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The Electronic Communication Committee (ECC) in Europe proposed a location-based transmission power allocation rule for secondary devices operating in the TV white space (TVWS). The further the secondary device is located from the TV cell border the higher transmission power level it can utilize. The Federal Communication Committee (FCC) in the US proposed a fixed transmission power allocation rule for all secondary transmitters. Both rules do not consider the secondary system's self-interference while setting the transmission power levels. In this paper, we propose a power allocation scheme for a cellular secondary system. Unlike the ECC and the FCC proposals we do the power allocation by considering the self-interference. We define the power allocation scheme as an optimization problem. The sum cell border data rate of the secondary network is selected to be the optimization objective. We observe that the optimal transmission power levels become approximately constant over the secondary deployment area. The FCC rule captures the general trend for cellular deployment in the TVWS, since it suggests the use of constant power. However, the transmission power should not be set equal to 4W but according to the allowable generated interference at the borders of the TV and secondary cells. [ABSTRACT FROM AUTHOR]