Master equations under appropriate assumptions are efficient tools for the study of open quantum systems. For many-body systems, subsystems of which locally couple to thermal baths and weakly interact with each other, the local approach provides a more convenient description than the global approach. However, these local master equations are believed to generate inconsistencies with the laws of thermodynamics when inter-subsystem interactions exist. Here we develop an alternative local master equation by applying the secular approximation to modify the inter-subsystem interactions. Our results show that violations of thermodynamic laws will be avoided after correcting inter-subsystem interactions. We finally use our results to investigate a two-qubit heat transfer network, and show the validity of our modified master equation.