Subject: Operational planning under joint uncertainties

Thesis supervisors:  Welington DE OLIVEIRA and Sophie DEMASSEY (CMA Mines Paris – PSL)

CIFRE thesis in collaboration with EDF

Abstract: The expansion of renewable energy sources (RES) leads to the growth of uncertainty in the power distribution network operation. The inherent variability and intermittency of RES present significant challenges to the efficient and reliable operation of power systems. To address these challenges, operational planning performed by distribution system operators should evolve, in particular, to allow the efficient utilization of different flexibility levers, such as active power modulation and reactive power management. Decisions on lever activation are based on the resolution of an alternating current optimal power flow problem (AC-OPF). This thesis develops algorithms for handling two stochastic AC-OPF models. These optimization problems are simultaneously nonconvex, nonsmooth, and discrete. The thesis aims to grasp these complexities accurately, by addressing the AC power flow equations without relying on convexification and by handling interdependent uncertainties either through a joint probability constraint or via scenario decomposition to cope with the discrete levers.

More specifically, the first proposed methodology addresses a continuous version of the joint chance-constrained AC-OPF. A first contribution of this work is the design of a numerical procedure (oracle) that enables the representation of the probability constraint as a difference of two convex functions. This step is followed by applying a known Difference-of-Convex (DoC) bundle method to the resulting continuous optimization problem. A second contribution concerns a new bundle algorithm with stronger convergence guarantees under weaker assumptions. For the chance-constrained AC-OPF, this algorithm provides a critical (generalized KKT) point. The work builds upon the employed DoC bundle and proposes a different master program and an original rule to update proximal parameter. The algorithm is capable of handling a broad class of nonsmooth and nonconvex optimization problems beyond the stochastic AC-OPF framework, provided the objective and constraint functions can be represented as differences of convex and weakly convex (CwC) functions. The practical performance of the algorithm is illustrated through numerical experiments on some nonconvex stochastic problems and is compared to the DoC bundle method for the chance-constrained AC-OPF in a 33-bus distribution network.

The second proposed methodology addresses operational planning rules for power modulation and curtailment, like priority and fairness, which result in logical and discrete formulations. The numerical results demonstrate the limitations of the bundle method for integrating integer variables. As an alternative, an optimization model is proposed that assigns a binary variable to each scenario and maximizes the number of satisfied scenarios within a limited budget. Applying penalization and block coordination allows separating those discrete considerations from the stochastic AC-OPF component, which is then decomposed into an individual deterministic AC-OPF for each scenario. Although it lacks theoretical convergence guarantees, the relevance of this approach is validated in practice.

Key words: Operational planning, Chance-constrained OPF, Discrete OPF, Stochastic optimization, Difference-of-convex, Weakly convex

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