Supplementary Document — Proofs of the Proposed Theorems

This paper proposes a distributed fixed-time multi-agent control strategy for the frequency restoration and voltage regulation, state of charge balancing and proportional reactive power sharing between photovoltaic-battery systems distributed in a microgrid with communication time delays. First, the feedback linearisation method is applied to find the direct relationships between explicit states and control inputs. Then, based on the model, the distributed fixed-time cooperative control system restores the frequency, regulates the average voltage to the nominal value and achieves accurate power sharing. For the state of charge balancing, a fixed-time observer is proposed to estimate the average state of charge of a battery using only information from neighbours. Based on the estimated value, a local fixed-time sliding mode control is applied to achieve the balanced state of charge. Due to robustness of the fixed-time control strategy, the balanced state of charge can be maintained despite intermittent photovoltaic generation and variable loads. The Artstein's transformation is applied to ensure the stability of the time delayed system. The dynamic performance is verified with an RTDS Technologies real-time digital simulator, using switching converter models, nonlinear lead-acid battery models, photovoltaic generation and communication delays in a European benchmark microgrid.