Decomposition Analysis of an Automotive Powertrain Design ...

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5) If the number of variables in y i is less than the upper bound, create y i+1 by appending the next highest ranked variable to y i and go to 3. Otherwise continue. 6) For each stored candidate vector, construct an additional set of candidate vectors, y ij , (j=1,.., e) by combinatorial enumeration of the set of variables in the candidate vector. Perform step 3 on each vector y ij (j=1,.., e). From those vectors with acceptable partitions select the one having the smallest number of linking variables. The first set of disjoint partitions satisfying the acceptability criteria is found with linking variables y = ( T e , F aero , p i , M, s, b, N e ). Subsequent enumeration reveals that y = ( T e , N e , p i ) gives the relatively balanced set of partitions in the FDT shown in Figure 10. Specification of engine torque, T e and engine speed, N e corresponds to a ‘cut’ at the crankshaft; specification of manifold pressure, p i , ‘cuts’ the engine relations from the transmission shift strategy which is specified as a function of manifold pressure. This partition is then accepted for coordination. Interestingly, Michelena and Papalambros (1995) using spectral partitioning obtained a bisection of this problem with one additional linking variable (the accessory torque), while Krishnamachari (1996) using integer programming with more relaxed constraints on relative size found the same partition as the heuristic search. The quality of the partition ultimately must be assessed by the suitability of an attendant coordination strategy. We now proceed to develop a coordination strategy based on the above partitions. 5 COORDINATION STRATEGY DEVELOPMENT The partition can be described generically as follows, Linking Variables y = ( T e , N e , p i ) Partition 1 (Engine relations) x 1 f(y, x 1 ) h 1 (y, x 1 ) = 0 h 1 ( x 1 )= 0 g 1 (y, x 1 ) ≤ 0 g 1 ( x 1 )≤ 0 Engine torque, engine speed, manifold pressure Engine variables Objective, dependent only on y and x 1 Defining equations dependent on linking variables, y, and x 1 Defining equations dependent only on x 1 Constraints dependent on linking variables, y, and x 1 Constraints dependent only on x 1 Partition 2 (Driveline relations)
x 2 h 2 (y, x 2 ) = 0 h 2 ( x 2 )= 0 g 2 (y, x 2 ) ≤ 0 g 2 ( x 2 )≤ 0 Driveline/vehicle variables Defining equations dependent on linking variables, y, and x 2 Defining equations dependent only on x 2 Constraints dependent on linking variables, y, and x 2 Constraints dependent only on x 2 and the generic hierarchical coordination strategy appropriate for this partition may be represented as follows. Master Problem Subproblems (i = 1, 2) min y f(y, x*(y)) s.to: hy(y) = 0 gy(y) ≤ 0 hi(y, xi*(y)) = 0 hi( xi*(y)) = 0 gi(y, xi*(y)) < 0 gi( xi*(y)) < 0 i = 1, 2 ⇒ y* ⇐ x*(y) min x i f(y*, x i) s.to: hi(y*, xi) = 0 hi( xi) = 0 gi(y*, xi) ≤ 0 (92) gi( xi) ≤ 0 i = 1, 2 The challenge is to develop a suitable approximation of x * (y). If the linking variables are geometric variables a gradient-based linear approximation or one based on parametric sensitivity analysis could be constructed. However, the linking variables are state variables (engine torque, engine speed, and manifold pressure) which take on discrete values as they are computed over their respective driving cycles. The fuel economy cycle of 1372 seconds implies at least as many discrete values of torque and speed to integrate the fuel flow over the cycle. Similarly, numerical integration of the performance constraints implies discrete values of torque and speed. A gradientbased approximation of x*(y) would imply 1372 representations of the geometric variables in x 1 to be represented in the master problem, a burdensome strategy at best. Moreover, the physical significance of the sensitivity of geometric variables to state variables is unclear. One might suggest a coordination strategy like those used in trajectory problems but such strategies are applied when the optimal variables sought are control variables explicitly dependent on the state variables. In the system model here, geometric and control variables were intentionally included to effect a problem of sufficient scope. This issue arises because we have made no aspect
Page 1 and 2: UNIVERSITY OF MICHIGAN DEPARTMENT O
Page 3 and 4: epresentation facilitates decomposi
Page 5 and 6: of changes in these parameters on t
Page 7 and 8: The U.S. metro-highway fuel economy
Page 9 and 10: Powertrain Relationships The princi
Page 11 and 12: N ds = ξ fd N d (23) dN ds /dt =
Page 13 and 14: to be stalled (Rs = 0). Equation (3
Page 15 and 16: the surface-to-volume ratio at a vo
Page 17 and 18: intake and exhaust valve, and acros
Page 19 and 20: driveability. These values are stor
Page 21 and 22: τ dV τ 0-60 = τ :⌡ ⌠ dt dt =
Page 23 and 24: There are also several geometric co
Page 25: g 13 : ξ 1 / ξ 2 - 2.0 ≤ 0 Step
Page 29 and 30: Master Problem Subproblem min f(y,
Page 31 and 32: 6 COORDINATED OPTIMIZATION SOLUTION
Page 33 and 34: educes fuel flow by 1.3%; the const
Page 35 and 36: Assanis, D.N., and Polishak, M. 198
Page 37 and 38: Patton, K.J., Nitschke, R.G., and H
Page 39 and 40: DECOMPOSITION ANALYSIS MODEL -BASED
Page 41 and 42: Master Problem Subproblem (i = 1, .
Page 43 and 44: 1 2 Impeller Stator Vortex flow dir
Page 45 and 46: Partitioned Graph Resultant FDT 111
Page 47 and 48: Table I Summary of Coordination Str
Page 49 and 50: Index Variable Partition Descriptio
Page 51 and 52: Index Variable Partition Descriptio
Page 53 and 54: 23 Fnd Fnd - ( Wd M g - ∆Wd) (fro
Page 55 and 56: 66 67 68 69 70 71 72 73 74 75 76 77
Page 57 and 58: Appendix B: Parameters in Optimizat
Page 59 and 60: Impeller Driving Table B.4: Base to
Page 61 and 62: Table B.8: Accessory losses. Amps @

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