# Power Flow Problem Formulation - PowerPoint PPT Presentation

1 / 18

Power Flow Problem Formulation. Lecture #19 EEE 574 Dr. Dan 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:.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Power Flow Problem Formulation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

## Power Flow Problem Formulation

Lecture #19

EEE 574

Dr. Dan 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 voltage/angle at a bus:

• Complex Power: S

• 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,

• 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.

450 MW

100 MW

50MW

Network

P=100 MW

Q=20 MVAR

P=300 MW

Q=100 MVAR

P=200 MW

Q=80 MVAR

Control

Center

Without knowledge of PLoss, PG cannot be determined a priori & vice versa.

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.)

• 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

X=3

X=1

• 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

• Formulating the Equation Set.

• Recall Nodal Analysis

• Multiplying both sides of above eqn. by E at the node and taking the complex conjugate,

• 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

Siq

i

q

SG

yiq

SL

Sir

r

yir

• The Power Balance Equation.

• 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:

• 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.

• There are three common ways of writing the power balance equation using real variables.

• Polar Form:

• 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.

• Hybrid Form:

• Individually show that starting with:

You obtain:

• We’ll use this form of the equation.

• 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)

• Slack Bus