Power Flow Problem Formulation

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# Power Flow Problem Formulation - PowerPoint PPT Presentation

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:.

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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 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.)

• 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)