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With the trend of multiple energies or flexible demand in power systems, binary variables appear in system-wide constraints, which are the foundation of marginal pricing currently in markets. An appropriate pricing method incentivizes compliance of market participants; otherwise, compliance can be incentivized by paying discriminatory uplift payments which jeopardize transparency of markets. This paper proposes two theorems to examine whether the binary variables brought by multiple energies and flexible demand will impact compliance under marginal pricing. The first theorem shows sufficient conditions with which marginal pricing with fixed binary variables incentivizes compliance, while the second theorem shows sufficient conditions to require uplift payments. To improve transparency by reducing uplift payments under cases which fall into the second theorem, this paper further proposes a pricing method by combining 1) designed constraints to price binary variables in system-wide constraints, and 2) convex hull pricing to price binary variables in private constraints. Effectiveness of the proposed theorems and pricing method is verified in an electricity-gas case (consisting of the IEEE 30-bus system and the NGS 10-node system) and the IEEE 118-bus test system.
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