Numerical algorithms for power system protection

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Numerical algorithms for power system protection

• Prof. dr. sc. Ante Marušic, doc. dr. sc. Juraj
  Havelka
• University of Zagreb
• Faculty of Electrical Engineering and Computing
• ante.marusic_at_fer.hr, juraj.havelka_at_fer.hr
• 2010/2011.

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Introduction

• Quality of digital relays depends on
• Numerical algorithm quality (software)
• Hardware quality
• General digital relay characteristics
  selectivity, stability, satisfactory trip time
  and sensitivity

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Lecture parts

• First part
• Types of signals
• Sampling theory
• Sampling and A/D circuits
• Numerical methods Interpolation formulas
  numerical integration and differentiation, curve
  fitting, Fourier analysis and digital filtering

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Lecture parts

• Second part
• Sinus wave based algorithms
• Fourier based algorithms
• Least squares based algorithms
• Differential equation based algorithms

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Lecture parts

• Third part Real time algorithm testing
• 50 Hz Signal
• Simulated short circuit
• Real short circuit

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Signal classification hierarchy

• Digital signals
• 1-0 (On Off or TTL) signal
• Pulse train (counters, timers)
• Analog signals
• DC signal (slow)
• Signal in time domain (fast)
• Signal in frequency domain

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Basic elements of digital protection

• AD converter resolution
• Nyquists theorem
• Analog filters
• Transducers
• Sample and hold circuit

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• Let us assume that numerical values of some
  function x(t) are given at equally spaced
  intervals every ?t seconds.
• 1/?t is then called sampling frequency.
• Signal can then be represented by discrete set
  of samples
• x(0), x(?t), x(2?t), , x(k?t),

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AD converter resolution

• Every sample of analog signal is converted in to
  digital value with final number of bits
• Conversion is preformed in AD converter

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• 3 bit resolution 238 combinations, which
  means 8 discrete divisions that analog signal can
  be represented with

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Nyquists theorem

• Sampling frequency
• (how often is AD conversion preformed)

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Nyquists theorem

• To avoid signal alias sampling frequency must be
  at least two times higher then maximum frequency
  component in analog signal
• For accurate waveform representation sampling
  frequency should be at least 5 to 10 times higher
  then maximum frequency component in analog signal

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Analog filtering
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Transducers and surge protection circuits

• Reduce voltages and currents (10 V and 20 to 40
  mA) to suit hardware requirements
• Protect hardware from overvoltages
• Signal distortion is the problem (current
  transducers saturation)

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Sample and Hold circuit
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Basic components of digital relay
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Numerical differentiation

• Derivatives in point k is

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Numerical integration
Lagrange interpolation formula
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Curve fitting

• Linear fit
• Exponential fit
• General polynomial fit
• General linear fit
• Levenberg-Marquardt fit

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Least square method
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Fourier analysis

• Fourier series
• Fourier transform

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Discrete Fourier transform DFT

• Samples of signals from AD time domain
• No need for curve fitting
• Use DFT frequency domain

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Smoothing Windows
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Smoothing Windows
n0, 1, , N-1
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Digital filters

• Input signal is discrete
• They are software programmable
• They are stable and predictable
• They do not drift with temperature or humidity
  and do not require precision components
• They have superior performance to cost ratio
• They do not age

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Digital filters
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Signal generator
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Control loop
?t1/fs
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Sine wave based algorithms

• Waveform is assumed to be sinusoidal
• They predict amplitude at every moment
• They can be used for impedance calculation
• Six are presented
• Sample and first derivative with two points
• Sample and first derivative with three points
• First and second derivative
• Two sample technique
• Three sample technique
• R i X calculation with three sample technique

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Sample and first derivative with two points
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Sample and first derivative with two points
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Sample and first derivative with three points
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Sample and first derivative with three points
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First and second derivative
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First and second derivative
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Two sample technique
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R i X calculation with three sample technique
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R i X calculation with three sample technique
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Fourier algorithms

• Waveform does not have to be pure sine
• The basic assumption is that the waveform is
  periodic
• The principle of work is moving frame
• Moving frame is constant in size which means that
  it always contains the same number of points

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Fourier series with whole period
If fs is 600 Hz then in one period of 20 ms there
are 12 samples. Weighting factors are calculated
in advance for fixed samplin frequency.
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Fourier series with whole period
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Fourier series with whole period third harmonic
n3
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Fourier series with whole period third harmonic
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Fourier series with half period
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FFT algorithm
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Least squares based algorithms

• All components of measured waveform must be
  predicted in mathematical model.
• After curve fitting data about amplitude,
  harmonics, angle, etc. are obtained
• Downside is large number of calculations
• They are complex
• Four of them are presented
• Algorithm with general polynomial fit
• LSQ 1, 3 multivariable algorithm
• LSQ 1, 3, 5 multivariable algorithm
• LSQ 1, 3, 5 ,7 multivariable algorithm

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General polynomial fit algorithm
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General polynomial fit algorithm
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LSQ 1, 3 multivariable algorithm
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LSQ 1, 3 multivariable algorithm
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LSQ 1, 3, 5 multivariable algorithm
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LSQ 1, 3, 5 multivariable algorithm
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LSQ 1, 3, 5, 7 multivariable algorithm
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LSQ 1, 3, 5, 7 multivariable algorithm
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Differential equation based algorithm

• There is no need to assume that the waveform is
  sine
• The fundamental approach is based on the fact
  that all protected equipment can be represented
  by differential equations of first or second
  order.
• The methods are described by reference to
  transmission line
• Three algorithms are presented
• Integration algorithm
• Third harmonic filtration algorithm
• Differential algorithm

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Integration algorithm
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Integration algorithm
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Third harmonic filtration algorithm
For fs600 Hz
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Third harmonic filtration algorithm
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Differential algorithm
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Differential algorithm
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Differential algorithm
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Algorithms and Real-Time operation

• System is operating in real-time if it can
  guarantee fulfillment of various tasks in
  specific time
• OS in real-time
• Hardware and software in real time
• Control loop time

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Algorithms and Real-Time operation
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Power system signal
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Sine wave based algorithms
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Fourier based algorithms
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Least squares based algorithms
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Differential equation based algorithms
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Short circuit
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Sinus wave based algorithms
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Fourier based algorithms
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Least squares based algorithms
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Differential equation based algorithms