A three-phase power of 3600 MW is to be transmitted

The per-phase impedance of a short threephase transmission line is 0:5 /53:15 ?W. The three-phase load at the receiving end is 900 kW at 0.8 p.f. lagging. If the line-toline sending-end voltage is 3.3 kV, determine (a) the receiving-end line-to-line voltage in kV, and (b) the line current. Draw the phasor diagram with the line current I , as reference.

Reconsider Problem 5.7 and find the following: (a) sending-end power factor, (b) sending-end three- phase power, and (c) the three-phase line loss.

The 100-km, 230-kV, 60-Hz three-phase line in Problems 4.18 and 4.39 delivers 300 MVA at 218 kV to the receiving end at full load. Using the nominal p circuit, calculate the: ABCD parameters, sending-end voltage, and percent voltage regulation when the receiving-end power factor is (a) 0.9 lagging, (b) unity, and (c) 0.9 leading. Assume a 50C conductor temperature to determine the resistance of this line.

The 500-kV, 60-Hz three-phase line in Problems 4.20 and 4.41 has a 180-km length and delivers 1600 MW at 475 kV and at 0.95 power factor leading to the receiving end at full load. Using the nominal p circuit, calculate the: (a) ABCD parameters, (b) sending-end voltage and current, (c) sending-end power and power factor, (d) full-load line losses and eciency, and (e) percent voltage regulation. Assume a 50 C conductor temperature to determine the resistance of this line.

A 40-km, 220-kV, 60-Hz three-phase overhead transmission line has a per-phase resistance of 0.15 W/km, a per-phase inductance of 1.3263 mH/km, and negligible shunt capacitance. Using the short line model, find the sending-end voltage, voltage regulation, sending-end power, and transmission line eciency when the line is supplying a three-phase load of: (a) 381 MVA at 0.8 power factor lagging and at 220 kV, (b) 381 MVA at 0.8 power factor leading and at 220 kV.

A 60-Hz, 100-mile, three-phase overhead transmission line, constructed of ACSR conductors, has a series impedance of (0:1826 + j0:784) W/mi per phase and a shunt capacitive reactance-to-neutral of 185:5 x 10_ / -90 W-mi per phase. Using the nominal p circuit for a medium-length transmission line, (a) determine the total series impedance and shunt admittance of the line. (b) Compute the voltage, the current, and the real and reactive power at the sending end if the load at the receiving end draws 200 MVA at unity power factor and at a line-to-line voltage of 230 kV. (c) Find the percent voltage regulation of the line.

Evaluate cosh (yl) and tanh(yl/2) for yl = 0:40/85 per unit.

A 400-km, 500-kV, 60-Hz uncompensated three-phase line has a positive-sequence series impedance z _ 0:03 + j0:35 W/km and a positive-sequence shunt admittance y = j4:4 x 10 S/km. Calculate: (a) Zc, (b) (yl), and (c) the exact ABCD parameters for this line.

At full load the line in Problem 5.14 delivers 1000 MW at unity power factor and at 475 kV. Calculate: (a) the sending-end voltage, (b) the sending-end current, (c) the sending-end power factor, (d) the full- load line losses, and (e) the percent voltage regulation.

The 500-kV, 60-Hz three-phase line in Problems 4.20 and 4.41 has a 300-km length. Calculate: (a) Zc, (b) (yl), and (c) the exact ABCD parameters for this line. Assume a 50C conductor temperature.

At full load, the line in Problem 5.16 delivers 1500 MVA at 480 kV to the receivingend load. Calculate the sending-end voltage and percent voltage regulation when the receiving-end power factor is (a) 0.9 lagging, (b) unity, and (c) 0.9 leading.

A 60-Hz, 230-mile, three-phase overhead transmission line has a series impedance z = 0:8431 79:04 W/mi and a shunt admittance y = 5:105 x 10 90 S/mi. The load at the receiving end is 125 MW at unity power factor and at 215 kV. Determine the voltage, current, real and reactive power at the sending end and the percent voltage regulation of the line. Also find the wavelength and velocity of propagation of the line.

Using per-unit calculations, rework Problem 5.18 to determine the sending-end voltageand current.

a) The series expansions of the hyperbolic functions are given by cosh y = 1 + y2 2 + y4 24 + y6 720 + . . . sinh y = 1 _ y2 6 + y4 120+ y6 5040+ . . . For the ABCD parameters of a long transmission line represented by an equivalent p circuit, apply the above expansion and consider only the first two terms, and express the result in terms of Y and Z. (b) For the nominal p and equivalent p circuits shown in Figures 5.3 and 5.7 of the text, show that A - 1/B = Y/2 and A-1 B = Y/2 hold good, respectively.

Starting with (5.1.1) of the text, show that A = VSIS + VRIR / VRIS + VSIR and B = V_S - V_R /VRIS + VSIR

Consider the A parameter of the long line given by cosh y, where y _ ffiffiffiffiffiffiffiffi ZY p . With x _ e?y _ x1 _ jx2, and A _ A1 _ jA2, show that x1 and x2 satisfy the following: x2 1 ? x2 2 ? 2A1x1 ? A2x2_ _ 1 _ 0 and x1x2 ? A2x1 _ A1x2_ _ 0:

Determine the equivalent p circuit for the line in Problem 5.14 and compare it with the nominal p circuit.

Determine the equivalent p circuit for the line in Problem 5.16. Compare the equivalent p circuit with the nominal p circuit.

Let the transmission line of Problem 5.12 be extended to cover a distance of 200 miles. Assume conditions at the load to be the same as in Problem 5.12. Determine the: (a) sending-end voltage, (b) sending-end current, (c) sending-end real and reactive powers, and (d) percent voltage regulation.

A 300-km, 500-kV, 60-Hz three-phase uncompensated line has a positive-sequence series reactance x _ 0:34 W/km and a positive-sequence shunt admittance y = j4:5 x 10 S/km. Neglecting losses, calculate: (a) Zc, (b) (yl), (c) the ABCD parameters, (d) the wavelength l of the line, in kilometers, and (e) the surge impedance loading in MW.

Determine the equivalent p circuit for the line in Problem 5.26.

Rated line voltage is applied to the sending end of the line in Problem 5.26. Calculate the receiving- end voltage when the receiving end is terminated by (a) an open circuit, (b) the surge impedance of the line, and (c) one-half of the surge impedance. (d) Also calculate the theoretical maximum real power that the line can deliver when rated voltage is applied to both ends of the line.

Rework Problems 5.9 and 5.16 neglecting the conductor resistance. Compare the results with and without losses.

From (4.6.22) and (4.10.4), the series inductance and shunt capacitance of a threephase overhead line are La = 2 x 10-7 ln(Deq/DSL) = m0/2p ln(Deq/DSL) H/m Can= 2pe0 ln(Deq/DSC) F/m where m0 = 4p x 10-7 H/m and e0 = 1/36p x 10-9 F/m Using these equations, determine formulas for surge impedance and velocity of propagation of an overhead lossless line. Then determine the surge impedance and velocity of propagation for the three-phase line given in Example 4.5. Assume positivesequence operation. Neglect line losses as well as the e_ects of the overhead neutral wires and the earth plane.

A 500-kV, 300-km, 60-Hz three-phase overhead transmission line, assumed to be lossless, has a series inductance of 0.97 mH/km per phase and a shunt capacitance of 0.0115 uF/km per phase. (a) Determine the phase constant b, the surge impedance ZC, velocity of propagation n, and the wavelength l of the line. (b) Determine the voltage, current, real and reactive power at the sending end, and the percent voltage regulation of the line if the receiving-end load is 800 MW at 0.8 power factorlagging and at500 kV.

The following parameters are based on a preliminary line design: VS = 1:0 per unit, VR _ 0:9 per unit, l = 5000 km, ZC = 320 W, d = 36:8 . A three-phase power of 700 MW is to be transmitted to a substation located 315 km from the source of power. (a) Determine a nominal voltage level for the three-phase transmission line, based on the practical line-loadability equation. (b) For the voltage level obtained in(a), determine the theoretical maximum power that can be transferred by the line.

Consider a long radial line terminated in its characteristic impedance ZC. Determine the following: (a) V1/ I1, known as the driving point impedance. (b) lV2l/lV1l, known as the voltage gain, in terms of al. (c) lI2l/lI1l, known as the current gain, in terms of al. (d) The complex power gain, -S21/S12, in terms of al. (e) The real power effciency, (-P21/P12) = n, in terms of al. [Note: 1 refers to sending end and 2 refers to receiving end. (S21) is the complex power received at 2; S12 is sent from 1.]

For the case of a lossless line, how would the results of Problem 5.33 change?In terms of ZC, which will be a real quantity for this case, express P12 in terms jI1j and jV1j.

For a lossless open-circuited line, express the sending-end voltage, V1, in terms of the receiving-end voltage, V2, for the three cases of short-line model, medium-length line model, and long-line model. Is it true that the voltage at the open receiving end of a long line is higher than that at the sending end, for small Bl.

For a short transmission line of impedance R _ jX_ ohms per phase, show that the maximum power that can be transmitted over the line is Pmax = V_ R / Z_ (ZVS / VR - R ) where Z = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R_ + X_ when the sending-end and receiving-end voltages are fixed, and for the condition Q = -V_RX/ R_ + X_ when dP/dQ = 0

(a) Consider complex power transmission via the three-phase short line for which the per-phase circuit is shown in Figure 5.19. Express S12, the complex power sent by bus 1 (or V1), and ?S21_, the complex power received by bus 2 (or V2), in terms of V1, V2, Z, Z, and y12 _ y1 ? y2, the power angle. (b) For a balanced three-phase transmission line, in per-unit notation, with Z _ 1 85?, y12 _ 10?, determine S12 and ?S21_ for

The line in Problem 5.14 has three ACSR 1113-kcmil conductors per phase. Calculate the theoretical maximum real power that this line can deliver and compare with the thermal limit of the line. Assume VS = VR = 1:0 per unit and unity power factor at the receiving end.

Repeat Problems 5.14 and 5.38 if the line length is (a) 200 km, (b) 600 km.

For the 500-kV line given in Problem 5.16, (a) calculate the theoretical maximum real power that the line can deliver to the receiving end when rated voltage is applied to both ends. (b) Calculate the receiving-end reactive power and power factor at this theoretical loading.

A 230-kV, 100-km, 60-Hz three-phase overhead transmission line with a rated current of 900 A/phase has a series impedance z = 0:088 _ j0:465 W/km and a shunt admittance y = j3:524 mS/km. (a) Obtain the nominal p equivalent circuit in normal units and in per unit on a base of 100 MVA (three phase) and 230 kV (line-to-line). (b) Determine the three-phase rated MVA of the line. (c) Compute the ABCD parameters. (d) Calculate the SIL.

A three-phase power of 460 MW is to the transmitted to a substation located 500 km from the source of power. With VS = 1 per unit, VR = 0:9 per unit, l = 5000 km, ZC = 500 W, and d = 36:87?, determine a nominal voltage level for the lossless transmission line, based on Eq (5.4.29) of the text.Using this result, find the theoretical three-phase maximum power that can be transferred by the lossless transmission line.

Open PowerWorld Simulator case Example 5_4 and graph the load bus voltage as a function of load real power (assuming unity power factor at the load). What is the maximum amount of real power that can be transferred to the load at unity power factor if we require the load voltage always be greater than 0.9 per unit?

Repeat Problem 5.43, but now vary the load reactive power, assuming the load real power is fixed at 1000 MW.

For the line in Problems 5.14 and 5.38, determine: (a) the practical line loadability in MW, assuming VS = 1:0 per unit, VRA0:95 per unit, and dmax = 35?; (b) the full-load current at 0.99 p.f. leading, based on the above practical line loadability; (c) the exact receiving-end voltage for the full-load current in (b) above; and (d) the percent voltage regulation. For this line, is loadability determined by the thermal limit, the voltagedrop limit, or steady-state stability?

Repeat Problem 5.45 for the 500-kV line given in Problem 5.10.

Determine the practical line loadability in MW and in per-unit of SIL for the line in Problem 5.14 if the line length is (a) 200 km, (b) 600 km. Assume VS = 1:0 per unit,VR = 0:95 per unit, dmax = 35, and 0.99 leading power factor at the receiving end.

It is desired to transmit 2000 MW from a power plant to a load center located 300 km from the plant. Determine the number of 60-Hz three-phase, uncompensated transmission lines required to transmit this power with one line out of service for the following cases: (a) 345-kV lines, Zc = 300 W, (b) 500- kV lines, Zc = 275 W, (c) 765-kV lines, Zc _=260 W. Assume that VS = 1:0 per unit, VR = 0:95 per unit, and dmax = 35.

Repeat Problem 5.48 if it is desired to transmit: (a) 3200 MW to a load center located 300 km from the plant, (b) 2000 MW to a load center located 400 km from the plant.

A three-phase power of 3600 MW is to be transmitted through four identical 60-Hz overhead transmission lines over a distance of 300 km. Based on a preliminary design, the phase constant and surge impedance of the line are b = 9:46 x 10 rad/km and ZC = 343 W, respectively. Assuming VS = 1:0 per unit, VR = 0:9 per unit, and a power angle d = 36:87, determine a suitable nominal voltage level in kV, based on the practical line-loadability criteria.

The power flow at any point on a transmission line can be calculated in terms of the ABCD parameters. By letting A _ jAj a, B _ jBj b, VR _ jVRj 0?, and VS _ jVSj d, the complex power at the receiving end can be shown to be PR + jQR = jVRj jVSj b ? a jBj ? jdj jV2 Rj b ? a jBj (a) Draw a phasor diagram corresponding to the above equation. Let it be represented by a triangle O0OA with O0 as the origin and OA representing PR _ jQR. (b) By shifting the origin from O0 to O, turn the result of (a) into a power diagram, redrawing the phasor diagram. For a given fixed value of jVRj and a set of values for jVSj, draw the loci of point A, thereby showing the so-called receiving-end circles. (c) From the result of (b) for a given load with a lagging power factor angle yR, determine the amount of reactive power that must be supplied to the receiving end to maintain a constant receiving-end voltage, if the sending-end voltage magnitude decreases from jVS1j to jVS2j.

(a) Consider complex power transmission via the three-phase long line for which the per-phase circuit is shown in Figure 5.20. See Problem 5.37 in which the short-line case was considered. Show that sending-end power _ S12 _ Y 0? 2 V2 1 _ V2 1 Z0? ? V1V2 Z0? e jy12 and received power _ ?S21 _ ? Y 0? 2 V2 2 ? V2 2 Z0? _ V1V2 Z0? e?jy12 where y12 _ y1 ? y2. (b) For a lossless line with equal voltage magnitudes at each end, show that P12 _ ?P21 _ V2 1 sin y12 ZC sin bl _ PSIL sin y12 sin bl (c) For y12 _ 45?, and b _ 0:002 rad/km, find P12=PSIL_ as a function of line length in km, and sketch it. (d) If a thermal limit of P12=PSIL_ _ 2 is set, which limit governs for short lines and long lines? PW

Open PowerWorld Simulator case Example 5_8. If we require the load bus voltage to be greater than or equal to 730 kV even with any line segment out of service, what is the maximum amount of real power that can be delivered to the load?

Repeat Problem 5.53, but now assume any two line segments may be out of service.

Recalculate the percent voltage regulation in Problem 5.15 when identical shunt reactors are installed at both ends of the line during light loads, providing 65% total shunt compensation. The reactors are removed at full load. Also calculate the impedance of each shunt reactor.

Rework Problem 5.17 when identical shunt reactors are installed at both ends of the line, providing 50% total shunt compensation. The reactors are removed at full load.

Identical series capacitors are installed at both ends of the line in Problem 5.14, providing 40% total series compensation. Determine the equivalent ABCD parameters of this compensated line. Also calculate the impedance of each series capacitor.

Identical series capacitors are installed at both ends of the line in Problem 5.16, providing 30% total series compensation. (a) Determine the equivalent ABCD parameters for this compensated line. (b) Determine the theoretical maximum real power that this series-compensated line can deliver when VS = VR = 1:0 per unit. Compare your result with that of Problem 5.40.

Determine the theoretical maximum real power that the series-compensated line in Problem 5.57 can deliver when VS = VR = 1:0 per unit. Compare your result with that of Problem 5.38.

What is the minimum amount of series capacitive compensation NC in percent of the positive- sequence line reactance needed to reduce the number of 765-kV lines in Example 5.8 from five to four. Assume two intermediate substations with one line section out of service. Also, neglect line losses and assume that the series compensation is suffciently distributed along the line so as to e_ectively reduce the series reactance of the equivalent p circuit to X' (1 - NC=100).

Determine the equivalent ABCD parameters for the line in Problem 5.14 if it has 70% shunt reactive (inductors) compensation and 40% series capacitive compensation. Half of this compensation is installed at each end of the line, as in Figure 5.14.

Consider the transmission line of Problem 5.18. (a) Find the ABCD parameters of the line when uncompensated. (b) For a series capacitive compensation of 70% (35% at the sending end and 35% at the receiving end), determine the ABCD parameters. Comment on the relative change in the magnitude of the B parameter with respect to the relative changes in the magnitudes of the A, C, and D parameters. Also comment on the maximum power that can be transmitted when series compensated.

Given the uncompensated line of Problem 5.18, let a three-phase shunt reactor (inductor) that compensates for 70% of the total shunt admittance of the line be connected at the receiving end of the line during no-load conditions. Determine the e_ect of voltage regulation with the reactor connected at no load. Assume that the reactor is removed under full-load conditions.

Let the three-phase lossless transmission line of Problem 5.31 supply a load of 1000 MVA at 0.8 power factor lagging and at 500 kV. (a) Determine the capacitance/phase and total three-phase Mvars supplied by a three-phase, D-connected shunt-capacitor bank at the receiving end to maintain the receiving-end voltage at 500 kV when the sending end of the line is energized at 500 kV. (b) If series capacitive compensation of 40% is installed at the midpoint of the line, without the shunt capacitor bank at the receiving end, compute the sending-end voltage and percent voltage regulation.

Open PowerWorld Simulator case Example 5_10 with the series capacitive compensation at both ends of the line in service. Graph the load bus voltage as a function of load real power (assuming unity power factor at the load). What is the maximum amount of real power that can be transferred to the load at unity power factor if we require the load voltage always be greater than 0.85 per unit?

Open PowerWorld Simulator case Example 5_10 with the series capacitive compensation at both ends of the line in service. With the reactive power load fixed at 500 Mvar, graph the load bus voltage as the MW load is varied between 0 and 2600 MW in 200 MW increments. Then repeat with both of the series compensation elements out of service.