Thermodynamic Analysis Simulation Computational Setup

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3.4.2 Artificial Neural Networks

Featuring a modular network representation for commonly used network architec- tures and a comprehensive set of training functions, the MATLAB Neural Network Toolbox 4.0.5 [66] is used to design and simulate the ANNs in this study. Based on the universal approximation theorem by Hornik et al. [51] and following the successful applications in IC engine combustion modeling e.g. [26][41], a multi- layer feed forward network architecture is chosen. Three different networks are designed to approximate the ROHR combustion characteristics and the specific nitrogen oxide and soot exhaust emissions as a function of seven key operating con- dition parameters. An outline of the configuration of the ANN, as well as the param- eters used is provided in Table 3.2 .

3.4.3 Optimization Algorithms

Given an engineering problem, the definition of an appropriate fitness or objective function, as well as the physical or technical constraints of the system parameter val- ues, are crucial to all optimization algorithms. Constraints Parameters in engineering systems, for example the valve timing, laminar flame speed or global AF-ratio in an IC engine, are subject to physical or technical con- straints. As the present study deals with phenomenological models based on physical and chemical parameters e.g. pressure values, temperatures, velocities, etc., the chosen optimization algorithms need to account for equality and inequality parame- ter constraints. Given these constraints, the variable size of the search dimensions asymmetric search space generally has an impact on the search strategy and accord- ingly the efficiency of the optimization algorithms. Details on the different model parameters and their corresponding size ranges are given in Chapter 4 et sqq. ARCHITECTURE Multi-layer feed forward network TRAINING Levenberg-Marquardt algorithm with back-propagation INITIALIZATION Nguyen-Widrow method ACTIVATION FUNCTION Sigmoid hidden neurons linear output neurons INPUTS c m , BMEP, p Inj , Δ t Inj , ϕ SOI , x EGR , λ global OUTPUTS ROHR Characteristics: ϕ SOC , ϕ 10 , ϕ 50 , ϕ 90 , Q max , ... NO x and soot emissions Tab. 3.2 Artificial Neural Network ANN Characteristics 23 FitnessObjective Functions In order to handle both single nitrogen oxide and soot emission and multiple rate of heat release objective optimization tasks with identical algorithms, a single objec- tive approximation function is used to describe the multi objective pareto optimal- ity 1 [90]. Given that the optimization task in the calibration part of the modelingoptimiza- tion scheme is the search for the “best” set of model parameters, that is the set of model parameters which produces the smallest deviations from the measurements e.g. least square errors, the single objective approximation function is defined as the weighted sum of the individual multiple objectives Table 3.3. To account for the accuracy of the experimental data used in the model calibra- tion, a tolerance value is assigned to each objective output of the model prior to the calculation of the objective function Figure 3.4. Genetic Algorithm The classic genetic algorithm GA [35] notwithstanding, the in-house developed GA uses real value parameter encoding, mutation, as well as a simulated annealing 2 mechanism in addition to the standard crossover mechanism for reproduction Table 3.4. 1. Pareto optimality - a.k.a. indifference curves, best solutions to a multi objective problem that could be achieved without disadvantaging at least one of the objectives ROHR NO X SOOT Tab. 3.3 Objective Functions used for the Model Calibration a b Fig. 3.4 Error Objective Function: a Standard Least Square Error LSE, b LSE Including Tolerance Value 2. Simulated Annealing - probabilistic “neighbourhood” search method, inspired by the annealing technique in metallurgy heating up and controlled cooling of a material f obj c i f obj i i ∑ c 1 Δ 2 ϕ SOC c 2 Δ 2 ϕ 10 c 3 Δ 2 ϕ 50 c 4 Δ 2 ϕ 90 … + + + + = = c 5 Δ 2 m prmx c 6 Δ 2 Q max c 7 Δ 2 ϕ Q ma x c 8 Δ 2 ε Q + + + f obj Δ 2 NO x = f obj Δ 2 m soot = x Obj ective Funct ion f obj Deviat ion Δ x-x x f obj x Obj ective Funct ion f obj Deviat ion Δ x-x x f obj x Obj ective Funct ion f obj Tolerance t x = x ± t Deviation Δ x-x x f obj x Obj ective Funct ion f obj Tolerance t x = x ± t Deviation Δ x-x x f obj