Fair Least Core: efficient, stable and unique game-theoretic reward allocation in Energy Communities by row-generation

Energy Communities are increasingly proposed as a tool to boost renewable penetration and maximize citizen participation in energy matters. These policies enable the formation of legal entities that bring together power system members, enabling collective investment and operation of energy assets. However, designing appropriate reward schemes is crucial to fairly incentivize individuals to join, as well to ensure collaborative and stable aggregation, maximizing community benefits. Cooperative Game Theory, emphasizing coordination among members, has been extensively proposed for ECs and microgrids; however, it is still perceived as obscure and difficult to compute due to its exponential computational requirements. This study proposes a novel framework for stable fair benefit allocation, named Fair Least Core, that provides uniqueness, replicability, stability and fairness. A novel row-generation algorithm is also proposed that allows to efficiently compute the imputations for coalitions of practical size. A case study of ECs with up to 50 members demonstrates the stability, reproducibility, fairness and efficiency properties of proposed model. The results also highlight how the market power of individual users changes as the community grows larger, which can steer the development of practical reliable, robust and fair reward allocations for energy system applications

A Bilevel Programming Approach to Price Decoupling in Pay-as-Clear Markets, with Application to Day-Ahead Electricity Markets

The Italian and European electricity markets are experiencing a crisis caused by sharp increases in gas prices, which are reflected in dramatically higher final cost of electricity for all consumers w.r.t. historical values. This, however, is only in part motivated by the higher price of the gas actually used for electricity production: a very significant contribution is instead due to the "Pay-as-Clear" (PaC) mechanism implemented in the Day-Ahead Electricity Market (DAM), whereby all producers are remunerated at the price of the most expensive--typically, gas-fired--unit. This has led to a surge of the interest in the development of mechanisms capable of decoupling the price paid to the units whose production cost depends on that of fuel--and that therefore must be able to track that to ensure the economic compatibility of their operations--from those for which this is not the case. However, since both types of units participate in satisfying the same demand, this is technically complex. Motivated by this highly compelling application we propose the concept of Segmented Pay-as-Clear (SPaC) market, introducing a new family of market clearing problems--in fact, a relatively straightforward modification of standard ones--that has the potential to achieve such a decoupling without losing the crucial features of the PaC, i.e., that of providing both long- and short-term price signals. The approach is based on dynamically partitioning demand across the segmented markets, where the partitioning is endogenous, i.e., controlled by the model variables, and is chosen to minimise the total system cost. The thusly modified model belongs to the family of Bilevel Programming problems with a non-linear non convex objective function, or more generally a Mathematical Program with Complementarity Constraints; these problems have a higher computational complexity than those corresponding to the standard PaC, but in the same ballpark as the models routinely used in real-world DAMs to represent "nonstandard" requirements, e.g., the unique buying price in the Italian DAM. Thus, SPaC models should still be solvable in a time compatible with market operation with appropriate algorithmic tools. Like all market models, SPaC is not immune to strategic bidding techniques, but some theoretical results indicate that, under the right conditions, the effect of these could be limited. An initial experimental analysis of the proposed models, carried out through Agent Based simulations, seems to indicate a good potential for significant system cost reductions and an effective decoupling of the two markets

Mercati dell’Energia Efficienti a Prezzi Marginali Disaccoppiati

Il mercato elettrico italiano ed europeo sta vivendo una situazione di stress causata dai forti incrementi del prezzo del gas, che si riflettono sul costo finale dell'energia elettrica per i consumatori. Tale incremento, però, è in effetti superiore a quanto strettamente dovuto al rincaro del gas effettivamente usato per la produzione elettrica. Questo in ragione del meccanismo Pay-as-Clear (PaC) implementato nel Mercato Day-Ahead dell'energia (DAM), per il quale tutti i produttori vengono remunerati al prezzo dell'unità più costosa---tipicamente quelle, appunto, a gas. Si rivela quindi necessario provare a disaccoppiare le unità di produzione (UP) il cui costo dipende da quello del combustibile, e che quindi devono poter seguire questo ai fini della compatibilit\`a economica delle loro operazioni, da quelle il cui costo di produzione è sostanzialmente costante e non dipende da tali dinamiche. Poiché entrambi i tipi di unità partecipano alla soddisfazione della stessa domanda, questo è però tecnicamente complesso. In questa nota si propone una famiglia di problemi di clearing del DAM (in effetti, una modifica di quelli esistenti) che ha la possibilità di ottenere un tale disaccoppiamento senza perdere le utili caratteristiche del PaC in termini di fornire segnali di prezzo indispensabili sia nel lungo che nel breve periodo. L'approccio si basa sul partizionare dinamicamente la domanda sui due mercati, ove la partizione è una variabile del modello e viene scelta per minimizzare il costo totale di sistema. Il problema così modificato risulta della famiglia dei problemi di Programmazione Bilevel con funzione obiettivo nonlineare nonconvessa, o più in generale un Mathematical Program with Complementarity Constraints; questi problemi hanno una complessità computazionale superiore a quelli attualmente utilizzati, ma sono ancora risolubili in tempi compatibili con l'operatività del mercato con opportuni strumenti algoritmici. Come tutti i modelli di mercato anche quelli proposti non sono immuni a tecniche di strategic bidding, ma alcuni risultati teorici indicano che, nelle giuste condizioni, l'effetto di questi potrebbe essere limitato. Una prima analisi sperimentale dei modelli proposti, effettuata attraverso simulazioni di tipo Agent Based, sembra indicare un buon potenziale per ottenere significative riduzioni del costo di sistema ed un efficace disaccoppiamento dei due mercati

Tape diagrams for rig categories with finite biproducts

Rig categories with finite biproducts are categories with two monoidal products, where one is a biproduct and the other distributes over it. In this report we present tape diagrams, a sound and complete diagrammatic language for rig categories with finite biproducts, which can be thought intuitively as string diagrams of string diagrams

Optimal sizing of energy communities with fair revenue sharing and exit clauses: value, role and business model of aggregators and users

Energy communities (ECs) are essential policy tools to meet the Energy Transition goals, as they can promote renewable energy sources, demand side management, demand response and citizen participation in energy matters. However, to fully unleash their potential, their design and scheduling requires a coordinated technical operation that the community itself may be ill-equipped to manage, in particular in view of the mutual technical and legal constraints ensuing from a coordinated design. Aggregators and Energy Service COmpanies (ESCOs) can perform this support role, but only provided that their goals are aligned to those of the community, not to incur in the agency problem. In this study, we propose a business model for aggregators of ECs, and its corresponding technical optimization problem, taking into account all crucial aspects: i) alleviating the risk of the agency problem, ii) fairly distributing the reward awarded to the EC, iii) estimating the fair payment for the aggregator services, and iv) defining appropriate exit clauses that rule what happens when a user leaves the EC. A detailed mathematical model is derived and discussed, employing several fair and theoretically-consistent reward distribution schemes, some of which are, to the best of our knowledge, proposed here for the first time. A case study is developed to quantify the value of the aggregator and compare the coordinated solution provided by the aggregator with non-coordinated configurations, numerically illustrating the impact of the reward distribution schemes. The results show that, in the case study, the aggregator enables reducing costs by 16% with respect to a baseline solution, and enables reaching 52.5% renewable share and about 46% self/shared consumption, whereas these same numbers are only 28-35% for the non-coordinated case. Our results suggest that the aggregator fair retribution is around 16-24% the added benefit produced with respect to the non-coordinated solution, and that stable reward distribution schemes such as Shapley/Core or Nucleolus are recommended. Moreover, the results highlight the unwanted effect that some non-cooperative ECs may have an added benefit without providing any positive effect to the power system. Our work lays the foundations for future studies on business models of aggregators for ECs and provides a methodology and preliminary results that can help policy makers and developers in tailoring national-level policies and market-offerings

Ellipsoidal Classification via SemiDefinite Programming

Separating two finite sets of points in a Euclidean space is a fundamental problem in classification. Customarily linear separation is used, but nonlinear separators such as spheres have been shown to have better performances in some tasks, such as edge detection in images. We exploit the relationships between the more general version of the spherical separation, where we use general ellipsoids, and the SVM model with quadratic kernel to propose a new classification approach. The implementation basically boils down to adding a SDP constraint to the standard SVM model in order to ensure that the chosen hyperplane in the feature space represents a non-degenerate ellipsoid in the input space; albeit being somewhat more costly than the original formulation, this still allows to exploit many of the techniques developed for SVR in combination with SDP approaches. We test our approach on several classification tasks, among which the edge detection problem for gray-scale images, proving that the approach is competitive with both the spherical classification one and the quadratic-kernel SVM one without the ellipsoidal restriction

Sequencing and Routing in a Large Warehouse with High Degree of Product Rotation

The paper deals with a sequencing and routing problem originated by a real-world application context. The problem consists in defining the best sequence of locations to visit within a warehouse for the storage and/or retrieval of a given set of items during a specified time horizon, where the storage/retrieval location of an item is given. Picking and put away of items are simultaneously addressed, by also considering some specific requirements given by the layout design and operating policies which are typical in the kind of warehouses under study. Specifically, the considered sequencing policy prescribes that storage locations must be replenished or emptied one at a time by following a specified order of precedence. Moreover, two fleet of vehicles are used to perform retrieving and storing operations, whose routing is restricted to disjoint areas of the warehouse. We model the problem as a constrained multicommodity flow problem on a space-time network, and we propose a Mixed-Integer Linear Programming formulation, whose primary goal is to minimize the time traveled by the vehicles during the time horizon. Since large-size realistic instances are hardly solvable within the time limit commonly imposed in the considered application context, a matheuristic approach based on a time horizon decomposition is proposed. Finally, we provide an extensive experimental analysis aiming at identifying suitable parameter settings for the proposed approach, and testing the matheuristic on particularly hard realistic scenarios. The computational experiments show the efficacy and the efficiency of the proposed approach

Solving Stochastic Hydrothermal Unit Commitment with a New Primal RecoverycTechnique Based on Lagrangian Solutions

The high penetration of intermittent renewable generation has prompted the development of Stochastic Hydrothermal Unit Commitmentc(SHUC) models, which are more difficult to be solved than their thermal-basedccounterparts due to hydro generation constraints and inflow uncertainties.cThis work presents a SHUC model applied in centralized cost-based dispatch, where the uncertainty is related to the water availability in reservoirs and demand. The SHUC is represented by a two-stage stochastic model, formulated as a large-scale mixed-binary linear programming problem. The solution strategy is divided into two steps, performed sequentially, with intercalated iterations to find the optimal generation schedule. The first step is the Lagrangian Relaxation (LR) approach. The second step is given by a Primal Recovery based on LR solutions and a heuristic based on Benders' Decomposition. Both steps benefit from each other, exchanging information over the iterative process. We assess our approach in terms of the quality of the solutions and running times on space and scenario LR decompositions. The results show the advantage of our primal recovery technique compared to solving the problem via MILP solver. This is true already for the deterministic case, and the advantage grows as the problem’s size (number of plants and/or scenarios) does

A Lagrangian approach to Chance Constrained Routing with Local Broadcast

Mobile cellular networks play a pivotal role in emerging Internet of Things (IoT) applications, such as vehicular collision alerts, malfunctioning alerts in Industry-4.0 manufacturing plants, periodic distribution of coordination information for swarming robots or platooning vehicles, etc. All these applications are characterized by the need of routing messages within a given local area (geographic proximity) with constraints about both timeliness and reliability (i.e., probability of reception). This paper presents a Non-Convex Mixed-Integer Nonlinear Programming model for a routing problem with probabilistic constraints on a wireless network. We propose an exact approach consisting of a branch-and-bound framework based on a novel Lagrangian decomposition to derive lower bounds. Preliminary experimental results indicate that the proposed algorithm is competitive with state-of-the-art general-purpose solvers, and can provide better solutions than existing highly tailored ad-hoc heuristics to this problem

A multiperiod drayage problem with customer-dependent service periods

We investigate a routing problem arising in the domain of drayage operations. to determine mimimum-cost vehicle routes in several periods. We adapt a set-covering model, which is solved either with all feasible routes by an off-the-shelf MIP solver, or by and a Price-and-Branch algorithm in which the pricing problem is a formulated as a collection of shortest path problems in tailor-made auxiliary acyclic networks. We propose a new arc-flow formulation based on the previous auxiliary networks and show that solving it by a MIP solver is usually preferable. Finally, we characterize how possible changes in flexibility levels affect routing costs