  In this paper, a transmission-distribution systems flexibility market is
introduced, in which system operators (SOs) jointly procure flexibility from
different systems to meet their needs (balancing and congestion management)
using a common market. This common market is, then, formulated as a cooperative
game aiming at identifying a stable and efficient split of costs of the jointly
procured flexibility among the participating SOs to incentivize their
cooperation. The non-emptiness of the core of this game is then mathematically
proven, implying the stability of the game and the naturally-arising incentive
for cooperation among the SOs. Several cost allocation mechanisms are then
introduced, while characterizing their mathematical properties. Numerical
results focusing on an interconnected system (composed of the IEEE 14-bus
transmission system and the Matpower 18-bus, 69-bus, and 141-bus distributions
systems) showcase the cooperation-induced reduction in system-wide flexibility
procurement costs, and identifies the varying costs borne by different SOs
under various cost allocations methods.
