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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:.

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power flow problem formulation

Power Flow Problem Formulation

Lecture #19

EEE 574

Dr. Dan Tylavsky

slide2
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
slide3
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,
slide4
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.
slide5
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.)

slide6
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
slide7
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
slide8
Formulating the Equation Set.
    • Recall Nodal Analysis
  • Multiplying both sides of above eqn. by E at the node and taking the complex conjugate,
slide9
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
slide10
Siq

i

q

SG

yiq

SL

Sir

r

yir

  • The Power Balance Equation.
slide11
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:
slide12
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.
slide13
There are three common ways of writing the power balance equation using real variables.
    • Polar Form:
slide14
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.

slide15
Hybrid Form:
  • Individually show that starting with:

You obtain:

  • We’ll use this form of the equation.
slide16
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)

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