  The focus on Renewable Energy Communities (REC) is fastly growing after the
European Union (EU) has introduced a dedicated regulation in 2018. The idea of
creating local groups of citizens, small- and medium-sized companies, and
public institutions, which self-produce and self-consume energy from renewable
sources is at the same time a way to save money for the participants, increase
efficiency of the energy system, and reduce CO$_2$ emissions. Member states
inside the EU are fixing more detailed regulations, which describe, how public
incentives are measured. A natural objective for the incentive policies is of
course to promote the self-consumption of a REC. A sophisticated incentive
policy is that based on the so called 'virtual framework'. Under this framework
all the energy produced by a REC is sold to the market, and all the energy
consumed must be paid to retailers: self-consumption occurs only 'virtually',
thanks a money compensation (paid by a central authority) for every MWh
produced and consumed by the REC in the same hour. In this context, two
problems have to be solved: the optimal investment in new technologies and a
fair division of the incentive among the community members. We address these
problems by considering a particular type of REC, composed by a representative
household and a biogas producer, where the potential demand of the community is
given by the household's demand, while both members produce renewable energy.
We set the problem as a leader-follower problem: the leader decide how to share
the incentive for the self-consumed energy, while the followers decide their
own optimal installation strategy. We solve the leader's problem by searching
for a Nash bargaining solution for the incentive's fair division, while the
follower problem is solved by finding the Nash equilibria of a static
competitive game between the members.
