A Real-Time Detection Algorithm for Sawtooth Crashes in Nuclear Fusion Plasmas Imke D
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A real-time detection algorithm for sawtooth crashes in nuclear fusion plasmas Imke D. Maassen van den Brink (0502234) DCT 2009.076 Master’s thesis Coaches: ir. G. Witvoet1 dr. ir. N.J. Doelman2 Supervisor: prof. dr. ir. M. Steinbuch1 Committee: Supervisor and coaches (see above) prof. dr. N. Lopes Cardozo1;3 dr. E. Westerhof3 1Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology Group 2TNO Science and Industry Department of Instrument Modeling and Control Delft, The Netherlands 3FOM Institute for Plasma Physics Rijnhuizen Department of Fusion Physics Nieuwegein, The Netherlands Eindhoven, August, 2009 2 Contents 1 Introduction 5 2 Introduction to nuclear fusion and tokamaks 7 2.1 Nuclear Fusion . 7 2.2 Tokamaks . 9 2.2.1 Magnetic field and flux . 10 2.2.2 Operating principle during a discharge . 11 2.3 Tokamak instabilities . 11 2.3.1 MHD instabilities . 11 2.3.2 Neoclassical tearing modes and magnetic islands . 12 2.3.3 Sawtooth oscillation . 12 2.4 Actuators and sensors . 13 2.4.1 Neutral Beam Injection . 14 2.4.2 Ohmic heating . 14 2.4.3 RF heating . 15 2.4.4 Electron Cyclotron Emission diagnostic . 16 2.5 Research objective . 17 2.5.1 Control of the sawtooth period . 17 2.5.2 TEXTOR . 17 2.5.3 Detection algorithm for sawtooth crashes . 18 3 Analysis methods of the sawtooth oscillation 19 3.1 Time domain analysis . 20 3.1.1 Difference filter . 20 3.1.2 Averaging filter + difference filter . 20 3.1.3 Standard deviation + difference filter . 21 3.2 Fourier transforms . 21 3.2.1 Fourier Series . 22 3.2.2 Continuous Fourier Transform . 24 3.2.3 Discrete-time Fourier Transform . 25 3.2.4 Discrete Fourier Transform . 26 3.2.5 Short-time Fourier Transform . 27 3.3 Wavelet analysis . 27 3.3.1 Continuous Wavelet Transform . 28 3.3.2 Discrete Wavelet Transform . 29 3.4 Cohen’s class of time-frequency distributions . 33 3.4.1 The energy distribution function . 33 3.4.2 Mathematical definition . 34 3.4.3 Discrete distributions of Cohen’s class . 34 3.4.4 Wigner-Ville Distribution . 36 3.4.5 Choi-Williams Distribution . 36 3.5 Application of the analysis methods on the sawtooth oscillation . 37 3 4 CONTENTS 3.5.1 Representation . 37 3.5.2 Clarity . 38 3.5.3 Computational complexity . 41 3.5.4 Concluding remarks . 44 4 Time domain sawtooth recognition algorithm 45 4.1 Noise filtering . 46 4.1.1 Frequency analysis of the sawtooth signal . 46 4.1.2 Filter design . 49 4.1.3 3rd order Butterworth filter . 55 4.1.4 Implementation structures . 55 4.2 Data processing method . 57 4.2.1 Statistical dispersion . 57 4.2.2 1st order difference filter . 60 4.3 Normalization . 62 4.3.1 Buffering the signals . 63 4.3.2 Noise level determination . 63 4.3.3 Normalizing the signal . 64 4.4 Determination crash times . 64 4.4.1 Combining channels . 64 4.4.2 Comparing with thresholds . 65 4.4.3 Hold subsequent exceeding times . 65 4.4.4 Determination exact crash-times . 66 4.5 Period . 66 4.6 Inversion radius . 67 4.7 Algorithm analysis . 68 4.7.1 Computational complexity . 68 4.7.2 Clarity . 70 5 Results 71 5.1 Operating regime . 71 5.2 Shot 107915 . 71 5.3 Shot 110051 . 74 5.4 Shot 110178 . 76 5.5 Shot 110185 . 78 5.6 Concluding remarks . 78 6 Conclusion 79 A Appendix I 81 A.1 Euler’s formula . 81 A.2 Interpretation negative frequencies . 81 A.3 Power spectrum . 82 A.4 Aliasing . 82 A.5 Computational complexities . 83 A.5.1 Time domain processing methods . 84 A.5.2 FFT algorithms . 84 A.5.3 STFT algorithms . 84 A.5.4 DWT algorithms . 85 A.5.5 Analytical signal computation methods . 85 CONTENTS 5 B Appendix II 87 B.1 algorithm . 87 B.2 filt............................................... 89 B.3 normal . 90 B.4 crashdet . 90 B.5 merg . 91 B.6 crashtime . 92 B.7 invrad . 93 B.8 periode . 94 References . 95 6 CONTENTS Abstract A new promising energy resource for the future is energy based on nuclear fusion in a tokamak reactor. Nuclear fusion in tokamaks suffer from a number of plasma instabilities, which significantly influence the efficiency of the nuclear reactions. One instability that occurs close to the center of the plasma is the sawtooth instability, named after the sawtooth shaped profile of the temperature and density at a fixed location in the plasma. The sawtooth instability involves both desired and undesired effects on the plasma and, therefore, a controller should be designed to control the sawtooth period real-time with an optimal trade-off be- tween these effects. The input of the controller is the present sawtooth period. Therefore, a reliable algorithm should be designed to determine accurately the sawtooth period for signals with a wide range of sawtooth periods, crash sizes and sawtooth shapes. Inaccurate determination of.