16
1.
Introduction
3. the choice of the sequence of construction of system elements, i.e.,
when each dam and canal should be constructed and how large they should
be
4. the choice of operating rules for the components of the river basin
Section 1.3 briefly describes some of the optimization techniques available
for use in the water resources area, while Section 1.4 examines the criteria
used for a water resources system.
1.3· Techniques for the Optimization of a Water Resources
System
Optimization problems arise because rarely does a mathematical description of a water resources system yields exactly the proper number of
independent equations to provide one and only one answer for the states
(values of the dependent variables) in the model. A problem that admits
of only one solution does not have to be optimized. The typical model is
under determined; that is, there are fewer independent equations than there
are variables whose values are sought. Such problems, in principle, have
an infinite number of solutions; the objective of optimization is to select
from the set of all possible solutions the best one(s) with respect to some
given criteria.
Optimization can be accomplished by many strategies, ranging from quite
sophisticated analytical and numerical mathematical procedures to the
intelligent application of simple arithmetic. Assuming that the problem to
be optimized is defined in some way, the two main methods of optimization
can be conveniently classified as follows:
1. Analytical methods that make use of the classical techniques of differential calculus and the calculus of variations [Beveridge and Schechter,
1970, pp. 508-522, 618-625]. These methods seek the extremum of a revenue or objective function f(x) by finding the values of χ = [x\, #2,. . . , x n ]
T
that cause the derivatives of /(x) with respect to χ to vanish. When the
extremum of /(x) is sought in the presence of constraints, techniques such
as Lagrange multipliers and constrained derivatives are used. For analytical
methods to be used, the problem to be optimized must be described in a
rather restricted way so that the functions and variables can be manipulated by known rules of mathematics. Analytical methods prove unsatisfactory for large, highly nonlinear problems, and will not be discussed in
this text.
1.
Introduction
3. the choice of the sequence of construction of system elements, i.e.,
when each dam and canal should be constructed and how large they should
be
4. the choice of operating rules for the components of the river basin
Section 1.3 briefly describes some of the optimization techniques available
for use in the water resources area, while Section 1.4 examines the criteria
used for a water resources system.
1.3· Techniques for the Optimization of a Water Resources
System
Optimization problems arise because rarely does a mathematical description of a water resources system yields exactly the proper number of
independent equations to provide one and only one answer for the states
(values of the dependent variables) in the model. A problem that admits
of only one solution does not have to be optimized. The typical model is
under determined; that is, there are fewer independent equations than there
are variables whose values are sought. Such problems, in principle, have
an infinite number of solutions; the objective of optimization is to select
from the set of all possible solutions the best one(s) with respect to some
given criteria.
Optimization can be accomplished by many strategies, ranging from quite
sophisticated analytical and numerical mathematical procedures to the
intelligent application of simple arithmetic. Assuming that the problem to
be optimized is defined in some way, the two main methods of optimization
can be conveniently classified as follows:
1. Analytical methods that make use of the classical techniques of differential calculus and the calculus of variations [Beveridge and Schechter,
1970, pp. 508-522, 618-625]. These methods seek the extremum of a revenue or objective function f(x) by finding the values of χ = [x\, #2,. . . , x n ]
T
that cause the derivatives of /(x) with respect to χ to vanish. When the
extremum of /(x) is sought in the presence of constraints, techniques such
as Lagrange multipliers and constrained derivatives are used. For analytical
methods to be used, the problem to be optimized must be described in a
rather restricted way so that the functions and variables can be manipulated by known rules of mathematics. Analytical methods prove unsatisfactory for large, highly nonlinear problems, and will not be discussed in
this text.