The performance of a cooperative non-orthogonal multiple access (NOMA) multi-relay system, where the communication between the base state (BS) and users is carried out through either the direct transmission or the assistance of energy harvesting (EH) relays, is evaluated in the presence of hardware impairment and imperfect channel state information (ICSI). A relay is selected among K EH relays by using partial relay selection (PRS) protocol to forward the received signal to a far user (FU) and a near user (NU). More specifically, exact analytical expressions for the outage probability are derived in closed-form. To obtain deeper insights, we present asymptotic expressions for the outage probability. The numerical results manifest that distortion parameter, κ, will jeopardize the performance of considered systems, and increasing the value of estimation parameter, δ, conduces to improve the outage performance.