where  mf is the injected fuel mass rate and the power output P has been computed from

In this expression, RPM is the engine speed in revolutions per minute, p is the cylinder pressure, dV is the volume change increment, IVC denotes the inlet valve closure and EVO is the exhaust valve opening。 L denotes the power losses due to friction, scavenging, and operation of peripherals such as the turbocharger and the injection system。 The power losses, L, have been estimated from the difference of the experimental power output of 157。62 KW and the tuning case integral RPM 120 EVO IVC pdV = 169。62 KW as L = 12 KW。 This value of L has been used in all the simulations in this study。 It should be noted that the computations have been performed at full engine load where changes to L are small in comparison to the power output。 Therefore, using a constant L in all the simulations is expected to have a negligible effect on the final optimization results。 Note that the above expression for the power output has been used throughout this study in the normalization of the emissions and the SFC with respect to KW-hr。

Normalization of the Parameter Space

The determination of the cost function gradients involves the computation of the partial derivatives。 The partial derivatives are approximated by difference quotients which requires an appropriate discretization of the parameters。 Therefore, to make this optimization method less dependent on the nature of the parameters, the optimization is performed on the unit cube in n-dimensional space

This means that the actual parameter set has been mapped onto X by the transformations

where pi is the i-th parameter bounded by pi,min and pi,max。 Therefore, a point x∈X has to be mapped into the true parameter space by the inverse transformation before an actual computer simulation can be performed。 This normalization has the additional advantage that the constants β and λ, which influence the error criterion and the backtracking step sizes, are less sensitive to changes in the optimization parameters or cost functions。 Thus, the normalization of the parameter space makes the optimization method more universal。 The parameter range together with the increments used in the computations of the difference quotients are listed in Table 7。

Temperature Constraint

A constraint on the average cylinder temperature at the exhaust valve opening has been used in analyzing the final optimum point predicted by the simulations。 This value has been set to 1250 K。 If, in a computation, this constraint is violated then that particular parameter set is marked and later eliminated as a possible optimum。 More precisely, at the end of each run, possible solution points which violate the temperature constraint are eliminated, and the optimum is chosen to be the best solution point which does not violate the constraint。 Note that in this study the constraints have never been violated。

Computational Costs

The computational costs are measured in terms of the number of cost function evaluations needed until the optimum or target point is reached。 In the steepest decent method, the total number of function evaluations consists of contributions from the gradient approximations and the line searches。 Once a new pivot is determined, in order to compute the gradients, one additional function evaluation per optimization parameter is needed。 The number of function evaluations along each line search varies from case to case。 The last line search yields a pivot which corresponds to the optimum point, and therefore, the gradients do not need to be computed there。 Consequently, the total number of cost function evaluations, #F, can be expressed as

where n is the dimension of the parameter space, m is the number of pivots (including the final point) and lk is the number of cost function evaluations in the k-th line search。

ZERO-DIMENSIONAL ENGINE CODE