To achieve the ambitious goal of carbon neutrality, the necessity of constructing a low-carbon power system is raising nowadays. The unit-commitment (UC) problem is fundamental for system planning and the classic operational models lead to serious computational problems in long-term system planning studies. Therefore, this paper proposes a heuristic unit-commitment approach for the low-carbon power system and applies it to the process of production simulation. The proposed approach effectively reduces the computing burden by avoiding solving the mix-integer optimization problem. Instead, specific rules at each time step are formulated based on the requirements of system, network, and unit levels, working as constraints in typical UC models. Also, the economical objective determines the sequence of units' turning on/off. From the perspective of a heuristic algorithm, the combination of the local optimal results at each time is considered the global optimal solution. Case studies show the effectiveness of the UC approach in improving calculational efficiency. Meanwhile, the simulation results of the modified IEEE 24-bus system demonstrate the feasibility of the heuristic UC approach in the production simulation for a low-carbon power system.