1. Numerical algorithms for power system protection Prof. dr. sc. Ante Marušić, doc. dr. sc. Juraj Havelka University of Zagreb Faculty of Electrical Engineering and Computing ante.marusic@fer.hrante.marusic@fer.hr, juraj.havelka@fer.hrjuraj.havelka@fer.hr 2010/2011.

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

3. 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

4. Lecture parts  Second part  Sinus wave based algorithms  Fourier based algorithms  Least squares based algorithms  Differential equation based algorithms

5. Lecture parts  Third part: Real time algorithm testing  50 Hz Signal  Simulated short circuit  Real short circuit

6. 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

8. Basic elements of digital protection  AD converter resolution  Nyquist’s theorem  Analog filters  Transducers  Sample and hold circuit

9. 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),…]

10. 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

11. 3 bit resolution; 2 3 =8 combinations, which means 8 discrete divisions that analog signal can be represented with

12. Nyquist’s theorem Sampling frequency (how often is AD conversion preformed)

13. Nyquist’s 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

14. Analog filtering

15. 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)

16. Sample and Hold circuit

17. transducer Surge protective circuit LP Filter AMUX Signal conditioning subsystem Sample and Hold circuit AD conv. DMUX Conversion subsystem Digital processing relay subsystem Basic components of digital relay

18. Numerical differentiation Derivatives in point k is

19. Numerical integration Lagrange interpolation formula Trapezoidal formula

20. Curve fitting  Linear fit:  Exponential fit:  General polynomial fit:  General linear fit:  Levenberg-Marquardt fit:

21. Least square method (k=0, 1, 2,..., m)

22. Fourier analysis Fourier series Fourier transform

23. Discrete Fourier transform DFT  Samples of signals from AD: time domain  No need for curve fitting  Use DFT: frequency domain k=0…N-1

24. Smoothing Windows

25. n=0, 1, …, N-1

26. 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

27. Digital filters

28. Signal generator

29. Control loop Δt=1/fs

30. 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

31. Sample and first derivative with two points

33. Sample and first derivative with three points

35. First and second derivative

37. Two sample technique

38. R i X calculation with three sample technique

40. 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

41. Fourier series with whole period j=0 do N 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.

42. Fourier series with whole period

43. Fourier series with whole period – third harmonic n=3 za f s =600 Hz

44. Fourier series with whole period – third harmonic

45. Fourier series with half period

46. FFT algorithm

47. 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

48. General polynomial fit algorithm

50. LSQ 1, 3 multivariable algorithm

52. LSQ 1, 3, 5 multivariable algorithm

54. LSQ 1, 3, 5, 7 multivariable algorithm

56. 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

57. Integration algorithm

59. Third harmonic filtration algorithm Elimination of m and n harmonics: For fs=600 Hz :

60. Third harmonic filtration algorithm

61. Differential algorithm

64. 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

65. Algorithms and Real-Time operation

66. Power system signal

67. Sine wave based algorithms

68. Fourier based algorithms

69. Least squares based algorithms

70. Differential equation based algorithms

71. Short circuit

72. Sinus wave based algorithms

73. Fourier based algorithms

74. Least squares based algorithms

75. Differential equation based algorithms