A technical survey on optimal power flow under
uncertainty: An extended out-of-sample analysis
With an increasing part of renewable energy generation connected into the grid, the share
of uncertainty and intermittency in the electricity generation mix is also increasing. In parallel,
the electricity sector liberalization, aiming at more investments into the power systems and
electricity price reductions, introduced competition at the generation as well as the supply
level. Each actor has now the economic objective of minimizing cost or maximizing profit.
The markets allow the different actors to exchange energy at different time scales. One goal
of the market operator is to minimize the social costs by dispatching the units in a convenient
way. Such a problem is called an optimal power flow (OPF). The overall system safe operation
is guaranteed only if consumption is equal to generation at any time. This constraint has to
be imposed when clearing the markets, in particular from day-ahead (DA) to real-time (RT).
However, it is harder to reach because of the increasing uncertainty part in the electricity
production.
The goal of this master thesis is to implement stochastic optimization techniques on the
OPF problem, aiming at improving the decisions with respect to the deterministic case, by
including information about the uncertain parameters. The problem clears energy and reserve
auctions simultaneously and takes into account the structure of the grid. Secondly, the implemented techniques are fairly compared through an out-an-sample analysis that assesses their
performance with respect to, e.g. the expected social cost.
The considered techniques are enumerated here and below. The structure of our problem is
two-stage (DA and RT) and uses historical data to describe the uncertain parameters.
Stochastic programming approximates the distribution function of the uncertainty by a finite set of scenarios and their corresponding occurrence probability. Risk management is also
incorporated to the formulation in order to account for worst-case realizations through the Conditional Value-at-Risk. However, stochastic programming comes with some drawbacks. First,
a high number of scenarios comes with a higher computational burden and second, the lack of
risk aversion (scenario-based decisions may result in very high cost in some extreme cases).
An alternative to stochastic programming is adaptive robust optimization. This technique
aims at optimizing the objective function under the worst-case realization within an uncertainty
set (that is, the space of plausible realizations) and ensures the feasibility for each realization
within it. The bigger the uncertainty set is, the higher is the immunity of the decisions against
extreme scenarios i.e. the more conservative are the decisions. Robust solutions, however, are
linked to an overly risk-averse behaviour leading to higher expected costs (the savings made in
case of the worst-case realization does not compensate the opportunity losses in other classical
cases).
To overcome the issue of conservativeness of the decisions, chance-constrained programming
represents a trade-off between scenario-based and robust optimization. In this method, the
system constraints are imposed with probability at least 1 − . There exist reformulations of
such a chance-constrained problem based on historical data. Among them, the scenario and
robust approaches are implemented and analyzed.
Finally, an out-of-sample analysis is ran on consistent data and allows to compare the
performances of each decision set (i.e. stochastic, robust and chance-constrained ones) in terms
of expected social cost. The out-of-sample analysis consists in solving a real-time market for each
computed day-ahead decision set and for each sample in the out-of-sample dataset, resulting in
a total cost distribution. A general conclusion is that the more conservative the decisions are,
the higher the expected cost is but the lower is its variation. The impact of the wind penetration
level on the undertaken decisions is also discussed.
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