[Paper Review] On the geometric convergence rate of distributed economic dispatch/demand response in power networks

Thinh T. Doan, Alex Olshevsky|arXiv (Cornell University)|Sep 21, 2016

Smart Grid Energy Management28 references18 citations

TL;DR

This paper proposes a fully distributed primal-dual algorithm for solving convex optimization problems in dynamic power networks where each node holds a local convex function and variables are coupled via a linear constraint. Under standard assumptions, the method achieves geometric convergence to the optimal solution, with convergence time scaling quartically in the number of nodes on time-varying undirected graphs and remaining constant for b-regular graphs with b ≥ 3.

ABSTRACT

Motivated by potential applications in power systems, we study a problem of optimizing a sum of $n$ convex functions on dynamic networks of $n$ nodes when each function is known to only a single node. The nodes' variables, while satisfy their local constraints, are coupled through a linear constraint. Our main contribution is to design a fully distributed primal-dual method for this problem. Under some fairly standard assumptions on objective functions, strong convexity and smoothness, we provide an explicit analysis for the convergence rate of our method on different networks. In particular, the nodes variables achieve a geometric convergence to the optimal with the associated convergence time scales quartically in the number of nodes on any sequence of time-varying undirected graphs satisfying a long-term connectivity condition. Moreover, this convergence time is constant independent on the number of nodes when the network is a b-regular simple graph with $b\geq 3$. Finally, to show the effectiveness of our method we also simulate a number of studies on economic dispatch problems and demand response problems in power systems.

Motivation & Objective

• To address distributed optimization in power networks where each node holds a local objective function and variables are coupled via a global linear constraint.
• To design a fully distributed algorithm that operates without a central coordinator, relying only on local computation and communication.
• To analyze the convergence rate of the proposed method under varying network topologies, particularly time-varying undirected graphs.
• To establish convergence time scaling with respect to network size and structure, especially for regular graphs.
• To validate the method’s effectiveness through simulations on economic dispatch and demand response problems in power systems.

Proposed method

• A fully distributed primal-dual algorithm is designed, where each node updates its local variables and dual multipliers using only local information and neighbor communication.
• The method leverages local gradient information of the convex objective functions and dual ascent steps to enforce the coupling linear constraint.
• Convergence analysis is conducted under assumptions of strong convexity and smoothness of the objective functions.
• The algorithm operates on time-varying undirected graphs satisfying a long-term connectivity condition, ensuring persistent communication over time.
• The convergence rate is analyzed using Lyapunov functions and network topology properties, particularly spectral gap and regularity.
• The method is implemented and tested in simulations on power system economic dispatch and demand response scenarios.

Experimental results

Research questions

• RQ1 How fast can a distributed primal-dual method converge to the optimal solution in a dynamic network of power system nodes?
• RQ2 What is the dependence of the convergence time on the number of nodes in time-varying undirected graphs?
• RQ3 Does the convergence time remain bounded as the network size increases for certain structured graphs, such as b-regular graphs with b ≥ 3?
• RQ4 How does the proposed method compare in performance to existing distributed optimization schemes for power system applications?
• RQ5 Can the method effectively solve real-world economic dispatch and demand response problems in power networks?

Key findings

• The proposed distributed primal-dual method achieves geometric convergence to the optimal solution under standard assumptions of strong convexity and smoothness.
• On any sequence of time-varying undirected graphs satisfying a long-term connectivity condition, the convergence time scales quartically with the number of nodes.
• For b-regular simple graphs with b ≥ 3, the convergence time is constant and independent of the number of nodes.
• The method effectively solves economic dispatch and demand response problems in power systems, as demonstrated by simulation studies.
• The convergence rate is analytically bounded using network topology properties, particularly the spectral gap of the graph Laplacian.