The rapid development of the Power Internet of Things (PIoT) enables power smart sensing devices to offload their computation tasks to nearby edges. However, due to the increasing demand for computing and the unbalanced spatial distribution of devices, it is imperative to seek cooperative computing and task scheduling optimization schemes at the edge for PIoT. In this paper, we develop a two-tier cooperative edge network paradigm in PIoT. Then, we define a novel fairness indicator based on the Theil index to measure the allocation balance of the system. We also formulate a fairness and delay guaranteed (FDG) task offloading and load balancing optimization problem, which aims to minimize the allocation difference of the edge network while satisfying the delay constraints for multiple tasks in PIoT. Moreover, we develop a Lyapunov optimization and whale optimization algorithm (LWOA) to solve the problem. The simulation results demonstrate that for two types of typical tasks in PIoT, compared with the NonB scheme, the proposed FDG scheme decreases the time-averaged allocation difference of the system by 10% and 35%, the time-averaged allocation difference within subsystems by 5% and 6%, the time-averaged delay by approximately 5% and 7%, and the time-averaged queue backlog by approximately 30% and 40%. Research, both theoretical and experimental, has demonstrated that cooperation at the edge can significantly improve the performance of PIoT.