Decomposition Analysis of an Automotive Powertrain Design ...

decomposition prior to examining the FDT; control, state, and geometric variables have been
included in the model, the FDT, and the partitioning.
We examine another strategy that addresses the physical significance of the sensitivity of
geometric variables to torque and speed. The key approximations of interest are the behavior
variables for fuel flow, (f(y, x 1 )), NOx, (g 1 (y, x 1 )) and acceleration calculations, (g 2 (y, x 2 ))
because they depend on linking variables and local variables. The challenge is to accommodate
torque, speed, and manifold pressure between the master problem and the subproblem.
When comparing engines of different displacement, it is common practice (Taylor 1985) to
compare torques normalized by displacement (bmep) and to compare flow rates normalized by
power (specific consumption) at equal bmep. With the same reasoning, displacement can be used
to scale engine behavior. We introduce engine displacement as a single variable, V d ; the subscripts
'y' and '1' indicate values of V d in the master problem and subproblem, respectively. We
introduce the equality
V d1 = n c π b 2 s/4 (93)
to hold displacement fixed in the subproblem and the equality
T ey
V dy
= T e1
V d1
(94)
to scale engine torque with displacement. This facilitates approximations for f(y, x 1 (y)) and g 1 (y,
x 1 (y)) with the following relationships:
N N
f(y, x 1 (y)) = ∑ ṁ f (T ey , N e ) = ∑ ṁ f (T e1 , N e ) V dy
V d1
i=1
i=1
(95)
N N
g 1 (y, x 1 (y)) = ∑ ṁ NOx (T ey , N e ) ηcat = ∑ ṁ NOx (T e1 , N e ) V dy
V d1
ηcat
i=1
i=1
(96)
These approximations allow us to formulate the coordination strategy given below.