Analyzing Large-Scale Electrical Systems: Nonlinear Equations Solutions

1. ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Lecture 1: Solution of Nonlinear Equations Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign overbye@illinois.edu Special Guest Lecture by (soon to be) Prof. Hao Zhu

2. Course Overview • Course presents the fundamental analytic, simulation and computation techniques for the analysis of large-scale electrical systems. The course stresses the importance of the structural characteristics of the systems, with an aim towards practical analysis. • Prof. Overbye will give a full introduction next lecture. Today Hao Zhu will jump into the analysis of nonlinear equations.

3. Linear Equations • Course assumes that students are familiar with the solution of linear equations expressed as • Here we will use the style of bolding matrices and vectors; another common style is to underline them • 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

4. Linear Equations A function fis linear if f(a1m1+ a2m2) = a1f(m1) + a2f(m2) That is 1) the output is proportional to the input 2) the principle of superposition holds Linear Example:y = f(x) = Ax y= A(x1+x2) = Ax1+ Ax2 Nonlinear Example: y = f(x) = c x2 y = c(x1+x2)2 ≠ (cx1)2 + (c x2)2

5. Linear Power System Elements

6. Nonlinear Equations • In this section 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. Nonlinear Equations • The nonlinear functions f(·) of interest include both algebraic and transcendental types • What we’ll find is the power flow problem becomes nonlinear when we consider constant power loads • We’ll first consider the problem in a single dimension, and then treat the more general case of n dimensions.

10. 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

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

12. Newton-Raphson Method, cont’d

13. Newton-Raphson Example

14. Newton-Raphson Example, cont’d

15. 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

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

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

18. Example 2 Graphical View

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

20. 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

21. 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

22. Normal Convergence f (x) desired root

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

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

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

26. Multi-Variable Newton-Raphson

27. Multi-Variable Case, cont’d

28. Multi-Variable Case, cont’d

29. Jacobian Matrix

30. Multi-Variable N-R Procedure

31. Multi-Variable Example

32. Multi-variable Example, cont’d

33. Multi-variable Example, cont’d

34. 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