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Addressing the urgent global challenge of man-made greenhouse gas emissions and climate change necessitates collaborative action between shipping lines and government regulatory agencies. Aligning with the International Maritime Organization’s emissions reduction strategy, this paper presents a novel bi-level programming model that unifies these stakeholders. On the upper level of the proposed bi-level model, a number of shipping lines optimize retrofitting plans for their vessels to maximize economic benefits. On the lower level, the regulatory agency responds to the carbon reduction efforts by setting retrofitting subsidies and emission penalty rates. This framework represents a multi-leader–single-follower game involving shipping lines and the regulatory agency, and its equilibrium is determined through an equilibrium problem with equilibrium constraints (EPEC). The EPEC comprises multiple single-leader–follower problems, each of which can be formulated as a mathematical program with equilibrium constraints (MPEC). The diagonalization algorithm (DM) is employed for its solution. Simulation studies performed based on a ten-year planning period show that the proposed approach can effectively promote vessel retrofitting and the use of green fuels, which leads to an annual emission reduction of over 50%. 

Abstract PurposeTo study the dosimetric impact of incorporating variable relative biological effectiveness (RBE) of protons in optimizing intensity‐modulated proton therapy (IMPT) treatment plans and to compare it with conventional constant RBE optimization and linear energy transfer (LET)‐based optimization. MethodsThis study included 10 pediatric ependymoma patients with challenging anatomical features for treatment planning. Four plans were generated for each patient according to different optimization strategies: (1) constant RBE optimization (ConstRBEopt) considering standard‐of‐care dose requirements; (2) LET optimization (LETopt) using a composite cost function simultaneously optimizing dose‐averaged LET (LET_d) and dose; (3) variable RBE optimization (VarRBEopt) using a recent phenomenological RBE model developed by McNamara et al.; and (4) hybrid RBE optimization (hRBEopt) assuming constant RBE for the target and variable RBE for organs at risk. By normalizing each plan to obtain the same target coverage in either constant or variable RBE, we compared dose, LET_d, LET‐weighted dose, and equivalent uniform dose between the different optimization approaches. ResultsWe found that the LETopt plans consistently achieved increased LET in tumor targets and similar or decreased LET in critical organs compared to other plans. On average, the VarRBEopt plans achieved lower mean and maximum doses with both constant and variable RBE in the brainstem and spinal cord for all 10 patients. To compensate for the underdosing of targets with 1.1 RBE for the VarRBEopt plans, the hRBEopt plans achieved higher physical dose in targets and reduced mean and especially maximum variable RBE doses compared to the ConstRBEopt and LETopt plans. ConclusionWe demonstrated the feasibility of directly incorporating variable RBE models in IMPT optimization. A hybrid RBE optimization strategy showed potential for clinical implementation by maintaining all current dose limits and reducing the incidence of high RBE in critical normal tissues in ependymoma patients. 

This paper discusses a market-based pool strategy for a microgrid (MG) to optimally trade electric power in the distribution electricity market (DEM). The increasing penetration levels of distributed energy resources (DERs) and MGs in distribution system (DS) stress distribution system operator (DSO) and require higher levels of coordinated control strategies. The distribution system operator has limited visibility and control over such distributed resources. To reduce the complexity of the system and improve the efficiency of the electricity market operation, we propose a decentralized pool strategy for an MG to integrate with a distribution system through a market mechanism. A market-based interactions procedure between MGs and DS is developed for MGs as price-makers to find an optimal bidding/offering strategy efficiently. To achieve a market equilibrium among all entities, we initially cast this problem as a bi-level programming problem, in which the upper level is an MG optimal scheduling problem and the lower level presents a DEM clearing mechanism. The proposed bi-level model is converted to a single mix-integer model which is easier to solve. Uncertainties associated with MG's rivals' offers and demands' bids are considered in this problem. The solution results from a modified IEEE 33-Bus distribution system are presented and discussed. Finally, some conclusions are drawn and examined.