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Industrial electricity consumers with flexible demand can profit by adjusting their load to short-term prices and by providing balancing services to the grid. Markets which support this kind of short-term position adjustment are the day-ahead market and balancing markets. We propose a formulation for a combined optimization model that computes an optimal distribution of flexibility between the balancing and day-ahead markets. The optimal solution also includes the specific bids for the day-ahead and balancing markets. Besides the expected profits of each market and their individual bidding languages, our model also takes their different roles in a continuous marketing of flexibility into account. To prevent overrating short-term profits we introduce a variable penalty term that adds a cost to unfavorable load schedules. We evaluate the optimization model in a rolling horizon case study based on the setting of a virtual battery at TRIMET SE, which is derived from a flexible aluminum electrolysis process. For such a battery we compute a daily optimal split of flexibility and trading decisions based on data in the period 04/2021 - 03/2022. We show that the optimal split is more profitable than using only one market or a fixed split between the markets., Comment: 29 pages, 13 figures, 1 table

In a combinatorial exchange setting, players place sell (resp. buy) bids on combinations of traded goods. Besides the question of finding an optimal selection of winning bids, the question of how to share the obtained profit is of high importance. The egalitarian allocation is a well-known solution concept of profit sharing games which tries to distribute profit among players in a most equal way while respecting individual contributions to the obtained profit. Given a set of winning bids, we construct a special network graph and show that every flow in said graph corresponds to a core payment. Furthermore, we show that the egalitarian allocation can be characterized as an almost equal maximum flow which is a maximum flow with the additional property that the difference of flow value on given edge sets is bounded by a constant. With this, we are able to compute the egalitarian allocation in polynomial time., Comment: 7 pages, 1 figure

The Set Cover Problem (SCP) and the Hitting Set Problem (HSP) are well-studied optimization problems. In this paper we introduce the Reward-Penalty-Selection Problem (RPSP) which can be understood as a combination of the SCP and the HSP where the objectives of both problems are contrary to each other. Applications of the RPSP can be found in the context of combinatorial exchanges in order to solve the corresponding winner determination problem. We give complexity results for the minimization and the maximization problem as well as for several variants with additional restrictions. Further, we provide an algorithm that runs in polynomial time for the special case of laminar sets and a dynamic programming approach for the case where the instance can be represented by a tree or a graph with bounded tree-width. We further present a graph theoretical generalization of this problem and results regarding its complexity., Comment: 24 pages, 8 figures, 1 table

The Egalitarian Allocation (EA) is a well-known profit sharing method for cooperative games which attempts to distribute profit among participants in a most equal way while respecting the individual contributions to the obtained profit. Despite having desirable properties from the viewpoint of game theory like being contained in the core, the EA is in general hard to compute. Another well-known method is given by Vickrey Payments (VP). Again, the VP have desirable properties like coalitional rationality, the VP do not fulfill budget balance in general and, thus, are not contained in the core in general. One attempt to overcome this shortcoming is to scale down the VP. This can be done by a unique scaling factor, or, by individual scaling factors. Now, the individual scaled Vickrey Payments (ISV) are computed by maximizing the scaling factors lexicographically. In this paper we show that the ISV payments are in fact identical to a weighted EA, thus exhibiting an interesting connection between EA and VP. With this, we conclude the uniqueness of the ISV payments and provide a polynomial time algorithm for computing a special weighted EA., Comment: 12 pages