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Lecture #19 EEE 574 Dr. Dan Tylavsky Power Flow Problem Formulation

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© Copyright 1999 Daniel Tylavsky –Rectangular Form Rep. of Phasor: 4 Notation: –Polar 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

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 Power Flow Problem Statement –Given: Network topology and branch impedance/admittance values, P L & Q L Values for all loads, Active power (P G ) at all generators (but one), V Sp =|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,

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 Power Flow Problem Statement –Find: V & at all load buses, V, Q G at all generator buses, (accounting for VAR limits) P G, Q G at the slack bus.

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky Network 50MW 450 MW 100 MW P=100 MW Q=20 MVAR P=200 MW Q=80 MVAR P=300 MW Q=100 MVAR Control Center Without knowledge of P Loss, P G cannot be determined a priori & vice versa. Defn: Slack Bus - That generator bus at which losses to the system will be provided. (Often the largest bus in the system.) Defn: Distributed Slack Bus - Losses to the system are supplied by several generators.

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 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: P L, Q L Find:V, –ii) Generator Bus (Type 2), a.k.a P-V bus. Given: P G,V G Find: Q, –iii) Slack Bus (Type 3) Given: V Sp, Sp Find: P G, Q G

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 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)=x 2 -4x+3=0 X=1 X=3

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 Formulating the Equation Set. –Recall Nodal Analysis –Multiplying both sides of above eqn. by E at the node and taking the complex conjugate,

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 Check necessary condition for unique solution. N=Total # of system buses n pq =# of load (P-Q) buses n pv =# of generator (P-V) buses 1=# of slack buses

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 The Power Balance Equation. SLSL SGSG i q r S iq S ir y iq y ir

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 Sometimes the power balance equation is written by taking the complex conjugate of each side of the equation. 4 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:

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 Theorem: If a function is analytic then it can be represented by a Taylor series. 4 Theorem: If the Cauchy- Rieman equationss hold and the derivatives of f are continuous, then the function is analytic. 4 Homework: Show that the Cauchy-Rieman equations are not obeyed by the power balance equation.

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky 4 There are three common ways of writing the power balance equation using real variables. –Polar Form:

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky –Rectangular Form: Solution is slightly slower to converge than polar form but, it is possible to construct a non-diverging iterative solution procedure. –Show for homework:

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky –Hybrid Form: –Individually show that starting with: –We’ll use this form of the equation. You obtain:

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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky –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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Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky Slack Bus

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