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In this paper, we consider timeliness and energy optimization in battery-powered dynamic embedded real-time systems, which must remain functional during an operation/mission with a bounded energy budget. We consider application activities that are subject to time/utility function time constraints, statistical assurance requirements on timeliness behavior, and an energy budget which cannot be exceeded at runtime. To account for the inevitable variability in activity arrivals in dynamic systems, we describe arrival behaviors using the unimodal arbitrary arrival model (UAM) . For such a model, we present a dynamic voltage scaling (DVS)-based CPU scheduling algorithm called the energy-bounded utility accrual algorithm (EBUA). Since the scheduling problem is intractable, EBUA allocates CPU cycles, scales clock frequency, and heuristically computes schedules using statistical estimates of cycle demands in polynomial time. We analytically establish EBUA's properties, including satisfaction of energy bounds, statistical assurances on individual activity timeliness behavior, optimal timeliness during underloads, and bounded time for mutually exclusively accessing shared non-CPU resources. Our simulation experiments validate our analytical results and illustrate the algorithm's effectiveness and superiority over past algorithms.