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Voltage and current profiles and low-order approximation of frequency-dependent transmission line parameters 1982
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Title | Voltage and current profiles and low-order approximation of frequency-dependent transmission line parameters |
Creator |
Marti, Luis |
Date Created | 2010-04-13T15:19:48Z |
Date Issued | 2010-04-13T15:19:48Z |
Date | 1982 |
Description | In this thesis project, two models related to the simulation of electromagnetic transients in power systems, have been developed. The first model has been designed to find voltages and currents at a number of intermediate, equally spaced points along the line ("profile"), from the solution at the end points (taking into account the frequency dependence of the line parameters). This "profile model" is derived from the cascade connection of n equivalent circuits, each one representing a segment of the line. The solution is carried out with an internal time step which is an exact submultiple of the travel time of the propagation mode in consideration. A series of tests shows that the results obtained with this model are more accurate than those obtained if the standard practice of segmenting the line is followed. With the aid of the profile routines, a movie which illustrates the propagation of transients along a line, has been produced as well. The second part of this thesis describes the development of a low-order approximation of the frequency-dependence of line parameters, from a reduced amount of information. Using the electrical parameters at power frequency and dc conductor resistances, the tower configuration of the line is reconstructed. This equivalent line configuration permits the evaluation of the line parameters over a wide frequency range. Rational functions are then used to approximate the frequency-dependence of these parameters over a limited frequency range with a reduced order model. Analytical tests and transient simulations, indicate that the model is reasonably accurate over the frequency range of interest in most practical applications. Both models have been incorporated into the UBC version of the Electromagnetic Transients Program (EMTP). |
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 | 2010-04-13T15:19:48Z |
DOI | 10.14288/1.0065490 |
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/2429/23403 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0065490/source |
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VOLTAGE AND CURRENT PROFILES AND LOW-ORDER APPROXIMATION OF FREQUENCY-DEPENDENT TRANSMISSION LINE PARAMETERS by LUIS MARTI Elec. Engr., Central University of Venezuela, 1979 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 A p r i l 1982 © Luis Marti, 1982 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 g a i n 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 o f E l e c t r i c a l Engineering 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 April 27th 1982 DE - 6 (.3/81) ABSTRACT In th is thes is p r o j e c t , two models re la ted to the s imulat ion of e lectromagnet ic t rans ien ts i n power systems, have been developed. The f i r s t model has been designed to f i n d voltages and currents at a number of in termediate , equa l ly spaced points along the l i n e ( " p r o f i l e " ) , from the s o l u t i o n at the end points ( taking in to account the frequency dependence of the l i n e parameters) . This " p r o f i l e model" i s der ived from the cascade connection of n equiva lent c i r c u i t s , each one represent ing a segment of the l i n e . The s o l u t i o n i s c a r r i e d out with an i n t e r n a l time step which i s an exact submult iple of the t r a v e l time of the propagation mode in c o n s i d e r a t i o n . A se r ies of tests shows that the r e s u l t s obtained with th is model are more accurate than those obtained i f the standard p rac t i ce of segmenting the l i n e i s fo l lowed. With the a id of the p r o f i l e rou t ines , a movie which i l l u s t r a t e s the propagation of t rans ients along a l i n e , has been produced as w e l l . The second part of th is thes is descr ibes the development of a low-order approximation of the frequency-dependence of l i n e parameters, from a reduced amount of in format ion . Using the e l e c t r i c a l parameters at power frequency and dc conductor r e s i s t a n c e s , the tower conf igura t ion of the l i n e i s recons t ruc ted . This equivalent l i n e con f igura t ion permits the eva luat ion of the l i n e parameters over a wide frequency range. Rat ional funct ions are then used to approximate the frequency-dependence of these parameters over a l i m i t e d frequency range with a reduced order model. A n a l y t i c a l tes ts and t rans ien t s imu la t ions , i n d i c a t e that the model i s reasonably accurate over the frequency range of i n t e r e s t i n most p r a c t i c a l a p p l i c a t i o n s . i i Both models have been incorporated in to the UBC vers ion of the Electromagnetic Transients Program (EMTP). i i i TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGEMENT v i i PART I: VOLTAGE AND CURRENT PROFILES ALONG A TRANSMISSION LINE INTRODUCTION 2 CHAPTER 1: THEORETICAL CONSIDERATIONS 4 1.1 Description of the P r o f i l e Model i n the Frequency Domain 4 1.2 Solution i n the Time Domain 10 1.3 I n i t i a l Conditions 13 CHAPTER 2: IMPLEMENTATION OF THE SOLUTION METHOD 15 2.1 General Cosiderations 15 2.2 Internal Time Step At' 17 2.3 Approximation of A-^(OJ) 19 2.4 Animated Motion Picture of T r a v e l l i n g Waves along the Line 22 CHAPTER 3: NUMERICAL RESULTS 28 3.1 Introduction 28 3.2 Truncation and Segmentation E f f e c t s 30 3.3 Energization of a Line Terminated with a Lightning A r r e s t e r 42 CONCLUDING REMARKS 51 i V Page PART I I : LOW-ORDER APPROXIMATION OF THE FREQUENCY DEPENDENCE OF TRANSMISSION LINE PARAMETERS INTRODUCTION 5 3 CHAPTER 1: THEORETICAL CONSIDERATIONS 5 5 1.1 Evaluation of the Transmission Line Parameters 1.1.1 Series Impedance Matrix 5 5 1.1.2 Carson's Correction Terms 52 1.1.3 Shunt Admittance Matrix 5 4 1.1.4 Comparison between Exact and Approximate Formulas 5 5 1.2 Evaluation of the Equivalent Line Configuration from the Parameters at Power Frequency 68 1.3 Skin E f f e c t Correction Factor 71 1.4 Correction of the Equivalent Line Configuration when R (j c i s Known 7 4 1.5 Evaluation of the C h a r a c t e r i s t i c Impedance and Propagation Function 7 7 CHAPTER 2: RATIONAL FUNCTIONS APPROXIMATION OF THE CHARACTERISTIC IMPEDANCE AND PROPAGATION FUNCTION 7 9 2.1 Rational Functions. General Considerations 7 9 2.2 Asymptotic Approximation when the Number of Poles i s Fixed 82 2.3 Approximation of the C h a r a c t e r i s t i c Impedance Zc(a>) 86 2.4 Approximation of the Propagation Function A^to) 88 2.5 Evaluation of C 9 3 2.6 Implementation of the Method 9 4 v Page CHAPTER 3: NUMERICAL RESULTS g 5 3.1 Recapitulation g 5 3.2 Evaluation of the Line Parameters from the Equivalent Line Configuration g 6 3.3 Evaluation of A^ (w) and Z c (to) 1 0 3 3.4 Low-Order Rational-Functions Approximations 1 1 2 3.5 Frequency Domain Response 3.5.1 Introduction 1 2 8 3.5.2 Open C i r c u i t Response 1 3 1 3.5.3 Short C i r c u i t Response ^ 3 9 3.6 Transient Simulation i n the EMTP 1 5 4 CONCLUSIONS APPENDICES APPENDIX I-B: JOSE MARTI'S FREQUENCY-DEPENDENCE MODEL APPENDIX I-C: USER'S GUIDE FOR THE VOLTAGE AND CURRENT PROFILE OPTION OF THE EMTP, AND FOR THE OUTPUT DISPLAY PROGRAM APPENDIX II-A: SKIN EFFECT CORRECTION FOR ROUND CYLINDRICAL CONDUCTORS 161 APPENDIX I-A: GENERAL SOLUTION OF THE LINE EQUATIONS IN THE FREQUENCY DOMAIN 1 6 2 167 179 187 APPENDIX II-B: CARSON'S CORRECTION TERMS FOR EARTH RETURN EFFECT 1 9 1 APPENDIX II-C: SIMULATION OF THE JOHN DAY- LOWER MONUMENTAL TRANSMISSION LINE USING THE LOW-ORDER APPROXIMATION PROGRAM 196 APPENDIX II-D: USER'S GUIDE FOR THE LOW-ORDER APPROXIMATION PROGRAM 205 BIBLIOGRAPHY 212 v i ACKNOWLEDGEMENT It i s impossible to thank a l l the i n d i v i d u a l s and i n s t i t u t i o n s that d i r e c t l y or i n d i r e c t l y collaborated i n the completion of this thesis p r o j e c t . However, r i s k i n g ingratitude towards those not mentioned below, I must give s p e c i a l thanks, To the " g i r l s " , i n the main o f f i c e of the Department of E l e c t r i c a l Engineering, for often going out of t h e i r way to give me a hand. To Dr. Dommel, for his insi g h t , help and unshakeable patience. To my brother, Jose, whose genius l i t the way of our shared dream. To Debby, for her love. To my parents. v i i PART I VOLTAGE AND CURRENT PROFILES ALONG A TRANSMISSION LINE 1 INTRODUCTION When a power system i s disturbed from i t s normal steady-state operation, a t r a n s i t i o n between the o r i g i n a l state and the new steady- state conditions (with possible changes i n the network) must occur. This t r a n s i t i o n period begins with very f a s t electromagnetic transients (in the order of milliseconds), which usually involve transient overvoltages or high transient currents. This i s followed by slower electromechanical transients (in the order of seconds), which are not discussed i n this t h e s i s . Protection devices are used to l i m i t the magnitude of transient overvoltages across expensive equipment i n order to prevent damage i n this equipment and subsequent substation f a i l u r e s . The s e l e c t i o n and the co-ordination of these devices require d e t a i l e d system studies. The e f f i c i e n c y and accuracy of computer programs used for such studies, become more important as systems grow i n size and complexity. Programs such as the Electromagnetic Transients Program (EMTP) are capable of simulating a large va r i e t y of transient conditions, for systems of varying size and complexity. This program, o r i g i n a l l y develop- ed by Dommel [1], [2], has evolved and grown as simulation models have been improved by other researchers. In the l a s t decade or so, much e f f o r t has been devoted to the accurate modelling of transmission l i n e s , taking into account the f r e - quency dependence of t h e i r parameters. One such model has recently been developed and implemented by J.R. Marti [ 3 ] . The s i m p l i c i t y and computa- t i o n a l f l e x i b i l i t y of this model have provided the basic tools for the development of other re l a t e d models (in p a r t i c u l a r , those discussed i n 2 3 t h i s t h e s i s ) . Most l i n e models, ei t h e r with frequency-dependent or constant parameters, give voltage and current information only at the end points of the l i n e . This information usually s a t i s f i e s the needs for most simula- t i o n s . There are s i t u a t i o n s , however, when the information at the end points of the l i n e may not be enough to give a clear picture of the over- a l l behaviour of the l i n e (e.g., when li g h t n i n g arresters are considered). The usual p r a c t i c e , when voltages and currents at intermediate places are needed, i s to segment the l i n e i n t o shorter sections, and to examine the response at the end points of each segment. This procedure, however, can be inaccurate and i s c e r t a i n l y time consuming. Very few attempts to determine the voltage and current p r o f i l e (without e x p l i c i t l y segmenting the line) can be found i n the l i t e r a t u r e . The only model found [4], which takes frequency-dependence into account, i s based on Semlyen's approach to the frequency-dependence problem [5]. In this model, the voltages and currents are found at only one intermedi- ate point along the l i n e . A p r o f i l e can thus be obtained by repeated c a l c u l a t i o n s with d i f f e r e n t segmentation r a t i o s . It i s the purpose of the f i r s t part of this project, to develop a p r o f i l e model for any number of equally spaced points along the l i n e . Such a model should meet the following requirements: 1. It should be more accurate than the conventional "external" segmentation of the l i n e . 2. The computer routines must be designed to permit a simple and e f f i c i e n t interconnection with the frequency-dependence version of the EMTP. 3. The routines should be easy to use, and should require a minimum amount of e f f o r t on the part of the user. CHAPTER 1 THEORETICAL CONSIDERATIONS 1.1 Description of the P r o f i l e Model i n the Frequency Domain Consider an m-phase transmission l i n e with ground return, as represented in Figure 1 .1 k i o- L<2 °- k m ° - g c- -o -O IDo -o m m -o g F i g . 1 . 1 : Representation of an m-phase transmission l i n e with ground return After the appropriate modal transformations have been made, as described i n appendix I-A, the behaviour of the l i n e can be analyzed i n the modal domain, where i t can be studied as a set of m, single-phase l i n e s . Each of these single-phase l i n e s can then be described in terms of i t s modal surge impedance Z c (u) and propagation function A-|(u)). The equations obtained from this single-phase model, are applicable to any l i n e mode, and therefore s u f f i c i e n t to describe the l i n e i n the phase domain. Consider then, the representation of one of the l i n e ' s propaga- tio n modes, shown i n Figure 1.2 4 5 iU(t) •m (t) vu(t) o m v m ( t ) F i g . 1.2: Modal representation. In the frequency domain, th i s l i n e can be described by the (frequency-dependent) model of Figure 1.3, as described i n appendix I-B. F i g . 1.3: Line model i n the frequency domain. where, Bk = V k - I K Z e q Im zeq Fk = V K + I K Z e q (backward t r a v e l l i n g functions) (forward t r a v e l l i n g functions) (1.1) (1 .2) The r e l a t i o n s h i p between forward and backward t r a v e l l i n g functions i s given by Bk = A 1 ,c Fm where, = e (1.3) (1 .4) Y(CO) = \ / z ' y Z' i s the modal series impedance (R'( w) + jwL'(w) ) per unit length (fi/km) Y' i s the modal shunt admittance (G1 + jaC 1) per unit length (mhos/km) &c i s the li n e length i n km If we connect i n cascade (n + 1) of the c i r c u i t s i n Figure 1.3, each representing a l i n e segment of length £, we obtain the equivalent c i r c u i t of Figure 1.4 -eq m o « F i g . 1.4: Cascade connection of (n + 1) equivalent l i n e c i r c u i t s . Note that to connect (n + 1) segments of length & i s equivalent to divide a l i n e of length £ c = £(n + 1) in t o (n + 1) segments of length I. This r e s u l t s i n n nodes with n intermediate voltages and currents. The following d e f i n i t i o n s can e a s i l y be extended from equations (1.1) to (1.4) for the forward and backward t r a v e l l i n g functions of each segment j . Bk = V k - I k Z e q &m - Ym _ Jm z e q (1 .5) B k , j ~ v j - ! j z e q Bm,j ~ v j + I j z e q (1 .6) Fk = V k + I k Z eq Fm _ vm + Im z e q (1 .7) F k , j = v j + x j z e q Fm,j = Vj * j zeq» (1 .8) where Vj and Ij (j = 1,2,..., n) are the voltages and currents at each of the n intermediate nodes. The relationships between forward and backward t r a v e l l i n g functions can be summarized as follows, Bm,1 = Ai 1=\ Bk,n = A1 F m B m,j = A1 Fk,j-1 (j = 2, n B k > j = A-j F m f j + 1 (j = 1, ..., n-1 ), (1.9) (1.10) (1.11) (1.12) where, and, Ai = e - Y ( u ) U (1.13) I = ftc/(n + 1) Note that I i s the length of a segment, and lc i s the length of the complete l i n e . Also note that the re l a t i o n s h i p s between the end 8 points of the l i n e (equations (1.1) through (1.4)) s t i l l hold true, regardless of the number of segments connected. From the equivalent c i r c u i t of Figure 1.4, Vj = Ij Z e q + B k f j Vj = - I j Z eq + Bm>j. Adding up these two equations, 2 Vj = B k > j + B n i / (1.14] Also, from the equivalent c i r c u i t , 2 Ij Z e q = B ^ j - B k f j . (1.15) Addition of (1.14) and (1.15) gives, Vj + Ij Z e q = Bjj^ j . Subtraction of (1.14) from (1.15) gives, Vj - U Z j Z e q - B k f j Comparing the l a s t two equations with (1.8) we f i n a l l y obtain, Fk,j = B m / j (1.16) Fm,j = B k , j . (1.17) Equations (1.11) and (1.12) can now be expressed i n terms of backward t r a v e l l i n g functions alone, Bm,j = A1 Bm,j-1 U = 2> "" n ) (1.18) B k , j = A1 Bk,j+1 (D = 1» n + 1 ) • (1.19) I f the intermediate backward t r a v e l l i n g functions Bj^j and Bj^j are known, the intermediate voltages and currents Vj and I j are uniquely determined by equations (1.14) and (1.15). Also, B k > j and B ^ j can be determined i f the forward t r a v e l l i n g functions l\ and F m (end points) are known. For example, with F k known from equation (1.9), B j ^ i can be obtained; with BJJ,^ known, B ^ j (j=2,...,n) i s determined by equation (1.18). An analogous procedure can be followed for B k f j , s t a r t i n g from Fm« 9 At this point i t can be seen that the only information needed to evaluate the i n t e r n a l voltages and currents i s given by and F m . This permits the evaluation of the intermediate voltages and currents i n a way which i s independent of the solution method being used for the complete l i n e . This presents c e r t a i n advantages that w i l l be discussed l a t e r . The propagation function A-|(td) i s i d e n t i c a l for each segment, and i t i s approximated by r a t i o n a l functions i n the same way as for the propagation function A - | f C ( tu) of the complete l i n e (see appendix I I - B ) . The p a r t i a l f r a c t i o n s expansion of the r a t i o n a l functions y i e l d s , A 1 U ) = _ J i l _ + _ J l V + . . . + _ J E V e " J U C ( K 2 0 ) s + B-| s + 3̂ s+Sm Since the surge impedance i s independent of the length of the l i n e , the same approximation used for the solution of the complete line i s used for the evaluation of the p r o f i l e , that i s , r, i \ v , k i A k T , , k m * (1 .21 ) Z c (to) = K o + 1 + * — + . . . + — s + a i s + a 2 s + a m ' 10 1 .2 Solution i n the Time Domain The intermediate voltages and currents are determined by the backward t r a v e l l i n g functions, and these can be sequentially determined from the forward t r a v e l l i n g functions at the end points of the l i n e . Consider equation (1.9) V i = A1 Fk« The time domain counterpart i s obtained using the Inverse Fourier Transform, b m > 1 ( t ) = a i * f k ( t ) . (1 .23) The convolution (indicated by "*") can be solved by recursive methods, as explained i n appendix II-B, giving, m bm,l(t> = . Z i b m / 1 > i ( t ) (1.24) with, b where, g^, c^ and d i are the integration constants from the convolution i n t e g r a l (1.23), and are given by, - S i At B , 1 , i ( t ) = 9 i V 1 f i ( t - A t ) + C i f k ( t - r ) + d i f K ( t - r - A t ) (1.25) 9i h i = d i = 1 - 9 i Si At k j (1 - hj) _ k j ( g j - hj) h k i and $j_ are obtained from the rational-functions approximation of A-] (u) for the length of one segment (see equation (1.20)). S i m i l a r l y , b k / n ( t ) i s obtained from equation (1.10), 11 b k , n ( t ) = a 1 ( t ) * f m ( t ) (1.26) m b k , n ( t ) = Z b k , n , i ( t ) (1.27) i=1 with, bk,n,i{t) = * i bk,n,i( t"At) + c ± f m ( t - r ) + d i f m ( t - r - A t ) . (1.28) The intermediate functions b j ^ j f t ) and b m ^ j ( t ) are obtained i n a si m i l a r fashion from equations (1.18) and (1.19). b m , j ( t ) = al<t > * V j-1 ( t ) ( 1- 2 9) m b m , j ( t ) = .1 b m , j , i ( t ) (1-30) 1-1 hm,j,i^^ = 9 i b m , j , i ( t - A t ) + C i b ^ j ^ t t - c ) + d ± b m # j _ } ( t - r -A t) , (1.31) and, b k / j ( t ) = a i ( t ) * b j ^ j + ̂ t ) (1.32) b k , j m (t) = .|1 b k , j , i ( t ) (1.33) b k . j . i t t ) = 9i b k , j , i ( t - A t ) + C i b k / j + 1 ( t - r ) + d i bkfj+1(t-r-At) (1.34) As mentioned i n the previous section, with b j ^ j and b ^ j known, the intermediate voltages are obtained using equation (1.14) i n the time domain, V j ( t ) = - | ( b k > j ( t ) + b m # j ( t ) ). (1.35) The intermediate currents can be obtained from equation (1.15) i n the time domain 2ej(t) = b m > j ( t ) - b k f j ( t ) , (1.36) where, e j ( t ) = i j ( t ) * z e q ( t ) . (1.37) 12 From equation (1.21), z e q ( t ) = [ k Q 6(t) + k, e " " 1 1 + ... + k m e " a m , t ] u ( t ) . (1.38) Introducing (1.38) into (1.37) m1 e j ( t ) = e j ? 0 ( t ) + | e j f i ( t ) , (1.39) where, e j f 0 ( t ) = k D i j ( t) (1 .40) e. .(t) = m, e. •(t-At) + p. i . ( t ) + g. i . ( t - A t ) . (1.41) The c o e f f i c i e n t s f o r the recursive convolution of equation (1.37) (see appendix I-B) mi, Pi and gi are given by, - a , - At = e ^ h 1 - mj P 1 aj_At = kj (1 - hj) g i a l Introducing equations (1.40) and (1.41) into (1.36) we obtain, after some algebraic manipulations, m' i . ( t ) = i r i ( b m i ( t ) - b k j ( t ) ) - q ijlt-At) - E mi e j i ( t - A t ) 3 p L 2 J i = 1 (1 .42) where, m* p = k 0 + Z p i 1 = 1 m" q = j S q ± 13 1 .3 I n i t i a l Conditions With equations (1.36) and (1.42), the current and voltage p r o f i l e s on a given l i n e can be obtained when the past h i s t o r y terms of the forward t r a v e l l i n g functions for the end points ( f j and f m ) are known. Note that a record of past h i s t o r y terms for the intermediate parameters must be kept for the s o l u t i o n . When t = 0̂ these h i s t o r y vectors must be evaluated from the i n i t i a l conditions of the simulation. If the currents and voltages at the ends of the l i n e are zero, a l l the h i s t o r y terms are set i n i t i a l l y to zero. If l i n e a r , ac steady-state conditions e x i s t p r i o r to t = 0, the past h i s t o r y terms can be evaluated from the phasor quantities associated with e j , t>m,j' bk,j a n c^ ^k» ej (t) <=J 1> Ej (co) b m , j ( t ) < a ^ V j ( c o ) b k # j ( t ) - — ^ V j ( w ) i j ( t ) «a o T j U ) From equation (1.20) m A 1 (co) = E A. . (co), 1 i=1 1 , 1 where, A .(co) = , k l , e 1»1 j co + 8^ Since F^ and F m are known from the steady-state s o l u t i o n of the system, B K > j -̂j_ ( a n d Bj^-j f-j_ (<»>) can be obtained from equations (1.9), (1.10), (1.18), and (1.19), V i , i ( a ) ) = A1 , i ( w ) *k ( U ) ) (1.43) B m . H(co) = AT Am) B . .(co) (j=2,...,n) (1.44) 14 V n , i ( " > = A 1 , i > > V U > <1'45> 5 k , j , i ( & ) ) = A 1 , i U ) ^ t , j + l , i ( a ) ) (j = 1i...»n-1)(1.46) The intermediate steady-state currents can be found using equation (1 .15) I, (co) = V i ' " ) - V j ( " > , 2 Z (to) eq (1.47) with Zeg((o) given by equation (1.21). Also, from equation (1.21) Z e q(to) = k o + 111 + ... + km' Z e q / i ( a i ) = Introducing Z (to) i n t o equation (1.37) i n the frequency domain E-(to) = Z (to) 7 . (to) (1.48) J et3 J results i n , E. .(to) = Z e„ H(to) 7 , ( 0 ) ) . (1.49) The time domain equivalents of these phasor quantities are then given by, b b , D ( t ) " • (to)| c o s (tot + a r g ( B • (to) ) ) V j ( t ) = w . (to)| c o s (tot 1J 1 + a r g ( B k j j (to)) ) i j ( t ) = (to)| cos(tot + a r g d j (to)) ) e 3 , i ( t ) = f i(to)| c o s (tot + a r g ( E ^ H (to)) ) j / -1- Note that i n the case of dc i n i t i a l conditions ( i . e . , trapped charge), to i s simply set to zero i n the equations above. CHAPTER 2 IMPLEMENTATION OF THE SOLUTION METHOD 2.1 General Considerations The model and equations described i n Chapter 1 have been imple- mented i n t o UBC's frequency-dependence version of the EMTP. Since the p r o f i l e model requires a r e l a t i v e l y small amount of information from the sol u t i o n of the l i n e at the end points, the interconnection with the EMTP i s very simple and straightforward. Less than 90 FORTRAN statements had to be added to the main program, and a l l p r o f i l e c a l c u l a t i o n s are made in subroutines. The p r o f i l e c a l c u l a t i o n s proceed simultaneously with the solu- tion for the end points (actually, there i s a delay of one time step i n order to avoid numerical i n s t a b i l i t y when the time step i s very close to the t r a v e l time of the complete l i n e ) . This method was preferred over the a l t e r n a t i v e of post-processing, in order to decrease memory requirements and to increase computational e f f i c i e n c y . The p r o f i l e model presents some important advantages over the external segmentation of the l i n e . The most obvious one i s that, from the user's point of view, i t i s easier and faster to request p r o f i l e c a l c u l a - tions with only one command, (see appendix I-C) than i t i s to set up several l i n e segments. The number of numerical convolutions (one of the most time consuming operations in the frequency-dependent solution of a line) i s reduced by a factor of (3n + 4)/(4n + 4), where n i s the number of i n t e r - mediate nodes. 15 16 The di f f e r e n c e , although not spectacular for a small number of intermediate points, can be considerable for r e l a t i v e l y large values of n. These savings range from 12.5% f o r 1 intermediate node, to 24.75% for 100 intermediate modes. The most important advantage, however, i s the increase i n accu- racy. Some of the computational aspects leading to this improved accuracy are discussed next. 17 2.2 Internal Time Step A t 1 In the solution of transmission l i n e s at d i s c r e t e time i n t e r v a l s , the main source of e r r o r i s the f a c t that the time step At i s not an exact f r a c t i o n of the t r a v e l time r of the l i n e [6]. This error, in the case of frequency-dependent solutions, occurs when the past hi s t o r y terms f ( t - r ) and f ( t - A t - r ) are evaluated. If r / A t i s not an integer number, li n e a r or higher-order i n t e r p o l a t i o n i s needed. For example, f(t-c) must be interpolated between f ( t - C - A t ) and f ( t - r + A t ) . The i n t e r p o l a t i o n error can be reduced by using a small A t , or i t can be eliminated by choosing A t to be an exact submultiple of C. The p r o f i l e routines s e l e c t an i n t e r n a l time step A t ' that i s the c l o s e s t (and largest) submultiple of I for the l i n e mode considered. Although there i s s t i l l a c e r t a i n amount of i n t e r p o l a t i o n involved when f ( t - A t ' - r ) and f ( t-C) are evaluated a t the end points, no a d d i t i o n a l i n t e r p o l a t i o n i s needed for the intermediate segments. The accuracy gained, in comparison with the external segmenta- t i o n method, increases with the number of sections considered. The improvement becomes more evident when the l i n e has more than one phase, and i t i s no longer possible to choose an external A t which i s a sub- multiple of l for a l l propagation modes. The use of an i n t e r n a l time step slows down the p r o f i l e c a l c u l a - tions because the number of time steps needed i s greater than, or equal to the number of time steps necessary to solve the l i n e at the end points. Note that the intermediate currents and voltages for each mode w i l l be evaluated at s l i g h t l y d i f f e r e n t points i n time, therefore, an a d d i t i o n a l i n t e r p o l a t i o n i s needed to obtain the phase domain q u a n t i t i e s . The i n t e r p o l a t i o n for phase voltages and currents introduces a minimal amount of error because i t i s not a cumulative process: the phase 18 voltages for one section are not needed to evaluate the phase voltages of the next section; and once a l l the voltages are obtained for one time step, they are no longer needed for any future c a l c u l a t i o n s . When the l i n e i s segmented externally, care must be taken to choose a time step that i s smaller than the tr a v e l time of the f a s t e s t propagation mode of the shortest section. Therefore, as the number of intermediate sections increases, At must be decreased accordingly. When the segmented l i n e i s part of a l a r g - er system, this reduction i s usually not j u s t i f i a b l e for the rest of the components of the network. This results i n an unnecessary (and often p r o h i b i t i v e ) increase i n computing costs. This problem i s a l l e v i a t e d with the p r o f i l e routines, because the decrease in A t i s confined to the p r o f i l e solution only, while the rest of the network can be solved with a larger A t . When (for a given mode) At i s larger than the tr a v e l time of the smallest section, the in t e r n a l time step A t 1 i s automatically set equal to i. If larger accu- racy i s desired, the p r o f i l e routines provide the option to decrease At' to At'/ m» where m i s an integer number supplied by the user (see appendix I-C) . 1 9 2 . 3 Approximation of A-| (co) The ra t i o n a l - f u n c t i o n s approximation of Ai(co) can be obtained with very high accuracy over the frequency range of i n t e r e s t i n transient studies. However, with the best approximations attainable with the (*) approximating routines presently available at UBC , peak errors of 0 . 5 % are not uncommon for values of | A-|(co)| between 1 . 0 and 0 . 1 (higher errors i n the region where | A-j(to)| < 0 . 1 are not s i g n i f i c a n t i n most a p p l i c a t i o n s ) . These errors tend to accumulate when the l i n e i s segmented. Suppose, for example, that a 5 0 0 km l i n e i s segmented into 1 0 sections. Let | A^co^)) and | A^co^] be the actual magnitudes of A-| (co) at co = COq for a 5 0 km and 5 0 0 km l i n e , r e s p e c t i v e l y . I W l 5 0 = 0 . 9 8 0 I A 1 5 0 0 = 0 . 8 1 7 0 7 Let us now suppose that the approximation of | A.| (to )| i s off by 0 . 5 1 % at CO = C0Q; IV'VUo = 0 . 9 8 5 If we calculate | A 1 1 (coQ)| ^Q Q from | • (O J 0)| ̂ Q we obtain, I VK>>I 5 0 0 = [ l A1<"o>l 5 0 ] 1 ° = ° ' 8 5 9 7 3 Comparing | Ai'(to 0)| 500 with | A-|(co0)| 5 0 Q we can see that an error of 0 . 5 1 % i n the approximation of the 5 0 km segment, yields an error of 5 . 2 2 % i n the estimated value of j A-| (wQ)| 500. (*) Jose R. Marti has been working (at the time t h i s thesis i s being written) on more refined versions of the approximating routines a v a i l a b l e at UBC. Preliminary r e s u l t s have been encouraging, and the new routines can be expected to y i e l d even more accurate approximations i n the near future. The preceding example i l l u s t r a t e s how the errors i n the approximation of A-|(io) accumulate when several lirtes are connected i n cascade. Note that the error increases with the number of sections, as the error of one section i s transmitted to the next. This i s an unavoidable source of error in any c a l c u l a t i o n i n v o l v i n g l i n e sectioning. The model developed i n Chapter 1 i s also a f f e c t e d by the q u a l i t y of the approximation of A-|(u>), but to a lesser extent than the external segmentation procedure. Since the evaluation of the convolutions a-| (t) * b k f j (t) and a-| ( t ) * b m # j (t) s t a r t at opposite ends of the l i n e , (see section 1.2) the error does not accumulate at one end only, but i t i s averaged along the l i n e . This can be more e a s i l y v i s u a l i z e d i n the frequency domain from equations (1.18) and (1.19) Bm,j " A l B m , j - l ( j = 2 n ) B k , j = A l B k , j + l ( j = 1 n - X ) Suppose that the l i n e has 10 intermediate nodes. Starting from node k , B B B B m, 1 " A l F k m, 2 " A l m,l A 2 F A l Fk ! , m, 3 = A l m, 2 A l 3 F k m,10 = A l Bm,9 = A 1 0 F A l F k S t a r t i n g from node m, B k,10 A l F m k,9 A l Bk,10 = A * F 1 m k,8 A l \,9 = A / F 1 m k , l A l Bk,2 10 = A. F 1 m It can then be seen that, when the error i s maximum i n i s minimum i n Bm/j and viceversa. The r e s u l t i s an approximately constant error l e v e l for a l l sections. 22 2.4 Animated Motion Picture of T r a v e l l i n g Waves Along the Line Suppose that the intermediate voltages (or currents) at a given time t are known. If the number of intermediate points i s large enough, i t i s possible to p l o t these voltages against the length of the l i n e , and obtain a smooth curve, as shown i n Figure 2.1 F i g . 2.1: Voltage p r o f i l e 3.6 ms a f t e r the energization of an open ended, single-phase l i n e . This p l o t of v(t) vs Jl can be v i s u a l i z e d as a s t i l l picture of a voltage wave at time t . If several of these s t i l l frames were displayed sequentially, for increasing values of t, the e f f e c t of "a moving wave" could then be created. To demonstrate the p o s s i b i l i t i e s of this procedure as a teaching t o o l , an animated movie of t r a v e l l i n g waves along a transmission l i n e was produced as part of this p r o j e c t . For the generation of this movie, the l i n e from John Day to Lower Monumental of the Bonneville Power Administration (BPA) was used as an example. To demonstrate the behaviour of the d i f f e r e n t propagation modes, the l i n e was assumed to be single-phase, either with zero or p o s i t i v e sequence parameters. Several transient sit u a t i o n s were simulated on the l i n e , and voltages and currents were obtained at 49 intermediate points (51 points per time step a f t e r including the end p o i n t s ) . These voltages and currents were plotted against the l i n e length (as shown i n Figure 2.1) at every time step on the IBM 3279 colour terminal, and photographed with a 16 mm movie camera. The number of exposures per p l o t was co-ordinated with the time step of the transient simulation to create a r e l a t i v e l y smooth motion at a projection speed of 18 frames per second. The r e s u l t i s a movie (13 min. long), i n which transient phenomena are seen as waves propagating on a l i n e (*). Figure 2.2 shows a few selected frames extracted from the movie. The s i t u a t i o n simulated i n this case was the i n j e c t i o n of a unit voltage pulse of 0.5 ms duration into the 500 km l i n e , with the receiving end open. Positive sequence parameters were used i n this p a r t i c u l a r case. This movie permits a v i s u a l i z a t i o n of the transient phenomena i n a way which i s very d i f f i c u l t to obtain from the usual p l o t s (voltage or current vs time at any fixed point on the l i n e ) . The p o s s i b i l i t i e s as a teaching aid are considerable, and the benefits as an analysis tool cannot be l i g h t l y disregarded. (*) Copies of this movie can be borrowed by contacting Dr. H.W. Dommel i n the Department of E l e c t r i c a l Engineering at UBC. 24 2.0 1 l.S 1.9 o.s-J 0 -0.3 -1.0 -1.5 -2.0 0 . 6 7 m s l.S- i.o - o.s- o • -0.5- -1.0- - l . 5 : 1.17 m s o so ico 1 5 0 z ; o 250 s : o 3 5 0 <ao <so 5 0 0 - 3 , _ _ OISr.lMCS (RILOKEIERSl ' 0 5 0 , C3 ! 5 D 203 2Zi 333 ?:0 <'.! "S3 5:3 OlS'S-iCt tK;:_5r.iTE*Si 2.3 1 1.5 1.9 0.5 -1.0-j -1.5 -2.0 1.64 m s 0 50 I C C 150 200 250 200 ;f,0 4150 450 500 OlSrSKCE IK:L3J- .£T£B5] 2.0 l.S-J 1.0 -o.s -1.0 -1.5 -2.0 4 1.85 m s 50 100 150 200 250 300 3:0 400 433 530 OISTRKCE I X : L E K £ T E S S ! -O.S- -1.0 - -1.5- -2.0 4——i r — . • i , , , 0 SO 100 ISO 200 250 300 353 430 450 500 DISTANCE (KILOMETERS) -0.5- -1.3 • -1.5- -2.9 1 , 1 , , , 0 SO 100 J50 200 253 333 353 4C3 450 505 OISIRSCE tPCLO-.ETERS) F i g . 2.2: Selected frames from the T r a v e l l i n g Waves Movie. 25 2.0- 15- '1 0.5 4 2 .45 m s -0.5- -1.0 • -1.5- S0 100 ISO 200 230 300 T.O 400 450 500 OISTSKCc IK.'LWETERSl 2.0 -j >S 1.0 0.5 -0.5 -1.0 -i.s-3 -2.0 2.92 m s 0 50 100 ISO 200 250 3:3 350 433 <50 50: OISTRNCt (KILCIETESS) 3.22 m s S 50 100 150 203 250 300 350 433 -50 500 0IS7RNCE IKILCMETERSI 2.0 : 1-5- 1.0 - 0.5 -0.5 -1.0 -1.5-3 3.63 m s 0 SO 100 ISO 200 25" 300 350 "SO *S0 500 DISTANCE (KILOMETERS: 2.0: I.S-j l.o • 0.S-; -0-5 - -2.0- 3.73 m s SC 100 ISO 200 2SC 300 350 400 4S0 500 oisrivcE UILGKETESSI 2.0 -j i.s-i 1.0- 0.5- 0- -0.5- -1.0- -1.5- -2.0 3.89 m s 0 SO 100 ISO 200 250 303 350 400 450 S00 OISTRNCE (KILOMETERS) F i g . 2.2: (continuation) 2.0 '1 0.5 4 .70 m s 0 50 100 150 200 2S0 300 350 «0 <S0 500 0ISTRNCE iKilOMETEnS) 2.0 i .H 1.0 o.H -0 .5 -1 .0 -i-H -2.0 5.17 m s SO !00 150 200 250 300 350 DISTANCE (KILOMETERS) I 1 1 -i.u-i "i i 1 i 1 0 50 100 150 200 250 300 350 CO OO 53J 0 50 100 ISO 200 250 300 350 DISTANCE (KIL3-.ETESS) DIS73MCE (KILC.1ETERS1 F i g . 2.2: (continuation) 27 For example, when a voltage l i m i t i n g device such as a lightning arrester i s connected to the end of the l i n e , i t i s sometimes erroneously assumed that the voltages along the l i n e w i l l not exceed the voltage determined by the c h a r a c t e r i s t i c of the l i g h t n i n g a r r e s t e r . To prove t h i s assumption wrong, i t i s s u f f i c i e n t to segment the l i n e and to observe some of the intermediate overvoltages. However, when the voltage wave i s seen r e f l e c t i n g back and f o r t h along the l i n e , the i n t e r p r e t a t i o n of the phenomena becomes much simpler. An animated movie can obviously not be produced every time a transient simulation i s performed, but such movies can be made for selected cases for teaching purposes. The best s o l u t i o n for routine v i s u a l i z a t i o n of power transients would be to display the wave motion on a suitable graphics terminal. At the time t h i s thesis i s being written, some work i s being done i n this d i r e c t i o n i n the Department of E l e c t r i c a l Engineering at UBC, using the Megatek 4000, f a s t - r e f r e s h , graphic s t a t i o n . I t can therefore be expected that in the near future, routine examination of wave motion w i l l become possible a f t e r the execution of a transient simulation. CHAPTER 3 NUMERICAL RESULTS 3.1 Introduction The r e s u l t s from a series of tests and comparisons w i l l be shown in this chapter i n order to assess the performance of the p r o f i l e model. In these tests the parameters of a t y p i c a l 500 KV l i n e w i l l be used ( i . e . , BPA's John Day to Lower Monumental transmission l i n e ) . The length of the l i n e i s assumed to be 500 km. The tower configuration i s shown i n Figure 3.1, and the physical c h a r a c t e r i s t i c s of the conductors are l i s t e d below: a) Phase conductors dc resistance = 0.032405 H/km tube thickness/outside diameter = 0.3636 (stranded conductor i s approximated as a tube, with the e f f e c t s of the s t e e l core ignored) diameter = 4.0691 cm b) Ground wires (assumed to be semented or "T-connected" ^*^) dc resistance = 1.6218 fi/km tube thickness/outside diameter =0.5 diameter = 0.98044 cm (*) Grounded at one tower, and insulated, as well as series interrupted at the adjacent towers. This arrangement provides e l e c t r o s t a t i c s h i e l d i n g (ground wires considered i n capacitance c a l c u l a t i o n s ) , but eliminates c i r c u l a t i n g currents (ground wires ignored i n impedance c a l c u l a t i o n s ) . 28 c) Ground r e s i s t i v i t y = 100 Q-m 29 3.93 m 3.93 m 1 c»|̂ c 30.02 m 23.62 m 6.55m | 6.55m c» Ua 0.457 m 15.24 m F i g . 3.1: Tower configuration of BPA's John Day to Lower Monumental 500 KV transmission l i n e . Height shown i s average height above ground. 30 3.2 Truncation and Segmentation E f f e c t s The two major sources of error i n the c a l c u l a t i o n of intermedi- ate voltages and currents are the truncation errors due to l i n e a r i n t e r - p o l ation i n the forward t r a v e l l i n g functions (caused when r/At i s not an integer number), and the segmentation errors caused by the accumulation of errors from the solution of one section to the next. These two e f f e c t s w i l l be i l l u s t r a t e d i n t h i s s e c t i o n . For t h i s purpose, a simple energization test w i l l be performed, with the receiving end being open (see Figure 3.2) t = 0 COS ( u j t ) 0 m -o F i g . 3.2: Energization t e s t . The l i n e w i l l be assumed to be single-phase with p o s i t i v e sequence parameters. The length of the l i n e i s 500 km, and 10 sections or 9 intermediate nodes w i l l be considered. In order to i s o l a t e truncation from segmentation e f f e c t s , for the f i r s t set of tests the time step At has been chosen to be exactly 1/20th of the t r a v e l time,- t h i s implies that A t =1^/2 (where r^0 i s the t r a v e l time for a 50 km section), and that no truncation errors w i l l be present i n the r e s u l t s . 31 It was mentioned i n Chapter 2 that the main source of segmenta- t i o n errors i s the approximation of the propagation function for each l i n e segment ( 5 0 km i n this case). Graph 1.1 shows the error function of the ratio n a l - f u n c t i o n s approximation of j A-j ( OJ)| and | A-] ( to)| (with the output from the l i n e constants program being used as the reference), f o r values of | A]_ (to) | between 1 . 0 and 0 . 1 . Lower values of |A1(to)| are not s i g n i f i c a n t f o r the purposes of t h i s p r o ject. Graph 1.2 shows the error function for [| A-|(to)| ] 1 0 compared to the rational-functions approximation of | A-j (to)| 500^* T n e t r a v e l time for the 5 0 0 km l i n e i s 1 . 6 7 9 8 ms, while r^Q i s 1 . 6 7 9 ms. The difference be- tween I O . C ^ Q and rE^QQ i s very small i n this case (approximately - 0 . 0 7 4 % ) . Graph 1 .3 shows the receiving end voltage when the l i n e i s ext e r n a l l y segmented (compared to the unsegmented l i n e ) . The assessment of the accuracy of the p r o f i l e model presents a p r a c t i c a l d i f f i c u l t y because i t i s not clear what reference or accurate results should be used for comparison purposes. If only single-frequency signals were injected into the l i n e , an exact t h e o r e t i c a l response could be obtained using the solution of the li n e equations in the frequency domain (see appendix I-A). However, this would be a rather impractical and time consuming process. A simpler a l t e r n a t i v e (although not as accu- rate) would be to use only two li n e segments and adjust the respective lengths to the intermediate point of i n t e r e s t . This should give a reason- ably accurate reference model, assuming that the p a r t i t i o n i n g of the li n e into two segments does not introduce s i g n i f i c a n t e r r o r s . In the following simulations such two-segment models w i l l be used as a reference for comparison purposes. 5.0 n FREQUENCY (HZ) ( a ) Graph 1.1: Error functions f o r the magnitudes of (co)|. (a) 50 km. (b) 500 km. FREQUENCY (HZ) Graph 1.3: Receiving end voltage. Externally segmented vs unsegmented l i n e . 34 Graphs 1.4 through 1.9 show the voltages at 450, 250, and 50 km from the receiving end, using both the p r o f i l e and the 10-segment models. Note that the error i n the 10-segment model decreases with the distance from the sending end, while the p r o f i l e presents a r e l a t i v e l y constant error (see section 2.3). Also note that the p r o f i l e model gives c o n s i s t e n t l y better r e s u l t s than the 10-segment model. Let us now consider truncation e f f e c t s . Graph 1.10 shows the e f f e c t (on the unsegmented line) of using a time step that i s not a sub- multiple of the t r a v e l time (in this case At = 0.1 ms, that i s , f ^ / A t = 1.68). The differences are not very large for the unsegmented l i n e , but when the l i n e i s segmented into 10 sections the error i s considerably larger (see Graph 1.11). The p r o f i l e model i s not very s e n s i t i v e to the external At; as a matter of fact i t i s only affected to the same extent the unsegmented l i n e i s affected (see Graphs 1.12 and 1.10; these graphs suggest that the trun- cation errors were introduced by the solution of the unsegmented l i n e and not by the p r o f i l e model). Graph 1.14, further i l l u s t r a t e s that even when the external At i s not a submultiple of £ (as i t occurs i n three-phase cases) the p r o f i l e model performs adequately. The running costs for the previous test are shown in Table 3.1 (based on UBC's rates of $1200 per hour of CPU time) Number of variables requested i n output 10 segments (cc $) P r o f i l e Model Main Program (cc $) Output Processing (cc $) Total (cc $) 1 0.67 0.68 0.08 0.74 9 0.76 0.68 0.10 0.78 18 0.86 0.68 0.18 0.86 Table 3.1: Computing costs for the single-phase energization t e s t . 35 Note that the p r o f i l e c a l c u l a t i o n s were only performed once (see appendix I-C) and a l l the voltages and currents were stored (written i n free format) i n an intermediate f i l e . The desired amount of intermediate variables are read from this f i l e and written i n a manageable form with the aid of a post-processing program. Also note, that i n the segmented case, a complete simulation has to be made every time the requested output changes. Also i n the segmented case, the i d e n t i f i c a t i o n (and manipulation for l a t e r p l o t t i n g ) of the output variables becomes d i f f i c u l t (at best) when more than 10 variables are printed i n the same run. When the number of l i n e s (or branches) increases, the p r o f i l e c a l c u l a t i o n s are compara- t i v e l y faster and the post-processing of the output almost becomes a necessity. The running costs for a two-phase, 10-section case are shown in Table 3.2 below. Number of variables 10 segments P r o f i l e Model requested i n output (cc $) Main Program (cc $) Output Processing (cc $) Total (cc $) 18 0.90 0.70 0.10 0.80 36 1 .00 0.70 0.15 0.85 Table 3.2: Running costs for the energization of a two-phase l i n e with 9 intermediate nodes. 36 4.00 -1 0.0050 0.0100 TIME (SECONDSI 0.0150 0.0200 Graph 1.4: Voltage at 450 km from the sending end. Two segments vs p r o f i l e model; At = r/20. o CM cr t— n UJ o L3 CO CO > CO I CNJ LU CJ d O > 0.0050 0.0100 TIME (SECONDS) 0.0150 0.0200 Graph 1.5: Voltage at 450 km from the sending end. Two segments vs 10 segments; At = r/20. 37 Graph 1.6: Voltage at 250 km from the sending end. Two segments vs p r o f i l e model; At = t/20. 4 . 00 - i ^ - 3 . 0 0 - 3 . 0 0 -\ - 4 . 0 0 0 . 0 0 5 0 0 . 0 1 0 0 TIME (SECONDS! 0 . 0 1 5 0 0 . 0 2 0 0 Graph 1.7: Voltage at 250 km from the sending end. Two segments vs 10 segments; At = r/20. 38 4.00 -i CM \ S 3 . 00 - i— II t— S 2 • 00 LU CJ £ - 3 . 0 0 : o > -4. 00 -j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j 1 , , , 1 •0 0.0050 0.0300 0.0150 0.0200 T J ME (SECONDS) Graph 1.8: Voltage at 50 km from the sending end. Two segments vs p r o f i l e model; At = r/20. Graph 1.9: Voltage at 50 km from the sending end. Two segments vs 10 segments; A t = r/20. „ 4.00-] 3 . 0 0 2 . 0 0 1 . 00 A 0 4 -l .oo H o - 2 . 0 0 -\ > - 3 . 0 0 - 4 . 0 0 -r 0 . 0 0 5 0 0 . 0 1 0 0 TIME (SECONDS) 1 1 ' 0 . 0 1 5 0 0 . 0 2 0 0 Graph 1.10: Receiving end voltage, unsegmented li n e , At = t/20 vs At = r/16.8. Graph 1.11: Receiving end voltage,10 segments vs -unsegmented l i n e (C=At/16.8). 40 o > - 4 . 0 0 . 0 0 5 0 0 . 0100 T IME (SECOND5I 0 . 0 1 5 0 0 . 0 2 0 0 Graph 1.12: Voltage at 450 km from the sending end, p r o f i l e model. At = r/20 vs At = f/16.8 x C O C O 2 M C O to n o CO o in LU CJ) cr o > 4 . 0 0 - , 3 . 0 0 H 2 . 0 0 H - 4 . 0 0 . 0 0 5 0 0 . 0 1 0 0 T I M E [SECONDSI 0 . 0 1 5 0 0 . 0 2 0 0 Graph 1.13: Voltage at 450 km from the sending end. Two segments vs p r o f i l e model. At = c/16.8 Graph 1.14: Voltage at 450 km from the sending end. Two segments (At = r/20) vs p r o f i l e model (At = r/16.8). 42 3.3 Energization of a Line Terminated With a Lightning Arrester Occasionally, the information at the end points of the l i n e gives i n s u f f i c i e n t i n s i g h t into the o v e r a l l performance of the l i n e . For example, i t i s sometimes assumed that the overvoltages at the receiving end are larger than at any intermediate point along the l i n e . This i s true as long as a l l the components i n the system are l i n e a r . When a non-linear, v o l t a g e - l i m i t i n g device such as a lightning arrester i s connected at the end of the l i n e , the voltages at the intermediate points can be s u b s t a n t i a l l y higher, and i n some lines with l i t t l e i n s u l a t i o n margin, flashover at some intermediate towers could occur. In this section, a s i t u a t i o n where intermediate voltages are higher than the receiving end voltages w i l l be simulated. Consider the c i r c u i t shown i n Figure 3.3 F i g . 3.3: Simulation of a three-phase l i n e terminated with l i g h t n i n g a r r e s t e r s . 43 The s i m p l i f i e d model used for the l i g h t n i n g arresters i s shown i n Figure 3.4 Figure 3.4: Lightning arrester model. (a) Equivalent c i r c u i t , (b) v - i c h a r a c t e r i s t i c . The voltage-controlled switch of figure 3.4 (a) closes when the absolute value of the receiving end voltage exceeds v„ ., and opens again S a t as soon as the current goes through zero. For this test, the parameters of the l i g h t n i n g arrester were chosen so that under normal switching operations, without trapped charge, the overvoltages were below v_ . ( i . e . , v_ . = 2.6 p.u.). The voltage s a L. sa u sources i n Figure 3.3 were set to 1.0 p.u. (peak) and the slope of the arr e s t e r ' s c h a r a c t e r i s t i c (for v>v sa^-) dv/di = 1.0 fi; the l i n e simulated i s BPA's John Day to Lower Monumental for a length of 500 km. Graph 1.15 shows the p r o f i l e of maximum overvoltages (absolute values) when there i s no trapped charge p r i o r to l i n e energization. 44 Graphs 1.16 to 1.18 show the receiving end voltages. Note that i n th i s case the maximum overvoltage at the receiving end i s larger than at any point along the l i n e . When the worst condition for trapped charge i s simulated, the li g h t n i n g arresters are triggered (see graphs 1.20 to 1.22). In this case the maximum overvoltages at several intermediate points along the li n e are higher than at the receiving end (see Graph 1.19). These voltages and the times at which they occur are l i s t e d i n Table 3.3. Distance from Phase A Phase B Phase C sending end Vmax Time Vmax Time Vmax Time (km) (p.u.) (ms) (p.u.) (ms) (p.u.) (ms) 50 -2.38 7.39 1 .89 7.12 -2.21 4.03 1 00 -2.79 7.36 2.05 6.94 -2.36 4.19 150 -2.87 7.53 2.20 6.78 -2.49 4.36 200 -2.92 7.51 2.37 6.60 -2.57 4.53 250 -2.92 7.56 2.51 6.53 -2.60 4.62 300 -2.90 7 .40 2.61 6.27 -2.61 4.84 350 -2.84 7.21 2.65 6.10 -2.64 4.99 400 2.80 • 16.73 2.72 5.93 -2.51 5.15 450 2.74 16.51 2.79 5.75 -2.65 5.30 500 2.73 2.31 2.87 5.59 -2.71 5.38 Table 3.3: Maximum overvoltages with trapped charge p r i o r to line energization. Graphs 1.23 and 1.24 show the voltages at 50 and 300 km from the sending end when trapped charge i s considered. Graph 1.16: Receiving end voltage. Phase A, no trapped charge. 46 Graph 1.17: Receiving end voltage. Phase B, no trapped charge. Graph 1.18: Receiving end voltage. Phase C, no trapped charge. 4 7 Graph 1.19: P r o f i l e of maximum overvoltages (absolute values). Trapped charge. Graph 1.20: Receiving end voltage. Phase A, trapped charge. Graph 1.21: Receiving end voltage. Phase B, trapped charge. Graph 1.22: Receiving end voltage. Phase C, trapped charge. 4.00 - i S 3.00 A S zooH i= \00 A OH -I .00 -2.00 A -3.00 H -4.00 0.0100 T i r t (SECONOSI 0.0)50 0.0200 4.00 T UJ ct 3 .00- o S 2-30 CL o_ £ i.oo H CD £ 0- cc r CL ^ -1.00-je: R -2.00 • U J o cc 5 -3.00 > -4.00 • 0.0050 0.0100 TIME (SECONDSI 0.0150 0.0200 4.00 • 3.00 S 2.00 A i .oo A 0 1 -1 .00 -2.00 -3.00 H -4.00 0.0050 0.0100 TIME (SECONDS) 0.0150 0.0200 Graph 1 .23: Voltage at 50 km from sending end. (a) Phase A. (b) Phase B. (c) Phase C. 4 . DO 0 . 0 ) 0 0 TIME (SECONDSI 0 2 0 0 1 . 0 0 U J ID £ 3 . 0 0 X § 2 . 0 0 - t i - e r o r 1 . 0 0 - <TJ UJ £ o- X a r ; - 1 . 0 0 - o " - 2 . 0 0 - U J LD <E g - 3 . 0 0 - - 4 . 0 0 0 . 0 0 5 0 0 . 0 1 O 0 TIME (SECONDSI •• 1 r 0 . 0 1 5 0 0 . 0 2 0 0 O.OOSO 0 . 0 ) 0 0 TIME (SECONDSI 0 . 0 1 5 0 0 . 0 2 0 0 Graph 1.24: Voltage at 300 km from sending end. (a) Phase A. (b) Phase B. (c) Phase C. CONCLUDING REMARKS In the f i r s t part of this thesis project a model for the evaluation of voltage and current p r o f i l e s along a transmission l i n e has been presented. The model and supporting routines have been designed to overcome the accuracy and data management d i f f i c u l t i e s encountered by standard segmentation methods for p r o f i l e c a l c u l a t i o n s . The main advantages achieved can be summarized as follows: i ) The model and routines are general, and lim i t e d only by the c a p a b i l i t i e s of the main (host) program. i i ) Truncation errors are e s s e n t i a l l y eliminated, and segmentation errors are minimized. i i i ) The routines are as f a s t or faster than standard segmentation procedures, and data management c a p a b i l i t i e s are c l e a r l y superior. The p o s s i b i l i t y of routine p r o f i l e c a l c u l a t i o n s , i s very useful i n some p r a c t i c a l applications (e.g., co-ordination of i n s u l a t i o n and protection devices). P r o f i l e related procedures such as the dynamic v i s u a l display of t r a v e l l i n g waves could prove to be an excellent teaching aid and permit a better understanding of transient phenomena. 51 PART I I LOW-ORDER APPROXIMATION OF THE FREQUENCY DEPENDENCE OF TRANSMISSION LINE PARAMETERS 52 INTRODUCTION Accurate models which take into account the frequency dependence of transmission l i n e parameters, have become very important tools i n the modern analysis of transient phenomena. Such accurate models usually require d e t a i l e d information about tower configuration, earth r e s i s t i v i t y , etc.; i f many li n e s have to be modelled, the approximation of the line parameters becomes a tedious and time consuming process. When the e f f e c t of one or more lin e s i s not c r i t i c a l to the outcome of a p a r t i c u l a r study, the accurate modelling of these lines i s neither j u s t i f i a b l e nor des i r a b l e . In such cases, a faster and simpler l i n e model would be preferred, i f i t s accuracy were not se r i o u s l y compromised. A s i m p l i f i e d l i n e model should meet the following requirements: a) It should be computationally faster, and easier to use than the ex i s t i n g accurate models. b) It should be more accurate than models that do not take into account the frequency dependence of l i n e parameters. c) It should require less information (or input data) than the more accurate models. d) It should be compatible with the host program of the more accurate models (in this case, the frequency-dependence version of UBC's EMTP c o d e ( * } ) • (*) This and a l l future references to the frequency-dependence model and/ or routines of the EMTP refer only to those developed by Jose R. Marti (as p a r t i a l requirement for his Ph.D. degree at the University of B r i t i s h Columbia [7 ] ) . It i s not intended to create the impression that t h i s i s the only frequency-dependence model available to EMTP users. 53 54 Such a s i m p l i f i e d l i n e model i s developed i n the second part of this thesis project, where the following assumptions have been made: (1) The l i n e i s balanced or p e r f e c t l y transposed, that i s , only two d i s t i n c t propagation modes are considered; zero sequence or ground mode, and p o s i t i v e sequence or sky mode. (2) The capacitances and shunt conductance of the l i n e are constant over the entire frequency range. (3) The ground r e s i s t i v i t y i s constant, and i t s value, as well as the values of the e l e c t r i c a l parameters R, L and C at power frequency, are known. CHAPTER 1 THEORETICAL CONSIDERATIONS 1.1 Evaluation of the Transmission Line Parameters 1.1.1 Series Impedance Matrix In the frequency domain, a transmission l i n e can be described i n terms of the following equations, d V p h ~3x d I dx = zph Iph P h = Y p h V p h , (1.1) (1 .2) where Z p n and Y p n are the series impedance and shunt admittance matrices, r e s p e c t i v e l y (subscript p ^ stands for phase q u a n t i t i e s ) . For an n-phase system (without ground wires), Z ,(to) ph Z 1 , 1 ( u , ) z 1 , 2 ( w ) Z 1 , n ( a ) ) z2/1(co) z n , 1 ( a , ) z (to) n, n (1.3) *1,1 ( U j ) y i , 2 ( u ) * 1 , n ( w ) '2,1 (to) y n , 1 U ) y (̂ ) -'njn (1 .4) The elements of Z ph(w) can be calculated from the physical c h a r a c t e r i s t i c s of the conductors and tower configuration: z k , k ( U ) = R k ( w ) + A R k , k ( u ) + ^ ( a ) L i n t , k < w ) + u L e x t , k + A xk,k< w )> i n fi/km (1 .5) 55 56 Z k , j ( u ) = Zj,k ( U } ) = A V j U ) + j ( U Lm,k,j + A X k , j ( u ) ) i n fi/kra (1.6) where, R^djj) i s the ac resistance of conductor k. Its dependency on the frequency i s due to skin e f f e c t (see appendix II-A) v'1"3) i S t h e i n t e r n a l inductance of conductor k. Its value also depends on the frequency because of skin e f f e c t . If skin e f f e c t i s ignored, v = 2*10-4 £n J 2 l _ i n H/km ( 1 ' 7 ) L . . i s the external inductance. It only depends on the geometry of the tower, and i t i s given by, L . = 2-10"4 In l i i i i i n H/km <1'8> ext,k r k L , i s the mutual inductance between conductors k and j . It only depends on the geometry of the tower and i t i s given by, V k , j = 2-10-4 . n ^ i i n H / k m d.9) k i 3 h^ i s the average height of conductor k i n m. GMRk i s the geometric mean radius of conductor k i n m. D, . i s the distance between conductor k and the image of conductor j in m. (see Fig.1.1) d v • i s the distance between conductors k and j i n m. r. i s the radius of conductor k i n m. k w = 2 T f " where f i s the frequency i n Hz. ARk,k ( U ) )' A R k , j ( a ) ) ' are Carson's correction terms for earth return e f f e c t s , i n /̂Km (see appendix II-B). 57 images F i g . 1.1: Tower geometry. In the case of bundled conductors ( i f the bundle i s symmetrical, and i f equal current d i s t r i b u t i o n i n the conductors of the bundle i s assumed), the bundle can be treated as a single equivalent conductor by replacing the GMRk and r k i n (1.7) and (1.8) by an equivalent GMR and an equivalent radius: N / _ G M R e q , k = v N* GMRk.AN 1 (1.10) N / r e q , k =yN-*k.A N _ 1 (1.11) where N = number of conductors in the bundle. GMRk = geometric mean radius of the i n d i v i d u a l conductor in m. A = radius of the bundle i n m. Note that equations (1.5) an (1.6) are general and account for the frequency dependence of the parameters. Let us now assume that a l l conductors are i d e n t i c a l . I t then follows that 58 *eq,k = r e q , j = r ^ ' 1 2 ) G M R P o k = G m < * n -i = GMR (1.13) Rk(co) =Rj(co) = Rs(co) (1.14). If the l i n e i s p e r f e c t l y transposed, we can assume that the impedance matrix Zp^ i s the average of the impedance matrices of the d i f f e r e n t transposition sections. This i s a p a r t i c u l a r l y good approxima- tio n i f the l i n e i s going to be studied at the end points only. This i s i l l u s t r a t e d , for a three-phase l i n e transposed i n two places, i n Figure 1 .2. A o- B o- C o- II III F i g . 1.2: Transposition for a single c i r c u i t . The impedance matrix for the entire l i n e i s , z P h = T<( zph i l + t > X I 1 + [ z p h I I X ] > z l l z12 z13 z21 z22 z23 z31 z32 z33 Js 2m *m z22 z23 z21 z32 z33 z31 z12 z13 z l l z33 z31 z32 z13 z l l z12 z23 z21 z22 m z s zm 59 with, 22 *33> zm = 2 ' z12 + z13 + z23> In an n-phase system, z s would be the average of a l l diagonal elements and z m the average of a l l off-diagonal elements. m 1 n - Z z n k=l k k 2 n-1 n Z Z z n(n-l) k = i j=2 kk for k<j (1.15) (1.16) Therefore, for a balanced (or p e r f e c t l y transposed l i n e ) , ph where, z (co) z (co) . z (co) z (co) m s z (co) m z (co) m z (co) s z (co) = R (co) + AR (co) + j(co L. (co) + co L ^ + AX (co) ) s s s i n t ext s z (co) = AR (co) + j(coL + AX (co) ) , m m m m (1.17) (1.18) (1.19) with, L e x t _ 2 i c f 4 in 2h r (1. 20) Lm = 2 i o " 4 In D d (1. 21) L i n t = 2 i o " 4 Zn r GMR (skin e f f e c t neglected) (1. 22) 1/n kSi h k n-1 n . n. .n o k k=l j=2 K ' J (geometric mean height) 1 2/(n-l)n n-1 n k=l j=2 k,: 2/(n-l)n k<j (geometric mean distance between k and and the image of j) k<j (geometric mean distance among the n conductors) (1.23) (1.24) (1.25) AR s(co), ARm(co) , Ax s(co) and AX m(co) , are the averages of Carson's correction te rms, 60 A R s(w) A x g ( c o ) A x m ( a ) ) = I E A R t ( c o ) . ' n k=l K n *=1 k .-, n-1 n , , t £ E A R t .(co) n(n-l) k=l j=2 k*D 9 n n . E 7 . E A x , . (co) n(n-l) k=l 3=2 k/D j>k j>k The balanced matrix of equation (1.17) leads to only two d i s t i n c t propagation modes (zero and p o s i t i v e sequence). Therefore the modal series impedance matrix has the form, z Q( co) Z (co) = m zA co) . o z (us) (1.26) where z ( c o ) = z ( c o ) + ( n - 1 ) z ( c o ) o s ° m z. (co) = z (co) - z ( co) 1 s m (1 .27) (1.28) To obtain the sequence impedances we introduce (1.18) and (1.19) into (1 .27) and (1 .28) z (co) = o R s + A R g + (n-1 ) A R s ( c o ) + j(co L. (co) + co L ^ + (n-1)co L + AX (co) + (n-1) A X ( c o ) ) , m t ext m s m (1 .29) zA co) but, R (co) + AR (co) - AR (co) s s m + j(co L. (to) + to L (co) — co L + AX (to) - AX ( c o ) ) , i n t ext m s m z Q ( c o ) = Ro(to) + j X 0 ( c o ) Z-| (to) = RT (to) + j X i (co), (1 .30) then, 61 R (co) = R (co) + AR (co) + (n-l)AR (co) (1.31) o s s m X (co) = co (L. (co) + L + (n-1) L ) + AX (co) + (n-1) AX (co) 0 m t ext m s m (1.32) R, (co) = R (co) + AR (co) - AR (co) (1.33) 1 s s m X (co) = co (L. (co) + L -L ) + AX (co) - AX (co). (1.34) 1 m t ext m s m 62 1.1.2 Carson's Correction Terms Carson's correction terms A R s , ARm, Ax s, and Axm account for the earth return e f f e c t (see appendix Il'-B) . These terms depend on the earth r e s i s t i v i t y p , on the frequency f, and on the r e l a t i v e p o s i t i o n of the conductors and earth plane. At power frequency (50 or 60 Hz) the term a i n equation (II-B.1] i s small, so that only the f i r s t term in equation (II-B.2) needs to be retained. If the l i n e i s p e r f e c t l y transposed, the averaging procedure described i n the previous section y i e l d s -4 A R S U ) = AR m(oj) = £ _ L _ L _ 1 2 — i n H/km (1.35) A X s ( c o ) = c o'2'10 - 4 [0.61 59315- In (2 h-k ./—) ] i n fy/km (1.36) A X m ( o j ) = ( j j'2'10 - 4 [0.6159315- £n ( D k- J~^) ] i n ft/km (1.37) where k = 4 TT fs • 10~ 4. Introducing (1.35) in t o (1.31) and (1.33) - 4 R 0 ( O J ) = R s(u>) + n V ' 1 0 — (1.38) R-l ( i o ) = R s ( c o ) (1 .39) If we assume that the skin e f f e c t influence on the i n t e r n a l inductance at power frequencies i s n e g l i g i b l e , then from equation (1.22) -4 „„ r L. . = 2*10 Hn m t GMR Introducing this value, and equations (1.36) and (1.37) into equation (1.32) we obtain, a f t e r some algebraic manipulations 63 X0(co) = 2 nu •' 10~ 4 in b ^ p / f (1.40) , ^ 1 ^ 1 " with VGMR d , 0.6159315 b = e S i m i l a r l y , introducing equations (1.36) and (1.37) into equation (1.34) X-i(co) = 2'co'10~4 Hn _JL_ (1.41) GMR 64 1.1.3 Shunt Admittance Matrix To evaluate Yp n (to) i t i s convenient to evaluate f i r s t Maxwell's p o t e n t i a l c o e f f i c i e n t matrix P p n Pph = P l , l P l , 2 u -1 •n,l run r n , n where 1 2 h i Pk,k = I T T £ n " r f i n km/F (1 .42) (1.43] p k , j =Pj,k 2^z~ l n l n M (1.44) If we assume that the shunt conductance matrix G i s zero, the inverse of Pph equals "the capacitance matrix K ph -1 ph ph ph Ph If the line i s p e r f e c t l y transposed we can average out the diagonal and off-diagonal elements of P p n to obtain a balanced matrix, whose elements are given by o = — " s n (1 .45) Pm n(n-l) n-1 n Lk=l j=2 ' k<j (1.46) Introducing (1.43) and (1.44), 1 „ 2 h p = -z An s 2TT£ r (1 .47) P = ~ An' ~ 2TTE d (1.48) with h, r, D and d defined as in the previous sections. The modal p o t e n t i a l c o e f f i c i e n t s are P Q = p s + (n-1) p m P l = P s " Pm Introducing equations (1.47) and (1.48), the modal, Maxwell's c o e f f i c i e n t s become, 1 „ 2 h D n - 1 .„> p = - Jin (1.49) o 2TTE ,n-l o r d -, n 2 h d P1 = 1 r p (1.50) 2 ue o The inverse of P m ode i s Kmode» a n d since P m ode i - s diagonal, the modal capacitances are simply the reciprocals of the modal p o t e n t i a l c o e f f i c i e n t s , 2 T r e 0 c = - (1 .51 ) o „ , n-1 Jin 2TTC c i = -rf7 (1-52) Jin r D 66 1.1.4 Comparison between Exact and Approximate Formulas In order to i l l u s t r a t e the accuracy of the approximate formulas for the sequence quantities, a t y p i c a l 500KV l i n e design w i l l be considered, 12.19 m 12.19m o o o o o o o o o o o o 15.24m O O O O C U 5 7 m F i g . 1.3: Tower configuration of reference l i n e , where, conductor diameter = 22.86 mm conductor GMR = 9.32688 mm conductor dc resistance = 0.104763 fy/km. Assuming that the l i n e i s p e r f e c t l y transposed, and that the bundle i s treated as a single equivalent conductor, we have r = 198.253 mm GMR = 188.427 mm d = 15.361 m D = 34.78 m Rdc = 0.02619 fi/km Table 1.1 shows the res u l t s of introducing these quantities i n the approximate formulas, compared to the "exact" parameters (at 60 Hz) obtained from the l i n e ' s tower geometry using UBC's Line Constants Program. Exact Approximate Error Formulas Formulas (%) R-l (fiAm) 0.02643 0.02643 L-i (mH/km) 0.8801 0.8802 0.009 C-| (yF/km) 0.0133 0.0132 0.84 R D (!)A"i) 0.1974 0.2041 -3.38 L Q (mH/km) 3.307 3.289 0.55 C Q (yF/km) 0.008361 0.008341 0.24 Table 1 . 1 : Comparison between exact and approximate parameters. These results i l l u s t r a t e that the approximate formulas are accurate enough at power frequency. However, i f the frequency were increased, these formulas would no longer be adequate. 68 1.2 Evaluation of an Equivalent Line Configuration Fran the Parameters at Power Frequency In order to evaluate the l i n e parameters over the frequency range of i n t e r e s t , the p h y s i c a l configuration of the l i n e must be known. If L-] , C-) , LQ and C Q are known at a s u f f i c i e n t l y low frequency (60 Hz, for instance), the approximate formulas developed in the previous sections can be used to f i n d some of the physical parameters of the l i n e , that i s r, d, GMR, h, and D . These quantities define an equivalent l i n e whose power frequency parameters are the same as those of the o r i g i n a l l i n e . If the physical parameters of the equivalent l i n e are close to those of the o r i g i n a l l i n e , the frequency v a r i a t i o n of i t s e l e c t r i c a l parameters should also be close to that of the o r i g i n a l l i n e . Consider the approximate equations for L-|, L Q / C-| and C Q L. = 2-10 - 4 An -5- GMR (1.53) L = 2n.10- 4£n _^JIIl <1'54> o n / n-1 k ,/ GMR • d r = 2 V Z ° (1.55) 1 ' 2lT~d £n — r D 2TTEO (1.56) 2 h TjH-1 £n n-1 r d where b = e 0 ' 6 1 5 9 3 1 5 k = 4 IT f~5 • 10~ 4 These are four equations and f i v e unknowns. The a d d i t i o n a l equation needed i s given by D 2 = ( 2 h ) 2 + d 2 . (1 .57) The difference between D calculated from equation (1.57) and the value obtained by finding the geometric mean distance between the conductors and the i r images, i s less than 2%. 69 Let us now define _ (Li/2-10 . a i " 6 ,_n, /^x n/ 2 a = b ( P/f) 2 a3 = S a4 = 6 k n e(Lo/2-10-*) ( 2TT EQ/C i) (2TT£ 0/C 0) Introducing these d e f i n i t i o n s into the equations above, we obtain d GMR GMR d' 2 h d r D n-1 2 h D' n-1 n T T r D 2 2 4 h + d After some algebraic manipulations we f i n a l l y obtain, GMR = — a l d = ( a 1 - a 2 ) 1 / n D = a 4 1/n a3 ^ 2 ,2 D - d 2 h d a 3D 1 .58) 1 .59) 1 .60) 1 .61 ) 1 .62) 1 .63) 1 .64) 1 .65) 1 .57) 1 .68) 1 .69) 1.70) 1 .71 ) 1 .72) Equations (1.68) to (1.72) give the physical configuration of an equivalent l i n e which has the same power frequency parameters as the l i n e we wish to simulate. A numerical example i s given i n table 1.2, where the tower configuration of the reference l i n e i s compared to the equivalent l i n e configuration obtained from equations (1.68) to (1.72) using 60 Hz parameters. 70 Reference Line (in m) Equivalent Line (in m) Error (%) d 15.361 14.334 6.69 GMR 0.188427 0.175276 6.98 D 34.78 32.66 6.09 h 15.24 14.675 3.71 r 0.198253 0.196490 0.89 Table 1.2: Comparison between the actual l i n e configuration and the equivalent l i n e c o n f i g u r a t i o n . Note that while the approximate equations for the e l e c t r i c a l parameters were very accurate (less than 1% error) the physical configura- tion obtained from the same equations shows errors up to 7%. This i s due to the logarithmic nature of the equations; small v a r i a t i o n s i n L and C, produce r e l a t i v e l y large changes in a-| , a2, and a^ (see equations (1.58) to (1.61)), which i n turn r e f l e c t on the equivalent l i n e configura- tion . 71 1 .3 Skin E f f e c t Correction Factor With the equivalent l i n e configuration obtained i n the previous section, i t i s now possible to use the more exact formulas (1.32) and (1.34) to evaluate the sequence reactances. However, we have not yet accounted for the v a r i a t i o n of Rg(a)) and L ^ n t (aj) due to skin e f f e c t . If we assume that the equivalent conductor i s tubular, skin e f f e c t depends only on the conductor's dc resistance and on the correction factor s (see appendix II-A), which i s defined as, i n t e r n a l radius q s = _ = -± external radius r F i g . 1.4: Tubular conductor Let us now r e c a l l equation (1.33) R^ t o ) = R s (co) + AR s( L O) - AR m ( to) Since AR s (co) and AR m ( to) can now be evaluated from the equivalent line configuration, and R-|(o)) i s assumed to be known, the s e l f - r e s i s t a n c e of the equivalent conductor can be determined, R s (co) = R-|(co) - A R s ( t o ) +AR m ( t o ) ( 1 . 7 7 2 If the dc resistance of the conductor i s a l s o known, i t i s possible to find (numerically, using the skin e f f e c t subroutine) the value of s that makes the s e l f - r e s i s t a n c e vary from R̂,-. to R s ( co). With s determined i n this manner, a l l the information needed to evaluate the e l e c t r i c a l parameters (over the entire frequency range) of the equivalent l i n e i s known. If the dc resistance i s not known, the value of s can be determined from the i n t e r n a l inductance I"i-n-t(to). From equation (1.34), L i n t ( i o ) i s given by L. Jio) = — [Xn(co) - AX (co) + AX (co)]-L + L (1.74) i n t co 1 s m ext m where a l l the quantities i n the ri g h t hand side can be evaluated with -the equivalent l i n e configuration. With Rg(co) and Lin^-(co) known, the values of and s can be determined numerically with the skin e f f e c t sub- routine . This a l t e r n a t i v e , although more desirable because less informa- tion needs to be known, i s more inaccurate than the determination of s with R^c given. The reason i s that since Li nt(co) i s a small f r a c t i o n of the t o t a l inductance, the difference between the actual and equivalent phy s i c a l parameters produce r e l a t i v e l y large errors i n the evaluation of L i n t ( c o ) . This can be i l l u s t r a t e d with a numerical example using the reference l i n e . Let us assume that at 60 Hz Li n t(u>) can be approximated' using equation (1.22) L. , = 2.10" 4 in r i n t GMR We now introduce the actual and equivalent l i n e parameters into this equation. The r e s u l t s are shown below. 73 Reference Line (in mH/km) Equivalent Line ( i n mH/km) Error (%) 0.01016667 0.0221495 -117.86 These re s u l t s i l l u s t r a t e that the r e l a t i v e l y small error i n GMR of 6.75% propagates exponentially because of the small magnitude of L i n t . It must be noted, however, that even though the r e l a t i v e error i s large, the difference i n magnitudes i s not, and the actual errors i n the determination of s and Rjj c are considerably smaller. For the reference l i n e , the values of s and calculated when R £ c i s not given are only 23.2% and 16.9% smaller, r e s p e c t i v e l y . 74 1 .4 Correction of the Equivalent Line Configuration when R̂,-, i s Known As mentioned i n the previous section, i f the dc resistance i s known, the value of s can be determined with good accuracy. The knowledge of an accurate value of s implies that an accurate value of the i n t e r n a l inductance i s also known. With this a d d i t i o n a l piece of information, a f i n e r approximation of the equivalent l i n e can be obtained from the f i r s t estimate given by equations (1.68) through (1.72). Consider equations (1.32) and (1.34), LQ(co) = L i n t(u>) + L e x t + (n-1) 1^+ (AXg(u) + (n-1) Ax^con/to L 1 ( U ) ) = L i n t ( W ) + L e x t " ^ + ( A x s ( w ) - AX m(w))/o>. In these two equations LQ(co) and L-](co) are the power frequency inductances and they are known data. The i n t e r n a l inductance L ^ n t (co) i s given by the skin e f f e c t subroutine at power frequency. Carson's correc- ti o n terms Ax s(co) and Ax m(co) are evaluated using the f i r s t approximation of the equivalent l i n e parameters. Therefore L ^ n t and L m can be deter- mined from equations (1.32) and (1.34) with a minimum (direct) knowledge of the tower configuration. Let a = L i (to) - L. Au) + (AX m(co) - Ax<,(co))/co (1.75) i i n t ° b = Ln(co) - L. (co) - (AX s(co) + (n-1) AXm(co))/co (1.76) ° i n t " 111 then T - (-b-a> (1.77) Un - n L e x t _ a + ha • From equation (1.20) A „ 2 h L e x t = 2.10-4 An _ (1.78) 75 from which = L J W / 2 - 1 0 ) (1 .79) where L e x ^ i s calculated from equation (1.78) and r i s the value obtained from equation (1.72). From equation (1.22) L i n t = 2.10~ 4 in l n T - GMR from which GMR = r e (Lint/2-10- 4) ( u 8 0 ) And f i n a l l y , d and D are obtained from equations (1.62) and (1.65) d = a i GMR (1.81) [ r a 4 ] l / ( n - l ) D = 1 • d (1.82) 2 h The corrected tower configuration (for the reference l i n e ) i s shown in table 1.3. Reference Line (in m) Equivalent Line (in m) Error (%) d 15.361 15.058 1 .97 GMR 0.188427 0.184440 2.12 D 34.78 34.01 2.22 h 15.24 14.99 1 .64 r 0.198253 0.196490 0.89 Table 1.3: Second approximation of the tower configuration when i s known. This new equivalent l i n e i s a better approximation of the o r i g i n a l l i n e . The improvement obtained using this procedure depends on the accuracy of the i n i t i a l estimate of the equivalent radius r . This i n i t i a l estimate of r i s usually very good because of the p a r t i c u l a r form of equations (1.68) through (1.72). It also depends on the value of 76 L^ n t(to) obtained using the skin e f f e c t subroutine, which i s also very accurate because l i t t l e error i s involved i n the evaluation of s when Rdc ^ s known. This f i n e adjustment procedure can be interpreted as the i t e r a t i v e solution of equations (1.53) through (1.57) exchanging, a f t e r the f i r s t i t e r a t i o n , equation (1.57) with the equation that calculates skin e f f e c t . If the dc resistance i s not known, the skin e f f e c t equation i s not available (actually not r e l i a b l e ) and only the f i r s t i t e r a t i o n can be performed. 77 1.5 Evaluation of the C h a r a c t e r i s t i c Impedance and Propagation Function With the equivalent l i n e configuration obtained i n the previous section, the modal parameters of the l i n e over the frequency range of i n t e r e s t can be evaluated, and with them, the c h a r a c t e r i s t i c impedance Z c and propagation function A-| for zero and p o s i t i v e sequence. These are defined as follows R + jcoL (1.83) Z c ( w ) = l/ G + jcoC A l(co) = e - ^ ( u , ) £ (1.84) Y (co) = \] (R + jcoL) (G + jcoC) . (1.85) where R, L, C are the sequence parameters defined i n the previous sections, & i s the l i n e length, and G i s the shunt conductance. The modal resistances and inductances are evaluated over the frequency range, using equations (1.31) through (1.34), with the equivalent l i n e configuration. The capacitances, since they are v i r t u a l l y constant over the frequency range of i n t e r e s t i n transient analysis, are those given as input data [10]. The shunt conductance, up to this point, had been assumed to be zero. However, i f G=0, the c h a r a c t e r i s t i c impedance goes to i n f i n i t y as the frequency goes to zero (see equation (1.83)). This s i t u a t i o n , apart from being p h y s i c a l l y impossible, i s computationally undesirable. The approximation of Zc(co) by r a t i o n a l functions would require a superfluous amount of poles and zeros to simulate the low frequency region. Therefore, a f i n i t e value of G i s assumed i n the evaluation of Zc(co.) (the same value of G i s used i n A-| (co ) f o r consistency). A t y p i c a l value f o r a 500 KV l i n e i s 0.3 • 1 0 - 7 S/km. I t i s to be noted that the magnitude of G does not a f f e c t 7 8 s i g n i f i c a n t l y the r e s u l t s of t r a n s i e n t simulations, however, past ejgpeiri- ence i n the simulation of HVDC l i n e s suggests that a f i n i t e value of Q contributes to the numerical s t a b i l i t y of the solution process during; r e l a t i v e l y long simulations (several seconds). Up to this point the l i n e under study has been assumed to fawe no ground wires. In general, ground wires are treated as normal confflaact- ors; in an n-phase system with m ground wires, Z^(OJ) and Y ^ ( 0 1 ) wo.uIkiS then be (n + m) x (n + m) matrices. Using appropriate reduction tecks- niques [11], Z p n(o)) and Y p h(w) can be reduced to (n x n ) . The ef fecit <of the m ground wires i s then i m p l i c i t i n these matrices, and equations {(11.5) and (1.6) for Zpj^to), a n d (1.43) and (1.44) for the fe-xwell's c o e f f i c i e n t matrix are no longer true. Although the o v e r a l l e f f e c t of ground wires on the l i n e paE.iaim- eters i s small, the equivalent l i n e configuration i s noticeably affeclbed. The accuracy of the approximation i s therefore reduced, although not foo the point of making i t useless (see appendix II-C). CHAPTER 2 RATIONAL FUNCTIONS APPROXIMATION OF THE CHARACTERISTIC IMPEDANCE AND PROPAGATION FUNCTION 2.1 Rational Functions. General Considerations In the frequency-dependence version of the EMTP, Z c (to ) and A^ (to) must be approximated by r a t i o n a l functions i n order to obtain closed mathematical forms which are compatible with the frequency-dependent solu t i o n algorithm. These r a t i o n a l functions have the general form B((o) = k 0 (s + z i ) ( s + Z2) ... (s + z m ) (2.1) (s + p]^) (s + P 2 ) ... (s + p n ) where, s = jto z^ = zero of B(to) P i = pole of B(to) k 0 = p o s i t i v e , r e a l constant n = m to approximate Z c ( t o ) n > m to approximate A-| (to ) A l l the zeros and poles i n equation (2.1) are re a l and l i e i n the left-hand side of the complex plane. Under these conditions, B(to) belongs to the class of minimum phase-shift functions, therefore, i t s magnitude function | B(to)| uniquely determines i t s phase function arg (B ( t o ) ) . This property reduces the approximating process to the l o c a l i z a t i o n of the poles and zeros of B(to) so that the magnitudes of B(to) and the function to be approximated are matched. 79 80 A very simple, and yet powerful, way to f i n d the poles and zeros of B(co) i s the use of Bode's asymptotes. Consider the magnitude function of B(co), and take i t s logarithm log | B(io)| = log k 0 + log | s + z-J + log | s + z j + ... + log | s + z j log | s + p j - log | s + p 2| - ... - log | s + p j . (2.2) For s = jco each one of these terms has an asymptotic behaviour with respect to co. Considering, for instance, the term y = -log | j w + p±\ , i t follows that for co << P l y = log p (2.3) for co >> P ; L y = - log co , (2.4) which, as a function of log co, define two s t r a i g h t l i n e s or asymptotes that i n t e r s e c t at co = p n . These are shown in Figure 2.1. I loglpj) l o g ( o j ) F i g . 2.1: Asymptotic behaviour of y = -log | s + p^j . Equation (2.3) defines a ho r i z o n t a l l i n e , and equation (2.4) defines a l i n e of slope - 1 units/dec, that i s , whenoj-i/o^ = 10, y decreases by 1 u n i t . 81 Equation (2.2) can then be v i s u a l i z e d as a set of building blocks or s t r a i g h t l i n e s with which the function to be approximated can be traced. I t should be noted that these l i n e segments do not represent the actual form of j B(ui)| , but an asymptotic guide or sketch that defines the boundaries where this function a c t u a l l y l i e s . The routines of the frequency-dependence option of the EMTP use this p r i n c i p l e to a l l o c a t e the poles and zeros (corners) of the approximating function [7]. For the purposes of this thesis project, these routines are not adequate because the user has no d i r e c t control over the order of the approximation used. Furthermore, for approximations of very low order (e.g., two or three poles), these routines can give e r r a t i c r e s u l t s . Therefore, a s i m p l i f i e d approach to the assignment of asymptotes has been developed in this thesis p roject. With this s i m p l i f i e d method,the number of poles i s fixed beforehand, and adequate approximations with very few corners are p o s s i b l e . The basic p r i n c i p l e s for this approach are discussed next. 8 2 2 . 2 Asymptotic Approximated on when the Number of Poles i s Fixed Consider H(aj), whose magnitude function (shown i n Figure 2 . 2 ) i s bounded and monotonically decreasing over the frequency range of i n t e r e s t ( 0 to 1 0 6 HZ). (Note that | H(co)| i s plotted on a log-log scale) |H(w)l F i g . 2 . 2 : Magnitude function of H ( L U ) As mentioned e a r l i e r , to approximate H(to) with a r a t i o n a l function B(co), i t i s s u f f i c i e n t to match their magnitude functions. Let B(co) be of order two, B(co) = k n (s + Z j ) (s + z 2 ) . (s + Pi ) (s + P 2 ) Taking the logarithm of | B(to)| , log | B(co)| = log k Q + log | s + z - J - log | s + Pi| + log | s + z 2 | - l o 9 " | s + p 2 l ( 2 . 5 ) (2.6) 83 Since | H(to)| i s bounded, i t can be subdivided into four equally spaced sections. The h o r i z o n t a l asymptotes of | B(co)| w i l l pass through h-) , n2 a n d T n e asymptotes with slope - 1 units/dec w i l l pass through h 2 and h^. The l o c a t i o n of the corners of the asymptotic approximation .will be at the i n t e r s e c t i o n of the asymptotes (see Figure 2.3). |H(LJ)1 , 4 F i g . 2.3: Asymptotic approximation of | H(O J ) | . For example, to f i n d p^ we i n t e r s e c t the s t r a i g h t l i n e s log | H(to )| = log h-, log | H( co)| = -log co + log h 2 + log to^. Subtracting these equations log h-| = - log OJ + log h 2 + log O J 2 log (ui) = - log h-| + log h 2 + log tu2 h2 0) = 84 This i s the value of to at which the i n t e r s e c t i o n occurs, therefore P i = "2 Proceeding analogously with the other corners, we f i n a l l y obtain h 3 P 2 = h4 t04 h 3 z2 = h4 0)4 h 5 The value of k Q can be found by matching | H(to)| and | B(to)| at to = 0 from which H(0)| = |B(0)l = k n z l z 2 r k Q = | H(0)| P l P 2 . Z l Z2 This procedure can be extended to an a r b i t r a r y number of poles and zeros n , by subdividing | H(to )| into 2n segments and then finding the int e r s e c t i o n of the hori z o n t a l asymptotes (which w i l l pass through n ^ , f o r i odd) with the non-horizontal asymptotes (which w i l l pass through h^, for i even). These int e r s e c t i o n s then define the positions of p^ and z^. p. = - ^ - " 2 i (2.7) h 2 i - l h 2 i .. <2'8> 1 W z, = , ^<*>2i '2i+l where i = 1,... ,n The constant k Q w i l l then by given by n p k 0 = H ( O ) n — i = l z i (2.9) 85 This method can be generalized to approximate functions with p o s i t i v e and negative slopes, provided they are single valued and can be broken down into segments of monotonically increasing and decreasing functions (see Figure 2.4). F i g . 2.4: A r b i t r a r y function with p o s i t i v e and negative slopes. 86 2.3 Approximation of the C h a r a c t e r i s t i c Impedance Zc(co) If G i s assumed to be zero, the c h a r a c t e r i s t i c impedance i s a monotonically decreasing function that becomes constant as to tends to i n f i n i t y . If G i s non-zero, | Zc(to)| can show two regions: a segment with p o s i t i v e slope for low frequencies and a segment with negative slope for the rest of the function (see Figure 2.5). 650 i FREQUENCY (HZ) F i g . 2.5: | Zc(to)| , zero sequence. From BPA's John Day- Lower Monumental 500 KV transmission l i n e . The presence of the peak in Figure 2.5 depends on the value of G. As mentioned e a r l i e r , the actual value of G does not a f f e c t s i g n i f i c a n t l y the r e s u l t s of a transient simulation, as long as i t i s non-zero. In the approximating routines developed i n this project, the peak i n j Zc(to)| i s eliminated, as shown by the dashed l i n e i n Figure 2.5 (this i s approximately equivalent to decreasing G). Since the p o s i t i v e slope region i s eliminated, fewer poles are needed i n the approximation, 87 and the r e s u l t i n g monotonically decreasing function can be approximated with the method described i n Section 2.2. The o v e r a l l accuracy of the approximation can be improved by s h i f t i n g pole-zero p a i r s around th e i r i n i t i a l p o s i t i o n s , u n t i l the area between | Z c(w)| and | B( LO)| i s minimized. When close matching at power frequency f Q i s preferred to a good o v e r a l l frequency response, the pole-zero p a i r that i s closest to fQ can be s h i f t e d u n t i l the difference between | B(ui 0)| and | Z c (co0 )| i s below an error l e v e l s p e c i f i e d by the user. This procedure, however, may d i s t o r t the o v e r a l l frequency response when the s p e c i f i e d number of poles i s very small (e.g., two or three poles). In the computer program developed as part of this thesis project, the use of either method ( s h i f t i n g or power frequency matching) i s o p tional. 2.4 Approximation of the Propagation Function A-] ( O J ) 88 The general shape of | A-|(u))| i s approximately the same for d i f f e r e n t l i n e lengths and propagation modes. Three d i f f e r e n t regions can be observed: I. A hor i z o n t a l region for low frequencies (low attenuation). I I . An elbow or t r a n s i t i o n zone for mid-frequencies. I I I . A high slope region for high frequencies (high attenuation). FREQUENCY (HZ F i g . 2.6: A-|(to) zero sequence. The high attenuation region (see Figure 2.6) presents an almost constant slope of 1 unit/dec. This suggests that | A-j(co)| could be approximated by a single-pole r a t i o n a l function B-|(O J ) , B 1 ( a ) ) = k i (s + P x) Such a single-pole approximation i s shown i n Figure 2.7. FREQUENCY (HZ) F i g . 2.7: Single-pole approximation of | Ai(co)| . In this single-pole approximation, two large-error regions can be observed; one at very high frequencies, and the other, i n the mid-frequency range. The error i n the high attenuation region i s not c r i t i c a l . High frequencies i n a t y p i c a l transient case have r e l a t i v e l y low magnitudes, and are r a p i d l y attenuated by the l i n e . Therefore, no a d d i t i o n a l poles w i l l be used to improve the approximation for values of j A-j (co )j below 0.4. The other high-error region i s in the t r a n s i t i o n zone. The accurate approximation of this zone i s very important because i t presents a r e l a t i v e l y low attenuation and i t usually l i e s close to the power frequency range, therefore, a d d i t i o n a l poles and zeros are needed to approximate t h i s region. One way to determine the a l l o c a t i o n of these a d d i t i o n a l corners 90 i s to examine the difference function between | A-^to)) and the single-pole approximation | B-j (cc)| . Consider the approximating function B(to) with n poles and n-1 zeros, B(co) = k Q (s + Z j ) ... (s + z n) (2.10) (s + p i ) ... (s + p n _ i ) (s + p ) Let B-|(u)) be the single-pole function that approximates | A-|(co)| B 1 ( a ) ) = k l (2.11) (s + p n ) ' with k! = | A-, {0)| P n . The difference function D(co) w i l l be defined as, log | A - | ( O J)| - log | Bi(o))| = log | D(co)| | D(O J ) | = | Ai (o))| . (2.12) j B1{m)\ Introducing (2.11) I D(w)l = J _ I A1 I s + Pn| | D( u)| = P n | A i ( M ) | | s + p n| . (2.13) Let us now define D ^ c u ) as the r a t i o n a l function that approximates D(to) ( s u b s t i t u t i n g A^ c o ) by B(ai) i n equation (2.12)) i | B (co) I D l ( c o ) = (2.14) |B1 ( C O ) J Introducing equations (2.10) and (2.11), we f i n a l l y obtain | D-, (to )| = kp I (s + z x ) ... ( S + z n_i)l (2.15) k-| | (s + P-)) ... (s + p n _ i )| The magnitude function of D(to) i s shown i n Figure 2.9(a) F i g . 2.9: Difference function |D(U>)| . Zero sequence. Reference l i n e (a) |D(to)| (b) |D(to)j , |B1(to)| and |A X (OJ)| . 92 The assignment of the ad d i t i o n a l corners of | B( to)| i s then reduced to the approximation of | D(to)| by | D-| (to)| . Since we are interested i n the accurate modelling of the t r a n s i t i o n zone, only the portion between f = 0 and f = f j_ w i l l be approximated. This segment represents a monotonically decreasing function, and i t i s approximated with n-1 poles and zeros, using the procedure described i n section 2.3. The accuracy of a transient simulation depends strongly on the approximation of A-| (to). i f an accurate power frequency response i s desired, an exact matching can be obtained by rep o s i t i o n i n g the c l o s e s t pole-zero p a i r u n t i l the difference l i e s within a p r e - s p e c i f i e d range. However, s h i f t i n g a single pole-zero p a i r may, in some instances, introduce d i s t o r t i o n i n the o v e r a l l frequency response. When o v e r a l l accuracy i s preferred to power frequency matching, a l l pole-zero p a i r s are s h i f t e d from their i n i t i a l positions u n t i l the area between | A-| (to)| and | B-j C cu)| i s minimized. 2.5 Evaluation of t In the previous section the magnitude function of A-| (co) has been approximated by a r a t i o n a l function B-j (co). I t i s shown i n appendix I-B that AT (co) = P( co) e " 3 ^ (2.16) Since B( co) i s a function of the class of minimum phase-shift, by approximating the magnitude function of A-| (co) with | B(co)| , the phase function of B( co) w i l l coincide with the phase function of P( to) . To approximate both magnitude and phase of A]_(co), c~ must be evaluated so that AT (co) = B(co) e " j W C (2.17) Taking the argument function of both sides of equation (2.17), arg (A-|(to)) = arg (B(co)) - cor from which, C = - i (arg (B(co)) - arg (A-,(co))). (2.18) Since B(co) i s not exactly equal to P(co), the value of c i n equation (2.18) w i l l vary with the frequency to a degree dictated by the accuracy of the approximation of the magnitude function. The frequency dependence version of the EMTP evaluates c by averaging the values obtained from equation (2.18) over a c e r t a i n frequency range ( i . e . , for values of | A-| (co )| between 0.9 and 0.7). When a high-order approximation i s used, this procedure i s j u s t i f i a b l e because the differences between | A-| (to )| and | B(co)| are small. In this project, however, A-) (co) i s not known, but rather the propagation function of an equivalent l i n e . Even i f th i s equivalent l i n e i s a good estimate of the o r i g i n a l l i n e , the low number of poles and zeros i n B(co) only guarantees a good f i t over a r e l a t i v e l y l i m i t e d frequency range. Because of these considerations, C i s evaluated at one point only, that i s , at the power frequency. 2.6 Implementation of the Method The evaluation of the equivalent l i n e representation and the approximation of A-|(to) and Z c(co), have been implemented i n a FORTRAN computer program. This program (with approximately 1300 FORTRAN statements) has been designed so that a future interconnection with the e x i s t i n g EMTP code can be made with minimum e f f o r t . A guide for i t s use i s attached i n appendix II-D. CHAPTER 3 NUMERICAL RESULTS 3.1 Recapitulation From the power frequency parameters (and dc resistance for higher accuracy) the physical configuration of an e l e c t r i c a l l y equivalent l i n e has been obtained. This permits the evaluation of R( to) and L(to), from which Z c(w) and A-] (w) can be calculated over the ent i r e frequency range. The frequency-dependence version of the EMTP requires that Zc(to) and A-|(co) be in a closed mathematical form. For this purpose, Zc(to)and A-| ( to) are approximated by r a t i o n a l functions using approximating techniques that permit the use of a very low number of poles and zeros. The transmission l i n e described i n Chapter 1 w i l l be simulated using the techniques described i n this thesis p r o j e c t . The results w i l l be compared with those obtained using the frequency-dependence version of the EMTP. 95 9.6 3.2 Evaluation of the Line Parameters Frcm the Equivalent Line Ccrifiguration The input data necessary to otain the equivalent l i n e configuration for the reference l i n e (see Figure 1.3) i s shown below: Power frequency = 60 Hz Line length Shunt conductance Earth r e s i s t i v i t y dc resistance (R d c = 500 km = 0.3-10 - 7 s / k m = 100fi-m = 0.02619 0,/km The e l e c t r i c a l parameters at 60 Hz (from the output of the Line Constants Program) are: Positive sequence: RT = 0.02643 n/km L-, = 0.8808 mH/km C} = 0.0133 UF/km Zero sequence: R Q = 0.1974 ft/km L Q = 3.3308 mH/km C Q = 0.008361 UF/km The r e s u l t i n g equivalent l i n e configuration i s summarized below. Reference Line Parameters (in m) Equivalent Line R d c given R d c not given Parameters . Error Parameters Error (in m) (%) (in m) (%) d 15.361 15.058 1 .97 14.334 6.69 GMR 0.18843 0.18444 2.12 0.17528 6.98 D 34.780 34.070 2.22 32.662 6.09 h 15.240 14.990 1 .64 14.675 3.71 r 0.19825 0.196490 0.89 0.19649 0.89 The v a r i a t i o n of R and L over the frequency range from 10~^ 1 : 0 Q 10 Hz, i s calculated from the equivalent l i n e configurations (see graphs 2.1 through 2.4). The output from the Line Constants Program i s used as reference. From these graphs, the following observations can be made: a) The estimated values of R and L are more accurate when i s known. b) Po s i t i v e sequence parameters present larger error l e v e l s than zero sequence parameters. c) The resistances present larger errors than the inductances. When R ^ C i s known, the skin e f f e c t correction factor s i s calcu- lated from the increase i n the s e l f - r e s i s t a n c e from i t s dc to i t s 60 Hz value R-l(oj) = R S ( O J ) + (ARs(co) - ARm(co)). Since the term ( A R S ( O J ) - A R^too)) i s ne g l i g i b l e at 60 Hz, the increase i n Rs(co) can be assumed to be caused by skin e f f e c t only. Therefore, the value of s estimated when R^c i s known is very accurate (see section 1.3). When R^ i s not known, s cannot be estimated accurately because i t i s calculated from the increase i n the i n t e r n a l inductance from i t s dc to i t s power frequency value; and as i t was shown i n section 1.3, the evaluation of L ^ n t using the equivalent l i n e configuration i s very inaccurate. Also, when R^ i s not known, the fi n e r adjustment of the line configuration explained i n section 1.4 cannot be made, and the evaluation of Carson's correction terms becomes less accurate. The p o s i t i v e sequence resistance i s more s e n s i t i v e to differences i n AR g and A R m , than the zero sequence resistance. The reason i s that R-| depends on (ARg-AR^, while R Q depends on (ARS + (n-1) AR m); 98 since AR S and AP^ are r e l a t i v e l y large, and of comparable magnitudes (for mid to high frequencies), small errors i n AR G and AR M produce r e l a t i v e l y large errors i n (AR S - AR^), while (AR S + (n-1 ) & R m ) i s affected to a lesser degree. The evaluation of the inductances i s more accurate than the evaluation of the resistances because skin e f f e c t only a f f e c t s L. ^, which i n t i s only a small f r a c t i o n of L-| and L Q . Furthermore, the magnitudes of Carson's correction terms are considerably smaller for the reactances. This can be e a s i l y v i s u a l i z e d i f we note that, for example, RQ increases over 10 4 times from i t s dc value, while L 0 decreases only to 1/4 t n of i t s dc value, when the frequency i s in the MHz range. 10* 10" 10* 10 10-' io-' i n u n i i in II i i HI M i i in II , i i in II , i i in 10 10 1 10 10 10 10 10 FREQUENCY (HZ) II .1 I ill ll ,1 I III II .1 m i 10 10 Ml II ,1 I ill n . 10 10" 'i iff 3 I f f 5= io-' 10-' j 111111 i m i l l I I I I I I I i i in II , i l in II , i i in II i l in n i i in II , i 10 10 1 10 10 10 10 10 10 10 FREQUENCY (HZ) M U M , i I I I I I I . 10 ti-er o e M i n i i 111111 I I I I I I I i i in II , i I n m , i i in II .1 i in II ,1 m i II .1 i in II J i in n , io" io i io io io io io io io i or FREQUENCY (HZ) Graph 2 . 1 : P o s i t i v e sequence resistance. (a) Rd cgiven (b) Rdc not given (c) Error function Graph 2 .2: Zero sequence resistance. (a) R^ given (b) Rdcnot given (c) Error function. ] .00 0.90 0.80 " 0.104 0 .60 0 . 5 0 I I I I I I I I I I I I I I I I I I I I I i I ill ll ,1 I ill ll .1 I in ll J Mil ll I I ill ll 1 1 , 1 1 1 1 1 1 1 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) 1 .00 0.90 H - 0 .80 H 0. )0 5 0 .60 A 0.50 1 I I I I I I I I I I I I I I I I I I I I I i n u n , i n u n , i M i n i i n n n , i M U M .I IIIIII . I I I I I I I , io io i io io io io io io io icr FREQUENCY (HZ) 5.0 4 .0 3.0 2 .0 i . H o -i.oA - 2 . 0 -3.0 A - 4 . 0 - 5 . 0 10 IIIIII J IIIIII i n u n i 10" 1 10 mil .1 IIIIII , n • ' 10 10 10 FREQUENCY (HZ) IIIIII ,1 IIIIII .1 n u n ,1 n u n 10 10 10 10 Graph 2.3: P o s i t i v e sequence inductance. (a) RcJc given (b) R(j c not given (c) Error function. 7.00 6.50 6.00 -f 5.50 5.00 4.50 4.00 3.50 3.00 2.50 2.00 1 50 1 .00 l l l l l l J l l l l l l M U M .1111111,1 1 .1 I III II ,1 I i l l II .1 I 111 i l , I . 10"" 10 1 10 10 10 10 10 10 10 10 FREQUENCY I HZ) imu I I I I I I I l im i t i m i n .i I I I I I I ,i i .11 in n ,i i in II .1 i in n ,i i nm . 10" 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) 5 0 4.0 3.0 2.0 1.0 0 -1.0 -Z .0H -3.0 -4.0 -5.0 I III II I I I I I I I I I I I I I I 1111111,1111111,1111111,1111111,1111111.1111111,1111111 . 10" 10" 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) Graph 2.4: Zero sequence inductance. (a) R^c given (b) Rdc not given (c) Error function. 103 3.3 Evaluation of A-| (co) and Zc(co) The magnitude and phase functions of A-| (co) and Zc(co) are shown i n graphs 2.5 through 2.12. Note that the error levels i n the evaluation of the p o s i t i v e sequence resistance, are also r e f l e c t e d i n Zc(co) and A-|(co). Also note that the high error region i n the phase function of Z c(to), p o s i t i v e sequence (for frequencies above 100 Hz), i s not very s i g n i f i c a n t because the phase angle i s very close to zero, and small differences between the phase angles of the r e a l function and the approximation, appear as large r e l a t i v e errors (see graph 2.9). S i m i l a r l y , when the magnitude of A-|(co) i s below 0.05 times i t s dc value, large r e l a t i v e errors lose their s i g n i f i c a n c e . In graph 2.5 th error function for | A-|(co)| <: 0.05 has been omitted. Note that the phase functions of Zc(co) have been plotted in degrees, while the phase functions of A-| (co) have been plotted i n radians (allowing arg (A-j (co)) >TT ) . (c) -5.0 1 I I I I I I I I I I I I I I I I I I I I I i I I I I I I ,1 I I I I I I , i I I I I I I .1 I I I I II ,i nun .1 I I I I I I , i I I I I I I . 10 10 1 10 ID 10 10 10 10 10 10 FREQUENCY (HZ) Graph 2.5: A-) magnitude function, p o s i t i v e sequence. (a) qiven (b) not given (c) Error function, dc ^ dc FREQUENCY (HZ) FREQUENCY (MZ) 5 . 0 - , 4.3 i 3.0 H 2.0j -2.0 -J -3.0 -4.0 -1 -5.0 I I I I I I I I 1 I I II 11 1 I I I I 11 , I I I II I I , I I I I I 11 , I I I I I 11 10 2 4 1 2 4 10 2 4 10 2 4 10 2 4 10 2 4 10 FREQUENCY (HZ) Graph 2.6: A-| phase function, p o s i t i v e sequence. (a) R̂, given (b) R, not given (c) Error function 106 1.0 -> FREQUENCY Graph 2.7: A-) magnitude function, zero sequence. (a) R, given (b) R, not given (c) Error function. 107 35-| (b) (c) - i — I I I I I I — i — I I I I I I — i — i M I I I , — i — i M I I I . — i — I I I I I I . 10"' 2 4 1 2 4 10 2 4 10 2 4 10 2 4 10 FREQUENCY I HZ) Graph 2.8: Ai phase function, zero sequence. (a) R d c given (b) R d c not given (c) Error function. 1 0 8 ( a ) 300 H 10 j i in II IIIIII i n u n i m i n ,i n u n ,i i m u .1 M i n i , i n u n ,i n u n ,i n u n # 10 ] 10 10 10 10 10 FREQUENCY (HZ) 10 10 10 1000 900 H (b) 200 -\ j i in II i i III II—i i in n—i i in II i m i II i m i II ,i i in II , i i in II , i I in n .1 i in n , 10 10' I 10 10 10 10 10 10 10 10 FREQUENCY (HZ) 5.0 4.0 3.0 2.0 1.0 0 -1.0 -2.0 -3.0 -4.0 -5.0 IIIIIII IIIIIII IIIIIII IIIIIII, I ,1 IIIIII . I llllll ,1 IIIIII . I IIIIII , I IIIIII , 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) (c) Graph 2.9: Z c magnitude function, p o s i t i v e sequence. (a) R d c given (b) R d c not given (c) Error function, 109 (a) -15 I IIIIIII IIIIIII IIIIIII i IIIIII .i HUM ,i IIIIII .i IIIIII ,i nun ,i IIIIII ,i IIIIII 10 10" 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ I 15-1 (b) -45 I IIIIIII IIIIIII IIIIIII i HUM .i m i n ,i IIIIII .i mil l ,I IIIIII i m m ,i IIIIII , 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) 110 1000 900 BOO 700 600 500 400 300 200 I I I I I I I I I I I I I I I I I I I I I I 11 inn ,i i inn ,i i in n .1 i in II ,i nun .1 mi I I - i nu II . 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (MZ) (a) 10 j i in II _) i in II—i i in II—i nun t i m m } i nun ,i nun ,1 m m ,i i nm ,i m m , 10 1 10 10 10 10 10' FREQUENCY (HZ) 10 10 10 (b) 5.0 4.0 -| 3.0 2.0 1.0 0 -1 .0 -2.0 -3.0 -4.0 -5.0 10 j i ill II j i ill II i mi ii i nu II ,i nu II ,i i in II j nu II ,i mi II .i nu II ,i mi II . 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) (O Graph 2.11: Z c magnitude function, zero sequence. (a) R d c given (b) R d c not given (c) Error function. I l l I5n o - ' 0 (b) 1 J I I I I I I I I I I I I I I I I I I I I I I I I III . I I I I III .1 I I I I I I .I I I I I I I , I I I I I I I I I I I I I I ,1 I I I I I I , 10 10 1 10 10 10 10 10 10 10 10 F R E Q U E N C Y (HZ) Graph 2 . 1 2 : Z c phase function, zero sequence. (a) R, given (b) R, not given (c) Error function. 112 3.4 Low-Order Rational-Functions Approximations In order to assess the performance of the r a t i o n a l - f u n c t i o n s approximating routines, A.] (to) and Zc(to) calculated from the equivalent l i n e configuration (of the reference l i n e ) w i l l be approximated with v a r i - ous numbers of poles and zeros. For these approximations R ^ c i s assumed to be known. Graphs 2.13 to 2.16 show the error functions of these approxima- tions with and without the pole-zero s h i f t i n g procedure explained i n Chapter 2. When no s h i f t i n g i s used, the error at 60 Hz has not been allowed to exceed 0.5%. The number of poles needed to approximate a given function de- pends on i t s p a r t i c u l a r shape and the accuracy desired. In the magnitude of the zero sequence propagation function, for instance, there i s l i t t l e improvement when the number of poles i s increased beyond f i v e . Note that for frequencies above 300 Hz, the error remains unchanged regardless the number of poles used because no a d d i t i o n a l poles are added to model this region. For frequencies above 1 KHz, that i s , for the high attenuation region (| A-] (to )| <0.1) the error exceeds 5%, but as mentioned i n Chapter 2 the accurate approximation of this region i s not important. Also note that the s h i f t i n g procedure reduces the o v e r a l l error, but slows considerably the f i t t i n g process (see table 3.1) as compared to the simple power frequency matching process. The approximation of A-) (to) for p o s i t i v e sequence, i s p a r t i c u - l a r l y d i f f i c u l t i n this l i n e because of the small depression i n | Ai(to)| i n the 1 to 100 Hz region (see graphs 2.13 and 2.17). This i r r e g u l a r i t y , u s u a l l y present for lengths of 500 km or more, i s responsible for the increase i n the error of the 5-pole approximation over the 4-pole one. However, as the number of poles increases beyond 5, the o v e r a l l error 113 decreases as expected. I t should be noted that the number of s h i f t i n g loops for these c a l c u l a t i o n s was limited to four (to decrease c o s t s ) . If the s h i f t i n g process had been allowed to continue automati- c a l l y ( u n t i l the improvement between i t e r a t i o n s were less than 1%), a better approximation would have been obtained. The approximation of Z c ( w ) for zero sequence i s comparatively more d i f f i c u l t because a large number of poles i s necessary to approximate the endpoints (see graphs 2.16 and 2.20). The EMTP routines needed 17 poles to accurately approximate the whole frequency range. However, errors i n excess of 1% only occur for frequencies above 20 KHz, therefore, a number of poles larger than eight i s not j u s t i f i a b l e i n the context of th i s p r oject. Note that i n this case, i f the s h i f t i n g process had been allowed to proceed beyond four i t e r a t i o n s , the error i n the low to mid frequencies would have increased s l i g h t l y to decrease the error at high frequencies. The approximation of Z c ( w ) for p o s i t i v e sequence, i s an in t e r e s t i n g case where the pole-zero s h i f t i n g procedure i s very important for the accurate matching of the function. When no s h i f t i n g i s used, an increase i n the order of the approximation j u s t accumulates the a d d i t i o n a l poles i n the low frequency region, d i s t o r t i n g the approximation rather than improving i t . In graphs 2.17 through 2.20, three sets of approximations are shown: a very low-order approximation (2 poles), a more accurate approxi- mation (10 poles ) , and a compromise between accuracy and number of poles (4 poles for Z,-.̂ ) and A-|(u>) p o s i t i v e sequence, 5 poles for A i ( w ) zero sequence, and 6 poles for Zc(co) zero sequence). One of the main advantages of the low-order approximation routines, i s that they are very inexpensive computationally. A high- 114 Graph 2.13: Error functions. Low-order approximation of A-| , pos i t i ve sequence. 115 Graph 2.14: Error functions. Low-order approximation of A-j , zero sequence. 116 Graph 2.15: Error functions. Low-order approximation of Z, p o s i t i v e sequence. Graph 2.16: Error functions. Low-order approximation of Z c, zero sequence. 118 accuracy approximation produced with the EMTP routines can cost over $10.00, while low-order approximation costs are usually under $3.00 (using 10 poles i n Zc(co) and A-|(co), and 4 s h i f t i n g loops). These running costs are based on UBC's rates of $1,200.00 per hour of CPU time under Terminal use. Table 3.1 shows the cost of a high accuracy simulation of the reference l i n e using the EMPT routines. The corresponding approximations are shown i n graphs 2.21 to 2.24, where the output from the Line Constants Program i s used as reference. UBC's l i n e constants program $ 4.50 TRANFLIN.0 (Calculation of Z c . (co ) and A-j (co)) $ 0.24 TRANA1.0 (approximation of A-, (co ) ) P o s i t i v e sequence (18 poles, 12 zeros) $ 1 .23 Zero sequence (21 poles, 15 zeros) $ 1.11 TRANZC.O (approximation of Zc(co)> P o s i t i v e sequence (9 poles, 9 zeros) $ 1.18 Zero sequence (17 poles, 17 zeros) $ 1 .75 Total $ 10.01 Table 3.1: Costs i n $cc of the frequency--dependence routines of the EMTP. 119 Table 3.2 shows the running costs of the low-approximation program for various numbers of poles and zeros. Number of Poles Cost $cc Zc(co) A-, (co) P o s i t i v e Zero P o s i t i v e Zero S h i f t i n g No s h i f t i n g Sequence Sequence Sequence Sequence 2 2 2 2 0.25 0.16 3 3 3 3 0.34 0.16 4 4 4 4 0.51 0.16 5 5 5 5 0.93 0.15 6 6 6 6 1 .27 0.17 7 7 7 7 1 .63 0.20 8 8 8 8 2.04 0.12 9 9 9 9 2.53 0.17 10 10 10 10 3.05 0.17 4 6 4 5 0.79 — Table 3.2: Costs i n $cc of the low-approximation program. It i s i n t e r e s t i n g to note that high and low-order approximations of A-|(co) have comparable error levels when | A^oo)] i s greater than 0.4. Therefore the higher computational e f f o r t i n the high-order approximation can be at t r i b u t e d to the modelling of the high attenuation region of Ai (co). 120 1.0 ~y FREOUENCr (HZ) Graph 2.17: Approximation of A-j, p o s i t i v e sequence. (a) 2 poles (b) 4 poles (c) 10 poles Graph 2.18: Approximation of Aj_, zero sequence. (a) 2 poles (b) 5 poles (c) 10 poles 1000 - i FREQUENCY (HZ) Graph 2.19: Approximation of Z c, p o s i t i v e sequence. (a) 2 poles (b) 4 poles (c) 10 poles Graph 2.20: Approximation of Z c, zero sequence. (a) 2 poles (b) 6 poles (c) 10 poles Graph 2.21 : A-] p o s i t i v e sequence, high-order approximation. (a) Magnitude function (b) Error function l . O - i FREOUENCT IH7J (b) Graph 2.22: Ai zero sequence, high-order approximation. (a) Magnitude function (b) Error function Graph 2.23: Z c p o s i t i v e sequence, high-order approximation. (a) Magnitude function (b) Error function J 000 - i FREQUENCY (HZ) (a) 5.0 1.0-1 3.0 2.0 1.0 0 4 -1.0 -2.0 -3.0 -j -4.0 -5.0 i I ill II ) i in II—i i in II—i I ill II ,1 I III II ,1 I III II J I III II .1 I III II .1 I III II ,1 I III II , 10 10 1 10 10 10 10 10 10 10 10 FREQUENCY (HZ) (b) Graph 2.24: Z c zero sequence, h igh-order approximation. (a) Magnitude funct ion (b) Error func t ion 128 3.5 Frequency Domain Response 3.5.1 Introduction The best t e s t for the v a l i d i t y of the approximating techniques described i n t h i s project, i s to simulate transient phenomena i n the EMTP, and to compare the r e s u l t s with those obtained using the available high- accuracy models. Unfortunately, the number of possible simulations i s very large and i t would not be p r a c t i c a l to present a large number of re- su l t s here. However, a small but wisely chosen number of tests can give a good idea of the r e l a t i v e accuracy of the models. Two such tests are the steady-state, open and short c i r c u i t responses to sinusoidal e x c i t a t i o n at one frequency; with the frequency varied over the frequency range of interest. From the solution of the l i n e equations i n the frequency domain (see appendix I-A), the open c i r c u i t response (see Figure 3.1), i s given by 2 V A (co) V (co) = — (3.1) m -y 1 + hf (co) where V Q i s the peak magnitude of the voltage source, and V m i s the receiving end voltage. S i m i l a r l y , the short c i r c u i t response, or receiving end current when the l i n e i s shorted, i s given by 2 V A (co) I (co) = ° 1 (3.2) m Z (co) (1 - A 2 (co)) c 1 These two tests present several advantages: (a) Open and short c i r c u i t terminations are extreme and t r y i n g cases for a l i n e model; i f the response i s good for these two cases, i t i s reasonable to assume that the response for any other termination w i l l a l so be good. 129 (b) The behaviour of the models over the whole frequency range can be r e a d i l y observed. (c) The r e s u l t s of the d i f f e r e n t rational-functions approximations can be compared to t h e o r e t i c a l r e s u l t s since A-| ( t o ) and Z c ( t o ) i n equations (3.1) and (3.2) can be evaluated at any given frequency from the out- put of UBC's Line Constants Program. cos ( w t ) o m V m COS (uot ) (a) (b) F i g . 3.1: (a) Open c i r c u i t and (b) Short c i r c u i t t e s t s . In these tests, the li n e w i l l be assumed to be single-phase, with p o s i t i v e or zero sequence parameters, i n order to i s o l a t e the e f f e c t s of the approximations involved i n each propagation mode. The following models have been tested: Model 1. - High-order approximation using the parameters calculated from UBC's Line Constants Program. Model 2. - High-order approximation using the parameters calculated from the equivalent l i n e representation with given. Model 3. - High-order approximation using the parameters calculated from the equivalent l i n e representation with R, not given. 130 Model 4. - Low-order approximation (2 poles, R^c given). Model 5. - Low-order approximation (10 poles, R^c given). Model 6. - Low-order approximation (4,5 and 6 poles, R^c given). Model 7. - Constant parameters option of the EMTP. Table 3.3 shows the number of poles and zeros used i n the d i f f e r e n t approximations. Z c ( co) AT (co) Zero Positive Zero Positive sequence sequence sequence sequence Model Poles Zeros Poles Zeros Poles Zeros Poles Zeros 1 17 17 9 9 21 15 18 12 2 17 17 9 9 21 15 18 1 2 3 18 18 9 9 25 19 17 11 4 2 2 2 2 2 1 2 1 5 10 10 10 10 10 9 10 9 6 6 6 4 4 5 4 4 3 Table 3.3: Number of poles and zeros used in the d i f f e r e n t models tested i n this section. o Models 2 and 3 are used to i s o l a t e the e f f e c t of the r a t i o n a l - functions approximation from the e f f e c t of the estimation of the line parameters (from the equivalent l i n e configuration). Models 4, 5 and 6 show the e f f e c t s of the order of the approximation i n the response of the l i n e . 131 3.5.2 Open C i r c u i t Response The open c i r c u i t response for the d i f f e r e n t l i n e models describ- ed above, i s shown i n graphs 2.25 through 2.31. Note that V m ( w ) depends on A-|(w) only (see equation (3.1)), therefore the e f f e c t s of the approxi- mation of Z c ( w ) are not present i n this t e s t . The response of model 2 for zero sequence i s v i r t u a l l y i d e n t i c a l to the t h e o r e t i c a l response; the p o s i t i v e sequence response of the same model shows s i g n i f i c a n t errors only for frequencies above 10 KHz. When R^c i s not known (model 3), the response i s s t i l l good for zero sequence, but presents considerable errors for p o s i t i v e sequence, r e f l e c t i n g the errors i n the simulation of A-|(co) (see graph 2.5). It i s i n t e r e s t i n g to note that the differences between the responses of models 5 and 6 are r e l a t i v e l y small, although the order of the approximation in model 6 i s almost twice that of model 5. The response of the 2-pole approximation (model 4) i s reasonably good in the low to mid frequencies range, and offers a d e f i n i t e o v e r a l l improvement over the constant parameters model (model 7). ~i—iMin—i i I I I I I — i — i I I I I I , — i — I I I I I I , i M U M . i — r m 10" 2 4 1 2 4 10 2 4 10 2 4 10 2 4 10 2 4 10' FREQUENCY (HZ) (a) 2.25: O/C response, model 1. (a) P o s i t i v e sequence (b) Zero sequence ~\ I I I I I I I I I 11 II 2 4 10 2 4 10" FREQUENCY (HZ) (a) i — i i u r 2 4 ID' 2.26: O/C response, model 2. (a) P o s i t i v e sequence (b) Zero sequence 134 F i g . 2.27: 0/C response, model 3. (a) P o s i t i v e sequence (b) Zero sequence 135 F i g . 2.28: 0/C response, model 4. (a) P o s i t i v e sequence (b) Zero sequence 136 CL o II LU D O o LU to Ln O O CL LO LU a: O CD ]0.o q 9.0 - 8.0 7.0 - 6.0 5.0 -j 4.0 - 3.0 : 2.0 - 1 .0 -i- I M U M 1 M U M 1 M U M . I I I I I I I 10"' 2 4 1 2 4 10 2 4 10 2 4 10' FREQUENCY (HZ) (a) IO.O q 9.0 - II F i g . 2.29: 0/C response, model 5. (a) P o s i t i v e sequence (b) Zero sequence lO.O-i 9.0 H F i g . 2.30: O/C response, model 6. (a) P o s i t i v e sequence (b) Zero sequence 138 FREQUENCY (HZ) (b) F i g . 2.31: 0/C response, model 7. (a) P o s i t i v e sequence (b) Zero sequence 139 3.5.3 Short C i r c u i t Response The short c i r c u i t responses of the d i f f e r e n t models are shown in graphs 2.32 through 2.45. In t h i s test, the re s u l t s are influenced by the approximation of Zc(co) (see equation (3.2)), and the errors i n the approximation of Ai(co) now combine (usually disfavourably) with the errors i n Z c(co). The r e l a t i v e l y large errors in the low frequency range are due 2 to the numerical s e n s i t i v i t y of the factor A-|/(1 - A-]) i n equation (3.2) at low frequencies, where the magnitude of A-|(u)) i s very close to 1.0. For example, a difference of 0.01% i n the approximation of A i ( w ) can produce errors up to 100% in the short c i r c u i t response [9]. These errors, however, are not very important as long as the dc response i s matched i d e n t i c a l l y , which i s the case when i s known. 0.050 -f FREQUENCY (HZ) Graph 2.32: S/C response, model 1. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence Graph 2.33: S/C response, model 1. Low frequencies, (a) P o s i t i v e sequence (b) Zero sequence Graph 2.34: S/C response, model 2. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence Graph 2.35: S/C response, model 2. Low frequencies. (a) P o s i t i v e sequence (b) Zero sequence O.OJO -j Graph 2.36: S/C response, model 3. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence Graph 2.37: S/C response, model 3. Low frequencies. (a) P o s i t i v e sequence (b) Zero sequence Graph 2.38: S/C response, model 4. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence 0.100 T .090 .080 .070 .060 .050 .040 .030 .020 .010 -0.000 10 H I I M i l l , 3 4 6 10 i i M I I I , — r 4 6 10" 2 FREQUENCY —I—I I M i l l 3 4 6 1 (HZ) -i—i I I I I I I 3 4 6 10 (a) Graph 2.39: S/C response, model 4. Low frequencies. (a) P o s i t i v e sequence (b) Zero sequence Graph 2.40: S/C response, model 5. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence 0.100 -i 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 H . 000 -f 1 — i — i I I I I I I , 1 — I — i i I 1111 , I—I—I I I I I I I I — I—I I I I I I I 10 2 3 4 6 10 2 3 4 6 10" 2 3 4 6 1 2 3 4 6 10 FREQUENCY (HZ) (a) Graph 2.41: S/C response, model 5. Low frequencies. (a) P o s i t i v e sequence (b) Zero sequence 0.050 H . ii _ 0.040 _ J UJ o o LU CO CO o Q- LU CO o co- co o - s CO 0.030 0.020 H « 0 . 0 1 0 H FREQUENCY (HZ) (a) 0.010 CL C J X CO Ixl O o UJ CO o cr. LU ( V I LU CO o 0_ CO LU 0.002 £ 0.001 -0.000 T 1 1—I M i l l 2 3 4 6 10" FREQUENCY (HZ) (b) Graph 2.42: S/C response, model 6. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence Graph 2.43: S/C response, model 6. Low frequencies. (a) P o s i t i v e sequence (b) Zero sequence 0.050 -* * 0.040 -̂ Graph 2.44: S/C response, model 7. Mid to high frequencies, (a) P o s i t i v e sequence (b) Zero sequence O.lOO-i 0.090 0.060 -1 ^ 0.070 -I LU o 0.060 -1 0.050 0.040 0.030 0.020 - 0.010 - 0.000 10' n—i I I I I I I , 3 4 6 10 -i—i i i 1111 3 4 6 10" n—I I I 11 ll 3 4 6 1 -|—I—I i I 11 II 2 3 4 6 10 FREQUENCY (HZ) (a) Graph 2.45: S/C response, model 7. Low frequencies, (a) P o s i t i v e sequence (b) Zero sequence 154 3.6 Transient Simulation i n the EMTP To complement the tests i n section 3.5, a transient simulation using the EMTP, w i l l be shown in this s e c t i o n . Consider the c i r c u i t i n Figure 3.2. t = 0 C O S (IAJI) 0 m - o F i g . 3.2: Line energization t e s t . The line w i l l be assumed to be single-phase with p o s i t i v e or zero sequence parameters. The 60 Hz sinusoidal voltage source (1.0 p.u. peak) i s connected to the c i r c u i t at t = 0, and the receiving end i s open. The receiving end voltages for the d i f f e r e n t models used i n section 3.5 are shown i n graphs 2.46 to 2.51. Note that the res u l t s from the high order approximation, obtained with the frequency-dependence routines of the EMTP (model 1), are used as a reference. Graph 2.46: Energization test, receiving end voltage. Model 2. (a) P o s i t i v e sequence (b) Zero sequence 156 a -4.00 or. 0.0050 0.0300 0.0150 0.0200 TIME (SECONDS) (a) 0.0250 0.0300 4.00 -i 0.0050 0.0100 0.0150 TIME (SECONDSI (b) 0.0200 0.0250 0.0300 Graph 2.47: Energization t e s t , receiving end voltage. Model 3. (a) P o s i t i v e sequence (b) Zero sequence 157 Graph 2.48: Energization t e s t , receiving end voltage. Model 4. (a) P o s i t i v e sequence (b) Zero sequence 158 Graph 2.49: Energization t e s t , receiving end voltage. Model 5. (a) P o s i t i v e sequence (b) Zero sequence 159 Graph 2.50: Energization t e s t , receiving end voltage. Model 6. (a) P o s i t i v e sequence (b) Zero sequence 160 Graph 2.51: Energization t e s t , receiving end voltage. Model 7. (a) P o s i t i v e sequence (b) Zero sequence CONCLUSIONS A low-order approximation for the frequency-dependent parameters of a transposed, overhead transmission l i n e , has been obtained i n the second part of this thesis p r o j e c t . The amount of information required for such an approximation i s r e l a t i v e l y small. Compared to the constant parameters option of the EMTP, the only a d d i t i o n a l parameters needed to approximate the frequency- dependent behaviour of the l i n e , are the earth r e s i s t i v i t y and the dc resistance of the conductors. The main advantages of this approximation can be summarized as follows: i ) It i s computationally faster than the e x i s t i n g frequency-dependence routines of the EMTP, both i n the line parameters approximating process and during transient simulations. i i ) It i s more accurate than the constant parameters model. i i i ) It gives very accurate r e s u l t s over the low to mid frequencies range. • Two main disadvantages can be pointed out: i ) The equivalent l i n e configuration i s only accurate when the transmission l i n e can be assumed to be p e r f e c t l y transposed. i i ) It loses accuracy when very f a s t transients are considered. 161 APPENDIX I-A GENERAL SOLUTION OF THE LINE EQUATIONS IN THE FREQUENCY DOMAIN 162 163 The equations r e l a t i n g voltages and currents i n the frequency domain are d V Dh - e— = Z , I , (series voltage drop equation) (I-A.1) dx Ph ph d Iph - — — — = Ypj^ (shunt current drop equation). (I-A.2) Where Zp n and Ypn are the series impedance and shunt admittance matrices, respectively, and the subscript p^ stands for phase q u a n t i t i e s . D i f f e r e n t i a t i n g (I-A.1) and (I-A.2) we obtain d V£h = ( Z ^ Y_J V_ u (I-A.3) d x2 Ph Ph ph d 2 I p h = (Y , Z . ) I . (I-A.4) 9 ph ph ph d x^ To proceed with the solution of equations (I-A.3) and (I-A.4), i t i s convenient to diagonalize ( Z ^ Y D n ^ a n d ^ Yph Zph^ i n o r d e r t o obtain a decoupled set of d i f f e r e n t i a l equations. Let P and Q be the matrices that diagonalize ( Z p h Yph^ a n d ( Y p h Z D h ) , r e s p e c t i v e l y . Then P"1 Zph Yph P = D z y (diagonal) (I-A.5) n-1 Y Z Q = D„„ (diagonal). (I-A.6) y ph ph y z Let us now define P"1 V p h = V m (I-A.7) Q-1 I = I m . (I-A.8) Introducing these d e f i n i t i o n s and equations (I-A.5) and (I-A.6) into (I-A.3) and (I-A.4). we obtain 164 (I-A.2) where, and, d 2 V, d x m 2 = Uzv V m = D_. V m (I-A.7) d2 I m = D._ I m (I-A.8) d x 2 yz Introducing the above d e f i n i t i o n s into equations (I-A.1) and d V, ii = i z m = D_ I m (I-A.9) d x d I. — = D y V m, (I-A.10) d x -1 'z P "ph; = p Z_KQ = Z m (diagonal) (I-A.11) D = Q~' Y , P = Y m (diagonal), (I-A.12) y ph Z m = modal impedance matrix Y m = modal admittance matrix D z y = D y z =y 2' Equations (I-A.7) to (I-A.10) can now be rewritten, d 2 vm 2 d x 2 d2 xm d x 2 d vm d x Y Z V m (I-A. 13) Y 2 I m (I-A.14) = Zm Im (I-A.15) 165 — = Y m V m (I-A.16) d x These equations define the behaviour of the l i n e i n the modal domain. Since they are uncoupled, the solution i s r e l a t i v e l y simple and given by V m(x,u» = A e " Y ( a ) ) X + B e Y ( w ) X (I-A.17) . » A - Y (O J ) x B Y(co) x . Im(x,co) = — e - - e ' (I-A.18) c c where, \l zm Ym (propagation constant) Zm (surge impedance) Y m Note that Z c and y are diagonal matrices; t h i s implies that equations (I-A.17) and (I-A.18) have the same form when Z c and y are either matrices or scalars ( i . e . , when only one propagation mode i s under consideration). A and B are also diagonal matrices and they are defined by boundary conditions. Consider, for example, the li n e below 'm -o m 'm k c- 1 6 6 where only one propagation mode w i l l be considered, and subscripts K and m now denote sending and receiving ends, r e s p e c t i v e l y . Taking, for instance, x = £ v -y-i Y£ m = A e + B e A -yl B yl _ Im - — e - — e z c Z c give A = vm zc Im yl 2 e vm + zc Im -yI B = ~2 e Therefore the voltage and current at the sending end become Vk = V m - Z c I m ^ v Y£ -yl yl -yl I = — e - e ' _ e' + e ' k Z c 2 Im 2 which i n terms of hyperbolic functions, give V k = V m cosh(yJc) - Z c I T a sinh(y£) (I-A.19) V m I k = — sinh(y^) - I m cosh(y^) (I-A.20) z c These represent the " c l a s s i c a l " solution f o r a single phase l i n e i n the frequency domain. For detailed proof of the matrix re l a t i o n s h i p s stated here, the reader should refer to [8]. APPENDIX I-B J O S E M A R T I ' S F R E Q U E N C Y - D E P E N D E N C E MODEL 167 168 What follows i s only a b r i e f overview of the basic concepts upon which the frequency-dependence model used i n this thesis project i s based. Obvious space l i m i t a t i o n s do not permit a detai l e d explanation of the model, but rather a reference guide for the reader who i s somewhat f a m i l i a r with the model. For a more complete explanation of the model the reader should refer to [3] and [7]. 169 Consider the l i n e represented (for a given mode) i n Figure I-B .1 below im(t) vm(t) F i g . I - B . 1 : Modal representation i n the time domain. The associated model for this l i n e in the frequency domain i s shown i n Figure I-B.2. F i g . I-B.2: Line model i n the frequency domain. Zeq(co) i s approximated as an R-C network which has, e s s e n t i a l l y , the same response as the surge impedance Z c(co). The backward propagation 170 functions B k( to) and B^i to) are defined as follows Bk(to) = V k(to) - Z e q(to) I*(to) (I-B.1) Bm(to) = V m(to) - Z e q(to) I m(to). (I-B.2) Let us now define the forward propagation functions F^to) and F m(to), F k(to) = V k(to) + Z e q(to) I k(to) (I-B.3) Fm(u) = Vm(to) + Z e q(to) I m(to). (I-B.4) The general solution of the l i n e i n the frequency domain (see appendix I-A) i s given by V k = cosh(Y&) V m - Z C sinh ( y l ) I m (I-B.5) I k = — sinh(Y)t) V m - cosh (YJi) I m . (I-B.6) Z c The r e l a t i o n s h i p between Bk(to) and F k(to) i s found by combining equations (I-B.1) to (I-B.6). After some algebraic manipulations, and taking i n t o account that Z eq(to) = Z c ( O J ) / w e obtain, -Y J Bk(to) = e F m(oj) (I-B.7) B^to) = e~yl F k(to). (I-B.8) Let us now define the propagation function A-| (̂ ) as -Y I? A^to) = e , (I-B.9) where I i s the l i n e length and Y = \/ Z' Y' (see appendix I-A). Equations (I-B.1) to (I-B.4) can now be rewritten as Bk(to) = V k(to) - Ek(to) = A-|Fm(to) (I-B.10) Bm(to) = Vm(to) - Emfto) = A-|Fk(to) (I-B.11) F k(to) = V k(to) + E k(to) (I-B.12) F m(to) = Vm(to) + Em(to) , (I-B.13) where, E k(to) = I k(to) Z e q(to) (I-B.14) EmJto) = I m(to) Z e q ( o ) ) . (I-B.15) 171 Since Zeg(co) i s the response of a l i n e a r R-C network, the time domain representation of equations (I-B.10) through (I-B.15) can be found by means of the Inverse Fourier Transform, that i s , b k ( t ) = vfc(t) - e k ( t ) = a i ( t ) * f m ( t ) (I-B.16) bm( f c) = vm(t) " em(t) = a , ( t ) * f k ( t ) , (I-B.17) where e k ( f c ) = z e q ( t ) * i k ( t ) (I-B.18) e m ( t ) = z e q ( t ) * i m ( t ) . (I-B.19) (lower case l e t t e r s are used to indicate time domain quantities, while upper case i s used i n the frequency domain) These equations define the equivalent c i r c u i t of Figure II-B.3 k o •m(t) o m v m ( t ) F i g . II-B.3: Frequency dependence model in the time domain. Note that i n the time domain, e k ( t ) i s the voltage across the equivalent R-C network that simulates Z c ( u ) . The propagation function a-|(t) i n the time domain can be expressed as a-,(t) = p(t-C), (I-B.20) 172 where p(t) has the same shape as a-|(t) but displaced-t time units from the o r i g i n . In the frequency domain A-|(to) can then be expressed as AT ( t o ) = P ( t o ) e - 3 ^ c (I-B.21 ) The function P(to) can be approximated by r a t i o n a l functions of the form P(s) = H (s + z i ) (s + Z2) ... (s + z n ) , (s + p-i ) (s + p 2) ... (s + p m) where z± and p^ are the zeros and poles of P(s) i n the complex plane; these s i n g u l a r i t i e s are r e a l and l i e in the left-hand side of the complex plane (m>n). P(s) can be expanded i n p a r t i a l f r a c t i o n s P(s) = k1 + k 2 + ... + km s + p-| s + p 2 s + p m (I-B.22) Therefore, in the time domain p(t) becomes p(t) = [k, e~plt + k 2 e _ P 2 t + ... + k m e"-^ 1] u ( t ) . From equation (I-B.20) we now obtain a-|(t) (I-B.23) a l ( t ) = [k! e - P l ( t " C ) + k 2 e-P2 ( t" c> + ... + k m e ^ m ^ O j u ( t _ c , With a-|(t) i n the form of equation (I-B.23), the convolutions i n equations (I-B.16) and (I-B.17) can be solved by recursive integration methods. Consider, for instance, the convolution i n t e g r a l of equation (I-B.16) OO bk(t> = f-co f m ( t _ u ) a l ( u ) d u ' since a i ( t ) = 0 f o r t<r t h i s i n t e g r a l becomes b k ( t ) = /" f m ( t - u ) a i ( u ) du. (I-B.24) Introducing equation (I-B.23) into (I-B.24), m b k ( t ) = E b. .(t) (I-B.25) i=l k , i where •D , * (I-B.26) °° ( u - r ) b k , l ( t ) = fr f m ( t - u ) k. e 1 du Note that i n equation (I-B.26) p i has been substituted by 6 i to avoid future confusion i n notation. The i n t e g r a l i n equation (I-B.26) can be broken i n t o two parts, **,i<t> = / , r + A t fm(t-u) k ± e " e i ( u - C ) d u + / ^ + A t f m ( t - u ) k ± e - e i ( u " C ) d u , (I-B.27) but the second i n t e g r a l i n equation (I-B.27) i s e _ ^ i A t b k i ( t - A t ) , and the f i r s t i n t e g r a l can be evaluated numerically using the trapezoidal r u l e . After some algebraic manipulations,, b, . (t) = g-; bv -j(t-At) + c. f (t-C) + d. f (t-C-At), (I-B.28) k, 1 J . JS. ,x i m i m where, 9i = e 1 1 - g ± h i BiAt k T7 (1 " *i d i = - — ( g i - h ±; and m b k ( t ) = I h .(t) k i = l k , i From equations (I-B.11) and (I-B.13) i t can be seen that f m ( t ) = 2 v m ( t ) - b m ( t ) , therefore, b k > i ( t ) i n equation (I-B.28) depends only on past h i s t o r y values of b k / i , b m and v m . The evaluation of e k ( t ) = i k ( t ) * z e q ( t ) proceeds i n a 174 s i m i l a r fashion as the evaluation of b k ( t ) . The r a t i o n a l f r a c t i o n s that approximate Z eq(io) are given by, - , . H ( s + 21) ( s + 22) ... ( s + zn) , Z (S) = n ; — j : ; r' eq (S + Pl) (s + p 2) .•. (s + p m) where, as i n the case of A-|(o>) the poles are r e a l and l i e in the left-hand side of the complex plane, but i n this case n = m. Expanding i n p a r t i a l f r a c t i o n s , k k Z e q ( s ) = k Q+ 1 + ... + m (I-B.29), s + p. s + p 1 m therefore, - a i t . . - r ^ t . z eq ( t ) = [k 0 6(t) + k! e 1 L + ... + k m e " 0 ^ ] u ( t ) , (I-B.30) where p^ has been substituted by a j_ to avoid ambiguity in the notation. From equation (I-B.18) CO ek = ' 0 i k ( t - u ) z e q ( u ) du, (I-B.31) where the lower l i m i t i n the i n t e g r a l i s zero because z e q ( t ) = 0 for t<0. Introducing equation (I-B.30) into (I-B.31), (I-B.32) where, ek<t> = e k , o ^ ) + J ^ k , ! ^ ' e k , o ( t ) = ±)<. { t ) k o = R o i k ( t ) (I-B.33) e k , i ( t ) = f Q i k ( t - u ) k i e d u - (I-B.34 Noting that the i n t e g r a l i n equation (I-B.34) i s analogous to the i n t e g r a l i n equation (I-B.24) with 1=0, e k j i ( t ) = ith e k > i ( t - A t ) + p ± i k ( t ) + q ± i k ( t - A t ) (I-B.35) where, -a.: At m̂ = e h - 1 ' 175 k. P i - M - h i ) <li ± (mi - hi) 1 Introducing equations (I-B.33) and (I-B.35) into (I-B.32) e k ( t ) = Rk L k { t ) + e k , e ( t _ A t ) + e k , c ( t _ A t ) ' (I-B.36) where, m Rk = R o + J x P i (equivalent constant resistance) e k , c ( t - A t ) = m i w ( t - A t ) (history of currents) m e k ( 6 ( t - A t ) = Z mi e k > i ( t - A t ) (history of p a r t i a l voltages e k ( i ) With equations (I-B.28) and (I-B.36), the equivalent c i r c u i t of Figure I-B.3 can be s i m p l i f i e d as shown in Figure I-B.4 i k ( t ) Rk k o vwr v k ( t ) e k c + e k e —(̂ )̂— b k ( „6 V m ( t ) F i g . I-B.4: Equivalent c i r c u i t i n the time domain This equivalent c i r c u i t (using elementary c i r c u i t transformations) can be transformed into the c i r c u i t of Figure I-B.5, which i s compatible with the EMTP; 176 'm(t) c m vm(t) F i g . I-B.5: Simplifed c i r c u i t i n the time domain. wherei , i s a past hi s t o r y current source, and Rv i s an equivalent eq,k constant resistance, ieq.k = f e k , c ( t " A t ) + e k , e ( t " A t ) + b k ^ ^ / R ] , (I-B.37) The equivalent c i r c u i t i n Figure I-B.5 can be solved recording the past h i s t o r y values of equation (I-B.37). At the beginning of the process these are zero i f the i n i t i a l conditions at the ends of the l i n e are zero. If these i n i t i a l conditions are not zero, that i s , i f they are given by pre-transient steady-state, or user-supplied dc i n i t i a l condi- tions ( i . e . , trapped charge), the past h i s t o r y current sources can be determined as follows. The equivalent c i r c u i t for steady-state conditions i s shown i n Figure I-B.6 177 F i g . I-B.6: Equivalent c i r c u i t for steady-state conditions. Consider for example node k. V k and I k are the phasor voltages and currents known from the i n i t i a l steady-state solution of the system. We must now find the steady-state equivalents of e k , i < • Ek,i b k , i < » \ , i fm < >• F m From equation (I-B.14) Ek = !k z e q but from equation (I-B.29) z e q = Z Q + Z 2 + ... + Z m, (I-B.38) where, Z D = k Q (I-B.39) z . = k i • (I-B.40) 1 jco + P i Introducing (I-B.39) and (I-B.40) into (I-B.14) _ _ m _ E K = Z Q I k + ( I Zi) I k 1=1 therefore, 178 E k , i = z i *k P a r t i a l h i s t o r y sources B k > i are obtained as follows: From equation (I-B.10) % = A l Fm-' but from equations (I-B.21) and (I-B.22) A l = A l , l + A l , 2 + ••• + A 1 > N where, k. 1 - ] C 0 C A-, • = — e 11 1 j co + p^ Introducing (I-B.42) i n t o (I-B.10), _ _ m *k = Fm A l , i > ' and, F i n a l l y , F m i s obtained from equation (I-B.4) Fm = vm + zeq •'-m The phasor quantities i n the time domain are given by e k , i ( t ) = | E k , i | cos (cut + a r g ( E k f i ) ) b ^ f t ) = |B k / i| cos (cot + a r g ( B k / i ) ) f m ( t ) = | F m | cos (cot + arg(F m)) . When the i n i t i a l conditions are supplied by the user, i . e . , trapped charge, the i n i t i a l values of e k /-j_, b k ^ and f m can be obtained as before by s e t t i n g co = 0. A s i m i l a r procedure i s followed for node m. APPENDIX I-C USER'S GUIDE FOR THE VOLTAGE AND CURRENT PROFILE OPTION OF THE EMTP, AND FOR THE OUTPUT DISPLAY PROGRAM 179 180 The routines that evaluate the e l e c t r i c a l p r o f i l e of a l i n e have been implemented as part of the frequency-dependence version of UBC's EMTP The object code of this version i s available to UBC users under the name LUI:MDTRAN3.0. The interconnection with the EMTP required approximately 90 FORTRAN statements, and the supporting subroutines required s l i g h t l y less than 660 FORTRAN statements. A t y p i c a l c a l l to this version of the EMTP i s shown below $ RUN LUI:MDTRAN3.0 0=PAR.INT 1=PAR 2=-2 3=-3 4=-4 5=TRAN.DA 6=-X The only differences between this command and the standard frequency-dependence command are l o g i c a l units 0 and 3. Logical unit 0 contains the constants k^ and 3̂ from the r a t i o n a l - functions approximation of A-|(co) for the length of one of the segments into which the l i n e has been subdivided. F i l e PAR.INT does not include the approximation for z c(w) since i t i s the same used for the complete l i n e ( l o g i c a l u nit 1 s t i l l contains the same parameters whether the pro- f i l e i s requested or not). The currents and voltages are written i n free format (to save computer costs and to ease the manipulation of the large amounts of data involved) i n t o l o g i c a l unit 3. To invoke the " p r o f i l e " option, the following modifications have been introduced i n t o the l i n e card (frequency-dependence option) i n f i l e TRAN.DA ( l o g i c a l u nit 5) I TY PE | NODE NAMES Freq. (Hz) ] M OD EL j 1 N IN T j | ID E L T j | 11 I TY PE | NODE k NODE m ] M OD EL j 1 N IN T j | ID E L T j 12 A6 A6 E62 12 12 12 1 2 3 i s t 7 e 9 10 11 U 13 14 15 IS 17 ia 19 20 21 22 2] 21 25 26 27 28 29 X 31 32 33 31 35 36 37 36 39 10 41 12 13 44 15 46 47 4B 49 so 5', 52 53 51 56 57 5* 5S 60 si 62 63 516556 15 70 pi n 73 71 15 77 It 73 - i 181 The a d d i t i o n a l parameters are explained below: NINT > 0 number of intermediate voltages and currents. The minimum i s 1 and the maximum i s 99. = 0 disables " p r o f i l e " option, IDELT > 0 number of times the i n t e r n a l At i s to be segmented. For example, IDELT=2 reduces the i n t e r n a l time step At' to At'/2. IDELT = 0 the i n t e r n a l At i s determined automatically by the program. (Note that even i f IDELT > 0, the voltages and currents w i l l s t i l l be printed with the same number of time steps defined i n the main program). A time saving option has been included i n the time card, A t ( s ) t m a x * s ) I M A X I P U N C H IS K IP £ § / ) 5 O CL A f ( H z ) * m o x t H z ) | 11 c r o iA E 1 0 . 6 E 1 0 . 6 13 13 13 E 8 0 [1 36" 11 E 8 0 E 8 0 E E 0 Ll 11 X i ? 3 i 6 7 B 9 1 0 ii I ? 13 U lb IC 17 18 19 70 ?i p ?3 ? 6 ? 7 7 3 1 0 31 3 ? 33 34 15 X J7 33 it ii 17 43 44 4b 45 47 ^ 0 y s i S7 S O ' 0 il 7(1 7* 77 1 1 - f T ! ! ISORT = 0 or 1 A l l voltages and currents w i l l be written on l o g i c a l u nit 3. ISORT = 2 Only the intermediate voltages w i l l be written on l o g i c a l u n i t 3. ISORT = 3 Only the intermediate currents, w i l l be written on l o g i c a l u n i t 3. A sample of the d i f f e r e n t input f i l e s i s shown below for the John Day - Lower Monumental l i n e . Only the intermediate voltages are requested, the l i n e w i l l be segmented into 9 sections, and the i n t e r n a l 182 time step w i l l be made two times smaller than the i n t e r n a l l y calculated time step. 1 INTERMEDIATE PARAMETERS ARRESTER TEST. 3PHASE MON LINE 2 .93322E-05.020 O 2 3 -1SWIA TWOA 60.00 - 1 9 4 -2SWIB TWOB 60.00 -1 9 5 -3SWIC TWOC 60.00 -1 9 6 7 ONEA SWIA .0005 1.0 8 ONEB SWIB .0003 1.0 9 ONEC SWIC .0006 1.0 10 92TW0A 1.0 2.6 2.0 11 92TW0B 1.0 2.6 2.0 12 92TW0C 1.0 2.6 2.0 13 14 140NEA 1 1.0 60. -10.8 O.O 15 140NEB 1 1.0 60. -130.8 0.0 16 140NEC 1 1.0 60. 109.2 0.0 17 18 TWOA TWOB TWOC 19 20 2 1 End of F i l e Sample input parameters f i l e of the p r o f i l e option of the EMTP To ease the manipulation of the large amounts of output generated during p r o f i l e c a l c u l a t i o n s , a separate program reads the output data from l o g i c a l u n i t 3 ( o r i g i n a l l y written i n free format and thus meaningless under MTS LIST OR COPY commands) and writes only the part of the output requested by the user. The object code of this a u x i l i a r y program i s available under the name LUI:OUTINTL.O. A t y p i c a l run command i s shown below $ RUN LUI:OUTINTL.O 3=-3 5=PAR.0UT 6=-A 7=-B The requested voltages are written on l o g i c a l u n i t 6 and the currents on l o g i c a l unit 7. P i l e PAR.OUT contains the output control parameters. An example 183 i s shown below (a sample control f i l e i s also a v a i l a b l e on "READ ONLY" basis under the name LUI:PAR.0UT). 1 +0 IALLV 2 -1 IALLI 3 1 1 (BRANCH.SECTION) (VOLTAGES) 4 2 1 (BRANCH,SECT ION) (VOLTAGES) 5 3 1 (BRANCH,SECTION) (VOLTAGES) 6 2 9 (BRANCH,SECTION) (VOLTAGES) 7 (BLANK CARD) (VOLTAGES) 8 (BLANK CARD) (CURRENTS) End of F i l e Sample control parameters f i l e The parameter IALLV controls the output mode: IALLV < 0 No voltages w i l l be p r i n t e d . IALLV = 1 A l l intermediate voltages w i l l be printed. IALLV = 0 Only the voltages s p e c i f i e d by the user w i l l be p r i n t e d . When IALLV = 0, the output i s selected by i n d i c a t i n g the branch(es) and intermediate section(s) for which the voltages are desired. The branch number i s determined by the order i n which the frequency dependent l i n e s were s p e c i f i e d i n the input f i l e for the EMTP (TRAN.DA i n th i s example). I f , for example, two three-phase li n e s are studied, phase B of l i n e 1 w i l l be branch number 2. If phase C of l i n e two i s desired, the branch number would be 6. The maximum number of voltages that can be sp e c i f i e d t h i s way i s 300. A blank card signals the end of the s p e c i f i e d voltages. I f IALLV i s less than zero or 1, a blank card must also be inser t e d , to s i g n a l the end of user s p e c i f i e d voltages. 184 IALLI i s analogous to IALLV, but used for the output of cur- rents. Note that i f , for example IALLV = IALLI = 1, two blank cards must be inserted a f t e r IALLI. A sample output from l o g i c a l units 6 and 7 , using the control f i l e shown above, i s l i s t e d below. Note that a reference table with the names of the l i n e nodes (as used i n TRAN.DA), and the code number of each branch i s printed for easy reference (also useful to v e r i f y that the out- put requested i s the output obtained). VOLTAGES (BRANCH,SECTION) REFERENCE TABLE LINE 1 FROM BUS SWIA TO BUS TWOA LINE 2 FROM BUS SWIB TO BUS TWOB LINE 3 FROM BUS TIME 0.0 0. 0.90589439E-05-0. 0.181 17888E-04-0. 0. 27 176832E-04-0. 0.36235776E-04-0 0.45294720E-04-0. 0.54353664E-04 0. 0.63412607E-04 0. 0.72471551E-04 O. 0.81530495E-04 0 0.90589439E-04 0 0.99648383E-04 O. 0. 10870733E-03 0. SWIC 1 1 0 67144878E 20728855E 71533231E 20253559E 83561079E 0 0 O 0 0 0 0 TO BUS TWOC 2 1 -0 .0 76-0.6714487BE- 76- 0.20728855E- 77- 0.71533231E- 77- 0.20253559E- 78- 0.83561079E- -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 3 1 -0 .0 76-0.67144878E 76- 0.20728855E 77- 0.71533231E 77- 0.20253559E 78- 0.83561079E -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 2 9 -0 .0 •76-0.67144878E-76 •76-0.20728855E-76 •77-0.71533231E-77 •77-0.20253559E-77 •78-0.83561079E-78 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 -0 .0 0.45294720E 0.46200614E 0.47106508E 0.48012403E 0.48918297E 0.49824192E 0.5073008GE O . 5 1635980E -03 0.0 -03 0.0 -03 0.64531823E-01- -03 0.13092492E+CO- -03 O.13217002E+OO- -03 0.97277730E-01- -03 0.44892915E-01- -03 0. 1 1844B03E-02- 0.0 0.0 0.12906365E+00 0.26216827E+00 0.27711089E+00 0.30802471E+00 0.35641999E+00 0.39645930E+OO •0.0 -0 .0 -0.0 -0 .0 0.64531823E-01-0.0 0.13092492E+0O-0.O 0.13217O02E+0O-0.0 0.97277730E-01-0.0 0.44892915E-01-0.0 O.11844803E-02-0.0 0 . 17574351E -02 0 o. 17664941E -02 0 0. 17755530E -02 0 0. 17846120E -02 0 0. 17936709E -02 0 0. 18027298E -02 0 0. 18 1 17888E -02 0 0. 18208477E -02 0 o. 18299067E -02 0 0. 18389656E -02 0 0. 18557322E -02 0 o. 18650575E -02 0 0. 18743827E -02 0 0. 18837080E -02 0 91578177E+00-0. 91442024E+00-0. 91304797E+00-0. 91166498E+O0-0. 91027129E+00-0. 90886692E+00-0. 90745188E+00-0. 90602620E+00-0. 90458989E+0O-0. 90314298E+O0-0. 90043776E+00-0. 89891700E+00-0. 89738506E+00-0. 89584197E+00-0. 11345900E+00- 11006518E+00- 10667015E+00- 10327395E+CO- 99876626E-01- 96478212E-01- 93078746E-01- 89678265E-01- 86276807E-01- 82874410E-01- 76574874E-01- 73069876E-01- 69564040E-01- 66057407E-01• 0.80320761E+00 0.80522853E+00 0.80724016E+00 O.80924245E+00 0.81123540E+O0 0.8132 1896E*00 0.81519312E+0O 0.81715785E+00 0.8191 1312E*0O 0.82105892E+00 0.82463582E+00 0.82661063E+00 O. 82857528E*00 0.83052975E+00 -0.0 •0.0 •0.0 0.0 •0.0 -0.0 -0.28648792E-01 •0.12252582E+00 •0.19735267E+00 -0.23084061E+00 -0.24633897E+00 -0.24802598E+00 •0.24856349E+00 -0.24852947E+00 Sample output from l o g i c a l u n i t 6. Output display program. 186 CURRENTS (BRANCH,SECTION) REFERENCE TABLE LINE 1 FROM BUS SWIA TO BUS TWOA LINE 2 FROM BUS SWIB TO BUS TWOB LINE 3 FROM BUS SWIC TO BUS TWOC NO CURRENT OUTPUT REQUESTED Sample output from l o g i c a l u n i t 7. Output display program. APPENDIX II-A SKIN EFFECT CORRECTION FOR ROUND CYLINDRICAL CONDUCTORS 187 188 As the frequency of the current c i r c u l a t i n g i n a conductor increases, the current density increases near the surface of the conductor. This phenomenon i s known as skin e f f e c t and i t aff e c t s the resistance and i n t e r n a l inductance i n large power conductors. The resistance increases with the frequency while the i n t e r n a l inductance decreases. The formulas shown below permit the c a l c u l a t i o n of skin e f f e c t for tubular conductors. Stranded conductors can be approximated as s o l i d conductors of the same cross-sectional area. Steel reinforced conductors can be approximated as tubular conductors when the influence of the s t e e l core i s ne g l i g i b l e (note that s o l i d conductors are only a p a r t i c u l a r case of tubular conductors). The formula used for the c a l c u l a t i o n of skin e f f e c t i s R s + J a ) L i n t _ ,1_ 2\ (ber mr + j b e i mr) + 0 (ker mr + jk e i mr) R d c ~ :2 m ~ S (ber' mr + j b e i ' mr) + 0 (ker' mr + j k e i ' mr) with _ ber' mq + j b e i ' mq ker' mq + j k e i ' mq ' and, R s = ac resistance (skin e f f e c t included) i n °./km R d c = ^ c resistance i n ^/km L i n t = i n t e r n a l inductance (skin e f f e c t included) i n H/km r = outside radius of conductor q = inside radius of conductor s = ^ • r and, (mr) 2 = k — - 1 - s 2 189 Fig . I I - A . 1 : Tubular conductor. The modified Bessel functions can be calculated with polynomial approximations. A subroutine that uses the above formulas i s used i n the Line Constants Program. The same subroutine i s used for a l l the skin e f f e c t c a l c u l a t i o n s i n this t h e s i s . A t y p i c a l c a l l to this subroutine has the form CALL SKIN (S, RDC, F, RS, XI) where S = q / r RDC = dc resistance i n fi/km F = frequency i n Hz 190 RS = ac resistance i n Q/km XI = ac reactance i n ft/km A sample c a l c u l a t i o n of R/R d c and L i n t / L i n t / d c f o r s = 0 , 5 a n d Rdc = 0 , 0 1 ft/km i s shown i n Figure II-A.2 10' = FREQUENCY (HZ) F i g . II-A.2: Va r i a t i o n of the resistance and i n t e r n a l inductance due to skin e f f e c t . APPENDIX II-B CARSON'S CORRECTION TERMS FOR EARTH RETURN EFFECT 191 192 Carson's correction terms account for the fa c t that the earth does not behave as an i n f i n i t e and p e r f e c t l y conducting plane for ground return currents. Carson's formula i s normally accurate enough for power systems studies. It i s based on the following assumptions: (a) The conductors are long enough so that three dimensional end-effects can be neglected (this permits the solution of the f i e l d problem on a plane perpendicular to the conductors). (b) The earth has uniform conductivity and can be represented as an i n f i n i t e plane, to which the conductors are p a r a l l e l . (c) The spacing between the conductors i s much larger than their radius, so that proximity e f f e c t s can be ignored. impedance and $ = 0. for mutual impedance), and on the parameter a, 1, K where Carson's correction terms depend on the angle $($ = 0 for s e l f a (II-B.1 ) wi th D 2 (in m) for s e l f impedance D D i , k ^ n m) f ° r mutual impedance f frequency in Hz P earth r e s i s t i v i t y i n fi«m. 193 h; images F i g . II-B.1: Tower geometry. The correction term for the resistance AR becomes i n f i n i t e when f -»• °°, while AX/to = AL goes to zero. If the earth r e s i s t i v i t y i s very low and f i s f i n i t e , AR and Ax are very small. In the l i m i t , i f earth i s a perfect conductor (p = 0), AR and Ax become zero. Carson's i n f i n i t e series when a < 5 can be written as, AR = 4to-10~4{^- -b]_a« cosi^ +b2[ (c2~lna) a2cos2<f>+4>a2sin2<t>] AX 4to- 10~4{-2-(0.6159315-lna) +b]_a- coscj) -d2a2cos2cf> +b3a3coscf> -b4[ (C4~lna) a4cos4<f)+<j>a4sin4cj>] +b7a 'cos7c|> -dga^cosScJ) •f-bgC (c6-lna)a6cos6<f>+cJ>a6sin6cf>] + 5a-'cos5cJ) -dga^cosScj) + b7a^cos7<f) - .. .} (II-B.3) -bg[ (cg-lna) â cos8<f>+<f>â sin8<}>] + .. .} (II-B.4) 194 Each 4 successive terms form a r e p e t i t i v e pattern. The c o e f f i c i e n t s b^, C i and d i are constants, which are obtained from the recursive formulas: b, = -— for odd subscripts. 1 6 sign . . b. = b. „ with the s t a r t i n g value < l i-2 i(i+2) 1 . b = — for even subscripts, <- 2 16 c- = c - + -T- + r ^ r with the s t a r t i n g value c = 1 .3659315, l i-2 l i+2 d • = T . b. , l 4 l with sign = ±1 changing af t e r each 4 successive terms (sign = +1 for i = 1,2,3,4; sign = -1 for i = 5,6,7,8, e t c . , ) . For a > 5, the following f i n i t e series i s used: A R ' A X ' = cos<j> a cos 6 2 cos2§ cos3<j> 3cos5<; a' cos3d 3cos 5<j> a-" 45cos7<f> 45cos7(j) a 7 4co-10"' 4LO- 10~ 4 (II-B.4) (II-B.5) The trigonometric functions are calculated d i r e c t l y from the geometry, cosd h i + n k . , x i k and sm<j> = ik D i k i k D i k and for higher terms in the ser i e s , from the recursive formulas a^cos (i<j>) = [a^^-cos ( i-1) ())• cos<f) - a 1 --'-sin (i-1) (j)* sinij)]. a a^cos (i<f>) = [ a 1 '''cos (i-1) (}>• sin<\> + a 1 "'"sin (i-1) <j>* cos<j)]-a For power c i r c u i t s at power frequency, a reasonable approxima- tio n can be obtained i f only two terms i n equation (II-B.2) are used. For higher frequencies, low r e s i s t i v i t y , and wider spacings, more terms need to be taken i n t o account as the parameter a becomes l a r g e r . The Line Con- stants Program uses these formulas to account for ground return e f f e c t s . 195 For the purposes of th i s thesis project, the section of the Line Constants Program that calculates Carson's terms has been rewritten as a subroutine. A t y p i c a l c a l l to th i s subroutine has the form CALL CARSON (HAV, DM, F, RHO, DRS, DRM, DXS, DXM) where, HAV = average height i n m. DM = geometric mean distance between conductors RHO = earth r e s i s t i v i t y i n °.«m. F = frequency i n Hz. DRS, DRM, DXS, DXM are the r e s u l t i n g s e l f and mutual correction terms i n ft/km. APPENDIX II-C SIMULATION OF THE JOHN DAY-LOWER MONUMENTAL TRANSMISSION LINE USING THE LOW-ORDER APPROXIMATION PROGRAM 196 197 BPA's John Day to Lower Monumental 500 KV transmission l i n e (see Figure 3 . 1 , part I) has been modeled with the low-order approximation routines developed i n part II of this thesis p roject. In order to i l l u s t r a t e the e f f e c t of ground wires i n the evaluation of the equivalent l i n e configuration, | A-|(to)| and | Zc(oo)| with segmented or T connected ground wires (see footnote, page 28) are shown in graphs II-C.1 and II-C.2. Graph II-C.3 and II-C.4 show | A-] (co) | and | Zc(co)| when the ground wires have been removed. It i s i n t e r e s t i n g to note in graphs II-C.3 and II-C.4, that the s l i g h t l y better approximation obtained for the reference l i n e (see Figure 1 .3, part II) i s probably due to the fact that the tower configuration of the reference l i n e i s h o r i z o n t a l , while the configuration of BPA's line i s t r i a n g u l a r . This would a f f e c t the accuracy of the estimation of Carson's correction terms at high frequencies, where the difference between considering an average height or the height of the i n d i v i d u a l conductors i s more marked. Graphs II-C.5 to II-C.7 show the open and short c i r c u i t responses of this l i n e when there are no ground wires. •198 Graph II-C .1 : A-| magnitude function. Segmented ground wires, R^c given. (a) Po s i t i v e sequence (b) Zero sequence. 199 9 5 0 TJ y 9 0 0 -1 FREQUENCY (HZ) 9 5 0 -q y 9 0 0 -i LU 8 5 0 -3 FREQUENCY (HZ) (b) Graph II-C.2: Z c magnitude function. Segmented ground wires, R$c given. (a) P o s i t i v e sequence (b) Zero sequence. Graph II-C.3: A-] magnitude function. No ground wires, Rd c given, (a) P o s i t i v e sequence (b) Zero sequence. Graph II-C.4: Z c magnitude function. No ground wires, R̂,-. given, (a) P o s i t i v e sequence (b) Zero sequence. 0 . 0 6 0 t Graph II-C.5: O/C response. No ground wires, R^c given. (a) P o s i t i v e sequence (b) Zero sequence. O.OJO H . y 0.009 - 11 L U 0.008 4 Graph II-C.6: S/C response. Mid to high frequencies. No ground wires, R$c given, (a) P o s i t i v e sequence (b) Zero sequence. 204 II C E Q. m CO O Q. L U C O z o C L C O L U C C (_) \ C O 0.150 -i 0. J20 H 0.030 0.060 0.030 ^ -0.000 JO" ~T I—I I I I I I I , 1 1—I I I I I I I , I 1—I I II I 11 2 3 4 6 JO 2 3 4 6 10" 2 3 4 6 J FREQUENCY (HZ) ~l 1—I I I I I IT 2 3 4 6 JO (a) 0.150 C L C J II 0.3 20 H cr CD 0.090 o L U C O o § 0.060 o C L C O U tn 0.030 -0.000 1 r 10" 2 I I I I I I I I , I 1 I I I I I I I , 1 1 I I I I I I I 1 1 I I I I I IT 3 4 6 JO 2 3 4 6 JO 2 3 4 6 J 2 3 4 6 JO FREQUENCY (HZ) (b) Graph II-C.7: S/C response. Low frequencies. No ground wires, R^c given. (a) Po s i t i v e sequence (b) Zero sequence. APPENDIX II-D USER'S GUIDE TO THE LOW-ORDER APPROXIMATION PROGRAM 205 206 The sequential f i l e LO.O contains the object code of the low-order approximation routines. LO.O i s av a i l a b l e with READ ONLY status to UBC's MTS users. A t y p i c a l run command i s shown below $ RUN LUI: LO.O 5 = DA.LO 6 = -6 7 = -7 8 = -8 9 = -9 10 = -10 11 = -11 12 = -12 DA.LO contains the input data and the output control parameters. Logical units 6 through 12 contain output information. A t y p i c a l data f i l e i s shown below (an example data f i l e has been permitted on a READ ONLY ba s i s . This f i l e can be read under the name LUI: DA.LOT) 1 3 2 60.00 3 500 . 0 4 .OOD00003 5 100.0 6 .88080000-03 3.330800D-03 7 .01330000000-06 .00836 10000-06 8 .026190000 .02643000 .19740000 9 .01 10 10 10 11 6 4 12 5 4 13 0 5 14 4 3 15 . 5 . 5 16 1 17 1 18 1 End of F i l e NUMBER OF PHASES FREQUENCY OF PARAMETERS LINE LENGTH CONDUCTANCE EARTH RESISTIVITY L K 6 0 H Z ) . L0I60HZ1 C 1 . CO RDC , RF ..RO FM I N NO OF DEC. NO OF POINTS/DEC NUMBER OF POLES IN ZCO AND ZC1 NUMBER OF POLES IN A10 AND A11 SHIFTING OF A10 AND A11 (0=N0) SHIFTING OF ZCO AND ZC1 (0=N0) ERROR (%) AT LINE FREQUENCY FOR ZC AND A1 WRITE R(W), L(W) (1=YES 0=N0) WRITE ZC(W), *ZC(W)* (1=YES 0=N0) WRITE A1. *A1* (1^YES 0=N0) Sample input parameters f i l e . The information from this f i l e i s read i n free format, therefore a blank or a semi-colon can be used as delimitator between q u a n t i t i e s . Line 1 i s the number of phases; the minimum i s 2 and the maximum i s 20. The v i o l a t i o n of these l i m i t s i s not detected i n t e r n a l l y by the program, but the system w i l l discontinue execution due to array overflow. 207 Lines 2 through 8 are self-explanatory. Note that i f the dc resistance i s not known ( l i n e 8) and the 'RDC not given' option i s required, a f l a g value of 0.0 must be introduced i n the place of R^c. Line 9 contains the s t a r t i n g frequency at which the frequency- dependent response of the parameters i s to be evaluated. Line 10 contains the number of decades and points per decade of the desired frequency band. Line 11 contains the number of poles required i n the simulation of Zc(to) for zero and p o s i t i v e sequences. The maximum number permitted i s 10 and the minimum i s 1. No warning messages detect the v i o l a t i o n of these l i m i t s , but an array overflow w i l l r e s u l t and the execution w i l l stop. Line 1 2 i s the number of poles for the approximation of the propagation function A-| (to); the maximum number i s 10 and the minimum i s 2. As in the case of Zc(to) no warning messages are provided, but i f the l i m i t s are vi o l a t e d , execution w i l l stop. Line 13 controls the s h i f t i n g option for the approximation of A-|(to). Zero disables this option and matching at l i n e frequency w i l l be assumed. Values greater than zero specify the maximum number of s h i f t i n g loops. If an improvement of more than 1% i n the area between | A-](to)| and | B(to)| i s not obtained, or i f the maximum number of loops i s met, the s h i f t i n g process stops and control i s directed to the next part of the program. Line 14 contains analogous information as l i n e 13 for the approximation of Z c(to). Line 15 i s the error i n (%) between the r e a l (obtained from R,L and C given by the user) and approximated functions Zc(co) and A-] (to) at the l i n e frequency ( l i n e 2). Note that the same error l e v e l w i l l be assumed for both zero and p o s i t i v e sequence. 208 If the s h i f t i n g option i s used, the matching procedure i s disabled. If no s h i f t i n g and no matching are desired, this error can be set to a large value (for example 50%). This w i l l override the matching procedure. Line 16 controls the information to be written i n l o g i c a l unit 7. An example i s shown below. 1 FREQUENCY RI (OHMS) L1 (MH) RO (OHM S J LO (MH) 2 0. 1000000E -01 0. 2619000E -01 0. 8808493E+00 0. 262 1959E -01 0. 5922672E+01 3 0. 1258925E -01 0. 26 19000E -01 0. 8808493E+00 0. 2622725E -01 0. 5853623E+01 4 0. 1584893E -01 0. 2619000E -01 0. 8808493E+00 0 . 2623690E -01 o. 5784579E+01 5 0. 1995262E -01 0. 26 19000E -01 0. 8808486E+00 0. 2624904E -01 0 .5715537E+01 6 0. 25 1 1886E -01 0. 2619000E -01 0. 8808486E+00 0. 2626431E -01 0 .5646500E+01 7 0. 3162278E -01 0. 26190COE -01 0. 8808493E+00 0. .2628355E -01 0. 5577470E+01 8 0. 3981072E -01 0. 2619000E -01 0. 8808493E+00 0 .2630776E -01 0 .5508444E+01 9 0 5011872E -01 0. 2619000E -01 0. 8808493E+00 0. 2633823E -01 0. , 5439424E + 01 10 0. 6309573E -01 0. 2619000E -01 0. 8808486E+00 0. .2637658E -01 0 . 537041 1E + 01 Sample output from l o g i c a l unit 7. Line 17 controls the output of the approximation process of Z c(io). Positive sequence i s written i n l o g i c a l unit 9 and zero sequence in l o g i c a l unit 8. 1 POSITIVE SEQUENCE SURGE IMPEDANCE 2 FREQUENCY ZC 1 *ZC 1 * ERROR (%) 3 0 .1000000E-01 o. 9341646E+03 0 9339930E+03 -0. 1836608E- 01 4 0 .1258925E-01 0. 9340593E+03 0. 9338927E+03 -0. 1783459E- 01 5 0 .1584893E-01 0. 9338926E+03 0. 9337339E+03 -0. 1699431E- 01 6 0 .1995262E-01 0. 9336287E+03 0. 9334824E+03 -0. 1566774E- 01 7 O .2511886E-01 o 9332112E+03 0. .9330845E+03 -0. 1357827E- 01 8 0 .3162278E-01 0. 9325514E+03 0 .9324553E+03 -0. 1029905E- 01 9 0 .3981072E-01 0. 9315105E+03 0. 9314622E+03 -0. 5182284E- 02 10 0 .5011872E-01 0. 9298727E+03 0 .9298981E+03 0. 2728810E- 02 Sample output from l o g i c a l unit 9. 1 2 3 4 5 6 7 8 9 10 ZERO SEQUENCE FREQUENCY 0.1000000E-01 0. 0.1258925E-01 0. 0.1584893E-01 0. 0.1995262E-01 0. 0.2511886E-01 0. 0.3162278E-01 0. 0.3981072E-01 0. 0.5011872E-01 0. SURGE IMPEDANCE ZCO 9363715E+03 9363715E+03 9363715E+03 9363715E+03 9363715E+03 9363715E+03 9363715E+03 9363715E+03 * ZCO* 0.9363422E+03 0.9363251E+03 0.9362980E+03 0.9362550E+03 0.9361868E+03 0.9360790E+03 0.9359083E+03 0.9356385E+03 ERROR (%) -0.3129076E-02 -0.4958766E-02 -0.7857896E-02 -0.1245087E-01 -0.1972561E-01 -0.3124371E-01 -0.4946962E-01 -0.7828299E-01 Sample output from l o g i c a l u n i t 8. 209 In the output for | Zc(co)| , zero sequence, ZCO i s the magnitude of the c h a r a c t e r i s t i c impedance generated from the equivalent l i n e c o n f i g - uration, *ZC0* i s the rat i o n a l - f u n c t i o n s approximation of | Zc(co)| and ERROR (%) i s the percentage error between them. Line 18 controls the output of | A-) ( w)| i n a s i m i l a r way as i n l i n e 17. Units 10 and 11 contain the zero and p o s i t i v e magnitude functions, r e s p e c t i v e l y . 1 ZERO S E Q U E N C E P R O P A G A T I O N F U N C T I O N 2 FREQUENCY A 1 0 * A 1 0 * ERROR {%) 3 O . I O O O O O O E - 0 1 0 . 9 8 6 0 7 4 8 E + 0 0 O . 9 8 6 0 7 4 7 E + 0 0 - O . 2 2 2 2 9 7 7 E - 0 5 4 0 . 1 2 5 8 9 2 5 E - 0 1 0 . 9 8 6 0 7 2 7 E + 0 0 0 . 9 8 6 0 7 4 7 E + 0 0 0 . 2 0 2 7 6 0 5 E - 0 3 5 0 . 1 5 8 4 8 9 3 E - 0 1 0 . 9 8 G 0 7 0 2 E + 0 0 0 . 9 8 6 0 7 4 7 E + 0 0 O . 4 G 1 0 1 6 3 E - 0 3 6 0 . 1 9 9 5 2 6 2 E - 0 1 0 . 9 8 6 0 6 6 9 E + 0 0 0 . 9 8 6 0 7 4 7 E + 0 0 0 . 7 8 6 5 4 5 9 E - 0 3 7 O . 2 5 1 1 8 8 6 E - 0 1 0 . 9 8 6 0 6 2 8 E + 0 0 0 . 9 8 6 0 7 4 6 E + 0 0 0 . 1 1 9 7 1 5 8 E - 0 2 8 O . 3 1 6 2 2 7 8 E - 0 1 O . 9 8 6 0 5 7 6 E + 0 0 0 . 9 8 6 0 7 4 5 E + 0 0 0 . 1 7 1 5 5 9 7 E - 0 2 9 0 . 3 9 8 1 0 7 2 E - 0 1 0 . 9 8 6 0 5 1 0 E + 0 0 0 . 9 8 6 0 7 4 4 E + 0 0 0 . 2 3 7 1 0 8 2 E - 0 2 10 0 . 5 0 1 1 8 7 2 E - 0 1 0 . 9 8 6 0 4 2 6 E + 0 0 0 . 9 8 6 0 7 4 2 E + 0 0 0 . 3 2 0 1 4 1 2 E - 0 2 Sample output from l o g i c a l u n i t 10. 1 P O S I T I V E SEQUENCE P R O P A G A T I O N F U N C T I O N 2 F R E Q U E N C Y A11 »A11 * ERROR (%) 3 0 . 1 0 0 0 0 0 0 E - 0 1 0 . 9 8 6 0 8 1 4 E + 0 0 0 . 9 8 6 0 8 1 O E + 0 0 - O . 4 8 6 3 5 9 1 E - 0 4 4 0 . 1 2 5 8 9 2 5 E - 0 1 0 . 9 8 6 0 8 0 8 E + 0 0 0 . 9 8 6 0 8 0 7 E + 0 0 - O . 9 2 2 6 4 8 7 E - 0 5 5 O . 1 5 8 4 8 9 3 E - 0 1 O . 9 8 6 0 7 9 7 E + 0 0 O . 9 8 6 0 8 0 2 E + 0 0 O . 5 3 1 9 2 5 4 E - 0 4 6 0 . 1 9 9 5 2 6 2 E - 0 1 0 . 9 8 6 0 7 8 0 E + 0 0 0 . 9 8 6 0 7 9 5 E + 0 0 0 . 1 5 2 0 1 8 0 E - 0 3 7 0 . 2 5 1 1 8 8 6 E - 0 1 0 . 9 8 6 0 7 5 4 E + 0 0 0 . 9 8 6 0 7 8 4 E + 0 0 0 . 3 0 8 3 8 9 7 E - 0 3 8 0 . 3 1 6 2 2 7 8 E - 0 1 0 . 9 8 6 0 7 1 2 E + 0 0 0 . 9 8 6 0 7 6 6 E + 0 0 0 . 5 5 5 5 8 1 2 E - 0 3 9 0 . 3 9 8 1 0 7 2 E - 0 1 0 . 9 8 6 0 6 4 5 E + 0 0 0 . 9 8 6 0 7 3 8 E + 0 0 0 . 9 4 5 7 5 1 8 E - 0 3 10 0 . 5 0 1 1 8 7 2 E - 0 1 0 . 9 8 6 0 5 4 0 E + 0 0 0 . 9 8 6 0 6 9 4 E + 0 0 0 . 1 5 6 0 1 4 0 E - 0 2 Sample output from l o g i c a l unit 1 1. Log i c a l unit 6 contains the record of the input parameters, equivalent l i n e representation, and approximation process. 210 1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 15 16 17 18 19 20 2 1 22 23 24 25 26 27 28 29 30 3 1 32 33 34 35 .36 37 38 39 40 4 1 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 6 1 62 63 64 65 66 67 68 69 End of RECORD OF INPUT PARAMETERS NUMBER OF PHASES* 3 LINE LENGTH* 500.0000000 LINE PARAMETERS PER UNIT LENGTH AT 60.000 HZ ARE: ZERO SEQUENCE RESISTANCE INDUCTANCE CAPACITANCE EARTH RESISTIVITY POSITIVE SEQUENCE RESISTANCE INDUCTANCE CAPACITANCE 0.1974000E+00 OHMS 0.3330800E-02 HENRYS 0.8361000E-08 FARADS O.1000000E+03 OHMS 0.2643000E-01 OHMS 0.8808000E-03 HENRYS O.133OOO0E-07 FARADS FROM THE GIVEN LINE PARAMETERS. THE ESTIMATED LINE CONFIGURATION IS: EQUIVALENT GEOMETRIC MEAN RADIUS RADIUS OF EQUIVALENT BUNDLED CONDUCTOR AVERAGE DISTANCE BETWEEN CONDUCTORS AVERAGE HEIGHT OF CONDUCTORS AVERAGE DISTANCE BETWEEN CONDUCTORS AND THEIR IMAGES O.1844397E+00 METERS O.1964899E+00 METERS •0.1508298E+02 METERS O.1498967E+02 METERS O.3400702E+02 METERS CORRECTION FACTOR FOR SKIN EFFECT S=(INTERNAL RADIUS)/!EXTERNAL RADIUS)* DC RESISTANCE 0.2619000E-01 OHMS **'• APROXIMATION PROCESS FOR ZCO **** NUMBER OF POLES* 6 NUMBER OF ZEROS= 6 SHIFTING OPTION SELECTED ERROR(%) AT 60.00000 HZ IS REACHED AFTER 4 SHIFTING LOOPS **** APROXIMATION PROCESS FOR A10 NUMBER OF POLES* 5 NUMBER OF ZEROS* 4 ERROR (%) AT 60.00000 HZ IS REACHED AFTER O ITERATIONS **** APROXIMATION PROCESS FOR ZC1 NUMBER OF POLES* 4 NUMBER OF ZEROS* 4 SHIFTING OPTION SELECTED ERROR(% ) AT 60.OOOOO HZ IS REACHED AFTER 3 SHIFTING LOOPS **** APROXIMATION PROCESS FOR A11 **** NUMBER OF POLES= 4 NUMBER OF ZEROS* 3 SHIFTING OPTION IS SELECTED ERROR AT 60.00 HZ IS 0.016O% REACHED AFTER 2 SHIFTING LOOPS O. 8095438E + 00 -O.7488213379 0. 2659203301 -O. 1 188697158 F i l e S_ample output from l o g i c a l u n i t 6. 211 Log i c a l unit 12 contains the c o e f f i c i e n t s of the p a r t i a l f r a c t i o n s expansion of the rational-functions approximation of the l i n e . The output format i s compatible with the frequency-dependence version of the EMTP, and i t i s the information to be introduced into l o g i c a l u n i t 1. It i s important to take into account that the formatted writing of information i s time consuming and computationally expensive. There- fore, for most applications i t i s recommended that the control parameters i n l i n e s 16 through 18 be set to zero. BIBLIOGRAPHY [I] H.W. Dommel, " D i g i t a l Computer Solution of Electromagnetic Transients i n Single - and Multiphase Networks". IEEE Trans., PAS-88, pp. 388-399, A p r i l 1969. [2] H.W. Dommel and W.S. Meyer, "Computation of Electromagnetic Transients". IEEE P r o c , V o l . 62 (7), pp. 983-993, July 1974. [3] J.R. Marti, "Accurate Modelling of Frequency-Dependent Transmission Lines i n Electromagnetic Transients Simulations". IEEE Power Industry Computer Applications (PICA) Conference, Philadelphia, PA, 9 pages, May 1981. [4] H.L. Leon, P r o f i l e s of Transient Voltages along Overhead Lines (M.A.Sc. t h e s i s ) . Department of E l e c t r i c a l Engineering, University of Toronto, December, 1979. [5] A. Semylen and A. Dabuleanu, "Fast and Accurate Switching Transient Calculations on Transmission Lines with Ground Return using Recursive Convolutions". IEEE Trans., PAS-94, pp. 561-571, March/ A p r i l 1975. [6] Line Constants of Overhead Lines User's Manual. Methods Analysis Group, Branch of System Engineering, Bonneville Power Administration, Portland, Oregon, Appendix 2 by H.W. Dommel, August 1977. [7] J.R. Marti, The Problem of Frequency Dependence in Transmission Line Modelling (PhD t h e s i s ) . Department of 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 1981. [8] Ibid., pp. 189-193. [9] Ibid., p. 106 [10] D.E. Hedman, "Propagation on Overhead Transmission Lines: I. Theory of Modal Analysis; I I . Earth Conduction Effects and P r a c t i c a l Results". IEEE Trans., PAS-84, pp. 200-211, March 1965. [II] H.W. Dommel, Notes on Advanced Power Systems Analysis. University of B r i t i s h Columbia, 1975. [12] UBC - Overhead Line Parameters Program. O r i g i n a l l y written by H.W. Dommel at the Bonneville Power Administration, Portland, Oregon. Modified at the Univ e r s i t y of B r i t i s h Columbia (UBC), Canada, by I.I. Dommel, K.C. Lee, and T. Hung. User's Manual, UBC, August 1980. 212 BIBLIOGRAPHY [13] UBC - Electromagnetic Transients Program (EMPT). O r i g i n a l l y written by H.W. Dommel at the Bonneville Power Administration, Portland, Oregon. Modified at the University of B r i t i s h Columbia (UBC), Canada, by H.W. Dommel. Frequency-dependence version o r i g i n a l l y written by J.R. Marti at UBC. Modified by L. Marti . User's Manual, UBC, August 1978 ( f i r s t published i n 1976). [14] J.P. Bickford, N. Mullineux, and J.R Reed, Computation of Power Systems Transients (book). Peregrinus (for the IEE), Herts (England), 1976. [15] L.V. Bewley, T r a v e l l i n g Waves on Transmission Systems (book). Dover, New York, 1963 ( f i r s t published i n 1933). [16] EHV Transmission Line Reference Book. Edison E l e c t r i c I n s t i t u t e , New York, N.Y., 1968. 213
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