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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 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,
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.)
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
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)
