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Heterojunction with intrinsic thin layer or HIT solar cells are considered favorable for large-scale manufacturing of solar modules, as they combine the high efficiency of crystalline silicon (c-Si) solar cells, with the low cost of amorphous silicon technology. In this article, based on experimental data published by Sanyo, we simulate the performance of a series of HIT cells on N-type crystalline silicon substrates with hydrogenated amorphous silicon (a-Si:H) emitter layers, to gain insight into carrier transport and the general functioning of these devices. Both single and double HIT structures are modeled, beginning with the initial Sanyo cells having low open circuit voltages but high fill factors, right up to double HIT cells exhibiting record values for both parameters. The one-dimensional numerical modeling program "Amorphous Semiconductor Device Modeling Program" has been used for this purpose. We show that the simulations can correctly reproduce the electrical characteristics and temperature dependence for a set of devices with varying I-layer thickness. Under standard AM1.5 illumination, we show that the transport is dominated by the diffusion mechanism, similar to conventional P/N homojunction solar cells, and tunneling is not required to describe the performance of state-of-the art devices. Also modeling has been used to study the sensitivity of N-c-Si HIT solar cell performance to various parameters. We find that the solar cell output is particularly sensitive to the defect states on the surface of the c-Si wafer facing the emitter, to the indium tin oxide/P-a-Si:H front contact barrier height and to the band gap and activation energy of the P-a-Si:H emitter, while the I-a-Si:H layer is necessary to achieve both high Voc and fill factor, as it passivates the defects on the surface of the c-Si wafer. Finally, we describe in detail for most parameters how they affect current transport and cell properties.