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ECE 530 – Analysis Techniques for Large-Scale Electrical Systems

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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

Special Guest Lecture by (soon to be) Prof. Hao Zhu

- 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.

- 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

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

- 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!

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

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

- 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.

- 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

Note, a priori we do NOT know x

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

rootx*

x (1)

x (4)

x

x (3)

x (2)

x (0)

f (x)

- Find the positive root of
using Newton’s method starting

- Computation must be done using radians!!!

- We continue the iterations to obtain the following set of results

- It is interesting to note that we get to the value of 1.89549 also if we start at 3.14159

- 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

f (x)

desired root

x(4)

x(2)

x(0)

x(3)

x(1)

f (x)

Note that we actuallyovershoot the solution

desired root

x(1)

x(0)

x

undesired root

f (x)

f (x)

x(1)

x(0)

x(2)

x

- 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