In Class Examples (ICE) 1
1. A 3φ 765kV, 60Hz, 300km, completely transposed line has the following positive-sequence impedance and admittance: z = 0.0165 + j0.3306 = 0.3310 87.14 o Ω/km y = j4.67 410-6 S/km Assuming positive-sequence operation, calculate the exact ABCD parameters of the line. Identical shunt reactors (inductors) are connected from each phase conductor to neutral at both ends of the 300km 765kV line in Example 7 during light load conditions, providing 75% compensation. The reactors are removed during heavy load conditions. Full load is l.9ka at u- nity p.f. and at 730kV. Assuming that the sending-end voltage is constant, determine the following: a. Percent voltage regulation of the uncompensated line. b. The equivalent shunt admittance and series impedance of the compensated line. c. Percent voltage regulation of the compensated line. 2. Identical series capacitors are installed in each phase at both ends of the 300km 765kV line in Example 7, providing 30% compensation. Determine the theoretical maximum power that this compensated line can deliver and compare with that of the uncompensated line. Assume V S = V R = 765kV. 2
3. A bolted short circuit occurs in a series R-L circuit with V = 20kV, X = 8 Ω, R = 0.8 Ω, and with maximum dc offset. The circuit breaker opens 3 cycles after fault inception. Determine (a) the rms ac fault current, (b) the rms momentary current at τ = 0.5 cycles, which passes through the breaker before it opens, and (c) the rms asymmetrical fault current that the breaker interrupts. 4. A 500 MVA 20 kv, 60 Hz synchronous generator with reactances X d = 0.15 pu, X d = 0.24 pu, X d = 1.1 pu and time constants Td = 0.035, Td = 2.0, T A = 0.2s is connected to a circuit breaker. The generator is operating at 5% above rated voltage and at no-load when a bolted three-phase short-circuit occurs on the load side of the breaker. The breaker interrupts the fault 3 cycles after fault inception. Determine (a) the subtransient fault current in pu and ka rms; (b) maximum dc offset as a function of time, and (c) rms asymmetrical fault current, which the breaker interrupts, assuming maximum dc offset. 5. The synchronous generator in the figure is operating 3
at rated MVA, 0.95 p.f. lagging and at 5% above rated voltage when a bolted three-phase short circuit occurs at Bus 1. Calculate the pu values of (a) subtransient fault current; (b) subtransient generator and motor currents, neglecting prefault current, and (c) subtransient generator and motor currents including prefault currents. 6. Faults at Bus 1 and Bus 2 in the power system of Problem 5 are of interest. The prefault voltage is 1.05pu and prefault load current is neglected. (a) Determing the 2x2 positive sequence bus impedance matrix. (b) For a bolted three-phase short-circuit at Bus 1, use Z bus to calculate the subtransient fault current and the contribution to the fault current from the transmission line. (c) Repeat part (b) for a bolted three-phase short-circuit at Bus 2. 7. A three-phase line feeding a balanced-y load has one of its phases (phase b) open. The load neutral is grounded, and the unbalanced line currents are Ia = 10 0 A, Ib = 0 A, and Ic = 10 120 A. Calculate the sequence currents and the neutral current. 4
8. A balanced-y load is in parallel with a balanceddelta-connected capacitor bank. The Y load has an impedance Z Y = (3 +j4) Ω per phase, and its neutral is grounded through an inductive reactance X n = 2 Ω. The capacitor bank has a reactance X c = 30 Ω per phase. Draw the sequence networks for this load and calculate the loadsequence impedances. 9. Draw the sequence networks for the given circuit and calculate the sequence components of the line current. Assume that the generator neutral is grounded through an impedance Z n = j10 Ω, and that the generator sequence impedances are Z g0 = j1 Ω, Z g1 = j15 Ω, and Z g2 = j3 Ω. 10. A Y-connected voltage source with the following unbalanced voltage (V ag = 277 0 V, V bg = 260-120 V, V cg = 295 115 V) is supplied to the balanced line and load of the given circuit. The source neutral is solidly grounded. Using the method of symmetrical components, calculate the source currents I a, I b, and I c. 5
11. A 75 kva, 480 V /208V Y transformer with a solidly grounded neutral is connected between the source and line of the previous example. The transformer leakage reactance is X eq = 0.1 pu; winding resistances and exciting current are neglected. Using the transformer ratings as base quantities, draw the pu sequence networks and calculate the phase a source current. 12. Three identical transformers are connected as a three-phase bank in order to feed power from a 900MVA, 13.8kV generator to a 345kV transmission line and to a 34.5kV distribution line. The transformer windings are connected as follows: 13.8kV windings (X):, to generator 199.2kV windings (H): solidly grounded Y, to 345kV line 19.92kV windings (M): grounded Y through Z n =j0.1ω, to 34.5kV line The positive sequence voltages and currents of the high and medium voltage Y windings lead the corresponding quantities of the low-voltage delta winding by 30 o. Draw the pu sequence networks, using a three-phase base of 900MVA and 13.8kV for terminal X. 6
13. Calculate Sp and Ss delivered by the three-phase source in Example 10. Verify that Sp = 3Ss. 7
14. Given a 9-bus power system. The P-Q load is known at each of the nine buses. Synchronous generators are connected at buses 1, 2, 5, and 7. For power flow s- tudy identify the P and Q mismatches, and the state variables associated with each bus. Choose bus 1 as the slack bus. 16. Figure shows the single-line diagram of a five-bus power system. Input data are given in Tables 6.1, 6.2, and 6.3. As shown in Table 6.1, bus 1, to which a generator is connected, is the swing bus. Bus 3, to which a generator and a load are connected, is a voltage-controlled bus. Buses 2, 4, and 5 are load buses. Note that the loads at buses 2 and 3 are inductive since Q 2 = -Q L2 = -2.8 and -Q L3 = -0.4 are negative. For each bus k, determine which of the variables V k, δ k, P k, and Q k are input data and which are unknowns. Also, compute the elements of the second row of Y bus. 8
17. For the power system of Example 14, use GaussSeidel to calculate V 2 (1), the phasor voltage at bus 2 after the first iteration. Use zero initial phase angles and 1.0 p.u. initial voltage magnitudes (except at bus 3, where V 3 = 1.05 p.u.) to start the iteration procedure. 18. Determine the Jacobian matrix for the power system in Example 14. Also calculate P 2 (0) in Step 1 and J1 24 (0) in Step 2 of the first Newton-Raphson iteration. Assume zero initial phase angles and 1.0 p.u. initial voltage magnitudes (except V 3 = 1.05 p.u.). 9
19. An interconnected 60Hz power system consists of one area with 3 turbine generator units rated 1000MVA, 750MVA, and 500MVA, respectively. The regulation constant of each unit is R = 0.05 p.u. based on its own rating. Each unit is initially operating at one-half of its own rating, when the system load suddenly increases by 200MW. Determine: a. the per-unit area frequency response characteristic β on a 1000MVA system base, b. the steady-state drop in area frequency, and c. the increase in turbine mechanical power output of each unit. Assume that the reference power setting of each turbinegenerator remains constant. Neglect losses and the dependence of load on frequency. 20. As shown in the figure, a 60Hz power system consists of two interconnected areas. Area 1 has 2000MW of total generation and an area frequency response characteristic β 1 = 700 MW/Hz. Area 2 has 4000MW of total generation and β 2 = 1400 MW/Hz. Each area is initially generating one-half of its total generation, at p tie1 = p tie2 = 0 and at 60Hz when the load in Area 10
1 suddenly increases by 100MW. Determine the steadystate frequency error f and the steady-state tie-line error p tie of each area for the following two cases: a. without LFC, and b. with LFC. Neglect losses and the dependence of load on frequency. 21. An area of an interconnected power system has two fossil-fuel units operating on economic dispatch. The variable operating costs of these units are given by: C 1 = 10P 1 + 8 10e-3P 2 1 $/hr C 2 = 8P 2 + 9 10e-3P 2 2 $/hr where P 1 and P 2 are in MW. Determine the power output of each unit, the incremental operating cost, and the total operating cost C T that minimizes C T as the total load demand P T varies from 500MW to 1500MW. Generating unit inequality constraints and transmission losses are neglected. 22. Do Example 19 if the units are subject to the following inequality constraints: 11
100 P 1 600 MW 400 P 2 1000 MW 21. Total transmission losses for the power system area given in Example 19 are given by: P L = 1.5e-4P 2 1 + 2 1O-5P 1 P 2 + 3 10e-5P 2 1 MW where P 1 and P 2 are given in MW. Determine the output of each unit, total transmission losses, total load demand, and total operating cost C T when the area λ = 16.00 $/MWhr. 12
23. The single-line diagram of a power system is shown in the figure, where negative and zero sequence reactances are also given. The neutrals of the generator and delta-y transformers are solidly grounded. The motor neutral is grounded through a reactance X n = 0.05 pu on the motor base. (a) Draw the pu zero, positive, and negative sequence networks on a 100 MVA, 13.8 kv base in the zone of the generator. (b) Reduce the sequence networks to their Thevenin equivalents, as viewed from Bus 2. Prefault voltage is V f = 1.05 0 pu. Prefault load current and delta-y transformer phase shift are neglected. 24. Calculate the pu subtransient fault currents in phases a, b, and c for a bolted three-phase-to-ground short circuit at Bus 2 in Example 23. 25. Calculate the subtransient fault current in pu and in ka for a bolted single line-to-ground short circuit from phase a to ground at Bus 2 in Example 23. Also calculate the pu line-to-ground voltages at faulted Bus 2. 13
26. Calculate the subtransient fault current in pu and in ka for a bolted line-to-line fault from phase b to c at Bus 2 in the Example 23. 27. Calculate (a) the subtransient fault current in each phase, (b) neutral fault current, and (c) contributions to the fault current from the motor and from the transmission line, for a bolted double-line-to-ground fault from phase b-to-c-to-ground at Bus 2 in Example 14. Neglect the delta Y transformer phase shifts. 28. Rework Example 27 with the -Y transformer phase shifts included. Assume American standard phase shift. 29. Faults at buses 1 and 2 for the three-phase power system given in Example 23 are of interest. The prefault voltage is 1.05 pu. Prefault load current is neglected. (a) Determine the pu zero, positive, and negative-sequence bus impedance matrices. Find the subtransient fault cur- 14
rent in pu for a bolted single line-to-ground fault current from phase a-to-ground (b) at Bus 1, and (c) at Bus 2. Find the pu line-to-ground voltages at (d) Bus 1 and (e) Bus 2 during the single line-to-ground fault at Bus 1. 15