POWERTRAIN DESIGN GROUP MEETING #5

1. POWERTRAIN DESIGN GROUP MEETING #5 Optimization of Hybrid Powertrain using DIRECT Algorithm Sachin Kumar Porandla Advisor Dr. Wenzhong Gao

2. Outline • What is Optimization • Local and Global Optimization • Optimization Process • Optimization Tools and Algorithms • DIRECT algorithm • Problem statement • Results

3. 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: where, are the design variables, is the objective function, are the constraint functions and are the lower and the upper bounds the design variable

4. 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.

5. 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

6. 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 Maximize mpgge (Objective) f(x) Optimization program ADVISOR 2.0 Optimization routine PSAT V-Elph Constraints g(x)

7. 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

8. 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. 

9. 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.

10. 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

11. DIRECT algorithm contd.. Figure showing three iterations in DIRECT algorithm

12. DIRECT algorithm contd.. Identifying potentially optimal rectangles Assuming that the unit hypercube with center is divided hyperrectangles, a hyperrectangle is said to be into if there exists rate-of-change constant potential such that the best value of the objective function where is positive constant and is the distance from the center point to the vertices

13. 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.

14. DIRECT algorithm contd.. • 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 ЄS 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.

15. 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 and and stop.

16. 1 = Composite F uelEconomy 0 . 55 0 . 45 + City _ FE Hwy _ FE • Problem Statement • A default ‘Parallel Hybrid Vehicle’ is optimized to maximize • the fuel economy on a composite of city and highway • driving schedule where City_FE is the city driving fuel economy and Hwy_FE represents the Highway fuel economy City (FTP-75) Highway (HWFET)

17. 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 : <= 0.5%

18. 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

19. Problem Statement contd.. Figure showing the parallel HEV Configuration before optimization in ADVSIRO 2.0

20. Results Simulation statistics Initial and the final values of the design variables

21. Results contd… Final fuel economy Initial fuel economy Initial and the final objective value

22. Results contd… Table showing performance before and after optimization

23. Results contd… Comparison of the emissions before and after optimization Initial and the final mass of the vehicle

24. Results contd… Figure showing the design variables and the objective function at each iteration

25. 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.