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Guðmundur Jón Ludvigsson
Stúdent frá MR 1988. C.S. í byggingaverkfræði frá HÍ1992. Vann hjá Lahmeyer International í Þýskalandi, frá 1992 til 1995
M.S. í byggingaverkfræði 1996. Starfandi hjá ICEconsult (LH-tækni) frá 1996
New Approach in Engineering: Global Optimisation of Design
Aiming for Maximum Profit of Investment
This paper introduces a new approach in development of complex project
schemes, with a special emphasis on the design of hydropower plants
(Elíasson & Ludvigsson 1996).
The goal of this new approach is
in all cases to maximise the ben-
efit of project investments, by
globally optimising the layout and
design parameters of the scheme in
question. This principle of economical-
ly optimal design is no new evidence. It
has however been difficult to use it, due
to its great complexity. The way out
has been to use local optimisation to
decide the design parameters. The best
known example in hydropower design,
is optimisation of tunnel diameters by
minimising the sum of construction
costs and power losses (Mosonyi 1991).
In local optimisation, dependent
design variables are optimised as if
independent. The weakness of this
approach is the substantial risk of miss-
ing the true global optimum.
In recent years evolutionary methods
have proved their worth in optimisa-
tion problems, where methods of
mathematical analysis are not applica-
ble due to system complexity. These
methods make it finally possible to
apply this new design method. A com-
puter model called Hydra, which uses
an evolutionary method called Genetic
Algorithm to globally optimise
hydropower plants is presented.
Objective function
The objective function is the profit,
also called the net present value of the
investment, H(xl,..xn):
H(x1,..x„) = T(xlv.xn) - K(xlv.xn) -
V(xlv.xn) - E(xj,..x„)
The function consists of following
parts. Each one is calculated for a
defined interest rate and lifetime of the
investment:
• Revenue T(xlv.xn) of the scheme
• Project investment K(xu..xn)
according to the design parameters
x,,..xn
• Operation and maintenance cost
V(xlv.xn)
• Environmental cost (either positive
or negative) connected with the
scheme E(xlv.xn)
The goal is to maximise the objective
function. Note that when the revenue
is fixed or zero then the goal is simply
to minimise the sum:
K(X|>..x„) + V(xlv.x„) + E(xlv.xn).
Optimisation Procedure
The logic of optimisation is mutual for
all problems. The goal is to find the
minimum or maximum of a defined
objective function, where the variables
can be subject to some constraints.
Mathematically put (maximisation
problem):
max f(x,, x2, ... xn)
a; < x^ < bj for i = 1 to n
gj(xu x2, ... x„) < Cj for all j
The problem is that the objective func-
tion is so complex that conventional
mathematical methods can’t solve it. A
method that, however, is able to find
this global optimum is called Genetic
Algorithm (see, e.g., Goldberg 1989;
Pirlot 1994). It is based on Charles
Darwin’s famous theory ‘Survival of the
fittest’. The logic is very briefly as fol-
lows:
• Each project arrangement is consid-
ered an individual (variables encod-
ed as genes in a chromosome) with a
fitness equal to the profit it pro-
duces
• Weak individuals tend to die before
reproducing, while the stronger
ones live longer and bear many off-
spring
• These offspring often inherit the
qualities that enabled their parents
to survive
• As time passes the individuals in
each generation become fitter, i.e.
the objective function comes closer
to its optimum value.
The main strength of the genetic
algorithm is that it is independent of
the behaviour of the objective function.
For example the function does not have
to be differentiable in the solution
space. Another strength is that the
number of optimisation parameters is
unrestricted. A minor weakness is that
the result is ‘close’ to the real optimum,
but this small difference is a negligible
factor compared with the unavoidable
inaccuracy in the assumptions and cal-
culations made in calculating the prof-
it.
The program ‘Hydra’ is a 32
bit Windows 95 application. £C±2?if
It includes a SQL connection to the
databases Access 97 or SQL server,
where all project data, unit prices,
results and documents stored. Also
included is a OLE connection to other
Windows applications, such as Excel
and Word, so results and other data can
easily be moved between programs.
The functionality is very briefly as
foliows (see www.lh.is/ICEconsult/
Hydra). A project is built up by creat-
ing and arranging stand alone structure
objects (which can be developed and
modified by the user outside the pro-
gram and "plugged" into Hydra)
together at defined coordinates (which
also can be free for optimisation). Main
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