Consensus has been widely used in distributed control

where distributed individuals need to share their states with their neighbors through communication links to achieve a common goal. However

the objectives of existing consensus-based control strategies for energy systems seldom address battery degradation cost

which is an important performance indicator to assess the performance and sustainability of battery energy storage (BES) systems. In this paper

we propose a consensus-based optimal control strategy for multi-microgrid systems

aiming at multiple control objectives including minimizing battery degradation cost. Distributed consensus is used to synchronize the ratio of BES output power to BES state-of-charge (SoC) among all microgrids while each microgrid is trying to reach its individual optimality. In order to reduce the pressure of commu-nication links and prevent excessive exposure of local information

this ratio is the only state variable shared between microgrids. Since our complex nonlinear problem might be difficult to solve by traditional methods

we design a compressive sensing-based gradient descent (CSGD) method to solve the control problem. Numerical simulation results show that our control strategy results in a 74.24 % reduction in battery degradation cost on average compared to the control method without considering battery degradation. In addition

the compressive sensing method causes an 89.12 % reduction in computation time cost compared to the traditional Monte Carlo (MC) method in solving the control problem.