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This paper develops energy management (EM) control for series hybrid electric vehicles (HEVs) that include an engine start-stop system (SSS). The objective of the control is to optimally split the energy between the sources of the powertrain and achieve fuel consumption minimization. In contrast to existing works, a fuel penalty is used to characterize more realistically SSS engine restarts, to enable more realistic design and testing of control algorithms. The paper first derives two important analytic results: a) analytic EM optimal solutions of fundamental and commonly used series HEV frameworks, and b) proof of optimality of charge sustaining operation in series HEVs. It then proposes a novel heuristic control strategy, the hysteresis power threshold strategy (HPTS), by amalgamating simple and effective control rules extracted from the suite of derived analytic EM optimal solutions. The decision parameters of the control strategy are small in number and freely tunable. The overall control performance can be fully optimized for different HEV parameters and driving cycles by a systematic tuning process, while also targeting charge sustaining operation. The performance of HPTS is evaluated and benchmarked against existing methodologies, including dynamic programming (DP) and a recently proposed state-of-the-art heuristic strategy. The results show the effectiveness and robustness of the HPTS and also indicate its potential to be used as the benchmark strategy for high fidelity HEV models, where DP is no longer applicable due to computational complexity.