Prove stability of the coupled second distributed ED scheme

Prove stability of the marginal-cost consensus and estimated average power-mismatch dynamics for the coupled discrete-time PI+Reset economic-dispatch scheme that incorporates the estimated average power mismatch into the marginal-cost closed-loop error, despite the nonlinear relationship between power mismatch variation and marginal cost.

Background

The second distributed economic-dispatch scheme introduces the estimated average power mismatch directly into the marginal-cost consensus error. Theorem 4 establishes boundedness of the marginal cost and estimated average power mismatch, but does not establish their stability because the power-mismatch increment depends nonlinearly on the marginal cost.

The unresolved stability question is important because boundedness alone does not provide the stronger guarantees needed for rigorous convergence and robust operation of the coupled isolated-BESS economic-dispatch controller. The paper notes that a decaying gain can resolve the issue, but that modification may compromise dynamic performance.

References

At present, it can only be proven that MC and the estimated average power mismatch are bounded. Unfortunately, the stability proofs are still pending, which is attributed to the nonlinear relationship between $\Delta P_ik$ and $\lambda_i$.

— Design of Economic Dispatch Schemes of An Isolated BESS Network Based on Distributed Discrete-time PI+Rest Consensus  (2609.19804 - Zhang et al., 17 Sep 2026) in Section 3, subsection “Further Discussion,” remark following Theorem 4