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

Page 1 of 14 Intelligent Powertrain Design Outline What is Optimization Local and Global Optimization Optimization Process Optimization Tools and Algorithms DIRECT algorithm Problem statement Results

Page 2 of 14 Intelligent Powertrain Design 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

Page 3 of 14 Intelligent Powertrain Design Constraints Acceleration ( 0 – 60 mph ≤ 11.2, mph: <= 4.4s,..etc) Gradeability ( ≥ 6.5% grade at 55 mph) Other Constraints ( difference in SOC ≤ 5% or final SOC > Initial SOC etc..) 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…) 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.

Page 4 of 14 Intelligent Powertrain Design Local minimum global minimum 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

Page 5 of 14 Intelligent Powertrain Design Optimization Process Optimizationroutine Maximize mpgge (Objective) f(x) Constraints g(x) 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 ADVISOR 2.0 PSAT V-Elph Optimization program Simulation tool

Page 6 of 14 Intelligent Powertrain Design 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 Optimization Tools and Algorithms

Page 7 of 14 Intelligent Powertrain Design 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. Tools and Algorithms contd…

Page 8 of 14 Intelligent Powertrain Design 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.

Page 9 of 14 Intelligent Powertrain Design 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 DIRECT algorithm contd..

Page 10 of 14 Intelligent Powertrain Design DIRECT algorithm contd.. Figure showing three iterations in DIRECT algorithm

Page 11 of 14 Intelligent Powertrain Design Identifying potentially optimal rectangles Assuming that the unit hypercube with center is divided hyperrectangles, a hyperrectangle such that into potential is said to be if there exists rate-of-change constant where the best value of the objective function is positive constant and is the distance from the center point to the vertices DIRECT algorithm contd..

Page 12 of 14 Intelligent Powertrain Design 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. DIRECT algorithm contd..

Page 13 of 14 Intelligent Powertrain Design 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. DIRECT algorithm contd..

Page 14 of 14 Intelligent Powertrain Design 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 Iterate. Report the results of this iteration, and then go to Step Terminate. The optimization is complete. Report the and and stop. DIRECT algorithm contd..

Page 15 of 14 Intelligent Powertrain Design 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 FEHwyFECity uelEconomyComposite F _ 45.0 _   City (FTP-75)Highway (HWFET) Problem Statement

Page 16 of 14 Intelligent Powertrain Design Objective: Maximize the composite fuel economy Constraints: mph : <= 11.2 s 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% Problem Statement contd..

Page 17 of 14 Intelligent Powertrain Design 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 VariableDescriptionLower boundUpper 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 1135 ess_cap_scale Battery Max. Ah capacity scale 0.333(8.3 Ah)1(25 Ah)

Page 18 of 14 Intelligent Powertrain Design Problem Statement contd.. Figure showing the parallel HEV Configuration before optimization in ADVSIRO 2.0

Page 19 of 14 Intelligent Powertrain Design DIRECT Statistics Iterations21 Simulation time~24 hrs Function Eval’s539 VariableDescription 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 (~15) ess_cap_scale Battery Max. Ah capacity scale 1(25 Ah)1.7496(43.7Ah) Results Initial and the final values of the design variables Simulation statistics

Page 20 of 14 Intelligent Powertrain Design Fuel Economy Initial ValueFinal Value 27.6 mpg37.9 mpg Results contd… Initial and the final objective value Initial fuel economy Final fuel economy

Page 21 of 14 Intelligent Powertrain Design Table showing performance before and after optimization ConstraintConstraint value Performance before optimization Performance after optimization mph<=11.2 s9 s8.87 s mph<= 4.4 s4.3 s4.39 s mph<= 20 s18.3 s17.84 s Gradeability at 55mph >= 6.5 %6.09%6.58% Difference in required and achieved speeds 2 mphn/a0mph Difference between initial and final SOC 0.5 %0.4%0.34 % Results contd…

Page 22 of 14 Intelligent Powertrain Design Comparison of the emissions before and after optimization Mass of the vehicle pre-optimizationpost-optimization 1545 kg1338 Initial and the final mass of the vehicle Results contd… Emissions before optimization Emissions after optimization CityHwy/City NOxCityHwy/City Nox HC CO NOx PM00

Page 23 of 14 Intelligent Powertrain Design Results contd… Figure showing the design variables and the objective function at each iteration

Page 24 of 14 Intelligent Powertrain Design 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 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 D.Jones,” DIRECT Global Optimization Algorithm,” Encyclopedia of Optimization, kluwer Academic Publishers, Report on “Optimal Design of Non-Conventional Vehicles” The University of Michigan, Dept. of Mechanical Engg., January 19, 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, Bjorkman, Mattias and Holmstrom, Kenneth, “Global optimization using the DIRECT Algorithm in MATLAB,” Advanced Modeling and optimization, Vol. 1, No. 2, 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 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.