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First we present an approach to formulate unbalanced three-phase power flow problems for general networks that explicitly separates device models and network models. A device model consists of (i) an internal model and (ii) a conversion rule. The conversion rule relates the internal variables (voltage, current, and power) of a device to its terminal variables through a conversion matrix Γ and these terminal variables are related by network equations. Second we apply this approach to balanced three-phase networks to formalize per-phase analysis and prove its validity for general networks using the spectral property of the conversion matrix Γ. 

Enabling participation of demand-side flexibility in electricity markets is key to improving power system resilience and increasing the penetration of renewable generation. In this work we are motivated by the curtailment of near-zero-marginal-cost renewable resources during periods of oversupply, a particularly important cause of inefficient generation dispatch. Focusing on shiftable load in a multi-interval economic dispatch setting, we show that incompatible incentives arise for loads in the standard market formulation. While the system's overall efficiency increases from dispatching flexible demand, the overall welfare of loads can decrease as a result of higher spot prices. We propose a market design to address this incentive issue. Specifically, by imposing a small number of additional constraints on the economic dispatch problem, we obtain a mechanism that guarantees individual rationality for all market participants while simultaneously obtaining a more efficient dispatch. Our formulation leads to a natural definition of a uniform, time-varying flexibility price that is paid to loads to incentivize flexible bidding. We provide theoretical guarantees and empirically validate our model with simulations on real-world generation data from California Independent System Operator (CAISO).