Let D be an integral domain and a star operation defined on D. We say that D is a -power conductor domain (-PCD) if for each pair and for each positive integer n we have We study -PCDs and characterize them as root closed domains satisfying for all nonzero a, b and all natural numbers . From this it follows easily that Prüfer domains are d-PCDs (where d denotes the trivial star operation), and v-domains (e.g. Krull domains) are v-PCDs. We also consider when a -PCD is completely integrally closed, and this leads to new characterizations of Krull domains. In particular, we show that a Noetherian domain is a Krull domain if and only if it is a w-PCD.