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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

  • 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,

  • 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



P=100 MW


P=300 MW

Q=100 MVAR

P=200 MW




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.)

X=3 types:


  • 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

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

S types:iq









  • The Power Balance 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:

  • Homework: Show that the Cauchy-Rieman equations are not obeyed by the power balance equation.

  • Show for homework:

Solution is slightly slower to converge than polar form but, it is possible to construct a non-diverging iterative solution procedure.

  • Individually show that starting with:

You obtain:

  • We’ll use this form of the equation.

  • 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 following set of equations for each bus type:

The end

The End following set of equations for each bus type: