The ongoing quest for transportation electrification with the massive proliferation of EV charging stations (EVCSs) will deepen the interaction and require the further coordination of coupled power and transportation networks (PTN). The individually-owned EVCSs located in an urban transportation network (UTN) will compete using price signals to maximize their respective payoffs. In this paper, a multi-agent deep reinforcement learning (MA-DRL) method is proposed to model the pricing game in UTN and determine the optimal charging prices for a single EVCS. The EVCS charging demand is first analyzed using a modified user equilibrium traffic assignment problem (UE-TAP) with elastic traveling demands and different charging prices. The price competition problem is then formulated as a game with incomplete information in which the market environment is complex due to nonlinear traffic assignments. Thus, the MA-DRL approach is proposed to learn the charging pricing strategies of multiple EVCSs and approximate the Nash Equilibrium (NE) of the pricing game using the incomplete information. The proposed solution will determine the optimal pricing strategies for an EVCS in UTN. The case studies on a 24-node Sioux-Falls network, and the real-world Xi'an and Hangzhou cities are conducted to verify the effectiveness and scalability of the proposed approach.

This paper develops a new radial super-efficiency data envelopment analysis (DEA) model, which allows input–output variables to take both negative and positive values. Compared with existing DEA models capable of dealing with negative data, the proposed model can rank the efficient DMUs and is feasible no matter whether the input–output data are non-negative or not. It successfully addresses the infeasibility issue of both the conventional radial super-efficiency DEA model and the Nerlove–Luenberger super-efficiency DEA model under the assumption of variable returns to scale. Moreover, it can project each DMU onto the super-efficiency frontier along a suitable direction and never leads to worse target inputs or outputs than the original ones for inefficient DMUs. Additional advantages of the proposed model include monotonicity, units invariance and output translation invariance. Two numerical examples demonstrate the practicality and superiority of the new model.

This paper discusses distributionally robust geometric programs with individual or joint chance constraints. Several groups of uncertainty sets are considered: uncertainty sets with first two order moments information; uncertainty sets with known first order or first two order moments information under nonnegative support; uncertainty sets constrained by the Kullback–Leibler divergence with a normal or discrete reference distribution; uncertainty sets constrained by the Wasserstein distance under discrete, full, or nonnegative real-space support; and joint uncertainty sets for the product of random variables. Under each group of uncertainty sets, we find deterministic reformulations of the distributionally robust geometric programs with individual or joint chance constraints. Convexity, solution methods, and relationships of the reformulation programs are discussed. Finally, numerical tests are carried out on a shape optimization problem.