Self-heating (SHE) TCAD numerical simulations have been performed, for the first time, on 30nm FDSOI MOS transistors at extremely low temperatures. The self-heating temperature rise dT$_{max}$ and the thermal resistance R$_{th}$ are computed as functions of the ambient temperature T$_a$ and the dissipated electrical power (P$_d$), considering calibrated silicon and oxide thermal conductivities. The characteristics of the SHE temperature rise dT$_{max}$(P$_d$) display sub-linear behavior at sufficiently high levels of dissipated power, in line with standard FDSOI SHE experimental data. It has been observed that the SHE temperature rise dT$_{max}$ can significantly exceed the ambient temperature more easily at very low temperatures. Furthermore, a detailed thermal analysis of the primary heat flows in the FDSOI device has been conducted, leading to the development of an analytical SHE model calibrated against TCAD simulation data. This SHE analytical model accurately describes the dT$_{max}$(P$_d$) and R$_{th}$(T$_a$) characteristics of an FDSOI MOS device operating at extreme low ambient temperatures. These TCAD simulations and analytical models hold great promise for predicting the SHE and electro-thermal performance of FDSOI MOS transistors against ambient temperature and dissipated power.