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Measurement of power system subsynchronous impedances and comparison with computer simulations 1984
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Title | Measurement of power system subsynchronous impedances and comparison with computer simulations |
Creator |
Hughes, Michael Brent |
Publisher | University of British Columbia |
Date Created | 2010-05-26T22:22:08Z |
Date Issued | 2010-05-26T22:22:08Z |
Date | 1984 |
Description | A test method is developed to measure the equivalent (driving point) positive sequence impedance, as a function of frequency, of an operating power system. The technique used is to apply a phase-to-phase fault at the system node of interest and record the transient voltage across and current in the fault. These quantities are then transformed into the frequency domain. The system driving point impedance is then taken as the ratio of the fault voltage to the current at each point in the frequency domain. Field results from a phase-to-phase fault at a central location on the B.C. Hydro 500 k-V network are presented and analysed to determine the system driving point impedance. The measured impedance versus frequency characteristic is compared with a predicted impedance characteristic based on an Electro-Magnetic Transient program study of a detailed model of the major B.C. Hydro transmission and generation. Correlation between the measured and calculated impedance is good, with explanations offered for any localized significant disagreement. The explanations offered are subsequently verified by Ontario Hydro's improvements to the analysis of the test data and additions to the computer model data. |
Subject |
Electric Power Systems |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2010-05-26T22:22:08Z |
DOI | 10.14288/1.0096292 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/25071 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0096292/source |
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MEASUREMENT OF POWER SYSTEM SUBSYNCHRONOUS IMPEDANCES AND COMPARISON WITH COMPUTER SIMULATIONS By MICHAEL BRENT HUGHES B.A.Sc, The University of British Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1984 © Michael Brent Hughes, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of (EufcT&CAi e?Hk<^6cf.g.AA The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date -IS P v 5 - ABSTRACT A test method i s developed to measure the equivalent (driving point) positive sequence impedance, as a function of frequency, of an operating power system. The technique used i s to apply a phase-to-phase f a u l t at the system node of interest and record the transient voltage across and current i n the f a u l t . These quantities are then transformed into the frequency domain. The system dr i v i n g point impedance i s then taken as the r a t i o of the f a u l t voltage to the current at each point i n the frequency domain. Fi e l d results from a phase-to-phase f a u l t at a central location on the B.C. Hydro 500 k-V network are presented and analysed to determine the system dr i v i n g point impedance. The measured impedance versus frequency c h a r a c t e r i s t i c i s compared with a predicted impedance c h a r a c t e r i s t i c based on an E l e c t r o - Magnetic Transient program study of a detailed model of the major B.C. Hydro transmission and generation. Correlation between the measured and calculated impedance i s good, with explanations offered for any l o c a l i z e d s i g n i f i c a n t disagreement. The explanations offered are subsequently v e r i f i e d by Ontario Hydro's improvements to the analysis of the test data and additions to the/ computer model data. i i Table of Contents 1 INTRODUCTION 1 2 MEASUREMENT TECHNIQUE 4 2.1 Theory 4 2.2 Application to Three Phase Network and Z 7 v v pos 2.3 Limitations and P r a c t i c a l Considerations 8 2.4 V a l i d i t y of Assumptions 11 2.5 Effect of Fault Duration 13 3 TEST EXECUTION AND RESULTS 15 3. 1 Test Setup 15 3.2 Recorded Results 17 4 ANALYSIS OF TEST RESULTS 19 4.1 Fourier Transform Calculation 19 4.2 Calculation of Z (jw) 19 pos J 4.3 Reduction of Gibbs O s c i l l a t i o n s 22 4.3.1 Analysis of Extended Signal 23 4.3.2 Weighting Function (Window) 24 5 COMPUTER SIMULATIONS 26 5.1 500 kV System Model and Simulation Results 26 5.2 Extended System Model and Simulation Results 29 6 EXPLANATION OF DISCREPANCIES BETWEEN MEASURED AND SIMULATED 32 7 IMPROVEMENTS TO MEASUREMENT TECHNIQUE 34 7.1 Computer Simulation Including Lower VqJ.tage Transmission 34 7.2 The E f f e c t of Including System Dynamics 34 7.3 Supersynchronous Impedance and Minimum Required Current 38 7.4 Modal (Sequence) Analysis of Test Results 41 8 CONCLUSIONS 43 9 REFERENCES 45 APPENDIX I - Derivation of Z (iw) for a Three Phase System 47 pos J APPENDIX II - Fourier Transform Calculation 49 APPENDIX III - Time Domain Window to Eliminate Gibbs O s c i l l a t i o n s 51 i i i L i s t of Figures 1. B.C. Hydro Major Transmission System 1 2. Impedance Measurement by Discrete Frequency Current Injection 4 3. Impedance Measurement by Noise Injection 5 4. Impedance Measurement by System Disturbance 6 5. Simulated Impedance Measurement Test Results Compared With Impedance Frequency Scan of the Same System Model 11 6. Fourier Transform of a Gated 60 Hz Cosine Waveform 14 7. W i l l i s t o n 500 kV Bus Fault Details 16 8. Measured Voltage Across the Fault and the Fault Current 18 9. Measured Fault Voltage 19 10. Fourier Transform of the Measured Fault Voltage 20 11. Fourier Transform of the Measured Fault Current 21 12. Measured System Equivalent Impedance Versus Frequency 22 13. Measured System Impedance - Reduction of Gibbs O s c i l l a t i o n By Analysis of 2 Second Time Period... 23 14. Frequency Spectrum of Gate Function . 25 15. Measured System Impedance - Reduction of Gibbs O s c i l l a t i o n By Use of Windowing Function 25 16. Computer Model of the B.C. Hydro Major Generation and 500 KV Transmission System 27 17. 500 kV System Model Impedance Frequency Scan and Comparison with the Measured System Impedance 28 18. More Detailed System Model Including Additional Nearby Generation and Some 230 KV Transmission 30 19. System Impedance Frequency Scan of Detailed EMTP Model and Comparison With The Measure System Impedance 31 20. Comparison of Simulated and Measured Voltage and Current for System Impedance Test Fault 31 21. Simplified System Model and It's Impedance 33 - iv - 22. System Impedance From Extensive Positive Sequence System Model 35 23. Change i n Open C i r c u i t Voltage Magnitude Due to the System Impedance Test Fault 36 24. Change i n Open C i r c u i t Voltage Phase Due to the System Impedance Test Fault 36 25. Open C i r c u i t Voltage Corrected for System Dynamic Swing 36 26. Fault Voltage Corrected for System Dynamic Swing 36 27. Measured System Impedance Corrected for System Dynamic Swing Compared With The System Impedance Frequency Scan of the Extensive Positive Sequence System Model 37 28. Measured System Impedance to 200 Hz 39 v ACKNOWLEDGEMENTS The author would l i k e to acknowledge his colleages Messrs. T.G. Martinich and R.W. Leonard as equal partners i n the work presented herein and express his gratitude for their generous consent to i t s use i n this thesis.' The author i s very grateful to the System Planning Department of B r i t i s h Columbia Hydro for i t s encouragement, cooperation and support for this project i n pa r t i c u l a r and for the Masters Degree program i n general. The f l e x i b i l i t y shown was instrumental i n allowing the author's studies to proceed while remaining employed, f u l l - t i m e , with B.C. Hydro. Special thanks are due to Messrs. Y. Mansour, M. Scott, J.H. Sawada, and M.G. Bradwell ( r e t i r e d ) , of B.C. Hydro for their valuable discussion and advice during the preparation of this thesis, and to Mr. Sawada and Dr. A. Morched of Ontario Hydro for permission to include their s i g n i f i c a n t improvements to the i n i t i a l work. The author would l i k e to thank Dr. L.M. Wedepohl as co-reader of this thesis. And f i n a l l y , the author would l i k e to express his sincere appreciation to Dr. H.W. Dommel for his most valued guidance and support as ins t r u c t o r , and M.A.Sc. program and thesis supervisor. - v i - 1 INTRODUCTION The present B.C. Hydro power system i s based almost e n t i r e l y on hydroelectric generation on the Peace River and Columbia River systems. This generation i s connected by a network of 500 kV transmission l i n e s to the lower mainland and Vancouver Island, the main load centers of the province (Figure 1) . In the case of the Peace River generation, series capacitors are used to increase the power transfer c a p a b i l i t i e s of the 500 kV transmission l i n e s . B.C. H Y D R O M A J O R T R A N S M I S S I O N GRID LEGEND: • GENERATING STATION a MAJOR SWITCHING STATION X SERIES CAPACITOR STATION 500 kV TRANSMISSION 230 S 360 kV TRANSMISSION 0 25 50 100 150 km WASHINGTON SELKJ8K-, — f - I jIDAHOJ MONTANA F I G U R E 1 - B . C . H Y D R O M A J O R T R A N S M I S S I O N S Y S T E M U n t i l recently, long range plans included the addition of a large c o a l - f i r e d thermal generating station at Hat Creek, that was to be incorporated into the e x i s t i n g 500 kV transmission system. Operating experience i n the southwestern United States [1] has shown that, under cer t a i n circumstances, there can be unstable e l e c t r i c a l o s c i l l a t i o n s i n the generation and transmission system that can induce destructive mechanical o s c i l l a t i o n s on the shafts of steam powered turbine generators. These o s c i l l a t i o n s can occur i f there i s a correspondence between an e l e c t r i c a l transmission system resonant frequency, due to the series capacitors i n series with the transmission l i n e reactance, and a mechanical torsi o n a l resonant frequency of the various turbine and generator components on a common shaft. This phenomenon i s now referred to as sub-synchronous resonance (SSR) and was responsible for two destructive events at the Mohave Generating Station i n the U.S.A. i n 1970 and 1971. It was therefore very important to assess the p o s s i b i l i t y of SSR occuring with the proposed Hat Creek generator connected to the center of a series compensated 500 kV transmission system. Various methods of analyzing and predicting the occurence of SSR had been developed and are reported i n the l i t e r a t u r e . The method adopted by B.C. Hydro i s explained in reference [2]. One of the key pieces of data required for this analysis was knowledge of the positive sequence equivalent impedance of the e l e c t r i c a l system as a function of frequency, Zp 0 g(jw), at the point of connection of the generating station. In p a r t i c u l a r , the system subsynchronous resonant frequencies (those below the 60 Hz system frequency) and the resistance offered at those frequencies, must be known. This data i s not generally available within power u t i l i t i e s . A detailed Electromagnetic Transients Program (EMTP) [3] model of the 500 kV transmission system was therefore constructed to derive this information. The system impedance frequency spectrum was derived for the EMTP model 2 developed however there was no measured data with which to validate i t . There were no industry standard tests that would measure Z (jw) and a pos l i t e r a t u r e search done i n early 1981 indicated that this had not previously been done. Given that subsynchronous resonance countermeasures would or would not be i n s t a l l e d based on the results of these SSR studies, i t was ess e n t i a l that the impedance data used was accurate. If erroneous impedance data led to a decision not to i n s t a l l countermeasures and they were in fact required, the consequences could be broken turbine generator shafts and a lengthy generating plant outage followed by output curtailments while countermeasures were being r e t r o f i t t e d . If erroneous impedance data led to generator design changes and i n s t a l l a t i o n of countermeasures that were i n fact not required, m i l l i o n s of do l l a r s would have been wasted. It was therefore decided to develop some means of measuring the system impedance as a function of frequency. 3 2 .MEASUREMENT TECHNIQUE 2.1 Theory Two general methods of ..measuring the system impedance were considered. These methods were e s s e n t i a l l y variations of methods which have been used successfully for measuring transfer functions of control systems such as i n exciters and power system s t a b i l i z e r s . The f i r s t was a discrete sinusoid or steady state technique [4], In theory, the impedance i s obtained by sequentially applying positive sequence currents at discret e frequencies into the system and recording the steady state voltages produced as a response to the input. The r a t i o of the magnitudes of voltage to current and the phase angle difference between the two at each frequency determines the d r i v i n g point impedance (Figure 2). The I ( j u s ) FIGURE 2 - IMPEDANCE MEASUREMENT BY DISCRETE FREQUENCY CURRENT INJECTION i n j e c t i o n could be done at the t e r t i a r y windings of a 500/230 kV auto-transformer. For an adequate frequency coverage, a large number of separate measurements would be necessary, thus making the technique time consuming. The idea was rejected because: 4 (1) i t would be d i f f i c u l t to couple a variable-frequency source into an energized high voltage transmission system, (2) power frequency voltages and currents would be so much greater than the injected signals that severe f i l t e r i n g and dynamic range requirements would be imposed on the test instrumentation, and, (3) the system configuration would have to remain constant during the course of the measurements. The second general method considered was the i n j e c t i o n of noise current signals [ 5 ,6 ,7 ,8 ] , with wide spectral content, into the system and the continuous recording of the voltage response along with the current input (Figure 3 ) . By d i g i t i z i n g the recorded waveforms and cal c u l a t i n g eU) (GO HO V Z<>»> FIGURE 3 - IMPEDANCE MEASUREMENT BY NOISE INJECTION t h e i r Fourier transforms, the system impedance, Z(jw), could then be found as before. The i n j e c t i o n point would again be at a transformer t e r t i a r y . In practice, however, the idea suffers from problems (1) and (2) mentioned above. There were no naturally occuring disturbances that could be used as the current input [5,7,8,9] and i t was estimated that for a 1 kW signal input (the maximum attainable with the test equipement on hand) the 5 voltage to be measured would be about 115 db down from the 60 Hz component and therefore extremely d i f f i c u l t to detect. Although the noise signal technique was rejected, a v a r i a t i o n of the idea seemed more promising and was studied. Abrupt system changes, such as applying and clearing system f a u l t s , generate transient voltages and currents having components over a wide range of frequencies including the subsynchronous region. The theory of compensation states that the app l i c a t i o n of a f a u l t could be represented as a change i n the system impedance or as an i n j e c t i o n of the fa u l t current into the unfaulted network (Figure 4). If the l a t e r view i s adopted then the system METHOD OF COfAPENSIVTION -o 1 / V V FIGURE 4 - IMPEDANCE MEASUREMENT BT SYSTEM DISTURBANCE 6 impedance can again be derived as the ra t i o of the Fourier Transforms of the voltage and current transient waveforms. 2.2 Application to Three Phase Network and Zpos In extending this concept to a three phase network and the positive sequence impedance, there are various types of f a u l t to be considered. Some generate positive sequence components; some generate positive and negative sequence; and others generate positive, negative and zero sequence [10]. A single phase-to-ground f a u l t was ruled out because i t produces equal amounts of p o s i t i v e , negative, and zero sequence currents. These currents e s s e n t i a l l y flow i n the series connection of the p o s i t i v e , negative and zero sequence equivalent networks. It would not be possible to deduce d i r e c t l y the positive sequence network from this type of f a u l t . A simultaneous three-phase f a u l t , i n theory, i s id e a l i n that only the positive sequence network and positive sequence current i s involved - no negative or zero sequence i s present. However, i n practice the three poles of the c i r c u i t breaker applying and clearing such a f a u l t w i l l not do so simultaneously. This i s due to mechanical constraints producing pole scatter and the fact that each pole w i l l interrupt at a current zero. Zero sequence currents would therefore be produced, making this type of f a u l t also unacceptable. The p o s s i b i l i t y of using a phase-to-phase f a u l t to stimulate subsynchronous currents was examined. Phase-to-phase f a u l t current consists of p r a c t i c a l l y equal amounts of positive and negative sequence currents. Appendix I shows that i f a balanced system i s assumed, the steady-state phasor relationship between the voltage across and the current i n a phase A to B f a u l t , at system frequency, i s : 7 V , - E = (Z + Z ) I (1) ab ab pos neg f If i t i s further assumed that the positive and negative sequence impedances are equal, which is true for a l l passive network components ( l i n e s , reactors, capacitors, transformers), then the above steady-state equation becomes: i V , - E , z = 1 A* ab pos 2 1 A phase-to-phase f a u l t i s therefore suitable as a positive sequence ex c i t a t i o n source and Z (iw) can be calculated as: pos J . F{v (t) - e (t)} z p o s ( j w ) = 2 "fTÎ ETT ( 3 ) 2.3 Limitations and P r a c t i c a l Considerations Application of a bolted phase-to-phase f a u l t i s a severe system disturbance. The f e a s i b i l i t y of using a limited f a u l t to reduce the f a u l t severity was investigated. The only equipment available to l i m i t the f a u l t current was a wye-connected, grounded neutral, three phase l i n e reactor. A limited phase-to-phase f a u l t could be created by applying only two phases of the reactor with the neutral point ungrounded. Unfortunately, this reactor arrangement would overstress the reactor neutral i n s u l a t i o n . A bolted phase-to-phase f a u l t was the only p r a c t i c a l disturbance l e f t available for this test. One l i m i t a t i o n that i s apparent from equation (3) i s that i f the f a u l t current I^(jw) i s zero at any frequency, then nothing can be said about Z (iw) at that frequency. To try and derive Z (iw) would be to pos J pos J divide by zero. 8 Another l i m i t a t i o n i s that open c i r c u i t voltage e , (t) i s not ab measurable once the f a u l t has been applied. e a> 0( t) ^ s e c l u a l t 0 v a i J ( t ) before the f a u l t i s applied. Based on the assumption that the f a u l t w i l l not be severe enough to cause e a ^ ) ( t ) t o change, the pre-fault ea^(t) i s continued (with the same frequency and magnitude) into the f a u l t and post-fault time periods. A further point to note about equation ( 3 ) i s the way the numerator, defined here as the f a u l t voltage V^(jw), i s calculated. The open c i r c u i t voltage can be subtracted from the phase-to-phase voltage i n the time domain V f(jw) = f { v a b ( t ) - e a b (t)} ( 4 ) or i n the frequency domain V f(jw) = F { v a b ( t ) } - F { e a b ( t ) } (5) Both are correct, however, the time-domain formulation w i l l y i e l d a better representation for V^(jw) for the following reasons. F i r s t , only the transient component of the f a u l t voltage contains information about Vf(jw). By doing the subtraction i n the time-domain i t should be readily apparent what portion of the recorded data should be transformed to the frequency domain. The minimum time w i l l be just before the f a u l t i s applied, which corresponds to steady-state conditions and v^(t) i s zero. The maximum time w i l l be when the transient has decayed to zero and steady-state conditions have returned - i . e . v^(t) i s zero again. This leads to the second advantage. Since v^(t) i s zero at the beginning and end of the time i n t e r v a l being transformed to the frequency domain, there i s no truncation of the time signal which could cause spurious ripples i n V f(jw) known as Gibbs o s c i l l a t i o n s [11]. 9 As a check on the impedance measurement technique, the equivalent impedance of an EMTP model of the B.C. Hydro 500 kV system was derived i n two d i f f e r e n t ways and compared. (1) A phase-to-phase f a u l t i n an EMTP power system model was simulated and Z p O S ( J w ) "measured" according to the impedance measurement technique. (2) A l l voltage sources (generators) i n the model were set to zero magnitude and one Amp of positive sequence current was injected into the system where the f a u l t had previously been. The resulting steady-state, positive sequence voltage at the point of current i n j e c t i o n i s a dir e c t measure of the system impedance at the current source frequency. Using the frequency scan feature of the EMTP the frequency of the current source was sequentially incremented and the steady-state solution recalculated to form the calculated system impedance. This process i s referred to hereafter as the system impedance frequency scan. Figure 5 compares the two system impedances derived as described above. The extremely close agreement confirms the v a l i d i t y of the impedance measurement technique. - 10 i — r"î e<?<iCoc.n sc4»o FREQUENCY (HZ) 6b. FIGURE 5 - SIMULATED IMPEDANCE MEASUREMENT TEST RESULTS COMPARED WITH IMPEDANCE FREQUENCY SCAN OF THE SAME STSTEM MODEL 2.4 V a l i d i t y of Assumptions The assumption that the positive sequence and the negative sequence network, d r i v i n g point impedances being equal i s reasonable. The negative sequence impedance i s i d e n t i c a l to the positive sequence for a l l non-rotating power system components. Even for synchronous machines, this i s a good approximation over the frequency range of int e r e s t . For synchronous machines, considering the 10 to 50 Hz frequency region, the po s i t i v e sequence operational inductance approximately corresponds to L"^, as shown by actual test measurements [12,13]. This i s p a r t i c u l a r l y true for salient-pole machines having pole face damper windings, such as for G.M. Shrum and Peace Canyon, where L"^ i s f a i r l y constant over these frequencies. The negative sequence inductance L n e g equals (L"^ + L"^)/2. By design, L"^ i s not greater than 1.3 L"^ which results in a maximum difference between the machine L and L of 15 percent. In the case pos neg of the Peace River system, this difference i s considerably lower. The e f f e c t of possible differences between L and L on the measured pos neg equivalent impedance, at most points i n the system, w i l l be very small due to other system components for which the positive and negative sequence impedances are i d e n t i c a l . Fourier analysis of the f a u l t current and voltage requires that the power system e l e c t r i c network, be l i n e a r and time-invariant [11]. Network l i n e a r i t y i s affected by saturation i n transformers and machines, and time-invariance by protective gap flashovers, l i n e tripouts, and so on. A phase-to-phase f a u l t does not produce overvoltages which.would cause s i g n i f i c a n t transformer saturation and as far as machine saturation i s concerned, the saturated and- unsaturated values of the d i r e c t axis subtransient reactance are almost i d e n t i c a l . Network time-invariance can be controlled by careful pre-test planning and set-up of the power system test conditions. The machine apparent D and Q-axis impedances change a f t e r f a u l t clearance. Provided that the post-fault period included i n the analysis i s not too long, less than 250 ms say, this change i s less than 15 percent - the difference i n going from the subtransient to the transient impedances. - 12 2.5 Effect of Fault Duration The f a u l t duration, and therefore the period of analysis, should be as short as possible. F i r s t l y , during the f a u l t the system generators accelerate causing an increase i n the system frequency. The shorter the f a u l t period the smaller the frequency departure from the pre-fault steady state value and therefore the smaller the error i n assuming the Thevenin or open c i r c u i t voltage to be constant. This i s required so that the time-invariance c r i t e r i o n of the impedance measurement technique i s s a t i s f i e d . Secondly, the subsynchronous content of V^(jw) and I^(jw) should be maximized. These are approximately gated 60 Hz cosine waveforms. The frequency spectrum amplitude of such a gated time function i s given by: s i n [(w - w ) T/2] s i n [(w + w Q) T/2] *{cos(w 0t) x g(t)} = ?w"-w-) + (w-"w-) = | {Sa [(w - wQ) T/2] + Sa [w + wQ) T/2] (6) where g(t) = 1 for 0<t<T = 0 elsewhere where T i s the f a u l t duration, Sa[] i s the sampling function, and W Q i s the 60 Hz fundamental frequency. This function goes to zero at c e r t a i n frequencies or nodes as shown i n Figure 6. These nodes are undesirable because the system impedance around these frequencies may not be determined accurately. The number of nodes occurring i n the subsynchronous region can be minimized by keeping the f a u l t duration small. As indicated i n Figure 6, f a u l t duration should consist of an odd number of half-cycles to maximize the subsynchronous spectrum magnitude. This i s because the two sampling functions centered around +60 and -60 Hz. i n t e r f e r e with each other constructively for T being an odd number of - 13 u vy u 4 .CYCLES. A / W ib. ib"! sSl! ib. iSI bo~. 7 b . sb. itTJ. 7537 FREQUENCY (HZ) FIGURE 6 - FOURIER TRANSFORM OF A GATED 60 HZ COSINE WAVEFORM no. 120. half cycles of wQ and destructively for T being an even number of half cycles. - 14 3 TEST EXECUTION AND RESULTS 3.1 Test Setup At the time the impedance measurement test was being developed plans were underway to apply phase-to-phase f a u l t s at W i l l i s t o n Substation (at Prince George) as part of acceptance tests for the new Kennedy //3 series capacitor stati o n . With minor modifications, the test setup could also be used for a system impedance test. W i l l i s t o n was not the id e a l impedance test location as far as the Hat Creek SSR study was concerned. However, i t could provide f i e l d test data with which to v e r i f y the system computer model and the required test equipment and personnel were already on s i t e . An additional f a u l t to determine the system impedance was therefore scheduled to follow the capacitor test. It was successfully carried out on 1 July, 1981. A phase A-to-phase B f a u l t was applied at W i l l i s t o n 500 kV substation by i s o l a t i n g a section of bus, applying a bolted f a u l t , and then i n i t i a t i n g the f a u l t by closing a c i r c u i t breaker. The f a u l t was cleared using the normal bus protection which, unfortunately, did not allow control over the f a u l t duration. Figure 7 shows the essentials of the bus arrangement. The phase-to-phase voltage appearing across the f a u l t was determined from the phase A and phase B 500 kV capacitive bushing taps on one of the st a t i o n 500/230 kV auto-transformers. The frequency response of the voltage-measuring c i r c u i t was f l a t from about 5 Hz to well beyond 60 Hz. The f a u l t current was measured with an existi n g 500 kV bus protection current transformer (CT). In order to ensure that the core of the CT was not magnetized prior to the test, a variable voltage source was used to desaturate the CT from the secondary side. This was considered advisable - 15 5CT2 5DICB2 (LOCKED OPEN) DISCONNECT BLADE REMOVED BOLTED FAULT 5CBI2 1 FAULT \ * E : E : > 5CTI2 FIGURE 7 - WILLISTQN 500 KV BUS FAULT DETAILS since the bus CT's did not have antiremanence gaps, and other f a u l t tests had taken place e a r l i e r on the same day. A l l signals were fed into the station control building via shielded control cables, and recorded on an instrumentation tape recorder with a bandwidth of 20 KHz and a resolution of 40 dB. The bandwidth was more than adequate for this application, but the rated resolution of 40 dB (1 part i n 100) was r e l a t i v e l y poor. The recorded quantities were d i g i t i z e d with a 12 b i t analog-to-digital converter at a sampling frequency of 6.6 kHz. A t o t a l of 525 milliseconds of each recorded waveform was d i g i t i z e d for computer analysis. - 16 3.2 Recorded Results The staged f a u l t was of three cycles duration. The only system change that occurred was, as expected, the automatic application of 200 MW of braking r e s i s t o r s at G.M. Shrum, four cycles a f t e r f a u l t incidence and t h e i r subsequent removal 44 cycles l a t e r . This violated the network time-invariance c r i t e r i o n required by the measurement technique. Braking r e s i s t o r a p p l i c a t i o n was anticipated and part of the pre-test planning investigated the effects of these r e s i s t o r s on the frequency scan of the system from the W i l l i s t o n 500 kV bus. As the studies indicated that the frequency scans with and without the brake were v i r t u a l l y i d e n t i c a l , i t was decided to allow these r e s i s t o r s to function normally and not compro- mise system s t a b i l i t y during the f i e l d t e s t . There was also a chance that the test proposal would be rejected i f the r i s k of system i n s t a b i l i t y were to be increased by d i s a b l i n g the braking r e s i s t o r s . Figure 8 shows the measured voltage appearing across the f a u l t (breaker contacts) and the f a u l t current. CT saturation can a f f e c t the accuracy of current measurement. The CT used was rated for a time-to-saturation of one cycle for a f u l l y o f f s e t 40 kA rms f a u l t current. Since the measured current was only about 10 kA rms and does not show an appreciable u n i d i r e c t i o n a l o f f s e t , i t i s expected that the CT accurately reproduced the f a u l t current. FIGURE 8 (fl) - MEASURED VOLTAGE ACROSS THE FAULT U J CC <-> o '0' . 000 0T0SO 0'. 100 0'. 150 0'. 200 o'.2SO o'. 300 o'. 350 o'. 400 o'.450 C?. SOO TIME (SEC0N051 FIGURE 8 IB) - MEASURED FAULT CURRENT - 18 4 ANALYSIS OF TEST RESULTS 4.1 Fourier Transform Calculation The Fourier Transform program used was the one written by Dr. H.W. Dommel as a supporting program for the EMTP. It had the advantage of being interfaced with the EMTP output data format and i t uses the c l a s s i c a l Fourier Transform formulation and numerical integration. The c l a s s i c a l Fourier Transform formulation assumes the time domain signal to be zero outside the time period given (which i t would be for this test) and w i l l therefore y i e l d a frequency spectrum down to 0 Hz (DC). Only the non-zero portion of v^(t) and i ^ ( t ) need therefore be analysed. Details of the Fourier Transform c a l c u l a t i o n [14] are presented i n Appendix I I . 4.2 Calculation of Zpos(jw) The function v ^ ( t ) , shown i n Figure 9, was created d i g i t a l l y by r a s PO. K 1, 1 i II M I ! ! i (V OL TS ) I : ! i 1 ! i j « , i i ! It ' V " \ s ~ ' 1 ^ ^ — \̂ •A r\ r\ f VO LT RG E 1 1 i 1 i 1 ! ! ! ! •/ .O OH -5] 11 j i j i i 1 i ! . i i : i i i i i : 1 i ! i 1 i i i i i 1 I '0.000 0.050 0.100 0.150 0.300 0.250 0.300 0.350 0.400 0.450 0.500 TIME (SECONDS) FIGURE 9 - MEASURED FAULT VOLTAGE extrapolating the measured pre-fault open c i r c u i t voltage, e K ( t ) , into the f a u l t and post-fault time period, and subtracting i t from the measured phase-to-phase f a u l t voltage, v K ( t ) . The exptrapolated e , ( t ) did not - 19 p e r f e c t l y cancel the measured phase-to-phase voltage i n the post-fault region. It was expected this was due to the effects of G.M. Shrum braking r e s i s t o r s being applied after the f a u l t , as well as a s l i g h t system swing i n response to the network disturbance. The voltage response due only to the applied f a u l t , and not to the effects previously mentioned, appears to be completely contained within the f i r s t 400 milliseconds. This led to an i n i t i a l choice of 400 ms as the period for the Fourier analysis. The Fourier transforms of the f a u l t voltage and current, v f(-t) and i f ( t ) are shown i n Figures 10 and 11 respectively. The positive sequence o FREQUENCY (HZ) FIGURE 10(B) - FOURIER TRANSFORM OF MEASURED FAULT VOLTAGE (ANGLE) - 20 so. °Io! 20. 25. W. 35. 40! M s ! s o . si? FREQUENCY (HZ) FIGURE 11(fl) - FOURIER TRANSFORM OF MEASURED FAULT CURRENT (MAGNITUDE) iT. 2o~. IT. 30. W. so"! 45"! si"! si7 FREQUENCY (HZ) FIGURE 11 IB) - FOURIER TRANSFORM OF MEASURED FAULT CURRENT (ANGLE) equivalent impedance was then formed i n accordance with equation (3). The r e a l and imaginary parts of this impedance are shown i n Figure 12 for the frequency range of 10 to 60 Hz; the frequency range of interest for subsynchronous resonance studies. Despite the prediction that a f a u l t duration of an even number of half cycles would be less desirable than one of an odd number of half cycles, the three cycle f a u l t produced an adequate current i n j e c t i o n i n terms of frequency content. The f a u l t current i s zero before and after the f a u l t , requiring no - 21 truncation for the Fourier analysis but, for the test results presented here, v a b ( t ) ~ e a b ^ t ^ does n o t return to zero soon a f t e r f a u l t clearance. Amplitude and phase errors [10] result from the truncation of this signal causing the spurious r i p p l e s , Gibbs o s c i l l a t i o n s , observed i n the f a u l t voltage spectrum (Figure 10) and the measured system impedance (Figure 12). Two system resonances, i . e . the frequencies where the equivalent reactance i s zero, are evident. The resonance near 30 Hz corresponds to a small equivalent resistance and w i l l be shown to be a series resonance. S i m i l a r i l y , the resonance near 20 Hz, having a large resistance, w i l l be shown to be a p a r a l l e l resonance. 4.3 Reduction of Gibbs O s c i l l a t i o n s Two methods to reduce the Gibbs o s c i l l a t i o n i n v,.(t) and Z (iw) f pos J were t r i e d . Both were successful i n the sense that they eliminated the Gibbs o s c i l l a t i o n , however neither was desirable i n that each appeared to - 22 roduce some loss of confidence and d e t a i l i n the smoothed Z (iw), p o s V J 4.3.1 Analysis of Extended Signal The f i r s t method was simply to extend the period of time being analysed from 400 msec to 2 seconds. This analysis was done using a spectrum analyser operating on the recordings of the impedance test data. Within 40 cycles of the f a u l t application the braking re s i s t o r s had been removed and within a further 4 cycles the system returned to i t s i n i t i a l condition. There was therefore, no f a u l t voltage truncation at t = 2 seconds. Figure 13 shows that the Gibbs o s c i l l a t i o n has been eliminated from the system impedance as desired. While there was no apparent way to remove the transient voltage signal isT 20. FIGURE 13 30. 3S7 40. FREQUENCY (HZ) MEASURED. SYSTEM IMPEDANCE - REDUCTION OF GIBBS OSCILLATION BY ANALYSIS OF 2 SECOND TIME PERIOD due to a p p l i c a t i o n of the braking r e s i s t o r s , i t was questionable whether adding a further external voltage transient, that of the braking r e s i s t o r clearing, would produce a more accurate Zp 0 g(jw). - 23 4.3.2 Weighting Function (Window) The second method used to reduce Gibbs o s c i l l a t i o n s , suggested by Dr. L.M. Wedepohl, was to average V^(jw) over the period of the Gibbs o s c i l l a t i o n . Appendix III shows that this averaging i n the frequency domain can be accomplished i n the time domain by multiplying the f a u l t voltage by a tapered window function of the form Sa [Pi (t-T/2)/2]. This function i s the major lobe of the sampling function centered i n the time sample T (Figure 14). It smoothly forces v^(t) to zero at the beginning and end of the analysis period of T seconds, thereby avoiding truncation and Gibbs o s c i l l a t i o n s . Figure 15 shows the new measured Z(jw) calculated using the averaged V^(jw). Again, the Gibbs o s c i l l a t i o n s has been eliminated as desired. The new Z (jw) i s indeed a smoothed version of the old. pos J However, comparing this "averaged" Z p O S ( J w ) with the Z p O S ( J w ) obtained by analysis of the longer duration signals (Figure 13), i t appears that the o s c i l l a t o r y c h a r a c t e r i s t i c of Z (jw) between 15 and 20 Hertz has J pos J now been smoothed out. Its existence i n Figure 13 would indicate that i t i s r e a l . Its absence from Figure 15 suggests a loss of data. Use of the averaging technique to smooth out Gibbs o s c i l l a t i o n s i s therefore not advisable here i f Z (jw) between 15 - 20 Hz. i s of pos J i n t e r e s t . - 24 „-40.0 IMPEDANCE (OHMS) -20.0 0,0 20.0 ho o cr 33 I com D D — C O l~" c r- TJTJ nm —io I — Q U I Z - < L O C D — I -< m cz C O " m J : TJ am -no D z: z •—<-) z m o Q I s : — • T J 0 z m C D C T J a - n o n o H Z Q O z "n o C D C D C O 60.0 ,-p.SOO 0 000 C D C co D AMPLITUDE (NO UNITS) •- I. SOO 0 r000 0 rS 1 .0 1 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) 1 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) -• 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) 0 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) - 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) ... 1.0 -0'.8 -0'.6 -0'.4 -0'.2 O'.O o'.2 o'.4 0'.6 0 TIME (SEC) CD Q 5 COMPUTER SIMULATIONS 5.1 500 kV System Model and Simulation Results With the phase-to-phase staged f a u l t test at W i l l i s t o n Substation having been completed, the known system parameters were used i n assembling an EMTP model of the system to represent the f i e l d test conditions. The model, shown i n the one-line diagram of Figure 16, represents almost a l l of the B.C. Hydro 500 kV system from the northern hydro plants on the Peace River - G.M. Shrum and Peace Canyon - to the main load centres i n the south - Ingledow and Meridian substati ons. Thus, the same EMTP model could be used for both frequency scan and transients studies. Transmission l i n e s were modelled by cascaded connections of multiphase p i - c i r c u i t s , observing proper phasing, transpositions, and physical layout with respect to other EHV c i r c u i t s on the same right-of-way. The frequency-dependence of the l i n e p o s i t i v e , negative and zero sequence operational impedances was not represented. Line parameters were evaluated at 60 Hz. No transformers were modelled as only the 500 kV transmission was e x p l i c i t l y represented and a l l generators and loads were represented as equivalents on 500 kV buses. System loads were represented by constant impedance equivalents. A l l series capacitor banks were represented by lumped capacitance and l i n e reactors by pure shunt reactance. Hydro, plants - s i x units at G.M. Shrum, two units at Peace Canyon and three units at Mica - were modelled as id e a l voltage sources behind source impedances. Machine dynamics were not modelled. The Ra + jX"d of the i n d i v i d u a l machines with the impedance of th e i r stepup transformers i n series were para l l e l e d to produce a source impedance behind each 500 kV - 26 G.M.SHRUM HI U T + TELKWA GLENANNAN \ SKEENA KELLY LAKE I MERIDIAN IPEACE CANYON = - © WILLISTON INGLEDOW MICA ASHTON CREEK NICOLA n SELKIRK 6 LEGEND: M O D E L O F J U L Y I, 1981 S Y S T E M FOR E M T P STUDIES 500 kV LINE (j LOAD X - N - INFINITE BUS BEHIND {Zr<^- SOURCE IMPEDANCE — I I — SERIES CAPACITOR i SHUNT REACTOR FIGURE -16 - COMPUTER MODEL OF THE B.C. HYDRO MAJOR GENERATION AND 500 KV TRANSMISSION SYSTEM generation bus. The t i e between Ingledow Substation and the Bonneville Power Administration (BPA) system was represented by an i n f i n i t e bus behind s e l f and mutual impedances derived from 1-phase and 3-phase fa u l t computations. S i m i l a r l y , an i d e a l source behind a source impedance was p l a c e d a t Skeena and S e l k i r k s u b s t a t i o n s . The r e s u l t s of a system impedance frequency scan from the W i l l i s t o n 500 kV bus, i s shown i n F i g u r e 17. The c a l c u l a t e d system impedance i s i n o FREQUENCY (HZ) FIGURE 17 - 500 KV SYSTEM MODEL IMPEDANCE FREQUENCY SCAN AND COMPARISON WITH THE MEASURED SYSTEM IMPEDANCE good agreement w i t h measurement around the system s e r i e s resonance. They agree to w i t h i n 0.3 Hz f o r the s e r i e s resonance frequence and 4% f o r the system r e s i s t a n c e a t s e r i e s resonance. The model, however, does not g i v e good agreement i n the area of system p a r a l l e l resonance. T h i s d i s c r e p a n c y , combined w i t h EMTP time domain s i m u l a t i o n s of the t e s t which s l i g h t l y underestimated the f a u l t c u r r e n t , i m p l i e d t h a t the system model - 28 was somewhat oversimplified. The model did not e x p l i c i t l y represent any lower voltage transmission or the seven 50 MV.A units at Bridge River. The above model assumed that the effects of the higher impedance paths through the underlying lower voltage transmission would be overwhelmed by the lower impedance 500 kV transmission. 5.2 Extended System Model and Simulation Results The EMTP 500 kV system model was progressively extended to include some 230 kV and 287 kV transmission as well as the Bridge River generation and i t s associated transmission. It was not feasible to increase further the complexity of the model because of size l i m i t a t i o n s imposed by the EMTP. The newly added buses were connected to the 500 kV buses by inductively coupled branches representing the equivalent of one or more actual transformers operating i n p a r a l l e l . Figure 18 shows the f i n a l d e t a i l e d system model, having 112 three-phase nodes. As more d e t a i l was added to the system model the agreement between the frequency scans and the measured system impedance progressively improved. The frequency scan of the f i n a l model compared to f i e l d r esults i s shown i n Figure 19. Although there are s t i l l differences i n the v i c i n i t y of p a r a l l e l resonance, the shape of the simulated impedance now more closely resembles that of the measured impedance. The impedance in the v i c i n i t y of the series resonance i s the same as i t was for the simpler 500 kV model (Figure 17). Figure 20 shows the simulated W i l l i s t o n f a u l t current and phase A-to-phase B voltage obtained by using the f i n a l detailed system model. Also shown for comparison are the f i e l d test measurements. - 29 G.M.SHRUMT RUPERT TELKWA GLENANAN \ SKENA 1 ©K ITIMAT KELLY LAKE MERIDIAN (PEACE CANYON o - 0 WILISTON IN6LED0W UT LEGEND: MICA ASHTON CREK 230 kV OR 360 kV LINE 500 kV LINE M O D E L O F J U L Y I, 1981 S Y S T E M F O R E M T P S T U D I E S () LOAD a _ INFINITE BUS BEHIND SOURCE IMPEDANCE —II— SERIES CAPACITOR ( SHUNT REACTOR TRANSFORMER F I G U R E 18 - M O R E D E T A I L E D S Y S T E M M O D E L I N C L U D I N G A D D I T I O N A L N E A R B Y G E N E R A T I O N A N D S O M E 2 3 0 K V T R A N S M I S S I O N - 30 -1 f^EQue^c-i iCAM FREQUENCY (HZ) io 157 FIGURE 19 - SYSTEM IMPEDANCE FREQUENCY SCAN OF DETAILED EMTP MODEL AND COMPARISON WITH THE MEASURED IMPEDANCE 6b. 'O.OOO 0.020 0.040 0.060 0.080 0.100 TIME (SECONDS) FIGURE 20(A) - COMPARISON OF SIMULATED AND MEASURED PHASE A TQ B VOLTAGE - 31 o UJ oz cr SvrAuv.HCT"tO e '0.000 0.020 0.040 0.060 0.080 0.100 TIME (SECONDS) 0.120 0.140 0.160 0.180 0.200 FIGURE 20 IB) - COMPARISON OF SIMULATED AND MEASURED FAULT CURRENT 6 EXPLANATION OF DISCREPANCIES BETWEEN MEASURED AND SIMULATED IMPEDANCE The discrepancy i n the impedance between simulation and f i e l d test i n the area of system p a r a l l e l resonance and the good agreement elsewhere may be explained. Figure 21 presents a reduced and much si m p l i f i e d system model from W i l l i s t o n where the Nicola, G.M. Shrum, Peace Canyon, Ingledow and Meridian buses are assumed to be short c i r c u i t s . The series RI, L l , CI branch b a s i c a l l y represents the equivalent for the main series compensated 500 kV transmission c i r c u i t s out of W i l l i s t o n . The R2, L2 branch i s the equivalent for the uncompensated r a d i a l 500 kV c i r c u i t from W i l l i s t o n to Skeena and the loads along that transmission (modelled in the EMTP), plus the higher impedance underlying subtransmission to the loads out of W i l l i s t o n (not f u l l y modelled). The frequency scan of the simple model [15], also shown i n Figure 21, i s s i m i l a r i n general shape to the frequency scan of the deta i l e d model. The pole of the impedance function i s the p a r a l l e l resonance between R2, L2 and RI, L l , CI. The zero of the impedance function, at a higher frequency, i s the series resonance of RI, L l , CI and i s , for a l l p r a c t i c a l purposes, independent of R2 and L2. - 32 FIGURE 21 - SIMPLIFIED SYSTEM MODEL AND IT'S IMPEDANCE The system series resonance involves, basically, only the main series compensated transmission, which can be quite accurately modelled in the EMTP - hence the good agreement with the measured Z p O S ( J w ) a C t n e series resonant frequency. Effects which were not accounted for in the simulations are high-gain exciter action, the slight time variance in machine reactances, and saturation. The influence of the combined effects is negligible around the series resonance frequency. The underlying subtransmission and the system loads whose frequency dependence is unknown could not be represented accurately. Both are factors in the behaviour of the system parallel resonance. These shortcomings in the system model account for the discrepancy between simulated and measured Z p O S ( j w ) near the parallel resonant frequency. - 33 7 IMPROVEMENTS TO MEASUREMENT TECHNIQUE 7.1 Computer Simulation Including Lower Voltage Transmission Dr. Atef Morched of Ontario Hydro subsequently confirmed the above explanations for the discrepancies around the p a r a l l e l resonance frequency. The author provided Dr. Morched with the test measurements as well as power flow and s t a b i l i t y data for the entire B.C. Hydro e l e c t r i c a l system down to the substation d i s t r i b u t i o n buses. The Ontario Hydro study [16] of the test data improved the calculated impedance (as opposed to the measured impedance) by c a l c u l a t i n g a system impedance frequency scan at the f a u l t location using frequency scaled power flow model data. This was a purely positive sequence model of p r a c t i c a l l y the entire B.C. Hydro transmission system including the lower voltage transmission for which there was not enough room to include i n the three phase EMTP model. Also, the simple constant impedance loads of the EMTP model were replaced with s t a t i c and dynamic loads representations (constant R, L and equivalent machine models) which more correc t l y accounted for the nature and frequency dependence of the loads. Figure 22 shows that these changes do improve the calculated impedance inasmuch as i t now more closely resembles the measured system impedance in the area of the system p a r a l l e l resonance. 7.2 The Effect of Including System Dynamics The Ontario Hydro study also improved the measured impedance by improving the system open c i r c u i t voltage used to derive the f a u l t voltage. It did so by running a transient s t a b i l i t y simulation of the f a u l t a p p l i c a t i o n , the f a u l t clearing, and the braking r e s i s t o r a p p l i c a t i o n to determine the way the system equivalence source changed - 34 FIGURE 22 - SYSTEM IMPEDRNCE FROM EXTENSIVE POSITIVE SEQUENCE SYSTEM MODEL during the test. This change i s shown i n terms of i t ' s 60 Hz magnitude (Figure 23) and phase (Figure 24). With the power system dynamics now taken into account, Figure 25 shows the new e , (t) and Figure 26 shows the ab r e s u l t i n g v ^ ( t ) . In addition to the above improvement, the Ontario Hydro study calculated the system equivalent positive sequence impedance as: 1 G v i ( J w ) Z W " 2 G"(jw) ( 7 ) i i J - 35 F I G U R E 2 3 - CHANGE IN OPEN C I R C U I T V O L T A G E M A G N I T U O E F I G U R E 2H - CHANGE IN OPEN C I R C U I T VOLTAGE P H A S E DUE TO S T S T E M I M P E D A N C E T E S T F A U L T DUE TO S T S T E M I M P E D A N C E TEST F A U L T F I G U R E 2 5 - OPEN C I R C U I T V O L T A G E C O R R E C T E O FOR S Y S T E M D Y N A M I C S H I N G F I G U R E 2fa - F A U L T VOLTAGE C O R R E C T E D FOR S t S T E M D Y N A M I C SWING where G^(jw) i s the auto power spectrum of the current and G v^(t) i s t n e cross power spectrum between the voltage and the current [7,8,14]. This technique reduces the effect of deterministic errors introduced by the measurements as well as random errors. The newly derived, measured system impedance i s shown i n Figure 27. The correspondence between the Ontario Hydro study's calculated and measured impedance i s very good throughout the entire subsynchronous frequency range. It represents a d e f i n i t e improvement over the results i n i t i a l l y obtained, espe c i a l l y i n the area of - 36 80 4 0 //M ' i / — • — . 1 0 - 4 0 10 ' " ' 2 0 . V y / V 1 A I /' 30 40 50 ' F R E Q U E N C Y C H Z ) ( S t f t ^ i 6 0 FIGURE 27 - MEASURED SYSTEM IMPEDANCE CORRECTED FOR SYSTEM DYNAMIC SWING COMPARED WITH THE SYSTEM IMPEDANCE FREQUENCY SCAN OF THE EXTENSIVE POSITIVE SEQUENCE MODEL the system p a r a l l e l resonance. In order to see the contribution made by each of the improvements, Z (jw) was recalculated using the above auto- and cross-correlation pos J 6 formula acting on the v^(t) signal derived using the o r i g i n a l (unimproved) e o K ( t ) « Th e r e s u l t i n g Z (jw) was i d e n t i c a l to the one derived without using c o r r e l a t i o n (Figure 12). It i s therefore concluded that the improvement Ontatio Hydro achieved i n t h e i r c a l c u l a t i o n of the measured Zp Q g(jw) i s e s s e n t i a l l y e n t i r e l y due to his i n c l u s i o n of the system dynamic swings i n e , ( t ) . Correlation did not help here because (1) there 3. D was an adequate signal-to-noise r a t i o to begin with and (2) the duration of the signal of 400 msec was not long enough to allow the c r o s s - c o r r e l a t i o n to reject any voltage signal not due to the f a u l t current s i g n a l , such as the system dynamic swing and the voltage transient due to the braking r e s i s t o r application. Correlation does help i n the - 37 i n t e r p r e t a t i o n of the system impedance i n that the cross and auto power spectrums can be used to provide an estimate of the signal-to-noise l e v e l of Z (iw) and the confidence i n t e r v a l associated with the measured pos impedance at each frequency. 7.3 Supersynchronous Impedance and Minimum Required Current While the main purpose of this test was to measure the system's subsynchronous equivalent impedance, the measurement technique i s equally v a l i d at supersynchronous frequencies. This data was also of interest i n that comparison of the measured and calculated z p o s ( J w ) could help determine the upper frequency l i m i t for which the EMTP system model was v a l i d . Figure 28 shows the frequency spectrums V^(jw), I ^ C j w ) , and the measured and calculated Z (iw) to 200 Hz. pos J Two c h a r a c t e r i s t i c s of the measured impedance are apparent. The f i r s t c h a r a c t e r i s t i c i s a noticable change i n slope i n the reactance curve above 60 Hz. This may be due to induction machine effects from motor loads i n the v i c i n i t y of the measurement point. Loads i n the EMTP model were s i m p l i s t i c a l l y represented by r e s i s t o r s which would explain why the calculated impedance does not exhibit this behaviour. The second and more l i m i t i n g c h a r a c t e r i s t i c of the measured impedance i s i t ' s i n s t a b i l i t y i n the v i c i n i t y of the I^(jw) nodes at 80 and 100 Hz. and the poor c o r r e l a t i o n between the measured and calculated impedance above 100 Hz. These are problems of an i n s u f f i c i e n t l e v e l of current i n j e c t i o n which was i d e n t i f i e d e a r l i e r as being a l i m i t a t i o n of this measurement technique. Refering to Figure 28 then, a minimum value of current i n j e c t i o n of about 20 Amps i s apparently required to determine the system impedance. It i s expected that with higher resolution recordings of the f i e l d data, the minimum current i n j e c t i o n required would decrease and the highest - 38 80. Ibo. 120. FREQUENCY (HZ) 200. FIGURE 28(A) - FREQUENCY SPECTRUM OF FAULT VOLTAGE (MAGNITUDE ONLY) 90~. lbo. 120. FREQUENCY (HZ) 200. FIGURE 28 18) - FREQUENCY SPECTRUM QF FAULT CURRENT (MAGNITUDE ONLY) T-40. 80. 100. 120. FREQUENCY (HZ) T" 140. 200. FIGURE 28(C) - MEASURED AND CALCULATED SYSTEM IMPEDANCE - 39 frequency at which the system impedance could determined would increase. Since reasonable measurements of the system impedance were obtained for current injections as low as 20 Amps, the d i g i t i z e d f a u l t current must be reasonably accurate to at least this resolution. The resolution of the tape recorder, for the impedance test, must therefore have been at least 20 Amps i n 20,000 (the peak signal value), or 60 dB which i s much greater than i t s 40 dB rating. It i s possible that the l i m i t i n g factor was not the resolution of the tape recording but instead the resolution of the 12 b i t analog-to-digital converter, which was 1 part i n 4096 or 72 dB. The current i n j e c t i o n levels obtained at frequencies close to the system p a r a l l e l resonance (15-20 Hz) are above the minimum acceptable l e v e l . The discrepancies between the measured and calculated system impedance at these frequencies are therefore more l i k e l y due to the EMTP system model d e f i c i e n c i e s previously discussed than to measurement inaccuracy. A recent Ontario Hydro test [17] demonstrated the value of high resolution recordings. System impedance tests were done on the Ontario Hydro d i s t r i b u t i o n system using the application and removal of shunt capacitors as the system disturbance. For this test the " f a u l t " voltage was derived i n real-time by removing the 60 Hz steady state voltage from the system node voltage before recording i t . The instrumentation scaling could then be done on the transient voltage rather than on the larger steady state voltage thereby making better use of the resolution of the recording device. A 14-bit resolution d i g i t a l storage oscilloscope was used^ to d i g i t i z e the data d i r e c t l y - i e : there was no intermediate tape recording. This improved resolution allowed good impedance measurements up to 5 KHz. The problem of low signal levels at the I^(jw) nodes was solved by repeating the test a number of times with d i f f e r e n t f a u l t - 40 durations and averaging the impedance spectrums obtained. Since i t i s unlikely that any two tests would have I^(jw) nodes at the same frequency, t h i s averaging tended to eliminate (d i l u t e ) the numerical i n s t a b i l i t y associated with these nodes. Another important aspect of Ontario Hydro's test was the use of a capacitor as a disturbing element. The capacitor's impedance at low frequencies, p a r t i c u l a r i l y the 60 Hz system frequency, i s high making i t equivalent to a r e l a t i v e l y mild system f a u l t . However, at higher frequencies the capacitor's impedance i s low. This frequency dependent f a u l t impedance w i l l therefore tend to accentuate the high frequency content of the transient current r e l a t i v e to i t s low frequency content, making i t better suited to a supersynchronous impedance test than a zero impedance bolted f a u l t would be. The capacitor a p p l i c a t i o n has only a small e f f e c t on the reactive power flow and p r a c t i c a l l y no e f f e c t on the system r e a l power flow. The system open c i r c u i t voltage i s almost c e r t a i n to be unaffected by this type of system disturbance and i s therefore l i k e l y to remain constant. This would eliminate the f a u l t voltage signal truncation and the r e s u l t i n g Gibbs o s c i l l a t i o n problems that arose i n the B.C. Hydro test. 7.4 Modal (Sequence) Analysis of Test Results It was previously stated that the type of f a u l t chosen for the impedance tests would have to provide only positive sequence excita t i o n , at subsynchronous frequencies, to the system. Section 2.2 showed that a phase-to-phase f a u l t would do t h i s , with c e r t a i n assumptions, and rejected a l l other types of f a u l t s because they also provided some zero sequence ex c i t a t i o n and response that could not be separated from the desired po s i t i v e sequence quantities. Mr. J.H. Sawada of B.C. Hydro has now - 41 extended the usefulness of the impedance measurement technique by applying the symmetrical component transformation (as a s p e c i a l modal transformation), i n the time or the frequence domain, to the f a u l t voltages and currents measured for a l l three phases and obtains the sequence impedances as [18]: Z 0(jw) = V f 0 ( j w ) / I f Q ( j w ) Z x(jw) = V f l ( j w ) / I f l ( j w ) (8) Z 2(jw) = V f 2 ( j w ) / I f 2 ( j w ) The above equations once again assume a balanced impedance matrix. The more general three phase impedance matrix [Z(jw)] can be derived using the matrix equation: Z Z , Z aa ab ac ba bb be Z Z . Z ca cb cc V . V , V , a l az a3 V V V b l b2 b3 V . V V c l c2 c3 •*"al *a2 *a3 I b l Xb2 Xb3 I c l Ic2 Xc3 -1 (9) where a l l quantities are functions of frequency and the above equation holds at each frequency, and where the numerical subscripts for the voltages and currents refer to data from d i f f e r e n t tests done with the same system conditions. For example, tests 1, 2 and 3 could be three separate line-to-ground f a u l t tests on phases a, b and c respectively. Such actual knowledge of [Z] as a function of frequency would be invaluable i n creating power system frequency dependent equivalents [19,20]. These would save having to model, e x p l i c i t l y , large parts of the power systems i n order to represent their frequency dependent c h a r a c t e r i s t i c . - 42 8. CONCLUSIONS This thesis describes a technique, and presents f i e l d test r e s u l t s , for measuring the subsynchronous positive sequence equivalent impedance, as a function of frequency, of an operating power system using Fourier analysis of the f a u l t current and phase-to-phase voltage obtained from a staged phase-to-phase f a u l t . As no external source of ex c i t a t i o n i s required, coupling problems are eliminated. The good c o r r e l a t i o n between EMTP simulation and the f i e l d test results j u s t i f y the approximations of network l i n e a r i t y , time invariance, and equality between the positive and negative sequence equivalent impedances, which are required by the technique. The EMTP, using a model of the present B.C. Hydro high voltage (500 kV) system, i s able to determine the equivalent network subsynchronous impedance i n r e l a t i o n to frequency i n good agreement with f i e l d measurements, except i n the v i c i n i t y of system p a r a l l e l resonance. This model provided the system series resonance within 0.3 Hz and equivalent resistance within 4 % of the measurement value, making i t s u f f i c i e n t l y detailed for use i n Hat Creek SSR studies. Reasons for the discrepancy between the measured and calculated system p a r a l l e l resonance impedance were given and subsequently confirmed by the work done at Ontario Hydro. Since SSR i s not l i k e l y to occur at a system p a r a l l e l resonance, the very detailed modelling required to correc t l y model the p a r a l l e l resonance behaviour was not considered necessary. I f , for some other app l i c a t i o n , the system p a r a l l e l resonance i s of in t e r e s t , the system model must be expanded to include more of the lower voltage transmission and perhaps frequency dependent equivalents of the l o c a l system loads. Two methods for eliminating the Gibbs o s c i l l a t i o n s from the measured f a u l t voltage and system impedance were t r i e d . Both were successful but they reduced confidence i n the result i n g system impedance and were therefore not - 43 used. Using a system disturbance less severe than a bolted f a u l t should leave the system open c i r c u i t voltage constant and eliminate the f a u l t voltage truncation that caused the Gibbs o s c i l l a t i o n s . While i t was primarily the system subsynchronous impedance that was of in t e r e s t , the impedance test i s equally v a l i d at supersynchronous frequencies and has been successfully applied by others at frequencies up to 5 kHz. It was also noted that a phase-to-phase f a u l t was used i n this test so that d i r e c t l y measurable voltage and current quantities could be used to determine the system p o s i t i v e sequence.impedance. The analysis technique has also been extended by others to a modal (eg: symmetrical components) transformation p r i o r to the spectral analysis so that, provided there i s enough current i n j e c t i o n , any modal impedance can be derived from the same modal f a u l t voltage and current. With the v a l i d i t y of the computer model of the present B.C. Hydro system established, EMTP impedance frequency scans of future transmission networks can therefore be used with confidence i n SSR investigations. - 44 9 REFERENCES [1] M.C. Hall,"Experience with 500 kV Subsynchronous Resonance and Resulting Turbine Generator Shaft Damage at Mohave Generating Station." Subsynchronous Resonance Symposium, IEEE Power Apparatus and Systems Summer Meeting, San Fransisco, C a l i f o r n i a , 24 July 1975 [2] L.A. Kilgore, D.G. Ramey and M.C. H a l l , "Simplified Transmission and Generation System Analysis Procedures for Subsynchronous Resonance Problems." IEEE Transactions on Power Apparatus and Systems, v o l . PAS-96, pp. 1840-1846, November/December 1977. [3] H.W. Dommel and I.I. Dommel, "Electromagnetic Transients Program User's Manual." University of B r i t i s h Columbia, August 1978 [4] H. Barnes and S. Kearley, "The Measurement of the Impedance Presented to Harmonic Currents by Power D i s t r i b u t i o n Networks." presented at the International Conference on E l e c t r i c i t y D i s t r i b u t i o n , Brighton, England, 1-5 June 1981. [5] D. Crevier and A. Mercier, "Estimation of Higher Frequency Network Equivalent Impedances by Harmonic Analysis of Natural Waveforms." IEEE Transactions on Power Apparatus and Systems, v o l . PAS-97, pp. 424-431, March/April 1978. [6] K.E. Bollinger, R. Winsor and D. Cotcher, "Power System I d e n t i f i c a t i o n Using Noise Signals." (SUMPWR 76 Abstr. A76 339-2); IEEE Transactions on Power Apparatus and Systems, v o l . 76, p. 1761, November/December 1976. [7] F.X. Macedo, "Monitoring of System Impedance Using Arc-Furnace Disturbances", U n i v e r s i t i e s Power Engineering Conference (1981) S h e f f i e l d [8] F.X. Macedo, "Power System Harmonic Impedance Measurement Using Natural Disturbances", UBV Proceedings of the Third International Conference, London, May 1982 [9] D. Crevier, A Mercier, "Estimation of Higher Frequency Network Equivalent Impedances by Harmonic Analysis of Natural Waveforms", IEEE Transactions on Power Apparatus and Systems, v o l . PAS-97, no. 2, March/April 1978. [10] E. Clarke, " C i r c u i t Analysis of A-C Power Systems, Volume 1." General E l e c t r i c Company; Schenectady, New York; March 1961. [11] B.P. La t h i , "Signals, Systems and Communication." John Wiley & Sons, New York, 1968 [12] T. Sugiyama, T. Nishiwaki, S. Takeda and S. Abe, "Measurements of Synchronous Machine Parameters Under Operating Condition." IEEE Transactions on Power Apparatus and Systems, v o l . PAS-101, pp. 895-904, A p r i l 1982. [13] R.P. Scbulz, C.E.J. Bowler and W.D. Jones, Discussion on "Generator Models Established by Frequency Response Tests on a 555 MV.A Machine." - 45 IEEE Transactions on Power Apparatus and Systems, v o l . PAS-91, pp. 2077-2084, September/October 1972. [14] K.C. Lee, "Propagation of the Wave Front On Untransposed Overhead and Underground Transmission Lines." M.A.Sc. Thesis, E l e c t r i c a l Engineering, University of B r i t i s h Columbia, A p r i l 1977 [15] General E l e c t r i c , " E l e c t r i c a l U t i l i t y System Engineering Seminar - Subsynchronous Resonance." San Diego, C a l i f o r n i a , August 1977 [16] Dr. A.S. Morched, " V e r i f i c a t i o n of the Frequency Domain Technique for Calculating Asymmetrical Fault Currents", Ontario Hydro, March 1984 [17] Dr. A.S. Morched, Private Communication, November 1984 [18] J.H. Sawada and M. Scott, "General Power System Network. Thevenin Equivalent Impedance Calculations Using Perturbation Method and Comparisons of Impedances Derived from F i e l d Measurement and EMTP Model." B.C. Hydro Internal Report, 18 October 1984 [19] A. C l e r i c i and L. Marzio, "Coordinated Use of TNA and D i g i t a l Computer for Switching-Surge Studies: Transient Equivalent of a Complex Network." IEEE Transactions on Power Apparatus and System, v o l . PAS-89, pp. 1717-1726, November/December 1970 [20] A.S. Morched and V. Brandwajn, "Transmission Network Equivalents for Electromagnetic Studies." IEEE Transactions on Power Apparatus and System, v o l . PAS-102, pp. 2984-2994, September 1983. - 46 APPENDIX I - Derivation of Zpos(jw) for a Three Phase System In phase quantities, the quantities i n Figure 4 are related by the matrix equation: [V] = [E] - [Z] [I] To convert to symmetrical component quantities, pre-multiply by the sequence transformation matrix [S]. [S][V] = [S][E] - [S][Z][I] Note that for a phase A to B f a u l t the symmetrical component transformation i s with phase C as reference. The transformation matrix [S] and i t ' s inverse [S]"" 1 are: [S] = 5 where a = e 1 1 1 a a 1 2 . a a 1 -J120 and [S] -1 1 a a , 2 1 a a 1 1 1 In order to express the current i n symmetrical component quantities, post-multiply [Z] by the i d e n t i t y [1] = [ S ] _ 1 [ S ] . [ S I M = [S][E] - [ S ] [ Z ] [ S ] _ 1 [S][I] [V ] = [E ] - [Z ] [I ] sym sym sym sym If we assume a balanced system, then the symmetrical component impedance matrix w i l l be: " Z 0 0 0 [z • ] -symJ 0 z l 0 0 0 Z 2 where: ZQ = zero sequence impedance Zy = positive sequence impedance, and Z2 = negative sequence impedance. - 47 Converting back to phase quantities, by pre-multiplying with [S] , we get: [ S ] ' l ^ s y J = l S r 1 [ EsyJ - [ S r 1 ^ ] [ I S Y J [V] = [E] - [ S ] _ 1 [ Z s y m ] [ S ] [I] Substituting the current vector for a phase A to B f a u l t : V a E a 1 a 21 a " Z 0 0 0 1 1 1 - X f Vb = E b 1 ~ 3 1 2 a a 0 z l 0 2 a a 1 h V c E c 1 1 1 0 0 Z 2 _ a 2 a 1 0 Calculating the voltage across the f a u l t , V - V , y i e l d s : 3. D <vv • <VV + < z i + V h O C Vab - Eab + ( Z 1 + Z 2 ) h F i n a l l y , defining V = V -E , and assuming Z, = -Z„, then: r ab ab 1 2 z - i • l£ ^1 2 I f - 48 APPENDIX II - Fourier Transform Calculation For a given time domain s i g n a l , say v ^ ( t ) , we can obtain i t s frequency domain representation V^(jw) by using the general form of the Fourier transform, V f(jw) = \ v f ( t ) e " j W t d t Assuming v^(t) i s zero outside the time period of interest (0<t<T) and separating the real,and imaginary components: Re{V f(jw)} = A(jw) = I v f ( t ) cos wt dt 0 c Im{Vf(jw)} = B(jw) = 1 v f ( t ) s i n wt dt ' 0 If the input quantity v^(t) i s known at closely spaced discrete time i n t e r v a l s between 0 and T, and i t i s reasonable to assume l i n e a r i n t e r p o l a t i o n between data points, then the equation for v^(t) between two adjacent points t l and t 2 i s : v f ( t ) = v i + ! i - r ' ' - V for t < t <t 2 and A t = t2~t\ Substituting this into the equations for V^(jw) we get: A 1 2(jw) V 2 " V 1 [ V 1 + "A~t~ ( t _ t l ) 1 c o s w t d t i • t • " TT h i ) cos wt dt + » 2-v 1 "AT ) t cos wt dt V — V 2 1 1 = [v, - —i t ] - s i n wt l A t l w ' 2 V 2 " V 1 1 1 + —; - (t s i n wt + - cos wt) A t w w : 1 - 49 1 V2" V1 V2" V1 = - s i n wt 2 [ ( v x + t 2 ) - - A ~ - t l ] s i n wt^ V2~ V1 V2~ V1 V2~ V1 [ ( V i . t i ) + + ( c o s w t z _ c o s w t i ) A tw 1 V 2 ~ v l = - [v„ s i n wt„ - v, s i n wt, + — (cos wt„-cos wt.)] w 2 2 1 1 A tw 2 1 Si m i l a r l y for the imaginary component, B^ 2, w e § e t : 1 v 2 " v l B 1 0 ( j w ) = - [~v„ cos wt + v. cos wt, + — ( s i n wt - s i n wt.)] iz w z Z J . J - A tw z l The calculations for A ^ ( j M ) and B ^ ( j w ) a^e repeated for a l l adjacent time samples over the entire time period under study (0<t<T). The re a l and imaginary part of the input quantity v(t) i n the frequency domain at any s p e c i f i c frequency w i s then the sum of a l l A^ 2(jw)'s and B^ 2(jw)'s, i . e . : V(jw) = A(jw) + jB(jw) i/At = £ = 0 A k , k + l ( J w ) + J\,k+1 The above calculations are then repeated over the frequency range of inter e s t ( i n this case 0<w<377 rad/sec. or 0<f<60 Hz) with whatever resolution, Aw, i s desired. This result i s then the Fourier transform of v ( t ) , V(jw). - 50 APPENDIX III - Time Domain Window to Eliminate Gibbs O s c i l l a t i o n s As derived so f a r , V^(jw) i s actually the Fourier transform of v^(t) multiplied by the gate function g ( t ) . V f(jw) = F {v f(t) x g(t)} where g(t) = 1 for -T/2 <t< T/2 = 0 elsewhere V f(jw) = F{v f(t)} * F{g(t)} = F{v f(t)} * Sa[wT/2] where Sa[x] = ( s i n x)/x i s the Sampling function. The calculated V^(jw) i s therefore equal to the r e a l Fourier transform of v^(t) convolved with the Sampling function Sa[wT/2]. It can be seen from Figure 13 that Sa[wT/2] has a major lobe at w = 0 and side lobe that o s c i l l a t e with a period of 2Pi/T (radians/sec.) in the frequency domain. These side lobes of F{g(t)} are what cause the Gibbs o s c i l l a t i o n i n V^(jw) and the Gibbs o s c i l l a t i o n has this same period. In order to reduce the Gibbs o s c i l l a t i o n V^(jw) can be averaged over the period +/-Pi/T (radians/sec.) to develop an improved V f(jw) defined as V' f(jw). ! w+Pi/T V(jv) dv w-Pi/T T/2 / w+Pi/T f(t-) \ e " j v t d v dt T/2 / w-Pi/T - 51 T 2~Pi T/2 •T/2 T/2 T/2 T/2 T/2 f ( t ) f ( t ) f ( t ) Sa(Pi t/T) e " j w t dt - j v t e J w-Pi/T w+Pi/T ] dt This averaging of V^(jw) over the period of the Gibbs o s c i l l a t i o n can therefore be achieved by simply multiplying (gating) v^(t) by Sa[Pi t/T] over the range -T/2 <t< T/2. This new shape of gate function i s the major lobe of Sa[Pi t/T] which goes smoothly to zero at the end of the time sample - i . e . t = +/- T/2. The time signal v^(t) i s therefore gradually attenuated to zero so that no truncation, and Gibbs o s c i l l a t i o n s , occurs. - 52
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