The Eects of Harmonics in Power Systems and Methods to Reduce or Eliminate Them

Transcription

1 The Eects of Harmonics in Power Systems and Methods to Reduce or Eliminate Them Wilbur N. Dale, Ph.D., P.E. March 15, 2012

2 Overview Introduction Analysis of Harmonics Sources of Harmonics Tolerating Harmonics Eliminating Harmonics 2

3 Introduction Fourier series properties Harmonic relationships Fundamental Theorem of Engineering 3

4 Fourier Series Properties v(t) = i(t) = 2Vk cos (kωt + θ k ) 2Ik cos (kωt + φ k ) Typically, we do not see a DC term. We see only odd harmonics (half-wave symmetry). 4

5 Fourier Series Properties V RMS = I RMS = P = Vk 2 I 2 k V k I k cos (θ k φ k ) 5

6 Harmonic relationships Theorem 1 The Fourier coecients for a piecewise continuous signal decrease at the rate of O ( 1 k ). Theorem 2 The Fourier coecients for a continuous signal decrease at the rate of at least O ( 1 k ). Theorem 3 Narrow current pulses are rich in harmonics. 6

7 Fundamental Theorem of Engineering Theorem 4 Follow the money. Corollary 1 Revenue is good. Corollary 2 Costs are bad. 7

8 Analysis of Harmonics + 2VRMS cos ωt Non-linear load 8

9 Analysis of Harmonics + 2VRMS cos ωt Non-linear load 9

10 Analysis of Harmonics p(t) = v(t)i(t) = 2V 1 cos(ωt) 2Ik cos(kωt + φ k ) = V 1 I k [cos((k + 1)ωt + φ k ) + cos((k 1)ωt + φ k )] = V 1 I 1 cos(φ 1 ) + V 1 I 2 cos(ωt + φ 2 ) + V 1 I k 1 cos(kωt + φ k 1 ) + k=2 V 1 I k+1 cos(kωt + φ k+1 ) = T(t) ω k=2 10

11 Analysis of Harmonics Average P = V 1 I 1 cos(φ 1 ). Harmonic currents do not transfer power from the generator to the load. Harmonic currents do generate losses in the distribution system. Harmonic currents cause torsional vibrations in the generator shaft. There is a danger of fatigue fracture. 11

12 Analysis of Harmonics In three phase systems, a balanced non-linear load will have generator vibrations only at the triplen frequencies. If the balanced load also has odd half-wave symmetry, the generator vibration will be only at the sextuplen frequencies. 12

13 Analysis of Harmonics + 2VRMS cos ωt Non-linear load Linear load 13

14 Analysis of Harmonics Transmission line + 2VRMS cos ωt Linear load Fundamental current Harmonic currents 14

15 Analysis of Harmonics Transmission line Linear load Harmonic currents 15

16 Analysis of Harmonics Harmonic currents cause losses throughout the network. Harmonic currents cause larger RMS voltage drops in the lines. 16

17 For inductors: Analysis of Harmonics V RMS = = Vk 2 = = ωl ωl I 2 k (kωl)2 = I 2 k k2 ( O ( 1 k I 2 k X2 L I 2 k k2 ω 2 L 2 )) 2 k 2 17

18 Analysis of Harmonics For 3-phase neutral currents: I A + I B + I C = 2I RMS cos (kωt + φ k ) + ( ( 2I RMS cos kω t T ) ) + φ k 3 + ( ( 2I RMS cos kω t + T ) ) + φ k 3 = [ ( 2πk 2I RMS cos (kωt + φ k ) cos 3 )] = 3 2I RMS cos (kωt + φ k ) if k mod 3 = 0 = 0 otherwise 18

19 Sources of Harmonics + 2VRMS cos ωt 19

20 Sources of Harmonics + 2VRMS cos ωt 20

21 Previously: Can we reduce L? Tolerating Harmonics V RMS = ωl Simple case of two parallel wires. L = µ 0 4π ( 2 ln ( ) d δ I 2 k k2 + 2 ln where d is the center to center distance between the wires and δ is the diameter of the wires. ) 21

22 Tolerating Harmonics Smallest realizable inductance occurs with d = δ. L min = µ 0 4π (2 ln 2 + 1) L min is not aected by wire size. Use several smaller cables and connect in parallel. 22

23 Tolerating Harmonics IEEE Recommended Practice for Applying Low Voltage Circ u i t B reakers Used in Industrial and C o m m e rcial Power S y s t e m s Published by the Institute of Electrical and Electronics Engineers, Inc. IEEE Std (Revision of IEEE Std ) Authorized licensed use limited to: Wilbur Dale. Downloaded on June 30,2010 at 04:20:08 UTC from IEEE Xplore. Restrictions apply. 23

24 Tolerating Harmonics Circuit breakers vs. fuses: I 2 t. Capacitor banks are a low impedance path. Resistance welding machines. Modern microprocessor controlled breakers. 24

25 Tolerating Harmonics IEEE Recommended Practice for Powering and G rounding E l e c t ronic E q u i p m e n t Published by the Institute of Electrical and Electronics Engineers, Inc. IEEE Std (Revision of IEEE Std ) Authorized licensed use limited to: Wilbur Dale. Downloaded on June 30,2010 at 04:09:21 UTC from IEEE Xplore. Restrictions apply. 25

26 Tolerating Harmonics K = k max (I k k) 2 15 < k max < 34 I k is in per-unit of transformer ratings. Non-linear load transformers with larger neutral conductors and temperature tolerance. 26

27 Eliminating Harmonics Delta-Wye eliminates triplen harmonics by absorption. Wye-Wye zigzag eliminates triplen harmonics and the 5th and 7th. 27

28 Change the load. Eliminating Harmonics Power factor correction power supplies are switching power supplies with near unit power factor and low harmonic content. In addition, the harmonics fall o at O ( 1 k 2 ). 28