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Since thermal energy generated by air conditionings (ACs) can be stored in the buildings providing the potential of shifting electricity consumption between different time periods, ACs are considered as an important demand response (DR) resource and attract extensive attentions. In order to achieve the compatibility of the inverter ACs with the current dispatch models, this paper attempts to model an inverter AC system as a thermal battery (TB). The comparisons between the TB and the lithium-ion battery are given. In order to protect the end-users' privacy and relieve the computational burden of the centralized control, a hierarchical control framework is designed and an aggregated TB model is proposed to handle the heterogeneity of the inverter ACs. A finite-horizon optimization model is used to compare the operating performances of the aggregated TBs and the lithium-ion batteries. Simulation results demonstrate that the aggregated TB model works well with the power dispatch model developed for the lithium-ion batteries. In other words, with the TB modeling of ACs, lithium-ion batteries can be replaced in current dispatch models providing service for the grid.