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ECE 530 – Analysis Techniques for Large-Scale Electrical Systems

ECE 530 – Analysis Techniques for Large-Scale Electrical Systems. Lecture 4 : Newton-Raphson Method. Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu. Power Flow Analysis.

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ECE 530 – Analysis Techniques for Large-Scale Electrical Systems

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  1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Lecture 4: Newton-Raphson Method Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu

  2. Power Flow Analysis • When analyzing power systems, we know the complex power being consumed by the load, and the power being injected by the generators plus their voltage magnitudes • Want to find voltage magnitude and angle at every bus

  3. Solving Nonlinear Equations • Most common technique for solving the nonlinear power flow is to use the Newton-Raphson method • Key idea behind Newton-Raphson is to use sequential linearization • More tractable for linear systems

  4. Linear Equations • Linear system of equations • Here we will use the style of bolding matrices and vectors • Later in course we’ll consider solution methods for sparse linear equations, which are quite common in electric power systems • Linear equations are conceptually easy to solve, provided A is nonsingular; then there is a single solution

  5. Linear Power System Elements

  6. Nonlinear Equations • Motivated by power flow analysis, we’ll consider the solution of nonlinear equations of the form: • Problem may be restated as finding a root x of f where both x and f(x) are n-vectors • A key challenge with nonlinear equations is there may be one, none or multiple solutions!

  7. Nonlinear Example of Multiple Solutions and No Solution Example 1: x2 - 2 = 0 has solutions x = 1.414… Example 2: x2 + 2 = 0 has no real solution f(x) = x2 - 2 f(x) = x2 + 2 no solution f(x) = 0 two solutions where f(x) = 0

  8. Nonlinear Equations • The notation f(x) is short-hand for the vector functionso the problem is to solve n equations for n unknowns

  9. Newton-Raphson Method • Newton developed his method for solving for the roots of nonlinear equations in 1671, but it wasn’t published until 1736 • Raphson developed a similar method in 1690; Raphson’s approach was actually simpler than Newton’s, and is what is used today • General form of scalar problem is to find an x such that f(x) = 0 • Key idea behind the Newton-Raphson method is to use sequential linearization

  10. Newton-Raphson Method (scalar) Note, a priori we do NOT know x

  11. Newton-Raphson Method, cont’d

  12. Newton-Raphson Example

  13. Newton-Raphson Example, cont’d

  14. Sequential Linear Approximations At each iteration the N-R method uses a linear approximation to determine the next value for x Function is f(x) = x2 - 2 = 0. Solutions are points where f(x) intersects f(x) = 0 axis

  15. rootx* x (1) x (4) x x (3) x (2) x (0) Newton’s Method for a Scalar Equation f (x)

  16. Example 2 • Find the positive root of using Newton’s method starting • Computation must be done using radians!!!

  17. Example 2 Graphical View

  18. Example 2 Iterations • We continue the iterations to obtain the following set of results

  19. Example 2, Changed Initial Guess • It is interesting to note that we get to the value of 1.89549 also if we start at 3.14159

  20. Newton-Raphson Comments • When close to the solution the error decreases quite quickly -- method has quadratic convergence • f(x(v)) is known as the mismatch, which we would like to drive to zero • Stopping criteria is when f(x(v))  <  • Results are dependent upon the initial guess. What if we had guessed x(0) = 0, or x (0) = -1? • A solution’s region of attraction (ROA) is the set of initial guesses that converge to the particular solution. The ROA is often hard to determine

  21. Normal Convergence f (x) desired root

  22. x(4) x(2) x(0) x(3) x(1) Oscillatory Convergence f (x) Note that we actuallyovershoot the solution

  23. desired root x(1) x(0) x undesired root Convergence to an Unwanted Root f (x)

  24. Divergence f (x) x(1) x(0) x(2) x

  25. Multi-Variable Newton-Raphson

  26. Multi-Variable Case, cont’d

  27. Multi-Variable Case, cont’d

  28. Jacobian Matrix

  29. Multi-Variable N-R Procedure

  30. Multi-Variable Example

  31. Multi-variable Example, cont’d

  32. Multi-variable Example, cont’d

  33. Stopping Criteria: Vector Norms • When x is a vector the stopping criteria is determined by calculating the vector norm. Any norm could be used, but the most common norm used is the infinity norm, , where • Other common norms are the one norm, which is the sum of the element absolute values and the Euclidean (or two norm) defined as = =

  34. Newton-Raphson Power Flow

  35. Power Flow Variables

  36. N-R Power Flow Solution

  37. Power Flow Jacobian Matrix

  38. Power Flow Jacobian Matrix, cont’d

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