This paper investigates a wireless-powered cooperative network (WPCN) consisting of one source-destination pair and multiple decode-and-forward (DF) relays. We develop an energy threshold based multi-relay selection (ETMRS) scheme for the considered WPCN. The proposed ETMRS scheme can be implemented in a fully distributed manner as the relays only need local information to switch between energy harvesting and information forwarding modes. By modeling the charging/discharging behaviours of the finite-capacity battery at each relay as a finite-state Markov Chain (MC), we derive an analytical expression for the system outage probability of the proposed ETMRS scheme over mixed Nakagami-$m$ and Rayleigh fading channels. Based on the derived expression, the optimal energy thresholds for all the relays corresponding to the minimum system outage probability can be obtained via an exhaustive search. However, this approach becomes computationally prohibitive when the number of relays and the associated number of battery energy levels is large. To resolve this issue, we propose a heuristic approach to optimize the energy threshold for each relay. To gain some useful insights for practical relay design, we also derive the upper bound for system outage probability corresponding to the case that all relays are equipped with infinite-capacity batteries. Numerical results validate our theoretical analysis. It is shown that the proposed heuristic approach can achieve a near-optimal system performance and our ETMRS scheme outperforms the existing single-relay selection scheme and common energy threshold scheme.