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

2. Power Flow Analysis 2 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 3

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 4

5. Linear Power System Elements 5

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! 6

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

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

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 9

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

11. Newton-Raphson Method, cont’d 11

12. Newton-Raphson Example 12

13. Newton-Raphson Example, cont’d 13

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

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

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

17. Example 2 Graphical View 17

18. Example 2 Iterations We continue the iterations to obtain the following set of results iteration number v 01.57079 12.00001 21.90100 31.89551 41.89549 18

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 iteration number v 03.14159 12.09440 21.91322 31.89567 41.89549 19

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 20

21. Normal Convergence desired root f (x) 21

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

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

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

25. Multi-Variable Newton-Raphson 25

26. Multi-Variable Case, cont’d 26

27. Multi-Variable Case, cont’d 27

28. Jacobian Matrix 28

29. Multi-Variable N-R Procedure 29

30. Multi-Variable Example 30

31. Multi-variable Example, cont’d 31

32. Multi-variable Example, cont’d 32

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 33 = =

34. Newton-Raphson Power Flow 34

35. Power Flow Variables 35

36. N-R Power Flow Solution 36

37. Power Flow Jacobian Matrix 37

38. Power Flow Jacobian Matrix, cont’d 38