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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

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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 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.

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

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

Linear Power System Elements

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!

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

Nonlinear Equations

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

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.

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

Newton-Raphson Method (scalar)

Note, a priori we do NOT know x

Newton-Raphson Method, cont’d

Newton-Raphson Example

Newton-Raphson Example, cont’d

Sequential Linear Approximations

At each

iteration the

N-R method

uses a linear


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


x (1)

x (4)


x (3)

x (2)

x (0)

Newton’s Method for a Scalar Equation

f (x)

Example 2

  • Find the positive root of

    using Newton’s method starting

  • Computation must be done using radians!!!

Example 2 Graphical View

Example 2 Iterations

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

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

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

Normal Convergence

f (x)

desired root






Oscillatory Convergence

f (x)

Note that we actuallyovershoot the solution

desired root




undesired root

Convergence to an Unwanted Root

f (x)


f (x)





Multi-Variable Newton-Raphson

Multi-Variable Case, cont’d

Multi-Variable Case, cont’d

Jacobian Matrix

Multi-Variable N-R Procedure

Multi-Variable Example

Multi-variable Example, cont’d

Multi-variable Example, cont’d

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

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