[Paper Review] Balancing time-varying demand-supply in distribution networks: an internal model approach

Claudio De Persis|arXiv (Cornell University)|Feb 4, 2013

Smart Grid Energy Management12 references14 citations

TL;DR

This paper proposes a distributed internal model control approach for balancing time-varying demand and supply in distribution networks using incremental passivity and internal model principles. By embedding a dynamic internal model of the exogenous disturbances at each edge, the controllers ensure asymptotic convergence of node imbalance (z) to zero, even under arbitrary time-varying inputs, with stability guaranteed via incremental passivity and Barbalat's lemma.

ABSTRACT

The problem of load balancing in a distribution network under unknown time- varying demand and supply is studied. A set of distributed controllers which regulate the amount of flow through the edges is designed to guarantee convergence of the solution to the steady state solution. The results are then extended to a class of nonlinear systems and compared with existing results. Incremental passivity and internal model are the main analytical tools.

Motivation & Objective

• To address load balancing in distribution networks under unknown, time-varying demand and supply signals.
• To design distributed edge-level controllers that achieve steady-state balancing without requiring global knowledge of the network.
• To extend the internal model principle and incremental passivity to cooperative control in flow networks with exogenous inputs.
• To compare the proposed method with existing robust control and saddle-point approaches in the literature.
• To establish conditions under which the closed-loop system guarantees asymptotic convergence of node imbalance to zero.

Proposed method

• The system is modeled as a linear flow network with state x, edge flows λ, and exogenous disturbances d generated by an exosystem w with dynamics ẇ = S^d w.
• An internal model is embedded in each edge controller, represented by a dynamic system η̇_k = Φ_k η_k + Λ_k z_k, λ_k = Ψ_k η_k + Γ_k z_k, to replicate the dynamics of the disturbance.
• Incremental passivity is used to analyze stability, with the closed-loop system shown to be incrementally passive from external inputs to outputs (z, u).
• A feedback law λ_ext = -Kz is applied, where K is a positive definite diagonal matrix, to ensure asymptotic convergence of z to zero.
• The stability proof relies on Barbalat’s lemma, requiring boundedness of Ẇ and ẇ, which holds due to the skew-symmetric nature of S^d.
• For nonlinear systems, the method is extended using a convex potential function F(x) with ∇F(x) = f(x), and the internal model is adapted accordingly.

Experimental results

Research questions

• RQ1 How can distributed controllers be designed to balance time-varying demand and supply in a networked system without centralized coordination?
• RQ2 What role does the internal model principle play in achieving robust regulation under time-varying exogenous inputs in flow networks?
• RQ3 How does incremental passivity contribute to stability analysis in the presence of time-varying disturbances?
• RQ4 In what ways does the proposed method differ from existing approaches based on saddle-point dynamics or robust control?
• RQ5 What conditions ensure asymptotic convergence of the node imbalance z to zero under general time-varying inputs?

Key findings

• The proposed distributed controllers guarantee lim_{t→∞} z(t) = 0 for all initial conditions, ensuring complete load balancing despite time-varying disturbances.
• The internal model principle is successfully applied in a state-space framework to handle general time-varying exogenous inputs, extending beyond constant disturbances.
• Incremental passivity ensures stability of the feedback interconnection, and Barbalat’s lemma confirms asymptotic convergence under bounded ẇ.
• For nonlinear systems, the method works when f(x) = ∇F(x) for a twice continuously differentiable convex function F, ensuring well-posedness and convergence.
• The controller dynamics include a dynamic internal model (Ṡη = S^d η - M₂ᵀz) that captures the disturbance dynamics, distinguishing it from simpler static feedback laws.
• The approach is shown to be more general than [2] and [9], as it handles time-varying inputs and avoids restrictive assumptions like strict passivity or saturation constraints.