Approaching Prosumer Social Optimum via Energy Sharing With Proof of Convergence

Abstract

With the advent of prosumers, the traditional centralized operation may become impracticable due to computational burden, privacy concerns, and conflicting interests. In this paper, an energy sharing mechanism is proposed to accommodate prosumers' strategic decision-making on their self-production and demand in the presence of capacity constraints. Under this setting, prosumers play a generalized Nash game. We prove main properties of the game: an equilibrium exists and is partially unique; no prosumer is worse off by energy sharing and the price-of-anarchy is 1 − O(1/I) where I is the number of prosumers. In particular, the PoA tends to 1 with a growing number of prosumers, meaning that the resulting total cost under the proposed energy sharing approaches social optimum. We prove that the corresponding prosumers' strategies converge to the social optimal solution as well. Finally we propose a bidding process and prove that it converges to the energy sharing equilibrium under mild conditions. Illustrative examples are provided to validate the results. Index Terms-Energy sharing, generalized Nash equilibrium, prosumer, bidding algorithm, distributed mechanism NOMENCLATURE A. Indices, Sets, and Functions i, I Index and set of prosumers. S iAction sets of prosumer i, and S = ∏ i∈I S i .Utility function of prosumer i J i (p i , d i ) Net cost of prosumer i, which equals to f i (p i ) − u i (d i ); and J(p, d) = ∑ i∈I J i (p i , d i ). Γ i (p, d, b) Total net cost of prosumer i with sharing, which equals tosatisfies the capacity constraint. B. Parameters I Number of prosumers. p i , p i Lower/upper bound of prosumer i's production. d i , d i Lower/upper bound of prosumer i's demand. a Energy sharing market sensitivity.

Approaching Prosumer Social Optimum via Energy Sharing with Proof of Convergence

“…The proof of Proposition 2 is in Appendix B of [43]. It verifies that the proposed mechanism can incentivize prosumers to join since no prosumer is worse off and at least one can benefit.…”

“…The proof of Proposition 1 is in Appendix A of [43]. Note that constraints (5b)-(5d) and (1b)-(1d) are identical, which implies the migration from centralized operation to the distributed mechanism does not sacrifice feasibility.…”

“…Even without assumptions A1 and A2, where prosumers might buy from the grid, prosumer i ∈ I can still obtain at least the same net utility in sharing as that in self-sufficiency by letting b i = ∑ j =ib j /(I − 1). In other words, Proposition 2 can be proved following a similar procedure to that in Appendix B of [43].…”

“…Particularly, a PoA equal to 1 implies the game achieves social optimal. The proof of Proposition 3 is in Appendix C of [43]. Note that PoA is conventionally larger than 1 with a positive cost at social optimal [45].…”

“…For proving convergence of the bidding process, we first give the following lemma with its proof in Appendix D of [43]. For conciseness, denote…”