This study is the first to adopt the Generalized Double-Moment scaling Normalization (GDMN) method to represent the rain Drop Size Distribution (DSD) in a bulk-type cloud microphysics scheme, specifically the Weather Research and Forecasting (WRF) Double-moment 6-class (WDM6) scheme. The modified scheme, referred to as WDM6-GDMN, is evaluated through simulations of an isolated summer convection case over the Korean Peninsula, using the universal double-moment normalized DSD function, <em>h</em>(<em>x</em>), derived from rain DSDs observed in the Boseong region during the summers of 2018 and 2019. WDM6-GDMN provides a more realistic spatial distribution of surface precipitation by better simulating convection-cell movement. Although none of the cloud microphysics parameterizations, including the bin-type scheme, reproduce the observed convection that developed in the southeast of the analysis domain, only WDM6-GDMN successfully captures this feature. Microphysical analysis demonstrates that, in WDM6-GDMN, enhanced cloud production due to stronger upward motion leads to the formation of more raindrops and, consequently, greater surface precipitation over southeastern region. Furthermore, the contoured frequency by altitude diagrams for the WDM6-GDMN reveals slower particle growth and weaker reflectivity in the lower atmosphere compared with the original scheme, in better agreement with observations.