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We perform a rare-event study on a simulated power system in which grid-scale batteries provide both regulation and emergency frequency control ancillary services. Using a model of random power disturbances at each bus, we employ the skipping sampler, a Markov Chain Monte Carlo algorithm for rare-event sampling, to build conditional distributions of the power disturbances leading to two kinds of instability: frequency excursions outside the normal operating band, and load shedding. Potential saturation in the benefits, and competition between the two services, are explored as the battery maximum power output increases. This article is part of the theme issue ‘The mathematics of energy systems’.