  We consider a network of prosumers involved in peer-to-peer energy exchanges,
with differentiation price preferences on the trades with their neighbors, and
we analyze two market designs: (i) a centralized market, used as a benchmark,
where a global market operator optimizes the flows (trades) between the nodes,
local demand and flexibility activation to maximize the system overall social
welfare; (ii) a distributed peer-to-peer market design where prosumers in local
energy communities optimize selfishly their trades, demand, and flexibility
activation. We first characterizethe solution of the peer-to-peer market as a
Variational Equilibrium and prove that the set of Variational Equilibria
coincides with the set of social welfare optimal solutions of market design
(i). We give several results that help understanding the structure of the
trades at an equilibriumor at the optimum. We characterize the impact of
preferences on the network line congestion and renewable energy waste under
both designs. We provide a reduced example for which we give the set of all
possible generalized equilibria, which enables to give an approximation of the
price ofanarchy. We provide a more realistic example which relies on the IEEE
14-bus network, for which we can simulate the trades under different preference
prices. Our analysis shows in particular that the preferences have a large
impact on the structure of the trades, but that one equilibrium(variational) is
optimal.
