This paper introduces a new energy management (EM) strategy for series hybrid electric vehicles (HEVs). Series HEVs operate in charge-depletion mode and then switch to the charge-sustaining mode in which the battery state of charge (SOC) is maintained within a certain range. The proposed EM strategy in this paper is a form of adaptive equivalent consumption minimization strategy (ECMS) that is designed for the charge-sustaining mode. The EM strategy defines soft bounds on the battery SOC and is penalized for exceeding these bounds. But, to catch energy-saving opportunities (CESOs), the EM strategy allows SOC to exceed the soft bounds. Thus, the introduced EM strategy is named ECMS-CESO. In addition, a range for the ECMS optimal equivalent factor is proposed for series HEVs. The proposed range is used in deriving the formula for calculating the adaptive equivalent factor. The main advantage of the proposed EM strategy is that ECMS-CESO can achieve close to optimal fuel economy without the need for predicting future driver demand. Since there is no need for prediction, the intensive calculations for finding the optimal control over the prediction horizon can be eliminated. Therefore, implementation of ECMS-CESO is easily feasible for real-time applications. Experimental powertrain data is collected to develop a powertrain model for a series HEV in this study. Simulation results on several drivecycles show that, on average, the fuel economy achieved by ECMS-CESO is within 6% of the maximum fuel economy. In addition, comparing ECMS-CESO with two existing adaptive ECMSs shows up to 5% improvement in fuel economy, on average.