1. Lecture #18 EEE 574 Dr. Dan Tylavsky Nonlinear Problem Solvers

2. © Copyright 1999 Daniel Tylavsky 4 Two traditional methods for solving simultaneous nonlinear eqautions: –Guass-Seidel Method (Overview - available in many text books and easy to understand.) –Newton Raphson Method (We’ll work through in detail).

3. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Gauss- Seidel Method –Suppose we wish to find solutions to equation: –Let’s write this equation in the following fixed- point form: –We can always get this form. If by no other means we can ways write:

4. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky –Then iterate as follows: –Where we start with an initial estimate, x 0. –Depending on the characteristics of the problem, the result may be fast or slow to converge, or the solution process may diverge. –With power flow problems that have been examined, the process usually converges. –Converged when, |x k+1 -x k |< .

5. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky Think-Pair-Square: Find a solution to the following problem starting with an initial estimate of x 0 =0.4. If x is a bus voltage, determine a reasonable convergence criteria and be able to justify it.

6. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky –Asymptotic Rate of Convergence: Largest integer r such that: –Gauss-Seidel method has linear convergence, i.e., r=1. –Define error as:

7. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 For linear convergence, asymptotic rate of convergence: 4 Incremental improvement in estimate is constant on a log scale.

8. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Gauss- Seidel Method –Advantages Simple conceptually Simple to program No LDU factorization. Sparsity is used simply Iterations take little execution time. Often more robust than Newton-Raphson –Disadvantages Sometimes not as robust. Slow linear convergence rate.

9. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Newton’s Method –Single Equation: Looking for the roots of f(x) Approach 1. Guess at solution x=x 0 x0x0 2. Represent f(x) as a linear approximation about x=x 0 using a Taylor series expansion: 3. Solve for a better estimate of x*: x1x1 x2x2 x* 4. Converged when: |f(x k )|< 

10. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky Think-Pair-Square: Using Newton’s method, find a solution to the following problem starting with an initial estimate of x 0 =0.4. If f(x) is sum of real/reactive power flow into a bus, and x represents bus voltage, determine a reasonable convergence criteria and be able to justify it.

11. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Compare iterates from G-S and Newton. G-S Newton Newton starts off worse because it is more sensitive to: An accurate initial guess. The “smoothness” of f(x) near x 0.

12. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky x1x1 x2x2 x0x0 -1.8 x3x3 x4x4 Newton x0x1x2x3x4x0x1x2x3x4

13. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky –Asymptotic Rate of Convergence: Largest integer r such that: –Newton’s method has quadratic convergence, i.e., r=2. Precision Limitation

14. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Newton-Raphson Method –Apply Newton’s Method to Simultaneous Equations: Looking for the roots of:

15. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Repeat same Newton-type steps: 1. Make ‘good’ guess at solution: 2. Approximate f(x) using a Taylor series expansion about point x 0 :

16. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 3. Solve linearized equations for a better estimate of x*:

17. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4. Check for convergence: 5. If converged, end, otherwise perform next iteration with i=2:

18. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Ex: Find the solution to the following 2-bus power flow problem:

19. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Bringing all terms to one side and multiplying numerator and denominator by (-j) gives: 4 Break equation into real and imaginary parts:

20. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Setting x 1 =  2,x 2 =V 2 gives: 1. Make a ‘good’ guess at solution: 2a. Evaluate f(x) at x 0 :

21. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 2b. Calculate analytical form of the Jacobian

22. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 2c. Evaluate the Jacobian at best estimate of solution:

23. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 3. Solve the linear approximation for a better estimate of x*. 4. Check for convergence.

24. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky 4 Iteration Results 012345012345

25. Nonlinear Problem Solvers © Copyright 1999 Daniel Tylavsky

26. The End