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Power Quality Harmonics Analysis and Real Measurements Data Part 2

Tham khảo tài liệu 'power quality harmonics analysis and real measurements data part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Electric Power Systems Harmonics - Identification and Measurements 9 I w1 I Note that w wk but w. I 1 I i 3 . N. 1 I i J The first bracket in Equation 19 presents the possible low or high frequency sinusoidal with a combination of exponential terms while the second bracket presents the harmonics whose frequencies wk k 1 . M are greater than 50 60 c s that contaminated the voltage or current waveforms. If these harmonics are identified to a certain degree of accuracy i.e. a large number of harmonics are chosen and then the first bracket presents the error in the voltage or current waveforms. Now assume that these harmonics are identified then the error e t can be written as N e t A1ea1t cosw1t Aie t cos w t Ộ 20 Fig. 1. Actual recorded phase currents. It is clear that this expression represents the general possible low or high frequency dynamic oscillations. This model represents the dynamic oscillations in the system in cases such as the currents of an induction motor when controlled by variable speed drive. As a special case if the sampling constants are equal to zero then the considered wave is just a summation of low frequency components. Without loss of generality and for simplicity it 10 Power Quality Harmonics Analysis and Real Measurements Data can be assumed that only two modes of equation 21 are considered then the error e t can be written as 21 e tj A e 71 cos w1tj A2ealt cos w2t ộ2j 21 Using the well-known trigonometric identity cos w2t ộ2j cos w2t cosộ2 - sinw2t sin 2 then equation 21 can be rewritten as e t j A1ea1t cos w1t ea2t cos w2t j A2cosộ2 e 1 sin w2tj A2sinộ2 22 It is obvious that equation 22 is a nonlinear function of A s rfs and Ộ s. By using the first two terms in the Taylor series expansion Aieơit i 1 2. Equation 22 turns out to be e tj A1 cosw1t tcosw1tj A1Ơ1 j cosw2tj A2 cosr y j tcosw2tj A2Ơ2 cosr y j - sinw2tj A2 sin 2 j tsinw2tj A2Ơ2 sin 2 j where the Taylor series expansion is given by A 1 ƠÍ Making the following substitutions in .