Achieving a specific solidification morphology during the directed energy deposition process can be key to desired properties of the deposited part. The careful control of thermal gradient and cooling rates can be utilized to achieve these grain morphologies. The heating of the build platform and post-deposition heat treatment are two ways in which we have shown to control these solidification parameters.Three dimensional heat transfer and material flow model along with columnar-to-equiaxed transition diagram have been used to model the process and the microstructure evolution during laser directed energy deposition of Al-0.5Sc-0.5Si powder. The computed results have been validated using the experimentally measured thermal cycles and melt pool shape and sizes. The computed thermal gradient and cooling rates are used to predict the solidification morphology compared with the experimentally observed microstructure. For a specific set of deposition parameters, the fraction of equiaxed grains is least in the bottom layers, highest in the middle layers, and reduced in the top layer. The increasing build platform temperature increases the fraction of equiaxed grains in any specific deposited layer. The microhardness of various layers in different specimens as a function of build platform temperature is also shown to be a function of the solidification microstructure. The developed and validated numerical model is able to predict the microstructure evolution and can be a tool for further microstructure engineering during the L-DED process.