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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

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Power flow problem formulation

Power Flow Problem Formulation

Lecture #19

EEE 574

Dr. Dan Tylavsky

Power flow problem formulation

  • 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 formulation

  • 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 formulation

  • 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.

Power flow problem formulation

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.)

Power flow problem formulation

  • 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

Power flow problem formulation



  • 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

Power flow problem formulation

  • Formulating the Equation Set.

    • Recall Nodal Analysis

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

Power flow problem formulation

  • 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

Power flow problem formulation










  • The Power Balance Equation.

Power flow problem formulation

  • 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:

Power flow problem formulation

  • 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.

Power flow problem formulation

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

    • Polar Form:

Power flow problem formulation

  • 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.

Power flow problem formulation

  • Hybrid Form:

  • Individually show that starting with:

You obtain:

  • We’ll use this form of the equation.

Power flow problem formulation

  • 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)

Power flow problem formulation

  • Slack Bus

The end

The End

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