We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byShea Heming
Modified about 1 year ago
Lecture #19 EEE 574 Dr. Dan Tylavsky Power Flow Problem Formulation
© 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
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,
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.
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.
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
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
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,
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
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
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:
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.
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:
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:
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:
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)
Power Flow Problem Formulation © Copyright 1999 Daniel Tylavsky Slack Bus
EE 369 POWER SYSTEM ANALYSIS Lecture 11 Power Flow Tom Overbye and Ross Baldick 1.
ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois.
TCIPG Reading Group – Fall 2012 Karl Reinhard & Ahmed Fawaz TCIPG 1 Trustworthy Cyber Infrastructure for the Power Grid University of Illinois.
Boyce/DiPrima 9 th ed, Ch 2.4: Differences Between Linear and Nonlinear Equations Elementary Differential Equations and Boundary Value Problems, 9 th edition,
Lecture 13 Newton-Raphson Power Flow Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS.
8.4 An Introduction to Functions: Linear Functions, Applications, and Models Part 1: Functions.
Optimal Power Flow Daniel Kirschen © 2011 D. Kirschen and the University of Washington 1.
M. Dumbser 1 / 23 Analisi Numerica Università degli Studi di Trento Dipartimento dIngegneria Civile ed Ambientale Dr.-Ing. Michael Dumbser Lecture on Numerical.
INEQUALITIES Brought To You By- Tutorial Services-The Math Center.
EE/Econ 458 PF Equations J. McCalley 1. Power system representation NODE or BUS (substation) BRANCHES (lines or transformers) NETWORK (but unloaded and.
1 Summary of Papers 1. P. Sauer and M. Pai, Power System Steady-State Stability and the Load Flow Jacobian, IEEE Transactions on Power Systems, Vol. 5,
© 2010 D. Kirschen and The University of Manchester1 New Formulations of the Optimal Power Flow Problem Prof. Daniel Kirschen The University of Manchester.
Page 25 Applications of the DerivativesFinding Local/Relative Extrema Local (or Relative) Extrema A function f has a local maximum at x = x 0 if locally,
Ch 3.2: Solutions of Linear Homogeneous Equations; Wronskian Let p, q be continuous functions on an interval I = ( , ), which could be infinite. For.
New Formulations of the Optimal Power Flow Problem Daniel Kirschen Close Professor of Electrical Engineering University of Washington © 2011 D. Kirschen.
Mathematics. Session Properties of Triangle - 2 Session Objectives.
Copyright © 2012, 2008, 2004 Pearson Education, Inc. Mrs. Rivas International Studies Charter School. Bell Ringer.
INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 2 Functions and Graphs.
Weve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential.
Section 2.1 Linear Equations in One Variable. Equation Equation- a mathematical expression that states that two (2) quantities are equal ** Use the equal.
Matrix Representation Matrix Rep.Same basics as introduced already. Convenient method of working with vectors. SuperpositionComplete set of vectors can.
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Numerical Solution of Ordinary Differential Equations.
Quadratic Equations, Solving a Quadratic Equation by factorization by graphical method by taking square roots by quadratic equation.
21-1 Copyright 2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 21 Hypothesis.
First Order Linear Differential Equations Any equation containing a derivative is called a differential equation. A function which satisfies the equation.
Ch 6.4: Differential Equations with Discontinuous Forcing Functions In this section, we focus on examples of nonhomogeneous initial value problems in which.
INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences 2007 Pearson Education Asia Chapter 0 Review of Algebra.
1.4 Inverses; Rules of Matrix Arithmetic. Properties of Matrix Operations For real numbers a and b,we always have ab=ba, which is called the commutative.
Chapter 2 Section 3. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. More on Solving Linear Equations Learn and use the four steps for.
© 2016 SlidePlayer.com Inc. All rights reserved.