1. (Problem 4.23) In Fig. 1, assume that V 1=V 2 =1 and Zline = 0.1∠ 850 .
1.1 For what nonzero θ12 is S12 purely real?
1.2 What is the maximum power, −P21, that can be received byV́ 2, and at what θ12 does this occur?
0
1.3 when θ12 =85 , what is the active power loss in the line?
1.4 For what θ12 is −P21 = 1?
Remarks: The problem helps you practice the technique of power flow computing with voltage phasors
known.
1.1)When S12 ( complex power , P12 + jQ 12 ) is purely real ,reactive power sent by V 1 is zero .
Therefore, Q 12=0
V 22 V1V2 0
Or, sinθ− sin ( 85 + θ12 ) =0
Z line Z line
12 0 1 x1 0
Or,
0.1
sin 85 −
0.1
sin ( 85 +θ12 ) [ ∵ sinθ =sin ( 1800−θ ) ]
Given, θ12 ≠ 0 0
∴ 850=1800 −(85 ¿ ¿ 0+θ12) ¿
Or, θ12=1800 −2 x 850=10 0
1.2) Active power received by V 2 is given by
v1 v2 v 22
−P21 = cos ( θ−θ12 )− cosθ
z line z line
0
At, θ−θ12=0 ∨,θ=θ12 active power received by v 2 , (−P12 ) is maximum.
Therefore , θ12=θ=850 , at maximum power received by V 2
v 1 ∠θ12−v 2 ∠00
1.3)Current flowing through the line is given by I 12=
z line
0
At, θ12 =85 ,
0 0
⃗I 12= 1 ∠ 85 −1 ∠ 0 0
=10 ∠0 −10∠−85
0
0
0.1∠ 85
Therefore , magnitude of current flowing through theline ,
112=√(10)2 +(−10)2 +2 x 10 x(−10) cos (85¿¿ 0)=13.51 pu¿
Therefore, power loss in the line = I 212 xZcosθ=13.512 x 0.1 x cos 850=1.59 pu
1.4)Given, −P21 =1
v1 v2 v 22
Or, cos (θ−θ 12 )− cos θ=1
z line z line
12 0 1 x1
Or, sin 85 − sin (85 ¿ ¿ 0−θ12)=1 ¿
0.1 0.1
0
Or, 10 sin 85 −10 sin (85 ¿ ¿ 0−θ12 )=1 ¿
Or,10 sin (❑❑−❑❑ ) =9.96−1=8.96
Or,850 −θ12=sin−1 0.896=63.660
Or,θ12=85 0−63.64 0=21.350
Power delivered to the line by v 2=P21 , i. e . power received by v 2=−P 21
2. (Example 10.8) A five-bus system is shown in Fig. 2. Its admittance matrix in per unit values Y is given.
Solve the power flow problem
2− j 20 −1+ j 10 0 −1+ j 10 0
y=
[
−1+ j 10 3− j 30 −1+ j10 −1+ j 10
0 −1+ j 10 2− j20
−1+ j 10 −1+ j 10
0 0
0
0
0
−1+ j10
3− j30 −1+ j10
−1+ j10 −1+ j 10 2− j 20
]
Figure 2: Problem
2 2.1. Identify the state variables and the equations that can be setup.
2.2 Use either method ( Newton-Raphson method and fast decoupled method) to solve the power flow
problem. Please copy paste your codes in your HW report. Put a box for the iteration procedure.
2.3. Please present the following:
• Plot the error versus iteration (use semilogy function). together in one plot
• Present a table of voltage phasors at each bus. results.
• Compute the total power loss in the system.
To identify the unknown voltage Phasors and Fourier injection equations
For bus 2:
0.883=|v 2||v 1|| y 12|cos ( θ 12+δ 1−δ 2 ) +|v 2|cos ( θ22 )+|v 2||v 3| cos ( θ23 +δ 3 −δ 2) +|v 2||v 4|cos ( θ 24+ δ 4 −δ 2 ) +|v 2|∨v 5∨cos(θ
For bus 3:
0.2076=¿ v 3||v 1|| y 31 ¿
For bus 4:
−1.7137=|v 4||v 1|| y 14|cos ( θ 14+ δ 1−δ 4 )+|v 4||v 2|| y 42|cos ( θ 92+δ 2−δ 4 ) +|v 4||v 3|| y 34|cos ( θ34 + δ 3−δ 4 ) +|v 4 ¿ 2| y 44 ¿
)
0.407=−¿
For bus 5:
−1.7355=¿
0.2504=¿
3. For the radial 400 V system in HW#1, assume that the loads’ power consumptions are known with
real power consumption marked in Fig. 4 of HW#1, and the power factor is 0.7 lagging. Summing all
loads leads to 92 kW power consumption while summing all DERs leads to 83 kW. Apparently, the
current assumed real power generation from DERs cannot meet the load demand. To accommodate the
power balance, please assume that DER1 can generate more than 30 kW and Bus 1’s voltage can be
controlled at 1 pu. For the rest of the DER buses (2, 3, 4, 5), please also assume that the voltages can be
controlled at 1 pu.
3.1 Please formulate the power flow problem for the 9-bus system. Which state variables in real domain
are to be identified? Express the set of equations F(x) = 0 explicitly using the identified state variables.
You may present codes to have a user-defined Matlab function to define F(x). Please also evaluate the
function using flat start (all voltage magnitudes are 1, all angles are 0) to find the values of F(x). The
function should be defined as: function Fx = myfun(x)
3.2 Assuming that the voltage phasors are given. Please find the following line flows measured at the
from bus side: Line 6-7, Line 7-1, Line 7-2, Line 7-8, Line 8-3, Line 8-9, Line 9-4, Line 9-5. Please provide
the MATLAB function code to have the voltage magnitude (pu) vector, voltage angle in degree vector,
Ybus matrix as the two input arguments, and the 8 branch flows as the output. The function should be
defined as:
function S flow = myfun(Vm, Va, Ybus)
Evaluate this function using flat start as well. Flow values should be zeros.
