Group Assignment 5: Optimization for an On Board Fuel Processing Unit to
Provide Hydrogen from Ammonia
Michael Huang, Zubin John, Bill Binder, Joel Toussaint
Task 1: Solving the Design Problem Deterministically
In the first part of this exercise expected utility of the ammonia cracker fuel cell car is
optimized (maximized).
The design of experiments performed in the previous assignment
showed that the design space had numerous local maxima making it difficult to find the
somewhat steep and thin global maximum. Many different algorithms of different types were
tested to see what could most efficiently and accurately calculate the global maximum. After
many tests it was found that only a genetic algorithm could find the correct peak the global
maximum was on. It was decided the most efficient way to accurately solve for the global
maximum was to use the Darwin algorithm to get close to the maximum and then acquire a finer
solution using Hooke-Jeeves algorithm. Once the Hooke-Jeeves algorithm had an appropriate
starting value it could very quickly converge to a maximum. Hooke-Jeeves was selected instead
of quicker more approximate gradient based algorithms because it consistently found values a
few percent higher which indicated the other methods were less accurate. Figure 1.1 shows the
design variables changing each run and eventually converging on the optimum.
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Figure 1.1: Design variables at each run. Blue is system voltage, red is stall torque, and green is
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It can be seen that the Darwin algorithm starts with the design variables spanning randomly
across their range. The voltage of the battery/PEMFC appears to converge to an optimum value
fastest after about 300 runs, most likely due to its smaller range. After about 500 runs stall
torque and ωmax start to converge to an optimum value although ωmax seems to slowly improve
after another 1000 runs. Figure 1.2 shows the design variable values vs. the expected utility.
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Figure 1.2: Design variables vs. expected utility. Blue is system voltage, red is stall torque, and
green is ωmax. Design variables at the same expected utility are from the same run.
Expected utility increases as the design variables narrow to their optimum values together. It is
interesting to see that the farther from optimum the larger the range of values that can be taken
by the design variables.
As the optimization begins to converge the design variables are
generated over a smaller and smaller range as they approach an optimum.
Once the Darwin algorithm nears a global maximum the run with design variable
values that give the highest expected utility is used to start the Hooke-Jeeves algorithm. Figure
1.3 shows the design variables vs. the run number.
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Figure 1.3: Design variable values vs. run number. The colors represent the same variables as in
previous figures.
The figure shows that the starting values generated from the Darwin algorithm are very close to
what the Hooke-Jeeves algorithm converges to. The Hooke-Jeeves algorithm only makes a very
small improvement but runs much faster than the Darwin algorithm. Figure 1.4 shows the design
variable values vs. expected utility.
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Figure 1.4: Design variable values vs. expected utility for Hooke-Jeeves algorithm. Design
variables at the same expected utility are from the same run.
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The expected utility nears the global maximum as the design variable values cluster near the
optimum. If even one of the design variables strays from their optimum value the expected
utility drops rapidly.
The global maximum found in this exercise is in agreement with what was found in the
previous assignment G4. It was found that the highest peak in the design of experiments from
G4 was at a voltage of 185 V. Similarly using the optimizer the optimal voltage found is at 179
V. Also the stall torque is found to be 367.6 Nm and ωmax is 1314.6 rad/s. This means that only
ωmax changed by a large amount from 900 Nm. This is not an issue since the space can have
rapidly shifting maxima as the voltage changes a small amount.
Task 2: Solving the Design Problem under Uncertainty
Due to the fact that our model requires approximately thirty seconds to calculate, we
decided that a surrogate model was necessary. In order to do this, a level 7 full factorial was
performed as data. This was then used to create the surrogate model with R2 values of 99.999%,
99.9311%, and 99.7135% for mileage, acceleration, and top speed, respectively. Since these
values were extremely close to one, it was concluded that the surrogate model would be
sufficient for use in performing the optimization under uncertainty. The Hooke-Jeeves algorithm
was used with five trials, starting at the optimized value that was achieved deterministically.
Figure 2.1 shows the model with the optimizer wrap.
Figure 2.1: Shows the model with the Optimizer
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Using the new optimal value from the five trial optimization, we then performed a ten trial and
consequently a twenty-five trial optimization. The change in the variables was less than one
percent and thus it was concluded that we were sufficiently close to the optimized value and that
additional iterations would be unnecessary due to the high cost of running the optimizer and low
return on values that did not change significantly. The final values were 186.1 V, 389 N-m, and
1287 rad/s for the system voltage, stall torque, and maximum angular velocity respectively.
Based on these numbers, the averages for the parameters were: zero to sixty acceleration time
was 8.463 seconds, the mileage was 73.76 mpg, and the top speed was 123.86 mph. This
provided an expected utility of 1.22*108. In order to further understand what the optimizer was
doing, we then decided to make plots of the variables against the expected utility as well as a plot
of the variables with respect to the run number, shown in Figure 2.2 and 2.3, respectively.
Figures 2.2 and 2.3 only show the case of the twenty-five trial optimization. The plots for the
five and ten trial optimization can be found in Appendix A and B.
25 Trial Parameters vs Uncertain E[U]
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Figure 1.2: Shows the design variables, i.e., the system voltage, stall torque, and maximum
angular velocity vs the expected utility.
It is clear from Figure 2.2 that the optimizer quickly located the general region of the
maximum expected utility. As it approached the optimal value of the expected utility, the number
of trials in that area increased. In addition, the closer to the maximum expected utility the
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optimizer was, the smaller the change in the variables. This is reasonable as it is expected that
the optimizer first finds the general area of the optimal value and then proceeds to refine its
search as it gets closer to the true optimal. Since Hooke-Jeeves algorithm uses the slope of the
solution it is also easy to understand that not many runs are necessary when it calculated a large
slope, only when the slope was approaching zero.
25 Trial Parameters vs Run Number
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Figure 2.3: Shows the design variables, the system voltage, stall torque, and maximum angular
velocity vs the run number.
Figure 2.3 helps to reinforce the conclusions made from Figure 2.2. In Figure 3 it is clear
that the optimizer changed the values of the variables considerably in the beginning of the
optimization but then the changes became very small as the run number increased. This is simply
the optimizer searching the space from the initial values. Of particular interest is that the
optimizer appears to avoid changing more than one variable at a time. By looking closely at the
very beginning of the optimization it is clear that the stall torque and maximum angular velocity
do not change under the optimizer has explored an area around the system voltage. It performed
a search in both directions from the initial value. Once this was done, the stall torque design
space was then explored in a similar fashion, with the other two variables remaining unchanged.
Lastly, the maximum angular velocity then did this in a similar fashion. This was then repeated,
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starting once again at the system voltage and appears to repeat indefinitely until the optimizer
has the optimal values.
The deterministic optimizer was able to achieve a higher value of utility as compared to
the expected utility from the uncertain optimizations. The utility from the deterministic optimizer
is 1.245*108 while the expected utility from the uncertain optimizer was 1.221*108. Without an
infinite number of trials it is not useful to compare the two values. This is due to the fact that the
uncertain optimizer was using a set of random entries that the deterministic optimizer is unable
to access and the uncertain optimizer only approaches this as the number of trials goes to infinite.
Task 3: Lessons Learned
The first issue that had to be addressed was our Dymola model's extremely long runtime
of nearly 30 seconds depending on what machine is used. Two attempts were made at producing
a surrogate model with the Krigging method. The first attempt, which used 1080 data points,
was most likely quite sufficient although there was some concern that the true optimum would be
outside of the range of points taken. A second attempt was made with nearly 2400 data points
over a larger range despite warnings that it wouldn't be feasible (it was mentioned that over 1000
would most likely not work). Delightfully, the surrogate model did not take long to generate and
gave an R2 value of no less than 99.7%. Figure 3.1 shows the RSM toolkit results that confirm
the 2392 point surrogate model.
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Figure 3.1: Surrogate model is successfully generated over 2392 data points with sufficient
accuracy.
A design of experiments was performed with the surrogate model to show that it matched with
the Dymola model design of experiments.
The next issue was selecting an appropriate optimization algorithm. Unfortunately the
design space was very bumpy most likely due to the demand model. This made using a less
efficient global optimization algorithm necessary. Fortunately since the surrogate model was
generated the run time was not unreasonable. After several practice runs it was determined that
the Darwin algorithm would work most efficiently with the data. After about an hour the Darwin
algorithm was close to converging to an optimum. Once it seemed that it found very close to a
global maximum and it wasn't going to go much higher it was decided that switching to a
gradient based algorithm, Hooke-Jeeves would be more efficient to further explore the peak.
In this assignment we were able to become familiar with many different optimization
algorithms and methods to explore our design space. The use of a surrogate model turned out to
be very helpful. The combination of a surrogate model and a genetic algorithm seem to be very
effective in optimizing a problem with many local maxima. If presented with another design
problem, we would like to do two things differently. Firstly, we would like to spend more time
planning and implementing the prospective system in Dymola so that the model reflects
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accurately the physical processes in the system. By making the Dymola model more modular and
efficient, it would hopefully reduce the run time to something more easily handled than 30
seconds per run. One of the trade-offs, which in this case was well worth it, was the surrogate
model. The creation and verification of the surrogate model required two designs of experiments,
both of which took a considerable amount of time. This time was definitely less than the amount
of time that would have been consumed had we tried running our optimizers with the Dymola
model. Secondly, we would strive to get more results for the design surveys. This would help us
develop a more accurate demand model that will reflect how desirable the design is to the target
market. The system had a trade-off where it seemed to sacrifice acceleration time in order to
increase the mileage to a certain level. This also had to be done with the top speed being a
sufficient level.
Appendices
Appendix A, Five Trials
5 Trial Parameters vs Uncertain E[U]
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5 Trial Parameters vs Run Number
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Appendix B, Ten Trials
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10 Trial Parameters vs Uncertain E[U]
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Parameters vs Run Number
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