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203

Optimization techniques in diesel engine

system design

Abstract: This chapter discusses the fundamental concepts, advanced

techniques, and theory in diesel engine system optimization to address the

complex interactions between engine subsystems, from single- and multi-

objective optimizations to deterministic and nondeterministic probabilistic

optimizations. It begins with an overview of system optimization theory by

introducing design of experiments (DoE), response surface methodology

(RSM) and the Monte Carlo simulation to address the optimizations in

design for target, design for variability and design for reliability. It then

elaborates the theory of using RSM, advanced DoE optimization, and Monte

Carlo simulation in engine system design. Finally, it details the optimizations

of robust design for variability and reliability.

Key words: system optimization, single- and multi-objective optimizations,

deterministic and nondeterministic probabilistic optimizations, design of

experiments (DoE), response surface methodology (RSM), Monte Carlo

simulation, Taguchi method, variability- and reliability-based optimizations,

robust design, statistical probability distribution.

3.1 Overview of system optimization theory

3.1.1 Introduction to optimization

Optimization is a process to search for solutions to reach the best objective

under certain constraints. Optimization techniques have been widely used

in all engineering areas to select the best design. Assume the total number

of factors is k. A single-objective optimization topic can be described as

follows.

Select factors from a k-dimensional vector X = [X

, X

, . . ., X

]

minimize the objective function f (X) = f (X

, X

, . . ., X

), subject to the

inequality constraint functions g

(X) ≤ 0, j = 1, 2, . . ., m, and the equality

constraint functions m

(X) = 0, u = 1, 2, . . ., p (p < k) within the factor

range

XX ≤XX

XX XX

≤

Xi, Xi

= 1, 2, . . .,

, where

and

XX and XX

represent the lower

and upper limits of the factor X

, respectively.

Factors, objective functions, and constraints are three key elements in

any optimization topic. The k-dimensional space formed by the k factors is

called factor space or design space. When k > 3, the design space cannot be

graphically displayed. The larger the number of factors, the more complex

the optimization topic.

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204 Diesel engine system design

The objective and constraint functions are mathematical models that describe

the relationships between the input factors and the output responses. The

function can be as simple as an explicit formula (e.g., engine cylinder heat

transfer area as a function of bore and stroke) or as complex as a surface- t

formula obtained by the regression of implicit numerical simulation results

(e.g., BSFC as a function of fuel injection timing and turbine area as in the

case of engine cycle simulation).

Usually, minimizing an objective function is equivalent to maximizing

the negative of the function. The objective can be either a single objective

function or multi-objective functions, or a weighted combination of multiple

objective functions in the form of

f(X) = w

· f

, X

, . . ., X

) + w

· f

, X

, . . ., X

) + . . .

+ w

· f

, X

, . . ., X

) 3.1

where w

, w

, . . ., w

are weighting factors (0 < w

< 1) and

w w

= 1.

The inequality and equality constraints are optional in the formulation of

an optimization topic. The number of equality constraints should be less than

the number of factors or the number of unknowns in the equality constraint

equations. Otherwise, all the factors will have a single set of deterministic

solutions. Then the optimization topic will become meaningless. However,

there is no limitation on the number of inequality constraints.

If all the objectives and constraints are linear functions, it is called a linear

optimization topic. Otherwise, it is a nonlinear one. The optimization topics

in diesel engine system design are usually multi-factor and multi-objective

nonlinear optimization. However, in many cases the topic is simpli ed

as a single-objective optimization to minimize BSFC by converting other

objectives to constraints.

The objective function may have multiple local optima (i.e., minimum or

maximum), especially when the number of factors is large and the objective

function is complex. The optimization seeks the global optima within the

design space or solution domain enclosed by the factor range and design

constraints (including the boundary). This can often be achieved by trying

many different and distant initial values of the factors during the optimization

search. In particular, it is worth noting that genetic algorithms have been

largely used in engine optimization recently as a robust method can avoid

premature convergence to a local minimum or maximum of the objective

function. Details on the advantages and drawbacks of genetic algorithms are

provided by Parmee and Watson (2000), Thiel et al. (2002), and Zottin et

al. (2008).

The optimization problem can be solved graphically or analytically. A

graphical example is to use a spreadsheet to plot the result and visually

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205Optimization techniques in diesel engine system design

select the optimum. An analytical example is to compute the gradient of the

objective function and search along the direction of the maximum gradient in

order to nd the optima. Detailed elaborations on the solution algorithms in

optimization are provided in numerous books and they are beyond the scope

of this book. The reader is referred to Siddall (1982), Montgomery (1991),

Myers and Montgomery (2002), Eriksson et al. (1999) and Edwards et al.

(2000) for more information. A comprehensive summary of the statistics for

engine optimization is provided by Edwards et al. (2000).

3.1.2 Design of experiments (DoE), response surface

methodology (RSM), and Monte Carlo simulation

Each factor in a design of experiments (DoE) is an input parameter. It can

be a control factor or a noise factor. It can be any design parameter (e.g.,

turbine area), calibration parameter (e.g., EGR valve opening) or attribute

parameter (e.g., intake manifold pressure). Each factor has two or more

discrete deterministic levels of value in the DoE. If each factor is viewed

as one ‘dimension’, multiple factors form a multi-dimensional factor space

or design space in the DoE. Each DoE run is a combination of factor levels

or a point in the factor space. The full list of all the DoE runs is called a

design matrix (or array). Each response in a DoE is an output parameter (e.g.,

BSFC, EGR rate, stress). The factors have their main effect and interaction

effect on a given response. The interaction effect refers to the fact that the

behavior of the response to one factor with respect to the factor’s levels

depends on the level of another factor.

DoE is a statistical design technique to design experiments or simulations.

It has been widely used to efciently identify design solutions in engine

development. Because the number of possible factor combinations as design

options increases exponentially with the number of factors, it is important to

use DoE to screen and identify the key factors. DoE is a systematic approach

to reduce the cost and time in nding the design solution. Instead of using a

trial-and-error or one-factor-at-a-time approach, DoE constructs a statistical

plan of the combination of factors. A DoE simultaneously varies the levels

of the factors by using fractional factorial designs instead of the full factorial

design to greatly reduce the number of runs by orders of magnitude without

losing the quality. The advantage of using DoE becomes very prominent

when the number of factors is much greater than three. The DoE design

methods include principally the Taguchi method and the response surface

methodology (RSM).

The counterpart of the DoE approach in the nondeterministic world is

the Monte Carlo simulation. The Monte Carlo simulation is a statistical

probabilistic technique to design experiments or simulations to study the

nondeterministic probability distribution of the factors and the responses.

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