Battery management systems (BMSs) are required to estimate many non-measurable values that describe the actual operating condition of batteries; such as the state of charge (SOC) or the state of health (SOH). In order to improve this evaluation, many physical states and parameters can be estimated using physics-based models (PBMs). These estimates could be used to improve the control and prognosis of batteries. In this series of papers we propose a new method to estimate the internal physical states, the SOC, the SOH, and the electrode-specific state of health (eSOH) parameters of a lithium-ion battery, using interconnected sigma-point Kalman filters (SPKFs) and a single-particle model with electrolyte dynamics (SPMe). This first paper focuses on state estimation for non-aged cells. To begin, we describe and validate our electrochemical model against a high-fidelity P2D model. After, the interconnected SPKF algorithm is described and the observability of our system is analyzed, showing that the interconnected estimator approach improves an observability measure of the system. Finally, the results of the estimator are discussed, comparing the estimated variables with the truth values under initialization, measurement and modeling uncertainties. The results show that the algorithm can estimate the internal battery states with high accuracy.