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Electric vehicles (EVs) will play an important role in the future smart grid because of their capabilities of storing electrical energy in their batteries during off-peak hours and supplying the stored energy to the power grid during peak hours. In this paper, we consider a power system with an aggregator and multiple customers with EVs and propose novel electricity load scheduling algorithms which, unlike previous works, jointly consider the load scheduling for appliances and the energy trading using EVs. Specifically, we allow customers to determine how much energy to purchase from or to sell to the aggregator while taking into consideration the load demands of their residential appliances and the associated electricity bill. We propose two different approaches: a collaborative and a non-collaborative approach. In the collaborative approach, we develop an optimal distributed load scheduling algorithm that maximizes the social welfare of the power system. In the non-collaborative approach, we model the energy scheduling problem as a non-cooperative game among self-interested customers, where each customer determines its own load scheduling and energy trading to maximize its own profit. In order to resolve the unfairness between heavy and light customers in the non-collaborative approach, we propose a tiered billing scheme that can control the electricity rates for customers according to their different energy consumption levels. In both approaches, we also consider the uncertainty in the load demands, with which customers' actual energy consumption may vary from the scheduled energy consumption. To study the impact of the uncertainty, we use the worst-case-uncertainty approach and develop distributed load scheduling algorithms that provide the guaranteed minimum performances in uncertain environments. Subsequently, we show when energy trading leads to an increase in the social welfare and we determine what are the customers' incentives to participate in the ener- y trading in various usage scenarios including practical environments with uncertain load demands.