A framework for joint management of regional water-energy ... - Orbit

where represents the standard deviation of the upper bound. The algorithm
iterates until the lower bound is inside the confidence interval of the upper
bound. The approximation is then considered statistically acceptable, the algorithm
stops, and the problem is solved.
4.3.5 Comparing SDP and SDDP
Now that both methods have been presented, a brief comparison is provided
following Pereira et al. (1999). Assume that the objective is to minimize the
operation costs of a hydro-thermal system from time step t to the end of the
planning horizon T, and that the storage state is discretized into 11 steps (0,
10, 20, …, 100%), here represented by the grid of black circles. In both
methods, the sum of immediate and future costs (as a function of storage) is
calculated at stage t, and then used at stage t – 1 as the future cost function.
In the version of SDP implemented here, the FCF is built by interpolating the
future costs that are calculated at every state discretization. Figure 12 shows
the calculation of the FCF from the first three reservoir levels at stage. On the
left panel, the thin gray lines represent optimal reservoir volumes considering
three inflow classes, and the red circles in the grid represent the reservoir
states that will be evaluated from stage t to the beginning of the planning period
at state t – 3. On the right panel, the three red circles start building the
approximation of future costs as a function of storage.
In SDDP, the FCF is built by linear extrapolation. Figure 13 shows how the
costs are only evaluated at a few reservoir levels that were identified in the
previous forward simulation (left panel). The dual of the reservoir balance
Storage at t
t − 3
t − 2
t −1 t
t +1 Costs from t to T
Figure 12. Approximation of the future cost function in SDP.
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