42
All four algorithms investigated have linear regression slopes m of approximately unity for the
ϕ
SOC
and
ϕ
10
characteristics, whereas the values for the
ϕ
50
and
ϕ
90
characteristics are significantly lower than 1, specifically for both the CMA-ES and GADS algorithm. As an example, Figure 4.6 b compares the
ϕ
50
data obtained from both the GA and GADS calibrated ROHR models, visualizing the effects mea-
sured by the linear regression slope m. A slope m which is lower than unity refers to a reduced sensitivity of the simulation output, i.e. while low measurement values are
over predicted by the simulation, higher values are under predicted.
4.3.3 Stochastic Initialization Evolution
In order to determine the influence of a stochastic initialization on the performance of evolutionary algorithms, 25 consecutive ROHR model calibrations are performed
using the EA algorithm. As shown in Figure 4.7 a, the initial variations caused by the stochastic initializa-
tion decrease with the number of function evaluations Δ f
Error
at initialization: 24’600; after 50’000 function evaluations: 1’700. Furthermore, neither the optimiza-
tion case with the best nor the worst stochastic parameter initialization remains the best nor worst case at the end of the optimization. Thus, although there is a signifi-
cant influence on the initial phase of the optimization, the stochastic manipulations used during the evolutionary processes i.e. recombination and mutation have a
larger impact on the optimization outcome.
To illustrate the influence of stochastic initialization on the individual model parameters, Figure 4.7 b shows the development of the combustion induced tur-
bulence scaling factor c
Comb
for the 25 consecutive optimization runs. Whereas ini- tial values are randomly distributed, the solutions tend to approach the best overall
value with an increasing number of function evaluations similar to the performance value variation decrease.
a b
Fig. 4.6 Comparative Algorithm Study Statistics: a Person’s Correlation
Coefficient Linear Regression Slope b ϕ
50
“1-to-1” Plot
P e
a rs
o n
s C
o rr
e la
ti o
n C
o e
ff ic
ie n
t r
[ -]
L in
e a
r R
e g
re ss
io n
S lo
p e
m [
-]
0. 5 0. 6
0. 7 0. 8
0. 9 1. 0
1. 1 1. 2
GA EA
CMA-ES GADS
SOC r
corr
m
reg
ϕ
50
r
corr
m
reg
ϕ
10
r
corr
m
reg
ϕ
90
r
corr
m
reg
ϕ
5
S im
u la
ti o
n s
[ °
C A
a T
D C
]
355 360
365 370
375 380
385 390
ϕ
50
Measurements [ ° CA aTDC] 355
360 365
370 375
380 385
390
EA GADS
m
GADS
m
EA
43
4.3.4 Summary
Although the four algorithms show similar global performances except for the minor deviations in
ϕ
50
and
ϕ
90
, the exponential temporal performance improve- ment of the EA makes it the preferred algorithm. Hence, the EA is used as parame-
ter calibration algorithm in all subsequent investigations and studies.
4.4 Model Study on Different Engine Sizes
To evaluate the general applicability of the proposed ROHR model, the three engines employed in this study cover both major application areas of modern
Common-Rail DI diesel engines automotive, heavy-duty and marine, specifications c.f. Section 3.5.1 and a wide range of operating conditions c.f. Appendix A. Using
the heavy-duty engine as the reference engine, in addition to the calibration of the model to each specific engine, the heavy-duty engine specific model is also applied to
the automotive and marine diesel engine, without any parameter modifications i.e. “bind try”.
4.4.1 “Heavy-Duty” Diesel
Figure 4.8 shows the results using an EA Evolutionary Algorithm with a popula- tion size n
pop
of 100 n
parent
= 50, n
offspring
= 150 to calibrate the 23 ROHR model
parameters given in Table 4.1. With a mean model calculation time of one-third of a second, approximately 12 hours are required to calibrate the ROHR model based on
the 19 operating conditions. Evident from an engineering point of view is the excellent prediction of the rela-
tive variations between two arbitrary operating conditions hereafter referred to as “trends” for the four ROHR characteristics, as well as the small deviations of the
a b
Fig. 4.7 Performance and Parameter Variations of 25 Optimization Runs:
a Error vs. Function Evaluations b Single Parameter Variation
E rr
o r
M e
a su
re m
e n
t -
S im
u la
ti o
n [
-]
10000 15000
20000 25000
30000 35000
40000 45000
Function Evaluations [ -] 10000
20000 30000
40000 50000
Maximum Error Init ializat ion Minimum Error Init ializat ion
Minimum Error End Ot her Opt imization Runs
c
c o
m b
[- ]
1 2
3 4
5 6
7 8
9 10
Function Evaluations [ -] 10000
20000 30000
40000 50000
Maximum Error Init ializat ion Minimum Error Init ializat ion
Minimum Error End Ot her Opt imization Runs
