Falk Hante, Michael Hintermüller, Sebastian Pokutta
Paulina Bock de Barillas
01.06.2021 − 31.05.2024
The move towards sustainable energy networks, with due consideration of the complexities arising from the integration of rather unsteady renewable resources, causes major operational challenges for network providers, cf. the Figure below. Mathematical control theory provides a key technology for optimization-guided dispatching processes, but is facing the challenges of large-scale dynamics from physical models such as partial differential equations (PDEs) linked with discrete decision models such as 0-1 switching for connecting/disconnecting capacitors, generating units and transmission lines. These problems are to be solved fast and repeatedly while gathering data to adapt.
The research agenda of this project tackles fundamental questions concerning control and optimization in context of this application. It concerns mathematics at the interface and beyond the state-of-the-art in combinatorial optimization and control theory for partial differential equations (PDEs). The main goal is to establish novel stationarity concepts for broad classes of PDE-dynamic mixed-integer programs based on exact and approximate relaxation techniques for extended formulations of such optimization problems, their primal-dual optimality conditions and their numerical solutions using semismooth Newton methods and thresholding strategies. A strong focus is given to treat combinatorial restrictions modelling minimal up/down time, ramping or switching order constraints.
The applicability of the theoretical achievements will be showcased on electricity distribution networks by means of case studies.
 A. Bärmann, A. Heidt, A. Martin, S. Pokutta and C. Thurner: Polyhedral Approximation of Ellipsoidal Uncertainty Sets via Extended Formulations – a computational case study. Computational Management Science, Vol. 13, Nr. 2, 2016.
 G. Braun, A. Roy and S. Pokutta. Stronger Reductions for Extended Formulations. arXiv preprint arXiv:1512.04932 (2018).
 M. Burger, Y. Dong, and M. Hintermüller: Exact relaxation for classes of minimization problems with binary constraints, arXiv preprint arXiv:1210.7507 (2012).
 S. Fiorini, S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf: Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds. Proceedings of STOC 2012. arXiv:1111.0837 (2012).
 S. Göttlich, F.M. Hante, A. Potschka, and L. Schewe: Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints. Mathematical Programming, Vol. 188, pp. 599–619 (2021).
 Hante, F.M. et al. Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications. In: Industrial Mathematics and Complex Systems, P. Manchanda, R. Lozi, A.H. Siddiqi (Eds.), Springer Singapore, pp. 77–122, 2017.
 F.M. Hante, and S. Sager: Relaxation methods for mixed-integer optimal control of partial differential equations, Computational Optimization and Applications, 2013.
 Hante, F.M.: Relaxation methods for hyperbolic PDE mixed-integer optimal control problems. Optimal Control Applications and Methods. Vol. 38, Nr. 6, pp. 1103–1110, 2017.
 F.M. Hante: Mixed-Integer Optimal Control for PDEs: Relaxation via Differential Inclusions and Applications to Gas Network Optimization. In: Industrial and Applied Mathematics, P. Manchanda, R. Lozi, A.H. Siddiqi (Eds.), pp. 157–171 Springer Singapore, 2020.
 M. Hintermüller, K. Ito, and K. Kunisch: The primal-dual active set strategy as a semismooth Newton method. SIAM Journal on Optimization, Vol 13, Nr. 3, 2002.