of the system
V high by the equation [16]: