Defining the multistage problem

The hydrothermal scheduling model itself is known from the literature (see paper for references). The specific model formulation and data is taken from the MSPLib. The deterministic demand data is stored in demand.csv.

The model is defined using the JuMP.jl and SDDP.jl packages in file hydrothermal_model.jl.

Compared to standard models solved with SDDP.jl the main difference is how the uncertain data in the model is parameterized:

• Recall that loglinearSDDP does not require the user to model an explicit state expansion for the given problem to take the history of the AR process into account.
• The realizations that are stored in node.noise_terms within SDDP and that we sample from in the forward pass are defined by the stagewise independent error term $\eta_t$ of the log-linear AR process only.
• However, with our adjustments in algorithm.jl and sampling_schemes.jl we make sure that in parameterize, the $\omega$ values that the inflow variables are actually fixed to are computed using the whole log-linear process formula given a particular realization of $\eta$.
• The first-stage data is considered deterministic, so there should be only one realization (for each dimension of the uncertainty). This can always be set to 0.
• Note that the parameterize statement contains some print commands. These are optional and can be removed. However, they can be helpful for comparing the actual inflows between different run configurations.

Importantly, as a special requirement, we have to store the references to the coupling constraints in subproblem.ext[:coupling_constraints], as this is required in the cut computation within dual.jl.

As we deal with a minimization problem, as usual in SDDP.jl, we specify a valid lower bound in the PolicyGraph definition.

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