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Automating the Lee Model Major Components • Simulator code – Verifying outputs – Verifying model equations – Graphical User interface • Auto-tuning the model parameters – Simple pre-processing of trace data – Optimization of fit of current curves – Tuning of genetic algorithm code Simulator Code • Rebuilding the simulation code – Port Excel VBA to C# – Object Oriented Design • 5 models – Corona, Axial, Radial, Radial RS, Radiative, Expanded Axial • Provide for future modifications – Runs in < 1sec – Export results to Excel – Graphical interface • Ease of use to run with new parameters and view results • Advanced graphing • Predefined tuned models Object Oriented Design Lee Model CoronaModel PlasmaFocus Machine Axial Model Constants Radiative Model ReflectedShock Model Radial Model ExpandedColumn Model 1 Base 1 1 MeasuredCurrent Metrics Simulator ModelResults 1 Views (GUI) ZedGraph Library MainForm 1 GAConfigPanel ConfigPanel ConfigIniFile ParametersIniFile Setup ResultsPanel TuningPanel GaphsPanel Fitting AutoFit GA IniFile GAFit Chromosome Graphical output NEW EXISTING Machine Configuration Simulation Results Detailed Graphs Auto-tuning problem • Defining what is a good-fit – Formulating a numerical problem – Coefficient of determination, R2 (inadequate) – Other visual cues of good fit • Finding the model parameters that “fits” – Optimization search algorithm • Local maxima problem • Genetic algorithm Goodness of fit • R2 definition R2 = 1- SSreg / SStot where SSreg = Sum (Imeas- Icomp )2 SStot = Sum ( I - Imean)2 i.e. sum of errors squared normalization factor • Curve features – Peak – Slope – End of radial phase Peak Slope of radial phase End of Radial Phase Locating features on current trace • Peak – easy • End of radial phase 2 1.5 1 0.5 Series1 0 -0.5 0 0.5 1 1.5 2 -0.5 -1 – Take 2nd difference – Take maximum difference as the point -1.5 -2 • Slope – Calculate slope from end of pinch to midradial phase Most linear portion from mid slope to end Fitness Function • Fitness, R2’ R2’ = w1 * R2 + w2 * ME + w3 * PE + w4 * SE w1, w2, w3, w4 are weights, w1 = 1 w3 = peak current / (peak current – pinch current) w4 = 2 * w3 w2 = 1.2 * w4 • ME is the maximum current errors (peaks), ME = (1 - ME1) + (1 - ME2) ME1 = computed peak current – measured peak current measured peak current – measured pinch current ME2 = computed peak time – measured peak time measured peak time – measured pinch time • • PE is similarly calculated. The radial slope error is calculated as follows:SE = 1 - computed peak current – mid-radial current pinch time – mid-radial time Genetic Algorithm Create Population of sets of parameters (genes) Rank the Population using Fitness function • • • • Create population Rank using fitness function Select parents for new population Create new population using Save the best gene (set with best fit) Generate new population using mutation and crossover – Mutation – Crossover • Add fittest genes from last pop Add best gene from previous pop Add a narrorw cluster of genes to new population GA Concepts – Population – Chromosome – Genes – Mutation – Cross-over – Elitism Optimization strategy Pre-process measured data Find the graph features to fit using the second order difference • Preprocess measured current • Local optimization stages – Axial for fitness F( massf, currf) – Radial for fitness F( massfr, currfr) Initial Guess massf, currf, massfr, currfr Fit Axial Phase by maximizing r2 using massf, currf • Global optimization stage massf, currf – Whole model for fitness F( massf, currf, massfr, currfr) massf, currf Fit Radial Phase by maximizing r2 using massfr, currfr massfr , currfr • Repeat above as “stages” Fit overall curve by maximizing r2 using all 4 parameters Tu ni ng R Tu 0 st ni ag ng e R Tu 0 st ni ag ng e R Tu 0 st ni ag ng e R Tu 0 st ni ag ng e R Tu 0 st ni ag ng e R 0 st ag Fi e na ls ta ge Fi na ls ta ge Fi na ls ta ge Fi na ls ta ge Fi na ls ta ge Fi na ls ta ge Fi na ls ta ge Fi na ls ta ge Fi na ls ta ge Tu ni n Tu g R 0 ni n g st R age 0 Fi sta na ge l Fi sta na ge ls F Tu in tage ni al s n Tu g R tage 0 ni n g st R age 0 Fi sta na ge l Fi sta na ge l Fi sta na ge ls ta ge Results 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 massf currf massfr currfr fitness 1.2 1 0.8 0.6 0.4 0.2 0 massf currf massfr currfr fitness