S2014 PCT Tutorial 1 Page 1 of 6 27/07/2014 Power Circuit Theory: Tutorial Set 1 Exercise 1 A balanced three-phase Y load has one phase voltage of Vcn =277∠450 V. If the phase sequence is negative sequence i.e. acb, calculate the line voltages Vca, Vab, and Vbc. Vca =480∠150 V, Vab =480∠1350 V, and Vbc =480∠−1050 V Exercise 2 What are the phase voltages for a balanced three-phase Y load, if Vba =12.47∠−350 kV? The phase sequence is positive sequence i.e. abc. Vbn =7.20∠−50 kV, Van =7.20∠1150 kV, and Vbc =720∠−1250 kV Exercise 3 A balanced Y load of 40 Ω resistors is connected to a balanced, three-phase, three-wire source. If Vcb =480∠−350 V Calculate the a phase line current. The phase sequence is negative sequence Ia =6.928∠550 A Exercise 4 In a three-phase, three-wire circuit, calculate the line currents to a balanced Y load for which ZY=60∠−300 Ω and Vcb=480∠650 V. The phase sequence is positive. Ia =4.619∠50 A, Ib =4.619∠−1150 A, and Ic =4.619∠1250 A Exercise 5 Calculate the total average power delivered by a three-phase source with the line to line voltage of 500 V to each of the following balanced Y connected loads with ZY equal to: a) 30 Ω; b) (30+ j 72) Ω; c) (30− j 12.5) Ω. (a) 8.333 kW, (b) 1.233 kW, (c) 7.101 kW Exercise 6 Calculate the magnitude of the line voltage (V Line- Line) at the source of the circuit in Figure E6. As shown, the load phase voltage is 100V and the impedance of each line is 2+j3 Ω. |VLine-Line|=179.5 VS2014 PCT Tutorial 1 Page 2 of 6 27/07/2014 10-j7 Ω 10-j7 Ω 10-j7 Ω Figure E6: Circuit for Exercise 6. Exercise 7 A 208 V three-phase circuit has two balanced loads, one a ∆ of 21∠300 Ω impedances and the other a Y of 9∠−600Ω impedances. Calculate the magnitude of the line current and also the total average power absorbed by the two loads. |Iline|=21.73 A, Pave=7.756 kW Exercise 8 In a 208 V three-phase circuit a balanced ∆ connected load absorbs 2 kW at a 0.8 leading power factor. Calculate Z∆. Z∆ =(41.51−j31.15) Ω Exercise 9 Two balanced three-phase motor loads comprising an induction motor and a synchronous motor are connected in parallel. The induction motor draws 400 kW at 0.8 power factor lagging and the synchronous motor draws 150 kVA at 0.9 power factor leading. Both motor loads are supplied by a balanced three-phase 4.160 kV source. If the cable impedance between the source and load is neglected, a) Draw the power triangle for each motor and for the combined-motor load. b) Determine the power factor of the combined-motor load. c) Determine the magnitude of the line current delivered by the source. d) A delta connected capacitor bank is now installed in parallel with the combinedmotor load. What value of capacitive reactance is required in each phase of the capacitor bank to make the source power factor unity? e) Determine the magnitude of the line current delivered by the source with the capacitor bank installed. (b) 0.916 lagging, (c) 81.1 A, (d) −j221.3 Ω, (e) 74.3 AS2014 PCT Tutorial 1 Page 3 of 6 27/07/2014 Exercise 10 A balanced 3 phase star connected 400 V system has a per phase impedance of 20 36.87 ∠ ° Ω . Assuming positive phase sequence and VAB as reference, determine the phase current and complex power. I S AN = 11.55 66.87 A, 8.00 36.87 kVA ∠ − ° = TY ∠ ° If the same impedances are connected in ∆ connection determine the line currents and the complex power assuming VAB as reference. I S AB T = 20 36.87 A, 24 36.87 kVA ∠ − ° = ∆ ∠ ° Exercise 11 Consider a single phase AC circuit shown in Figure E11. Figure E11: Circuit for Exercise 11. The voltage and impedance values are given in Figure E11. Determine i) the branch complex powers S1, S2 and S3 and total complex power. S S S S 1 = 7.900 80.91 kVA, 2.287 59.04 kVA, 6.000 36. ∠ ° = 2 ∠ − ° = 3 ∠ 87 kVA, 11.89 52.57 kVA. ° = T ∠ ° ii) the supply current and overall power factor. I p f S = 59.43 52.57 A, . . 0.608 lagging ∠ − ° = iii) the capacitance value that is required to be connected across the load to improve the power factor to unity, assuming a 50 Hz supply. C = 751.2 F µ Exercise 12 Consider a single phase AC circuit shown in Exercise E12. Figure E12: Circuit for Exercise 12 The instantaneous power is given by, p t t ( ) = + 960 1200cos 628 36.87 W ( − °) Determine the value of rms current supplied to the load, the complex power supplied to the load and the load impedance. I S Z rms = = 7.714 A, 1.200 36.87 kVA, 20.17 36.87 . ∠ ° = ∠ ° ΩS2014 PCT Tutorial 1 Page 4 of 6 27/07/2014 Exercise 13 A 100 kVA, 11000:2200 V, 60 Hz, single-phase transformer has the following equivalent circuit parameters referred to the high-potential side: R1 = 6.1 Ω R'2 = 7.2 Ω Xl1 = 31.2 Ω X'l2 = 31.2 Ω Xm = 57.3 kΩ R c = 124 kΩ The transformer is supplying at 2.20 kV a load circuit of 50 Ω impedance and a leading power factor of 0.7. Draw a phasor diagram (not to scale) showing the various phasor magnitudes and angles for this operating condition. Determine the potential difference and power factor at the high-potential terminals of the transformer. 10.707 kV, 0.7458 leading Exercise 14 A 20 kVA, 2200:220 V, 60 Hz, single-phase distribution transformer gave the following test results: i) Open-circuit test, low-potential winding excited: Voc = 220 V, Ioc = 1.52 A, Poc = 161 W ii) Short-circuit test, high-potential winding excited: Vsc = 205 V, Isc = 9.1 A, Psc = 465 W iii) Direct-current winding resistances: R lp = 31.1 mΩ, Rhp = 2.51 Ω Determine the equivalent circuit of the transformer referred to the high-potential side. (Rc =30.06 kΩ, Xm = 16.51 kΩ, Xl1 & X'l2 =10.91 Ω, R1 = 2.51= Ω, R'2 = 3.11 Ω) Exercise 15 Consider the one-port network shown in Figure E15: R C V C V R I V=|V| 0° 50 Hz V one-port Figure E15: Circuit for Exercise 15 Show that the graph of the locus of the port impedance, Z, as the resistance R is varied from 0 to ∞ Ω , is a straight line. Show also that the locus of the port admittance, Y, as the resistance R is varied from 0 to ∞ Ω , is the arc of circle. Determine the centre and radius of the circle.S2014 PCT Tutorial 1 Page 5 of 6 27/07/2014 Show that the graph of the loci of I, VC and VR as the resistance R is varied from 0 to ∞ Ω , are arcs of circles. Determine the centre and radius of each circle. Exercise 16 Consider the one-port network shown in Figure E16: V L VR R I V=|V| 0° 50 Hz V one-port RL L V R L V' L Figure E16: Circuit for Exercise 16 Show that the graphs of the loci of VRL , VL′, VR and VL as the resistance R is varied from 0 to ∞ Ω , are arcs of circles. Determine the centre and radius of each circle. Determine the transformer tap ratios when the receiving end voltage is equal to the sending end voltage, the high voltage line operates at 230 kV and transmits 80 MW at 0.8 p.f. and the impedance of the line is (40 + j 150) Ω. Assume tstr = 1. The arrangement is shown in Figure E17. Load tr Transmission line Vs Vr 1: ts :1 Figure E17: Circuit for Exercise 17 1.1401 s t = Exercise 18 A wattmeter is connected in a single-phase circuit to measure the average power. Show that the average power is V I v t V t cos , given 2 cos and (θ θ ω θ v i − ) ( ) = + ( v ) i t I t ( ) = + 2 cos(ω θi ) .S2014 PCT Tutorial 1 Page 6 of 6 27/07/2014 Exercise 19 From a consideration of the instantaneous voltage and currents in a three-phase system, both star and delta, show that the total power can be measured by means of two wattmeters. Exercise 20 Show that the power in a three-phase, three-wire system with balanced loads is constant at every instant. Deduce an expression for the power in terms of the line voltage, line current, and the power-factor. Exercise 21 Show that in a balanced system in which the power is measured by the two-wattmeter method ( ) ( ) ( ) 1 2 1 2 cos 30 and cos 30 . Prove also, 3 cos where is the power factor angle and denotes a line quantity. l l l l T l l P V I P V I P P P V I l φ φ φ φ = + = − + = = Exercise 22 Given a balanced, positive-sequence, three-phase set of voltages in which Van ph = V ∠0, show analytically that V V ab an ph = 3 30 3 30 ∠ ° = V ∠ ° . Exercise 23 Given a balanced system of three phase voltages, show analytically vab + vac = 3van Exercise 24 The maximum power entering a series RL circuit is 500 W and the minimum power is −180 W. The voltage is 230 V, 50 Hz. Determine i. The values of R and L, R L = 73.22 437.0 mH Ω = ii. The value of capacitance, connected across the terminals of the network, if the maximum power is to be 360 W. C =10.83 F µ Exercise 25 The power in a balanced three-phase system in which the power is measured by the twowattmeter method to be P1 =100 W and P2 = 50 W. Calculate the power factor of the load. The sequence is positive and wattmeter 1 measures the current in the a phase and wattmeter 2 measures the current in the c phase. Power factor 0.866 leading =S2014 PCT Tutorial 2 Per Unit Values and Power Flow.doc Page 1 of 4 30/07/2014 Power Circuit Theory: Tutorial Set 2 Exercise 1 A 100 kVA, 11000:2200 V, 50 Hz, single-phase transformer has the following equivalent circuit parameters referred to the high-potential side: R1 = 6.1 : R'2 = 7.2 : Xl1 = 31.2 : X'l2 = 31.2 : The magnetising reactance and the core loss resistance may be neglected. The transformer is supplying at 2.20 kV a load circuit of 50 :impedance and a leading power factor of 0.7. Draw a phasor diagram (not to scale) showing the various phasor magnitudes and angles for this operating condition. Determine the potential difference and power factor at the highpotential terminals of the transformer. Compare your answers with Problem 13 of Tutorial 1. (10.700 kV, 0.7306 leading) Exercise 2 Repeat Exercise 1, assuming the winding resistances may be neglected. Compare your answers with Exercise 13 of Tutorial 1. (10.615 kV, 0.7254 leading) Exercise 3 The reactance X '' of a generator is given as 0.20 p.u. based on the generator's nameplate rating of 13.2 kV, 30 MVA. Calculate X '' on a new base of 13.8 kV, 50 MVA. (0.305 p.u.) Exercise 4 A single phase transformer is rated 110/440 V, 2.5 kVA. Leakage reactance, measured from the low potential side, is 0.6 :. Determine the p.u. leakage reactance. (0. 124 p.u.) Exercise 5 Repeat Exercise 13 of Tutorial 1 using per unit values. Exercise 6 A three phase alternator rated at 50 MVA 12.8 kV has a synchronous reactance of 2.0 p.u.. It supplies an 11 kV overhead line 5 km long each phase of which has an impedance of (0.22 + j 0.51) :/km. Calculate the total impedance on a base of 100 MVA and 11 kV. Z pu  0.9091 7.524 p.u. jS2014 PCT Tutorial 2 Per Unit Values and Power Flow.doc Page 2 of 4 30/07/2014 Exercise 7 Consider a power system with two parallel connected generators and three parallel connected motor loads connected through transformers T1 at the sending end and T2 at the receiving end of a transmission line. The transmission line has an impedance of j60 :. Assuming 100 MVA and 33 kV at the generators as base values obtain all the reactances in per unit values and draw the reactance diagram. The equipment data are shown in Table E7. Equipment Power Rating (MVA) Voltage (kV) Reactance G1 100 33 12% G2 50 33 10% T1 100 33/220 0.08 p.u. T2 100 220/33 0.08 p.u. M1 30 30 18% M2 20 30 20% M3 40 30 15% Table P7 Exercise 8 Consider a power system as shown in Figure E8. The manufacturer's data, the line and load details are indicated in Table E8. Select a common base of 100 MVA and 11 kV on the generator side, obtain all the reactances in per unit values and draw the reactance diagram. The motor M operates at 60 MVA at 0.8 pf leading at a terminal voltage of 10.5 kV and the load is 70 MVA at 0.6 p.f. lagging, 10.5 kV. Transmission lines 1 and 2 have impedances of j35 : and j50 : respectively. Determine i) the voltage at the generator bus bar 1. VBus 1 10.942 7.967 kV ‘ q ii) the internal emfs of the generators (assume the current in each generator is equal). E E G G 1 2 11.416 13.595 kV, 11.330 12.691 kV ‘ q ‘ q M G 1 T2 T1 Load G 2 T4 T3 Line 1 Line 2 Bus 1 Figure E8: Circuit for Exercise 8. Equipment Power Rating (MVA) Voltage (kV) Reactance G1 80 11 18% G2 80 11 15% T1 120 11/220 0.08 p.u. T2 120 220/11 0.06 p.u. T3 60 11/110 0.08 p.u. T4 60 110/11 0.06 p.u. M 60 10.5 16% Table E8S2014 PCT Tutorial 2 Per Unit Values and Power Flow.doc Page 3 of 4 30/07/2014 Exercise 9 Frequently the p.u. value of impedance Z p.u. given is known, based on S p.u. given , usually the equipment rating, but we need Z p.u. on a different VA base Sbase new . Show that:             2 . . . . base given base new p u p u new given base base new given V S Z Z V S § · § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ © ¹ Exercise 10 Three zones of a single-phase circuit are shown in Figure E10 below. Transformer T1 is rated 30 kVA, 240/480 V with per-unit leakage reactance of 10%. Transformer T2 is rated 20 kVA, 440/110 V with per-unit leakage reactance of 12/121 p.u. The transmission line has an impedance of j2.4 : The load has an impedance of (0.6 + j0.3) :The generator is ideal and has a terminal voltage of 228 ‘00 VUsing base values of 30 kVA and 240 V in Zone 1, a) Draw the per unit circuit; and b) Calculate the per unit current. I pu 0.5565 42.92 p.u. ‘ q Load G T1 T2 Transmission line Zone 1 Zone 2 Zone 3 Figure E10: Circuit for Exercise E10 Exercise 11 A three phase interconnector has resistance per conductor of 10 : per phase and an inductive reactance of 30 : per phase and sending and receiving end voltages of 140 and 120 kV respectively. The sending end voltage leads the receiving end voltage by 5º. Calculate the receiving end power and power factor. S R R  67.29 55.44 MVA, 0.7718 lagging, j pf Exercise 12 A three phase interconnector has resistance per conductor of 4 : per phase and an inductive reactance of 10 : per phase. When the load is adjusted such that the terminal voltage at both ends is 33 kV the power loss in the interconnector is 600 kW. Calculate the sending and receiving end powers and power factors. 12.14 3.987 MVA, 11.54 5.487 MVA,  0.9501 leading, 0.9031 leading, S R S R j j pf pf S S  S2014 PCT Tutorial 2 Per Unit Values and Power Flow.doc Page 4 of 4 30/07/2014 Exercise 13 To analyse the power flow between two nodes, consider the nodes 1 and 2 as shown in Figure E13. V1 I Y =g+jb V2 S12 S21 12 Figure E13: Circuit for Exercise 13: Power flow between two nodes If T12 1 2 ‘  ‘ V V , show that, the real and reactive power per phase respectively are given by, P g V V V b V V 12 1 1 2 12 1 2 12    2 cos sin T T and Q g V V b V V V 12 1 2 12 1 2 12 1    sin cos T T  2 Show also the real and reactive power at node 2 are given by, P g V V V b V V 21 1 2 12 2 1 2 12    cos sin T 2 T and Q g V V b V V V 21 1 2 12 1 2 12 2    sin cos T T  2 Show that the maximum P21 occurs when tan 12 b X g R T  Exercise 14 A three phase interconnector has a resistance of 10 : per phase and an inductive reactance of 30 : per phase. Determine the maximum power that can be delivered to a load at the receiving end if the terminal voltages at both sending and receiving ends are maintained at 132 kV. P max 376.8 MWS2014 PCT Tutorial 3 Inductance Page 1 of 4 4/08/2014 Power Circuit Theory: Tutorial Set 3 Exercise 1 A three phase transmission line supplies a load of 350 MVA at 275 kV through a transmission line of length 80 km. Find the transmission line conductor diameter, if the line losses are not to exceed 3 % of rated line MVA. Assume the conductor resistivity as 2.84 × 10−8 Ω.m. d = 21.12 mm Exercise 2 A three phase fully transposed transmission line has solid conductors of 2 cm diameter, spaced at distances between centres of 3.65 m, 5.5 m and 8.2 m. Determine the inductance of the complete line per km. L = 1.31127 mH/km Exercise 3 A single phase transmission line consists of 5 conductors as shown Figure E3. 0.75 m 0.75 m 1.5 m 0.75 m A1 A2 A3 B1 B2 Figure E3: Conductor arrangement for Exercise 3. Conductors A1 to A3 are solid with a radius of 2.5 cm each and conductors B1 and B2 are the return conductors, which are also solid, with a radius of 3 cm each. Determine the inductance of the complete line per km. L = 1.044 mH/km Exercise 4 A three phase fully transposed transmission line with horizontal conductor arrangement is shown in Figure E4a. 7 m A B C 7 m Figure E4a: Conductor arrangement for Exercise 4a. The GMR of each conductor is 1.465 cm. a) Calculate the inductance per phase per km of the line. L = 1.280 mH/km b) The line has to be replaced with a two conductor bundle system as shown in Figure E4b, with spacing between conductors in the bundle of 30 cm. If the new line inductance is 80 % of the original line inductance in part (a). Find the new GMR of each bundle conductor. GMR = 9.249 mmS2014 PCT Tutorial 3 Inductance Page 2 of 4 4/08/2014 7 m A B C 7 m 30 cm 30 cm 30 cm Figure E4b: Conductor arrangement for Exercise 4b. Exercise 5 Determine the inductance per phase per km of the three phase transposed line as shown in Figure E5. The diameter of each conductor is 2.532 cm. 7.5 m a b c a' b' c' 9 m 4 m 4 m Figure E5 Conductor arrangement for Exercise 5 L = 606.81 H/km µ Exercise 6 Consider a three phase double circuit transposed line as shown in Figure E6 with bundled conductors. a 20 m 30 cm 30 cm 25 m 30 cm 30 cm 30 cm 30 cm 18 m 10 m 9 m b c c' b' a' Figure E6: Conductor arrangement for Exercise 6.S2014 PCT Tutorial 3 Inductance Page 3 of 4 4/08/2014 The details of spacing between conductors are given in Figure E6. The conductors have a GMR of 1.75 cm. The circuit arrangement is a, b, c, c', b', a' as shown. Calculate the inductance per phase per km of the line. L = 497.19 H/km µ Exercise 7 A 50 Hz, 3 phase transmission line is 16 km long and has solid conductors 2.5 cm in diameter with a geometric mean spacing of 4.2 m. The line may be considered to be transposed. Calculate the sending end voltage when 100 MVA is delivered at zero power factor lagging and at a voltage of 132 kV. The line resistance and capacitance may be neglected. 136.62 kV s V = Exercise 8 A single phase power transmission line has its conductors 3.05 m apart and at the same height above earth. The two conductors of a telephone circuit are 0.61 m apart and they are both at the same height above earth but 2.8 m below the power line. The horizontal distance between the central plane of the power line and the central plane of the telephone line is 6.1 m. If the lines run parallel to each other for 1 km, calculate the 50 Hz voltage induced in the telephone circuit when the power line carries 500 A. V = 5.262 V Exercise 9 A three phase 50 Hz power transmission line has its conductors at the corners of an equilateral triangle 3.05 m apart. The two lower conductors are at the same height above earth and 3.65 m above the two conductors of a telephone line. These latter two conductors are 0.61 m apart and they are both at the same height above earth. The central plane of the power line and the central plane of the telephone line is coincident. If the lines run parallel to each other for 1 km, calculate the magnitude of the voltage induced in the telephone circuit when the power line carries 500 A symmetrical currents. V = 3.219 V Exercise 10 The law of Biot-Savart is H l R = ∫ C 4πiR2 (d × ˆ ) where the symbols have their usual meaning. Using this formula derive an expression for the flux linkage external to a long straight current carrying conductor.S2014 PCT Tutorial 3 Inductance Page 4 of 4 4/08/2014 Exercise 11 Show that the total internal flux linkage per unit length in a circular conductor with uniform current density is constant and is independent of the radius of the conductor. Exercise 12 Derive an expression for the inductance of a single-phase two-wire system, having conductors of radius r and separation d between centres, remote from earth. Exercise 13 Derive an expression for the flux linkages of one conductor in a group of n conductors carrying currents whose sum is zero. Hence derive an expression for the inductance per unit length of composite conductors of a one phase line consisting of m strands in one conductor and n strands in the other conductor. Exercise 14 Derive an expression for the inductance of a symmetrically spaced, three-phase, three-wire system having conductors or radius r and separation d between centres, remote from earth. Exercise 15 Derive an expression for the inductance of an unsymmetrically spaced, three-phase, threewire system having conductors or radius r, remote from earth. The lines may be assumed to be transposed at regular intervals.S2014 PCT Tutorial 4 Capacitance Page 1 of 4 4/08/2014 Power Circuit Theory: Tutorial Set 4 Exercise 1 Given a balanced system of three phase voltages, show analytically vab + vac = 3van Exercise 2 In a three phase transmission line the conductors are equilaterally spaced 7 m apart. The conductor radius is 2.5 cm. Find the charging current for a line length of 1 km, if the transmission line voltage is 132 kV at 50 Hz, given ε 0 = 8.85419 10 F/m. × −12 236.4 mA I j charging = Exercise 3 A three phase transmission line has conductors arranged horizontally as shown Figure E3. h A B C D D Figure E3: Conductor arrangement for Exercise 3. The line is fully transposed and the diameter of each conductor is 3.5 cm, h = 16 m, D = 7 m. Find the capacitive reactance of the transmission line and the sending end line charging current if the line length is 200 km with and without taking the effect of ground into account. The transmission line voltage is 132 kV at 50 Hz and ε 0 = 8.85419 10 F/m. × −12 Without ground: 1780 , 42.81 A X I j c charging = Ω = With ground: 1767 , 43.12 A X I j c charging = Ω =S2014 PCT Tutorial 4 Capacitance Page 2 of 4 4/08/2014 Exercise 4 Calculate the capacitance per metre of a single-phase 50 Hz transmission line remote from the earth using four copper conductors, each 16 mm diameter, arranged at the corners of a 500 mm square with polarities as shown in Figure E4. C =14.68 pF/m Figure E4: Conductor arrangement for Exercise 4. Exercise 5 A three phase transmission line has its conductors arranged in a triangular formation so that two of the distances between the conductors are 8 m and the third distance is 14 m. The conductor radius is 2.5 cm. If the length of the line is 180 km and the normal operating voltage is 220 kV determine the capacitive reactance to neutral for the entire length of the line, the sending end charging current per km and the total charging MVA at 50 Hz, given 12 ε 0 = 8.85419 10 F/m. × − X I j Q c charging =1893 , 372.8 mA/km, 25.57 MVAr Ω = = Exercise 6 Determine the capacitance per phase per km of the three phase transposed line as shown in Figure E6 given ε π 0 =10 /(36 ) F/m. −9 The diameter of each conductor is 2.532 cm. 7.5 m a b c a' b' c' 9 m 4 m 4 m Figure E6 Conductor arrangement for Exercise 6 = 19.0976 nF/km n CS2014 PCT Tutorial 4 Capacitance Page 3 of 4 4/08/2014 Exercise E7 Consider a three phase double circuit transposed line as shown in Figure E7 with bundled conductors. The details of spacing between conductors are given in Figure E7. The conductors have a diameter of 5 cm. The circuit arrangement is a, b, c, c', b', a'. Determine the capacitance per phase per km of the line and the sending end charging current per km of the line if the line voltage is 220 kV and 50 Hz frequency. Figure E7: Conductor arrangement for Exercise 7. C I j n charging = = 23.2655 nF/km, 928.376 mA/km Exercise 8 Derive an expression for the capacitance per unit length of a two-wire transmission line with conductors of radius r m, spaced D m apart, remote from earth. Exercise 9 Derive an expression for the capacitance per unit length of a three-wire transmission line with conductors of radius r m, symmetrically spaced D m apart, remote from earth. Exercise 10 Derive an expression for the capacitance per unit length of a three-wire transmission line with conductors of radius r m, unsymmetrically spaced Dab, Dbc, and Dca, m apart, remote from earth. Exercise 11 Derive an expression for the capacitance per unit length of a three-wire transmission line with conductors of radius r m, unsymmetrically spaced Dab, Dbc and Dca, m apart, located at heights ha, hb and hc m above a perfectly conducting earth plane.S2014 PCT Tutorial 4 Capacitance Page 4 of 4 4/08/2014 Exercise 12 Derive an expression for the capacitance per unit length of a single-wire transmission line with conductor of radius r m, located D m above a perfectly conducting earth plane earth. Show that the loci of the equipotentials form circles and determine the centre and radius of each circle.S2014 PCT Tutorial 5 Transmission Lines Page 1 of 4 4/08/2014 Power Circuit Theory: Tutorial Set 5 Exercise 1 A 20km, 50 Hz, single circuit three phase line delivers 2.4 MW at 11 kV to a balanced load. The resistance of the line is 236 mΩ/km and the inductance of the line is 1.32 mH/km. a) Determine the per phase series impedance of the line, Z j l = + (4.720 8.294) Ω b) What must the magnitude of the sending end voltage be when the power factor is i) 0.8 lagging, V =13.43 kV ii) Unity, V =12.17 kV iii) 0.6 leading V =10.13 kV c) Determine the percentage voltage regulation at the above power factors, i) Regulation 22.06 %, ii) Regulation 1 = = = 0.59 %, iii) Regulation 7.909 %. − d) Draw phasor diagrams depicting the operation of the line in each case. Exercise 2 A 400 kV, 50 Hz, 3 phase transmission line is 160 km long. The series impedance of the line is (0.10+j 0.35) Ω per km and the shunt admittance is j 3.75 × 10−6 S per km. Obtain the ABCD constants of the line using the nominal π -model. A j B j C j S D j = + (983.2 4.800 10 , 16 56 , 1.44 595 10 ; 983.2 4.800 10 )× −3 = + ( )Ω = (− + )× −6 = + ( )× −3. Exercise 3 The line in Problem 2 delivers 300 MVA at 0.8 power factor lagging at a voltage of 400 kV. Assuming Van as reference and positive phase sequence determine the a) sending end voltage, Vab, Vab = 429.0 33.78 kV ∠ ° b) sending end current, Ia = 360.8 18.82 A ∠ − ° c) voltage regulation, Regulation 9.085 % = d) sending end power and Sending end power 247.5 103.0 MVA = + ( j ) e) transmission efficiency. η = 96.97 % Exercise 4 The line in Exercise 2 delivers 240 MW at unity power factor. Assuming Van as reference and negative phase sequence a) sending end voltage, Vab, Vab = 404.4 24.96 kV ∠ − ° b) sending end current, Ia = 367.6 22.22 A ∠ ° c) voltage regulation, Regulation 2.837 % = d) sending end power and Sending end power 246.0 76.11 MVA = ( − j ) e) transmission efficiency. η = 97.56 %S2014 PCT Tutorial 5 Transmission Lines Page 2 of 4 4/08/2014 Exercise 5 Consider a 3 phase, 275 kV, 50 Hz transmission line, 320 km long. The parameters of the line are: Line resistance r = 0.04 Ω /km Line inductance L = 0.80 mH/km Line capacitance C = 140 pF/km The line delivers 100 MW power at 0.8 power factor lagging at a voltage of 275 kV. Using the long line transmission line model and assuming Van as reference and positive phase sequence calculate the a) sending end voltage, Vab, Vab = 302.5 34.89 kV ∠ ° b) sending end current, Ia = 260.9 36.47 A ∠ − ° c) sending end power and Sending end power 102.6 90.35 MVA = + ( j ) d) transmission efficiency. η = 97.43 % Exercise 6 Derive the ABCD parameters for the nominal π representation of the medium length transmission line and show that AD – BC = 1. 1 , , 1 , 1 . 2 4 2 Z Y Z Y Z Y = + = = + = + Z Y     A B C D   Exercise 7 Derive the ABCD parameters for the nominal T representation of the medium length transmission line and show that AD – BC = 1. 1 , 1 , , 1 . 2 4 2 Z Y Z Y Z Y = + = + = = + Z Y     A B C D   Exercise 8 Determine the efficiency and regulation of a three phase, 100 km, 50 Hz transmission line delivering 20 MW at a p.f. of 0.8 lagging and 66 kV to a balanced load. The conductors are of copper, each having resistance of 0.1 Ω per km, 1.5 cm outside diameter, spaced equilaterally 2 m between centres. Use (a) the nominal π and Regulation 18.11 %, 93.51 % = = η (b) the nominal T Regulation 18.04 %, 93.54 % = = η models for the line.S2014 PCT Tutorial 5 Transmission Lines Page 3 of 4 4/08/2014 Exercise 9 The constants of a three phase transmission line are A= 0.92 2 ∠ ° and B= 140 70 ∠ ° Ω per phase. At the receiving end the voltage is 132 kV and the load is 60 MVA at 0.8 power factor lagging. Calculate the sending end voltage. 178.962 12.593 kV s V = ∠ ° Exercise 10 A short three phase transmission line has negligible resistance and a series reactance of 16 Ω per phase. The input power to the line is (60 +j 48) MVA. If Vr = 100 kV per phase calculate Vs and Is. V I s = + = (102.4 3.2 kV / phase 200 150 A j j ) s ( − ) Exercise 11 A 132 kV three phase transmission line has a resistance of 12.5 Ω and a reactance of 33.5 Ω per phase. For a voltage drop of 10% of rated voltage, calculate the receiving end power if its power factor is a) Unity; b) 0.8 lagging Calculate also the power limit of the line if the voltages at the two ends are both equal to rated voltage. ( ) max a) 110.515 MW b) 45.4745 34.1058 MVA 316.945 MW r r j P = = + = S S Exercise 12 A three phase transmission line has a resistance of 10 Ω and a reactance of 30 Ω per phase. Calculate the maximum power that can be transmitted when the receiving end and sending end voltages are 132 kV and 135 kV respectively. P max = 385.32 MW Exercise 13 A 132 kV three phase transmission line has a resistance of 10 Ω and a reactance of 40 Ω per phase. The delivered load is 200 MW at 0.8 power factor lagging. If the voltages at the ends of the line are both 132 kV, calculate the loading of the required compensation plant and the load angle. QC = = ° 264.878 MVAr export, 31.6728 δS2014 PCT Tutorial 5 Transmission Lines Page 4 of 4 4/08/2014 Exercise 14 A three phase transmission line supplies 450 MW to a load at 0. 8 power factor lagging. The reactance of the circuit is 20 Ω per phase and its resistance may be neglected. Determine the reactive power supply required at the receiving end if the voltage at both sending and receiving ends is to be maintained at 275 kV. QC = 364.372 MVAr export Exercise 15 A three phase transmission line has a reactance of 24 Ω per phase and its series resistance and shunt capacitance be neglected. If the magnitude of the voltage at the receiving end is 132 kV and that at the sending end is 140 kV when the magnitude of the transmitted current is 450 A, calculate the power delivered at the receiving end. Sr = (95.5918 38.0433 MVA − j ) Exercise 16 A 66 kV short three phase transmission line has a series impedance of (0.075 +j 0.2) Ω per phase and a route length of 100 km. It is operating with 66 kV at each end and a load angle of 10º. Calculate the receiving end power and power factor. Sr = (32.0699 15.3351 MVA, 0.90216 leading − j pf ) = Exercise 17 A three phase transmission line has a series impedance of 300 78 ∠ ° Ω per phase and a total shunt admittance of j24 ×10−4 S per phase. The voltage at the receiving end is 220 kV but there is no load at that end. A load of 100 MW at unity power factor is connected at the midpoint of the line. Using a nominal π representation, calculate the magnitude of the sending end voltage. 184.267 kV VS =S2014 PCT Tutorial 6 Symmetrical Faults Page 1 of 5 4/08/2014 Power Circuit Theory: Tutorial Set 6 Exercise 1 An alternator and a synchronous motor are rated 30 MVA, 13.2 kV and each has a subtransient reactance of 20%. The line connecting them has a reactance of 10% on the base of the machine ratings. The motor is drawing 25 MW at 0.8 power factor leading and a terminal voltage of 12.8 kV when a symmetrical bolted three phase fault occurs at the motor terminals. Calculate the subtransient current in the alternator, motor and the fault using the internal voltage of the machines. I j I j I j fault g = − 10.60 kA, 1128 3396 A, 1128 7207 A = ( − ) m = (− − ) Exercise 2 Solve Problem 1 by the use of Thévenin's theorem. Z j V I j th fault fault = = 0.12 p.u., 0.97 0 p.u., 8.08 p.u. ∠ = −   Exercise 3 Consider a 2 generator power system shown in Figure E3. The impedance data of the network are given in Figure 1. If a symmetrical three phase fault with fault impedance of j0.08 p.u. occurs at bus 1, calculate the fault current and the change in bus voltages due to the fault current. Assume the prefault bus voltages as 1 p.u.. Figure E3: Power system for Exercise 3. I j V V fault = − ∆ 4.531 p.u., 0.6374 p.u., 0.4086 p.u.. 1 2 = − ∆ = − Exercise 4 Three 6.6 kV alternators of ratings 2 MVA, 5 MVA and 8 MVA and per unit reactances 0.08, 0.12 and 0.16 respectively are connected to a common busbar. From the busbar, a feeder cable of reactance 0.125 Ω connects to a substation. Calculate the fault MVA, if a three phase symmetrical fault occurs at the substation. S = 87.405 MVAS2014 PCT Tutorial 6 Symmetrical Faults Page 2 of 5 4/08/2014 Exercise 5 Consider a 3 generator power system feeding a load through 220 kV transmission line as shown in Figure 2. The impedance data of the network is given in Figure E5. A symmetrical bolted three phase fault occurs at load bus 2. Calculate the fault current and fault short circuit MVA. Assume the prefault bus voltages as 1 p.u.. Figure E5: Power system for Exercise 5. I j S fault fault = − 477.3 A, 181.9 MVA. = Exercise 6 Two synchronous motors having subtransient reactances of 0.8 and 0.25 p.u., respectively, on a base of 480 V 2 MVA are connected to a bus. This motor bus is connected by a line having a reactance of 23 mΩ to a bus of a power system. At the power system bus the short circuit MVA of the power system is 9.6 MVA for a nominal voltage of 480 V. When the voltage at the motor bus is 440 V, calculate the fault current when a symmetrical bolted three phase fault occurs at the motor terminals. Neglect the load current. 16.98 kA I j fault = − Exercise 7 A three phase transformer is connected star/star. It supplies a star connected load of (400 + j 600) Ω/ phase through a transmission line, each conductor of which has impedance (4 + j 6) Ω. The secondary winding of the transformer has three times as many turns as the primary. The transformer has parameters Primary R = 0.5 Ω/phase X = 2.5 Ω/phase Secondary R = 5.0 Ω/phase X = 25.0 Ω/phaseS2014 PCT Tutorial 6 Symmetrical Faults Page 3 of 5 4/08/2014 The transformer is fed from an alternator rated at 1.5 MVA, 11 kV, 0.2 p.u. transient reactance. Calculate the magnitude of the transformer secondary terminal voltage when the primary terminal voltage is 11 kV. Vsec = 32.088 kV If a bolted symmetrical fault occurs half way along the transmission line, calculate the magnitude of the alternator current. The alternator e.m.f. behind its transient reactance can be assumed constant. 339.09 A Ialt = Exercise 8 Derive the delta-star and star-delta transformations. Exercise 9 Consider a simple power system shown in Figure E9. The generators are represented using transient reactance and other reactances are expressed in p.u. values. Shunt capacitances and resistances are neglected. A symmetrical three phase fault is initiated with fault impedance Zfault = j 0.12 p.u. at a) Bus 1, b) Bus 2 and c) Bus 3. Assume the prefault bus voltages as 1 p.u..Calculate the fault current, bus voltages and line currents in each case. z = j 0.6 12 Figure E9: Network for Exercise 9S2014 PCT Tutorial 6 Symmetrical Faults Page 4 of 5 4/08/2014 1 2 3 12 23 13 1 2 a) Fault at bus 1 3.346 p.u., 0.4015 p.u., 0.6934 p.u., 0.5474 p.u., 0.4866 p.u., 0.3650 p.u., 0.3650 p.u.. b) Fault at bus 2 2.919 p.u., 0.7325 p.u., 0.3 fault fault I j V V V I j I j I j I j V V = − = = = = = − = = − = = 3 12 23 13 1 2 3 12 23 503 p.u., 0.5414 p.u., 0.6369 p.u., 0.4777 p.u., 0.4777 p.u.. c) Fault at bus 3 2.145 p.u., 0.7098 p.u., 0.6630 p.u., 0.2574 p.u., 0.0780 p.u., 1.014 p. fault V I j I j I j I j V V V I j I j = = − = = − = − = = = = − = − u., 1.131 p.u.. I j 13 = − Exercise 10 Three star connected 11 kV alternators are connected each in series with a similar current limiting reactor to a common busbar. The alternators each have a rating of 10 MVa and sub transient reactance/phase of 0.06 p.u.. Two 11/33 kV transformers of 15 MVA rating, 0.03 p.u. reactance and 10 MVA rating, 0.02 p.u. reactance respectively, connected in parallel to this busbar, supply a transmission line of impedance (0.2 + j 0.7) Ω/ km/phase. At a substation 10 km from the generating station is a 25 MVA 33/11 kV transformer of 0.06 p.u. reactance. Calculate the reactance of the current limiting reactors if each alternator is not to carry more than 7/3 times full load current, when a symmetrical short circuit occurs on the 11 kV busbars in the sub- station. 0.84915 r X = Ω Exercise 11 A power station contains three identical three phase 50 Hz alternators each of 200 MVA and 0.9 p.u. synchronous reactance. These machines are running equally loaded and each is connected to an 11/132 kV transformer of 200 MVA and 0.09 p.u. reactance. The 132 kV busbar feeds a short transmission line having a resistance of 0.5 Ω and an inductance of 8 mH for each phase conductor. The resistance of the transformers and the alternators may be neglected. A load of 400 MW at 0.8 p.f. lagging is supplied by the line. If one of the phase voltages at the load end of the line is taken as 1.0 0 ∠ per unit, calculate the e.m.f. of the corresponding phase in each of the alternators. E =1.7043 24.586 p.u. ∠ ° If a three phase short circuit occurs at the load end of the transmission line, calculate the magnitude of the steady state fault current. 4.1540 kA I f =S2014 PCT Tutorial 6 Symmetrical Faults Page 5 of 5 4/08/2014 Exercise 12 Three 11 kV alternators rated at 10 MVA, with a resistance of 0.02 and subtransient reactance of 0.15 p.u., have their busbars connected by three three-phase mesh connected reactors, each having 0.015 per unit resistance and 0.09 per unit reactance/phase on a 10 MVA rating. A 20 MVA transformer of 0.01 per unit resistance and 0.10 reactance is connected to the busbars of one machine and feeds at 132 kV, a transmission line that has 9 Ω resistance and 30 Ω reactance per phase. Calculate the magnitude of the currents that flow in each of the machines when a symmetrical fault occurs at the end of the transmission line. If an oil circuit breaker to be installed at the point of the fault is to have a rating determined by this fault condition, what must be its minimum rating? I I I S 1 2 3 = = = = 1076.76 kA, 1076.76 kA, 1725.76 kA, 73.9088 MVA Exercise 13 A power station has three section busbars, with 11 kV generators, connected on the ring system as follows: Section 1, two generators of 30 and 20 MVA with 0.15 and 0.12 per unit reactances respectively; Section 2, one generator of 60 MVA and 0.2 per unit reactance; Section 3, two generators of 30 and 60 MVA with 0.18 and 0.2 per unit reactances respectively; The sections 1 and 2, 2 and 3, and 3 and 1 are connected through reactors of 60, 90 and 80 MVA with reactances 0.2, 0.3 and 0.24 respectively. Determine the magnitude of the fault current when a three phase symmetrical fault occurs at the far end of a feeder with a 0.05 per unit reactance and 0.01 per unit resistance per phase and rating 30 MVA connected to number three section busbar, using a) Mesh analysis; b) Nodal analysis; c) Thévenin's theorem and d) The delta-star transformation. Calculate the magnitude of the current supplied by the 60 MVA generator connected on section three busbar under this condition. I I fault G MVA = = 17.7967 kA, 6.7467 kA 3_ 60S2014 PCT Tutorial 7 Zbus Symmetrical Faults Page 1 of 3 4/08/2014 Power Circuit Theory: Tutorial Set 7 Exercise 1 A generator and a synchronous motor are rated 30 MVA, 13.2 kV and each has a subtransient reactance of 20%. The line connecting them has a reactance of 10% on the base of the machine ratings. The motor is drawing 20 MW at 0.8 power factor leading and a terminal voltage of 12.8 kV when a symmetrical bolted three phase fault occurs at the motor terminals. Calculate the subtransient current in the fault using the Zbus matrix. Z j V I j 22 0 = = 0.12 p.u., 0.97 0 p.u., 8.08 p.u. m faultm ∠ = − Exercise 2 Solve problem 1 when a bolted fault occurs at the terminals of the generator. Z j V I 11 0 = = 0.12 p.u., 0.9207 4.282 p.u., 7.673 85. g ∠ ° = faultg ∠ − 72 p.u.. ° Exercise 3 Consider a 2 generator 4 bus power system shown in Figure E3. The impedance data of the network in p.u. are given in Figure E3. A bolted symmetrical three phase fault occurs at bus 4. Assuming the prefault bus voltages as 1 p.u., calculate the fault current and the current delivered by the generators during the fault and the bus voltages during the fault. Figure E3: Power system for Exercise 3. _ . . 1_ _ . . 2_ _ . . 1_ _ . . 2_ _ . . 3_ _ . . 4_ _ . . 4.005 p.u., 2.0314 p.u., 1.9732 p.u., 0.6547 p.u., 0.6646 p.u., 0.6598 p.u., 0.4005 p.u.. fault p u G fault p u G fault p u fault p u fault p u fault p u fault p u I j I j I j V V V V = − = − = − = = = =S2014 PCT Tutorial 7 Zbus Symmetrical Faults Page 2 of 3 4/08/2014 Exercise 4 Consider a 2 generator 4 bus power system shown in Figure E4. The impedance data of the network are given in Figure 4. A symmetrical three phase fault occurs at bus 4 through a fault impedance of j 0.06 p.u., using the Zbus matrix method calculate the fault current, the bus voltages and currents between bus sections during the fault. Assume the prefault bus voltages as 1 p.u.. Figure E4: Power system for Exercise 4. _ . . 1_ _ . . 2 _ _ . . 3_ _ . . 4 _ _ . . 12 _ _ . . 13_ _ . . 14 _ 3.6847 p.u., 0.5538 p.u., 0.6154 p.u., 0.6673 p.u., 0.2189 p.u., 0.1540 p.u., 0.5158 p.u., fault p u fault p u fault p u fault p u fault p u fault p u fault p u fault I j V V V V I j I j I = − = = = = = = _ . . 24 _ _ . . 34 _ _ . . 0.6698 p.u., 1.983 p.u., 0.9964 p.u.. p u fault p u fault p u j I j I j = − = − = − Exercise 5 Consider a 3 generator 9 bus power system shown in Figure E5. The impedance data of the network are given in Figure 3. Determine the admittance matrix of the system. Figure E5: Power system for Exercise 5.S2014 PCT Tutorial 7 Zbus Symmetrical Faults Page 3 of 3 4/08/2014 8.403 1.736 6.976 1.6 6.706 1.706 0.5192 3.118 0.2445 1.630 0.2747 1.488 1.736 0.2445 1.630 0.4890 4.996 0.2445 1.630 0.2445 1.630 0.5192 3.118 0.2747 1.488 1.6 0.2747 1.488 0.5113 j j j j j j j j j Y j j j j j j j j j − − − − − + − + = − + − − + − + − − + − + − 5.059 0.2366 1.972 0.2366 1.972 0.5443 4.433 0.3077 2.462 1.706 0.2747 1.488 0.3077 2.462 0.5824 5.656 j j j j j j j j j                     − +    − + − − +     − + − + −  Exercise 6 Three 11 kV alternators rated at 10 MVA, with a resistance of 0.02 and subtransient reactance of 0.15 p.u., have their busbars connected by three three-phase mesh connected reactors, each having 0.015 per unit resistance and 0.09 per unit reactance/phase on a 10 MVA rating. A 20 MVA transformer of 0.01 per unit resistance and 0.10 reactance is connected to the busbars of one machine and feeds at 132 kV, a transmission line that has 9 Ω resistance and 30 Ω reactance per phase. Calculate the magnitude of the currents that flow in each of the machines when a symmetrical fault occurs at the end of the transmission line using the Zbus method. If an oil circuit breaker to be installed at the point of the fault is to have a rating determined by this fault condition, what must be its minimum rating? I I I S 1 2 3 = = = = 1076.76 kA, 1076.76 kA, 1725.76 kA, 73.9088 MVA Exercise 7 A power station has three section busbars, with 11 kV generators, connected on the ring system as follows: Section 1, two generators of 30 and 20 MVA with 0.15 and 0.12 per unit reactances respectively; Section 2, one generator of 60 MVA and 0.2 per unit reactance; Section 3, two generators of 30 and 60 MVA with 0.18 and 0.2 per unit reactances respectively; The sections 1 and 2, 2 and 3, and 3 and 1 are connected through reactors of 60, 90 and 80 MVA with reactances 0.2, 0.3 and 0.24 respectively. Determine the magnitude of the fault current when a three phase symmetrical fault occurs at the far end of a feeder with a 0.05 per unit reactance and 0.01 per unit resistance per phase and rating 30 MVA connected to number three section busbar, using the Zbus method. Calculate the magnitude of the current supplied by the 60 MVA generator connected on section three busbar under this condition. I I fault G MVA = = 17.7967 kA, 6.7467 kA 3_ 60S2014 PCT Tutorial 8 Symmetrical Components.doc Page 1 of 3 12/09/2014 Power Circuit Theory: Tutorial Set 8 Exercise 1 Calculate the symmetrical components of the following voltages and currents: i) 240 35 V, 210 259 V, 200 135 V ii) 10 35 A, 14 260 A, 12 110 A a b c a b c V V V I I I ‘  q ‘ q ‘ q ‘  q ‘ q ‘ q 0 1 2 0 1 2 i) 67.65 85.73 V, 196.2 1.706 V, 64.52 94.73 V ii) 2.804 78.64 A, 11.10 5.222 A, 3.946 149.9 A a a a a a a V V V I I I ‘  q ‘  q ‘  q ‘  q ‘  q ‘  q Exercise 2 Calculate the phase voltages and currents from the symmetrical components given below: 0 1 2 0 1 2 i) 60 30 V, 130 120 V, 110 30 V ii) 3 30 A, 8 90 A, 6 30 A a a a a a a V V V I I I ‘  q ‘ q ‘ q ‘  q ‘ q ‘ q i) 160.3 59.14 V, 90.23 16.09 V, 252.9 92.96 V ii) 12.29 50.63 A, 5 30 A, 12.29 110.6 A a b c a b c V V V I I I ‘ q ‘ q ‘  q ‘ q ‘  q ‘  q Exercise 3 A three phase star connected load of (4 + j 2) : per phase is connected across an unbalanced 3 phase system with the following line voltages: V V V ab bc ca ‘  q ‘ q ‘  q 400.5 32.45 V, 420.5 125.54 V, 157.9 126.3 V Calculate the following: i) The symmetrical components of the line voltages; V V V ab ab ab 0 1 2 0, 154.7 77.51 V, 311.1 11.85 V ‘  q ‘  q ii) The phase voltages using the answers to i); V V V an bn cn ‘  q ‘ q ‘  q 146.8 11.48 V, 268.6 136.3 V, 164.4 72.17 V iii) The line currents; I I I a b c ‘  q ‘ q ‘  q 32.81 38.05 A, 60.07 109.7 A, 36.75 98.74 A iv) The complex power (compute using both phase and sequence quantities). S 26.99 26.57 kVA ‘ qS2014 PCT Tutorial 8 Symmetrical Components.doc Page 2 of 3 12/09/2014 Exercise 4 Given 2 2 3 3 2 2 2 2 3 3 1 1 1 1 1 1 1 1 1 1 j j j j e e a a a a e e S S S S   ª º « » ª º « » « » « » « » « » « ¬ » ¼ ¬ ¼ A show 1 2 2 1 1 1 1 1 3 1 a a a a  ª º « » « » « ¬ » ¼ A Exercise 5 Given 2 2 1 1 1 1 1 a a a a ª º « » « » « ¬ » ¼ A Show 1 3 3 1 0 0 0 1 0 0 0 1  ª º « » « » « » ¬ ¼ A U A U Exercise 6 Given 2 2 1 1 1 1 1 a a a a ª º « » « » « ¬ » ¼ A Show 1 1 1 1 3 0 0 1 1 1 0 0 0 1 1 1 0 0 0  ª º ª º « » « » « » « » « ¬ » « » ¼ ¬ ¼ A AS2014 PCT Tutorial 8 Symmetrical Components.doc Page 3 of 3 12/09/2014 Exercise 7 Derive the impedance matrix, in the 012 domain, for the circuit shown in Figure E7. Ia Z ng Va Ib Ic V Vc b In Za Zb Zc Vbn Vcn V ng V an Figure E7 Exercise 8 Show that if s M M abc M s M M M s Z Z Z Z Z Z Z Z Z Z       ª º « » « » « ¬ » ¼ Z 012 is diagonal. Exercise 9 Show analytically the sum of a balanced set of phasors is zero. Exercise 10 Show analytically the sum of two balanced, but unequal, sets of phasors of the same sequence is also a balanced set of phasors. Exercise 11 Show the sum of two balanced phasors of opposite sequence is an unbalanced set of phasors that sums to zero.S2014 PCT Tutorial 9 Unsymmetrical Faults Page 1 of 3 17/10/2014 Power Circuit Theory: Tutorial Set 9 Exercise 1 A single line to ground fault occurs on the terminals of an 11 kV, 50 MVA three phase generator with its neutral solidly grounded. The sub-transient reactance of the generator is 0.22 p.u. and the negative and zero sequence reactances are 0.34 p.u. and 0.12 p.u. respectively. Determine the fault current and line to line voltages if the generator is not loaded. Assume the fault impedance as zero and 11 kV and 50 MVA as base quantities. Assume the fault occurs between the a phase and ground and positive sequence exists. _ . . _ . . _ . . 4.412 p.u., 0, 0 11.58 kA 6.685 75.44 kV, 12.94 90 kV, 6.685 104.6 kV a fault p u b p u c p u fault ab bc ca I I j I I I j V V V = = − = = = − = ∠ ° = ∠ − ° = ∠ ° Exercise 2 Solve Problem 1 above for a line to line fault between phases b and c with zero fault impedance. _ _ . . _ _ . . _ 0, 3.093 p.u., 3.093 p.u. 8.117 kA 11.57 0 kV, 0, 11.57 180 kV a b fault p u c fault p u b fault ab bc ca I I I I V V V = = − = = − = ∠ = = ∠ ° Exercise 3 Solve Problem 1 above for a double line to ground fault involving phases b, c and ground with zero fault impedance. _ _ . . _ _ . . _ _ . . 0, 5.04 134.6 p.u., 5.04 45.44 p.u. 18.85 kA, 5.474 0 kV, 0, 5.474 180 kV a fault p u b fault p u c fault p u fault b c ab bc ca I I I I I I j V V V = = ∠ ° = ∠ ° = + = = ∠ = = ∠ °S2014 PCT Tutorial 9 Unsymmetrical Faults Page 2 of 3 17/10/2014 Exercise 4 A three phase synchronous generator is star connected and grounded through a reactance of 0.12 p.u.. The generator is running at no-load generating rated voltage. An unbalanced fault occurs leading to a fault current of I I a b = = 0 p.u., 3.93101 162.858 p.u., ∠ ° and Ic = 3.93101 17.1422 p.u. ∠ ° Assume the sequence reactance of the machine as X X 0 1 = = 0.09 p.u., 0.21 p.u., and X 2 = 0.26 p.u. . Calculate the following during the fault: i) The terminal voltages of the machine with respect to ground. V V V a b c = = = 0.9964 p.u., 0.3939 p.u. − ii) The neutral voltage of the machine with respect to ground Vn = 0.2781 p.u. Exercise 5 A two source utility model is shown in Figure E5. G 1 T2 T1 G 2 A B Line 1 C Line 2 D E Figure E4: Two source utility model The generator and transformer configurations are indicated in the diagram. The neutrals of the generators are grounded through j0.3 p.u. reactors, on the machine base. The generators are not loaded and are running at rated frequency and voltage with their emfs in phase. The equipment data are given in Table E5. Equipment MVA Voltage (kV) X1 X2 X0 G1 30 25 0.22 p.u. 0.36 p.u. 0.08 p.u. G2 40 13.2 0.15 p.u. 0.25 p.u. 0.06 p.u. T1 40 25/115 8.0 %. 8.0 %. 8.0 %. T2 50 13.2/115 8.5 %. 8.5 %. 8.5 %. Line 1 115 8 Ω 8 Ω 16 Ω Line 2 115 6 Ω 6 Ω 15 Ω Table E5: Power system parameters.S2014 PCT Tutorial 9 Unsymmetrical Faults Page 3 of 3 17/10/2014 If the following different types of fault appear at bus C with Zf = j 16 Ω, determine the fault current If. a) Balanced three phase fault _ _ _ I I I f a f b f c =1022 1022 ∠ − 90 A, 150 A, 1022 ° = ∠ ° = ∠30 A. ° b) Single line, phase a, to ground fault _ _ _ I I I f a f b f c = 984.8 90 A, 0, 0. ∠ − ° = = c) Line to line, b-c phases, fault _ _ _ 0, 848.5 180 A, 848.5 0 A. I I I f a f b f c = = ∠ ° = ∠ d) Double line, b-c phases, to ground fault I f =1053 90 A. ∠ ° Exercise 6 Derive the equation for the fault current when a single line to ground fault occurs on a balanced network. State any assumptions made. Exercise 7 Derive the equation for the fault current when a double line fault occurs on a balanced network. State any assumptions made. Exercise 8 Derive the equation for the fault current when a double line to ground fault occurs on a balanced network. State any assumptions made.PCT Tutorial 10 Power System Transients Page 1 of 2 23/10/2014 Power Circuit Theory: Tutorial Set 10 Exercise 1 Consider the RLC circuit shown in Figure E1. The switch S is closed at t = 0. Determine the instantaneous current at t = 0.4 s, given, R = 2 Ω, L =0.2 H, C = 5/9 F, V = 200 V. Plot the current wave form. If the value of the capacitance is changed to 0.5 mF sketch the resulting waveform. V S i R L C Figure E1: Circuit for Exercise 1. i (0.4) = 80.37 A Exercise 2 For the circuit shown in Figure E2 the switch S is closed at t = 0. Show the current is ( ) ( ) ( ) ( ) 2 2 2 sin sin R t i t t e u t Vm L R L ω φ θ φ θ ω   − = + − − −   +   , where tan 1 L ω R θ = − , given v t V t ( ) = + m sin (ω φ ) v i L S R Figure E2: Circuit for Exercise E2 Exercise 3 The RL circuit shown in Figure E3 is supplied with a voltage v t t ( ) = − 310sin 100 V. ( π φ) The circuit S is closed at t = 0. Determine the instantaneous current at t = 69 ms if φ = 45°. Draw the current waveform. Sketch the current waveform for the conditions tan and tan . 1 1 2 L L R R ω π ω φ φ = = − − − −PCT Tutorial 10 Power System Transients Page 2 of 2 23/10/2014 Figure E3: Circuit for Exercise E3. i (0.069) = 4.070 A Exercise 4 The LC circuit shown in Exercise E4 is supplied with a voltage v t t ( ) = − 310sin 100 V. ( π φ) A fault occurs when the voltage wave reaches a maximum. Draw the re-striking voltage waveform across the circuit breaker, CB. If the load side of the fault has an inductance of 2 H and a capacitance of 0.8 F determine the double frequencies of oscillations. Figure E4: Circuit for Exercise E4. f f 1 2 = = 503.3 mHz , 125.5 mHz Exercise 5 Show that the energy stored in the electric field of a capacitor is ½ C V2, where C is the capacitance in farads and V is the potential difference in volts between the plates of the capacitor. Exercise 6 Show that the energy stored in the magnetic field of an inductor is ½ L I2, where L is the inductance in henries and I is the current in amperes in the inductor. Exercise 7 Show that the energies in the electric and magnetic fields of the initial forward travelling waves of voltage and current on a transmission line are equal.