Modelling and Analysis of Power Networks
Prof. Dr. Göran Andersson
HS 08/09
24. 09. 2008
Set 1
Example 1.
Consider two parallel connected transformers according to the figure below.
Transformer (a) has a turns ratio of 1:1 (p.u./p.u.) while the turns-ratio of transformer
(b) will be varied as described below. The transformers are feeding a load at bus 2,
Iload = 1.05 ∠ -450 p.u. and voltage E2 = 1∠00 p.u. The reactances of the transformers
are given in the figure. Calculate the complex power through the transformers when
the turns-ratios of transformer b are (t12 = a12ejφ12)
(a) a12 = 1, φ12 = 0
(b) a12 = 1.05, φ12 = 0
(c) a12 = 1, φ12 = 30
Comments!
Example 2.
In Figure 2. V1 and S2 are given, and S1 and V2 are asked for. Find the solution as a
function of PD2 for PD2≥0.
V1=1.0 0
o
SD1
jQG2=j0.1
SG1
ZL=j0.5
SD2=P D2+j1.0
Figure 2. (All powers,voltages and impedances are in p.u.)
Papaemmanouil Antonis
papaemmanouil@eeh.ee.ethz.ch
Example 3.
Find θ2, |V3|, θ3, SG1, and QG2 for the system shown in Figure 3. In the transmission
system all the shunt elements are capacitors with an admittance yC = j0.01 p.u., while
all the series elements are inductors with an impedance of zL = j0.1 p.u.
SG1
V 1=1.0 0
o
V3
SD3=2.8653+j1.2244
Figure 3. (All powers and voltages are in p.u.)
Example 4.
Consider the power system in Figure 4. Assume that the series line impedances are
z L = rL + jxL = 0.0099 + j 0.0995 = 0.0995∠84.2894o p.u. Neglect the capacitive
(shunt) impedances. All powers and voltages are in p.u.
a) Verify that a solution of the power flow equations is given by
o
⎡θ 2 ⎤ ⎡ −5 ⎤
⎥
⎢θ ⎥ ⎢
−10o ⎥
3⎥
⎢
⎢
=
θ=
⎢θ 4 ⎥ ⎢ −10o ⎥
⎥
⎢ ⎥ ⎢
⎣θ5 ⎦ ⎢⎣ −15o ⎥⎦
PG2=0.883
PG1
V1=1.0 0
⎡ V4 ⎤ ⎡1.0 ⎤
V =⎢ ⎥=⎢ ⎥
⎣⎢ V5 ⎥⎦ ⎣1.0 ⎦
PG3=0.2076
o
SD3=0.2+j0.1
j1.0
V4
SD4=1.7137+j0.5983
j0.8
V5
SD5=1.7355+j0.5496
Figure 4. (All powers and voltages are in p.u.)
b) Calculate the slack bus power S1=SG1.
c) Calculate the total line losses.
d) Show that the (complex) load demand may be met with lower line losses by
shifting generation to generator 3.
Papaemmanouil Antonis
papaemmanouil@eeh.ee.ethz.ch