2.9 SHORT-CIRCUIT MVA AND EQUIVALENT IMPEDANCE
2.9.2.1 If Single-Phase Short-Circuit MVA Is Already Known
Then the single-phase-to-ground short-circuit fault current can be determined from
If kV
f
( ) ( )
( )
SLG SLG
L L
MVA A
=
−
1000 3
SC (2.131)
But,
If(SLG) = Ia0 + Ia1 + Ia2 (2.132)
or
I V
Z Z Z
f(SLG)= L N
+ −+ 3
0 1 2
(2.133)
or
I V
f Z
G (SLG)=3 L N−
(2.134)
where
ZG = Z0 + Z1 + Z2 (2.135)
From Equations 2.125c and 2.127,
Z kV
G
f
= 3 L L2−
MVA (SLG)SC Ω (2.136)
and
ZG B
f
= 3MVA
MVA pu
(SLG)SC
(2.137) From
Z0 = ZG − Z1 − Z2 (2.138)
or in general,
X0 = XG − X1 − X2 (2.139)
since the resistance involved is usually very small with respect to the associated reactance value.
EXAMPLE 2.7
A short-circuit (fault) study shows that at bus 15 in a 132-kV system, on a 100 MVA base, short- circuit MVA is 710 MVA and the single-line-to-ground short-circuit MVA is 825 MVA. Determine the following:
(a) The positive and negative reactances of the system (b) The XG of the system
(c) The zero-sequence reactance of the system
Solution
(a) The positive- and negative-reactances of the system are
X1=X2=100 MVA=
710 MVA 0.1408 pu (b) The XG of the system is
XG=300 = 825MVA 0 3636
MVA . pu
(c) The zero-sequence of the system is
X0 = 0.3636 − 0.1408 = 0.2228 pu Note that, all values above are on a 100-MVA, 132-kV base.
REFERENCES
1. Elgerd, O. I., Electric Energy Systems Theory: An Introduction. McGraw-Hill, New York, 1971.
2. Institute of Electrical and Electronics Engineers, Graphic Symbols for Electrical and Electronics Diagrams, IEEE Std. 315-1971 [or American National Standards Institute (ANSI) Y32.2-1971]. IEEE, New York, 1971.
3. Stevenson, W. D., Elements of Power System Analysis, 4th ed. McGraw-Hill, New York, 1981.
4. Anderson, P. M., Analysis of Faulted Power Systems. Iowa State Univ. Press, Ames, IA, 1973.
GENERAL REFERENCES
AIEE Standards Committee Report, Electr. Eng. (Am. Inst. Electr. Eng.) 65, 512 (1946).
Clarke, E., Circuit Analysis of AC Power Systems, Vol. 1, General Electric Co., Schenectady, New York, 1943.
Eaton, J. R., Electric Power Transmission Systems. Prentice-Hall, Englewood Cliffs, NJ, 1972.
Elgerd, O. I., Basic Electric Power Engineering. Addison-Wesley, Reading, MA, 1977.
Gönen, T., Electric Power Distribution System Engineering, 2nd ed., CRC Press, Boca Raton, FL, 2008.
Gross, C. A., Power System Analysis. Wiley, New York, 1979.
Gross, E. T. B., and Gulachenski, E. M., Experience of the New England Electric Company with generator protection by resonant neutral grounding. IEEE Trans. Power Appar. Syst. PAS-92 (4) 1186–1194 (1973).
Neuenswander, J. R., Modern Power Systems. International Textbook Co., New York, 1971.
Nilsson, J. W., Introduction to Circuits, Instruments, and Electronics. Harcourt, Brace & World, New York, 1968.
Skilling, H. H., Electrical Engineering Circuits, 2nd ed. Wiley, New York, 1966.
Travis, I., Per unit quantities. Trans. Am. Inst. Electr. Eng. 56, 143–151 (1937).
Wagner, C. F., and Evans, R. D., Symmetrical Components. McGraw-Hill, New York, 1933.
Weedy, B. M., Electric Power Systems, 3rd ed. Wiley, New York, 1979.
Zaborsky, J., and Rittenhouse, J. W., Electric Power Transmission. Rensselaer Bookstore, Troy, New York, 1969.
PROBLEMS
1. Assume that the impedance of a line connecting buses 1 and 2 is 50∠90° Ω and that the bus voltages are 7560∠10° and 7200∠0° V per phase, respectively. Determine the following:
(a) Real power per phase that is being transmitted from bus 1 to bus 2 (b) Reactive power per phase that is being transmitted from bus 1 to bus 2 (c) Complex power per phase that is being transmitted
2. Solve Problem 1 assuming that the line impedance is 50∠26° Ω/phase.
3. Verify the following equations:
(a) Vpu(L–N) = Vpu(L–L)
(b) VApu(1ϕ) = VApu(3ϕ)
(c) Zpu(Y) = Zpu(Δ)
4. Verify the following equations:
(a) Equation 2.24 for a single-phase system (b) Equation 2.80 for a three-phase system 5. Show that ZB(Δ) = 3ZB(Y).
6. Consider two three-phase transmission lines with different voltage levels that are located side by side in close proximity. Assume that the bases of VAB, VB(1), and IB(1) and the bases of VAB, VB(2), and IB(2) are designated for the first and second lines, respectively. If the mutual reactance between the lines is XmΩ, show that this mutual reactance in per unit can be expressed as
X X
kV kV
m m B
B B
pu( )
(1) (2)
= (physical ) MVA
[ [ ]
7. Consider Example 2.3 and assume that the transformer is connected in delta–wye. Use a 25-MVA base and determine the following:
(a) New line-to-line voltage of low-voltage side (b) New low-voltage side base impedance (c) Turns ratio of windings
(d) Transformer reactance referred to the low-voltage side in ohms (e) Transformer reactance referred to the low-voltage side in per units 8. Verify the following equations:
(a) Equation 2.92 (b) Equation 2.93 (c) Equation 2.94 (d) Equation 2.96
9. Verify the following equations:
(a) Equation 2.100 (b) Equation 2.102
10. Consider the one-line diagram given in Figure P2.1. Assume that the three-phase trans- former T1 has nameplate ratings of 15,000 kVA, 7.97/13.8Y − 69Δ kV with leakage imped- ance of 0.01 + j0.08 pu based on its ratings, and that the three-phase transformer T2 has nameplate ratings of 1500 kVA, 7.97Δ kV − 277/480Y V with leakage impedance of 0.01 + j0.05 pu based on its ratings. Assume that the three-phase generator G1 is rated 10/12.5 MW/MVA, 7.97/13.8Y kV with an impedance of 0 + j1.10 pu based on its ratings, and that three-phase generator G2 is rated 4/5 MW/MVA, 7.62/13.2Y kV with an imped- ance of 0 + j0.90 pu based on its ratings. The transmission line TL23 has a length of 50 mi and is composed of 4/0 ACSR (aluminum conductor steel reinforced) conductors with an equivalent spacing (Dm) of 8 ft and has an impedance of 0.445 + j0.976 Ω/mi. Its shunt susceptance is given as 5.78 μS/mi. The line connects buses 2 and 3. Bus 3 is assumed to be an infinite bus, that is, the magnitude of its voltage remains constant at given values and its phase position is unchanged regardless of the power and power factor demands that may be put on it. Furthermore, it is assumed to have a constant frequency equal to the nominal frequency of the system studied. Transmission line TL14 connects buses 1 and 4. It has a line length of 2 mi and an impedance of 0.80 + j0.80 Ω/mi.
1 2
4 5
3
G1
T1
T2 TL23
TL14 I5
G2
Load 1
Load 5
FIGURE P2.1 One-line diagram for Problem 2.10.
Because of the line length, its shunt susceptance is assumed to be negligible. The load that is connected to bus 1 has a current magnitude |I1| of 523 A and a lagging power factor of 0.707. The load that is connected to bus 5 is given as 8000 + j6000 kVA. Use the arbi- trarily selected 5000 kVA as the three-phase kVA base and 39.84/69.00 kV as the line-to- neutral and line-to-line voltage base and determine the following:
(a) Complete Table P2.1 for the indicated values. Note the IL means line current and Iϕ means phase currents in delta-connected apparatus.
(b) Draw a single-line positive-sequence network of this simple power system. Use the nominal π circuit to represent the 69-kV line. Show the values of all impedances and susceptances in per units on the chosen bases. Show all loads in per unit P + jQ.
11. Assume that a 500 + j200-kVA load is connected to a load bus that has a voltage of 1.0∠0° pu.
If the power base is 1000 kVA, determine the per-unit R and X of the load:
(a) When load is represented by parallel connection (b) When load is represented by series connection
TABLE P2.1
Table for Problem 10
Quantity Nominally 69-kV Circuits Nominally 13-kV Circuits Nominally 480-V Circuits
kVAB(3ϕ) 5000 kVA 5000 kVA 5000 kVA
kVB(L–L) 69 kV
kVB(L–N) 39.84 kV
IB(L)
IB(ϕ)
ZB YB
51