Environmental Engineering Reference
In-Depth Information
Appendix H
Newton's method
Newton's method is a well-known algorithm for finding roots of equations in one or
more dimensions, and popularly used in solving optimal power flow problems [181].
For instance, the solution of Lagrangian function equation (G.1) can be obtained
by employing the Newton's method to the following system of linearised equations:
W
z
=−
y
(H.1)
⎧
⎨
W
is the matrix of derivatives for Lagrangian function
L
z
is the vector of correction terms
y
is the gradient vector
where
⎩
Newton's method is an iterative form that uses the non-linear equations of the
Lagrangian function to calculate a correction
z
for the
x
m
values.
x
m
+
1
x
m
=
+
z
(H.2)
The following tests are conducted to guarantee an adequate solution [213]:
All power mismatches are within a prescribed tolerance;
●
The inequality constraints are satisfied;
●
The gradient vector is 0;
●
Further reductions in the objective function are possible only if constraints are
violated.
●
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