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AppendixA
Solving Systems of Nonlinear Equations
Chapter 4 of this book describes and analyzes the power flow problem. In its ac
version, this problem is a system of nonlinear equations. This appendix describes
the most common method for solving a system of nonlinear equations, namely, the
Newton-Raphson method. This is an iterative method that uses initial values for the
unknowns and, then, at each iteration, updates these values until no change occurs
in two consecutive iterations.
Forthesakeofclarity,wefirstdescribetheworkingofthismethodforthecaseof
just one nonlinear equation with one unknown. Then, the general case of n nonlinear
equations and n unknowns is considered.
We also explain how to directly solve systems of nonlinear equations using
appropriate software.
A.1 Newton-RaphsonAlgorithm
TheNewton-Raphsonalgorithm is described in this section.
A.1.1 OneUnknown
. /
Consider a nonlinear function f x W R ! R. We aim at finding a value of x so that:
. /
f x D0: (A.1)
To do so, we first consider a given value of x, e.g., x.0/. In general, we have that
f x.0/ ¤ 0. Thus, it is necessary to find x.0/ so that f x.0/ C x.0/ D 0.
©Springer International Publishing AG 2018 271
A.J. Conejo, L. Baringo, Power System Operations, Power Electronics and Power
Systems, https://doi.org/10.1007/978-3-319-69407-8
272 A Solving Systems of Nonlinear Equations
Using Taylor series, we can express f x.0/ C x.0/ as:
. / .0/ x.0/ 2 2 . / .0/
f x.0/ C x.0/ D f x.0/ Cx.0/ df x C d f x C:::
dx 2 dx2
(A.2)
Considering only the first two terms in Eq. (A.2) and since we seek to find x.0/
so that f x.0/ C x.0/ D 0, we can approximately compute x.0/ as:
.0/
x.0/ f x : (A.3)
. / .0/
df x
dx
Next, we can update x as:
x.1/ D x.0/ C x.0/: (A.4)
Then, we check if f x.1/ D 0. If so, we have found a value of x that satisfies
f .x/ D 0. If not, we repeat the above step to find x.1/ so that f x.1/ C x.1/ D 0
and so on.
In general, we can compute x./ as:
./
x.C1/ D x./ f x ; (A.5)
. / ./
df x
dx
where is the iteration counter.
Considering the above, the Newton-Raphson method consists of the following
steps:
• Step 0: initialize the iteration counter ( D 0) and provide an initial value for x,
i.e., x D x./ D x.0/.
• Step 1: compute x.C1/ using Eq. (A.5).
• Step 2: check if the difference between the values of x in two consecutive
ˇ .C1/ ./ˇ
ˇ ˇ
iterations is lower than a prespecified tolerance , i.e., check if x x
<.Ifso,thealgorithmhasconvergedandthesolutionisx.C1/.Ifnot,continue
at Step 3.
• Step 3: update the iteration counter C 1 and continue at Step 1.
Illustrative Example A.1 Newton-Raphsonalgorithmforaone-unknownproblem
Weconsider the following quadratic function:
. / 2
f x Dx 3xC2;
A Solving Systems of Nonlinear Equations 273
whosefirst derivative is:
df.x/ D 2x3:
dx
TheNewton-Raphsonalgorithm proceeds as follows:
• Step0:weinitialize the iteration counter ( D 0) and provide an initial value for
x, e.g., x./ D x.0/ D 0.
• Step 1: we compute x.1/ using the equation below:
. /2 . /
x 0 3x0 C2 2
. / . / 0 30C2
x 1 D x 0 D0 D0:6667:
. /
2x 0 3 203
. / . /
• Step 2: we compute absolute value of the difference between x 1 and x 0 , i.e.,
0:66670 D0:6667.Sincethisdifferenceisnotsmallenough,wecontinueat
j j
Step 3.
• Step 3: we update the iteration counter D 0 C 1 D 1 and continue at Step 1.
• Step 1: we compute x.2/ using the equation below:
. / 2 . /
x 1 3x1 C2 2
. / . / 0:6667 30:6667C2
x 2 D x 1 D0:6667 D0:9333:
. /
2x 1 3 20:66673
. / . /
• Step 2: we compute the absolute value of the difference between x 2 and x 1 ,
i.e., 0:9333 0:6667 D 0:2666. Since this difference is not small enough, we
j j
continue at Step 3.
• Step 3: we update the iteration counter D 1 C 1 D 2 and continue at Step 1.
This iterative algorithm is repeated until the difference between the values of x in
two consecutive iterations is small enough. Table A.1 summarizes the results. The
4
algorithm converges in four iterations for a tolerance of 1 10 .
Note that the number of iterations needed for convergence by the Newton-
Raphson algorithm is small.
Table A.1 Illustrative Iteration x
ExampleA.2:results 0 0
1 0.6667
2 0.9333
3 0.9961
4 1.0000
274 A Solving Systems of Nonlinear Equations
A.1.2 ManyUnknowns
The Newton-Raphson method described in the previous section is extended in this
section to the general case of a system of n nonlinear equations with n unknowns,
as the one described below:
8
ˆ . /
f x ;x ;:::;x D0;
ˆ1 1 2 n
ˆ
ˆ
ˆ
< . /
f x ;x ;:::;x D0;
2 1 2 n (A.6)
ˆ :
ˆ :
ˆ :
ˆ
ˆ
: . /
f x ;x ;:::;x D0;
n 1 2 n
. / n
where f x ;x ;:::;x W R !R,iD1;:::;n,arenonlinearfunctions.
i 1 2 n
Thesystemofequations (A.6) can be rewritten in compact form as:
. /
f x D0; (A.7)
where:
> n n
. / Π. / . / . /
• f x D f x f x :::f x D0:R !R ,
1 2 > n
Œ
• x D x x :::x ,
1 2 n
Π>
• 0 D 00:::0 ,and
• >denotesthetranspose operator.
.0/ .0/
Givenaninitial value for vector x, i.e., x , we have, in general, that f x ¤0.
.0/ .0/ .0/
Thus,weneedtofindx sothatf x Cx D0.Usingthefirst-orderTaylor
.0/ .0/
series, f x Cx can be approximately expressed as:
.0/ .0/ .0/ .0/ .0/
f x Cx f x CJ x ; (A.8)
where J is the n n Jacobian:
2@f . / . / . / 3
x @f x @f x
1 1 1
6 @x @x @x 7
6 1 2 n 7
. / . / . /
@f x @f x @f x
6 2 2 2 7
6 7
6 @x @x @x 7
J D 6 1 2 n 7: (A.9)
6 : : : : 7
: : :: :
6 : : : 7
4 . / . / . / 5
@f x @f x @f x
n n n
@x @x @x
1 2 n
.0/ .0/ .0/
Since we seek f x Cx D0,fromEq.(A.8)wecancomputex as:
.0/ .0/ 1 .0/
x J f x : (A.10)
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