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Power Flow Problem Formulation

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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:

- 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,

- Given:

- 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.

- Find:

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

- i) Load Bus (Type 0), a.k.a. P-Q bus.

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

- Necessary (but not sufficient) condition for a unique solution is that the number of equations is equal to the number of unknowns.

- 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

The End