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Due to the threats of climate change and global warming, carbon emissions are becoming a new concern during power system operation. This paper proposes a consumption-side carbon emission penalty scheme, where consumers are penalized based on their individual carbon emission responsibilities and penalty rates. Firstly, carbon emission responsibilities (CER) of consumers are determined after allocating the generators' carbon emission responsibilities to consumers through power flow tracing. Then, a low-carbon economic dispatch (LCED) model is developed with incorporation of the emission penalty scheme, in which the penalty-related cost is considered as a part of the objective function. Moreover, the consumers' differentiated penalty rates used in the LCED model are determined based on a two-level optimization model. The high-level problem determines the consumers' penalty rates which can minimize the negative impact of the penalty scheme on the social welfare while cutting the carbon emissions to a certain level. The low-level problems represent a set of LCED models to dispatch the generators' output at the given penalty rates endogenously generated within the high-level problem. Evidenced by both the theoretical analysis and simulation results, the proposed technique provides a more flexible and effective tool for the carbon emissions control compared with the traditional generation-side penalty scheme (such as carbon tax). The electricity prices derived from the proposed LCED are more stimulating for consumers to alter their electricity consumption behaviors to participate in carbon emission mitigation. Consequently, the carbon emissions can be well mitigated with less social welfare losses.