## Section: New Results

### Short and middle term electricity production planning

#### Efficient optimization technique for weekly unit commitment

Participants : R. Apparigliato, J.P. Vial, R. Zorgati.

The Unit Commitment Problem (UCP) consists of defining the minimal-cost power generation schedule for a given set of power plants. Due to many complex constraints, the deterministic UCP, even in its deterministic version, is a challenging large-size, non-convex, non-linear optimization problem, but there exist nowadays efficient tools to solve it. For a very short term horizon, the deterministic UCP is satisfactory; it is currently used for the daily scheduling in an industrial way. For the two/four-week time horizon which we are concerned with, uncertainty become significant and cannot be ignored anymore, making it necessary to treat the UCP as a stochastic problem. Dealing with uncertainty introduces a level of complexity that is of an order of magnitude higher than in the deterministic case. Thus, there is a need to design new stochastic optimization techniques and models, that are implementable in an industrial context.

*Contribution*. Among possible tools, robust optimization offers promising opportunities. It has the very attractive
property of leading to computationally tractable problems. To investigate this new approach, we focused our investigations
on integrating the uncertainty on water inflows in the management of a hydraulic valley. To account for the progressive
unfolding of uncertainty and the opportunity to take corrective, or recourse, actions, we modeled future decisions
as linear functions of observed past inflows. These so-called *linear decision rules* restrict the field of possible
future recourse but still capture a good deal of the adaptive feature of real-time management. We implemented
the robust optimization approach on a small but representative valley with a one week horizon. We simulated the
performance of the obtained on a large sample of randomly generated scenarios. In view of the lack of readily available
alternatives, we benchmarked the robust optimization against a simple enough a deterministic policy with daily revision.
The latter approach is quite close to operational practice: the one-day ahead controls are those obtained by optimizing a
one-week deterministic model; the model is revised on a daily basis to account for the actual water levels in the reservoirs.
We could have used a similar periodic review scheme with the robust optimization scheme, but, even though it is
perfectly implementable at the operation level, it turned out to be computationally too expensive in the extensive
simulation runs. This has put the robust optimization in a clear disadvantage with respect to the deterministic policy
with periodic review. Nevertheless, the robust approach reduces violations of the volume constraints from 75%
to 95%. This improvement is obtained at the expense of modest 0.5% increase
of the production cost in comparison with the cost of the deterministic technique. Two EDF's internal reports are in progress
([25] ,[26] ).

In the newt year, we'll study the problem of management of the margin of production, defined as the assessment between the offer and the demand. This problem can be formulated as finding optimal decisions, according to an economical criterion, for hedging against supply shortage or for selling the positive margin of production. A EDF's report is in progress ([27] ).

#### Solving the French optimal power planning problem in the midterm via dynamic bundle methods

Participants : G Emiel [edf, impa], C. Sagastizábal.

We are interested in the use of bundle methods to solve large-scale mixed-integer problems, possibly nonconvex. Specifically, we consider optimization problems arising in electrical power management. Given an electric generation mix, the aim is to minimize production costs subject to operating constraints of generation units and other external constraints, like network flow capacities. There are many different problems fitting such a large framework. In particular, the time horizon chosen for the scheduling highly determines the specificity of problems. Short, middle and long term decisions have their own peculiarities that need to be reflected in the modeling. While short term problems are generally modeled in a deterministic framework, for longer terms, inherent uncertainties may result in poor solutions if a deterministic model is still used. Consider for instance the French case, where winter demand has uncertainties reaching up to several thousands of MW. When comparing this value to typical peak loads (70000 MW), we see that for the modeling to yield any significant values, it must explicitly incorporate the stochastic nature of the problem. For the mid-term problem we are interested to solve, uncertainty is represented by a scenario tree, composed by many nodes representing all possible values of the demand, at each given time step.

A solution approach introduced in [29] based on stochastic Lagrangian relaxation, can be applied to solve this problem. However, since the relaxed constraints involve satisfaction of demand at each node,
the dimension of the dual problem increases with the size of the scenario tree.
In addition, demand constraints are formulated as 3 different equations at each node, corresponding to 3 blocks: base, average, and peak demand. As a result, the dual dimension becomes very soon too big to be dealt with (from the nondifferentiable point of view, a problem with more than 10^{5} variables is considered ``large scale''.

Instead of dualizing all the constraints at once, an alternative
approach is to choose at each iteration *subsets* of
constraints to be dualized. In this dynamical relaxation, subsets
J have cardinality |J| much smaller than the original one. As a result, the
corresponding dual function is
manageable from the nondifferentiable optimization point of view.

Such dynamic relaxation could be done according to some rule depending on which multipliers are active, for example (i.e., analyzing at which nodes and blocks the demand is not satisfied). From the dual point of view, this approach yields multipliers with varying dimensions and a dual objective function that changes along iterations.

Based on [31] , we discuss how to apply a bundle dynamic methodology to solve this kind of dual problems, putting a particular emphasis on the specific structure of the considered power management problem. We also investigate alternative approaches, related to the so-called *incremental* methods.

Work in progress.