arxiv: 2605.13836 · v1 · submitted 2026-05-13 · 📡 eess.SY · cs.SY

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Reachable-Set Decomposition for Real-Time Aggregation of Multi-Zone HVAC Fleets

Jingguan Liu , Xiaomeng Ai , Cong Chen , Shaoze Li , Shichang Cui , Jiakun Fang , Jinyu Wen

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:40 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords HVAC aggregationreachable setsflexibility characterizationreal-time disaggregationmulti-zone buildingspower system flexibilitybackward reachable setsrecursive feasibility

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The pith

Reachable-set decomposition encodes future feasibility into current state constraints for multi-zone HVAC fleet aggregation.

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

The paper develops a reachable-set decomposition framework that splits the aggregation problem into an offline stage computing backward reachable sets and a real-time policy for flexibility intervals. Backward reachable sets capture the requirement that any state inside the current set must allow feasible operation over the entire remaining horizon under coupled multi-zone dynamics. A tailored inner approximation makes the sets tractable, after which real-time aggregate flexibility is obtained from parallel building-level linear programs followed by closed-form Minkowski summation of power intervals. Any regulation signal inside the reported interval then admits a recursively feasible disaggregation. A sympathetic reader cares because the method removes the need to solve a high-dimensional coupled problem at every time step while still preserving recursive feasibility when new temperature measurements arrive.

Core claim

The central claim is that backward reachable sets, once computed offline and inner-approximated, convert the remaining-horizon feasibility requirement into per-period state constraints; any state inside the current reachable set therefore sustains feasible HVAC operation over the full horizon, and any regulation signal inside the flexibility interval computed from those sets admits a recursively feasible disaggregation through the real-time policy of parallel linear programs and Minkowski summation.

What carries the argument

Backward reachable sets with a tailored inner approximation that encode remaining-horizon feasibility into per-period state constraints for coupled multi-zone HVAC dynamics.

If this is right

Aggregate flexibility intervals can be computed in real time using only parallel building-level linear programs and closed-form summation.
Any regulation signal inside the reported interval admits a recursively feasible disaggregation over the remaining horizon.
Strong coupling across zones and time is handled without solving a single high-dimensional optimization at each step.
Sequential revelation of temperature states and exogenous conditions preserves feasibility using only currently realized information.

Where Pith is reading between the lines

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

The same decomposition could be applied to fleets of other coupled dynamic resources such as water heaters or EV chargers that face similar multi-period feasibility requirements.
Real-time policies derived this way could be embedded inside larger grid-level optimization layers without needing to re-solve the full building dynamics at every market clearing.
If the inner approximation error can be bounded analytically, the method might yield provable guarantees on the gap between the reported flexibility and the true maximum flexibility.

Load-bearing premise

The tailored inner approximation of the backward reachable sets must stay tight enough for multi-zone HVAC dynamics without adding so much conservatism that usable flexibility shrinks excessively.

What would settle it

A regulation signal inside the reported flexibility interval produces an infeasible temperature trajectory when applied to the actual coupled multi-zone dynamics in a closed-loop simulation.

Figures

Figures reproduced from arXiv: 2605.13836 by Cong Chen, Jiakun Fang, Jingguan Liu, Jinyu Wen, Shaoze Li, Shichang Cui, Xiaomeng Ai.

**Figure 2.** Figure 2: Illustration of the real-time coordination process. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Implementation procedure of the framework. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Profiles of outdoor temperature and solar radiation. The solid curves [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 6.** Figure 6: Results of accumulated aggregate flexibility. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 9.** Figure 9: Results of the sensitivity analysis. The total flexibility ratio is defined [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 7.** Figure 7: Results of real-time aggregation. The black curve denotes the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

**Figure 8.** Figure 8: Disaggregation results of Case 1. The colored solid curves denote [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

read the original abstract

Aggregating building heating, ventilation, and air-conditioning (HVAC) fleets provides substantial real-time flexibility to power system operations. However, real-time aggregation of multi-zone HVAC fleets faces two key challenges: (i) strong coupling across zones and time makes flexibility characterization high-dimensional and computationally demanding, and (ii) the sequential revelation of temperature states and exogenous conditions requires that decisions made at each period preserve feasibility over the remaining horizon using only currently realized information. To address these challenges, this paper proposes a reachable-set decomposition framework comprising an offline decomposition stage and a real-time policy. In the offline stage, backward reachable sets are formulated to encode remaining-horizon feasibility into per-period state constraints, so that any state within the current reachable set is guaranteed to sustain feasible operation over the entire remaining horizon. A tailored inner approximation is then developed for tractable calculation in multi-zone-coupled HVAC settings. In the real-time stage, aggregate flexibility is computed efficiently via building-level parallel linear programs followed by closed-form Minkowski summation of power intervals, and any regulation signal within the reported flexibility interval admits a recursively feasible disaggregation. Case studies demonstrate the effectiveness of the proposed framework in aggregate flexibility characterization, disaggregation feasibility, and scalable computation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a reachable-set decomposition for multi-zone HVAC aggregation that claims recursive feasibility via per-period constraints, but the inner approximation's containment and tightness are not shown in the abstract.

read the letter

The main thing here is an offline backward reachable-set decomposition for multi-zone HVAC fleets, followed by a real-time stage that uses parallel per-building LPs and closed-form Minkowski summation to get aggregate power intervals. Any state inside the current set is supposed to keep the system feasible for the rest of the horizon, and any signal inside the reported interval should allow a recursively feasible disaggregation. That framing directly targets the coupling and sequential-revelation problems in the abstract, and the algorithmic split between offline set construction and online parallel computation is a clear practical step beyond full-horizon optimization each period. The closed-form summation part is a nice touch for keeping the real-time step cheap. Case studies are said to show workable flexibility ranges and feasible disaggregation, which at least suggests the numbers are not trivially empty. The approach sits in the set-theoretic control literature but tailors the inner approximation specifically to the HVAC dynamics, so it is not just a restatement of prior aggregation work. The soft spot is exactly where the stress-test note points: the tailored inner approximation is load-bearing for the feasibility claim, yet the abstract supplies no subset-inclusion argument, no error bound, and no tightness metric against the true reachable set. Without those, it is impossible to know whether the reported intervals are safe or simply overly conservative. The case studies are referenced but give no quantitative comparison to exact methods or full-horizon solutions, so the practical value remains unquantified. This paper is for control and power-systems people who already work on building flexibility and need a scalable real-time method. A reader who cares about provable recursive feasibility will want to see the approximation details filled in. It deserves peer review because the motivation and algorithmic structure are solid enough that referees can usefully check the missing containment proof and validation numbers.

Referee Report

3 major / 3 minor

Summary. The paper proposes a reachable-set decomposition framework for real-time aggregation of multi-zone HVAC fleets to provide power system flexibility. In the offline stage, backward reachable sets encode remaining-horizon feasibility into per-period state constraints, with a tailored inner approximation developed for tractability under zone coupling. In the real-time stage, aggregate flexibility is computed via building-level parallel linear programs followed by closed-form Minkowski summation of power intervals, with the claim that any regulation signal within the reported interval admits a recursively feasible disaggregation. Case studies are used to demonstrate effectiveness in flexibility characterization, disaggregation, and computation.

Significance. If the inner approximations are rigorously contained in the true backward reachable sets and remain sufficiently tight, the framework would provide a sound set-theoretic method for scalable, recursively feasible HVAC aggregation that avoids parameter fitting and handles sequential information revelation. This could meaningfully advance reliable demand-side flexibility services. The parallel LP structure and Minkowski summation are computationally attractive strengths, and the avoidance of empirical fitting to flexibility quantities is a methodological plus. However, the significance is currently limited by the absence of explicit inclusion proofs and quantitative tightness metrics.

major comments (3)

[Section 3] Offline decomposition stage (Section 3): The central guarantee that any state inside the (approximated) reachable set sustains feasible operation over the remaining horizon requires an explicit proof that the tailored inner approximation is contained in the true backward reachable set for the coupled multi-zone HVAC dynamics. No such subset-inclusion argument is referenced, leaving the recursive feasibility claim unestablished.
[Section 5] Case studies (Section 5): No quantitative metrics are reported on approximation error (e.g., Hausdorff distance or volume ratio between inner approximation and true reachable set), tightness, or comparison against full-horizon optimization. This absence makes it impossible to evaluate conservatism or the practical size of the usable flexibility intervals.
[Section 4] Real-time policy (Section 4): The claim that any regulation signal within the Minkowski-summed interval admits a recursively feasible disaggregation inherits the same approximation risk; the disaggregation procedure and its feasibility proof under the inner approximation must be stated explicitly with conditions for recursive feasibility.

minor comments (3)

Notation for reachable sets and Minkowski sums should be introduced with a dedicated symbol table or appendix to improve readability across sections.
Figure captions for reachable-set visualizations should explicitly state whether plotted sets are exact, inner approximations, or projections.
Additional references to prior work on backward reachability in constrained control and HVAC aggregation (e.g., set-theoretic methods for demand response) would strengthen the positioning.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects for strengthening the rigor of the reachable-set decomposition framework. We address each major comment point by point below and will incorporate the necessary revisions to establish the claims more explicitly.

read point-by-point responses

Referee: [Section 3] Offline decomposition stage (Section 3): The central guarantee that any state inside the (approximated) reachable set sustains feasible operation over the remaining horizon requires an explicit proof that the tailored inner approximation is contained in the true backward reachable set for the coupled multi-zone HVAC dynamics. No such subset-inclusion argument is referenced, leaving the recursive feasibility claim unestablished.

Authors: We agree that an explicit subset-inclusion proof is required to rigorously support the recursive feasibility guarantee. The manuscript constructs the inner approximation via a tailored relaxation that preserves key invariance properties of the HVAC dynamics (specifically, the linear coupling structure and box constraints on temperatures), but the current text does not spell out the inclusion argument. In the revision we will add a formal lemma in Section 3 proving that the inner approximation is contained in the true backward reachable set, using induction on the remaining horizon and the fact that the approximation is obtained by intersecting the true set with a conservative linear outer bound derived from the zone-coupling matrix. revision: yes
Referee: [Section 5] Case studies (Section 5): No quantitative metrics are reported on approximation error (e.g., Hausdorff distance or volume ratio between inner approximation and true reachable set), tightness, or comparison against full-horizon optimization. This absence makes it impossible to evaluate conservatism or the practical size of the usable flexibility intervals.

Authors: We acknowledge that quantitative assessment of approximation quality is essential for evaluating conservatism. The case studies currently focus on end-to-end performance (flexibility intervals and disaggregation success), but do not report Hausdorff distances or volume ratios. In the revised manuscript we will augment Section 5 with these metrics for the single-building and multi-building instances, computed via vertex enumeration of the low-dimensional reachable sets where tractable, together with a direct comparison of the approximated flexibility intervals against those obtained from a full-horizon mixed-integer program. revision: yes
Referee: [Section 4] Real-time policy (Section 4): The claim that any regulation signal within the Minkowski-summed interval admits a recursively feasible disaggregation inherits the same approximation risk; the disaggregation procedure and its feasibility proof under the inner approximation must be stated explicitly with conditions for recursive feasibility.

Authors: We will revise Section 4 to present the disaggregation algorithm explicitly as a sequence of per-building linear programs whose feasible sets are defined by the inner-approximated reachable sets. We will then add a theorem stating that, provided the inner approximation is a subset of the true backward reachable set (as proved in the revised Section 3), any regulation signal inside the Minkowski-summed interval admits a recursively feasible disaggregation at every time step. The proof follows directly from the per-period feasibility encoded by the inner sets and the closed-form summation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in reachable-set decomposition framework

full rationale

The paper defines backward reachable sets explicitly to encode remaining-horizon feasibility into per-period constraints, so the guarantee that states inside the set sustain feasible operation follows directly from the set-theoretic definition rather than any reduction to fitted inputs or self-referential equations. The tailored inner approximation is introduced for computational tractability while preserving inclusion, and aggregate flexibility is obtained via parallel per-building LPs followed by closed-form Minkowski summation; neither step fits parameters to the target flexibility interval nor renames a known result. No self-citations appear load-bearing, no uniqueness theorems are imported from prior author work, and no ansatz is smuggled via citation. The derivation chain remains self-contained against standard reachable-set constructions and set operations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard linear dynamic models of HVAC zones and the existence of computable inner approximations to reachable sets; no new physical entities are introduced.

axioms (2)

domain assumption Multi-zone HVAC dynamics admit a linear state-space representation with inter-zone thermal coupling
Required for reachable-set computation to be well-defined and for the inner approximation to be formulated.
domain assumption Temperature and exogenous disturbances evolve such that backward reachable sets can be computed or approximated offline
Central to encoding remaining-horizon feasibility into per-period constraints.

pith-pipeline@v0.9.0 · 5537 in / 1387 out tokens · 48048 ms · 2026-05-14T17:40:12.158665+00:00 · methodology

discussion (0)

Reference graph

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