**Project Heads**

*Michael Hintermüller, Volker Mehrmann, Carlos Rautenberg*

**Project Members**

Marcelo Bongarti

**Project Duration**

01.06.2021 – 31.12.2021

**Located at**

WIAS

Motivated by efficient energy distribution, we develop theory and solution algorithms for a new class of generalized Nash equilibrium problems (GNEPs) arising in game theoretic formulations of energy markets. As agents of the GNEP measure data along the underlying equilibrium process, we establish model predictive control (MPC) and closed loop strategies to target realistic control scenarios. The pertinent transport physics together with additional system constraints are considered within a hierarchy of models and with possible stochastic perturbations.

A main challenge is the shift from current energy carrier portfolios to renewable ones with an optimal distribution. Thereby, of interest are energy carriers transportable over long distance and traded on markets. The mathematical modelling of the dynamics of these modern energy markets leads to complex intertwined scientific features. We have non-cooperative producers / wholesalers which are subject to the market dynamics and the network transport and operations depend on underlying physics associated with the energy carrier. Moreover, demand, network operation, and other exogenous quantities add uncertainties to price development and transport volume of energy carriers.

From a mathematical perspective the problem will be analysed in a game theoretical setting. The mathematical research focus will be on *Generalized Nash Equilibrium Problems (GNEP)* with a shared PDE constraint. If the GNEP is jointly convex, then one can look for a variational equilibrium of the GNEP. Furthermore, under a constraint qualification, smoothness and convexity assumptions, a GNEP can be related to a Quasi-Variational Inequality (QVI).

**Pricing:** In the context of goods being exchanged on a market between consumer and producer, a pricing mechanism comes from considering the price π as the multiplier associated with the constraint linking supply s and demand d.

d ≤ s ⊥ π ≥ 0 or d = s ⊥ π ∈ R.

This will lead to Multiple Objectives Problem with Equilibrium Constraints (MOPEC).

**Challenges**:

- Splitting strategies (Jacobi, Gauss-Seidel) usually have limited effectiveness due to the coupling between players.
- Even if each player’s optimization problem is convex, the (Q)VI associated mapping is likely not monotone which will require a non-monotone (Q)VI solver.
- Multiplicity of global optimal solutions, Pareto fronts.

The developed methods will lead to software packages and be transferred to applications.

**Project Webpages**

**Selected Publications
**

- A. Alphonse, M. Hintermüller, C.N. Rautenberg, Existence, iteration procedures and directional differentiability for parabolic QVIs, Calculus of Variations and Partial Differential Equations, 59:95/1–95/53, 2020.
- D. Gahururu, M. Hintermüller, S.M. Stengl, and T.M. Surowiec. Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion, preprint, 2019.

**Selected Pictures
**

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