Chen, M.S., Lai, K.C., Thallam, R.S., El-Hawary, M.E., Gross, C, Phadke, AG Gungor, R B, Glover, J D. Transmission The electrical Engineering Handbook Ed. Richard C. dorf Boca Raton CRC Press llc. 2000
Chen, M.S., Lai, K.C., Thallam, R.S., El-Hawary, M.E., Gross, C., Phadke, A.G., Gungor, R.B., Glover, J.D. “Transmission” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
61 Transmission 61.1 Alternating Current Overhead: Line Parameters Models, Standard Voltages, Insulators Line Parameters.Models. Standard Voltages.Insulators Mo-Shing Chen 61.2 Alternating Current Underground: Line Parameters, Models, Standard Voltages, Cables University of Texas at Arlington Cable parameters· Models· Standard Voltag K C. Lai 61.3 High-Voltage Direct-Current Transmission University of Texas at Arlington Configurations of DCTransmission. Economic Comparison of Ac and DC Transmission. Principles of Converter Rao s. hallam Operation· Converter Control· Developments Mohamed E. El-Hawary Series Capacitors. Synchronous Compensators. Shunt Capacitors. Shunt Reactors. Static VAR Compensators(SVC) Technical University of Nova Scotia 61.5 Fault Analysis in Power Syster Charles gross Simplifications in the System Model. The Four Basic Fault Auburn University Types.An Example Fault Study Further Considerations 61.6 Protection Arun g. phadke Fundamental Principles of Protection.Overcurrent Virginia Polytechnic Institute and ProtectionDistanceProtection.pilotProtection.computer State University Relaying R B. Gungo 61.7 Transient Operation of Power Systems University of South Alabama Stable Operation of Power Systems 61. 8 Planning Duncan glover Planning Tools. Basic Planning Principles. Equipment FaAAElectrical Corporation atings. Planning Criteria. Value-Based Transmission Planning 61.1 Alternating Current Overhead: Line Parameters, Models Standard Voltages, Insulators Mo-Shing Chen The most common element of a three-phase power system is the overhead transmission line. The interconnec tion of these elements forms the major part of the power system network. The basic overhead transmission lines consist of a group of phase conductors that transmit the electrical energy, the earth return, and usually one or more neutral conductors(Fi Line parameters The transmission line parameters can be divided into two parts: series impedance and shunt admittance. Since these values are subject to installation and utilization, e.g., operation frequency and distance between cables, the manufacturers are often unable to provide these data. The most accurate values are obtained through measuring in the field, but it has been done only occasionally. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 61 Transmission 61.1 Alternating Current Overhead: Line Parameters, Models, Standard Voltages, Insulators Line Parameters • Models • Standard Voltages • Insulators 61.2 Alternating Current Underground: Line Parameters, Models, Standard Voltages, Cables Cable Parameters • Models • Standard Voltages • Cable Standards 61.3 High-Voltage Direct-Current Transmission Configurations of DCTransmission • Economic Comparison of AC and DC Transmission • Principles of Converter Operation • Converter Control • Developments 61.4 Compensation Series Capacitors • Synchronous Compensators • Shunt Capacitors • Shunt Reactors • Static VAR Compensators (SVC) 61.5 Fault Analysis in Power Systems Simplifications in the System Model • The Four Basic Fault Types • An Example Fault Study • Further Considerations 61.6 Protection Fundamental Principles of Protection • Overcurrent Protection • Distance Protection • Pilot Protection • Computer Relaying 61.7 Transient Operation of Power Systems Stable Operation of Power Systems 61.8 Planning Planning Tools • Basic Planning Principles • Equipment Ratings • Planning Criteria • Value-Based Transmission Planning 61.1 Alternating Current Overhead: Line Parameters, Models, Standard Voltages, Insulators Mo-Shing Chen The most common element of a three-phase power system is the overhead transmission line. The interconnection of these elements forms the major part of the power system network. The basic overhead transmission lines consist of a group of phase conductors that transmit the electrical energy, the earth return, and usually one or more neutral conductors (Fig. 61.1). Line Parameters The transmission line parameters can be divided into two parts: series impedance and shunt admittance. Since these values are subject to installation and utilization, e.g., operation frequency and distance between cables, the manufacturers are often unable to provide these data. The most accurate values are obtained through measuring in the field, but it has been done only occasionally. Mo-Shing Chen University of Texas at Arlington K.C. Lai University of Texas at Arlington Rao S. Thallam Salt River Project, Phoenix Mohamed E. El-Hawary Technical University of Nova Scotia Charles Gross Auburn University Arun G. Phadke Virginia Polytechnic Institute and State University R.B. Gungor University of South Alabama J. Duncan Glover FaAAElectrical Corporation
vvv Ib Ig=Ia+Ib+Ic+In FIGURE 61.1 A three-phase transmission line with one FIGURE 61.2 Geometric diagr onductors a and b Though the symmetrical component method has been used to simplify many of the problems in power system analysis, the following Paragraphs, which describe the formulas in the calculation of the line parameters, are much more general and are not limited to the application of symmetrical components. The sequence impedances and admittances used in the symmetrical components method can be easily calculated by a matrix transformation[Chen and Dillon, 1974. A detailed discussion of symmetrical components can be found in Clarke(1943 Series Impedance The network equation of a three-phase transmission line with one neutral wire(as given in Fig. 61. 1)in which only series impedances are considered is given as follows: an-g V Zbc-g Zm-gIbLvb (61.1) vcIZa-g Zob-g Za-g Zmn-gIve 8 where Zi-g= self-impedance of phase i conductor and Zi-g= mutual impedance between phase i conductor and phase j conductor. The subscript g indicates a ground return. Formulas for calculating Zi-g and Zi-g were developed by J.R. Carson based on an earth of uniform conductivity and semi-infinite in extent [Carson, 1926]. For two con- ductors a and b with earth return, as shown in Fig. 61.2, the self-and mutual impedances in ohms per mile are jo In b+oo(p+jq) (61.3) where the"prime"is used to indicate distributed parameters in per-unit length; z,=r+ jx,= conductor a internal impedance, Q2/mi; h,=height of conductor a, ft; r,=radius of conductor a, ft; da,=distance between conductors a and b, ft; So,= distance from one conductor to image of other, ft; @=2Tff= frequency, cycles/s, Ho= the magnetic permeability of free space, Ho= 4T X 10-X 1609.34 H/mi; and P, g are the correction terms for earth return effect and are given later The conductor internal impedance consists of the effective resistance and the internal reactance. The effective resistance is affected by three factors: temperature, frequency, and current density In coping with the temper ature effect on the resistance, a correction can be applied. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Though the symmetrical component method has been used to simplify many of the problems in power system analysis, the following paragraphs, which describe the formulas in the calculation of the line parameters, are much more general and are not limited to the application of symmetrical components. The sequence impedances and admittances used in the symmetrical components method can be easily calculated by a matrix transformation [Chen and Dillon, 1974]. A detailed discussion of symmetrical components can be found in Clarke [1943]. Series Impedance The network equation of a three-phase transmission line with one neutral wire (as given in Fig. 61.1) in which only series impedances are considered is given as follows: (61.1) where Zii–g = self-impedance of phase i conductor and Zij–g = mutual impedance between phase i conductor and phase j conductor. The subscript g indicates a ground return. Formulas for calculating Zii–g and Zij–g were developed by J. R. Carson based on an earth of uniform conductivity and semi-infinite in extent [Carson, 1926]. For two conductors a and b with earth return, as shown in Fig. 61.2, the self- and mutual impedances in ohms per mile are (61.2) (61.3) where the “prime” is used to indicate distributed parameters in per-unit length; za = rc + jxi = conductor a internal impedance, W/mi; ha = height of conductor a, ft; ra = radius of conductor a, ft; dab = distance between conductors a and b, ft; Sab = distance from one conductor to image of other, ft; w = 2pf; f = frequency, cycles/s; m0 = the magnetic permeability of free space, m0= 4p ¥ 10–7 ¥ 1609.34 H/mi; and p, q are the correction terms for earth return effect and are given later. The conductor internal impedance consists of the effective resistance and the internal reactance. The effective resistance is affected by three factors: temperature, frequency, and current density. In coping with the temperature effect on the resistance, a correction can be applied. FIGURE 61.1 A three-phase transmission line with one neutral wire. FIGURE 61.2 Geometric diagram of conductors a and b. V V V V ZZZZ ZZZZ ZZZZ ZZZZ I I I I A B C N aa g ab g ac g an g ba g bb g bc g bn g ca g cb g cc g cn g na g nb g nc g nn g a b c n È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ = È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ È Î Í Í Í Í ˘ ˚ ˙ – – – – – – – – – – – – – – – – ˙ ˙ ˙ + È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ V V V V a b c n Z ¢ = + z j + + h r p jq aa g a a a – w ln ( ) m p w m p 0 0 2 2 Z j ¢ = + + S d p jq ab g ab ab – w ln ( ) m p w m p 0 0 2
TABLE 61.1 Electrical Properties of Metals Used in Transmission Lines Electrical Metal ( Copper=100)20g·m(10-5) Resistance (per℃C) Copper(HC, annealed) 0.0039 Copper(HC, hard-drawn Aluminum (EC grade, 1/2 H-H Mild steel 13.80 0.0040 Rnew= Roo[1+ a(Tnew-20)] 61.4 where Rnew=resistance at new temperature, Tnew=new temperature inC, R2oo=resistance at 20oC (Table 61.1), and a= temperature coefficient of resistance(Table 61.1) An increase in frequency causes nonuniform current density. This phenomenon is called skin effect. Skin effect increases the effective ac resistance of a conductor and decreases its internal inductance. The internal dance of a solid round conductor in ohms per meter considering the skin effect is calculated by 2r I,(mr where p=resistivity of conductor, Q2. m;r=radius of conductor, m; Io=modified Bessel function of the first kind of order 0; I ,= modified Bessel function of the first kind of order 1; and m= vjou/p= reciprocal of complex depth of penetration The ratios of effective ac resistance to dc resistance for commonly used conductors are given in many handbooks [such as Electrical Transmission and Distribution Reference Book and Aluminum Electrical Conductor Handbook]. A simplified formula is also given in Clarke[ 1943] p and q are the correction terms for earth return effect. For perfectly conducting ground, they are zero. The determination of p and q requires the evaluation of an infinite integral. Since the series converge fast at power frequency or less, they can be calculated by the following equations k cos e 0.6728+In=cos 20+0 sin 20 k cos 30 k cos 40 45√2 153 q=-0.0386+mn21 k- cos 20 k cos 30 k cos e- √2 ln2+1.0895|cos40+θsin40 384 k=8.565×10-4D
© 2000 by CRC Press LLC Rnew = R20°[1 + a (Tnew – 20)] (61.4) where Rnew = resistance at new temperature, Tnew = new temperature in °C, R20° = resistance at 20°C (Table 61.1), and a = temperature coefficient of resistance (Table 61.1). An increase in frequency causes nonuniform current density. This phenomenon is called skin effect. Skin effect increases the effective ac resistance of a conductor and decreases its internal inductance. The internal impedance of a solid round conductor in ohms per meter considering the skin effect is calculated by (61.5) where r = resistivity of conductor, W · m; r = radius of conductor, m; I0 = modified Bessel function of the first kind of order 0; I1 = modified Bessel function of the first kind of order 1; and = reciprocal of complex depth of penetration. The ratios of effective ac resistance to dc resistance for commonly used conductors are given in many handbooks [such as Electrical Transmission and Distribution Reference Book and Aluminum Electrical Conductor Handbook]. A simplified formula is also given in Clarke [1943]. p and q are the correction terms for earth return effect. For perfectly conducting ground, they are zero. The determination of p and q requires the evaluation of an infinite integral. Since the series converge fast at power frequency or less, they can be calculated by the following equations: (61.6) (61.7) with TABLE 61.1 Electrical Properties of Metals Used in Transmission Lines Relative Electrical Temperature Conductivity Resistivity at Coefficient of Metal (Copper = 100) 20°C, W · m (10–8) Resistance (per °C) Copper (HC, annealed) 100 1.724 0.0039 Copper (HC, hard-drawn) 97 1.777 0.0039 Aluminum (EC grade, 1/2 H-H) 61 2.826 0.0040 Mild steel 12 13.80 0.0045 Lead 8 21.4 0.0040 z m r I mr I mr = r 2p 0 1 ( ) ( ) m = jwm/r p k k k k k = + + Ê Ë Á ˆ ¯ ˜ + È Î Í ˘ ˚ ˙ + p q q q q q p q 8 1 3 2 16 0 6728 2 2 2 3 45 2 4 1536 2 3 4 – cos . ln cos sin cos – cos q k k k k k k = - + + - + - + Ê Ë Á ˆ ¯ ˜ + È Î Í ˘ ˚ ˙ 0 0386 1 2 2 1 3 2 2 64 3 45 2 384 2 1 0895 4 4 2 3 4 . cos cos cos . cos sin ln ln q p q q q q q k D f = 8 565 ¥ 10 4 . – r
where D- 2h;(ft),0=0, for self-impedance; D=S,(ft), for mutual impedance(see Fig 61.2 for 0); and Shunt admittance The shunt admittance consists of the conductance and the capacitive susceptance. The conductance of a transmission line is usually very small and is neglected in steady-state studies. A capacitance matrix related to phase voltages and charges of a three-phase transmission line is Caa -Cab -Cac V Qabc Cabc. Vabc or Q= -Cba Cbb (61.8) The capacitance matrix can be calculated by inverting a potential coefficient matrix. Qabc= Pabc· Abc or abc=Pabc·Qabc Paa Pab Pac lQ Pa Pbb Pu ll Qu VPa Pc pQ P In In where d;=distance between conductors i and j, h,=height of conductor i, Si,= distance from one conductor to the image of the other, r; = radius of conductor i, E= permittivity of the medium surrounding the conductor, and 1= length of conductor. Though most of the overhead lines are bare conductors, aerial cables may consist of cable with shielding tape or sheath. For a single-core conductor with its sheath grounded, the capacitance Ci in per-unit length can be easily calculated by Eq (61. 12), and all Cis are equal to zero 61.12 In(,/r) where E,= absolute permittivity (dielectric constant of free space), E,- relative permittivity of cable insulation r= outside radius of conductor core, and r= inside radius of conductor sheath. Models In steady-state problems, three-phase transmission lines are represented by lumped-Tt equivalent networks, series resistances and inductances between buses are lumped in the middle, and shunt capacitances of the c 2000 by CRC Press LLC
© 2000 by CRC Press LLC where D= 2hi (ft), q = 0, for self-impedance; D = Sij (ft), for mutual impedance (see Fig. 61.2 for q); and r = earth resistivity, W/m3 . Shunt Admittance The shunt admittance consists of the conductance and the capacitive susceptance. The conductance of a transmission line is usually very small and is neglected in steady-state studies. A capacitance matrix related to phase voltages and charges of a three-phase transmission line is (61.8) The capacitance matrix can be calculated by inverting a potential coefficient matrix. Qabc = Pabc–1 · Vabc or Vabc = Pabc · Qabc or (61.9) (61.10) (61.11) where dij = distance between conductors i and j, hi = height of conductor i, Sij = distance from one conductor to the image of the other, ri = radius of conductor i, e = permittivity of the medium surrounding the conductor, and l = length of conductor. Though most of the overhead lines are bare conductors, aerial cables may consist of cable with shielding tape or sheath. For a single-core conductor with its sheath grounded, the capacitance Cii in per-unit length can be easily calculated by Eq. (61.12), and all Cij’s are equal to zero. (61.12) where e0 = absolute permittivity (dielectric constant of free space), er = relative permittivity of cable insulation, r1 = outside radius of conductor core, and r2 = inside radius of conductor sheath. Models In steady-state problems, three-phase transmission lines are represented by lumped-p equivalent networks, series resistances and inductances between buses are lumped in the middle, and shunt capacitances of the Qabc Cabc Vabc Q Q Q CCC CCC CCC V V V a b c aa ab ac ba bb bc ca cb cc a b c = × È Î Í Í Í ˘ ˚ ˙ ˙ ˙ = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ È Î Í Í Í ˘ ˚ ˙ ˙ ˙ or – – – – – – V V V PPP PPP PPP Q Q Q a b c aa ab ac ba bb bc ca cb cc a b c È Î Í Í Í ˘ ˚ ˙ ˙ ˙ = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ È Î Í Í Í ˘ ˚ ˙ ˙ ˙ P l h r ii i i = 2 2 pe ln P l S d ij ij ij = 2pe ln C r r r = 2 0 2 1 pe e ln / ( )
Impedance /2 Shunt /2 Shunt Admittance FIGURE 61.3 Generalized conductor model TABLE 61.2 Standard System Voltage, kv Category 4.5 15的2 tra-high voltage(EHV) 345 (principally in Europe) 765 Ultra-high voltage(UHV) 1100 transmission lines are divided into two halves and lumped at buses connecting the lines(Fig. 61.3). More discussion on the transmission line models can be found in El-Hawary [1995 Standard Voltages Standard transmission voltages are established in the United States by the American National Standards Institute (ANSI). There is no clear delineation between distribution, subtransmission, and transmission voltage levels Table 61.2 shows the standard voltages listed in ANSI Standard C84 and C92. 2, all of which are in use at present. Insulators The electrical operating performance of a transmission line depends primarily on the insulation Insulators not only must have sufficient mechanical strength to support the greatest loads of ice and wind that may be reasonably expected, with an ample margin, but must be so designed to withstand severe mechanical abuse, lightning, and power arcs without mechanically failing. They must prevent a flashover for practically any frequency operation condition and many transient voltage conditions, under any conditions of humidity, temperature, rain, or snow, and with accumulations of dirt, salt, and other contaminants which are not periodically washed off by The majority of present insulators are made of glazed porcelain Porcelain is a ceramic product obtained by the high-temperature vitrification of clay, finely ground feldspar, and silica. Porcelain insulators for transmission may be disks, posts, or long-rod types. Glass insulators have been used on a significant proportion of trans- mission lines. These are made from toughened glass and are usually clear and colorless or light green. For transmission voltages they are available only as disk types. Synthetic insulators are usually manufactured long-rod or post types. Use of synthetic insulators on transmission lines is relatively recent, and a few questions c 2000 by CRC Press LLC
© 2000 by CRC Press LLC transmission lines are divided into two halves and lumped at buses connecting the lines (Fig. 61.3). More discussion on the transmission line models can be found in El-Hawary [1995]. Standard Voltages Standard transmission voltages are established in the United States by the American National Standards Institute (ANSI). There is no clear delineation between distribution, subtransmission, and transmission voltage levels. Table 61.2 shows the standard voltages listed in ANSI Standard C84 and C92.2, all of which are in use at present. Insulators The electrical operating performance of a transmission line depends primarily on the insulation. Insulators not only must have sufficient mechanical strength to support the greatest loads of ice and wind that may be reasonably expected, with an ample margin, but must be so designed to withstand severe mechanical abuse, lightning, and power arcs without mechanically failing. They must prevent a flashover for practically any powerfrequency operation condition and many transient voltage conditions, under any conditions of humidity, temperature, rain, or snow, and with accumulations of dirt, salt, and other contaminants which are not periodically washed off by rains. The majority of present insulators are made of glazed porcelain. Porcelain is a ceramic product obtained by the high-temperature vitrification of clay, finely ground feldspar, and silica. Porcelain insulators for transmission may be disks, posts, or long-rod types. Glass insulators have been used on a significant proportion of transmission lines. These are made from toughened glass and are usually clear and colorless or light green. For transmission voltages they are available only as disk types. Synthetic insulators are usually manufactured as long-rod or post types. Use of synthetic insulators on transmission lines is relatively recent, and a few questions FIGURE 61.3 Generalized conductor model. TABLE 61.2 Standard System Voltage, kV Rating Category Nominal Maximum 34.5 36.5 46 48.3 69 72.5 115 121 138 145 161 169 230 242 Extra-high voltage (EHV) 345 362 400 (principally in Europe) 500 550 765 800 Ultra-high voltage (UHV) 1100 1200
TABLE 61.3 Typical Line Insulation Line Voltage, kV No of Standard Disks Controlling Parameter(Typical Lightning or contamination 2306 Lightning, switching surge, or contamination 24-26 Switching surge or contamination Switching surge or contamination about their use are still under study. Improvements in design and manufacture in recent years have made synthetic insulators increasingly attractive since the strength-to-weight ratio is significantly higher than that of porcelain and can result in reduced tower costs, especially on EHV and UHV transmission lines. NEMA Publication"High Voltage Insulator Standard"and AIEE Standard 41 have been combined in ANSI Standards C29.1 through C29. 9. Standard C29.1 covers all electrical and mechanical tests for all types of insulators. The standards for the various insulators covering flashover voltages(wet, dry, and impulse; radio influence; leakage distance; standard dimensions; and mechanical-strength characteristics)are addressed. These standards should be consulted when specifying or purchasing insulators The electrical strength of line insulation may be determined by power frequency, switching surge, or lightning typical line insulation levels and the controlling ges, different parameters tend to dominate. Table 61.3 shows performance requirements. At different li Defining Term Surge impedance loading(SIL): The surge impedance of a transmission line is the characteristic impedance with resistance set to zero(resistance is assumed small compared to reactance). The power that flows in a lossless transmission line terminated in a resistive load equal to the lines surge impedance is denoted as the surge impedance loading of the line. Related Topics 3.5 Three-Phase Circuits 55.2 Dielectric Losses References Aluminum Electrical Conductor Handbook, 2nd ed. Aluminum Association, 1982. J.R. Carson, Wave propagation in overhead wires with ground return, " Bell System Tech J, vol 5, Pp. 539-554 S. Chen and w. E. Dillon,"Power system modeling, Proc. IEEE, vol 93, no. 7, Pp. 901-915, 1974. E. Clarke, Circuit Analysis of A-C Power Systems, vols. I and 2, New York: Wiley, 1943. Electrical Transmission and Distribution Reference Book, Central Station Engineers of the Westinghouse Electric Corporation, East Pittsburg .E. El-Hawary, Electric Power Systems: Design and Analysis, revised edition, Piscataway, N J. IEEE Press, 199 Further Information Other recommended publications regarding EHV transmission lines include Transmission Line Refe 345 and Above 2nd ed. 1982. from Electric Power Research Institute, Palo Alto, Calif and the ieee Group on Insulator Contamination publication "Application guide for insulators in a contaminated ment, IEEE Trans. Power Appar Syst, September/October 1979 Research on higher voltage levels has been conducted by several organizations: Electric Power Research Institute, Bonneville Power Administration, and others. The use of more than three phases for electric transmission has been studied intensively by sponsors such as the U.S. Department of Energy c 2000 by CRC Press LLC
© 2000 by CRC Press LLC about their use are still under study. Improvements in design and manufacture in recent years have made synthetic insulators increasingly attractive since the strength-to-weight ratio is significantly higher than that of porcelain and can result in reduced tower costs, especially on EHV and UHV transmission lines. NEMA Publication “High Voltage Insulator Standard” and AIEE Standard 41 have been combined in ANSI Standards C29.1 through C29.9. Standard C29.1 covers all electrical and mechanical tests for all types of insulators. The standards for the various insulators covering flashover voltages (wet, dry, and impulse; radio influence; leakage distance; standard dimensions; and mechanical-strength characteristics) are addressed. These standards should be consulted when specifying or purchasing insulators. The electrical strength of line insulation may be determined by power frequency, switching surge, or lightning performance requirements. At different line voltages, different parameters tend to dominate. Table 61.3 shows typical line insulation levels and the controlling parameter. Defining Term Surge impedance loading (SIL): The surge impedance of a transmission line is the characteristic impedance with resistance set to zero (resistance is assumed small compared to reactance). The power that flows in a lossless transmission line terminated in a resistive load equal to the line’s surge impedance is denoted as the surge impedance loading of the line. Related Topics 3.5 Three-Phase Circuits • 55.2 Dielectric Losses References Aluminum Electrical Conductor Handbook, 2nd ed., Aluminum Association, 1982. J. R. Carson, “Wave propagation in overhead wires with ground return,” Bell System Tech. J., vol. 5, pp. 539–554, 1926. M. S. Chen and W. E. Dillon, “Power system modeling,” Proc. IEEE, vol. 93, no. 7, pp. 901–915, 1974. E. Clarke, Circuit Analysis of A-C Power Systems, vols. 1 and 2, New York: Wiley, 1943. Electrical Transmission and Distribution Reference Book, Central Station Engineers of the Westinghouse Electric Corporation, East Pittsburgh, Pa. M. E. El-Hawary, Electric Power Systems: Design and Analysis, revised edition, Piscataway, N.J.: IEEE Press, 1995. Further Information Other recommended publications regarding EHV transmission lines include Transmission Line Reference Book, 345 kV and Above, 2nd ed., 1982, from Electric Power Research Institute, Palo Alto, Calif., and the IEEE Working Group on Insulator Contamination publication “Application guide for insulators in a contaminated environment,” IEEE Trans. Power Appar. Syst., September/October 1979. Research on higher voltage levels has been conducted by several organizations: Electric Power Research Institute, Bonneville Power Administration, and others. The use of more than three phases for electric power transmission has been studied intensively by sponsors such as the U.S. Department of Energy. TABLE 61.3 Typical Line Insulation Line Voltage, kV No. of Standard Disks Controlling Parameter (Typical) 115 7–9 Lightning or contamination 138 7–10 Lightning or contamination 230 11–12 Lightning or contamination 345 16–18 Lightning, switching surge, or contamination 500 24–26 Switching surge or contamination 765 30–37 Switching surge or contamination
61.2 Alternating Current Underground: Line Parameters, Models Standard Voltages. Cables Mo-Shing Chen and K.C. Lai Although the capital costs of an underground power cable are usually several times those of an overhead line ual capacity, installation of undergroun is continuously increasing for reasons of safety, security reliability, aesthetics, or availability of right-of-way. In heavily populated urban areas, underground cable ystems are mostly preferred. Two types of cables are commonly used at the transmission voltage level: pipe-type cables and self-contained oil-filled cables. The selection depends on voltage, power requirements, length, cost, and reliability. In the United States, over 90% of underground cables are pipe-type design. Cable parameters A general formulation of impedance and admittance of single-core coaxial and pipe-type cables was proposed by Prof. Akihiro Ametani of Doshisha University in Kyoto, Japan [Ametani, 1980]. The impedance and adm tance of a cable system are defined in the two matrix equations d(v) =-[Z]·(D) 61.13) d(D)=-y]·(V) 61.14) where(V)and(n) are vectors of the voltages and currents at a distance x along the cable and [z and [y are square matrices of the impedance and admittance. For a pipe-type cable, shown in Fig. 61.4, the impedance and admittance matrices can be written as Eqs. (61. 15)and(61. 16) by assuming: 1. The displacement currents and dielectric losses are negligible. 2. Each conducting medium of a cable has constant permeability. 3. The pipe thickness is greater than the penetration depth of the pipe wall. [Z=[Z]+[zp (61.15) [Y=jo[P]-1 (61.16) [P]=[P]+[P where [P] is a potential coefficient matrix. [Zl= single-core cable internal impedance matrix Za O 0] [Zil (61.17) [O][0 [ZI I impedance mat c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 61.2 Alternating Current Underground: Line Parameters, Models, Standard Voltages, Cables Mo-Shing Chen and K.C. Lai Although the capital costs of an underground power cable are usually several times those of an overhead line of equal capacity, installation of underground cable is continuously increasing for reasons of safety, security, reliability, aesthetics, or availability of right-of-way. In heavily populated urban areas, underground cable systems are mostly preferred. Two types of cables are commonly used at the transmission voltage level: pipe-type cables and self-contained oil-filled cables. The selection depends on voltage, power requirements, length, cost, and reliability. In the United States, over 90% of underground cables are pipe-type design. Cable Parameters A general formulation of impedance and admittance of single-core coaxial and pipe-type cables was proposed by Prof. Akihiro Ametani of Doshisha University in Kyoto, Japan [Ametani, 1980]. The impedance and admittance of a cable system are defined in the two matrix equations (61.13) (61.14) where (V) and (I) are vectors of the voltages and currents at a distance x along the cable and [Z] and [Y] are square matrices of the impedance and admittance. For a pipe-type cable, shown in Fig. 61.4, the impedance and admittance matrices can be written as Eqs. (61.15) and (61.16) by assuming: 1. The displacement currents and dielectric losses are negligible. 2. Each conducting medium of a cable has constant permeability. 3. The pipe thickness is greater than the penetration depth of the pipe wall. [Z] = [Zi ] + [Zp] (61.15) [Y] = jw[P]–1 (61.16) [P] = [Pi ] + [Pp] where [P] is a potential coefficient matrix. [Zi ] = single-core cable internal impedance matrix (61.17) [Zp] = pipe internal impedance matrix d V dx Z I ( ) = × –[ ] ( ) d I dx Y V ( ) = × –[ ] ( ) = ××× ××× ××× È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Z Z Z i i in 1 2 0 0 0 0 0 0 M M O M
FIGuRE 61.4 A pipe-type cable system FIGURE 61.5 A single-core cable cross section. mll…pl (61.18) Lz21z1…zn The diagonal submatrix in [Z] expresses the self-impedance matrix of a single-core cable. When a single core cable consists of a core and sheath( Fig. 61.5), the self-impedance matrix is given by [Z= where Zssi= sheath self-impedance sheath-outt sheath/pipe-insulation (61.20) Zsi= mutual impedance between the core and sheath 'sheath-mutual (61.21) Zoi= core self-impedance (Zcore Zcorelsheath-insulation Zsheath-in )+ Z-Z 'sheath-mutu (61.2) where pm Io(mr) I1(m)
© 2000 by CRC Press LLC (61.18) The diagonal submatrix in [Zi ] expresses the self-impedance matrix of a single-core cable. When a singlecore cable consists of a core and sheath (Fig. 61.5), the self-impedance matrix is given by (61.19) where Zssj = sheath self-impedance = Zsheath-out + Zsheath/pipe-insulation (61.20) Zcsj = mutual impedance between the core and sheath = Zssj – Zsheath-mutual (61.21) Zccj = core self-impedance = (Zcore + Zcore/sheath-insulation + Zsheath-in) + Zcsj – Zsheath-mutual (61.22) where (61.23) (61.24) FIGURE 61.4 A pipe-type cable system. FIGURE 61.5 A single-core cable cross section. = ××× ××× ××× È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Z Z Z Z Z Z Z Z Z p p p n p p p n p n p n pnn 11 12 1 12 22 2 1 2 M M O M [Z ] Z Z Z Z ij ccj csj csj ssj = È Î Í Í ˘ ˚ ˙ ˙ Z m r I mr I mr core = r 2p 1 0 1 1 1 ( ) ( ) Z j r r core/sheath-insulation = wm p 1 2 1 2 ln
sheath-in [o(mr2)K,(mr,)+ Ko(mr,)I(mr) (61.25) sheath-mutual (61.26) 2Tr,,D Lo(mr3)K,(mr,)+ Ko(mr)I(mr2) (61.27) joHo. cos -{+R-4 sheath/pipe-insulation (61.28) where p= resistivity of conductor, D=I(mr3)K,(mT2)-I(mr2)K,(mr3),Y= Eulers constant =1.7811, I modified Bessel function of the first kind of order K= modified Bessel function of the second kind of order jou/p= reciprocal of the complex depth of penetration A submatrix of [Zp] is given in the following form ZniI (61.29) Zpit in Eq(61.29)is the impedance between the jth and kth inner conductors with respect to the pipe inn surface. When j= k, Zpik= Zpipe-in otherwise Zpit is given in Eq (61.31) =pmkm)+y‖ Kn(mq) (61.30) 2πaK,(m q)nu, Kn(mq)-mqKn(mq) In 4+H Ko( mq) K(mq) (61.31) d k cos(nek) 2u nu, K, (mq)-mgkn(mg) n where q is the inside radius of the pipe(Fig. 61.4) The formulation of the potential coefficient matrix of a pipe-type cable is similar to the impedance matrix. 0][P2l 00:P c 2000 by CRC Press LLC
© 2000 by CRC Press LLC (61.25) (61.26) (61.27) (61.28) where r = resistivity of conductor, D = I1(mr3)K1(mr2) – I1(mr2)K1(mr3), g = Euler’s constant = 1.7811, Ii = modified Bessel function of the first kind of order i, Ki = modified Bessel function of the second kind of order i, and m = = reciprocal of the complex depth of penetration. A submatrix of [Zp] is given in the following form: (61.29) Zpjk in Eq. (61.29) is the impedance between the jth and kth inner conductors with respect to the pipe inner surface. When j = k, Zpjk = Zpipe-in; otherwise Zpjk is given in Eq. (61.31). (61.30) (61.31) where q is the inside radius of the pipe (Fig. 61.4). The formulation of the potential coefficient matrix of a pipe-type cable is similar to the impedance matrix. (61.32) Z m r D I mr K mr K mr I mr sheath-in = + r 2p 2 0 2 1 3 0 21 3 [ ( ) ( ) ( ) ( )] Z rrD sheath-mutual = r 2p 2 3 Z m r D I mr K mr K mr I mr sheath-out = + r 2p 3 0 3 1 2 0 31 2 [ ( ) ( ) ( ) ( )] Z j qRd qR i i i sheath/pipe-insulation = Ê + - Ë Á ˆ ¯ ˜ wm - p 0 1 2 22 2 2 cosh jw r m/ [ ] Z Z Z Z Z pjk pjk pjk pjk pjk = È Î Í Í ˘ ˚ ˙ ˙ Z m q K mq K mq j d q K mq n K mq mqK mq i n n n rn n pipe-in = + Ê Ë Á ˆ ¯ ˜ ¢ È Î Í Í ˘ ˚ ˙ ˙ = • Â r p wm 2 p m 0 1 2 1 ( ) ( ) ( ) ( )– ( ) Z j q S mq K mq K mq d d q n K mq n K mq mqK mq n pjk jk r j k n jk r n n rn n = + + Ê Ë Á ˆ ¯ ˜ ¢ È Î Í ˘ ˚ ˙ Ï Ì Ô Ô Ó Ô Ô ¸ ˝ Ô Ô ˛ Ô Ô = • Â wm p m q m m 0 0 1 2 1 2 2 1 ln ( ) ( ) cos( ) ( ) ( )– ( ) – [ ] [ ] [] [] [] [ ] [] [] [] [ ] P P P P i i i in = ××× ××× ××× È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ 1 2 0 0 0 0 0 0 M MOM
