Analysis of chaos in an electric network using Lorenz system

1
Analysis of chaos in an electric network using
Lorenz system

• presented by
• Sandeep Arora
• Graduate Student (EE)
• Arizona State University

2
Contents

• Chaos in electric networks
• Lorenz system
• Passive filters
• Tests performed
• Results
• Conclusions

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Chaos in electric networks

• Main reason for chaos in electric networks is
  highly varying and unpredictable loads. Typical
  examples of such erratic loads
• Arc loads ( arc furnace/welding, plasma torch)
• Power electronic loads
• Steel rolling mill

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Chaos in electric networks

• These loads are problematic because they cause
• Voltage flicker
• Unbalance of power in the circuit
• Stress in the circuit

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Outline of this project

• Load voltage analysis of an R-L circuit using a
  clean current waveform at 60 Hz
• Frequency and magnitude scaling of Lorenz system
  components x, y, z
• Repetition of the above analysis with a chaotic
  current wave using Lorenz x
• An attempt to attenuate chaos using a shunt
  capacitance and a tuned R-L-C filter

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R-L circuit used in test

• Parameter values used
• R 0.2(ohms) L 0.001(H) RL 5(ohms)
• Vs 5000cos(377t) Is 1000cos(377t)

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R-L circuit analysis

• Writing differential equations
• Vs(t) - Vload(t) L(dI/dt) RI(t)
• I(t) Is(t) Iload(t) (Is current source)
• Vload IloadRL. (Voltage between 1-2)
• For clean wave analysis
• Is 1000cos(377t)
• For chaotic wave analysis
• Is m(Lorenz(x))

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Analysis with clean wave

• Load voltage vs time Load current vs time

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FFT of load voltage using clean wave

• N 2517
• T 0.25 sec
• DT 0.25/2517
• Max at points(p)
• 16 and 2503
• DF 1/(NDT)
• fmax (p-1)DF
• 60 Hz

• FFT

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Lorenz System

• Lorenz system is a chaotic attractor, defined by
  a set of three ordinary differential equations
• (dx/dt) sigma(-x y)
• (dy/dt) -xz gammax -y
• (dz/dt) xy - bz.
• The values chosen for these parameters
• sigma 10 b 8/3 gamma 28

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Lorenz System(contd.)

• Graphical display of Lorenz system

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FFT of Lorenz System

• N 297
• T 5 sec
• DT 5/297
• Max at point(p)
• 7
• DF 1/(NDT)
• fmax (p-1)DF
• 3 Hz

• FFT

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Modified Lorenz System

• Frequency scaling factor k 30
• Magnitude scaling factor m 250
• Modified differential equations
• (dx/dt) ksigma(-x y)
• (dy/dt) k(-xz gammax -y)
• (dz/dt) k(xy - bz).
• Current source modification
• Is m(Lorenz(x))

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Lorenz System(contd.)

• Graphical display of Lorenz system (k 30)

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Lorenz System(3-D plot)

• A plot between x, y and z for t5 s

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FFT of modified Lorenz System

• N 3553
• T 2 sec
• DT 2/3553
• Max at point (p)
• 6 or 3549
• DF 1/(NDT)
• fmax (p-1)DF
• 2.5 Hz

• Current FFT (k30)

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Chaos in the circuit

• Load voltage(T .25s) k 30, m 250

• Load Voltage(T0.1s) k 1250, m 250

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Passive Filters

• Readily designed for harmonics
• Usually based on the tuned resonance principal
  (providing low impedance path)
• Types of filters
• Shunt capacitor
• R-L-C tuned filter
• Band pass filter
• High pass filter

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Load voltage waveform using shunt capacitance

• Parameter values used
• R 0.2(ohms) L 0.001(H) RL 5(ohms)
• C 0.025(F) Vs 5000cos(377t)

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Circuit analysis

• Writing differential equations
• Vs(t) - Vload(t) L(dI/dt) RI(t)
• I(t) Is(t) Iload(t) Ic(t)
• (Is current source, Ic Current thru C)
• Vload IloadRL (Voltage between 1-2)
• Ic(t) C(dVload/dt)
• Is(t) m(Lorenz(x))
• Differential equations for modified LS

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Load voltage waveform using tuned RLC filter

• Rf, Lf, Cf are resistance, inductance and
  capacitance of the RLC tuned filter

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Parameter selection

• Based on dominant chaotic frequency and Quality
  Factor (QF) of RLC filter
• QF OmegaLf/Rf should be kept low for a
  broadband filter/ (here QF 2, Rf 0.5)
• Fd Dominant frequency in FFT of chaotic wave
• Fr (Res. Freq.) Fd 1/(2pi(LfCf)1/2)
• Lf 1/(2piFd), Cf 1/(Lf(2piFd)2)

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Circuit analysis

• Writing differential equations
• Vs(t) - Vload(t) L(dI/dt) RI(t)
• I(t) Is(t) Iload(t) If(t)
• (Is current source, If Filter current)
• Vload IloadRL (Voltage between 1-2)
• Vload If(t)Rf Lf(dIf/dt) Vc
• dVc/dt (1/Cf)If
• Is(t) m(Lorenz(x))

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An outline of the tests performed

• Steps followed in tests performed
• Voltage analysis of clean current wave
• Voltage analysis of chaotic current wave
• Detection of dominant chaotic frequency
• Designing an appropriate filter to remove chaos
• Comparison of filtered output with original clean
  wave

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Test 1

• Load voltage Vs time
• (t 0.5 sec)

• FFT of Is(chaotic)
• K30 m250 fm2Hz

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Load voltage Vs time (using filters)

• Using shunt capacitor
• C 100 micro Farads

• Using a tuned filter
• Res F 2 Hz, QF 2

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Comparison of FFTs

• FFT of load voltage without using filter

• FFT of filtered output
• load voltage

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Test 2

• Load voltage Vs time
• (t 0.1 sec)

• FFT of Is (k1250 m250 fm290Hz)

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Load voltage Vs time (using filters)

• Using shunt capacitor
• C 0.025 Farads

• Using a tuned filter,
• Res F290 Hz, QF 2

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Comparison of FFTs

• FFT of load voltage without using filter

• FFT of filtered output
• load voltage

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THD comparisons
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Some more results

• No chaos m250
• k1000, Gamma23

• Transient chaos
• m 2500, Gamma23

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Conclusions

• As the frequencies of the chaotic component
  currents increase, it becomes relatively easier
  to filter chaos
• A shunt capacitor performs better than a tuned
  filter if the chaotic component frequencies are
  high.
• Broadband filters perform better than narrow
  tuned filters for chaotic signals