Question bank (Part 1)
Power System (3)
Problem 1
For the circuit shown in figure 1, find the following;
1. The bus admittance matrix assuming no mutual coupling between any
of the branches
2. Using the admittance matrix modification procedure and assuming nomutual coupling between branches, modify the admittance bus obtained
in 1 to reflect removal of two branches 1-3 and 2-5 from the circuit
Fig. 1
Problem 2
For the linear graph shown in figure 2 for the circuit shown in figure 1, find
the following disregarding all mutual coupling between branches;
1. The branch-to-node incidence matrix A for the circuit with node 0 as
reference.
2. Find the circuit admittance matrix using the network incidence matrix
Problem 3
For the circuit shown in figure 1, considering that only two branches 1-3 and
2-3 are mutually coupled with mutual impedance is 𝑗0.15 per unit, find the
following;
1. Branch admittance matrix.
2. Branch-to-node incidence matrix A wit node 0 reference. Then find the
circuit admittance matrix using the network incidence matrix
Problem 4
For the circuit shown in figure 1, solve;
1. The nodal equation to find the voltages at the four buses of Prob. 2.
2. The nodal equation to find the voltages at the four buses of Prob. 3.
2
Using both gaussian elimination and Kron reduction methods.
Problem 5
- Determine the bus admittance matrix (𝑌𝑏𝑢𝑠 ) for the following power
three phase system (note that some of the values have already been
determined for you). Assume a three-phase 100 MVA per unit base.
- Assume that a 75 Mvar shunt capacitance (three phase assuming one
per unit bus voltage) is added at bus 4. Calculate the new value of Y44.
Bus input data
Problem 6
- Form the impedance matrix for the circuit shown in Fig. after removing
node 5 by converting the voltage source to a current source, Determine the
voltages with respect to reference node at each of four other nodes when
𝑉 = 1.2∠0° and the load currents are 𝐼𝐿1 = −𝑗0.1, 𝐼𝐿2 = −𝑗0.1, 𝐼𝐿3 =
−𝑗0.2, and 𝐼𝐿4 = −𝑗0.2, all in per unit.
- Then draw the Thevenin equivalent circuit at bus 4 and use it to determine
the current drawn by a capacitor of reactance 5.4 per unit connected
between bus 4 and reference.
- Calculate the voltage changes at each of the buses due to the capacitor.
3
- Calculate the total reactive power loss in the system.
Problem 7
- Modify the impedance matrix of the Prob. 1 to include a capacitor of
reactance 5.4 per unit connected from bus 4 to reference.
- Calculate the new bus voltages using the modified impedance bus.
Problem 8
For the reactance network of the Fig. find;
- The impedance matrix by direct formulation
- The voltage at each bus
- The current drawn by a capacitor having a reactance of 5.0 per unit
connected from bus 3 to neutral.
- The change in voltage at each bus when the capacitor is connected at
bus 3.
- The voltage at each bus after connecting the capacitor.
The magnitude and angle of each of the generated voltages may be assumed to
remain constant.
4
Problem 9
Find the impedance bus for the three-bus circuit of the Fig. by the impedance
matrix building algorithm.
Problem 10
Consider the simplified electric power system shown in Figure for which the
power flow solution can be obtained without resorting to iterative techniques.
a. Compute the elements of the bus admittance matrix 𝑌𝑏𝑢𝑠 .
b. Calculate the phase angle 𝛿2 by using the real power equation at bus 2
(voltage-controlled bus).
c. Determine |𝑉3 | and 𝛿3 by using both the real and reactive power
equations at bus 3 (load bus).
d. Find the real power generated at bus 1 (swing bus).
e. Evaluate the total real power losses in the system.
Problem 11
- Assume a 0.8 + 𝑗0.4 per unit load at bus 2 is being supplied by a
generator at bus 1 through a transmission line with series impedance of
0.05 + 𝑗0.1 per unit. Assuming bus 1 is the swing bus with a fixed per
unit voltage of 1.0∠0, use the Gauss-Seidel method to calculate the
voltage at bus 2 after three iterations.
5
- Repeat the above problem with the swing bus voltage changed to 1.0 ∠30°
per unit.
Problem 12
- For the three-bus system whose 𝑌𝑏𝑢𝑠 is given below, calculate the second
iteration value of 𝑉3 using the Gauss-Seidel method. Assume bus 1 as the
slack (with 𝑉1 = 1.0∠0), and buses 2 and 3 are load buses with a per unit
load of 𝑆2 = 1 + 𝑗0.5 and 𝑆3 = 1.5 + 𝑗0.75. Use voltage guesses of
1.0∠0 at both buses 2 and 3. The bus admittance matrix for a three-bus
system is
- Repeat the first problem except assume the bus 1 (slack bus) voltage of
𝑉1 = 1.05∠0.
Problem 13
The bus admittance matrix for the power system shown in Figure is given by
With the complex powers on load buses 2, 3, and 4 as shown in Figure,
determine the value for 𝑉2 that is produced by the first and second iterations of
the Gauss–Seidel procedure. Choose the initial guess 𝑉2 (0) = 𝑉3 (0) =
𝑉4 (0) = 1.0 ∠0 per unit.
6
Problem 14
The bus admittance matrix of a three-bus power system is given by
with 𝑉1 = 1.0 ∠0 per unit; 𝑉2 = 1.0 per unit; 𝑃2 = 60 MW; 𝑃3 = −80 MW;
𝑄3 = −60 MVAR (lagging) as a part of the power-flow solution of the
system. Find 𝑉2 and 𝑉3 within a tolerance of 0.01 per unit. Start with 𝛿2 = 0,
𝑉3 = 1.0 per unit, and 𝛿3 = 0. By using
- Gauss-Seidel iteration method
- Newton-Raphson method use a maximum power flow mismatch of 0.1
MVA.
- Fast decoupled method
- DC load flow
Problem 15
A generator bus (with a 1.0 per unit voltage) supplies a 150 MW, 50 Mvar
load through a lossless transmission line with per unit (100 MVA base)
impedance of 𝑗0.1 and no line charging. Starting with an initial voltage guess
7
of 1.0 ∠0, iterate until converged using the Newton–Raphson power flow
method and fast decoupled method. For convergence criteria use a maximum
power flow mismatch of 0.1 MVA.
Problem 16
For a three bus power system assume bus 1 is the swing with a per unit
voltage of 1.0 ∠0, bus 2 is a PQ bus with a per unit load of 2.0 + 𝑗0.5, and
bus 3 is a PV bus with 1.0 per unit generation and a 1.0 voltage setpoint. The
per unit line impedances are j0.1 between buses 1 and 2, 𝑗0.4 between buses 1
and 3, and 𝑗0.2 between buses 2 and 3. Using a flat start, use the Newton–
Raphson approach and fast decoupled method to determine the first iteration
phasor voltages at buses 2 and 3.
Problem 17
Figure shows the one-line diagram of a simple power system. Generators are
connected at buses 1 and 4 while loads are indicated at all four buses. Base
values for the transmission system, are 100 MVA, 230 kV. The 𝑄 values or
load are calculated from the corresponding 𝑃 values assuming a power factor
or 0.85. The net scheduled values, 𝑃𝑘 and 𝑄𝑘 are negative at the load buses 2
and 3. Generated 𝑄𝐺𝑘 is not specified where voltage magnitude is constant. In
the voltage column the values for the load buses are flat start estimates. The
slack bus voltage magnitude |𝑉1 | and angle 𝛿1 , and magnitude |𝑉4 | at bus 4,
are to be kept constant at the values listed.
1. Calculate the value of 𝑉2 for the first iteration using Gauss-Seidel
method.
2. Calculate the value of 𝑉2 for the first iteration using Gauss-Seidel
method with considering acceleration factor equal to 1.6.
3. Calculate the voltage at bus 4 with the originally estimated voltages at
buses 2 and 3 replaced by the accelerated values indicated in (2).
4. Calculate the active and reactive power at the slack bus.
8
Problem 18
The small power system of Ex. 17, power flow study of the system is to be
made by the Newton-Raphson method using the polar form of the equations
for P and Q.
1. Determine the number of rows and columns in the jacobian.
(0)
2. Calculate the initial mismatch ∆𝑃3
jacobian elements.
9
and the initial values of the
3. Calculate the state variables required in the power flow solution using
Newton Raphson method.
4. Calculate the active and reactive power at the slack bus.
5. Calculate the state variables required in the power flow solution using
Fast decoupled method.
Problem 19
Two transformers are connected in parallel to supply an impedance to neutral
per phase of 0.8 + 𝑗 0.6 per unit at a voltage of 𝑉2 = 1 .0∠0° per unit.
Transformer Ta has a voltage ratio equal to the ratio of the base voltages on
the two sides of the transformer. This transformer has an impedance of 𝑗0.1
per unit on the appropriate base. The second transformer Tb also has an
impedance of 𝑗0.1 per unit on the same base but has a step-up toward the load
of 1.05 times that of Ta (secondary windings on 1.05 tap). Figure 4.10 shows
the equivalent circuit with transformer Tb represented by its impedance and
the insertion of a voltage ∆𝑉.
1. Find the complex power transmitted to the load through each
transformer using 𝑌𝑏𝑢𝑠 model for each of two parallel transformers.
2. Repeat (1) except that Tb includes both a transformer having the same
turns ratio as Ta and a regulating transformer with a phase shift of 3° (t
= 1 .0∠3° ). The impedance of the two components of Tb is j0.1 per unit
on the base of Ta.
Problem 20
For each bus k, determine which of the variables 𝑉𝑘 , 𝛿𝑘 , 𝑃𝑘 , and 𝑄𝑘 are input
data and which are unknowns.
10
Table. Bus input data
Table. Line input data
Table. Transformer input data
11
Table. Input data and unknowns
1. Compute the elements of the 𝑌𝑏𝑢𝑠 .
2. Determine the DC power-flow solution for the five-bus system.
12