# RZ2010 Numerical Algorithms

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• Numerical algorithms for power system protectionProf. dr. sc. Ante Marui, doc. dr. sc. Juraj HavelkaUniversity of ZagrebFaculty of Electrical Engineering and Computingante.marusic@fer.hr, juraj.havelka@fer.hr

2010/2011.

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

• Lecture partsFirst partTypes of signalsSampling theorySampling and A/D circuitsNumerical methods: Interpolation formulas numerical integration and differentiation, curve fitting, Fourier analysis and digital filtering

• Lecture partsSecond partSinus wave based algorithmsFourier based algorithmsLeast squares based algorithmsDifferential equation based algorithms

• Lecture partsThird part: Real time algorithm testing50 Hz SignalSimulated short circuitReal short circuit

• Signal classification hierarchyDigital signals1-0 (On Off or TTL) signalPulse train (counters, timers)Analog signalsDC signal (slow)Signal in time domain (fast)Signal in frequency domain

• Basic elements of digital protectionAD converter resolutionNyquists theoremAnalog filtersTransducersSample and hold circuit

• 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(2t), , x(kt),]

• AD converter resolutionEvery sample of analog signal is converted in to digital value with final number of bitsConversion is preformed in AD converter

• 3 bit resolution; 23=8 combinations, which means 8 discrete divisions that analog signal can be represented with

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

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

• Analog filtering

• Transducers and surge protection circuitsReduce voltages and currents (10 V and 20 to 40 mA) to suit hardware requirementsProtect hardware from overvoltagesSignal distortion is the problem (current transducers saturation)

• Sample and Hold circuit

• Basic components of digital relay

• Numerical differentiationDerivatives in point k is

• Numerical integrationLagrange interpolation formula

• Curve fittingLinear fit:

Exponential fit:

General polynomial fit:

General linear fit:

Levenberg-Marquardt fit:

• Least square method

• Fourier analysis

Fourier series Fourier transform

• Discrete Fourier transform DFTSamples of signals from AD: time domainNo need for curve fittingUse DFT: frequency domain

• Smoothing Windows

• Smoothing Windowsn=0, 1, , N-1

• Digital filtersInput signal is discreteThey are software programmableThey are stable and predictableThey do not drift with temperature or humidity and do not require precision componentsThey have superior performance to cost ratioThey do not age

• Digital filters

• Signal generator

• Control loopt=1/fs

text

text

ALGORITHM

ALGORITHM

Measure

Calculate

Measure

Calculate

Measure

• Sine wave based algorithmsWaveform is assumed to be sinusoidalThey predict amplitude at every momentThey can be used for impedance calculationSix are presented:Sample and first derivative with two pointsSample and first derivative with three pointsFirst and second derivativeTwo sample techniqueThree sample techniqueR i X calculation with three sample technique

• Sample and first derivative with two points

• Sample and first derivative with two points

• Sample and first derivative with three points

• Sample and first derivative with three points

• First and second derivative

• First and second derivative

• Two sample technique

• R i X calculation with three sample technique

• R i X calculation with three sample technique

• Fourier algorithmsWaveform does not have to be pure sineThe basic assumption is that the waveform is periodicThe principle of work is moving frameMoving frame is constant in size which means that it always contains the same number of points

• Fourier series with whole periodIf 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.

• Fourier series with whole period

• Fourier series with whole period third harmonicn=3

• Fourier series with whole period third harmonic

• Fourier series with half period

• FFT algorithm

• Least squares based algorithmsAll components of measured waveform must be predicted in mathematical model.After curve fitting data about amplitude, harmonics, angle, etc. are obtainedDownside is large number of calculationsThey are complexFour of them are presented:Algorithm with general polynomial fitLSQ 1, 3 multivariable algorithmLSQ 1, 3, 5 multivariable algorithmLSQ 1, 3, 5 ,7 multivariable algorithm

• General polynomial fit algorithm

• General polynomial fit algorithm

• LSQ 1, 3 multivariable algorithm

• LSQ 1, 3 multivariable algorithm

• LSQ 1, 3, 5 multivariable algorithm

• LSQ 1, 3, 5 multivariable algorithm

• LSQ 1, 3, 5, 7 multivariable algorithm

• LSQ 1, 3, 5, 7 multivariable algorithm

• Differential equation based algorithmThere is no need to assume that the waveform is sineThe 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 lineThree algorithms are presented:Integration algorithmThird harmonic filtration algorithmDifferential algorithm

• Integration algorithm

• Integration algorithm

• Third harmonic filtration algorithmFor fs=600 Hz :

• Third harmonic filtration algorithm

• Differential algorithm

• Differential algorithm

• Differential algorithm

• Algorithms and Real-Time operationSystem is operating in real-time if it can guarantee fulfillment of various tasks in specific timeOS in real-timeHardware and software in real timeControl loop time

• Algorithms and Real-Time operation

• Power system signal

• Sine wave based algorithms

• Fourier based algorithms

• Least squares based algorithms

• Differential equation based algorithms

• Short circuit

text

P

380 V

380 V

Z=R+jX

N1:N2=2

SC

• Sinus wave based algorithms

• Fourier based algorithms

• Least squares based algorithms

• Differential equation based algorithms

***************************************************************************