Hedging Strategy and Electricity Contract Engineering - IFOR

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6.6 Power portfolio optimization with CVaR 159
and vice versa, a linear penalty function could be introduced to the loss function
- Û V end V Kã jÝ , where
-
is a positive constant, representing the value of water.
Another possibility to avoid that some scenarios leave a low amount of water
for coming periods is to sharpen the constraint on the end-water level and state
that V K j V ã end j 1Ü Ü Ü J where every trajectory of the water level, V Kã j
has to exceed the threshold V
á?á@á
end . The threshold V end can again be determined
by a one-year optimization as described for the static case.
6.6.7. Size of dynamic problem
Compared to the static model, the dynamic model will typically have fewer
variables describing the dispatch. We here only need one weighting factor per
exercise function, which will give us r variables, compared to twice ô the
number of periods in the static case. Normally the number of periods will exceed
the number of exercise functions, why one could expect this optimization
to be faster. However, whereas the constraints on the water level in the static
case were scenario independent, in the dynamic case they are not, why the size
of the dynamic problem is typically substantially larger. To get an idea on the
size of the problem we again study the A matrix. The size of A is given by
A
2K J J 4 r n 2K J ë ë
K J ë
1
. For large problems, i. e. many periods
¢ ë ÿ ë ÿ ë ë ì
and many scenarios, the number of variables will essentially be given by K J
and the number of constraints by 2K J and we can immediately conclude that
the size of the dynamic problem will then be substantial. ¢ The density of the
matrix, defined as the percentage non-zero elements is given by
Û 2Û 3Ý ôkÝ
Û 4Ý?Û J ô 1Ý
K 2 r K 4r n J 4r 1
2K J J r n 2K J K
and will for large problems be determined by the ratio
K 2 J
2K 2 J 2 1
2J á
Even though the size of the matrix grows fast for large problems, the density
will obviously be fairly low.