Household energy consumption is growing continuously, with uncertainties arising from photovoltaic (PV) generation and household loads. To address these issues, an optimal scheduling model for household energy management was developed. The energy storage capabilities of electric vehicle (EV) were leveraged, while battery degradation costs caused by frequent charging/discharging were accounted for. The stochastic nature of PV generation and the flexibility in power consumption of smart appliances were also synergized. A real-time energy scheduling algorithm under time-varying tariffs was proposed based on the improved Lyapunov optimization theory. EV charging/discharging was controlled, the operation of different household appliances was scheduled, and the bidirectional power transactions between households and the grid were optimized. The time-varying nature of PV generation and grid tariffs was effectively coped with. Energy utility was maximized while the waiting time delay for each household power demand was ensured not to exceed its tolerable period. Theoretical analysis showed that the proposed algorithm could drive the optimization objective to converge to the optimal value without relying on prior statistical information of the system. The effectiveness and economic efficiency of the proposed optimization strategy were verified through comparisons with existing algorithms and performance analyses under various conditions.