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A critical enabling technology for electrified vehicles and renewable energy resources is battery energy storage. Advanced battery systems represent a promising technology for these applications, however their dynamics are governed by relatively complex electrochemical phenomena whose parameters degrade over time and vary across manufacturer. Moreover, limited sensing and actuation exists to monitor and control the internal state of these systems. As such, battery management systems require advanced identification, estimation, and control algorithms. In this paper we examine a new battery state-of-charge (SOC) estimation algorithm based upon the backstepping method for partial differential equations (PDEs). The estimator is synthesized from the so-called single particle model (SPM). Our development enables us to rigorously analyze observability and stability properties of the estimator design. In a companion paper we examine state-of-health (SOH) estimation, framed as a parameter identification problem for parabolic PDEs and nonlinearly parameterized output functions.