The electrical engineering handbook
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 © 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 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 (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 © 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 Z ii – g = self-impedance of phase i conductor and Z ij – g = mutual impedance between phase i conductor and phase j conductor. The subscript g indicates a ground return. Formulas for calculating Z ii – g and Z ij – 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 (61.2) (61.3) where the “prime” is used to indicate distributed parameters in per-unit length; z a = r c + jx i = conductor a internal impedance, W /mi; h a = height of conductor a , ft; r a = radius of conductor a , ft; d ab = distance between conductors a and b , ft; S ab = distance from one conductor to image of other, ft; w = 2 p f ; f = frequency, cycles/s; m 0 = the magnetic permeability of free space, m 0 = 4 p ´ 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 temper- ature 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 aag abg acg ang bag bbg bcg bng cag cbg ccg cng nag nbg ncg nng a b c n é ë ê ê ê ê ù û ú ú ú ú = é ë ê ê ê ê ê ù û ú ú ú ú ú é ë ê ê ê ê ù û ú –––– –––– –––– –––– úú ú ú + é ë ê ê ê ê ù û ú ú ú ú V V V V a b c n ¢ =+ + +Zzj h r pjq aag a a a – ln ( )w m p w m p 00 2 2 ¢ =++Zj S d pjq abg ab ab – ln ( )w m p w m p 00 2 â 2000 by CRC Press LLC R new = R 20 [1 + a ( T new 20)] (61.4) where R new = resistance at new temperature, T new = new temperature in C, R 20 = resistance at 20 C (Table 61.1), and a = temperature coefcient 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 ã m; r = radius of conductor, m; I 0 = modied Bessel function of the rst kind of order 0; I 1 = modied Bessel function of the rst 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 simplied 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 innite 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 Coefcient of Metal (Copper = 100) 20 C, W ã 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 Imr Imr = r p2 0 1 () () mj=mwr/ pk k k kk =++ ổ ố ỗ ử ứ ữ + ộ ở ờ ự ỷ ỳ + p qqqq qp q 8 1 32 16 06728 2 22 3 452 4 1536 2 34 cos . ln cos sin cos cos q k k kk k k =- + + - + -+ ổ ố ỗ ử ứ ữ + ộ ở ờ ự ỷ ỳ 00386 1 2 2 1 32 2 64 3 452 384 2 10895 4 4 23 4 . cos cos cos . cos sin ln ln q pq q qqq kD f =8565 10 4 . r © 2000 by CRC Press LLC where D = 2 h i (ft), q = 0, for self-impedance; D = S ij (ft), for mutual impedance (see Fig. 61.2 for q); and r = earth resistivity, W/m 3 . 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 d ij = distance between conductors i and j, h i = height of conductor i, S ij = distance from one conductor to the image of the other, r i = 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 C ii in per-unit length can be easily calculated by Eq. (61.12), and all C ij ’s are equal to zero. (61.12) where e 0 = absolute permittivity (dielectric constant of free space), e r = relative permittivity of cable insulation, r 1 = outside radius of conductor core, and r 2 = 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 =× é ë ê ê ê ù û ú ú ú = é ë ê ê ê ù û ú ú ú é ë ê ê ê ù û ú ú ú 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 é ë ê ê ê ù û ú ú ú = é ë ê ê ê ù û ú ú ú é ë ê ê ê ù û ú ú ú P lh r ii i i = 2 2 pe ln P l S d ij ij ij = 2pe ln C rr r = 2 0 21 pe e ln /() © 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 power- 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 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 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 as long-rod or post types. Use of synthetic insulators on transmission lines is relatively recent, and a few questions FIGURE 61.3Generalized conductor model. TABLE 61.2Standard 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 © 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 environ- 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 power transmission has been studied intensively by sponsors such as the U.S. Department of Energy. TABLE 61.3Typical 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 © 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 admit- tance 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] = [Z i ] + [Z p ] (61.15) [Y] = jw[P] –1 (61.16) [P] = [P i ] + [P p ] where [P] is a potential coefficient matrix. [Z i ] = single-core cable internal impedance matrix (61.17) [Z p ] = pipe internal impedance matrix dV dx ZI () –[]()=× dI dx YV () –[]()=× = ××× ××× ××× é ë ê ê ê ê ù û ú ú ú ú [][] [] [][] [] [] [] [] Z Z Z i i in 1 2 00 00 00 MMOM © 2000 by CRC Press LLC (61.18) The diagonal submatrix in [Z i ] 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 (61.19) where Z ssj = sheath self-impedance = Z sheath-out + Z sheath/pipe-insulation (61.20) Z csj = mutual impedance between the core and sheath = Z ssj – Z sheath-mutual (61.21) Z ccj = core self-impedance = (Z core + Z core/sheath-insulation + Z sheath-in ) + Z csj – Z sheath-mutual (61.22) where (61.23) (61.24) FIGURE 61.4A pipe-type cable system. FIGURE 61.5A single-core cable cross section. = ××× ××× ××× é ë ê ê ê ê ê ù û ú ú ú ú ú [][] [] [][][] [][] [] ZZ Z ZZ Z ZZ Z pp pn pp pn pn pn pnn 11 12 1 12 22 2 12 MMOM []Z ZZ ZZ ij ccj csj csj ssj = é ë ê ê ù û ú ú Z m r Imr Imr core = r p2 1 01 11 () () Z jr r core/sheath-insulation = wm p 12 1 2 ln â 2000 by CRC Press LLC (61.25) (61.26) (61.27) (61.28) where r = resistivity of conductor, D = I 1 (mr 3 )K 1 (mr 2 ) I 1 (mr 2 )K 1 (mr 3 ), g = Eulers constant = 1.7811, I i = modied Bessel function of the rst kind of order i, K i = modied Bessel function of the second kind of order i, and m = = reciprocal of the complex depth of penetration. A submatrix of [Z p ] is given in the following form: (61.29) Z pjk in Eq. (61.29) is the impedance between the jth and kth inner conductors with respect to the pipe inner surface. When j = k, Z pjk = Z pipe-in ; otherwise Z pjk 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 coefcient matrix of a pipe-type cable is similar to the impedance matrix. (61.32) Z m rD ImrKmr KmrImr sheath-in =+ r p2 2 0213 0213 [( ) ( ) ( )( )] Z rrD sheath-mutual = r p2 23 Z m rD ImrKmr KmrImr sheath-out =+ r p2 3 0312 0312 [( ) ( ) ( )( )] Z jqRd qR ii i sheath/pipe-insulation = +- ổ ố ỗ ử ứ ữ - wm p 0 1 222 22 cosh jwrm/ []Z ZZ ZZ pjk pjk pjk pjk pjk = ộ ở ờ ờ ự ỷ ỳ ỳ Z m q Kmq Kmq jd q Kmq n K mq mqK mq i n n rn n n pipe-in =+ ổ ố ỗ ử ứ ữ Â ộ ở ờ ờ ự ỷ ỳ ỳ = Ơ ồ r p wm pm2 0 1 2 1 () () () () () Z j q Smq Kmq Kmq dd q n Kmq n K mq mqK mq n pjk jk r jk n jk r n rn n n = + + ổ ố ỗ ử ứ ữ Â ộ ở ờ ự ỷ ỳ ỡ ớ ù ù ợ ù ù ỹ ý ù ù ỵ ù ù = Ơ ồ wm p m qm m 0 0 1 2 1 2 2 1 ln () () cos( ) () () () [] [ ] [] [] [] [ ] [] [] [] [ ] P P P P i i i in = ììì ììì ììì ộ ở ờ ờ ờ ờ ự ỷ ỳ ỳ ỳ ỳ 1 2 00 00 00 MMOM . the same current. One converter sets the current that will be common to all converters in the system. Except for the converter that sets the current, the. 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