A stochastic design optimization methodology to reduce emission spread in combustion engines
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ORIGINAL PAPER
A stochastic design optimization methodology to reduce emission spread in combustion engines Kadir Mourat1,3 · Carola Eckstein2 · Thomas Koch3 Received: 11 August 2020 / Accepted: 6 November 2020 © The Author(s) 2020
Abstract This paper introduces a method for efficiently solving stochastic optimization problems in the field of engine calibration. The main objective is to make more conscious decisions during the base engine calibration process by considering the system uncertainty due to component tolerances and thus enabling more robust design, low emissions, and avoiding expensive recalibration steps that generate costs and possibly postpone the start of production. The main idea behind the approach is to optimize the design parameters of the engine control unit (ECU) that are subject to uncertainty by considering the resulting output uncertainty. The premise is that a model of the system under study exists, which can be evaluated cheaply, and the system tolerance is known. Furthermore, it is essential that the stochastic optimization problem can be formulated such that the objective function and the constraint functions can be expressed using proper metrics such as the value at risk (VaR). The main idea is to derive analytically closed formulations for the VaR, which are cheap to evaluate and thus reduce the computational effort of evaluating the objective and constraints. The VaR is therefore learned as a function of the input parameters of the initial model using a supervised learning algorithm. For this work, we employ Gaussian process regression models. To illustrate the benefits of the approach, it is applied to a representative engine calibration problem. The results show a significant improvement in emissions compared to the deterministic setting, where the optimization problem is constructed using safety coefficients. We also show that the computation time is comparable to the deterministic setting and is orders of magnitude less than solving the problem using the Monte-Carlo or quasi-Monte-Carlo method. Keywords Diesel engine · Emission modeling · Design of experiments · Experimental · Gaussian process · Stochastic modeling · Stochastic optimization · Engine calibration · Monte Carlo Abbreviations ARD Automatic relevance determination CAD a. TDCf Crank angle degree after top dead center firing DoE Design of experiments ECU Engine control unit EGR Exhaust gas recirculation GP Gaussian process ICE Internal combustion engine IQ Injected quantity LOOCV Leave-one-out cross-validation * Kadir Mourat [email protected] 1
Oberrahserstraße 53, 41748 Viersen, Germany
2
Robert Bosch GmbH, Powertrain Solutions, Stuttgart‑Feuerbach, Stuttgart, Germany
3
Institut für Kolbenmaschinen (IFKM), Karlsruhe Institute of Technology, Karlsruhe, Germany
MI Main injection MINLP mixed-integer nonlinear programming MC Monte Carlo NOx Nitrogen oxides NRMSE Normalized root-mean-square error OP Operation point PI Pilot injection PoI Post injection QMC Quasi-Monte Carlo R2 Co
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