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Full frequency-dependent modelling of underground cables for electromagnetic transient analysis Yu, Ting-Chung 2001

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Full Frequency-Dependent Modelling of Underground Cables for Electromagnetic Transient Analysis by Ting-Chung Yu B.E., Feng-Chia University, 1988 M. Sc., University of Missouri-Columbia, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A September 2001 ©Ting-Chung Yu, 2001 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 of B r i t i s h Columbia, I agree that 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of 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 of 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 that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of l^lfir^/fiLcJl OMA frn/njajfov hyj/heerfelj: The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date j & f t , 2? ; Jcrtrf A B S T R A C T „ A major difficulty in multiphase cable modelling with traditional electromagnetic transient program like the EMTP is the synthesis of the frequency-dependent transformation matrices that relate modal and phase domain variables. Although much effort has been devoted in the last several decades to solve this problem, we believe that the best solution is to completely eliminate the necessity of the frequency-dependent transformation matrices. The purpose of this thesis is to develop an accurate and stable model to simulate the behavior of underground cable systems under transient conditions. This thesis presents a new cable model (zCable model), which separates the representation of the wave propagation into two parts: a constant ideal propagation, which parameters depend only on the geometry of the cable configurations, and a frequency-dependent distortion, which depends on the skin effect. This approach permits the representation of the frequency.-dependent part of the parameters directly in phase coordinates and avoids the difficulties related to frequency-dependent transformation matrices. A simultaneous curve-fitting procedure is introduced to synthesize each element of the phase-domain frequency-dependent loss impedance matrix with rational function approximations in the frequency domain. After synthesizing each element of the loss impedance matrix by the proposed procedure, the fitted functions show a very good agreement with the original ones. By synthesizing each element of the impedance matrix n simultaneously, the new technique avoids the numerical stability issues of traditional procedures. A "pi-circuit" correction is proposed to solve the problem of different travelling times in the ideal line propagation. This approach makes the travelling times of all modes identical to one another in the modal domain and avoids the linear interpolation process used in the traditional multiphase line and cable models; By avoiding this interpolation process, a much larger integration time step At can be used without loss of accuracy, which results in considerable savings in computational time. The thesis presents a number of simulations where the behavior of the new zCable model is compared with that of the established JMARTI (FD line) and LMARTI (FDQ cable) models. These simulations show a very good agreement between the zCable model and the very accurate cable model - LMARTI model. The main advantage of the proposed model compared to existing full frequency-dependent transformation matrix models is the new model's absolute numerical stability for any kind of asymmetrical cable configurations and for arbitrary fault conditions. In addition, the new model parameters can be obtained with robust algorithms and the model can be efficiently implemented in the context of the real-time PC-cluster simulator developed by the power systems research group at UBC. iii T A B L E O F C O N T E N T S ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vi LIST OF FIGURES vii ACKNOWLEDGEMENTS xi Chapter 1 Introduction 1 1.1 Overview 1 1.2 Background and Motivation 3 1.3 Full Frequency Dependent Underground Cable Modelling (zCable) 5 1.4 About This Thesis 6 Chapter 2 Literature Review 7 2.1 Modal Domain Models 7 2.2 Phase Domain Models 11 2.3 Sectionalized Line Models 13 Chapter 3 Multiphase Underground Cable Modelling 15 3.1 Frequency Dependent Parameters of Underground cables 15 3.1.1 Series Impedance Matrix 16 3.1.2 Shunt Admittance Matrix 28 3.2 Development of zCable Model 31 iv 3.3 Modelling of the Ideal Line Section 36 3.3.1 Pi-Circuit Correction Technique 39 j 3.4 Modelling of the Loss Section 44 3.4.1 Curve-Fitting Procedure 47 3.4.2 Numerical Stability 60 3.4.3 Voltage Drop Formulation 63 3.5 Phase Domain Underground Cable Solution 67 3.5.1 Determination of the Length of the Cable Segments 70 Chapter 4 Transient Simulation and Validation 72 4.1 Energization into a Line-to-Ground Short Circuit Simulation 73 4.2 Open Circuit Simulation 81 4.3 Cable System under Load Simulation 88 4.4 Impulse Response Simulation 95 4.5 Computational Speed 103 Chapter 5 Conclusions 105 Chapter 6 Suggestions for Future Work 108 BIBLIOGRAPHY 110 Appendix I Error Analysis of Linear Interpolation Process 115 Appendix II Discretization Formulas for a Simplified Equivalent Circuit of the Ideal Line Section with Slower Propagation Mode 118 Appendix III Discretization Formulas for a Parallel RL Circuit 124 v L I S T O F T A B L E S Table 1 Physical data ofthe 230KV cable for the test case 20 Table 2 Constant and poles of the synthesized functions for the self-impedance of the core 51 Table 3 Constant and poles of the synthesized functions for the mutual impedance between core and sheath 52 Table 4 Constant and poles of the synthesized functions for the self-impedance of the sheath 52 Table 5 Constant and poles of the synthesized functions for the mutual impedance between cable a and b 52 Table 6 Constant and poles of the synthesized functions for the mutual impedance between cable a and c 53 Table 7 Comparison of relative CPU times (zCable with 10 segments) 103 Table 8 Comparison of relative CPU times (zCable with 50 segments) 104 vi L I S T O F F I G U R E S Figure 3.1 Basic construction of a single-core coaxial underground cable 16 Figure 3.2 Conductor and ground return representation of a single-core underground cable 17 Figure 3.3 Underground cable arrangement for the test case 21 Figure 3.4 Resistance and inductance of core's self-impedance 23 Figure 3.5 Resistance and inductance of mutual-impedance between core and sheath 24 Figure 3.6 Resistance and inductance of sheath's self-impedance 25 Figure 3.7 Resistance and inductance of mutual-impedance between cables a and b 26 Figure 3.8 Resistance and inductance of mutual-impedance between cables a and c 27 Figure 3.9 Representation of distributed lines in the frequency domain 32 Figure 3.10 Separation of basic effects in each zCable model segment 34 Figure 3.11 Transformation between phase and modal domain of the ideal line section ... 37 Figure 3.12 Ideal line section model of multiphase cables in the modal domain 38 Figure 3.13 The representation of the pi-circuit correction for the slower propagation modes (correction is made in each line segment) 41 Figure 3.14 Equivalent circuit of the ideal line section with slower propagation mode .... 41 Figure 3.15 Discrete-time representation ofthe ideal line section with slower propagation mode 41 Figure 3.16 Reduced equivalent circuit of the ideal line section for a slower propagation mode 42 Figure 3.17 Equivalent synthesized circuit for one segment of a single-phase underground cable 47 vii Figure 3.18 Block diagram of the fitting and optimization procedures 54 Figure 3.19 Comparison of the original and fitted of self-impedance of the core 55 Figure 3.20 Comparison of the original and fitted of mutual-impedance between core and sheath 56 Figure 3.21 Comparison of the original and fitted of self-impedance of the sheath 57 Figure 3.22 Comparison of the original and fitted mutual-impedance between cables a and b 58 Figure 3.23 Comparison of the original and fitted mutual-impedance between cables a and c 59 Figure 3.24 Comparison of eigenvalue 1 of the original and fitted matrices 61 Figure 3.25 Comparison of eigenvalue 4 of the original and fitted matrices 62 Figure 3.26 Comparison of eigenvalue 6 of the original and fitted matrices 62 Figure 3.27 Energization into a line-to ground short circuit simulation: Voltage at the receiving end of core 2(pure trapezoidal integration rule) 64 Figure 3.28 Energization into a line-to ground short circuit simulation: Voltage at the receiving end of core 2(trapezoidal with damping integration rule, a = 0.15) 65 Figure 3.29 Proposed cable segment for the zCable model 67 Figure 3.30 Equivalent network for the full-length of cable in the discrete-time domain ... 68 Figure 3.31 Relationship between segment length and frequency 70 Figure 4.1 System configuration for energization into a line-to ground short circuit simulation 73 Figure 4.2 Energization into a line-to ground short circuit simulation: Voltage at the receiving end of core 1 75 Figure 4.3 Energization into a line-to ground short circuit simulation: Voltage at the receiving end of core 2 76 viii Figure 4.4(Energization into a line-to ground short circuit simulation: Voltage at the receiving end of core 3 77 Figure 4.5 Energization into a line-to ground short circuit simulation: Receiving end current of sheath 1 , 78 Figure 4.6 Energization into a line-to ground short circuit simulation: Receiving end current of sheath 2 79 Figure 4.7 Energization into a line-to ground short circuit simulation: Receiving end current of sheath 3 : 80 Figure 4.8 System configuration for an open circuit simulation 81 Figure 4.9 Open-circuit simulation: Voltage at the receiving end of core 1 82 Figure 4.10 Open-circuit simulation: Voltage at the receiving end of sheath 1 83 Figure 4.11 Open-circuit simulation: Voltage at the receiving end of core 2 84 Figure 4.12 Open-circuit simulation: Voltage at the receiving end of sheath 2 85 Figure 4.13 Open-circuit simulation: Voltage at the receiving end of core 3 86 Figure 4.14 Open-circuit simulation: Voltage at the receiving end of sheath 3 87 Figure 4.15 System configuration for a cable system under load simulation 88 Figure 4.16 Cable system under load simulation: Voltage at the receiving end of core 1.'... 89 Figure 4.17 Cable system under load simulation: Voltage at the receiving end of sheath 1.. 90 Figure 4.18 Cable system under load simulation: Voltage at the receiving end of core 2.... 91 Figure 4.19 Cable system under load simulation: Voltage at the receiving end of sheath 2.. 92 Figure 4.20 Cable system under load simulation: Voltage at the receiving end of core 3 ... 93 Figure 4.21 Cable system under load simulation: Voltage at the receiving end of sheath 3.. 94 Figure 4.22 System configuration for an impulse response simulation 95 Figure 4.23 Impulse response simulation: Voltage at the receiving end of core 1 97 ix Figure 4.24 Impulse response simulation: Voltage at the receiving end of sheath 1 98 Figure 4.25 Impulse response simulation: Voltage at the receiving end of core 2 99 Figure 4.26 Impulse response simulation: Voltage at the receiving end of sheath 2 100 Figure 4.27 Impulse response simulation: Voltage at the receiving end of core 3 101 Figure 4.28 Impulse response simulation: Voltage at the receiving end of sheath 3 102 Figure 1.1 Discrete-time EMTP model of a single-phase ideal line system 115 Figure 1.2 Frequency response of the ratio of the interpolated values to the exact values for the history voltage source 117 Figure II. 1 Discrete-time equivalent circuit of the ideal line section with slower propagation mode 118 Figure II.2 Reduced discrete-time circuit for the ideal line section with slower propagation mode 121 Figure III. 1 Discrete-time representation of a parallel RL circuit 124 x A C K N O W L E D G E M E N T S I wish to express my sincere gratitude to all the people who have helped me in the fulfillment of this thesis: I would like to express my deepest gratitude to my thesis supervisor, Dr. Jose R. Marti, whom I respect and admire tremendously. His expertise in the area of power system, his broad and unlimited technical guidance, his support, advice, and especially his invaluable friendship not only enriched my personal and professional life but also made this thesis possible. To Dr. H. W. Dommel, for whom I have equal respect and admiration, for his great advice, encouragement, his useful discussion, information, and his outstanding course and seminar at UBC. Many thanks to Dr. Luis Marti and Dr. Fernando Castellanos, who supported me with their suggestions, advice, and research results for the model development and validation of results in this thesis work. To all the other former and current members of UBC power group for their support, cooperation, encouragement, and kind assistance. xi To all the professors and staff in the Department of Electrical and Computer Engineering who taught, guided and helped me constantly. To my dear parents and parents in-law for their infinite love, patience, support, and encouragement through all my life. Special thanks to my lovely and magnificent wife Chin-Hui for her never-ending love, patience, sensitive understanding, and constant encouragement. I am grateful for your love and support in these years. Thank you very much for the great time and memories we share. Finally, I would like to thank UBC, Dr. Jose R. Marti, the National Sciences and Engineering Research Council of Canada (NSERC), my parents and parents in-law for providing the financial support for my graduate studies. xii Chapter 1 Introduction 1.1 O v e r v i e w Overhead transmission lines and underground cables are widely used for distribution and transmission of electrical power. For economic reasons, overhead lines are used extensively in rural areas. However, in urban "areas where overhead construction is impractical, unsafe, costly or environmentally undesirable, it is usual to install insulated cables buried underground. Even in rural areas there are many situations where underground cables may be practical for reasons of safety (airports), reliability (areas with vulnerability to damage by natural forces) or aesthetics (areas with unusual or invaluable scenery). The insulation design of overhead lines and cables relies strongly on the knowledge of the transient voltages. Transient overvoltages are either of external origin, such as lightning strokes, or generated internally by switching operations. Underground cables are affected by lightning strokes, which strike adjacent overhead lines. Switching overvoltages are produced both by opening and closing switching devices at various locations in the cable system as well as by electrical faults. Voltages and currents in the transient period can reach their magnitude peaks very fast and can have arbitrary waveforms. In order to avoid equipment damages in underground cable system, appropriate protective devices and insulating level must be used. Since during transient periods the signals usually contain a wide range of frequencies, the ability of the 1 equipment to resist the overvoltages or overeurrents and the effectiveness of the protective devices in this period can only be appraised if the phenomena and behavior of the system are accurately described and predicted. 2 1.2 B a c k g r o u n d a n d M o t i v a t i o n A number of analytical methods based on the model developed for overhead lines are used in the transient analysis of underground cables. However, underground cables have significant different electrical characteristics than overhead lines, and these differences must be taken into account in cable modelling. The criteria and assumptions originally used in overhead lines are not always directly applicable to the case of underground cables. The EMTP (Electro-Magnetic Transients Program) developed at BPA by H. W. Dommel [1,2] is probably the most extensively used program for transients analysis in power systems. It represents each component (resistor, inductor, and capacitor) as an equivalent resistance and history source in the time domain, and then solves the equations of the whole electrical system using algebraic numerical methods [1,2]. A number of papers can be found in the literature [3,4,5,6] for the modelling of overhead lines and underground cables. The majority of them use modal decomposition theory [7,8] to decouple the physical system (phase domain) into a mathematically equivalent decoupled system (modal domain) in order to solve a multiphase line as if it consisted of a number of single-phase lines. Frequency-independent real and constant transformation matrices, which relate the phase and modal domains, are generally used in the present EMTP frequency-dependent line models [5]. However, the frequency dependence of the matrices cannot be ignored in some situations, especially for asymmetrical lines and cables. In this thesis, the term "full frequency-dependent model" will be used to signify line or cable models that take into account the frequency dependence of the modal transformation matrices. 3 L. Marti [6] introduced a full frequency-dependent underground cable model, the LMARTI model. However, direct synthesis of frequency-dependent transformation matrices is a difficult and time-consuming process. - Some methods have been proposed [9,10,11,12] to find line solutions directly in the phase domain rather than in the modal domain. This avoids convolution operations due to modal transformations, and possible numerical instabilities due to mode crossing. However, these techniques still require recursive convolution for the frequency dependence of the characteristic admittance matrix and the propagation matrix. To overcome the problems of frequency-dependent transformation matrices, Castellanos and Marti proposed a new model (zLine model) for overhead transmission lines [14,15,16,17]. The approach of the zLine model is to split the propagation wave of the line into two parts: a) constant ideal line section, and b) frequency-dependent loss section. With this approach, zLine avoids the use of frequency-dependent transformation matrices and can be formulated directly in the phase coordinates. This model is efficient, accurate, and especially more stable than the other full frequency-dependent models for overhead lines. Given the flexibility, accuracy, efficiency, and stable properties of the zLine model and considering the different electrical characteristics between underground cables and overhead lines, it has been the purpose of this thesis to extend this approach to underground cable modelling (zCable model). 4 1.3 F u l l F r e q u e n c y D e p e n d e n t U n d e r g r o u n d C a b l e M o d e l l i n g ( z C a b l e ) The basic principles of the zCable model were proposed as early as [13,14,15,16,17,18] in 1987. This thesis, however, is the result of a number of years of experiences and refinements that make the zCable model a valid production-model alternative to existing EMTP frequency-dependent line and cable models. The approach of the zCable model is to separate the representation of the wave propagation phenomena into two parts: a) the ideal line section with constant parameters, and b) the loss section with frequency-dependent parameters. This approach permits the representation of the frequency-dependent part of the cable parameters directly in phase coordinates thus avoiding the need for frequency-dependent transformation matrices. A major difference from [14,15,16,17] is that for the cable case the ideal constant-parameter line section in the zCable model is not solved directly in the phase domain as in the case of the zLine model [14,15,16,17], but it is solved in the modal domain to account for the different modal velocities in cable propagation. Nonetheless, since the parameters of the ideal line section are not frequency-dependent, the transformation matrix used in this part of the model is still frequency-independent. In this thesis, an improved curve-fitting procedure is proposed to synthesize the frequency-dependent loss impedance matrix [Zioss(a>)] with rational functions in phase coordinates. An important new contribution introduced in the present work is the concept of a "pi-circuit correction" to replace the traditional interpolation method in the EMTP line models when the integration step is not an exact submultiple of the travelling time T . This correction overcomes the need to have a Ar several times smaller than x in order to compensate for interpolation errors and results in considerable solution time savings. 5 1.4 A b o u t This Thesis The present work introduced in this thesis has been divided into: • Chapter 1: Briefly presents an overview of underground cable systems and transient problems, previous transient analysis works, the motivation of this thesis, and the present work (zCable model). • Chapter 2: Summarizes some previous important contributions to the areas of modelling and transient phenomena analysis on overhead lines and underground cables. • Chapter 3: Describes in detail the basic theory and the developing process of the zCable model. In this chapter the author also introduces an important new contribution - the concept of the "pi-circuit correction". • Chapter 4: Gives comparisons for the results of a number of simulation cases to validate the accuracy and stability of the zCable model. • Chapter 5: Summarizes the main features and contributions of this thesis. • Chapter 6: Gives some recommended future research topics. 6 Chapter 2 Literature Review One of the problems considered in power systems transients modelling is how to accurately include frequency-dependent effects in time domain simulation. These effects usually arise from eddy currents in conducting materials. For the past several decades, much effort has been devoted to this problem. This chapter gives a literature survey to introduce some previously presented methods and models, which made significant contributions in this area. The methods and models reviewed here are grouped into three categories as follows, 2.1 M o d a l D o m a i n M o d e l s The Electro-Magnetic Transient Program (EMTP) developed at the Bonneville Power Administration (BPA) by H. W. Dommel [1,2] is the most widely used time domain transient analysis program. Modal decomposition theory [7,8] is usually used in the EMTP for the distributed representation of transmission system. Since the EMTP is a time domain method, it has difficulty to include frequency-dependent effects in transient analysis. Thus, many researchers have devoted their efforts to incorporate this effect in the EMTP. Dommel and Meyer [2] included the frequency dependence of transmission line parameters by traditional convolution theory. Theoretically, the frequency dependence of the matrices in the frequency domain can be introduced into the time domain simulation by 7 numerical convolutions [19]. A full numerical convolution is always possible, but this leads to a high computational burden because of the many time steps in the convolution integral. A. Semlyen and A. Dabuleanu [3] promoted an efficient convolution method named the "recursive convolution" technique. This technique can be achieved by approximating the needed functions by exponential functions directly in the time domain [3] or by rational functions in the frequency domain [20]. The convolution integrals can be obtained recursively by using previous convolution results and a limited amount of inputs contained in the integration. This recursive convolution method reduced computer simulating times and storage requirements. One of the most widely used overhead line models is the JMARTI (FD line) model proposed by J. Marti [5]. This model includes the frequency dependence of the line parameters and the distributed nature of losses. Marti applied Semlyen's recursive convolution technique to his FD line model and proposed better approximating routines based on Bode's asymptotic approximation method to synthesize the frequency-dependent functions with rational functions in the frequency domain [21]. This model has been very reliable and accurate for most of the overhead line cases. In Semlyen's and Marti's works [3,5,20,21] the modal transformation matrices that relate phase and modal domains ignore the frequency dependence of the line parameters and are assumed to be real and constant. The benefit of this assumption is to gain high computational efficiency, but it can lead to erroneous results in the cases of strongly 8 asymmetrical overhead lines or underground cables because in cases of strong asymmetry the elements of the modal transformation matrices can depend drastically on frequency. The full frequency-dependent FDQ cable model developed by L. Marti [6,22] solves the problem of a strongly frequency-dependent transformation matrix by synthesizing the elements of this matrix with rational functions in the frequency domain. This model gives very good results for cable simulation in both low and high frequency phenomena. However, in order to carry out the recursive convolutions for the purpose of updating the history sources ofthe equivalent model circuits, all elements of the frequency-dependent propagation matrix, characteristic admittance matrix, and modal transformation matrices must be synthesized with rational functions in the frequency domain. This results in a large number of operations in the time domain convolutions. Also, it is difficult to guarantee for all cases the numerical stability of the synthesis functions for the frequency-dependent transformation matrices. A new vector-fitting technique has been recently introduced by Gustavsen and Semlyen [23] to increase the stability of the transformation matrices synthesis for overhead line models. This method is based on a technique that allows each vector of the matrix to be fitted with the same set of poles. The vector-fitting technique is based on doing the approximation in two stages: a) In the first stage the fitting is performed with real poles distributed over the entire frequency range of interest, and b) In the second stage a frequency-dependent scaling parameter is introduced that permits the scalar functions to be accurately fitted with the prescribed poles. This procedure is very accurate and is successful in fitting 9 the transformation matrix functions in most of the cases. However, as reported in [23] the fitting technique is not always possible to obtain an accurate fit with stable poles only when relying on modal decomposition with a frequency-dependent transformation matrix. Therefore, despite the very accurate results reported with this technique, the technique cannot still guarantee the absolute numerical stability of the frequency-dependent modal domain functions. 10 2.2 P h a s e D o m a i n M o d e l s Gustavsen, Sletbak, and Henriksen [9] have presented a travelling wave model for overhead lines and cables in which the frequency dependence of the modal transformation matrices is accurately taken into account by performing the convolution in the phase domain. In this model, they fitted only the tail portion of the time domain step responses with piecewise linear functions, and represented the front portion of the step responses in its detailed form in order to ensure an accurate fit. The propagation and characteristic admittance matrices, which are approximated with piecewise constant-valued functions in the time domain, can be found by differentiating the front portion numerically, and the tail portion analytically. Then, these two matrices can be adopted in the convolution operations. The convolution integrals used in this model are evaluated using a combination of numerical convolution and recursive convolution. The results of this model are quite good. However, as mentioned in [9], this model is less efficient than modal domain models using recursive convolutions. Angelidis and Semlyen [10] proposed a new methodology for the direct phase domain calculation of transmission line transient. The method they presented is called "Two-Sided Recursions (TSR)", which performs short convolutions with both input and output variables instead of using convolutions of the input variables only. The TSR coefficients are identified directly from transfer functions over a wide range of frequencies without any intermediate procedures, such as rational approximations. However, the stability of the TSR coefficients is affected by the selection of the fitting frequencies and the weighting factors at each frequency. Despite the accurate results reported with this method, the phase domain TSR 11 method increases the stability concerns and requires greater computational effort than a low-order traditional technique, using rational fitting approximation of the modal transfer matrices. Noda, Nagaoka and Ametani [11] introduced a new approach to model the overhead lines and cables in the phase domain rather than in the modal domain. In this approach, time domain convolutions are replaced by an ARMA (Auto-Regressive Moving Average) model. Using Z-transform theory, an ARMA model can economically express a phase domain response, which has discontinuities due to modal travelling time difference. Despite the very accurate results indicated with this technique, the resulting model is dependent on time step At and is not applicable for an arbitrary time step. The authors have recently solved this restriction by introducing an IARMA (Interpolated ARMA) model [24], which allows each ARMA model to use a different time step according to its desired frequency response. Nguyen, Dommel, and J. Marti [12] recently introduced a new overhead line model based on synthesizing all elements of the characteristic admittance and propagation matrices directly in the phase domain. Because the synthesis of the frequency-dependent modal transformation matrix is avoided, the model requires fewer convolutions at each time step than the modal domain models. The main assumption of this model is that the elements in the phase domain propagation matrix can be accurately fitted using stable minimum-phase rational functions. This is possible for most practical cases of overhead line configurations. 12 2.3 S e c t i o n a l i z e d L i n e M o d e l s A sectionalized line concept was proposed by Semlyen and W. G. Huang [13] for three-phase overhead lines with frequency-dependent parameters and corona. In their model the overhead line is subdivided into a number of short segments, each with a small travelling time. The model contains two different types of blocks for each segment: a) a longitudinal block for the modelling of frequency dependence, and b) a transversal block for the modelling of the nonlinear effect of corona. The computations are carried out in the modal domain for the longitudinal block and in the phase domain for the transversal branches, which include corona in each phase. The travelling time delay in the longitudinal block decouples the nodes for the corona branches, and it allows the representation of the line segment by Norton equivalents at its ends. Semlyen's and Huang's model gives good simulation results. Nonetheless, there are two drawbacks indicated in [13]: a) the model will introduce errors due to the use of a constant transformation matrix, and b) the line discretization and corona modelling used in this model requires approximately three order of magnitude more computational time than that of usual frequency-dependent line representation. Ametani et al. [18] proposed a simple and efficient frequency-dependent distributed-parameter line model by combining an ideal lossless distributed-parameter line and a lumped impedance circuit which takes into account the frequency-dependent effects. The impedance circuit is derived from four-terminal parameters theory. A lattice circuit with two or three branches is used to represent the frequency-dependent effect. The distributed line can be a frequency-independent lossless line of which the parameters are given at very high frequency. 13 The lumped impedance circuit Zn may be subdivided into: Zn /4 connected to the sending and receiving ends, and Zn 12 connected at the center of the ideal distributed line in order to represent the distributed nature of the losses. The distributed line can also be subdivided into more segments with more complicated analysis in order to obtain higher accuracy results. This model shows fairly good accuracy in the whole time period for a single-phase underground cable system. However, as reported in [18] this model needs a better approximation of the oscillating waveform during the transient period especially for the sheath voltage. F. Castellanos and J. Marti proposed the zLine model [14,15,16,17] for frequency-dependent overhead lines. Similarly to [13], the basic concept of the zLine model is to subdivide the representation of the total length line into a number of segments, each segment containing two parts: a) the ideal line section with constant parameters, and b) the loss section with frequency-dependent parameters. This model can be formulated directly in the phase domain thus avoiding the use of modal transformation matrices. Its only limitation is that the line has to be sectionalized into a reasonable number of segments so as to simulate the actual distributed nature of losses. This model is accurate, efficient, and especially stable. The basic theory of the zCable model is based on the sectionalized line concept and the work of [14,15,16,17]. The developing process and validation of the zCable model will be presented in detail in the following chapters. 14 Chapter 3 Multiphase Underground Cable Modelling This chapter is the main part of this thesis. First, we introduce the basic theory of frequency-dependent parameters for underground cables and show some analytical results. Then, the process of developing the zCable model will be presented in detail and we will show how the zCable model works. The important new concept of the "pi-circuit correction" is introduced to solve the problem of different travelling times in the ideal line section. A modified curve-fitting procedure is also introduced to synthesize the series loss impedance matrix [Zioss (a>)] (per unit length) in such a way as to fully assure the numerical stability of the model solution. 3.1 F r e q u e n c y D e p e n d e n t P a r a m e t e r s o f U n d e r g r o u n d C a b l e s Theoretically, the transmission system is defined by the series impedance Z and the shunt admittance Y. Before we start the development of the new cable model (zCable model), impedances and admittances ofthe cable system have to be known. Much effort has been made to evaluate the frequency-dependent parameters of underground cables [25,26,27,28,29,30]. For viability of demonstration purpose in this thesis, we choose single-core cables with concentric cylindrical conductors as the test case in order to obtain the parameters easily from analytical formulas [25,27]. 15 Consider an underground cable system consists of N single-phase cables, each with a cross-section of the type shown in Figure 3.1. Each cable has two metallic conductors, of which one is the central core and the other is a conducting sheath. The formulas used in this thesis are based on the following assumptions: • The cable is of circular symmetry type and the axes of its conductors are mutually parallel and are parallel to the surface of the ground. • The cable system has longitudinal homogeneity. • Displacement currents in the air, conductor, and ground are assumed to be negligible. • Each conducting medium of the cable has constant permeability. 3.1.1 Series Impedance Matrix The series impedance matrix per unit length of an TV-phase cable system in loop quantities consists of N submatrices as follows: 16 [ Z / n ] [ Z / 1 2 ] [-2^ /21 ] [Z/22] [ZI\N] IN\. (3-1) The diagonal submatrix [ZHI] represents the self-impedance matrix per unit length of the cable /, and the off-diagonal submatrix [ Z ^ ] describes the mutual impedances matrix per unit length between cable i and cable j. (a) Diagonal Impedance Submatrix [ZUI ] For the case of a single-core coaxial cable, an equivalent circuit for the series impedances per unit length is given in Figure 3.2. Axis Core Vlloop Z'cre Z'shi Z'ins 1 )\ihop+d\lloop Sheath Outer insulation Earth / -.hm Z'ins ' V2hop + d \2loop 11 loop Vlloop Figure 3.2 Conductor and ground return representation of a single-core underground cable The analysis is based on the loop equation dWuoop] dx - [Z!ii]UHoop] (3-2) The self-impedance matrix per unit length of cable /' is represented as follows in. Zcc(i) Zcs(Q Zsc(i) Z S S ^ (3-3) 17 where t Zcc(i ) — Zcre(i) + Zins\(i) + Zshi(i) Zcs(i ) = Zsc(i) = ~Zshm(i) Zss{i ) ~ Zshe(i) + Zins2{i) + Zes(i) with Zcre^ : internal impedance per unit length of the core (Q/m) Zws\{i)>Zins2(Q '• m e impedance per unit length of the inner and outer insulations (Q/m) Zshj^ : internal impedance per unit length of the inner sheath (Q/m) Zshe(i) '• internal impedance per unit length of the outer sheath (Q/m) Zshm{i) '• mutual impedance per unit length of the sheath (Q/m) Zes(i): self-impedance per unit length of the earth return path (Q/m) The simpler term to calculate is the impedance of the insulation, given by [31]. 4 s = f ^ - (3-4) where ju = permeability of the insulation r = outside radius of the insulation q = inside radius of the insulation The complete classical formulas for the series impedance of underground cables have been developed by S. A. Schelkunoff [33]. For a general approach, L. M. Wedepohl [25], H. W. Dommel [31], and A. Ametani [27] summarize the internal and mutual impedances of a 18 tubular conductor based on the work of [33]. These formulas are all based on modified Bessel functions [34]. (b) Off-diagonal Impedance Submatrix [Znj] It is apparent that only the sheath-earth loops of each single-phase cable are coupled, and submatrix [Znj] is given by [^] = 0 0 o zeij (3-5) The analytical formula of Zeij is given by [25,27,31]. L. M. Wedepohl has also developed approximate formulas for the series impedances of cables with hyperbolic cotangent functions, which are generally more suitable for digital computation. However, there are some restrictions that limit the applicability of these approximate formulas [31]. Since loop equations are not suitable for the EMTP solution model, the loop voltages and currents need to be replaced by voltages to ground and currents in the core and sheath. The transformation from loop quantities to node quantities can be achieved by the manipulation of the rows and columns of submatrices [Zlu] and [Z/;y][31]. After the manipulation, the transformed submatrices [ZnU] and [ZniJ] in node quantities become Z 11(0 A|2(0 '21(0 ^22(0, 7 7 (3-6) 19 where Z 1 1 ( / ) = Z c c ( / ) + 2ZC i S ( / ) + Zss^ •^ 22(0 = ZSs(i) (c) Evaluation of Series Impedances It is very easy to obtain all the analytical solutions for all the parameters ^ e(/) 5^ ;( 0 ,Z;, e ( ( . ) ,Z;,m ( ( . ) ,Z;, ( , . ) ,Z^ 1 ( , . ) ,Z;„ i 2 ( , ) , and Zeij using the analytical formulas mentioned before [25,27,31 ]. In this thesis, we chose a 230KV three-phase single-core coaxial cable system as the test case. The physical data and arrangement of this underground cable system are shown in Table 1 and Figure 3.3. Table 1 Physical data of the 230KV cable for the test case Name Value Inner radius of the core (cm) 0.0 Outer radius of the core (cm) rl 2.34 Inner radius of the sheath (cm) r2 3.85 Outer radius of the sheath (cm) r3 4.13 Outer insulation radius (cm) r4 4.84 Core resistivity (Q-m) 0.0170 x IO"6 Sheath resistivity (Q-m) 0.2100 x 10'6 Inner insulation tan5 0.001 Outer insulation tan5 0.001 Inner insulation permittivity E r 3.5 Outer insulation permittivity sr 8.0 All relative permeability u.r 1.0 Earth resistivity (Q-m) 100 20 The results of evaluating the diagonal self-impedance submatrix [ Z M i i ] , the off-diagonal mutual impedance submatrices [Z„ ] 2] and [Z„1 3] in nodal quantities over an extend frequency range are shown in Figures 3.4 to 3.8. Observing Figures 3.4 to 3.8, the behavior of resistances and inductances for each element of the submatrices clearly indicates that resistances increase and inductance decrease with increasing frequency due to the skin effect. 1.2m 0.25m-4*-0.25m-H Figure 3.3 Underground cable arrangement for the test case We also found that the inductances of the diagonal self-impedance submatrices decrease to a constant value, and those of the off-diagonal mutual impedance matrices decrease to zero at very high frequencies. The reasons for these phenomena are described next, For the diagonal self-impedance submatrices, the value of the parameters L\re{i)^shi^,Lshe(i),Lshm(i) a n d decreases to zero at very high frequencies due to the skin effect. Therefore, there are only the L j n s X ( i ) and Lins2(i) items left in the submatrices. From electromagnetic theory, only external flux exists for a tubular 21 conductor at very high frequencies. Therefore, we can easily understand that the external inductances per unit length of the diagonal self-impedance submatrix can be described as [ C l = Lins\{i) + Lins2{j) Ljns2(i) and these external inductances are independent of frequency from equation (3-4). For the off-diagonal mutual impedance submatrices, as described above, only the most outer current loops (sheath with earth return) through which the phases become coupled. As shown in Figure 3.2, the majority of earth return current flows very close to the touching surface of ground and outer insulation of the cable at very high frequencies due to the skin effect. Because the field created by I2I in the outer sheath is cancelled by the one created by returning current I2l in the earth, there are no mutual fluxes and mutual inductances to be generated between cable / and cable j. Therefore, the external inductances of the off-diagonal mutual impedance submatrices are all zeros. 22 Frequency (Hz) (a) Resistance of the core's self-impedance Frequency (Hz) (b) Inductance of the core's self-impedance Figure 3.4 Resistance and inductance of the core's self-impedance 23 Frequency(Hz) (a) Resistance of the mutual-impedance between core and sheath 10 Frequency (Hz) (b) Inductance of the mutual-impedance between core and sheath Figure 3.5 Resistance and inductance of the mutual-impedance between core and sheath 24 Frequency (Hz) (a) Resistance of the sheath's self-impedance Frequency (Hz) (b) Inductance of the sheath's self-impedance Figure 3.6 Resistance and inductance of the sheath's self-impedance 25 10." 10" 10" 10 10" 10 Frequency(Hz) (a) Resistance of the mutual-impedance between cables a and b 10 10 10" ,10 Frequency(Hz) 10 10 (b) Inductance of the mutual-impedance between cables a and b Figure 3.7 Resistance and inductance of the mutual-impedance between cables a and b 26 10 E I 10° a> o £= ro -»—' to t/> (D 10 10 10 10 10 Frequency (Hz) 10 10 (a) Resistance of the mutual-impedance between cables a and c 10 ^ 1 0 E X_ ro o gio J a ro ' o T3 C 10 10 10 10 10 10 Frequency(Hz) 10 10 (b) Inductance of the mutual-impedance between cables a and c Figure 3.8 Resistance and inductance of the mutual-impedance between cables a and c 27 3.1.2 Shunt Admittance Matrix The shunt admittance matrix per unit length of an TV-phase cable system in loop quantities is given by Pin] P/,2] [YnN] •• [YlNN] (3-7) Because there is no coupling between the TV phases of the cable in the shunt admittances, all the off-diagonal submatrices [YL] are null matrices. Since the current changes along the longitudinal direction of the cable, the loop equations for the shunt admittance of a single-phase cable are not coupled. Therefore, the diagonal submatrices \Ylu ] then can be defined as [Y'ui] = Ycs{0 0 (3-8) 0 Ysg(0_ where Ycs^ = shunt admittance per unit length for the insulation layer between core and sheath (S/m) y ^ = shunt admittance per unit length for the insulation layer between sheath and ground (S/m) From basic electrical theory, the shunt admittance Y' of the tubular insulation layer with inside radius r0 and outside radius r\ is Y = G + jcoC . The shunt capacitance C' per unit length is defined as [31] 28 c = 27T£0£r (3-9) ln(r, / r 0 ) where £-Q = absolute permeability of free space sr = relative permeability of the insulation layer In the literature on electromagnetics [31,32], the shunt conductance G' per unit length is determined by the dielectric dissipation power factor, capacitance and frequency according to the relationship where tan 8 is the dielectric dissipation factor. In this thesis a constant relative permittivity is assumed for all insulation layers which makes the shunt capacitance C' constant, and frequency-dependent effects of the conductance G' are ignored. As in the case of the series impedance, equation (3-8) is also not in a suitable form for EMTP solution. The transformation from loop quantities to node quantities can be achieved through a manipulation similar to [31] for the series impedance. The off-diagonal submatrices [Ynjj] are still null matrices and the diagonal submatrix [YnU] in node quantities G = co C • tan S (3-10) becomes [Ynil] = T •sg(') (3-11) -Y, 29 After calculation using very common formulas [31], the shunt conductance matrix [G ] and shunt capacitance matrix [C ] in node quantities for the test case defined in the previous section are as follows, 2.457059e-6 -2.457059e-6 0.0 0.0 0.0 0.0 -2.457059e-6 0.0 0.0 1.985828e-5 0.0 0.0 0.0 2.457059e-6 -2.457059e-6 0.0 - 2.457059e - 6 1.985828e - 5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0. 0.0 0.0 0.0 2.457059e-6 -2.457059e-6 -2.457059e-6 1.985828e-5 (mho/km) [C ] 3.910531e-7 -3.910531e-7 0.0 0.0 0.0 0.0 - 3.910531e-7 3.160543e-6 0.0 0.0 0.0 0.0 0.0 0.0 3.910531e-7 - 3.910531e-7 0.0 0.0 0.0 0.0 - 3.910531e-7 3.160543e-6 0.0 0.0 0.0 0.0 0.0 0.0 3.910531e-7 - 3.910531e-7 0.0 0.0 0.0 0.0 - 3.910531e - 7 3.160543e-6 (F/km) 30 3.2 D e v e l o p m e n t o f T h e z C a b l e M o d e l The zCable model presented in this thesis is based on the approach of [14,15,16,17], which subdivides the total line length into a number of segments and then splits the representation of the wave propagation phenomena in each segment into two parts: • Ideal line section: this section represents the electric and magnetic fields outside the conductor. Assuming the permittivity and permeability of the external media are constant, the parameters of the ideal line section are constant with frequency and only depend on the geometry of the system of conductors. • Loss section: this section includes the losses and internal inductances inside the conductor and ground that are frequency-dependent due to the skin effect. In physical meaning, the ideal line section corresponds to the ideal wave propagation due to the external magnetic field [L] and electric field [C], while the loss section corresponds to the wave distortion due to the resistance and internal inductance of the conductor and ground. These two different kinds of sections clearly separate the parameters of cables into two groups: a) constant parameters, which only depend on the cable geometry, and b) frequency-dependent parameters due to the skin effect. The general line equations for frequency-dependent distributed lines can be easily expressed in the frequency domain as (Figure 3.9) dV -2L = [zyi (3-12) dx dl -— = [Y\V (3-13) dx 31 I I where [Z ] is the series impedance matrix per unit length (Q /m) and [Y ] is the shunt admittance matrix per unit length (S/m). — • x / • v v v "dl In ix J dx Figure 3.9 Representation of distributed lines in the frequency domain After differentiating voltage and current equations with x a second time, the wave propagation equations in the frequency domain are obtained: d2V < < d2I < ' ^ - = [Z7 ] -P- and ^ = [YZ]-I (3-14) dx dx where [Z Y ] and [Y Z] are full matrices that couple the wave propagation of voltage and current in every phase. The element of the series impedance matrix [Z ] can be written as z'^ = Ry(co) + joj-[L^(jD) + Lft] (3-15) where Ry (co) is the resistance(per unit length) of the conductor and ground return; Ly is the external inductance(per unit length) associated with external flux outside the conductor ' and ground; and ^^(co) is the internal inductance(per unit length) related to the internal flux inside the conductor and ground. With equation (3-15), the series impedance matrix [Z ] can be expressed as 32 [Z(co)] = ([R(co)] + jco[Lint(co)]) + jco[Lext] = [Zhss(co)] + jco[Lext] (3-16) i i = t 2loss O ) ] + [Zideal] wit" Z l o s s { i j ) (co) = Ry (co) + jco • Lf (co) 7 - ,v, r'ext zideal(ij) ~ J® " Lij The elements of the series loss impedance matrix [Z(oss(co)] (per unit length) are frequency-dependent due to the skin effect. In the zCable model, the total [Zioss(co)] in each segment is lumped into a loss section arid can be synthesized with rational approximations. The series loss impedance matrix [Z/ 0 5 5 (co)] has stronger physical meanings and is simpler to synthesize than "desired" frequency-dependent matrices such as the characteristic admittance matrix, the modal transformation matrices, and the propagation matrix which are the functions synthesized in other frequency-dependent line models. The shunt admittance matrix [Y ] per unit length can be described as [Y] = [G] + j<D[C] (3-17) where [C ] is the shunt capacitance matrix(per unit length) that permits the conductor to maintain potential across the insulation, and [G ] is the shunt conductance matrix(per unit length) representing dielectric losses. The elements of [C ] are dependent on the permittivity of the insulation and the diameter of the conductors and insulation layers. In this thesis the elements of [C ] are assumed to be constant by assuming constant permittivity of the 33 insulation. The frequency dependence of [G ] is also ignored in this work because its effect is usually not significant in transient simulations. The consideration of frequency-dependent t i [G ] or [C ] matrices would require an extension to the presented work and it is suggested as future research. In the zCable model, the total cable length (Lcabie) is first subdivided into a number of shorter cable segments (/ = Lcabie IN) in order to distribute the lumped loss sections of the model. Each segment is then modelled as consisting of two parts: an ideal line section and a loss section (Figure 3.10). ideal line section loss section external flux & _( "electric T r c t d - ^ --i y [Lext][C] i I dielectric loss [G]/2 resistance& A A ^ n W T f h ^ A / V V A / W [Z,oss(w)] dielectric loss [G]/2< . J L_~ Figure 3.10 Separation of basic effects in each zCable model segment The ideal line section contains the representation of the external magnetic field [Lext ] and the electric field [C], which parameters only depend on the cable geometry and are constant and independent of frequency. This section corresponds to a lossless constant-parameter distributed line which can be represented easily using Dommel's model [1]. Since the parameters of this section are constant, even though modal transformation matrices are 34 needed to relate phase and modal quantities, the elements of the transformation matrices are constant and frequency-independent. f The loss section consists of two subsections. The first subsection includes the resistance of the conductor [R(co)] and their internal inductance [Lmt (co)], which parameters are frequency-dependent due to the skin effect and are slightly affected by the cable geometry. This subsection is lumped into the series impedance matrix [Zioss(co)] and is synthesized with rational functions in phase coordinates. The other subsection in the model is the constant dielectric losses matrix [G]. 35 3.3 M o d e l l i n g o f the Idea l L i n e S e c t i o n . In the work of [14,15,16,17], after subdividing each segment into the loss and the ideal line sections and extracting the parameters of loss section from the total line parameters, the solution of the ideal line section in the time domain for each segment is directly formulated in phase coordinates. Since the propagation velocity is the speed of light for all modes of an overhead line, the travelling time is the same for all modes and there is only one time delay. Therefore, the equations for solving the multiphase ideal line can be easily extended from Dommel's single phase lossless line case [1] in the phase domain by replacing the voltage and current variables with vectors and the impedance with an impedance matrix. This avoids the difficulties associated with frequency-dependent transformation matrices. The equations can then easily be solved in the time domain. However, the construction and electrical properties of underground cables are different from those of overhead lines. The permittivities of the inner and outer insulation layers of underground cables are usually different from each other. This results in different propagation velocities for each mode of the ideal line section in each segment. Therefore, we have different travelling times and, of course, different time delays for each mode of underground cables. Because of the characteristic of different time delays, it is more complicated for cables to formulate the equations for solving the multiphase ideal line sections directly in phase coordinates, as in the case of overhead lines [14,15,16,17]. This difficulty can be overcome by decoupling the phase variables into mode variables. Each of the independent equations in the modal domain can then be solved as single-phase lines using modal travelling time and modal surge impedances. 36 The transformation between the phase domain and the modal domain of the ideal line section for the multiphase cable is shown in Figure 3.11. k, phase 1 K, s phase n K n s core sheath co sheath phase domain modal domain * phase domain 4 • q . • w Figure 3.11 Transformation between phase and modal domain of the ideal line section In EMTP modelling [31], the modal domain end-node voltage vectors [vmode(7)] and [ vw° d e(01 c a n be expressed as functions of the modal characteristic impedance [Z™ o d e(7)], the modal current into the end-node vectors [z™ode(0] a n d [C° d e(01 > a n u t n e modal history voltage sources vectors [efnode(t)] and [e™%ie(t)]. The system equations ofthe ideal line section for the zCable model in the modal domain can then be described as: [v,mode(0] = [zcmoae][rae(0] + t>JBT(0] ^mod -ir-mode mode. [vrde(o]=[zcmode][Cde(o]+tcode(o] (3-18) (3-19) The equivalent circuit for equations (3-18) and (3-19) is shown in Figure 3.12. 37 [ikmo (V ] [Z m o d e ] [z>»°"e] vmmode(tn k o-[v."""le(t)] -o m o-[vmoUe(t)] Figure 3.12 Ideal line section model of multiphase cables in the modal domain The history voltage source vectors [e™de(0] a n a* [e™£de(0] a r e updated at each time step of the solution as a function of past voltage and current values, and have the form [efh°de(0] = [Zcmode][Cde(^ - */)] + [ v r d e C - r,)] (3-20) « d 6 ( 0 ] = [Zrde][/f°de(> - TO] + [ v r d e ( > - r,-)] (3-21) with Tt : travelling time of mode i and m o d e : characteristic impedance matrix in modal quantities ( Q ) [Cmode] where [^ mode] ^s m e external inductance matrix per unit length in modal quantities, and [Cm o (j e] is the shunt capacitance matrix per unit length in modal quantities. After solving each equation of the ideal line section for all segments in the modal domain, the modal quantities have to be transformed back to phase quantities in order to solve the line equations together with the rest of the network, which is normally defined in the phase domain. For example, equation (3-19) is interfaced with the rest of the network by transforming it from modal quantities to phase quantities as follows, 38 [vPMhase (0] = [Zcphase ][iPMhaS£ (01 + [effir'it)] (3-22) with voltage and current vectors in phase quantities, [vjEfW5e(0] = [7 ' /r 1[v™ o d e(0] (3-23) [i jP*^(0] = [7-][i™ode(0] • (3-24) the characteristic impedance matrix in phase quantities, [zphase] = [ 7 , ; r l [ Z r n o d e ] [ r . r l ( 3 _ 2 5 ) and the history voltage source vectors in phase quantities, KtSe(0) = [T/r\eTkde(0] (3-26) where [Tj] is the transformation matrix. This matrix is the eigenvector matrix of the product [T'][Z'](in this thesis the product is [C ][L ] for the ideal line section). This matrix will be different for each particular cable type and configuration, but it is always real and constant for a lossless line with constant inductance and capacitance. 3.3.1 Pi-Circuit Correction Technique As indicated, for underground cables there are different travelling times for each mode of the ideal line section. When different propagation modes have different travelling times, traditional EMTP models use a linear interpolation process to obtain precise history values located in-between discretization steps. However, this process limits the accuracy of the model. To minimize these interpolation errors (See Appendix I), the integration step in~ the simulation has to be decreased to usually five to ten times smaller than the travelling time 39 T . This increases the solution time to five to ten times longer than it would be needed if there were no interpolation errors. In this thesis we present a new technique, the "pi-circuit correction" technique to avoid the linear interpolation process. The idea of the pi-circuit correction is described as follows, • For the mode with longer travelling time, we split the modal domain representation of the ideal line part in each segment of the model into two parts (Figure 3.13). • The first part is still an ideal line section, and we choose the travelling time of this part to be the same as the travelling time of the ideal line section with the fastest propagation mode. In multiphase cables case, the fastest propagation model has the smallest travelling time T . Therefore, the length of this part should be less than the total length of the ideal line section in each segment. • The second part is represented as a pi-circuit with constant inductance L and capacitance C. Because the length of this pi-circuit is short compared with the length of the ideal line section, and it is even much shorter compared to the total length of the cable, the error due to this lumping is very small. • Since the pi-circuit correction is applied in the modal domain, there is no coupling between pi-circuits of different modes, and this makes the computational process of this pi-circuit correction very easy. • The purpose of the pi-circuit is not to act as a low-pass filter, and indeed it does not filter out high frequencies in the simulation, but to provide an extra time delay A r for the slower propagation mode (T s i o w e r _propagat ion_mode = Tfastest_propagation mode + A r ) . 40 o £ fastest propagation mode ) o slower propagation mode O -same travelling time as fastest propagation mode > - O pi-circuit correction Figure 3.13 The representation of the pi-circuit correction for the slower propagation modes (correction is made in each line segment) Figure 3.14 is the equivalent circuit of the ideal line section for a slower propagation mode. After the time discretization of the pi-circuit, we have Figure 3.15 k o — + 2 mode modei ~kh '(0 7 mode £c m' + o -O p modet e...,:—\t) v' it) C 12 - o -O m + C 12 .. ^ vm(0 - o Figure 3.14 Equivalent circuit of the ideal line section with slower propagation mode Where Lpi — -£rnode(;') 'Ipi » ^pi ~ Cmode(i) 'Ipi Ipi: length of the pi-circuit Figure 3.15 Discrete-time representation of the ideal line section for a slower propagation mode 41 where R L , RQ are the discrete-time equivalent resistances of the inductance and capacitance of the pi-circuit, and ei (r), eQ\ (t), e^2 (0 are the history voltage sources of the inductance and capacitance. After performing Thevenin and Norton manipulations, the equivalent circuit in Figure 3.15 can be simplified to the one in Figure 3.16. (See Appendix II) k o— + o-2" mode -AAA-mode. 'kh X!) 2"' mode -AAA o m + -o Figure 3.16 Reduced equivalent circuit of the ideal line section for a slower propagation mode where / vmoden . 7 m o d e p , p p \ p ,'mode = \ Z c R C + Z c KL+KCKLJ-KC Rl + 2ZfodtRc + Zf°^RL + RCRL (3-27) and emh (0 = R2c R% + IZf^'Rc + Zf^RL + RCRL 7 mode [ « d e ( 0 + c R, C (Zfod* + Rc)-Rc Rl + 2Z™deRc + Z™deRL + RCRL •eLif) Z™deRc+Z™deRL+RcRL Rc + 2Z™dQRc + Z™deRL + RCRL (3-28) 42 In the reduced equivalent circuit of Figure 3.16, the modal characteristic impedance Z c m o d e and history voltage source e^ode(0 for a slower propagation mode can be obtained easily using equations (3-27) and (3-28). After obtaining the modified equivalent characteristic impedance matrix and history voltage source vectors, they are transformed from modal quantities to phase quantities in order to be combined with the rest of the network. After adding the pi-circuit correction to the model of the ideal line section in each segment, all propagation modes have now the same travelling times and therefore the same time delays in the modal domain. Because no interpolation errors are now present, the integration time step At can be chosen as large as the travelling time x of the fastest propagation mode. Compared with the traditional interpolation process, the proposed technique results in considerable savings in computational time without sacrifice in the accuracy of the model. 43 3.4 M o d e l l i n g o f the L o s s Sec t ion In theory [19], input-output relationships in the frequency domain become numerical convolution in the time domain. However, traditional convolution methods are time consuming. Semlyen and Dabuleanu [3] promoted an efficient method called the recursive convolution technique, which speeds up traditional convolution and obtains significant computer timesaving. Recursive convolution requires the approximation of all needed functions with exponential functions in the time domain [3] which can also be attained by approximation with rational functions in the frequency domain [5]. Recursive convolution is based on recursive evaluation of the convolution integral using the previous, and already calculated, convolution results and a very limited amount of new input information pertaining to the last time interval only. J. Marti applied this recursive convolution technique to his JMARTI FD line model [5,21] in the EMTP. In this model he proposed a frequency domain rational-function approximating routine based on Bode's asymptotes [35] of the magnitude functions. This procedure avoided the problems of illconditioning and inaccuracy inherent to high-order approximations in a wide frequency band [19]. The elements of the series loss impedance matrix [Zfoss (co)] (per unit length) are frequency-dependent functions due to the skin effect. In the zCable model they are synthesized with rational approximations in the frequency domain which permits their representation with equivalent circuits in the discrete-time EMTP solution. 44 When synthesizing the elements of the [Zioss(co)] matrix, one has to be concerned with the numerical stability of the resulting synthesis for a coupled system. From network synthesis theory [36], for a multi-input multi-output system there are a number of requirements and conditions related to poles and zeros of the elements of the system matrix that need to be met in order to assure the numerical stability of the synthesis. As mentioned in [15,16,17], for the case of the zLine model, if each element of [Zioss(a>)] is synthesized independently of the other elements, even though stable poles are used in theses individual synthesis the model may produce unstable solutions in the time domain simulation. In network theory, the finite poles of the network functions correspond to the natural frequencies of the network. Since the natural frequencies are a property of the network and are independent of the excitation, we would expect the form of the natural response to be the same for all excitations of the network. One way to satisfy this condition is to synthesize all elements of the series loss impedance matrix [Z/ 0 5 i (co)] using the same set of poles. These poles can then be factored out of the matrix giving a single scalar transfer function with stable poles. This procedure not only guarantees stability but also results in more efficient coding of the model. Since the poles of all elements must be the same, each element of [Zioss(co)] must be synthesized simultaneously or in coordination with each other. In this thesis, a modification of the curve-fitting procedure originally introduced in [15,17] is proposed to synthesize all elements of the series loss impedance matrix [Z[oss(co)] simultaneously. In the vector-fitting 45 procedure of [23], Gustavsen and Semlyen let the fitted elements of each vector in the matrix share the same set of poles, and this accurate fitting can be achieved with a relatively low number of poles. However, in this fitting procedure the requirement of all elements of the matrix to have the same set of poles is not strictly enforced, which may explain the reported possibility of numerical oscillations in this otherwise very accurate model. In the synthesized functions for the zCable model, each element of the series loss i impedance matrix [Z{oss(co)] has the following properties: • The basic component of the fitting blocks is a. first order term of the form S^ (s =jco) s + P in the frequency domain. This simple term corresponds to a parallel RL circuit in the time domain. • Each element of [Z[oss(co)] is represented as a number of the above fitting blocks in series. t • For the diagonal elements of [Zioss(co)], an additional DC resistance(per unit length) is added to the series of fitting blocks to satisfy the DC condition. • Each element of [Zioss(a>)] uses the same number of fitting blocks. • The general expression for the [Z[oss (co)] matrix are as follows, ™ sKHl for the diagonal elements (3-29) ZhssfOj)^) - zZ l=l ™ sK, 'j(l) (3-30) S + P; for the off-diagonal elements m where subscript"/' indicates "fitted" function 46 The synthesized equivalent circuit for one segment of a single-phase underground cable is shown in Figure 3.17. The fitting and subsequent optimization procedures to obtain the above synthesized functions are described next. G'*l/2 core ideal line section sheath R' * I - A A A H R' * I In G'*l/2 /I1(1) •L/U(2) ' loss section R' * I - A A A H pi * i pi * ; ' A/22(l) ' /22(2) ' ri * I fi * I ^fi2(\) ' 7^22(2) ' Dl * I "/22(m) Figure 3.17 Equivalent synthesized circuit for one segment of a single-phase underground cable 3.4.1 Curve-Fitting Procedure The basic form of the synthesized functions used in the zCable model has been outlined in previous section. The fitting and optimization procedures are described next. A. Fitting Procedure First, we assume a particular frequency a>\. Then the constants , i ^ i ) and poles Pa(\),Pij(\) of the first fitting block in equations (3-29) and (3-30) are calculated using equations (3-31) to (3-34) below to exactly match the analytical cable data [Z/ 0 i y j (<»i)] at the 47 frequency co\. This will give the first RL fitting block for each element of the loss impedance matrix. % ( 1 ) - P + Rij{a>\) (includes i =j) (3-31) 4Vo L fijQ) =2~' + LU ^  ^ ( i n c l u d e s ' =7') ®1 A/(^l) (3-32) % 1 ) = Rfij(Y) (includes / =j) Pm = % d ) 7 % ( 1 ) ( i n c l u d e s *' =7) (3-33) (3-34) where /?jy(t»l),Z^-(fi>j) = resistance and inductance (per unit length) of the cable data at co\ Rfij(\)Xflj(\)= resistance and inductance (per unit length) of the first RL fitting block The same procedure is repeated for the next frequency <x>2 • The constants and poles of the second RL fitting block for each element of loss impedance matrix are now calculated to exactly match not the original cable data at co2 [Zjoss (co2)], but the difference between this value and the influence of the first block: [Zloss(G>2)] = [Zloss(OJ2)] ~ [— , ] % ( 1 ) + J^Lfijil) t first block evaluated at co-, 48 In detail, for all element i,j, including i =j, with . _ (co2Lij(co2))2 p , , n * M 2 ) - Rij(co2) + R l j ( 0 ) 2 ) ( 3 " 3 5 ) L m ) = ^ ^ + Lij(co2) ^ (3-36) co2Lij(co2) Kij(2) = Rfij{2) (3-37) ^•(2)=%(2) /%(2) (3-38) R y ^ R y W - i ^ ^ l f (3-39) Rjim+a)2Lftm Lytatf-Lyte)- ,^fg (3-40) Rfij(i)+a)2LMi) where % (^2)' Ay (^2 ) = m e resistance and inductance (per unit length) of cable data at frequency co2 Rij(co2),Lij(co2) = the resistance and inductance (per unit length) of cable data subtracted by the influence of the first RL block at frequency a>2 Ryj/(2) > Lyjy(2) = the resistance and inductance (per unit length) of the second RL fitting block for all including i =j. The step to obtain the third RL fitting block is similar to the second step, except that the block's constant and poles for each element of the loss impedance matrix should now 49 match the original analytical cable data [Z ioss(co-f)\ subtracted by the influences of previous two RL fitting blocks at frequency <y3. The procedure is then repeated for the total m frequency points, and at the end of the procedure we will have m RL fitting blocks for each of the elements in the loss impedance matrix. After completing this process, the addition of the m RL fitting blocks for each element of the synthesized loss impedance matrix [Ziossj-(co)\ is the complete expression of equations (3-29) and (3-30), and represents a series of RL parallel circuit as shown in Figure 3.17. In the algorithm developed in this thesis, the evaluation of the current RL fitting block depends on the influence of the previous fitting blocks. This is different from the original zLine fitting procedure of [15,17], where each block was calculated to match the line data independently of the other blocks. B. Optimization Procedure After evaluating the synthesized functions in the previous procedure, some apparent differences between the original and fitted functions can still be observed. An optimization procedure based on a Gauss-Seidel iteration and the work of [13] is applied to minimize these differences. This procedure is similar to the previous fitting procedure except that each RL fitting block for each element in the loss impedance matrix is calculated to match the analytical cable data subtracted by the influences of all other "/w-1" fitting blocks at that frequency. After obtaining a complete set of matching RL blocks (m blocks) for each element of [Ziossf(co)], the accuracy of the fitted functions is checked. If the differences are within 50 an acceptable specified limit, the optimization procedure can be ceased and the final fitted functions are obtained. Otherwise, more iterations are needed until the differences are within the specified level. The number of iterations needed in this optimization procedure usually depends on the number of RL fitting blocks used, however, in general acceptable differences are obtained with ten iterations or less. A flow chart illustrating the fitting and optimization procedures is shown in Figure 3.18. The results for the synthesis of the cable system presented in section 3.1 using eight RL blocks for a frequency range from 10"1 to 106 Hz are listed in Tables 2 to 6. Figures 3.19 to 3.23 compare the obtained fitted curves [Ziossf(a>)] and the original data curves [Zioss(a>)]. Table 2 Constants and poles of the synthesized functions for the self-impedance of the core Constant K PoleP 2.135288072711953e-004 7.550581161951129e-001 2.462702525162753e-003 1.014017306416074e+001 2.795699660620309e-002 1.136081554383791e+002 2.743422535994113e-001 1.073641854135195e+003 2.701759130172722e+000 1.116487814689622e+004 3.055546275027630e+001 1.208963326359363e+005 3.831337449722134e+002 1.414635014050347e+006 4.336044835980410e+004 5.335665724376278e+007 51 Table 3 Constants and poles of the synthesized functions for the mutual impedance between core and sheath Constant K PoleP 2.116749154346271e-004 7.429559821181907e-001 2.417228176436445e-003 1.001082144374518e+001 2.523260344836368e-002 1.079337508238402e+002 2.575769695964021e-001 1.109100082513990e+003 2.653537067559744e+000 1.135123464315377e+004 2.955444384892309e+001 1.206285389778882e+005 3.819821714885786e+002 1.422065129458736e+006 4.348991049446897e+004 5.358219466914544e+007 Table 4 Constants and poles of the synthesized functions for the self-impedance of the sheath Constant K PoleP 2.207392341571748e-004 7.784028694670802e-001 2.174955755865113e-003 9.420261512547574e+000 2.470453962248652e-002 1.041380643218053e+002 2.558782208253824e-001 1.096678811917996e+003 2.645511099853513e+000 1.130503779733941e+004 2.910479077924362e+001 1.199104648226937e+005 3.822200501009348e+002 1.421719389100061e+006 4.349488139320968e+004 5.358871563807655e+007 Table 5 Constants and poles of the synthesized functions for the mutual impedance between cable a and cable b Constant K Pole P 2.113458915342068e-004 7.421350756491967e-001 2.414155553584520e-003 9.992353935800145e+000 2.520722247459428e-002 1.078368871114252e+002 2.568868021223690e-001 1.107134775342093e+003 2.630865815770030e+000 1.128996762302871e+004 2.809329233672224e+001 1.178536978723049e+005 3.525838353752401e+002 1.35856511839821le+006 4.530629799241756e+004 5.280979763026466e+007 52 Table 6 Constants and poles of the synthesized functions for the mutual impedance between cable a and cable c Constant K Pole P 2.113331692403233e-004 7.421024323437496e-001 2.413664595379630e-003 9.991033950760279e+000 2.519091403901001e-002 1.077922631241064e+002 2.563621051645398e-001 1.105694696831621e+003 2.613807081780734e+000 1.124350452955758e+004 2.749967785962933e+001 1.162846755670261e+005 3.259478598642644e+002 1.295137185330023e+006 3.039469754456299e+004 5.299473118121008e+007 From the comparisons in Figures 3.19 to 3.23, we can see that the fitted curves match very closely the original curves for the self-impedances of core and sheath, the mutual-impedances between core and sheath, and the mutual-impedances between cables at all frequencies. The maximum error in these figures is in the order of 5% in magnitude. In our experience, the average number of fitting blocks used for an accurate synthesis and curve-fitting is about one block per decade. Generally, the total number of fitting blocks is usually Q less than ten (for a fitting up to 10 Hz) in most of the simulation cases. 53 2 x i w 03 e-i—i O g e o -X i 8-a oo o 13 x i 13 "d w £ B o C * 2 * +j 4) cd O oo J3-S 8 o .a i £ -a P2 x i s a .2 X> 13 I ^ g « 11 xs *d U o X5 X ! X) o O Cd a> o "X! += 3 fl cd 3 x> 1 ft s « oo X ! < H ^ ci o a cr kH cd +-; O Ki oo 03 " H "fl . .a ft Cd .9 § o o •5 1) o l fl <D s J D 13 0) X ! .t—> o 3 o <D < £ fl C T u o X) fl 3 O T 3 X^ fl ~ o E <2 a oo oo <D "3 x! < H ^ CN •9 -S cd « X ! O 1 <a o o ts g .2 c eg — o IU oo Xj 1 1 3 >1 C g fl (D cd O 2 £ 5, CS M <U c*> cd t s 1 3 o o • 3 •a A tN X ! X ! a b X ! 2 ° fl cd o a> cr o CD X <u fl » <u X ! •4—* <D S A oo oo < X I £3 t5 C u s CJ <D oo X ! 3 X ! o t d o o fl C l i o o «J fl ts 2 ^ fl cd o u oo Xj fl u ^ fl X >> cr C x> <u ^ ts -s * XI H CN cj cd >B fl X) 03 cd ^ r „ 03 O B tS © .5 X ) a o X ! Figure 3.18 Block diagram of the fitting and optimization procedures 54 6 10"2 10° io2 io4 io6 Frequency (Hz) (a) Magnitude of the original and fitted self-impedance of the core 1.5 -2 0 2 4 I 10 10 10 10 10 Frequency (Hz) (b) Phase angle of the original and fitted self-impedance of the core gure 3.19 Comparison of the original and fitted of self-impedance ofthe 55 10 10 ^10 E l i o 1 CD c CT) CD sio" 1 10 10 original function — fitted function 10 10 10 Frequency (Hz) 10 10 (a) Magnitude of the original and fitted mutual-impedance between core and sheath 1.8 1.7 C M i\i 1/T c co T 3 2_.1.5 0 CT) c CO 0 1.4 co CO 1.3 1.2 original function fitted function 10 10 10 Frequency (Hz) 10 10 (b) Phase angle of the original and fitted mutual-impedance between core and sheath gure 3.20 Comparison of the original and fitted mutual-impedance between core and sheath 56 io4 10 I i ' • ' • ' ' 1 -2 0 2 4 , 6 10 10 10 10 10 Frequency (Hz) (a) Magnitude of the original and fitted self-impedance of the sheath 1.5 Frequency (Hz) (b) Phase angle of the original and fitted self-impedance ofthe sheath gure 3.21 Comparison of the original and fitted mutual-impedance ofthe sheath 57 -2 0 2 4 6 10 10 10 10 10 Frequency (Hz) (a) Magnitude of the original and fitted mutual-impedance between cables a and b — original function — fitted function \ ' -2 0 2 4 1 10 10 10 10 10 Frequency (Hz) (b) Phase angle of the original arid fitted mutual-impedance between cables a and b gure 3.22 Comparison of the original and fitted mutual-impedance between cables a and CD N 1.6 c 1.5 CD T 3 CD _CD CT) C CO CD ttt CO SZ Q. 1.4 1.3 •1 o 58 10 io 31 1021 co o N 1 "5 10 CD •a B o c 10 CT) CO .-1 10 10 10 original function — fitted function 10 10 10 Frequency (Hz) 10 10 (a) Magnitude of the original and fitted mutual-impedance between cables a and c 1.7 1.6 co o N c co T 3 2H.4 af cn e CO a) 1.3 CO CO 1.2 1.1 original function fitted function 10 10 10 Frequency(Hz) 10 10 (b) Phase angle of the original and fitted mutual-impedance between cables a and c gure 3.23 Comparison of the original and fitted mutual-impedance between cables a and 59 3.4.2 Numerical Stability The stability of the fitted functions can be studied using the concept of input-output control theory. In theory [36,37,38], the stability of a system can be defined in terms of its input and output quantities. In other words, a system is said to be stable if for every bounded input (excitation) there corresponds a bounded output (response). The above concept can be applied equally well to a general multi-input multi-output. In this case, the system can be • described by the vector state equation x(t) = [A]x(t) + [B]u(t) where x(t) and u(t) are vectors and [A] and [B] are constant matrices. More generally, in the frequency domain the time domain vector differential equation becomes (oX{a>) = [A]X(co) + [B]U(co) where [A] = [A(a))] [B] = [B(co)} For the series loss impedance system in each segment of the zCable model, the state equation in the frequency domain for a state vector I(a>) with an input vector V(co) becomes: co • I(a>) = -[Li0ss(«>)Y][Rioss(«>)] • m + t W ^ F 1 • V{a) (3-41) with matrix [A(a>)]--[Lioss(co)][Rioss(co)]. If all the eigenvalues of matrix [A(a>)] have negative real parts at all frequencies, then, since the cable model is linear we can apply superposition of frequency responses and the time domain response will be stable for an arbitrary input w(t) and state vector jc(t). In other words, if all the eigenvalues of the matrix -[Liossj-(co)][Riossf(o})] for the fitted functions have negative real parts for all frequencies, then the response of the system using the fitted loss impedance matrix [Z[ossy(co)] will be 60 stable. Figures 3.24 to 3.26 show a comparison of the eigenvalues of the original matrix -[LlossWMlossi.®)] ^ i t s f l t t e d m a t r i x -[Llossf(°>)]lRlossf(<»)] f o r Ae example of the three-phase cable system studied. In this case all eigenvalues are real. -2 0 2 4 1 10 10 10 10 10 Frequency (Hz) Figure 3.24 Comparison of eigenvalue 1 of the original and fitted matrices 61 -2 0 2 4 6 10 10 10 10 10 Frequency (Hz) Figure 3.25 Comparison of eigenvalue 4 of the original and fitted matrices -io1 -2 0 2 4 I 10 10 10 10 10 Frequency (Hz) Figure 3.26 Comparison of eigenvalue 6 of the original and fitted matrices 62 As shown in Figures 3.24 to 3.26, the eigenvalues of the fitted matrix closely match those of the original matrix and have negative real values at all frequencies. These results show that the system will be stable when the fitted matrix [Ziossf(co)] takes the place of the original matrix [Zioss (&>)], and will assure the stability of the synthesized model. 3.4.3 Voltage Drop Formulation After obtaining the synthesized functions for each element of the loss impedance matrix, the system equations for voltage drop across the multiphase loss section of each segment in the zCable model can be formulated in the discrete time domain using a suitable integration rule. In this thesis, we use trapezoidal integration with damping [39,40,41] in order to avoid the numerical oscillations due to trapezoidal integration. The equation of this integration rule has been described in [41] as follows, y n + \ = y n + Y [ ( l + a ) f n + l + ( l ~ a ) f n ] ( 3 " 4 2 ) where a = damping coefficient It can be shown that the net effect of applying the equation of trapezoidal integration with damping (equation (3-42)) to an ideal inductor or capacitor is equivalent to using ordinary trapezoidal integration in an inductor with a shunt conductance G = (aAt) 1 2 L , and in a capacitor with a series resistance R = (aAt) 12C. It may be noted that a = 0 results in ordinary trapezoidal integration, while a = 1 results in Backward Euler integration. As stated in [41], trapezoidal integration with damping is more accurate than Backward Euler integration by choosing appropriate damping coefficient a, and was able to eliminate the 63 numerical problems due to trapezoidal integration. As the above description, we know that the damping coefficient a should be chosen as small as possible in order to have the smallest phase distortion. In this thesis, we choose a = 0.15 because it is the smallest a that is sufficient to dampen out the oscillations in all test cases shown in Chapter 4. Figures 3.27 and 3.28 are the simulation results for a line-to ground short circuit simulation (Figure 4.1) using pure trapezoidal and trapezoidal with damping integration rules. The accumulation of oscillations displayed in Figure 3.27 is related to the reflected waves in each segment of the model combining with each other along the total cable length. From the comparison in Figures 3.27 and 3.28, we can see that the numerical oscillations are successfully eliminated using the trapezoidal with damping integration rule with a damping coefficient a = 0.15. -1.6" ' 1 1 -i 1 0 0.2 0.4 0.6 0.8 1 Time (Sec.) x 1 0 - 3 Figure 3.27 Energization into a line-to-ground short circuit, simulation: Voltage at the receiving end of core 2 (pure trapezoidal integration rule) 64 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 Figure 3.28 Energization into a line-to-ground short circuit simulation: Voltage at the receiving end of core 2 (trapezoidal with damping integration rule, a = 0.15) The general form of the voltage drop equations across the loss section in each segment of the zCable model in the time domain is as follows, AV(t) = [Req]- I(t) + H(t) (3-43) where [Req] is the equivalent resistance matrix of the loss section in each segment. The elements of [Req ] are given (see Appendix III) by 2K m Reqii =iRUDC + At -] • / for the diagonal elements (3-44) /=1 — + ( a + l ) - i > 7 ( / ) At 2K m ij(0 Reqij At ^eqij - 9 l=\ — + {a + \).Pm At ] • / for the off-diagonal elements (3-45) 65 where /: the length of the segment The equivalent history voltage source vector H(t) is given (see Appendix III) by n m j=U=l (3-46) with n the number of conductors (a single-phase underground cable can be represented with two phase conductors) and From the above formulas, we can easily obtain the recursive discrete-time solution for the loss section of each segment in the zCable model using digital computation. Then, the results of this part can be combined with the results ofthe ideal line section for each segment. Finally, the complete solution for whole cable system can be obtained by combing the solutions of all segments. l + (g-l)./fr(/) ! + ( « + ! ) . AKW)PW)'1 heqij(l)(0 -• 1j(t -At) (3-47) 66 3.5 P h a s e D o m a i n U n d e r g r o u n d C a b l e S o l u t i o n As discussed earlier in this thesis, in order to represent the distributed nature of the losses, the full cable length is first subdivided into a number of short-length segments. As it was shown in Figure 3.10, earlier in this Chapter, each short segment consists of a distributed ideal line section and a lumped loss section. The solution of each short cable segment is obtained by combining the solutions of the ideal line section and the loss section. For distributed-parameter reasons, as above, the loss section of each segment is divided into two halves, and each half is placed at both ends of the ideal line section, as shown in Figure 3.29. ideal line section I HI ) 4 Z D — ^ — [ 2 u > » ] / 2 — [Iex,]and[q I loss section/2 I (external magnetic • (losses and and electric fields) . internal flux) • Figure 3.29 Proposed cable segment for the zCable model The solution for the full-length of cable is achieved by combining the solutions of all zCable segments. All the short-length cable segments are lined up in cascade and the equivalent network for the full-length cable in the discrete-time domain is shown in Figure 3.30. [Z,oss(C0)]/2 loss section/2 (losses and internal flux) 67 Figure 3.30 Equivalent network for the full-length of cable in the discrete-time domain 68 From the circuit in Figure 3.30, the full matrix of the cable system can be realized as a number of diagonal submatrices. These diagonal submatrices are independent of each other due to the decoupling delay introduced by the ideal line section of each segment. For example, if the full-length cable is.subdivided into N segments, then we will have a total of N+l blocks in the equivalent network (Figure 3.30) and correspondingly N+l independent diagonal submatrices in the full system matrix (equation 3-48). [1] 0 0 0 0 0 0 [2] 0 0 0 0 0 0 [3] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [N + l] (3-48) When we find the solution of the full-length cable system we don't need to build the full matrix but only JV+1 diagonal submatrices corresponding to the N+l blocks of the equivalent network. Actually, all the diagonal submatrices are the same except the first and last block-diagonal submatrices corresponding to the sending and receiving end of the cable. There are, therefore, only three different types of block-diagonal submatrices to be built. When the full-length cable is solved together with the whole power system network, the power network can only see the first and last segments of the zCable system during the power system solution. The other segments are solved internally by themselves. Using this technique, we don't need to deal with the complicated and vary large matrix of the full cable system but only the simpler and smaller diagonal submatrices. The total cable solution is computationally very efficient. 69 3.5.1 Determination of the Length of the Cable Segments In order to estimate the accuracy of the zCable model and the effect of the lumped losses, Dai Nan [42] introduced an algorithm that automatically chooses the maximum short segment length for a maximum frequency of interest and the desired model accuracy. In this algorithm, an exact solution is compared with a solution based on the zCable model at a number of specified frequencies in order to obtain a relationship between the maximum segment length and the maximum frequency of interest. The specification and arrangement of the underground cable used in the tests of [42] is the same as those used in section 3.1 in this work. The curve in Figure 3.31, derived from the work of [42], gives the relationship between the maximum zCable segment length and a given maximum frequency of interest for a maximum error of 5% and a ground resistivity of 100 Q-m. Frequency (Hz) Figure 3.31 Relationship between segment length and frequency 70 The code for simulating and testing the zCable model has been written in the C++ object-oriented language and is part of the development project for UBC's OVNI real-time simulator [43]. A number of transient analysis simulations of the zCable model were carried out for asymmetrical cable configurations, and the results were compared with those of the JMARTI line model (FD line) [5] and the LMARTI cable model (FDQ cable) [6] as shown in the next chapter. 71 Chapter 4 Transient Simulation and Validation In this chapter, the results of transient simulations for the zCable model are compared with those of the frequency-dependent JMARTI (FD line) line model [5,21] and the frequency^ dependent LMARTI (FDQ cable) cable model [6]. The FD line model is a full frequency-dependent model using the constant modal transformation matrix technique. The FDQ cable model is a very accurate model in both low and high frequency phenomena for symmetrical and asymmetrical cable configurations. It has been validated against field tests [6,22] and is used as the reference model for validation ofthe proposed model. The underground cable used in the following tests is a 230KV three-phase single-core coaxial cable. The physical data and cable arrangement were shown in Table 1 and Figure 3.3 in Chapter 3. The total length of the cable is 5km. The modal transformation matrix used in the FD line model was calculated at 1000Hz. Eight RL parallel blocks were used to fit each element of the loss impedance matrix [Zioss(co)] accurately up to 1MHz. The results of the curve-fitting of these frequency-dependent functions in the matrix [Z/0 iW (a>)] for the test cable were shown in Figures 3.19 to 3.23 in Chapter 3. From Figure 3.31 in Chapter 3, we choose a segment length of 0.1km to simulate accurately up to 20KHz. After introducing the pi-circuit correction, all modes for the three phases have the same travelling time T, which is 0.624ps. The time step At of all 72 simulations was chosen as large as this value (0.624u,s) in order to get the smallest possible in computational times. The complete zCable model for the following tests consists of 50 segments of 0.1km each for a total cable length of 5km. 4.1 Energization into a Line-to-Ground Short Circuit Simulation The system configuration for energization into a line-to-ground short-circuit simulation is shown in Figure 4.1. A three-phase 60Hz balanced sinusoidal voltage source with peak magnitude of 1 .Op.u. is connected and switched on at time zero (t = 0), energizing the sending end of each core. The receiving end of each core is connected to a resistive load of 500Q. The sending and receiving ends of the sheaths are directly connected to ground. A single-phase short-circuit fault is applied at the receiving end of core 1 through a resistance of 0.05Q at time zero. The main goal of this simulation is to evaluate the induced voltage in the unfaulted phases when a line-to-ground fault is applied to one of the phases. 5km M 1.0 p.u si ^ ^ ty> c2 C3 8. / = 0.0 s a o S S o 5 0 . 0 5 0 Figure 4.1 System configuration for energization into a line-to-ground short circuit simulation The comparison of the voltages at the receiving end of cores 1, 2, 3 and the receiving end currents of sheaths 1, 2, 3 for the zCable model, FD line model, and FDQ cable model are shown in Figures 4.2 to 4.7. 73 The results of Figures 4.2(a) to 4.7(a) show noticeable differences on both magnitude and phase shift between the results with FD line model versus those with the FDQ cable and zCable models. From the enlargements for the initial transient phenomena (Figure 4.2(b) to 4.7(b)), the good behavior of FD line within the first 100~200 us is probably related to the fact that the constant modal transformation matrix for this model was calculated at the high frequency of 1 KHz. As the high frequency transient attenuates, the differences between FD line and zCable and FDQ cable increase because the main frequency after the fault is back to 60Hz. In this study, the results with the zCable model present a very good agreement with those of the FDQ cable model over the entire simulation range. 74 0.035 0.03 0.025 0.02 CD cn ™ 0.015 o > 0.01 0.005 zCable model FDQ cable model FD line model 0.2 0.4 0.6 T ime (Sec.) 0.8 (b) Initial transient of the receiving end voltage x 10 Figure 4.2 Energization into a line-to-ground short circuit simulation: Voltage at receiving end of core 1 75 1.5 5 I 1 ' ' 1 ' o 0.005 0.01 0.015 0.02 Time (Sec.) (a) Receiving end voltage -0.1 -0.2 -0.3 -0.4 CL CD -0.5 O) CD O -0.6 > -0.7 -0.8 -0.9 -1 zCable model FDQ cable model FD line model n 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage ;ure 4.3 Energization into a line-to-ground short circuit simulation: Voltage at the receiving end of core 2 76 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 -0.1 -0.2 -0.3 d cL -0.4 <D O) £ -0.5 o > -0.6 -0.7 -0.8 -0.9 0.2 zCable model FDQ cable model FD line model 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage rare 4.4 Energization into a line-to-ground short circuit simulation: Voltage at the receiving end of core 3 77 x 10 — zCable model — F D Q cable model - - F D line model 0.005 0.01 Time (Sec.) (a) Receiving end current 0.015 0.02 x 10 3 ri. ¥ - 3 0 L _ o zCable model F D Q cable model F D line model 0.2 0.4 0.6 Time (Sec.) 0.8 (b) Initial transient of the receiving end current x 10 Figure 4.5 Energization into a line-to-ground short circuit simulation: Receiving end current of sheath 1 78 X 10 3 I i i i 1 0 0.005 0.01 0.015 0.02 Time (Sec.) (a) Receiving end current x 10 Time (Sec.) (b) Initial transient of the receiving end current ;ure 4.6 Energization into a line-to-ground short circuit simulation: Receiving end current of sheath 2 79 x 10 2.5 i Time (Sec.) (a) Receiying end current x 10 0.5 I - i 1 r i i 1 1 1 1 0 0.2 0.4 0.6 0.8 1 Time (Sec.) x 1 0 3 (b) Initial transient of the receiving end current gure 4.7 Energization into a line-to-ground short circuit simulation: Receiving end current of sheath 3 80 4.2 O p e n - C i r c u i t S i m u l a t i o n The simulation system for an open-circuit simulation is shown in Figure 4.8. A three-phase 60Hz balanced sinusoidal voltage source with peak magnitude of l.Op.u. is connected to the sending end of each core and is switched on at the beginning of the simulation (t = 0). The receiving end of all cores and sheaths are left open. The sending end of all sheaths are directly connected to ground. The purpose of this study is to observe the phenomena at the receiving end of all cores and sheaths, which are left open. 5 km l.Op.u. Cp Cp Cp o c l -o Si i-o-o c2 -o s2 •O C3 -o S3 Figure 4.8 System configuration for an open-circuit simulation The voltages at the receiving end of cores 1, 2, 3 and sheaths 1, 2, 3 for the compared models are shown in Figures 4.9 to 4.14. From Figures 4.9 to 4.14, we can see that the results of FD line model match closely those of the other two models for all cores, but strongly deviate from those of the other two models for all sheaths, not only in magnitude but also in phase shift. The results of the zCable model and FDQ cable model are practically superimposed at all places for the entire simulation. 81 2 1.8 1.6 1.4 zi 1.2 CL <D 1 D) ro o 0.8 0.6 0.4 0.2 zCable model FDQ cable model FD line model 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 zCable model FDQ cable model FD line model 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage Figure 4 . 9 Open-circuit simulation: Voltage at the receiving end of core 1 82 0.015 0.01 0.005 3 CL a> O) ro +-* o > -0.005 -0.01 -0.015 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 0.015 zCable model FDQ cable model FD line model -0.005 -0.015 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage Figure 4.10 Open-circuit simulation: Voltage at the receiving end of sheath 1 83 1.5 zCable model FDQ cable model — FD line model -1.5 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 -0.1 -0.2 -0.3 r s -0.4 cL cu -0.5 cn ro .*-» o > -0.6 -0.7 -0.8 -0.9 -1 VJ 0.2 zCable model FDQ cable model FD line model 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage Figure 4.11 Open-circuit simulation: Voltage at the receiving end of core 2 84 x10 -6 -8 -10 — zCable model • - FDQ cable model - FD line model 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 x 10 3 cL d) co ro -*-» o > -10 zCable model FDQ cable model FD line model 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage Figure 4.12 Open-circuit simulation: Voltage at the receiving end of sheath 2 85 1.5 -1.5 0 -0.1 -0.2 -0.3 zi -0.4 cL CD -0.5 cn ro o -0.6 -0.7 -0.8 -0.9 -1 zCable model FDQ cable model FD line model 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 zCable model FDQ cable model FD line model 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) initial transient of the receiving end voltage Figure 4.13 Open-circuit simulation: Voltage at the receiving end of core 3 86 0.015 0.01 3 d. cu co CD o > 0.005 -0.005 -0.01 i I I zCable model FDQ cable model FD line model 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 3 d 0 CD CO -»—* o > 0.01 0.008 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 -0.008 -0.01 zCable model FDQ cable model FD line model 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) initial transient of the receiving end voltage Figure 4.14 Open-circuit simulation: Voltage at the receiving end of sheath 3 87 4.3 C a b l e S y s t e m u n d e r L o a d S i m u l a t i o n The system configuration for a cable system under load simulation is shown in Figure 4.15. The sending end of all cores are energized at the beginning of the simulation (t = 0) with a three-phase 60Hz balanced sinusoidal voltage source with a peak magnitude of 1 .Op.u.. The receiving end of each core is connected to a resistance load of 500Q. The sending end of all the sheaths are directly connected to ground; however, the receiving ends of the sheaths are left open. The purpose of this study is to observe the voltages induced in the sheaths by coupling from the cores under loads. 5km M 1.0 p.u si C2 i - O -S2 C3 s3 ^1 ° f ° i Figure 4.15 System configuration for a cable system under load simulation The voltages at the receiving ends of all cores and sheaths for the zCable, FD line, and FDQ cable models are shown in Figures 4.16 to 4.21. The results of Figures 4.16 to 4.21 show that the zCable model closely agrees with the FDQ cable model at all places. Similarly to the results of the open-circuit simulation, the results of the FD line model match closely those of the zCable and FDQ cable models at the receiving end of all cores, but deviate at the receiving end of all sheaths. The only difference with respect to the open-circuit simulation is that the degree of deviation at the receiving end of the sheaths for the load simulation is less than that of the open-circuit simulation. 88 1.5 ri. 0.5 0) £ 0 o > -0.5 -1.5 zCable model FDQ cable model FD line model 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 0.4 0.6 Time (Sec.) x10 (b) Initial transient of the receiving end voltage ;mre 4.16 Cable system under load simulation: Voltage at the receiving end of core 1 89 0.04 zCable model FDQ cable model FD line model -0.04 -0.06 JL 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.06 0.04 0.02 Z3 CL cu cn co o > -0.02 -0.04 -0.06 T zCable model FDQ cable model FD line model 0.02 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage jure 4.17 Cable system under load simulation: Voltage at the receiving end of sheath 1 90 1 zCable model FDQ cable model FD line model 0.5 3 u 6. cu cn ro o -0.5 -1.5 -0.1 -0.2 -0.3 d -0.4 Q. CD -0.5 cn ro +-* Vo -0.6 -0.7 -0.8 -0.9 -1 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 zCable model FDQ cable model FD line model 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage gure 4.18 Cable system under load simulation: Voltage at the receiving end of core 2 91 -0.02 h -0.03 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 0.03 0.02 0.01 3 Q. <D 0 cn co o > -0.01 -0.02 -0.03 zCable model FDQ cable model FD line model 0.2 0.4 0.6 Time (Sec.) 0.8 x 10 (b) Initial transient of the receiving end voltage gure 4.19 Cable system under load simulation: Voltage at the receiving end of sheath 2 92 1.5 zCable model FDQ cable model FD line model 0.005 0.01 Time (Sec.) (a) Receiving end voltage 0.015 0.02 0.4 0.6 Time (Sec.) x 10 (b) Initial transient of the receiving end voltage ?ure 4.20 Cable system under load simulation: Voltage at the receiving end of core 93 -0.06 0 0.005 0.01 0.015 0.02 Time (Sec.) (a) Receiving end voltage 0.06 0.04 0.02 3 ri. cu cn ro "5 > -0.02 -0.04 -0.06 zCable model FDQ cable model FD line model 0.2 0.4 0.6 0.8 x 10 Time (Sec.) (b) Initial transient of the receiving end voltage gure 4.21 Cable system under load simulation: Voltage at the receiving end of sheath 3 94 4.4 I m p u l s e R e s p o n s e S i m u l a t i o n The system configuration for an impulse response simulation is shown in Figure 4.22. A 1.2 x 5.0jus voltage impulse is connected to the sending end of core 1, and switched on at time zero. The receiving ends of all cores and sheaths are left open. Except for the sending end of corel, all other sending ends of all other cores and sheaths are directly connected to ground. The main goal of this simulation is to observe the natural response of the system when a sudden shock is applied to a cable system. 5 km 1.2 x 5.0ns voltage impulse - o c l -Os! • O c2 -O S2 •O C3 -Os3 Figure 4.22 System configuration for an impulse response simulation The voltages at the receiving end of all cores and sheaths for the zCable, FD line, and FDQ cable model are shown in Figures 4.23 to 4.28. From the results of the simulation, we can observe that the zCable model closely matches the results of the FDQ cable model in magnitude and phase. The results of the FD line model are only superimposed on those of the zCable and FDQ cable model at the receiving end of corel; however, the results of FD line strongly deviate from those of the other two models at all other places. Since underground cables have strongly asymmetrical configurations, we believed that the deviations of the FD line model with respect to the 95 zCable and FDQ cable models are due to its use of a real and constant transformation matrix. The imaginary part of this matrix (ignored by FD line) is very important in strongly asymmetrical configurations. 96 1.5 Z3 a. Q> 0.5 cn co -«-> o > -0.5 zCable model FDQ cable model FD line model 0.002 0.004 0.006 Time (Sec.) (a) Receiving end voltage 0.008 0.01 1.5 3 cL cu 0.5 cn ro •4—* o > -0.5 zCable model FDQ cable model FD line model 0.5 1.5 Time (Sec.) (b) Initial transient of the receiving end voltage x 10 Figure 23 Impulse response simulation: Voltage at the receiving end of core 1 97 10 X 1 0 Sir 6 — zCable model • - FDQ cable model - FD line model 0.002 0.004 0.006 Time (Sec.) (a) Receiving end voltage 0.008 0.01 10 8 6 <~? 4 cL 0) 2 cn ro S o X 10 zCable model FDQ cable model FD line model 1 . Time (Sec.) x 10 (b) Initial transient of the receiving end voltage gure 4.24 Impulse response simulation: Voltage at the receiving end of sheath 1 98 x 10 zCable model FDQ cable model FD line model 0.002 0.004 0.006 Time (Sec.) (a) Receiving end voltage 0.008 0.01 x 10 zCable model FDQ cable model FD line model -4 0.5 1.5 Time (Sec.) (b) Initial transient of the receiving end voltage x 10 Figure 4.25 Impulse response simulation: Voltage at the receiving end of core 2 99 x10 zCable model FDQ cable model FD line model 0.002 0.004 0.006 Time (Sec.) (a) Receiving end voltage 0.008 0.01 x 10 -1 zCable model FDQ cable model FD line model 0.5 1.5 Time (Sec.) (b) Initial transient of the receiving end voltage x 10 ?ufe 4.26 Impulse response simulation: Voltage at the receiving end of sheath 2 100 zCable model FDQ cable model FD line model 0.5 1.5 Time (Sec.) (b) Initial transient of the receiving end voltage x 10 ?ure 4.27 Impulse response simulation: Voltage at the receiving end of core 3 101 x 10 3 r ! A 2fr Z3 3 0 cu CD CO .1 o > -2 zCable model FDQ cable model FD line model 0.002 0.004 0.006 Time (Sec.) (a) Receiving end voltage 0.008 0.01 x 10 zCable model FDQ cable model FD line model Time (Sec.) (b) Initial transient of the receiving end voltage x 10 gure 4.28 Impulse response simulation: Voltage at the receiving end'of sheath 3 102 4.5 C o m p u t a t i o n a l S p e e d The main goal df this thesis has been to develop a very accurate and stable frequency-dependent underground cable model. In other words, computational speed has not been our main concern of this thesis. However, we still need to know this property in order to assess the practicability of the zCable model. To measure the relative computational speed of the zCable model with that of the FD line and FDQ cable models, the computational time of the single line-to-ground fault test presented in section 4.1 of this Chapter was measured. These measurements were performed on an AMD K6-2 366Hz personal computer. Due to the number of segments required to simulate the distributed nature of the losses, the proposed zCable model is computationally less efficient than the FD line and FDQ cable models. Tables 7 and 8 show the relative computational time for the FD line, FDQ cable, and zCable models when 10 and 50 segments are used in the zCable model. The indicated computational time corresponds to lxlO" 3 sec. of transient simulation. As indicated earlier, the simulation comparisons presented in Section 4.1 to 4.4 were performed with zCable using 50 segments. Table 7 Comparison of relative CPU times (zCable withlO segments) Model Relative computational time (sec.) FD line 3.0 FDQ cable 5.0 zCable (10 segments) 6.0 103 Table 8 Comparison of relative CPU times (zCable with 50 segments) Model Relative computational time (sec.) FD line 3.0 FDQ cable 5.0 zCable (50 segments) 28.0 From the results of Tables 7 and 8 we can see that the number of segments used in the zCable model is the main factor that determines its computational time. For example, using eight RL blocks per element of [Zioss(co)] and a total of 10 segments, the speed of the zCable model is competitive with that of the FDQ cable model. However, using the same number of blocks but 50 segments, the speed of the zCable model is about 6 times slower than that of the FDQ cable model. It is expected that these results will improve as the programming code for zCable is optimized in the future. Since the computational speed of the zCable model is not orders of magnitude lower than that of the existing very accurate cable model (FDQ cable model), and the zCable model guarantee the numerical stability ofthe solution for all cases, it is expected that this model will represent a good practical alternative for the reliable modelling of fast transients in underground cable systems. 104 Chapter 5 Conclusions A new model to simulate electromagnetic transients on frequency-dependent underground cables has been presented in this thesis. This new model (zCable model) is based on Castellanos and Marti's work of [14,15,16,17], which separates the representation of the wave propagation along an underground cable segment into two parts: a) constant ideal propagation due to the external magnetic field [Lext] and electric field [C], which depend only on the geometry of the cable configuration, and b) frequency-dependent distortion due to the resistance [R(co)], dielectric loss [G], and internal flux [Lmt(co)] inside the conductor and ground. Since the model separate the external effects, which depend on the cable geometry, from the internal ones, it can represent exactly the geometry of any arbitrary cable configuration. A difference between the cable model developed here and the zLine model in [14,15,16,17] is that the different permittivities in the cable result in different modal velocities. It is then more convenient to solve for the wave propagation of the ideal line section in the modal domain rather than in the phase domain. Nonetheless, the modal transformation matrix associated with this part of the zCable model is still frequency-independent and real due to the property of constant inductance L and constant capacitance C for the lossless ideal line section parameters. An improved coordinated curve-fitting procedure is also proposed in this thesis to synthesize and fit all elements of the frequency-dependent loss impedance matrix [Zioss(a>)] 105 V simultaneously with rational functions. To guarantee the numerical stability of the [ZIOSS(G))] matrix in the zCable model, all elements of this matrix are approximated using the same set of poles. These poles can then factored out of the matrix giving a single scalar transfer function with stable poles. This procedure avoids the possible numerical stability problems of conventional frequency-dependent transformation matrix line and cable models. As presented in Chapter 3, the results of the fitted functions for each element of the loss impedance matrix [Z/0 i M (<w)] show a good agreement with those of the original functions. In addition, the eigenvalues of the fitted matrix - [Liossj-(cv)][Riossf(a))] have negative real parts at all frequencies, and closely match those of the original matrix -[Li0SS(a))][Ri0SS(a>)]. The above description shows that the modified curve-fitting procedure proposed in this thesis is very accurate and also numerically stable for all system solutions. An important contribution of this thesis is the concept of a "pi-circuit correction" to make the travelling time of all modes identical to one another in the modal domain. This avoids the interpolation process of traditional line and cable models with different modal travelling times, and a much larger integration step, five to ten times larger than that of traditional models, can be used without loss of accuracy. This procedure results in considerable savings in computational time. Compared to the industry-standard FD line and FDQ cable models, the results of the zCable model show a very good agreement with the FDQ cable model and both models are 106 more accurate than the constant transformation matrix FD line model. The main advantages of the zCable model over the FDQ cable model are: a) the zCable model avoids the difficulties of synthesizing the frequency-dependent transformation matrix that relates phase and modal domain quantities, and b) the fitting procedure of the zCable model is very robust and efficient, and leads to absolutely numerically stable synthesized functions. Although at this point of initial programming the zCable model is slower than the FDQ cable model, its high accuracy and absolute numerical stability for all possible cable configurations, make this new model a valuable contribution to transient analysis simulations over a wide frequency range. It is expected that the computational efficiency of the model will improve with more optimized programming but more significantly with the use of latency technique [44] currently under development in our real-time power system simulation group. Additional comments on this latency development are made under suggestions for future work. 107 Chapter 6 Suggestions for Future Work The work presented in this thesis is based on the accumulated work of many previous researchers and engineers, as listed in the bibliography. The future research is always based on the previous ones. In this thesis, the discrete-time solution of the zCable model has been successfully verified by proving the correctness of the time domain solution. We hope that the contribution of this work can be used as the basis of future researches. Some suggested works are listed as follows, • Efficient implementation in the context of the O V N I real-time simulator Incorporate the zCable model efficiently into UBC's OVNI (Object Virtual network Integrator) real-time power system simulator for efficient implementation of a cable system in the context of a complete power system representation. • Implementation of latency techniques in a complete power system network In order to represent the distributed nature of the losses, the underground cable made proposed needs to subdivide the total cable length into a number of short segments. This forces the zCable model to use a smaller integration time step At than that of the external network. If the external network uses the same time step as that of the zCable model, it unnecessarily increases the computational burden. The latency technique being advanced in [44] could be implemented in a future work to solve this problem, so that a larger 108 integration time step At can be used in the external network (theoretically up to the travelling time T of the full line/cable length). Implementation of frequency-dependent shunt admittances in the zCable model Because the variation of the shunt conductance and capacitance with frequency is difficult to obtain, and in order to more easily assess the viability of the zCable model, the shunt conductance and capacitance were assumed io be constant in this thesis. However, it is possible to include the frequency dependence of shunt admittances in the loss section of the zCable if more precise information can be obtained in the future. 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[43] L. R. Linares, "OVNI (Object Virtual Network Integrator) A New Fast Algorithm for The Simulation of Very Large Electric Networks in Real Time", Ph.D. Thesis, Department of Electrical and Computer Engineering, The University of British Columbia, August 2000. [44] F. A. Moreira, J. A. Hollman, L. R. Linares, J. R. Marti, "Network Decoupling by Latency Exploitation and Distributed Hardware Architecture", Proc. International Conference on Power System Transients (IPST'01), Rio de Janeiro, Brazil, June 2001, pp. 317-321. 114 Appendix I Error Analysis of Linear Interpolation Process In a single-phase ideal line system, the discrete-time EMTP model of this system is shown in Figure 1.1. h(t) k o-vk(t) o-e«Jt) Ut) -o m + -o Figure 1.1 Discrete-time EMTP model of a single-phase ideal line system As mentioned in Chapter 3, the end node voltage vk(t) and vm(t) are described as functions of the characteristic impedance Zc, current into the end-node ik (t) and (t), and history voltage sources ekh(t) and emn(t). The history voltage sources ekh(t) and emh(t) are updated at each time step of solution as a function of past voltage and current values, and have the form ekh (0 = Zc-im(t-r) + vm (t - T) emh(t) = Zc -ik(t-T) + vk(t-T) (1-2) The solutions of ekh (t) and emh (t) are exact as long as the travelling time x is an integer multiple of the integration time step Ar. If this is not the case, then linear interpolation is used in traditional EMTP solutions in order to obtain precise history values. 115 Since due to the interpolation process the history voltage sources are only obtained approximately, it is important to have some understanding of the errors caused by this, process. One technique to assess these errors is to transform equation (1-1) from the time domain into the frequency domain, Ekh={Zc-lm+Vmye-^ (1-3) Equation (1-3) provides the exact solution if there are no interpolation errors. If linear interpolation is used, and we assume the interpolated value lies in the middle of the time step interval At, then equation (1-3) becomes I -ja>(r+—) -jco(t ) ^(interpolated) = ( Z c ' 4* +Vm)-^<e ? + e 2 ) ( M ) Therefore, the ratio of the interpolated value to the exact value becomes -^^(interpolated) , T c . = cos(ci) ) (1-5) -^^(exact) ^ The frequency response of equation (1-5) is shown in Figure 1.2 as a function of the Nyquist frequency fNyquist fNyquist = ~ (I_6) The Nyquist frequency is the theoretically highest frequency of interest in the simulation for an integration time step Ar, amounting to 2 samples/cycle. From Figure 1.2, we can see that the interpolation error increases as the frequency increases. We may have errors in the order of almost 100% in magnitude as the frequency goes up to the highest frequency of interest in the simulation. 116 Figure 1.2 Frequency response ofthe ratio of the interpolated values to the exact values for the history voltage source In the zCable model, the total cable length is subdivided into a number of short segments to simulate the distributed nature of losses. The interpolation error in each segment can accumulate and lead to a severe error for the complete cable solution. 117 Appendix II Discretization Formulas for a Simplified Equivalent Circuit of the Ideal Line Section with Slower Propagation Mode As indicated in chapter 3, after introducing the concept of the "pi-circuit correction" the ideal line section for the slower propagation mode can be modelled as an ideal line section with shorter length plus a pi-circuit with constant L and C. In order to include this section into the system matrix, it is necessary to modify its discrete-time equivalent circuit (Figure II. 1) to a simplified circuit (Figure II.2) through Thevenin and Norton theorem manipulations, as shown in the following steps. Figure II. 1 Discrete-time equivalent circuit of the ideal line section with slower propagation mode where ^rnode . c n a r a c t e r i s t i c impedance in modal quantities emhAe(0>ekhde(l) '• history voltage sources of the ideal line section Rl,R(j: equivalent discretized resistances of the inductance and capacitance 118 ei (t), erji (r), eQ2 (0 : history voltage sources of the inductance and capacitance Step 1: Convert Z™odeand e^de(t) to a Norton equivalent circuit as in the following diagram. ^ mode k o V A — i o -0. m - O -' mode \modi hcffl -o~ R. e,(t) + -.0 I*. m - O + - o with mode/o, i mode _ ^mh \l) nmh V) m o d e hC\(j) = «Cl(0 C ec2(0 (II-1) C Step 2: Combine the parallel components h™d* (t),Zfod<i ,hci(t) and J? c, and the equivalent circuit becomes ^7 mode k o V A — i o -.0* m Z'. h'...,:"M'%t] •R, - O --O + v,„(0 - o 119 with -7 mode n ^'mode _ Lc ' KC c 7 mode + R, c * d e ( 0 - « d e ( 0 + ^ i (0 (II-2) (II-3) Step 3: Convert (t), Z c m o d e to a Thevenin equivalent representation as follows, with 'mode/.\ v'mode j ' m o d e / . N emh \t) = lc -nmh (0 (II-4) Step 4: Combine e™d<t(t),z'™de,RL,eL(t) and convert them to a Norton equivalent circuit as in the following diagram. o -g mode -AAAr-v*(0 O-7" modd -O m h"mrcl\ty •Rcv'"{t) - o 120 with "mode / .N ^ M - ^ s r ( I I - 5 ) Zlm0de =Zcmode + RL (II-6) ^Sode(o = i o d e (o-^(o (II-7) Step 5: Combine the last four parallel components hm™ode(t),Zcmode,hC2(t),Rc, a n d then convert the circuit to the final Thevenin equivalent representation shown in Figure II.2. £ mode k o W V + O -ekrdV) gin mode - A A / V - - O m mh it) v,„(0 - o Figure II.-2 Reduced discrete-time circuit for the ideal line section with slower propagation mode with "'modez-o. 7 " 'mode j , ' "mode/,\ emh V) = z c -nmh (0 y m o d e n '"mode _ z c ' % z ; m o d e + i ? c (II-8) (II-9) (11-10) 121 In order to obtain the discretization formulas for the simplified characteristic impedance Zc m o d e and history voltage source e w m o d e (t), we have to derive the formulas for Z ^ m o d e and hmfodQ(t) first. From equations (II-6) and (11-10), we have ^"mode _ ^'mode + j r j c c L 2 mode ^ C z™de + Rc Z c m o d e ^ c + Z™odeRL + £C£L vmode , D (11-11) and 'mode l4Tde(0- ci(0l+Ac2(0-1 /'mode z ;m o d e ^ T d e ( o - «i(oj+Ac2(o-z 'mode /'mode L C £d e ( 0 + ^ ci(0 1 /'mode eL(0 + hC2(t) Z 'mode z , 'mode 1 mode/A , 1 „ / A ^mode ""' v ' RC 1 ,s 1 e i(0 + — eC2(0 ^"mode ^ " ' Z™deRc+Z™deRL+RcRL „ mode/ A rtnode + z ™ o d e t f c + z ™ o d e ^ + i ? c . R / , *Cl(0 Z c m o d e + f l c eZ,(0 + -7T-^C2(0 z c m o d e ^ c + z r d e ^ + ^ ^ ' *C (11-12) 122 Substituting equations (II-11) and (11-12) into equations (II-8) and (II-9), we have y"mode = (ZfODSRC + Z™ODERL + RCRL)' rC (11-13) Z D „ " ' m o d e ^ _ Rc „ m o d e / A , Z C KC „ (T\ L D L D _ ( Z £ _ ± « C ) * C . , i ( 0 + ^ — . « ( , ) (11-14) Z D K C where Z D = R C + 2Z c m o d e /? c + Z?OD*RL + RCRL (11-15) Equations (11-13) and (11-14) are the discrete-time formulas for the characteristic impedance and history voltage source of the ideal line section with slower propagation mode and can be easily included in the system matrix. 123 Appendix III Discretization Formulas for a Parallel RL Circuit Consider Dommel's discrete-time representation [1,2] for a parallel RL circuit, shown i Figure III. 1 Ut) „ + + (I) (II) (HI) Figure III. 1 Discrete-time representation of a parallel RL circuit i i Where R , L : resistance and inductance per unit length of the RL circuit /: length of the RL circuit From Figure III. 1 (I), the system equations can be described as follows, v(f) = Vfl (0 = vL (0 /(/) = iR (0 + iL (0 iR(t) = From equation (III-2) R VL{l)=LiM = v{t) at Tdij(t) T d r . . . . . X1 T d r . . . v(r)n v(0 = L = L-[i(t)-iR (0] = I - [/(0 -dt dt dt R v(t)dt = Ld{i(t)-V-^-) K (III-l) (III-2) (III-3) 124 Taking the integral on both sides, equation (III-3) becomes \v(t)dt = L\d[i(t)-^] (IH-4) Using the trapezoidal with damping integration rule, equation (III-4) becomes [(l + a)v(t) + (l-a)v(t-At)]. A , = m ) _ i { t _ A 0 ] _ £ [ v ( 0 _ v ( , _ AO] (a + l + — ).v(t) = ^ [i(t)-i(t-At)] + (-^- + a-\)-v(t-At) R- At At R- at v(t) = 2L At 2L RAt -[i(t)-i(t-At)] + 2L RAt + a-l + a + l 1 2L R-At •v(t-At) + a + l 1 a-At At — + + — R 2L 2L •i(t) + ~ + R(a-\) At 2L At v(t - At) • 1 + R(a +1) 1 a-At At — + + — R 2L 2L • i(t - At) (III-5) For a parallel RL circuit, the impedance Z' (per unit length) of this RL circuit is given by 1 sR.' SK Z = 1 1 R S+P R sL i with s = jco (III-6) From equation (III-6), we have R=K ; L=K/P R = R l = Kl ; L = L • I = (K-l)/P Substituting equation (III-7) into (III-5), the voltage equation becomes v(0 = Req-i(t) + heq(t) (III-7) (III-8) 125 with Reg = equivalent resistance of the discretized parallel RL circuit heq (t) = equivalent history source of the discretized parallel RL circuit where 2K-1 R = L = lEd = M (Hi-9) V L + a ^ t + A t _ P^t + Pa-At + 2 l + p ( a + l ) R 2L 2L At and 2L + R(a-\) M O - •«~A'>" 1 a .A , A , ' " A ' ) — + R(a + 1) — + + — Ar R 2L 2L 2 „, „ 2K-I — + P(a-1) At . „ / - , _ A A Ai v(t-At)-- i(t-i) (III-10) 2 v ' 2 — + P(a +1) — + P(a +1) At At — + P(a-\) M : v(t-At)-Req-i{t-A) — + P(a + l) At — + P(a-\) 4f v(t-At)-veq(t-At) — + P(a + \) At — + P(a-l) 4/ v(t - At) - [v(t - At) - heq (t - At)} (III-11) — + P(pc + Y) At 126 

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