Abstract
This paper shows that the compositional rule of inference reduces to the compatibility modification inference, when the antecedent of a fuzzy rule and its input fulfil some properties. Two main types of implications are investigated, those generalizing the classical material conditional (residuated and strong implications), called in this paper m-implications, and those generalizing the classical Cartesian product (t-norms and pseudo-conjunctions), called t-implications. For m-implications, when their maximum modus ponens generating function is used, the compatibility measure depends obviously on the relationship between input and antecedent, but also on the t-norm the implication comes from. Similarly, for t-implications, also used as their own modus ponens generating function, the compatibility measure depends again on the input and antecedent, but also on the t-norm they come from. As it could be expected, in the first case the compatibility measure is a degree of the inclusion of the input into the antecedent, while in the second one it is a degree of the intersection of both.

Abstract
The new energy policies adopted by the European Union are set to help in the decarbonization of the energy system. In this context, the share of Variable Renewable Energy Sources is growing, affecting electricity markets, and increasing the need for system flexibility to accommodate their volatility. For this reason, legislation and incentives are being developed to engage consumers in the power sector activities and in providing their potential flexibility in the scope of grid system services. This work identifies energy and cross-sector Business Models (BM) centered on or linked to the provision of distributed flexibility to the DSO and TSO, building on those found in previous research projects or from companies' commercial proposals. These BM are described and classified according to the main actor. The remaining actors, their roles, the interactions among them, how value is created by the BM activities and their value propositions are also described.

Abstract
In Walrasian markets, an auctioneer proposes a price to the market participants, who react by revealing the quantities they are willing to buy or sell at this price. The auctioneer then proposes new prices to improve the demand and supply match until the equilibrium is reached. This market, common for stock exchanges, has also been proposed for electricity markets like power electricity exchanges, where iterations among auctioneer and market participants take place before the interval settlement period (ISP) until supply and demand match and a stable price is reached. We propose a Walrasian design for local electricity markets where the iterations between auctioneer and market participants happen in real time, so previous imbalances are used to correct the proposed price for the next ISP. The designs are simulated to test convergence and their capability of achieving efficient dynamic prices.

Abstract
This work presents an approach to the flexibility of energy consumption in Renewable Energy Communities (RECs). A two-stage model for quantifying the flexibility provided by the domestic energy resources operation and its negotiation in a market platform is proposed. In stage 1, the optimal consumption of each prosumer is determined, as well as the respective technical flexibility of their resources, namely the maximum and minimum resource operation limits. In stage 2, this technical flexibility is offered in a local flexibility-only market structure, in which both the DSO and the prosumers can present their flexibility needs and requirements. The flexibility selling and buying bids of the prosumers participating in the market are priced based on their base tariff, which is the energy cost of the prosumers corresponding to their optimal schedule of the first stage when no flexibility is provided. Therefore, providing flexibility is an incentive to reduce their energy bill or increase their utility, encouraging their participation in the local flexibility market.

Abstract
When the price of the energy shared within an energy community is based on a local energy market, it is the responsibility of each participant to bid adequately so that participating provides a larger benefit than not participating. Alternatively, centralized energy community bill minimization may be an option, but a mechanism to share the collective benefits among the members is needed. This mechanism should be fair and easy to explain, no members should be harmed with respect to their individual optimal behavior and should provide the right economic signal. This paper analyses and compares some common pricing mechanisms for the internal compensation for the energy shared among the members of an energy community centrally managed. Simple case examples are used to identify those pricing mechanisms that are fairer and provide the righter economic signals to the participants.

Abstract
This paper presents an innovative digital platform for managing energy communities with self-consumption and energy trading in a local electricity market. Its architecture is based on micro-services, such as the energy transaction service, the settlement service to compute the financial compensations among community members for the energy transacted, or a resource sizing service. This approach enables the platform to be more efficient and scalable, making easier to incorporate new functionalities while maintaining a secure community and energy transactions management. The transactions and settlement procedures, adapted to the Portuguese regulation, are described, and the results of the platform operating a post-delivery pool market are presented and analyzed. This paper contributes to the understanding and improvement of renewable energy communities' business models and management, offering insights for policymakers, researchers, and practitioners in the field.