This article is the first of a two-part series presenting the thermodynamic evaluation and techno-economics of developing negative-emission power plants. The aim of this research is to evaluate the potential of biochar co-production in negative-emission power plants based on biomass-fed integrated gasification solid oxide fuel cell systems with carbon capture and storage (BIGFC/CCS) units. The influence of two gasification agents, namely, air and steam-oxygen, on the proposed system is investigated. In Part I, we present the thermodynamic models. A sensitivity analysis is carried out to investigate the system response to stepwise increase in biochar co-production (up to 10% by weight). Providing a secondary oxy-combustor in the steam-oxygen gasification case has been shown to be a solution to meet the heat requirements of the allothermal gasification process. A comprehensive exergy analysis indicated significant efficiency improvement for the steam-oxygen gasification case. The results show that the biomass steam-oxygen gasification yields the higher electrical exergy efficiency (48.3%) and combined heat and power (CHP) exergy efficiency (54.6%) for the similar rates of biochar co-production. The specific power output per unit of CO2 stored is 2.65 MW/(kg/s) and 3.58 MW/(kg/s) for the air and steam-oxygen gasification cases, respectively, when the biochar is co-produced at 10% by weight for the given biomass flow of 20 kg/s. Moreover, the total CO2 stored due to the proposed system is calculated as 133.9 t/h, and it is estimated to remove 1.17 Mt of CO2 from the atmosphere annually (when the biochar-based carbon storage is also considered). The models are used for the techno-economic analysis presented in Part II of the series.