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We analyze the problem of partial shading in monolithically integrated thin-film photovoltaic (TFPV) modules, and explore how the shape and size of the shadows dictate their performance and reliability. We focus on the aspects of the shading problem unique to monolithic TFPV, arising from thin long rectangular series-connected cells, with partial shadows covering only a fraction of the cell area. Using calibrated 2-D circuit simulations, we show that due to the cell shape, the unshaded portion of partially shaded cell experiences higher heat dissipation due to redistribution of voltages and currents across the cells. We then use thermal imaging techniques to compare our results with module behavior under shade in realistic situations. We also analyze the effect of shadow size and orientation by considering several possible shading scenarios. We find that thin edge shadows can cause potentially catastrophic reverse bias damage, depending on their orientation. Finally, we show that external bypass diodes cannot protect the individual cells from shadow-induced reverse stress, but can limit the string output power loss for larger shadows.