Stochastic Programming - Index of

150 STOCHASTIC PROGRAMMING
2.8.2 Further developments
The model shown above is very simplified. Modeling of real systems must take
into account a number of other aspects as well. In this section, we list some
of them to give you a feeling for what may happen.
First, these models are traditionally set in a context where the major goal is
to meet demand, rather than maximize profit. In a pure hydro based system,
the goal is then to obtain as much energy as possible from the available water
(which of course is still uncertain). In a system with other sources for energy
as well, we also have to take into account the cost of these sources, for example
natural gas, oil or nuclear power.
Obviously, in a model as simple as ours, maximizing the amount of energy
obtained from the available water resources makes little sense, as we have
(implicitly) assumed that the amount of energy we get from 1m 3 of water
is fixed. The reality is normally different. First, the turbines are not equally
efficient at all production levels. They have some optimal (below maximal)
production levels where the amount of energy per m 3 water is optimized.
Generally, the function describing energy production as a result of water usage
in a power plant with several turbines is neither convex nor monotone. In
particular, the non-convexity is serious. It stems from physical properties of
the turbines.
But there is more than that. The energy production also depends on the
head (hydrostatic pressure) that applies at a station during a period. It is
common to measure water pressure as the height of the water column having
the given pressure at the bottom. This is particularly complicated if the
water released from one power plant is submerged in the reservoir of the
downstream power plant. In this case the head of the upper station will depend
on the reservoir level of the lower station, generating another source of nonconvexities.
Traditionally, these models have been solved using stochastic dynamic
programming. This can work reasonably well as long as the dimension of
the state space is small. A requirement in stochastic dynamic programming
is that there is independence between periods. Hence, if water inflow in one
period (stage) is correlated to that of the previous period(s), the state space
must be expanded to contain the inflow in these previous period(s). If this
happens, SDP is soon out of business.
Furthermore, in deregulated markets it may be necessary to include price
as a random variable. Price is correlated to inflow in the present period, but
even more to inflow in earlier periods through the reservoir levels. This creates
dependencies which are very hard to tackle in SDP.
Hence, researchers have turned to other methods, for example scenario
aggregation, where dependencies are of no concern. So far, it is not clear
how successful this will be.