Stacked plate heat exchangers are widely used in thermal energy storage systems and a comprehensive and accurate analysis is necessary for their application and optimization. The fluid flow distribution between the plates is important to ensure even and full usage of the thermal energy storage potential. However, due to the complex topography of the plate surface, it would be computationally expensive to simulate the flow distribution in the multiple channels using a full three-dimensional model, so this work applies a reduced-dimensional model to significantly reduce the computational cost of the simulation and provides a comprehensive analysis of the effect of the internal structure on the internal flow distribution. The work extends a previously presented model to consider transient flow and a multichannel height distribution strategy to allow for simulating multiple channels between stacks of plates. Based on fully-developed flow assumptions, the three-dimensional model is reduced to a planar model, thus obtaining simulation results with satisfactory accuracy at a significantly lower computational cost. The model is verified by a three-dimensional simulation of a sliced two-channel model representing the considered system. The reduced-dimensional model gives similar results to the three-dimensional model for different geometrical and physical parameters. Lastly, the extended reduced-dimensional model is used to simulate the flow of a full two-channel model and the influence of the plate topography on the internal flow distribution is investigated through a comprehensive parametric analysis. The analysis shows that the complex topography of the plate surface eliminates the variation in inlet velocity and significantly changes the internal fluid flow, eventually resulting in a consistent velocity distribution.