6 . 1 Optimality in a Stochastic Sett ing
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Extending the constraint set to include stochastic elements is a reasonably simple exercise. We merely need to define which values are stochastic and allow some of the variables to depend on the observed value of various stochastic variables. We can model correlations and dependencies by allowing some of the stochastic variables to be given by a weighted sum of various random �ariables. For instance, suppose there is a correlation between the controlled and-uncontrolled inflows of a particular river chain. If we let �' ( and '1/; be random variables from appropriate distributions, then the correlation can be modelled by setting
J
= �+AI( and[;
= '1/; + >.u(, wherethe >. 's represent correlation factors. In general each inflow would be a function of a random variable corresponding uniquely to that inflow, and various other random variables (of which other inflows are also functions) corresponding to correlations bought in by various environmental effects (local weather patterns, for instance).
A difficulty which arises when specifying the objective function is that of deter mining exactly what we are trying to optimize-the answer is not at all obvious. The usual objective used for Stochastic Programming is that of minimizing the expected cost (or maximizing the expected benefit) ; the actual objective depends on the intention of the model and what is reasonably achievable.
The "tails" of random variable distributions present a difficulty when using expected cost, since these tails are often not well approximated due to a lack of information about this area of the distribution (consider approximating a statis tically 1 in 1 00 year drought using only 50 years of past data). However, these
tails may actually drive the solution, since the costs associated with such tails are often large and could swamp the data which would otherwise lead to more rea
sonable solutions. Another difficulty with these extreme values is that, generally, approximations and constraints of the model are based on near-average values of the stochastic variables and such approximations and constraints may break down, or become unreasonable, in the extreme. Also, some constraints are not hard, but are more easily modelled in this way, especially in the face of extreme conditions; given a serious drought, noone can expect minimum river levels to be maintained. Therefore, the difficulty may be the way the problem is modelled. However, in this situation, it may
not
be desirable to re-model such constraints, as this re-modelling may cause the model to become computationally intractible.CHAPTER 6. MODELLING STOCHASTIC INFLOWS
the tails of the underlying distribution (e.g. a discrete approximation). Such a situation can be thought of as optimizing over some set of "reasonable" values, and treating extreme situations as "acts-of-god" , for which special actions will be taken which are not (or can not be) modelled explicitly. By specifying where the truncations are made, we are defining such unreasonable situations.
Minimizing the expected cost is not the only objective t hat could be used; it assumes that the intention is to do well in the long-term. To do better in the short-term, one could include criteria for taking risks on the forecasts, or, risk trying to do better in an average year by foresaking security in an extreme year. Given the unpredictability of inflows into hydro reservoirs in New Zealand and the lack of imported power, such risk taking is most likely untenable. Because of this, and the difficulties inherent in specifying other objectives, we use the minimization of expected cost, or an approximation thereof, as the objective for this model. By changing the random variable distributions used, or incorporating a weighting function into the objective function, we can change the importance given to various probable futures and hence include some flexibility in the definition of the objective.
6 . 2 The General P roblem
Recall (from Table 5.2) that the stochastic variables in the model are present only as right-hand sides of some constraints and bounds. Hence, the general stochastic program can be written as the multi-period stochastic program
Min E [Z(x, y)]
(IY
where Z is the cost of generation, e = [6 6 6]T is the set of random variables (with possible correlations) , y is some type of state or history information from previous
time steps upon which our decisions (and some random variables) are contingent, and x represents the decision variables. Here the expectation is calculated as an integral over all possible random variable values which form some multi-dimensional
CHAPTER 6. MODELLING STOCHASTIC INFLOvVS 1 1 0
set. It is the calculation of this integral that creates the difficulty, since, as pointed out by Wets [22] , while there are adequate ways of numerically computing integrals over one (and possibly two) dimensions, there are no reasonable ways for doing so over three or more dimensions.
It is well known that when the probability distributions underlying the stochas tic variables are discrete, the formulation can be written as an equivalent large-scale deterministic program (with a stochastic interpret.ation) . Furthermore, when the model of the physical system underlying the stochastic problem is linear (as it is in this case) , the equivalent large-scale deterministic problem obtained is an LP. Many methods discretize the probability distributions underlying the stochastic variables (either before or during solution) to take advantage of this property, however the large-scale nature of the equivalent deterministic problem means that even then one must limit the size of "local" searches (possibly by limiting the size of the problem investigated) so as to make the problem tractable.
In Section 6.3 we consider using a continuous approximation, via a fixed basis,
to the distributions underlying the stochastic variables, in a similar manner to the approximation of Electricity Curves. The benefit of using a fixed basis is that it eliminates the difficulty of calculating expected values, because one can obtain a fixed polynomial expression.
A difficulty which arises when discretizing time is that of deciding exactly when the random variable is observed (and when this information can be used) . Such knowledge at the beginning of the week assumes perfect foresight over the week, while allowing this knowledge only at the end of the week assumes that we cannot react to knowledge gained during the week. For the latter, however, some state variables (in our case, storage at the end of the week) must depend upon the actual value of the random variables. To approximate a limited use of the knowledge before
the end of the week, we could assume, say, that the Contract Curve of the station needs to be fixed at the beginning of the week; however, the release, storage and spill can all depend on the value of the random variable, allowing hydro generation to be replaced by non-supply (or re-supply) depending on the observed value of the random variable, simulating an ability to react to information gained over the week.
For the working model we assume, for simplicity, that all decisions must be made at the beginning of the week. This allows the possible first week's inflows