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 / ( )