TUTORIAL SHEET-1
Subject: Power System Analysis & Stability (UEE602) 1. A 230 KV transmission line has a series impedance of (4+j60) Ohms and a shunt admittance -3
of j2×10
S. Using 100 MVA and the line voltage as base values, calculate per unit
impedance and per unit admittance of the line. 2. A three-phase star-connected system is rated as 50MVA and 110kV. Express 40,000 kVA of three-phase apparent power as a per-unit value referred to (a) the three-phase system kVA as base, and (b) the per phase KVA as base. 3. A 5KVA 400/200 V, 50 Hz single-phase transformer has primary and secondary leakage reactance each of 2.5 Ω. Determine the total reactance in per unit. 4.
Assuming S base of 25 and 60 MVA calculate the through impedance in ohms between the generators and the output terminals of the transformer for the system shown in fig. The specifications of the generators and transformer are given in table:Generators1 30 MVA 11 kV X’’ = 0.20 pu
Generators 2 25 MVA 11 kV X’’ =0.25 pu
Transformer 60 MVA 11 kV (delta), 66 kV (star) X = 0.10 pu
G1 (Transformer in delta star) G2
5. A 50 MVA 15 KV three-phase generator has a sub transient reactance of 0.20 per unit. The generator supplies two motors over a transmission line having transformer at both ends, as shown on the one-line diagram of fig. The motors have rated inputs of 30 MVA and 20 MVA, both 30 KV with 0.15 pu sub transient reactance. The rating of the sending-end transformer T1 is 50 MVA 11∆-132Y 11∆ -132Y with leakage reactance of 0.10 per unit. Transformer T2 at the receiving end has three single phase transformers connected as a three-phase unit. The rating of each individual transformer is 20 MVA, 33/76 kV with leakage reactance of 0.12 per unit. Series impedance of the line is (25+j75)ohms. Draw the impedance diagram with all impedances marked in per unit. Select the generator rating as the base in the generator circuit.
(45.1 KV) (15 KV)
11/132 KV 50 MVA
(180 KV) 25+j75 Ω
131.6/33 KV 60 MVA
50 MVA 30KV, j0.15
50 MVA 15KV j0.20
j0.10
j0.12
20 MVA 30 KV, j0.15
6. Fig. shows a two machine system. The rating are as follows: Synchronous generator 20 MVA, 11 kV, X’’ =0.5 pu Synchronous motor 15 MVA, 11 kV, X’’ = 0.15 pu Transformer T1 25 MVA, 12.5 ∆/132Y kV Transformer T2 20 MVA, 132Y/11∆ kV Line (200+j500)Ω Static load 5 MVA 0.8 power factor lagging Draw the impedance diagram for the system. Choose a base voltage of 132 kV for the transmission line and a base volt-ampere of 20 MVA If the motor is a synchronous machine drawing 15 MVA at 0.9 power factor (leading) and the terminal voltage 1.1 pu find the generator bus voltage.
TUTORIAL SHEET-2
Subject: Power System Analysis & Stability (UEE602) 1) Compute the Y-bus matrix for the following bus configurations by singular transformation method and verify your answer by direct inspection method. 2
j0.2
j0.25
1
3
j0.5 j0.3 j0.2
4
j0.25
5
Fig 1 2
0 . 0 8 + j 0 . 2 4
8 1 0 . j 6 + 0 0 .
1
3 0.03+j0.09
Fig 2
2
0 . 2 0 + j 0 . 8
0 1 0 . j + 5 2 0 .
0.20+j0.8
1
0 . 3 + j 1 . 2
3
6 0 . j 5 + 1 . 0
4
Fig 3
TUTORIAL SHEET-3
Subject: Power System Analysis & Stability (UEE602) 1) For the sample system of Fig.1 the generator are connected at all the four buses, while load are at buses 2, 3 and 4. The values of real and reactive power listed in table 1. All buses other than slack are of PQ-type. Line data are given in table 2. Find the voltages and the bus angles at the three buses using the Gauss iterative and Gauss Seidel methods (Solve upto 2 iterations). Using the results obtained, find slack bus power injections and line flows.
Fig.1: Sample System. Table 1:- Input Data Bus 1 2 3 4
Pi (p.u) -0.45 -0.51 -0.60
Q i (p.u) -0.15 -0.25 -0.30
Vi (p.u) o 1.050 -
Type of bus Slack PQ PQ PQ
Table 2:- Line Data Line No. 1 2 3 4
Bus Code (p-q) 1-2 1-4 2-3 3-4
Line impedance 0.08+j0.20 0.05+j0.10 0.04+j0.12 0.04+j0.14
2) In question 1, let second be a PV bus now with |V 2 |= 1.00 .Find the voltages and bus angles at the three buses by using Gauss iterative and Gauss Seidel methods (solve upto 2 iterations). Using the results obtained, find slack bus power injections and line flows. Table 3 Input Data Bus 1 2 3 4
Pi (p.u) -0.45 -0.51 -0.60
Q i (p.u) -0.25 -0.30
Vi (p.u) o 1.050 1.00 -
Type of bus Slack PV PQ PQ
TUTORIAL SHEET-4
Subject: Power System Analysis & Stability (UEE602) __________________________________________________________________________________ 1) In a system of four buses generator is connected at all the buses, while load are at buses 2, 3 and 4. The values of real and reactive power listed in table 1. All buses other than slack are of PQ-type. Line data are given in table 2. Find t he voltages and the bus angles at the three buses using Newton-Raphson method (Solve upto 2 iterations). Using the results obtained, find slack bus power injections and line flows. Table 1:- Input Data Bus 1 2 3 4
Pi (p.u) -0.45 -0.51 -0.60
Q i (p.u) -0.15 -0.25 -0.30
Vi (p.u) o 1.050 -
Type of bus Slack PQ PQ PQ
Table 2:- Line Data Line No. 1 2 3 4
Bus Code (p-q) 1-2 1-4 2-3 3-4
Line impedance 0.08+j0.20 0.05+j0.10 0.04+j0.12 0.04+j0.14
2) The data for the three bus system is given in table (a) & table (b). Controllable reactive power is available at bus 3 with the constraint,
Table (a):- Bus data BUS
REAL LOAD DEMAND
1 2 3
2.0 0.0 1.5
REACTIVE LOAD DEMAND 1.0 0.0 0.6
REAL POWER GENERATION Unspecified 0.5 0.0
REACTIVE POWER GENERATION Unspecified 1.0 Q G3=?
VOLTAGE SPECIFICATION
V1 = 1.04+j0 (slack Bus) Unspecified (PQ Bus) V3=1.04
Table (b):- Line Data Line No. Bus Code (p-q) Line impedance Shunt admittance 1 1-2 0.02+j0.08 j0.02 2 2-3 0.02+j0.08 j0.02 3 3-1 0.02+j0.08 j0.02 Find the load flow solution using the NR method. Use a tolerance of 0.01 for power mismatch.
TUTORIAL SHEET-5
Subject: Power System Analysis & Stability (UEE602)
1) Consider three bus systems. The line reactances are given in the table ( a) the line resistances are neglected. The magnitude of all the three bus voltages is specified as 1.00 pu. The power at the buses is listed in table (b). Table (a):- Line Data Line No.
Bus Code (p-q)
Line impedance (pu)
1
1-2
j0.2
2
2-3
j0.2
3
3-1
j0.2
Table (b):- Input data (in pu) BUS
REAL LOAD DEMAND
REACTIVE REAL POWER REACTIVE VOLTAGE SPECIFICATION LOAD GENERATION POWER DEMAND GENERATION 1 2.0 1.0 Unspecified Unspecified V1 = 1.04+j0 (slack Bus) 2 0.0 0.0 0.5 1.0 Unspecified (PQ Bus) 3 1.5 0.6 0.0 Q G3=? V3=1.04 Obtain the load flow solution by Decoupled Newton ’s and Fast Decoupled Load Flow methods after 3 iterations. 2) If a capacitor of – j0.005 pu reactance is added in a line 2-3, obtain the load flow solution using FDLF method after 3 iterations. Compare the results obtained with those in Q1 using FDLF method.
TUTORIAL SHEET-6
Subject: Power System Analysis & Stability (UEE602)
1) Consider a three bus system. The line data and bus data are given in the tables below. Obtain load flow solution using FDLF method up to 3 iterations. Table (a):- Line Data Line No.
Bus Code (p-q)
Line impedance
1
1-2
j0.25
2
2-3
j0.25
3
3-1
j0.25
Table (b):- Input data BUS
BUS TYPE
Vi(PU)
i
Pi
Q i
1 2
Slack bus PQ bus
1.00 ?
0 ?
-0.5
-0.4
3 PV bus 0.90 ? 0.5 Now, a capacitor of reactance – j0.10 is connected between PV bus and the ground. Obtain load flow solution using FDLF method up to 3 iterations, and compare the results. 2) A delta connected balanced resistive load is connected across unbalanced three-phase supply as shown in fig (a). With current in line A and B specified, find the symmetrical components of line currents. Also fine the symmetrical components of the delta currents. The value of resistor is R. The currents in phase A and phase B are 5-j2.5 and 7.5+j10 Amperes, respectively. A
B
C
3) A 25 MVA, 11kV, three-phase generator has a sub transient reactance of 20%. The generator supplies two motor over a transmission line with transformers at both the ends. The motors have inputs of 15 MVA and 7.5 MVA, both 10 kV with 25% sub transient reactance. Three phase transformer are both rated 30 MVA, 10.8/121 kV connection delta-star with leakage reactance 10% each. The series reactance of line 100 ohms. Draw positive and negative sequence of the network of the system in per unit.
TUTORIAL SHEET-7
Subject: Power System Analysis & Stability (UEE602) 1) Determine Z-bus for the following system building algorithm. Verify your result using -1
the relation ZBUS = YBUS .
2)
Consider the network and evaluate Z-bus by c urrent injection method.
3) Determine Z-bus for the following system building algorithm. Verify your result using -1
the relation ZBUS = YBUS .
TUTORIAL SHEET-9
Subject: Power System Analysis & Stability (UEE602) 1) For the radial network shown in Fig 9.6, a three-phase fault occurs at F. Determine the fault current and the line voltage at 11 kV bus under fault conditions. 10 MVA 15%
G1
G2
10 MVA 12.5%
11 kV
10 MVA, 10%
33 kV
5 MVA, 8%
6.6 kV 0.135+j0.08 F
3 km cable
2) Three 6.6 kV generators A, B and C, each of 10% leakage reactance and MVA rating 40, 50 and 25, respectively are interconnected electrically, as shown in Fig, by a tie bar through current limiting reactors, and each of 12% reactance based upon the rating of machine to which it is connected. A three-phase feeder is supplied from the bus bar of generator A at a line voltage of 6.6 kV. The feeder has a resistance of 0.12 ohms/phase. Estimate the maximum MVA that can be fed into a symmetrical short circuit at the far end of the feeder. G1
12%
G2
12%
G3
12%
3) Consider the four bus system of Fig. Buses 1 and 2 are generator buses and 3 and 4 are load buses. The generator are rated 11 kV, 100 MVA, with transient reactance of 10% each. Both the transformers are 11/110 kV, 100 MVA with a leakage reactance of 5%. The reactances of line to the base of 100 MVA, 110 kV are injected on the figure. Obtain the short circuit solution for the three phase solid fault on bus 4. G1
3 1
j0.15
j0.2
j0.1
j0.1
j0.15
4
2
G2
4) A synchronous generator and a synchronous motor each rated 25 MVA, 11 kV having 15 % sub transient reactance are connected through transformers and a line as shown in Fig. The transformers are rated 25MVA, 11/66 kV with leakage reactance 10%. The line has a reactance of 10 % on the base of 25 MVA, 66 kV. The motor is drawing 15 MW at 0.8 pf leading and a terminal voltage of 10.6 kV when a symmetrical 3-phase fault occurs at the motor terminals. Find the substrasient current in generator motor and fault.
G1
Motor
T1
T2
Line
TUTORIAL SHEET-10
Subject: Power System Analysis & Stability (UEE602) 1) Two 11 kV, 20 MVA, three-phase, star connected generation operate in parallel, the positive, negative and zero sequence reactance of each being, respectively, j 0.18, j 0.15, j 0.10 pu. The star point of one generator is isolated and that of the other is
earthed through 2.0 ohm resistor. A line-to-line fault occurs at the terminals of the generators. Estimate (i) the fault current, (ii) current in grounding resistor, and (iii) the voltage across grounding resistor. 2) On the base of 25 MVA and 11 kV in generator circuit, draw the positive, negative and zero sequence network of the system shown in fig. before occurrence of solid LG at bus g the motors are loaded to draw 15 and 7.5 MW at 10 kV, 0.8 leading pf. If prefault current is neglected, calculate the fault current and sub transient current in all parts of the system.
Three phase generator has sub transient reactance of 20% its zero sequence reactance is 0.06. The generator is solidly grounded with reactance 2.5 ohms. Both the motors are rated with 10 kV with 25% reactance. The transformers are rated as 30MVA, 10.8/121 kV connected in star delta each having reactance of 10%, star side is grounded which is on generator side. The series reactance of the line is 100 ohms. 3) For the system shown in question 2) assume the LLG fault and calculate fault current and voltage of the healthy bus. 4) A 10MVA, 6.9kV star connected generator has positive, negative and zero sequence reactance of 25, 20 and 7% respectively. An inductive reactor with 6% reactance based on the rating of the generator is placed in the line from neutral to ground. For a line to line fault at the generator terminals when it is operating at rated voltage and is disconnected from the system determine, (a) initial symmetrical r.m.s. line and neutral currents and (b) line to line and line to neutral voltages. 5) Repeat Q4 for a LLG fault. 6) A small generating station has two alternators, each 5MVA with a reactance of 8%, connected to a bus bar. The circuit breakers at the out-feed are rated 150MVA. A new supply is added rated 10MVA with a reactance of 7.5%. Can the same breakers be used? If not, what is the value of the reactance to be connected in series with the bus bar section, if the bus bar voltage is 3.3kV?
TUTORIAL SHEET-11
Subject: Power System Analysis & Stability (UEE602) 1) The
ABCD
parameters of a nomimal- network representing a three phase
transmission line are: A=D=0.97
0.6 , B=60 70 , C=0.001 91 S.
find the steady state stability limit if both the sending end and receiving end voltages are held constant at 132 kV: (a) with the
ABCD
constant as given, (b) with
shunt admittance neglected, and (c) with both series resistances and shunt admittances neglected. 2) Develop an expression for the maximum steady state power which can be transmitted over the line if the voltage at each end is maintained constant. Neglect the effect of capacitance of the line. Show that if the reactance X of the line could be varied, the resistance remaining constant, the maximum steady state power that could be transmitted over the line is greatest when X=R. 3) For the system shown in fig, an inductor of reactance 0.6 pu per phase is connected at the mid-point of the transmission line. Determine the steady state power limit under the following condition. (a) With inductor switch S open; (b) With inductor switch S closed; (c) With inductor replaced by capacitor of same per unit reactance; (d) With inductor replaced by resistor of 1.5 resistance.
j0.3 pu E= 1.2 pu
j0.3 pu
j0.1
V=1.0 pu infinite bus
X= j0.8 pu J0.6 pu
3
4) A two pole 50 Hz 60 MVA turbo generator has a moment of inertia of 9x 10 Kg2
m . Calculate a) The kinetic energy in MJ at rated speed, b) The inertia constant M and H. c)
The inertia constant on 50 MVA base.
5) A 500 MVA synchronous machine has
H 1
= 4.6 MJ/MVA, and a 1500 MVA machine
has H 2 =3.0 MJ/MVA. The two machines operate in parallel in power station. What is the equivalent H constant for the two, relative to a 100 MVA base.
