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In the course of the energy transition, load and supply centers are growing apart in many electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. In this context, the establishment of price zones in order to at least partially price grid bottlenecks is under discussion. This paper addresses the key question of how to configure price zones on a network in order to optimally govern investment and production decisions in the long run. We extend the multilevel equilibrium model from Grimm et al. (2017a) to endogenously determine welfare-maximizing price zones for a given electricity market and analyze their impact on market outcomes. This mixed-integer nonlinear model contains a graph partitioning problem on the first level to model the zoning of the network. Using a generalized Benders decomposition and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that already a few optimally chosen zones lead to significant welfare gains.

In this work we analyze the structural properties of the set of feasible bookings in the European entry-exit gas market system. We present formal definitions of feasible bookings and then analyze properties that are important if one wants to optimize over them. Thus, we study whether the sets of feasible nominations and bookings are bounded, convex, connected, conic, and star-shaped. The results depend on the specific model of gas flow in a network. Here, we discuss a simple linear flow model with arc capacities as well as nonlinear and mixed-integer nonlinear models of passive and active networks, respectively. It turns out that the set of feasible bookings has some unintuitive properties. For instance, we show that the set is nonconvex even though only a simple linear flow model is used.

We show that mixed-integer control problems for evolution type partial differential equations can be regarded as operator differential inclusions. This yields a relaxation result including a characterization of the optimal value for mixed-integer optimal control problems with control constraints. The theory is related to partial outer convexification and sum-up rounding methods. The results are applied to optimal valve switching control for gas pipeline operations. A numerical example illustrates the approach.

We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved.

This paper mainly studies two topics: linear complementarity problems (LCPs) for modeling electricity market equilibria and optimization under uncertainty. While there have been quite some attempts to deal with uncertain LCPs in a stochastic - i.e., distributional - sense, robust LCPs have only gained attention very recently. In this paper, we consider both perfectly competitive and Nash-Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition this particularly means that the two classical welfare theorems also hold in both considered robust cases. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of Γ-robustifications of Nash-Cournot models. Thus, an analogue of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.

Pricing of access to energy networks is an important issue in liberalized energy sectors because of the natural monopoly character of the underlying transport infrastructures. We introduce a general pricing framework for potential-based energy flows in arbitrarily structured transport networks. In different specifications of our general pricing model we discuss first- and second-best pricing results and compare different pricing outcomes of potential-free and potential-based energy flow models. Our results show that considering nonlinear laws of physics leads to significantly different pricing results on networks and that these differences can only be seen in sufficiently complex, e.g., cyclic, networks as they can be found in real-world situations.

We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.

The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities
(2018)

It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function - in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute global optimal solutions in finite time and we also provide a worst-case iteration bound. However, first numerical experiences reveal that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal design of zonal pricing electricity markets. This together with nonlinearities due to economic modeling yields extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof, whereas the second is a tailored generalized Benders decomposition. We prove for both methods that they yield global optimal solutions. Finally, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation.

We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing's optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern Germany.