POWERTRAIN DESIGN GROUP MEETING #5
Optimization of Hybrid Powertrain using DIRECT
Algorithm
Sachin Kumar Porandla
Advisor
Dr. Wenzhong Gao
Outline
• What is Optimization
• Local and Global Optimization
• Optimization Process
• Optimization Tools and Algorithms
• DIRECT algorithm
• Problem statement
• Results
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Optimization
• Optimization is the process of minimizing or
maximizing a desired objective function using a set of
design variables while satisfying the prevailing
constraints.
A general optimization problem can be defined as:
min
s.t.
f  x1 , x 2 ,..., x n 
g1  x1 , x 2 ,..., x n   0,

g m  x1 , x 2 ,..., x n   0,
l i  xi  u i ,
where,
x1 ...x n
is the objective function,
g1.. m x1 , x2 ,..., xn 
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Integer
are the design variables,
f x1 , x2 ,..., xn 
li , u i
iI
are the constraint functions and
are the lower and the upper bounds the design variable
xi
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Optimization contd…
• Objective function
• Maximize fuel economy
• Minimize emissions or weight or cost etc..
• Maximize Performance limits
• Design Variables
• power ratings of vehicle components ( ICE, motor, battery)
• Control strategy parameters
• Other variables ( drive ratio, battery SOC, mass of the vehicle, weight, cost…)
• Constraints
• Acceleration ( 0 – 60 mph ≤ 11.2, 40- 60 mph: <= 4.4s,..etc)
• Gradeability ( ≥ 6.5% grade at 55 mph)
• Other Constraints ( difference in SOC ≤ 5% or final SOC > Initial SOC etc..)
•The optimization problem tries to minimize/maximize the objective function
by searching the multidimensional parameter space for the various
combinations of the design variables and selecting the best combination at
each iteration.
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Local and Global optimization
• Local optimization are good at finding local minima, use
Derivatives of the objective function to find the path of
greatest improvement, fast convergence. Doesnt work for
noisy and discontinuous functions
• Global Finds global minimum, derivative-free, slow
convergence because of larger design space
global minimum
Local minimum
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Optimization Process
• Forms an optimization loop
• Optimization program calls Simulation/Analysis tool with
new design points
• The Simulation tool calculates the objective function and
verifies the constraint functions
Simulation tool
Optimization program
Optimization
routine
ADVISOR 2.0
Maximize mpgge
(Objective)
f(x)
PSAT
V-Elph
Constraints
g(x)
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Optimization Tools and Algorithms
Gradient-Based
FMINCON
• MATLAB Optimization Toolbox
• Non-linear bounded and constraint problems
• Sequential Quadratic Programming methods
VisualDOC RSA (Response Surface Approximations)
• Generates surface approximation based on DOE
• Estimates optimum based on surface gradients
• Updates surface based on exact function value
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Tools and Algorithms contd…
VisualDOC DGO (Direct Gradient Optimization)
Applies Sequential Quadratic Programming methods
to function values to determine gradients and search
direction
iSIGHT
• Offers a wide variety of algorithms and solution methods to
choose from. Two key features of this tool are 1) its
flexibility in defining linkages between multiple programs,
and 2) the ability to combine multiple solution methods in
series or parallel to solve a specific problem.
• Provides response surface visualization tools that allow the
user to explore the impacts of design parameters manually
based on design-of-experiments based approximation.
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DIRECT algorithm
DIRECT : DIvided RECTangles
• a global optimization algorithm
• a modification of the standard Lipschitzian approach
that eliminates the need to specify the Lipschitz constant
•Lipschitz constant is a weighing parameter, which
decides the emphasis on the global and the local search
•eliminates the use of Lipschitz constant by searching all
possible values for the Lipchitz constant thus putting a
balanced emphasis on both the global and local search.
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DIRECT algorithm contd..
•The algorithm begins by scaling the design box to a
n-dimensional unit hypercube. DIRECT initiates its search
by evaluating the objective function at the center point of
the hypercube
• DIRECT then divides the potentially optimal
hyperrectangles by sampling the longest coordinate
directions of the hyperrectangle and trisecting based on
the directions with the smallest function value until the global
minimum is found
• Sampling of the maximum length directions prevents
boxes from becoming overly skewed and trisecting in the
direction of the best function value allows the biggest
rectangles contain the best function value. This strategy
increases the attractiveness of searching near points with
good function values
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DIRECT algorithm contd..
Figure showing three iterations in DIRECT algorithm
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DIRECT algorithm contd..
Identifying potentially optimal rectangles
Assuming that the unit hypercube with center ci is divided
into m hyperrectangles, a hyperrectangle j is said to be
~
potential if there exists rate-of-change constant K  0
such that
f c j   K d j  f ci   K d i
~
~
for all i  1,..., m
f c j   K d j  f min   | f min |
~
where f min
the best value of the objective function
  0 is positive constant and d is the distance from the
center point to the vertices
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DIRECT algorithm contd..
•The first equation forces the selection of the
rectangles in the lower right convex hull of dots and
the second equation insists that the obtained function
value exceeds the current best function value by a
nontrivial amount.
•This prevents the algorithm from becoming too local,
wasting precious function evaluations in search of
smaller function improvements. The parameter 
introduced balances the local and global search.
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DIRECT algorithm contd..
1. Normalize the search space to be the unit hypercube.
Let c1 be the center point of this hypercube and
evaluate f(c1).
2. Identify the set S of potentially optimal rectangles
(those rectangles defining the bottom of the convex
hull of a scatter plot of rectangle diameter versus
f(ci) for all rectangle centers ci)
3. Choose any rectangle r Є S
4. For the rectangle r:
4a. Identify the set I of dimensions with the maximum
side length. Let δ equal one-third of this maximum side length.
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DIRECT algorithm contd..
4b. Sample the function at the points c±δei for all i ∈ I, where c is
the center of the rectangle and ei is the ith unit vector.
4c. Divide the rectangle containing c into thirds along the
dimensions in I, starting with the dimension with the lowest value
of f(c ± δei) and continuing to the dimension with the highest f(c ±
δei).
5. Update S. Set S = S – {r}. If S is not empty, go to Step 3.
Otherwise go to Step 6.
6. Iterate. Report the results of this iteration, and then go
to Step 2.
7. Terminate. The optimization is complete. Report the xmin
and f and stop.
min
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Problem Statement
• A default ‘Parallel Hybrid Vehicle’ is optimized to maximize
the fuel economy on a composite of city and highway
driving schedule
Composite F uelEconomy

1
0 . 55
0 . 45
+
City _ FE Hwy _ FE
where City_FE is the city driving fuel economy and Hwy_FE represents the Highway fuel economy
City (FTP-75)
Highway (HWFET)
Vehicle Speed (mph)
60
50
40
30
20
10
0
0
500
1000
1500
2000
2500
Time (s)
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Problem Statement contd..
• Objective: Maximize the composite fuel economy
• Constraints:
0 -60 mph : <= 11.2 s
40-60 mph: <= 4.4s
0-85 mph : <= 20s
Greadability : >=6.5% grade at 55 mph
Difference in required and achieved speeds : <= 3.2 km/h
Difference between initial and final SOC
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: <= 0.5%
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Problem Statement contd..
• Design Variable: The design variables for this study
consists of 4 variables, two variables defining the size of
the fuel converter and motor, one representing the number
energy modules and the fourth representing the maximum
Ampere hour (Ah) capacity
• Upper and lower bounds of the variables are listed below
Variable
Description
Lower bound
Upper bound
fc_pwr_scale
Fuel converter
power scale
1(41kW)
3(123kW)
mc_trq_scale
Motor/controller
peak power scale
0.8(60kW)
2.5(187.5kW)
ess_module_num
Battery’s number
of modules
11
35
ess_cap_scale
Battery Max. Ah
capacity scale
0.333(8.3 Ah)
1(25 Ah)
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Problem Statement contd..
Figure showing the parallel HEV Configuration before optimization in ADVSIRO 2.0
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Results
DIRECT Statistics
Iterations
21
Simulation time ~24 hrs
Function Eval’s
539
Simulation statistics
Variable
Description
Initial value (before
optimization)
Final value( after
optimization)
fc_pwr_scale
Fuel Converter
power scale
2(82kW)
1.037(42.5kW)
mc_trq_scale
Motor/controller
peak power scale
1.25(93kW)
0.8035(60.2kW)
ess_module_num
Battery’s number
of modules
30
14.4 (~15)
ess_cap_scale
Battery Max. Ah
capacity scale
1(25 Ah)
1.7496(43.7Ah)
Initial and the final values of the design variables
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Results contd…
Final fuel economy
Initial fuel economy
Fuel Economy
Initial Value
Final Value
27.6 mpg
37.9 mpg
Initial and the final objective value
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Results contd…
Constraint
Constraint value
Performance before
optimization
Performance after
optimization
0 - 60 mph
<=11.2 s
9s
8.87 s
40 - 60 mph
<= 4.4 s
4.3 s
4.39 s
0 - 85 mph
<= 20 s
18.3 s
17.84 s
Gradeability at
55mph
>= 6.5 %
6.09%
6.58%
Difference in
required and
achieved speeds
2 mph
n/a
0mph
Difference between
initial and final SOC
0.5 %
0.4%
0.34 %
Table showing performance before and after optimization
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Results contd…
Emissions before
optimization
City
HC
0.388
CO
2.286
NOx
0.395
PM
0
Hwy/City NOx
Emissions after optimization
City
Hwy/City Nox
0.386
0.56
2.044
0.376
0.57
0
Comparison of the emissions before and after optimization
Mass of the vehicle
pre-optimization
post-optimization
1545 kg
1338
Initial and the final mass of the vehicle
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Results contd…
Figure showing the design variables and the objective function at each iteration
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References
• Ryan Fellini, Nestor Michelena, Panos Papalambros, and Michael Sasena, “Optimal Design of Automotive
Hybrid Powertrain Systems,” Proceedings of EcoDesign 99 - First Int. Symp. On Environmentally Conscious
Design and Inverse Manufacturing (H. Yoshikawa et al., eds.), Tokyo, Japan, February 1999, pp. 400-405.
• Wipke, K., and Markel, T., ”Optimization Techniques for Hybrid Electric Vehicle Analysis Using ADVISOR,”
Proceedings of ASME, International Mechanical Engineering Congress and Exposition, New York, New York.
(11/11/01-11/16/01)
• M.J.Box, ”A new method of constrained optimization and a comparison with other methods,” Imperial Chemical
Industries Limited, Central Instrument Reseacrh Laboratory, Bozedown House, Whitchurch Hill, Nr. Reading,
Berks 1965.
• D.Jones,” DIRECT Global Optimization Algorithm,” Encyclopedia of Optimization, kluwer Academic Publishers,
2001.
• Report on “Optimal Design of Non-Conventional Vehicles” The University of Michigan, Dept. of Mechanical
Engg., January 19, 2001.
• Haskell R.E., and Jackson C.A., “Tree-Direct: An Efficient Global Optimization Algorithm,” Proc. International
ICSC Symposium on Engineering of Intelligent Systems, University of La Laguna, Tenerife, Spain, February 11-13,
1998.
• Bjorkman, Mattias and Holmstrom, Kenneth, “Global optimization using the DIRECT Algorithm in MATLAB,”
Advanced Modeling and optimization, Vol. 1, No. 2, 1999.
• Jones, D.R., Perttunen, C.D., Stuckman, B.E. ,”Lipschitzian Optimization without Lipschitz Constant,” Journal of
Oprtimization Theory and Applications, Vol. 79, No. 1, October 1993.
• Finkel D.E., and Kelley C.T., “Convergence Analysis of the DIRECT Algorithm,” N. C. State University Center for
Research in Scientific Computation Tech Report number CRSC-TR04-28, July, 2004.
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