1. INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS by Bruce A. Mork Michigan Technological University Presented for EE 5320 Course Michigan Technological University March 27, 2001

2. NONLINEARITIES - EXAMPLES CHEMISTRY AND PHYSICSCHEMISTRY AND PHYSICS –CHEMICAL REACTIONS (KINETICS) –THIN FILM DEPOSITION (LASER DEPOSITION) –TURBULENT FLUID FLOWS –CRYSTALLINE GROWTH MEDICAL & BIOLOGICALMEDICAL & BIOLOGICAL –HEART DISORDERS (ARRHYTHMIA, FIBRILLATION) –BRAIN AND NERVOUS SYSTEM –POPULATION DYNAMICS –EPIDEMIOLOGY FORECASTING (FLU, DISEASES) –FOOD SUPPLY (VS. WEATHER & POPULATION) ECONOMYECONOMY –STOCK MARKET –WORLD ECONOMY

3. NONLINEARITIES - EXAMPLES ENGINEERINGENGINEERING –STRUCTURES: BUILDINGS, BRIDGES –HIGH VOLTAGE TRANSMISSION LINES - WIND-INDUCED VIBRATIONS - GALLOPING –MAGNETIC CIRCUITS –COMMUNICATIONS SCRAMBLING - CHAOS GENERATOR –POWER SYSTEMS LOAD FLOW - VOLTAGE COLLAPSE –NONLINEAR CONTROLS –SIMPLE PENDULUM - LINEAR FOR SMALL SWINGS, BECOMES NONLINEAR FOR LARGE ANGLE OSCILLATIONS KEY REALIZATIONSKEY REALIZATIONS –NONLINEARITY IS THE RULE, NOT THE EXCEPTION ! –LINEARIZATION OR REDUCED-ORDER MODELING MAY ONLY BE VALID FOR A SMALL RANGE OF SYSTEM OPERATION (I.E. SMALL-SIGNAL RESPONSE) ……… BE CAREFUL !

4. DUFFING’S OSCILLATOR - A SIMPLE NONLINEAR SYSTEM

12. FERRORESONANCE IN WYE-CONNECTED SYSTEMS A B C H1 H2 H3 VAVAVAVA VCVCVCVC VBVBVBVB X1 X2 X3 X0

13. BACKFED VOLTAGE DEPENDS ON CORE CONFIGURATION 3-LEG STACKED CORE TRIPLEX WOUND OR STACKED SHELL FORM 5-LEG STACKED CORE 5-LEG WOUND CORE 4-LEG STACKED CORE

14. NONLINEAR DYNAMICAL SYSTEMS: BASIC CHARACTERISTICS MULTIPLE MODES OF RESPONSE POSSIBLE FOR IDENTICAL SYSTEM PARAMETERS.MULTIPLE MODES OF RESPONSE POSSIBLE FOR IDENTICAL SYSTEM PARAMETERS. STEADY STATE RESPONSES MAY BE OF DIFFERENT PERIOD THAN FORCING FUNCTION, OR NONPERIODIC (CHAOTIC).STEADY STATE RESPONSES MAY BE OF DIFFERENT PERIOD THAN FORCING FUNCTION, OR NONPERIODIC (CHAOTIC). STEADY STATE RESPONSE MAY BE EXTREMELY SENSITIVE TO INITIAL CONDITIONS OR PERTURBATIONS.STEADY STATE RESPONSE MAY BE EXTREMELY SENSITIVE TO INITIAL CONDITIONS OR PERTURBATIONS. BEHAVIORS CANNOT PROPERLY BE PREDICTED BY LINEARIZED OR REDUCED ORDER MODELS.BEHAVIORS CANNOT PROPERLY BE PREDICTED BY LINEARIZED OR REDUCED ORDER MODELS. THEORY MATURED IN LATE 70s, EARLY 80s.THEORY MATURED IN LATE 70s, EARLY 80s. PRACTICAL APPLICATIONS FROM LATE 80s.PRACTICAL APPLICATIONS FROM LATE 80s.

15. EXPERIMENTAL PROCEDURE: CATEGORIZATION OF MODES OF FERRORESONANCE BEHAVIOR FULL SCALE LABORATORY TESTS.FULL SCALE LABORATORY TESTS. 5-LEG WOUND CORE, RATED 75-kVA, WINDINGS: 12,470GY/7200 - 480GY/277 (TYPICAL IN 80% OF U.S. SYSTEMS).5-LEG WOUND CORE, RATED 75-kVA, WINDINGS: 12,470GY/7200 - 480GY/277 (TYPICAL IN 80% OF U.S. SYSTEMS). RATED VOLTAGE APPLIED.RATED VOLTAGE APPLIED. ONE OR TWO PHASES OPEN-CIRCUITED.ONE OR TWO PHASES OPEN-CIRCUITED. CAPACITANCE(S) CONNECTED TO OPEN PHASE(S) TO SIMULATE CABLE.CAPACITANCE(S) CONNECTED TO OPEN PHASE(S) TO SIMULATE CABLE. VOLTAGE WAVEFORMS ON OPEN PHASE(S) RECORDED AS CAPACITANCE IS VARIED.VOLTAGE WAVEFORMS ON OPEN PHASE(S) RECORDED AS CAPACITANCE IS VARIED.

16. VOLTAGE X1-X0 PHASE PLANE DIAGRAM FOR V X1 C = 9  F X2, X3 ENERGIZED X1 OPEN “ PERIOD ONE ”

17. VOLTAGE X1-X0 DFT FOR V X1 C = 9  F X2, X3 ENERGIZED X1 OPEN “ PERIOD ONE ” ONLY ODD HARMONICS

18. VOLTAGE X1-X0 PHASE PLANE DIAGRAM FOR V X1 C = 10  F X2, X3 ENERGIZED X1 OPEN “ PERIOD TWO ”

19. VOLTAGE X1-X0 DFT FOR V X1 C = 10  F X2, X3 ENERGIZED X1 OPEN “ PERIOD TWO ” HARMONICS AT MULTIPLES OF 30 Hz.

20. VOLTAGE X1-X0 PHASE PLANE DIAGRAM FOR V X1 C = 15  F X2, X3 ENERGIZED X1 OPEN “ TRANSITIONAL CHAOS ” CHAOS ” TRAJECTORY DOES NOT REPEAT.

21. VOLTAGE X1-X0 DFT FOR V X1 C = 15  F X2, X3 ENERGIZED X1 OPEN “ TRANSITIONAL CHAOS ” CHAOS ” NOTE: DISTRIBUTED DISTRIBUTED SPECTRUM. SPECTRUM.

22. VOLTAGE X1-X0 PHASE PLANE DIAGRAM FOR V X1 C = 17  F X2, X3 ENERGIZED X1 OPEN “ PERIOD FIVE ”

23. VOLTAGE X1-X0 DFT FOR V X1 C = 17  F X2, X3 ENERGIZED X1 OPEN “ PERIOD FIVE ” HARMONICS AT “ODD ONE-FIFTH” SPACINGS. i.e. 12, 36, 60, 84...

24. VOLTAGE X1-X0 PHASE PLANE DIAGRAM FOR V X1 C = 18  F X2, X3 ENERGIZED X1 OPEN “ TRANSITIONAL CHAOS ” CHAOS ” NOTE: TRAJECTORY TRAJECTORY DOES NOT DOES NOT REPEAT. REPEAT.

25. VOLTAGE X1-X0 DFT FOR V X1 C = 18  F X2, X3 ENERGIZED X1 OPEN “ TRANSITIONAL CHAOS ” CHAOS ” NOTE: DISTRIBUTED DISTRIBUTED SPECTRUM. SPECTRUM.

26. VOLTAGE X1-X0 PHASE PLANE DIAGRAM FOR V X1 C = 25  F X2, X3 ENERGIZED X1 OPEN “ PERIOD THREE ”

27. VOLTAGE X1-X0 DFT FOR V X1 C = 25  F X2, X3 ENERGIZED X1 OPEN “ PERIOD THREE ” HARMONICS AT “ODD ONE-THIRD” SPACINGS. i.e. 20, 60, 100...

28. VOLTAGE X1-X0 POINCARÉ SECTION FOR V X1 C = 40  F X2, X3 ENERGIZED X1 OPEN “ CHAOS ” ONE POINT PER CYCLE SAMPLED FROM PHASE PLANETRAJECTORY.

29. VOLTAGE X1-X0 DFT FOR V X1 C = 40  F X2, X3 ENERGIZED X1 OPEN “ CHAOS ” NOTE: DISTRIBUTED DISTRIBUTED FREQUENCY FREQUENCY SPECTRUM. SPECTRUM.

30. SPONTANEOUSTRANSITION BETWEEN MODES OF PERIOD ONE. C = 20  F X1, X3 ENERGIZED X2 OPEN BLURRED AREAS SHOW MODE TRANSITION.

31. SPONTANEOUSTRANSITION BETWEEN MODES OF PERIOD ONE. C = 14  F X1 ENERGIZED X2 OPEN BLURRED AREAS SHOW MODE TRANSITION.

32. SPONTANEOUSTRANSITION BETWEEN MODES OF PERIOD ONE. C = 14  F X1 ENERGIZED X2 OPEN BLURRED AREAS SHOW MODE TRANSITION.

33. INTERMITTENCY X2 & X3 ENERGIZED, X1 OPEN, C = 45  F

34. GLOBAL PREDICTION OF FERRORESONANCE PREDICTION APPEARS DIFFICULT DUE TO WIDE RANGE OF POSSIBLE BEHAVIORS.PREDICTION APPEARS DIFFICULT DUE TO WIDE RANGE OF POSSIBLE BEHAVIORS. A TYPE OF BIFURCATION DIAGRAM, AS USED TO STUDY NONLINEAR SYSTEMS, IS INTRODUCED FOR THIS PURPOSE.A TYPE OF BIFURCATION DIAGRAM, AS USED TO STUDY NONLINEAR SYSTEMS, IS INTRODUCED FOR THIS PURPOSE. MAGNITUDES OF VOLTAGES FROM SIMULATED POINCARÉ SECTIONS ARE PLOTTED AS THE CAPACITANCE IS SLOWLY VARIED (BOTH UP AND DOWN).MAGNITUDES OF VOLTAGES FROM SIMULATED POINCARÉ SECTIONS ARE PLOTTED AS THE CAPACITANCE IS SLOWLY VARIED (BOTH UP AND DOWN). POINTS ARE SAMPLED ONCE EACH 60-Hz CYCLE.POINTS ARE SAMPLED ONCE EACH 60-Hz CYCLE. AN “ADEQUATE ” MODEL IS REQUIRED.AN “ADEQUATE ” MODEL IS REQUIRED.

35. BIFURCATIONDIAGRAMS: ENERGIZE X2, X3. X1 LEFT OPEN. CAPACITANCE VARIED 0 - 30  F CAPACITANCE VARIED 30 - 0  F MODES:1-2-C-5-C-3-C

36. CONCLUSIONS FERRORESONANT BEHAVIOR IS TYPICAL OF NONLINEAR DYNAMICAL SYSTEMS.FERRORESONANT BEHAVIOR IS TYPICAL OF NONLINEAR DYNAMICAL SYSTEMS. RESPONSES MAY BE PERIODIC OR CHAOTIC.RESPONSES MAY BE PERIODIC OR CHAOTIC. MULTIPLE MODES OF RESPONSE ARE POSSIBLE FOR THE SAME PARAMETERS.MULTIPLE MODES OF RESPONSE ARE POSSIBLE FOR THE SAME PARAMETERS. STEADY STATE RESPONSES CAN BE SENSITIVE TO INITIAL CONDITIONS OR PERTURBATIONS.STEADY STATE RESPONSES CAN BE SENSITIVE TO INITIAL CONDITIONS OR PERTURBATIONS. SPONTANEOUS TRANSITIONS FROM ONE MODE TO ANOTHER ARE POSSIBLE.SPONTANEOUS TRANSITIONS FROM ONE MODE TO ANOTHER ARE POSSIBLE. WHEN SIMULATING, THERE MAY NOT BE “ONE CORRECT” RESPONSE.WHEN SIMULATING, THERE MAY NOT BE “ONE CORRECT” RESPONSE.

37. CONCLUSIONS (CONT’D) BIFURCATIONS OCCUR AS CAPACITANCE IS VARIED UPWARD OR DOWNWARD.BIFURCATIONS OCCUR AS CAPACITANCE IS VARIED UPWARD OR DOWNWARD. PLOTTING Vpeak vs. CAPACITANCE OR OTHER VARIABLES GIVES DISCONTINUOUS OR MULTI-VALUED FUNCTIONS.PLOTTING Vpeak vs. CAPACITANCE OR OTHER VARIABLES GIVES DISCONTINUOUS OR MULTI-VALUED FUNCTIONS. THEREFORE, SUPPOSITION OF TRENDS BASED ON LINEARIZING A LIMITED SET OF DATA IS PARTICULARLY PRONE TO ERROR.THEREFORE, SUPPOSITION OF TRENDS BASED ON LINEARIZING A LIMITED SET OF DATA IS PARTICULARLY PRONE TO ERROR. BIFURCATION DIAGRAMS PROVIDE A ROAD MAP, AVOIDING NEED TO DO SEPARATE SIMULATIONS AT DISCRETE VALUES OF CAPACITANCE AND INITIAL CONDITIONS.BIFURCATION DIAGRAMS PROVIDE A ROAD MAP, AVOIDING NEED TO DO SEPARATE SIMULATIONS AT DISCRETE VALUES OF CAPACITANCE AND INITIAL CONDITIONS.

38. CONCLUSIONS (CONT’D) DFTs ARE USEFUL FOR CATEGORIZING THE DIFFERENT MODES OF FERRORESONANCE.DFTs ARE USEFUL FOR CATEGORIZING THE DIFFERENT MODES OF FERRORESONANCE. PHASE PLANE DIAGRAMS PROVIDE A UNIQUE SIGNATURE OF PERIODIC RESPONSES.PHASE PLANE DIAGRAMS PROVIDE A UNIQUE SIGNATURE OF PERIODIC RESPONSES. PHASE PLANE TRAJECTORIES PROVIDE THE DATA FOR POINCARÉ SECTIONS, FRACTAL DIMENSION, AND LYAPUNOV EXPONENTS.PHASE PLANE TRAJECTORIES PROVIDE THE DATA FOR POINCARÉ SECTIONS, FRACTAL DIMENSION, AND LYAPUNOV EXPONENTS. CATEGORIZATION OF RESPONSES CAN BE MORE CLEARLY DONE USING THE SYNTAX OF NONLINEAR DYNAMICS.CATEGORIZATION OF RESPONSES CAN BE MORE CLEARLY DONE USING THE SYNTAX OF NONLINEAR DYNAMICS.

39. CONTINUING WORK THE AUTHORS ARE CONTINUING THIS WORK UNDER SEPARATE FUNDING.THE AUTHORS ARE CONTINUING THIS WORK UNDER SEPARATE FUNDING. –IMPROVEMENT OF LUMPED PARAMETER TRANSFORMER MODELING. –CONTINUED APPLICATION OF NONLINEAR DYNAMICS AND CHAOS TO LOW-FREQUENCY TRANSIENTS. ACKNOWLEDGEMENTSACKNOWLEDGEMENTS –National Science Foundation RIA Grant –BONNEVILLE POWER ADMINISTRATION. –AREA UTILITIES, NRECA, AND CONSULTANTS.

40. COMMENTS? QUESTIONS?