Introduction to the Power
Flow Problem
Smith College, EGR 325
April 3, 2006
1
Objectives
• Introduce the power flow problem
– Numerical solution – issues?
• Investigate two power flow programs
– PowerWorld
– MATPower
2
Complex Power – Phasor Domain
IMPORTANT
 is the power factor angle
S V I
V

*
I
I  I  
S  V I V (cos 0  j sin 0)  I (cos   j sin  )
*
S  V I cos  j V I sin 
S  P j Q
Real Power
Reactive Power
3
Power Flow Analysis
• We know
– The system topology (the circuit diagram)
– The impedance of each line
– The load at each load bus (S = P + jQ)
– The capability of each generator (P, V)
• We want to know
– The output of each generator (S)
– The voltage at each bus (V = V)
– The power flow on each line (Pflow)
4
Real Power Flow Equations
n
Si  Pi  jQi  Vi  Yik*Vk* 
k 1

n
 Vi Vk
k 1
n
j ik
V
V
e
(Gik  jBik )
 i k
k 1
(cos ik  j sin  ik )(Gik  jBik )
Resolving into the real and imaginary parts
Pi 
Qi 
n
 Vi Vk (Gik cosik  Bik sinik )  PGi  PDi
k 1
n
 Vi Vk (Gik sinik  Bik cosik )  QGi  QDi
k 1
5
Power Flow Analysis
• Data needs?
• Uses?
6
Real Power Flow Equations
• How many equations and how many
unknowns?
• Numerical methods
– Lack of convergence
– Slack bus
• Definition
• Mathematical and physical role
7
Running a Power Flow Program
• Programs created in the days of
FORTRAN programming
• Input data one of two standard formats
– IEEE
– PTI
• Output data format designed by the
programmer
• ** Cases may or may not converge! **
8
Running a Power Flow Program
• We will use two popular programs
– MATPower
• tabular data input and output
• relatively easy to use
– PowerWorld
• visual
• more difficult to use
• Terminology
– One-line diagrams
– per unit system (normalize all values)
9
Power System Diagrams
• Circuit vs. one-line diagram
17.6 MW
16.0 MW
28.8 MVR
-16.0 MVR
59.7 kV
17.6 MW
28.8 MVR
Generators are
shown as circles
40.0 kV
16.0 MW
16.0 MVR
Transmission lines
are shown as a
single line
Arrows are
used to
show loads
10
Power System Diagrams
• Circuit vs. one-line diagram
16.8 MW
16.0 MW
6.4 MVR
0.0 MVR
44.94 kV
16.8 MW
6.4 MVR
40.0 kV
16.0 MW
16.0 MVR
16.0 MVR
11
12
13
14
15
16
Summary
• Introduction to the power flow problem
and programs
• Numerical methods
– Significance of lack of convergence
– Slack bus
17