1. Meeting of the Steering Group on Simulation (SGS) Defining the simulation plan in the Kriging meta-model development Thessaloniki, 07 February 2014

2. Need for a simulation plan
Once the simulation models are ready a plan is required in order to cover all the vehicles of a specific segment with all the necessary technology combinations
It requires
Knowing the parameters of the model that characterize a vehicle in terms of CO2 difference between WLTP and NEDC
Knowing the technology packages to involve in the simulation
Define a strategy to combine parameters and technologies to cover the market

3. Parameter selection Expert judgment
Pros
Cheap and fast
Cons
Leave some uncertainties (especially for rather unexplored issues like the WLTP/NEDC correlation)
Difficult to foresee interactions
Global sensitivity analysis of the model
All the inputs are taken into consideration
All their interactions are checked
Computationally expensive (it requires many simulation)

4. Global sensitivity analysis
Family of theories and techniques aimed at defining how “the uncertainty in the model outputs can be apportioned to the different sources of uncertainties in the model inputs”
Important to:
uncover technical errors in the model
identify critical regions in the inputs domain
simplify models
Variance-based methods
Output variance as a proxy of output uncertainty
Variance decomposition formulas as sensitivity analysis tools

5. Variance-based methods (Cukier, 1973; Sobol, 1990; Homma and Saltelli, 2000)

6. First order effect

7. Variance decomposition

8. Sobol sensitivity indices
An efficient (numerical) evaluation of Sobol indices requires the application of techniques based on QMC methods (Saltelli et al 2012)

9. Sensitivity analysis of Cruise
Select a vehicle configuration (e.g. Opel Astra)
Identify all the Cruise parameters that characterize the model
Define ranges of variability for each parameter (in order to consider the variability in the segment)
Design a simulation plan for the evaluation of the sensitivity indices
Perform the simulations
Check for unfeasible combinations
Evaluate the sensitivity indices
Rank the parameters on the basis of their effect
Define a threshold before influential vs. non-influential

10. Definition of representative vehicle segments
Sensitivity Analysis of model outputs
Two validated vehicle configurations modeled with PHEM
Vehicle parameters are then set to change in 60% (on average ±30%) range
From these ranges, more that parameters combinations have been evaluated in order to evaluate first order and total order sensitivity indices

11. Definition of representative vehicle segments
Sensitivity Analysis results
Gasoline
Diesel
Power
Power
Gear-box
Gear-box
Mass
Mass
Can be fixed to whatever value

12. Quasi-Monte Carlo Simulation Plan
Monte Carlo framework using Sobol quasi-random sequences: “Quasi Monte Carlo” methods
If the sequence of random numbers keeps “low-discrepancy” whatever the number of dimensions, the size of the Monte Carlo experiment decreases
Discrepancy: a measure of deviation from uniformity
Prof. I.M.Sobol has devoted big part of his life in studying LDS properties

13. Some properties of Sobol sequences (1)

14. Some properties of Sobol sequences (2)
In integral calculation, error e:
Sobol sequence (for N=2k)
Other LDS
Random numbers

15. Conclusions
Quasi-random experimental design can be used in building the simulation plan
Coverage of the input-space depends on the number of inputs
Global Sensitivity Analysis can be used to reduce the number of inputs in an effective way
The number of simulations to estimate each meta-model is defined by the accuracy level required in the study