Source-linked AI summary
Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment
Ashish Cherukuri, Jorge Cortes
TL;DR
The paper addresses distributed economic dispatch for generators that must meet demand, respect capacity limits, and minimize total cost. It combines dynamic average consensus with distributed nonsmooth-gradient dynamics to estimate load mismatch and adjust generation. The resulting strategy converges to dispatch solutions from arbitrary initial allocations and is analyzed for time-varying loads and intermittent generation.
Problem
The paper asks how distributed generators can meet demand while minimizing total generation cost and respecting individual capacity constraints in decentralized power networks.
Method
The strategy couples dynamic average consensus for estimating load mismatch with Laplacian-nonsmooth-gradient dynamics for distributed generation allocation.
Results
The distributed dynamics converges to the economic-dispatch solution set from any initial power allocation, with analysis covering time-varying loads and generator addition or deletion.
Takeaways & Limitations
The coordination strategy provides initialization-free distributed dispatch with robustness to changing loads and intermittent power generation.
Takeaways & Limitations
The paper leaves convergence analysis for combinations with traditional primary and secondary generator controllers for future work.
Abstract
from arXiv · showhide
This paper considers the economic dispatch problem for a network of power generating units communicating over a strongly connected, weight-balanced digraph. The collective aim is to meet a power demand while respecting individual generator constraints and minimizing the total generation cost. We design a distributed coordination algorithm consisting of two interconnected dynamical systems. One block uses dynamic average consensus to estimate the evolving mismatch in load satisfaction given the generation levels of the units. The other block adjusts the generation levels based on the optimization objective and the estimate of the load mismatch. Our convergence analysis shows that the resulting strategy provably converges to the solution of the dispatch problem starting from any initial power allocation, and therefore does not require any specific procedure for initialization. We also characterize the algorithm robustness properties against the addition and deletion of units (capturing scenarios with intermittent power generation) and its ability to track time-varying loads. Our technical approach employs a novel refinement of the LaSalle Invariance Principle for differential inclusions, that we also establish and is of independent interest. Several simulations illustrate our results.
1 Introduction
The paper motivates distributed economic-dispatch algorithms for decentralized, dynamic grids and introduces a two-part strategy designed to converge from arbitrary initial allocations while handling changing loads and generator availability.
- Economic dispatch coordinates generators to meet demand, minimize summed generation cost, and respect individual capacity constraints.
- The approach addresses limitations of prior methods involving nonconvex constraints, all-to-all communication, omitted individual constraints, or suboptimal solutions.
- The proposed distributed dynamics combines dynamic average consensus for load-mismatch estimation with Laplacian-nonsmooth-gradient generation allocation.
- The algorithm is designed to solve economic dispatch from any initial power allocation, including under time-varying loads and unit addition or deletion.
2 Preliminaries
The preliminaries establish graph-theoretic, dynamic-average-consensus, and nonsmooth-analysis tools used to analyze distributed coordination over strongly connected, weight-balanced digraphs.
- Graph theory: A strongly connected digraph has a path between every vertex pair, while weight balance means each vertex’s weighted in-degree equals its weighted out-degree.
- Dynamic average consensus: Dynamic average consensus makes each agent’s state asymptotically track the average of distributed input signals over the communication graph.
- Dynamic average consensus: With bounded input derivatives and an initialization condition on the auxiliary state, dynamic-consensus error is ultimately bounded and vanishes for inputs converging to constants.
- Nonsmooth analysis and differential inclusions: Differential inclusions describe absolutely continuous trajectories whose derivatives belong almost everywhere to a set-valued map.
- Nonsmooth analysis and differential inclusions: The paper develops a refinement of LaSalle’s Invariance Principle for differential inclusions to analyze the coordination algorithms.
3 Problem statement
The problem statement formulates economic dispatch as convex cost minimization subject to total-load equality and generator-specific box constraints, with a modified nonsmooth formulation supporting the proposed analysis.
- Problem formulation: Economic dispatch minimizes total generation cost while requiring total generation to equal the positive load P_l.
- Problem formulation: Each generator is restricted by lower and upper production limits, yielding individual box constraints on its allocation.
- Problem formulation: The feasible set intersects the box-constrained allocation set with the hyperplane satisfying the load condition.
- Scope: The design assumes, for simplicity, that transmission losses, line capacities, valve-point effects, and other additional practical constraints are not modeled.
- Reformulation: A modified economic-dispatch formulation uses a convex, locally Lipschitz, regular total cost while preserving the original problem’s solutions under a condition on ϵ.
4 Robust centralized algorithmic solution
The centralized load-mismatch plus Laplacian-nonsmooth-gradient dynamics combines cost optimization with feedback that drives total generation toward the load from any initial allocation. Its trajectories converge to the economic-dispatch solution set, while the feedback mismatch dynamics converge exponentially.
- Algorithm design: The dynamics combines cost optimization at fixed total generation with feedback correction driven by the load-satisfaction mismatch.The feedback term drives the generation-load mismatch toward zero independently of the initial power allocation.
- Algorithm design: The feedback term requires aggregated network information, so the centralized dynamics is not directly implementable in distributed form.This limitation motivates the distributed strategy developed in the following section.
- Convergence analysis: Trajectories starting from any point in R^n converge to the set of economic-dispatch solutions.The proof uses a refined LaSalle Invariance Principle for differential inclusions and establishes boundedness before applying the convergence argument.
- Robustness analysis: The load-satisfaction mismatch converges exponentially, supporting robustness to time-varying loads and generator outages or returns when changes are sufficiently slow.The paper relates these robustness properties to the exponential convergence rate of the mismatch dynamics.
5 Robust distributed algorithmic solution
The distributed strategy replaces aggregated mismatch feedback with dynamic average consensus, allowing neighboring generators to coordinate economic dispatch from arbitrary initial allocations. The analysis establishes convergence and robustness to bounded load variation and finite unit addition or deletion events.
- Distributed strategy: Dynamic average consensus lets each generator estimate the average load-satisfaction mismatch when the total load is known to only one unit.The estimate replaces the centralized feedback term in the generation dynamics.
- Distributed strategy: The dac+L∂ dynamics is distributed because each agent needs only information from its communication neighbors.The generation, estimator, and auxiliary-state dynamics are coupled through the weight-balanced communication graph.
- Convergence analysis: For admissible positive design parameters, trajectories from any initial state converge to the economic-dispatch solution set.The convergence proof uses omega-limit-set characterization, coordinate changes, and a refined LaSalle principle for differential inclusions.
- Convergence analysis: No specific initialization preprocessing is required: each generator may select its generation level independently while convergence remains guaranteed.The parameter condition for convergence can also be replaced by a distributedly checkable sufficient condition.
- Robustness analysis: The mismatch dynamics is exponentially stable and input-to-state stable, making the strategy robust to arbitrary bounded perturbations.The paper uses this property to analyze dynamic loads and intermittent power generation.
- Robustness analysis: For twice continuously differentiable loads with bounded first and second derivatives, the generation-load mismatch remains bounded and has an ultimate bound.The result applies the exponential mismatch estimate to time-varying load inputs.
- Robustness analysis: With finite unit addition and deletion events, the mismatch effect vanishes exponentially and the generators converge to the economic-dispatch solution set for the final generator group.Addition and deletion are modeled by time-varying strongly connected, weight-balanced communication digraphs.
6 Simulations in a IEEE 118 bus system
Simulations on the IEEE 118-bus system illustrate convergence under load changes, tracking of time-varying demand, and robustness to generator additions and deletions.
- System and setup: The IEEE 118-bus example contains 54 generators with quadratic individual generation costs.The communication digraph is G, and parameters are selected as ν1 = 1, ν2 = 1.3, α = 10, β = 40, and ϵ = 0.0086.
- Stepwise load changes: With load changing from 4600 to 4200, generation converges to cost-minimizing allocations satisfying each successive demand.The first load is applied for 150 seconds and the second for the next 150 seconds.
- Time-varying load: For Pl(t) = 4300 + 100 sin(0.05t), total generation tracks the time-varying load while the mismatch remains within an ultimate bound.This behavior is established as a robustness property of the distributed dynamics.
- Intermittent generation: The strategy remains robust to intermittent generation when units leave and rejoin the network, with remaining agents applying a trajectory invariance routine.The simulated network changes are represented by the digraphs listed in Table 1.
7 Conclusions
The paper concludes that its distributed strategy solves economic dispatch from arbitrary initial allocations and remains robust to changing loads and intermittent generation. It also identifies extensions involving additional generator and network constraints as future work.
- Conclusions: The proposed strategy solves economic dispatch from any initial power allocation.Its design combines dynamic average consensus for mismatch estimation with distributed optimization for generation allocation.
- Conclusions: The mismatch dynamics are input-to-state stable, supporting robustness to initialization errors, time-varying loads, and intermittent power generation.The conclusion connects these properties to the algorithm's distributed coordination design.
- Conclusions: Simulations illustrate changing-load and generator-commitment scenarios, including unit additions and deletions.The IEEE 118-bus experiments include time-varying demand and changing communication topologies.
- Future work: Future work includes preserving generator box constraints and extending the method to transmission losses, line capacities, ramp limits, prohibited zones, and valve-point effects.The paper also proposes studying combinations with primary and secondary generator controllers.
A Refined LaSalle Invariance Principle for differential inclusions
This appendix establishes a refined LaSalle Invariance Principle for differential inclusions. Under regularity, invariance, and strict-descent conditions, bounded trajectories have omega-limit sets contained in the zero-Lie-derivative set.
- Motivation and proof: The result provides convergence-analysis tools for the coordination algorithms while extending earlier differential-equation results to differential inclusions.Figure A.1 illustrates the neighborhood construction used in one proof case.
- Proposition: The refinement applies to bounded solutions of differential inclusions with upper semicontinuous, nonempty, convex, compact-valued dynamics.The omega-limit set must lie in a closed embedded submanifold where a locally Lipschitz regular function is defined.
- Proposition: The invariant set E consists of points in S where the set-valued Lie derivative contains zero.The proposition assumes E belongs to a level set of the Lyapunov-like function W.
- Proposition: Away from E, every compact subset of S admits a compact neighborhood on which the maximum set-valued Lie derivative is uniformly negative.This supplies the strict decrease condition required by the refinement.
- Proof strategy: The appendix first proves that the omega-limit set intersects E, then uses weak positive invariance and descent of W to show the entire omega-limit set lies in E.The contradiction argument relies on W being bounded below on the compact omega-limit set.
B Continuity properties of set-valued Lie derivatives
This appendix establishes a continuity property for set-valued Lie derivatives. The property converts pointwise nonpositivity and an equilibrium characterization into uniform strict negativity away from the invariant set.
- Setup: For a locally Lipschitz regular W and continuous g, the set-valued map is formed from g(x, ζ) over ζ ∈ ∂W(x).The construction underlies the continuity argument for set-valued Lie derivatives.
- Assumptions: On the submanifold S, every generalized gradient satisfies ζ⊤g(x, ζ) ≤ 0, and equality for some ζ implies x ∈ E.These assumptions identify where the Lyapunov-like derivative can fail to be strictly negative.
- Result: Every compact M ⊂ S disjoint from E has a compact neighborhood where the maximum set-valued Lie derivative is bounded above by some δ < 0.The proof obtains this by contradiction using upper semicontinuity and compactness of the generalized-gradient values.