Electricity storage is likely to be an important factor in balancing fluctuations in renewable generators' output, but concentrated ownership could lead to market power. We model this for short-term (daily) storage in the British electricity wholesale market, with generating companies acting as either price-takers or Cournot competitors. We discuss how competitive storage charging and discharging behaviour depends on the balance between the market price and shadow price of stored electricity. Electricity storage raises welfare, consumer surplus and renewable generators’ revenues, while reducing revenues for conventional generators. Market power in storage slightly reduces the welfare gains; Cournot behaviour by generators reduces welfare but has relatively little impact on the incremental effect of storage. Market power in electricity storage is undesirable, but market power in generation is much worse. The interactions between market power in generation and in storage are complex, suggesting that predictions from one market may not apply elsewhere and context-specific modelling will be valuable.