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

values θ (€/MWh) for each inflow scenario and week of the year are calculated
by taking the derivative of the total costs with respect to the reservoir level:
F DF
q = »
(10)
E DE
where the optimal value function F has units of € and the energy storage E
has units of MWh.
The water values represent the expected value of a marginal amount of water
(energy) if it is stored for later use. They are used as the marginal costs of the
releases, and added to the objective function of a linear problem used to simulate
the system:
F
é c p f w ù
ú
(11)
û
*
T T T
t
= min - + qt,
kDEt
wp , , DE
êë
The term f T w is not included in (11) if irrigation is considered as a constraint.
This problem is subject to the same constraints as (1) or (8), depending on
whether irrigation is included as a constraint or a decision variable, and an
additional constraint on reservoir releases ΔE t :
£ å D £ H
(12)
0 Et
The linear problem (11) was solved forwards on weekly time steps using
simulated control (1961–1990) or climate change (2036–2065) data of energy
inflows, irrigation demand and power demand. The results of this simulation
include time series of irrigation allocation, power production levels, equivalent
energy storage and reservoir releases.
4.3 Stochastic dual dynamic programming
The SDDP algorithm consists of two phases: a backward optimization phase
that generates cuts to approximate the future cost function (FCF) from the
last time step to the first; and a forward simulation phase that uses the representation
of the FCF to simulate the system. The backward and forward phases
provide lower and upper bounds to the true function, respectively; and the
algorithm iterates until the convergence criterion is met.
4.3.1 Inflow: continuous Markov process
In SDDP, instead of representing inflow uncertainty through a discrete Markov
chain as it was done in the SDP method (Section 4.2.1), the inflow uncer-
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