The gravitational-wave (GW) events, produced by the coalescence of binary neutron-stars (BNS), can be treated as the standard sirens to probe the expansion history of the Universe, if their redshifts could be determined from the electromagnetic observations. For the high-redshift ($zgtrsim 0.1$) events, the short $gamma$-ray bursts (sGRBs) and the afterglows are always considered as the primary electromagnetic counterparts. In this paper, by investigating various models of sGRBs and afterglows, we discuss the rates and distributions of BNS mergers multi-messenger observations with GW detectors in second-generation (2G), 2.5G, 3G era with the detectable sGRBs and the afterglows. For instance, for Cosmic Explorer GW detector, the rate is about (300-3500) per year with GECAM-like detector for $gamma$-ray emissions and LSST/WFST detector for optical afterglows. In addition, we find these events have the redshifts $zlesssim 2$ and the inclination angles $iotalesssim 20^{circ}$. These results justify the rough estimation in previous works. Considering these events as standard sirens to constrain the equation-of-state parameters of dark energy $w_{0}$ and $w_{a}$, we obtain the potential constraints of $Delta w_{0}simeq 0.02-0.05$ and $Delta w_{a}simeq 0.1-0.4$.
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