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Additional time-coupled constraints restore recursive feasibility in economic NMPC for microgrids when generator runtime and startup limits are enforced.

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2026-06-29 15:24 UTC pith:6FP3MEUA

load-bearing objection The paper shows a practical fix for recursive feasibility in economic NMPC when generator runtime and startup limits are added, using time-coupled constraints on top of a quadratic power-flow MIQCP, but stability still rests on an unchecked periodic dissipativity assumption. the 2 major comments →

arxiv 2605.27553 v1 pith:6FP3MEUA submitted 2026-05-26 math.OC cs.SYeess.SY

Economic Nonlinear Model Predictive Control for Microgrids with Generator Up and Downtime Constraints

J\"urgen Gutekunst , Armin Nurkanovic , Ekaterina Kostina , Hans Georg Bock , Robert Scholz , Amer Mesanovic This is my paper
classification math.OC cs.SYeess.SY
keywords economic NMPCmicrogrid controlmixed-integer quadratic programmingrecursive feasibilitygenerator runtime constraintsperiodic dissipativity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces full AC power flow equations with convex quadratic approximations inside an economic NMPC formulation that also tracks discrete generator on/off decisions and their associated runtime and startup-count limits. Simple terminal constraints prove insufficient to keep the resulting MIQCP feasible from one step to the next. The authors therefore introduce extra time-coupled constraints that link decisions across the entire horizon and prove that these constraints restore recursive feasibility. Under the further assumption of periodic dissipativity they establish closed-loop stability. Numerical tests on a six-bus microgrid confirm that the scheme remains solvable in real time by standard MIQCP solvers.

Core claim

When generator runtime and startup-event constraints are present, standard terminal conditions no longer guarantee recursive feasibility of the economic NMPC scheme; a set of additional time-coupled constraints restores the feasibility property and, together with periodic dissipativity, yields asymptotic stability of the closed-loop system.

What carries the argument

The additional time-coupled constraints that enforce generator minimum up-time, minimum down-time, and maximum startup counts across the prediction horizon inside the MIQCP.

Load-bearing premise

The underlying system must satisfy periodic dissipativity for the stability proof to hold.

What would settle it

A concrete counter-example or simulation in which the proposed time-coupled constraints are active yet the closed-loop trajectory loses feasibility or diverges would falsify the recursive-feasibility and stability claims.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The resulting MIQCP remains solvable in seconds by off-the-shelf solvers such as CPLEX for realistic microgrid sizes.
  • Recursive feasibility holds for any initial state that satisfies the tightened constraint set at the first step.
  • Stability follows directly once periodic dissipativity is granted, without requiring a terminal cost or terminal region.
  • The same constraint-augmentation technique applies to any economic NMPC problem whose discrete dynamics include minimum dwell-time rules.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same time-coupling idea could be tested on larger networks or on problems that include battery degradation or demand-response decisions.
  • If periodic dissipativity fails in practice, one could still retain feasibility by periodically resetting the constraint set, at the price of possible performance loss.
  • The convex quadratic power-flow surrogate may introduce steady-state bias; comparing closed-loop cost against a full AC solver on the same trajectories would quantify that bias.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper develops an economic NMPC scheme for microgrid multistage OPF that incorporates generator runtime and startup constraints. It replaces the AC power-flow equations with convex quadratic approximations, yielding MIQCPs that are solved by off-the-shelf solvers such as CPLEX. The authors show that standard terminal constraints fail to guarantee recursive feasibility when runtime and startup limits are present, introduce additional time-coupled constraints to restore recursive feasibility, and prove this property. Stability is established under the external assumption of periodic dissipativity of the closed-loop system. The approach is illustrated on a 6-bus microgrid under varying demand profiles.

Significance. If the periodic dissipativity assumption can be verified or replaced by verifiable conditions, the work supplies a practical, computationally lighter route to economic NMPC for microgrids that respects discrete generator logic while retaining recursive-feasibility and stability guarantees. The explicit treatment of time-coupled runtime/startup constraints and the demonstration that simple terminal sets are insufficient are useful contributions for the mixed-integer economic MPC literature.

major comments (2)
  1. [Abstract / stability theorem] Abstract and the stability section: the asymptotic-stability claim is obtained only after invoking periodic dissipativity of the underlying system (with the added runtime/startup constraints) as an assumption; no verification, sufficient condition, or numerical check is supplied showing that the 6-bus quadratic power-flow model plus integer generator dynamics actually satisfies the required dissipativity inequality for the chosen stage cost and storage function. Because this assumption is load-bearing for the stability theorem, its lack of substantiation weakens the central theoretical guarantee.
  2. [Recursive feasibility section] The recursive-feasibility proof (section following the terminal-constraint discussion) establishes feasibility under the proposed time-coupled constraints, yet the manuscript does not quantify how the added constraints affect the feasible set size or the resulting sub-optimality gap relative to the original MINLP formulation.
minor comments (2)
  1. The quadratic power-flow approximation error is not quantified for the 6-bus test cases; a brief comparison of the approximated versus exact AC solutions at the reported operating points would strengthen the practical-tractability claim.
  2. Solver runtimes and iteration counts for the MIQCP instances are mentioned only qualitatively; tabulating these quantities across the demand scenarios would allow readers to assess real-time feasibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the positive assessment of the paper's contributions. We address each major comment below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract / stability theorem] Abstract and the stability section: the asymptotic-stability claim is obtained only after invoking periodic dissipativity of the underlying system (with the added runtime/startup constraints) as an assumption; no verification, sufficient condition, or numerical check is supplied showing that the 6-bus quadratic power-flow model plus integer generator dynamics actually satisfies the required dissipativity inequality for the chosen stage cost and storage function. Because this assumption is load-bearing for the stability theorem, its lack of substantiation weakens the central theoretical guarantee.

    Authors: We agree that the stability theorem rests on the periodic dissipativity assumption, which is stated explicitly in the manuscript as an external hypothesis (standard in economic NMPC literature to obtain stability without terminal penalties). No general sufficient condition is provided because dissipativity depends on the specific stage cost, storage function, and closed-loop dynamics, making a universal verifiable condition difficult to derive. For the 6-bus example, a numerical check could be performed by simulating the closed-loop trajectories under the proposed NMPC and verifying the dissipativity inequality over one or more periods, but this was omitted to keep the focus on the feasibility result and the mixed-integer aspects. We will revise the stability section and abstract to clarify the assumption's role, add a short remark on how it can be checked numerically for concrete instances, and note that verification for the reported simulations is feasible but left implicit. revision: partial

  2. Referee: [Recursive feasibility section] The recursive-feasibility proof (section following the terminal-constraint discussion) establishes feasibility under the proposed time-coupled constraints, yet the manuscript does not quantify how the added constraints affect the feasible set size or the resulting sub-optimality gap relative to the original MINLP formulation.

    Authors: The recursive-feasibility proof is deliberately focused on showing that the proposed time-coupled constraints restore the property that standard terminal sets lose when runtime and startup limits are present. Quantifying the reduction in feasible-set volume or the increase in cost relative to the original (non-convex) MINLP is a separate question that would require either set-theoretic volume estimates or extensive Monte-Carlo comparisons of optimal values; such analysis is outside the scope of the current contribution, which centers on feasibility guarantees for the MIQCP approximation. The quadratic relaxation already introduces its own sub-optimality relative to the full AC-MINLP, and the added constraints are a minimal augmentation needed for feasibility. We will add a brief remark in the recursive-feasibility section acknowledging the potential conservatism without providing a full quantification, as the latter would substantially expand the paper. revision: partial

Circularity Check

0 steps flagged

No significant circularity; stability conditional on external assumption

full rationale

The paper derives recursive feasibility of the NMPC scheme via added time-coupled constraints on generator runtime and startups, proving the property directly from the MIQCP formulation. Stability is stated under the separate assumption of periodic dissipativity of the underlying system (quadratic power-flow plus integer dynamics), without deriving that property from the controller equations, fitting parameters to data, or reducing to self-citations. No load-bearing step renames a fit as a prediction or imports uniqueness via author-overlapping citations. The central claims remain independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central stability claim rests on one domain assumption; the convex quadratic replacement is a deliberate modeling choice rather than a derived result. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption periodic dissipativity of the underlying system
    Invoked explicitly to prove stability of the NMPC controller.

pith-pipeline@v0.9.1-grok · 5766 in / 1349 out tokens · 36886 ms · 2026-06-29T15:24:11.656700+00:00 · methodology

0 comments
read the original abstract

Recently there has been a lot of progress in the development of economic nonlinear model predictive control (NMPC) schemes for multistage optimal power flow (OPF) problems. However, the additional inclusion of discrete decision variables to model generator runtimes and generator startup costs can amount to large scale mixed-integer nonlinear programs (MINLPs) that are computationally very challenging. This work investigates the practical approach that replaces the nonlinear AC power flow equations by convex quadratic approximations. In combination with the discrete generator dynamics this leads to a mixed-integer quadratically constrained program (MIQCP) which is of significantly lower complexity and can be solved in reasonable time by off-the-shelf solvers such as CPLEX. We further show that simple terminal constraints are not sufficient to guarantee recursive feasibility of the NMPC scheme if constraints on generator runtime and on the number of generator startup events are present. To address this challenge we propose the use of additional time-coupled constraints and prove the resulting recursive feasibility property. Based on the assumption of periodic dissipativity of the underlying system we can prove stability of the proposed controller. To illustrate our results, we present simulations of a realistic 6-bus microgrid under different demand scenarios.

Figures

Figures reproduced from arXiv: 2605.27553 by Amer Mesanovic, Armin Nurkanovic, Ekaterina Kostina, Hans Georg Bock, J\"urgen Gutekunst, Robert Scholz.

Figure 1
Figure 1. Figure 1: Illustration of the product xy and the McCormick tetrahedron shaped envelope defined by the four linear in￾equalities with the bounds x, y P r0, 2s. • Square terms For pure quadratic terms x 2 the lower bounds of the McCormick envelope can be improved by a stronger convex quadratic constraint leading to the convex envelope defined by x|2 ě x 2 , x|2 ď px ` xqx ´ xx. • Trigonometric terms The trigonometric … view at source ↗
Figure 2
Figure 2. Figure 2: Topology of the test microgrid. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Closed-loop trajectory in the nominal case. The generator and batteries follow the periodic reference solu [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Closed-loop trajectory for the varying solar input scenario. The dashed lines represent the periodic reference [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

discussion (0)

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Reference graph

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