$w_a$ is proposed to describe the cosmological evolution of the EoS of dark energy. We will also consider the holographic dark energy (HDE) model in this work, which is a dynamical dark energy model based on the consideration of quantum effective field theory and holographic principle of quantum gravity [@Li:2004rb; @Cohen:1998zx]. In the HDE model, the type (quintessence or quintom) and the cosmological evolution of dark energy are solely determined by a dimensionless constant $c$ (note that this is not the speed of light). For more detailed studies on the HDE model, see e.g. Refs. [@Li:2009jx; @Li:2009bn; @Li:2004rb; @Zhang:2005hs; @Zhang:2007sh; @Gao:2007ep; @Cui:2014sma; @Zhang:2014sqa; @Zhang:2019ple; @Zhang:2006av; @Zhang:2006qu; @Zhang:2007es; @Zhang:2007an; @Zhang:2008mb; @Zhang:2009un; @Cui:2009ns; @Feng:2009hr; @Wang:2012uf; @Zhang:2014ija; @Zhang:2015rha; @Zhao:2017urm; @Feng:2018yew; @Zhang:2015uhk; @Wang:2016tsz; @Geng:2014hoa; @Geng:2015ara]. In this work, we use these four typical, simple dark energy models, namely, the $\Lambda$CDM, $w$CDM, CPL, and HDE models, as examples to make an analysis for the real-time cosmology.

The structure of this paper is arranged as follows. In Sect. \[Method and data\], we present the analysis method and the observational data used in this work. In Sect. \[sec:Results and Discussions\], we report the constraint results of cosmological parameters and make some relevant discussions. In Sect. \[sec:Conclusion\], the 