Despite the importance of conceptual formulations for exploratory projects, the variety and evolution of concepts generated in an exploratory project to describe the innovative purpose across the whole innovation process is still misunderstood. In this paper, we propose to address this issue of multiplicity and coherence of concept formulation in exploratory projects at three levels - cognitive, managerial and strategic - in order to describe the dynamics of conceptual works. We rely on a on a longitudinal study (7 years) of the innovation capability management in a large established firm, SNCF, the French railroad company. Our main results are i) to give a typology of seven concept formulations; ii) to show that these are interdependent and part of a structured process of building a "desirable unknown" to impact three dimensions: cognitive generative power, collaborative attractiveness for new organizations experimentation, and strategic positioning renewal of the firm in quickly evolving environments; and iii) to explicit specific conceptual formulation patterns that can improve their performance. We thereby contribute to guide practitioners on building conceptual formulations to reach their innovation goals.