Xin Q. Diesel Engine System Design
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246 Diesel engine system design
© Woodhead Publishing Limited, 2011
the fractional factorial design. Due to confounding some interaction terms
have to be removed from the emulator model. In this example, a maximum
of eight terms in the model is allowed for eight DoE runs, for example,
Y = C
0
+ C
1
X
1
+ C
2
X
2
+ C
3
X
3
+ C
4
X
4
+
C
5
X
1
X
2
+ C
7
X
1
X
4
+ C
10
X
3
X
4
3.22
Keeping which terms in the model depends on the importance of the term
for a particular problem.
Confounding is measured by ‘DoE design resolution’. Low resolution
means strong (or complicated) confounding of effects and hence poor estimate
of the affected model coefcients. On the other hand, high resolution means
weak confounding and an increase in the number of DoE runs. According to
Eriksson et al. (1999), the resolution III design corresponds to a design in
which the main effects are confounded with the two-factor interactions, and
the two-factor interactions confounded with each other. This is undesirable
in factor screening. Therefore, the resolution III designs should be used with
great care. The resolution IV designs have the main effects unconfounded
with the two-factor interactions, but the two-factor interactions are still
confounded with each other. The resolution V or higher has the two-factor
interactions unconfounded with each other. This means that the resolutions
V and above are almost as good as the full factorial designs with very little
confounding effects, but they require a large number of runs. Eriksson
et al. (1999) concluded that ‘the resolution IV designs are recommended
for screening, because they offer a suitable balance between the number of
factors screened and the number of experiments needed’.
Different fractional reduction patterns yield different confounding patterns.
Confounding pattern affects the selection of the terms in the emulator model
structure. In other words, if the number of terms in the emulator model has
been determined, an appropriate confounding pattern can be selected in order
to make the number of DoE runs greater than the number of model terms.
Table 3.4 summarizes such a relationship. For example, if a fractional DoE
of 2
k–1
is used, the eight DoE runs for k = 4 cannot t a complete model
that has eleven terms. Either some polynomial terms have to be dropped
from the model or the number of DoE runs has to be increased. Table 3.5
summarizes the resolutions of fractional factorial designs. Compared with
the full factorial design, fractional factorial designs may greatly reduce the
number of DoE runs for ‘factor screening’ at a penalty of confounding effects
and possibly associated loss of accuracy or predictability in the emulator
models. Good fractional factorial designs can give low uncertainty in the
model coefcients and small prediction errors with a small number of DoE
runs.
The second guideline for statistical DoE design is to differentiate the use
of orthogonal designs and non-standard designs. In fractional factorial design,
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© Woodhead Publishing Limited, 2011
Table 3.4 Relationship between confounding in fractional factorial design and emulator model structure
Number
of
factors
(k)
Required
emulator
terms for
first-order
with
interaction
Required
DoE runs
of 2-level
full
factorial
2
k
Required
DoE runs
of 2-level
fractional
factorial
2
k–1
Required
DoE runs
of 2-level
fractional
factorial
2
k–2
Required
DoE runs
of 2-level
fractional
factorial
2
k–3
Required
emulator
terms for
second-
order with
interaction
Required
DoE runs
of 3-level
full
factorial 3
k
Required
DoE runs
of 3-level
fractional
factorial
3
k–1
Required
DoE runs
of 3-level
fractional
factorial
3
k–2
Required
DoE runs
of 3-level
fractional
factorial
3
k–3
2 4 4 6 9
3 7 8 4 (III) 10 27 9
4 11 16 8 (IV) 15 81 27 9
5 16 32 16 (V) 8 (III) 21 243 81 27 9
6 22 64 32 (VI) 16 (IV) 8 (III) 28 729 243 81 27
7 29 128 64 32 (IV) 16 (IV) 36 2,187 729 243 81
8 37 256 128 64 32 (IV) 45 6,561 2,187 729 243
9 46 512 256 128 64 55 19,683 6,561 2,187 729
10 56 1,024 512 256 128 66 59,049 19,683 6,561 2,187
11 67 2,048 1,024 512 256 78 177,147 59,049 19,683 6,561
12 79 4,096 2,048 1,024 512 91 531,441 177,147 59,049 19,683
13 92 8,192 4,096 2,048 1,024 105 1,594,323 531,441 177,147 59,049
14 106 16,384 8,192 4,096 2,048 120 4,782,969 1,594,323 531,441 177,147
15 121 32,768 16,384 8,192 4,096 136 14,348,907 4,782,969 1,594,323 531,441
Note: The Roman numeral in parentheses indicate the DoE design resolution.
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248 Diesel engine system design
© Woodhead Publishing Limited, 2011
Table 3.5 Resolution of fractional factorial designs (based on the analysis by Eriksson
et al., 1999)
Number of
DoE runs
Number of factors
3 4 5 6 7 8
4 2
3-1
Resolution
III
Not
available
Not
available
Not
available
Not
available
Not
available
8 2
3
Full
factorial
2
4-1
Resolution
IV
2
5-2
Resolution
III
2
6-3
Resolution
III
2
7-4
Resolution
III
Not
available
16 Not
available
2
4
Full
factorial
2
5-1
Resolution
V
2
6-2
Resolution
IV
2
7-3
Resolution
IV
2
8-4
Resolution
IV
32 Not
available
Not
available
2
5
Full
factorial
2
6-1
Resolution
VI
2
7-2
Resolution
IV
2
8-3
Resolution
IV
as pointed out by Myers and Montgomery (2002), standard designs, with a
dominant preferable property of orthogonality, are the optimal design for
rst-order models or rst-order-with-interaction models. But in the case of
second-order models, the standard RSM designs are rarely optimal designs.
The famous Plackett–Burman designs are special fractional factorial designs
of resolution III (e.g., many used by the Taguchi DoE method in its inner
array for control factors), and hence they can only be used to estimate
main effects rather than interaction effects. They also noted that for the
purposes of RSM analysis, the standard central composite (CCC/CCF) or
the Box–Behnken (BB) designs should be used whenever possible, and the
central composite design is the most popular class of second-order designs.
However, it is found that the non-standard design of space-lling is also very
effective for engine applications. The space-lling design is also known as
Latin hypercube sampling, a random sampling of a prescribed number of DoE
points (runs), n, to ll the factor space. The n points (runs) project onto n
different levels in each factor, as shown in Fig. 3.16. The space-lling design
is especially useful in the following situations: (1) a large design space; (2)
large factor intervals; (3) a large number of factors; (4) many interactions;
(5) the need to build highly accurate third-order emulator models. Figure
3.16 also shows several other DoE designs that have three factors and three
levels. Note that the DoE design points (runs) of the CCC expand to ve
levels of factor settings residing on a sphere and hence the factor region is
symmetrical. The CCF design has strictly three factor levels in a cube or
hypercube. The CCC design is slightly better than the CCF design (Eriksson
et al., 1999).
The third guideline for statistical DoE design is to pay attention to rotatability
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© Woodhead Publishing Limited, 2011
3-factor 3-level full
factorial (27 cases)
3-factor 3-level CCF (central
composite face-centered, 15
cases)
3-factor 3-level CCC (central
composite circumscribed, 15
cases)
3-factor 3-level BB
(Box–Behnken, 13 cases)
3-factor 3-level
D-Optimal (14 cases)
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
X
2
X
2
X
2
X
2
X
2
X
3
X
3
X
3
X
3
X
3
X
1
X
1
X
1
X
1
X
1
0
0
0
0
0
0
0
0
0
Factorial
point Star (axial)
point
0
0
0
–1
–1
–1
–1
–1
–1
–1
–1
–1
–1
5-factor 100-case space fill DoE matrix design
DoE Factor X
2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
DoE Factor X
1
3.16 DoE matrix designs.
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250 Diesel engine system design
© Woodhead Publishing Limited, 2011
and assign factors in proper directions if needed. Rotatability means the
statistical design provides equal precision of prediction in all directions. It
is a very important property because the nature of the response surface is
usually unknown prior to running the DoE cases. Rotatability in composite
designs can be controlled by adjusting the distance of the ‘star points’ (axial
points) to the center point. According to Montgomery (1991), any rst-order
orthogonal design is rotatable; the face-centered central composite designs
are not rotatable, and this could be a serious disadvantage; Box–Behnken
design is rotatable or nearly rotatable.
The fourth guideline for statistical DoE design is to tailor the non-reachable
‘dead corners’ in the DoE factor space with proper designs. In the DoE
designs with an irregular factor space or polyhedron (e.g., corners missing
in a cubic space of standard design, or the region is constrained by certain
factor relationships), the Box–Behnken or non-standard D-Optimal design
should be considered. The Box–Behnken design does not contain any points
at the vertices of the cubic region that is created by the upper and lower
limit levels of each factor (i.e., no corner points, Fig. 3.16). This feature
could be advantageous when the points on the corners of the cube represent
the factor-level combinations that are impossible to test or simulate due to
certain constraints. The D-Optimal designs are constructed by including some
of the extreme vertices of the constrained region, centers of edges between
the hyper-cube corners, or the centers of the faces of the hyper-cube, and
so on. The D-Optimal design makes efcient use of the entire factor space,
and it can be a preferred choice when there are no classical designs that
can well investigate the irregular region, and when the number of DoE runs
that can be afforded is smaller than the number of runs of any available
classical design. The D-Optimal method may tailor the corners of the factor
space, and it also provides as much orthogonality as possible between the
columns in the design matrix, hence maximizes the output information for a
given number of DoE runs. In fact, the D-Optimal was regarded as the most
suitable statistical design method by some authors for engine calibration due
to its capability of handling the constraints on factor combinations (Roepke
and Fischer, 2001). It should be noted that the irregular factor space occurs
frequently in diesel engine system design. For example, a factor combination
of a very high exhaust restriction, a very large VGT vane opening, and a
very large EGR valve opening will result in an extremely low air–fuel ratio
in some DoE runs, which belong to the impossible ‘corners’.
Figure 3.17 shows an example of DoE input factor set-up used in diesel
engine system design. In summary, the practical rules of RSM DoE design
for diesel engine system design include the following:
1. The rst-order model with two factor levels can be used to preliminarily
screen the factors. The second-order or even third-order with more factor
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© Woodhead Publishing Limited, 2011
DoE input factor range (Note: The bold data means the default values, i.e., ‘L-def’.)
DoE input factor set-up with ‘one-factor-at-a-time’ method (Note: ‘L1’ means Level 1; ‘L-def’ means default value.)
DoE input factor set-up with the D-Optimal method (Note: –1 means Level 1; –0.3333 means Level 2; 1 means Level 5)
Input DoE simulation case Æ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
X
1
Overall turbine efficiency L1 L2 L3 L4 L5 L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def …
X
2
Overall compressor efficiency L-def L-def L-def L-def L-def L1 L2 L3 L4 L5 L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def …
X
3
Turbine size L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L1 L2 L3 L4 L5 L-def L-def L-def L-def L-def …
X
4
EGR circuit flow restriction (mm) L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L1 L2 L3 L4 L5 …
X
5
EGR cooler outlet temperature (K) L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def …
X
6
CAC cooling heat transfer coeffi (W/m
2
.K) L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def …
X
7
Exhaust restriction (CDPF flow design) L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def L-def …
Input DoE simulation case Æ 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
X
1
Overall turbine efficiency 1 –1 1 1 1 1 1 1 1 1 1 1 1 –1 –1 –1 –1 …
X
2
Overall compressor efficiency 1 1 1 1 1 –1 1 1 1 –1 1 1 –1 –1 –1 –1 –1 …
X
3
Turbine size –1 –1 1 –1 –1 1 1 –1 1 –1 1 –1 –1 –1 –1 –1 –1 …
X
4
EGR circuit flow restriction –1 1 1 1 1 1 1 1 –1 –1 –1 –1 1 –1 1 1 0.3333 …
X
5
EGR cooler outlet gas temperature –1 –1 –1 1 –1 –1 1 –1 1 –1 –1 1 1 1 –1 0.3333 –1 …
X
6
CAC cooling heat transfer coefficient –1 –1 –1 –1 1 1 1 –1 –1 1 1 1 1 –1 –1 1 1 …
X
7
Exhaust restriction (CDPF flow design) –1 –1 –1 –1 –1 –1 –1 1 1 1 1 1 1 –0.3333 0.3333 –1 1 …
3.17 Illustration of DoE input factor set-up.
DoE input parameter index Input parameter name level 1 (= –1) level 2 (= –0.333) level 3 (= 0) level 4 (= 0.333) level 5 (= +1)
X
1
Overall turbine efficiency 57% 60% 63% 66% 69%
X
2
Overall compressor efficiency 64% 67% 70% 73% 76%
X
3
Ovberall turbine size
0.36 0.4 0.44 0.48 0.52
(turbine area or VGT vane opening)
X
4
EGR circuit flow restriction (mm) 10 13.23 15.81 18.03 20
X
5
EGR cooler outlet gas
450 475 500 525 550
temperature (K)
X
6
Charge air cooler cooling heat
550 650 750 850 950
transfer coefficient (W/m
2
.K)
X
7
Exhaust restriction (CDPF flow
1.52 1.37 1.22 1.07 0.92
restriction parameter)
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252 Diesel engine system design
© Woodhead Publishing Limited, 2011
levels should be used to build the emulator regression model for the
optimization in engine system design.
2. For a very small number of factors (e.g., two or three), the full factorial
design may be used to conduct the DoE runs and build the emulator
models.
3. For a large number of factors, design resolution of the confounding
effect should be checked before a standard fractional factorial design
and emulator model structure are selected.
4. Normally, the central composite designs with three to ve factor levels
should be used for the second-order emulator models. Sometimes, as
many as ten factor levels can be used or realized by combining two
ve-level DoE that have two different factor ranges in order to build
emulators.
5. The nonstandard space- ll design is very effective for the third-order
emulator models.
6. For the irregular or constrained factor space where the ‘corner’ points
are undesirable or impossible to run, or when the region is non-cubic,
the Box–Behnken or the D-Optimal designs can be considered.
7. Adding complementary runs for unconfounding is a common technique
to improve the model accuracy iteratively. Adding supplementary DoE
runs to upgrade the emulator model (e.g., to cubic order) is also common,
and this can be achieved with the D-Optimal supplementary design.
3.2.4 Analysis and optimization with response surface
models
Canonical analysis
Once the emulator model is obtained by surface t, it can be used to predict
responses inside the factor space. It should be noted that the regression model
should not be used for extrapolation outside the factor range. The model can
also be used to analyze the sensitivity characteristics of the factor–response
relationship. Most importantly, the model can be used to conduct optimization
to search for the optima located on the response surface. A theoretical analysis
of the optimization is reviewed below on the basis of canonical analysis.
The general form of a second-order response surface emulator model is
given by
YC
CX
CX
i
k
ii
CX
ii
CX
i
k
ii
CX
ii
CX
i
ii
jj
YC =YC
YC YC
+
CX CX
ii
ii
CX
ii
CX CX
ii
CX
CX CX
CX
ii
CX CX
ii
CX
i
i
+
0
0
=1
=1
2
,<
ii,<ii
=2
SS
CXSSCX
ii
SS
ii
CX
ii
CXSSCX
ii
CX
+ SS+
SS
kkk
ij
ij
CX
ij
CX
ij
X
ij
X
ij
3.23
In the optimization, the stationary point refers to the point of factor settings
corresponding to zero partial derivatives of the response with respect to all
the factors ∂Y/∂X
1
= ∂Y/∂X
2
= . . . = ∂Y/∂X
k
= 0. The stationary point may
Diesel-Xin-03.indd 252 5/5/11 11:46:04 AM
253Optimization techniques in diesel engine system design
© Woodhead Publishing Limited, 2011
represent a maximum response, a minimum response, or a saddle point (i.e.,
neither a maximum nor a minimum in a hyperbolic response). Denote Y as
the approximation of Y
obs
by ignoring the error u. According to Montgomery
(1991), equation 3.24 can be written in matrix notation as follows with the
vectors
ˆˆ
XC
and
XC and XC
,
and a k ¥ k matrix
C
:
YC
XC
XC
X
XCXXC
X
X
X
X
k
YC =YC
+
YC +YC
XC XC
+
wh
er
e
ere er
=
0
+
0
+
ˆˆ
ˆˆ
ˆˆ
ˆ
¢¢
XC
¢¢
XC
XC
¢¢
XC
+
¢¢
+
◊
◊
◊
È
Î
1
2
ÍÍÍ
È
Í
È
Í
È
Í
È
Í
ÍÍÍ
Í
ÍÍÍ
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Î
Í
Î
Í
Í
Í
˘
˚
˙
˘
˙
˘
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˚
˙
˚
˙
˙
˙
◊
◊
◊
È
Î
Í
È
Í
È
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
=
ˆ
C
C
C
C
k
1
2
ÍÍÍ
Í
Í
Í
Í
Í
Í
Í
Í
ÍÍÍ
Í
ÍÍÍ
Í
Í
Í
Í
Í
Í
Í
Í
Í
Î
Í
Î
Í
Í
Í
˘
˚
˙
˘
˙
˘
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˚
˙
˚
˙
˙
˙
◊◊◊
◊◊◊
=
2
1
22
2
C
C
CC
C
C
k
k
11
12
CC
12
CC
2
222
◊◊◊
È
Î
Í
È
Í
È
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Í
Î
Í
Î
Í
Í
Í
˘
˚
˙
˘
˙
˘
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˙
˚
˙
˚
˙
˙
˙
Ï
Ì
Ô
Ï
Ô
Ï
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ì
Ô
Ì
Ô
Ô
Ô
Ó
Ô
Ì
Ô
Ì
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
Ô
symmetric
C
kk
ÔÔÔ
Ó
Ô
Ó
Ô
Ó
Ô
Ó
Ô
Ô
Ô
Ô
Ô
Ô
Ô
3.24
The optima can be found by setting the derivative of Y with respect to the
elements of the vector
ˆ
X
equal to zero. This is expressed as:
∂
∂
Y
X
∂X∂
CC
ˆ
ˆˆ
ˆˆ
=
CC CC
CC+ CC
2
CC2 CC
X2 X
CCXCC2 CCXCC
=
0
3.25
The stationary point is the solution of equation 3.25, and given by
ˆˆ
ˆˆ
ˆˆ
–
XC
XC
CY
CX
CX
C
0
XC
0
XC
XC = – XC
1
ˆˆ
1
ˆˆ
XC
1
XC
2
XC
2
XC
and
CY and CY
=
CX CX
CX+ CX
1
CX
1
CX
2
CX
2
CX
1
ˆˆ
1
ˆˆ
00
CY
00
CY
CX
00
CX
=
00
=
00
CX CX
00
CX CX
0
CX
0
CX
¢
ˆˆ
¢
ˆˆ
3.26
Canonical analysis is used to analyze the nature of the stationary point.
Canonical analysis refers to a coordinate translation and rotation from the
original coordinate system to be anchored at the stationary point, as shown
in Fig. 3.2c earlier. The polynomial model can be transformed to a new
coordinate system with the origin at the stationary point (X
10
, X
20
) and with
the principal axes of the new coordinate system (w
c,1
and w
c,2
) coming from
the tted response surface. This coordinate transformation results in the
tted emulator model to be transformed into the canonical form as follows
(Montgomery, 1991):
YY
ww
w
cc
cc
ww
cc
ww
ck
c
YY =YY
YY YY
+
ww ww
cc
cc
ww
cc
ww ww
cc
ww
+ . . . +
0,
YY
0,
YY
cc0,cc
0,
YY YY
0,
YY YY
+
0,
+
1,
cc1,cc
1
2
,2
ww
,2
ww
cc,2cc
ww
cc
ww
,2
ww
cc
ww
ww
cc
ww ww
cc
ww
,2
ww
cc
ww ww
cc
ww
,2
,2
2
,
ck,ck
ll
ww
ll
ww
cc
ll
cc
+
ll
+
ww +ww
ll
ww +ww
ww ww
ll
ww ww
0,
ll
0,
cc0,cc
ll
cc0,cc
1,
ll
1,
ww
1,
ww
ll
ww
1,
ww
cc1,cc
ll
cc1,cc
ww
cc
ww
1,
ww
cc
ww
ll
ww
cc
ww
1,
ww
cc
ww
1
ll
1
ww
1
ww
ll
ww
1
ww
2
ll
2
ww
2
ww
ll
ww
2
ww
l
ck
l
ck
,,
2
=1
,,
=
k
i
k
ci
,,ci,,
ci
,,ci,,
Yw
Yw
+ Yw+
Yw
ci
Yw
ci
0
Yw
0
Yw
Yw
0
Yw
2
S
YwSYw
l
Yw
l
Yw
ci
Yw
ci
l
ci
Yw
ci
3.27
where w
c,1
, w
c,2
, . . ., w
c,k
are the transformed factors called canonical
variables, and l
c,1
, l
c,2
, . . ., l
c,k
are the eigenvalues or the characteristic
roots of the matrix
C
.
The nature of the stationary point on the response surface (maximum,
Diesel-Xin-03.indd 253 5/5/11 11:46:06 AM
254 Diesel engine system design
© Woodhead Publishing Limited, 2011
minimum, or saddle point) can be determined from the signs and magnitudes
of the eigenvalues. If the eigenvalues or the roots of the determinant
equation of the canonical analysis are all positive, then the stationary point
is a minimum. If the eigenvalues are all negative, then the stationary point
is a maximum. If the eigenvalues have different signs, the stationary point
is a saddle point. A very small (essentially zero) eigenvalue results in a
ridge system where the response surface is elongated along the direction
of that canonical eigenvalue. Ridge analysis of the response surface is very
important, and is explained in detail by Myers and Montgomery (2002). All
these theoretical analyses may help locate the optima (or stationary) point of
the response surface, or point out the most efcient direction (along certain
canonical axis) to revise the DoE factor range in statistical DoE design in
order to approach the optima efciently. These analyses may avoid the blind
trial-and-error adjustment in the DoE factor range. The theoretical analysis
may also replace the laborious graphical search for the global optima when
the number of factors is large.
Optimization search algorithms
Many methods can be used to search for the optima, such as the simple but
powerful method of gradient search. The gradient search includes steepest
ascent search (in the direction of the maximum increase in the response to
reach its highest peak) and steepest descent search (in the direction of the
maximum decrease in the response to arrive at the bottom of its valley).
Basically, the direction in gradient search is perpendicular to the parallel lines
of the response contour plot. Gradient-based search algorithms sometimes
have problems with numerical errors or oscillations around the local optima.
Genetic algorithms (Parmee and Watson, 2000) or other search algorithms
such as SIMPLEX do not require evaluating the derivatives of the model
function (Zottin et al., 2008). Therefore, sometimes they can be more robust
than the gradient-based algorithms, although they still can converge to the
local optima rather than the global optima.
The required optimization technique for engine system design is the one
which can handle the following: complex trade-offs; constraints and the
responses having high nonlinearity (e.g., the behavior of engine soot and
BSFC); or multiple local optima (e.g., badly behaved responses or oscillating
numerical errors). For details of different optimization search algorithms,
the reader is referred to Eriksson et al. (1999) and Thiel et al. (2002).
Advanced data display for sensitivity and optimization
In engine system design, due to the complexity of many different engine
responses and the large amount of sensitivity data produced by the emulator
Diesel-Xin-03.indd 254 5/5/11 11:46:06 AM
255Optimization techniques in diesel engine system design
© Woodhead Publishing Limited, 2011
models, presenting the analysis results of sensitivity study and optimization in
an accessible and concise format is very important. In the RSM optimization,
using a two-dimensional contour plot on a two-factor or two-response domain
is a very effective method of graphical display for understanding system
sensitivity and locating the optima.
The most basic method to construct the contour plot is to compute
response contours with the emulator model by using constant intervals in
the X
1
vs. X
2
factor domain while holding all other factors constant. This
method becomes awkward when the number of factors is much larger than
three because many factors must be held constant in turns to construct such
contours.
A four-dimensional response contour plotting method to handle four
factors was proposed by Eriksson et al. (1999). They expanded a single two-
dimensional plot to an array of grids of contour plots along the horizontal
and vertical directions so that two more factors (dimensions) can be added
as an outer array of plot. The two-dimensional contour plots for sweeping
the rst two factors are rst created at each factor setting of the other two
factors. For example, three two-dimensional plots are created side-by-side
horizontally at three factor levels of the third factor. Then they are cloned
vertically row-by-row at four factor levels of the fourth factor. In this way,
a grid block of 3 ¥ 4 = 12 two-dimensional contour plots can be created
so that the parametric trends and the optimal design point can be observed
conveniently for four factors in a one-factor-at-a-time parametric manner.
A much more powerful graphical approach proposed earlier (Fig. 3.6) is
to generate the optimum contour maps by using constrained optimization
where all the points in the factor domain are optimized, for example, in the
sense of minimum fuel consumption. In such optimization using the emulator
model, the factor settings at each point are no longer constant. Instead, they
are optimized to correspond to the minimum fuel consumption subject to
the constraints of each particular pair of X
1
and X
2
values. An even more
powerful optimum contour map can be generated by mapping the factor
domain (X
1
vs. X
2
) to a response domain (Y
1
vs. Y
2
). In this case, every
point in the response domain is obtained by constrained optimization subject
to the constraints of each particular pair of Y
1
and Y
2
values. The required
factor settings for achieving the minimum fuel consumption map and the
corresponding other responses can also be plotted as contour maps in the Y
1
vs. Y
2
response domain. The factor boundary can be computed by a search
algorithm on such a response domain. The factor boundary is bounded by
both the DoE factor range and constraints used in the optimization. The
factor boundary in the response domain may give a Pareto frontier for the
multi-objective optimization problem to optimize the trade-off between two
responses on the horizontal and vertical axes of the map. In diesel engine
system design, a frequently used response domain is the air system capability
Diesel-Xin-03.indd 255 5/5/11 11:46:06 AM