1. Power Flow Problem Formulation
Lecture #19
EEE 574
Dr. Dan Tylavsky
Instructional Objectives
At the end of this lecture:
a) you will be able to describe in lay terms the information contained in an electrical schematic diagram,
b) you will be able to describe the difference between a discrete and a continuous system,
c) you will be able to describe the difference between a digital or binary discrete system and an arbitrary discrete system.

2. Notation: Polar Form Rep. of Phasor: Rectangular Form Rep. of Phasor:
© Copyright 1999 Daniel Tylavsky
Notation:
Polar Form Rep. of Phasor:
Rectangular Form Rep. of Phasor:
Specified generator power injected at a bus:
Specified load power drawn from a bus:
Specified load/generator reactive power:
Specified voltage/angle at a bus:
Complex Power: S

3. Power Flow Problem Statement
© Copyright 1999 Daniel Tylavsky
Power Flow Problem Statement
Given:
Network topology and branch impedance/admittance values,
PL & QL Values for all loads,
Active power (PG) at all generators (but one),
VSp=|E| at all generator buses,
Maximum and minimum VAR limits of each generator,
Transformer off-nominal tap ratio values,
Reference (slack, swing) bus voltage & angle,

4. Power Flow Problem Statement
© Copyright 1999 Daniel Tylavsky
Power Flow Problem Statement
Find:
V &  at all load buses,
V, QG at all generator buses, (accounting for VAR limits)
PG, QG at the slack bus.

5. Control Center Network
© Copyright 1999 Daniel Tylavsky
450 MW
100 MW
Control
Center
50MW
Network
P=100 MW
Q=20 MVAR
P=300 MW
Q=100 MVAR
Without knowledge of PLoss, PG cannot be determined a priori & vice versa.
P=200 MW
Q=80 MVAR
Defn: Distributed Slack Bus - Losses to the system are supplied by several generators.
Defn: Slack Bus - That generator bus at which losses to the system will be provided. (Often the largest bus in the system.)

6. From IEEE bus input data we must model the following 3 bus types:
© Copyright 1999 Daniel Tylavsky
From IEEE bus input data we must model the following 3 bus types:
i) Load Bus (Type 0), a.k.a. P-Q bus.
Given: PL, QL
Find:V, 
ii) Generator Bus (Type 2), a.k.a P-V bus.
Given: PG,VG
Find: Q, 
iii) Slack Bus (Type 3)
Given: VSp, Sp
Find: PG, QG

7. Formulating the Equation Set.
© Copyright 1999 Daniel Tylavsky
Formulating the Equation Set.
Necessary (but not sufficient) condition for a unique solution is that the number of equations is equal to the number of unknowns.
For linear system, must additionally require that all equations be independent.
For nonlinear systems, independence does not guarantee a unique solution, e.g., f(x)=x2-4x+3=0
X=1
X=3

8. Formulating the Equation Set.
© Copyright 1999 Daniel Tylavsky
Formulating the Equation Set.
Recall Nodal Analysis
Multiplying both sides of above eqn. by E at the node and taking the complex conjugate,

9. Check necessary condition for unique solution.
© Copyright 1999 Daniel Tylavsky
Check necessary condition for unique solution.
N=Total # of system buses
npq=# of load (P-Q) buses
npv=# of generator (P-V) buses
1=# of slack buses

10. The Power Balance Equation.
© Copyright 1999 Daniel Tylavsky
The Power Balance Equation. SL SG i q r
Siq
Sir
yiq
yir

11. Can we apply Newton’s method to these equations in complex form?
© Copyright 1999 Daniel Tylavsky
Sometimes the power balance equation is written by taking the complex conjugate of each side of the equation.
Can we apply Newton’s method to these equations in complex form?
Recall Newton’s method is based on Taylor’s theorem, which is complex form is:

12. © Copyright 1999 Daniel Tylavsky
Theorem: If a function is analytic then it can be represented by a Taylor series.
Theorem: If the Cauchy- Rieman equationss hold and the derivatives of f are continuous, then the function is analytic.
Homework: Show that the Cauchy-Rieman equations are not obeyed by the power balance equation.

13. © Copyright 1999 Daniel Tylavsky
There are three common ways of writing the power balance equation using real variables.
Polar Form:

14. Rectangular Form: Show for homework:
© Copyright 1999 Daniel Tylavsky
Rectangular Form:
Show for homework:
Solution is slightly slower to converge than polar form but, it is possible to construct a non-diverging iterative solution procedure.

15. Individually show that starting with:
© Copyright 1999 Daniel Tylavsky
Hybrid Form:
Individually show that starting with:
You obtain:
We’ll use this form of the equation.

16. © Copyright 1999 Daniel Tylavsky
For our power flow problem formulation we’ll need the following set of equations for each bus type:
P-Q Bus
P-V Bus (not on VAR limits)
(Important: When on VAR limits, the PV bus equations are the same as the PQ bus equations)

17. © Copyright 1999 Daniel Tylavsky
Slack Bus

18. The End