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

Source: arXiv:2605.13836 · Published 2026-05-13 · By Jingguan Liu, Xiaomeng Ai, Cong Chen, Shaoze Li, Shichang Cui, Jiakun Fang et al.

TL;DR

This paper addresses the challenge of real-time aggregation of multi-zone HVAC fleets to provide flexible demand-side resources for power system operations. The key problem is how to characterize and coordinate the aggregate flexibility of a fleet given strong temporal and spatial coupling in thermal dynamics, and the sequential revelation of system states and exogenous factors that requires time-causal and recursively feasible operation. Prior approaches either lack scalability, do not guarantee recursive feasibility, or assume full-horizon information. The authors propose a novel reachable-set decomposition framework that transforms the full multi-period coupling into tractable per-period state constraints by leveraging backward reachable sets computed offline. They introduce a tailored inner polyhedral approximation of these high-dimensional reachable sets for computational tractability and scalability. The resulting framework enables efficient real-time computation of aggregate HVAC flexibility via parallel building-level linear programs and closed-form Minkowski summations of power intervals, while guaranteeing that any regulation signal within the reported flexibility admits a recursively feasible disaggregation to zone controls. Extensive case studies validate the method’s effectiveness in flexibility characterization, recursive feasibility, and computational scalability, representing the first such scalable real-time framework for multi-zone HVAC fleets under sequential information.

Key findings

• The reachable sets recursively characterize all indoor temperature states at period t from which a feasible trajectory over the remaining horizon exists under any admissible exogenous realization (Definition 1, Eq. 9).
• A polytopic inner approximation of the reachable set defined as an affine image of the unit ℓ∞-ball enables a tractable linear program formulation for reachable set computation (Section III-C).
• The offline linear program (21) to compute the affine inner approximation is solved backward in time for each building; solutions encode remaining-horizon feasibility into per-period constraints (Eq. 22).
• Real-time aggregate flexibility is computed as a closed-form Minkowski sum of building-level power intervals obtained by solving parallel linear programs (Eq. 27 and 28), avoiding expensive high-dimensional Minkowski sum computations.
• Any regulation signal within the reported aggregate power interval admits a closed-form recursively feasible disaggregation to zone-level commands as a convex combination of lower and upper endpoint profiles (Proposition 2, Eq. 30).
• The framework simultaneously achieves time-causality, recursive feasibility, high-dimensional multi-zone coupling handling, and scalability for large HVAC fleets (Table I comparison).
• Parallelization across buildings ensures linear scaling of computational complexity, making the method suitable for large-scale real-time HVAC fleet operation.
• Case studies demonstrate improved accuracy of aggregate flexibility characterization and guaranteed feasibility of disaggregated control trajectories under sequential information.

Threat model

The threat model centers on the sequential revelation of uncertain exogenous conditions such as outdoor temperature and solar radiation, which are modeled as bounded sets but are unknown before each time period. The aggregator must make HVAC control decisions based only on current and past information while guaranteeing feasibility over all possible future exogenous realizations. The adversary cannot manipulate the system arbitrarily but represents uncertainty and time-causality constraints. Thus, recursive feasibility must be ensured against all admissible exogenous trajectories.

Methodology — deep read

1. Threat Model & Assumptions: The adversary is essentially the uncertainty in future exogenous inputs (outdoor temperature, solar radiation) revealed sequentially during real-time operation. The system must maintain feasible HVAC operation and indoor comfort despite unknown future conditions, enforcing recursive feasibility under time-causal information.

2. Data: The model uses physical parameters for multi-zone buildings (thermal capacitances, resistances, comfort bounds) and exogenous conditions specified as compact uncertainty sets (polytopes) characterized by vertex scenarios. No explicit data-driven model; dynamics modeled via discrete-time thermal difference equations.

3. Architecture / Algorithm: The key algorithmic innovation is applying backward reachable set theory to encode multi-period multi-zone thermal dynamics and constraints into per-period reachable sets describing feasible indoor temperature states sustaining future feasibility. The reachable sets are implicitly defined by nested quantifiers over state and input feasibility under all admissible exogenous conditions.

To enable tractable computation of these high-dimensional reachable sets, the authors propose an inner polyhedral approximation as an affine map of a unit ℓ∞-ball (Eq. 11). This yields a linear programming (LP) problem (Eq. 21) to maximize the affine map's size under containment constraints (Proposition 1) that are expressed via linear inequalities and auxiliary nonnegative matrices.

The LP encodes recursive feasibility by checking that all reachable sets satisfy thermal dynamics, comfort, power constraints, and future feasible sets under all extremal exogenous scenarios (vertices of uncertainty sets).

4. Training Regime: The offline LPs to compute reachable sets are solved recursively backward in time for each building separately, enabling parallel computation. There is no stochastic learning, but rather deterministic constrained optimization respecting worst-case exogenous bounds. Hyperparameters include discretization period, time horizon length.

5. Evaluation Protocol: Flexibility sets are validated on well-known building thermal models with realistic parameters, comparing computational complexity, flexibility characterization, and recursive feasibility guarantees. Aggregation quality is compared with existing methods qualitatively (Table I). The disaggregation policy correctness is formally proven (Proposition 2).

6. Reproducibility: The methodology is explicitly formulated with linear programs and affine transformations detailed. However, code or datasets are not publicly referenced. The approach requires building-specific physical parameters and uncertainty sets but otherwise is generalizable.

Example end-to-end: Starting at the final time period with no terminal constraint, the reachable set is initialized as the entire temperature space. Then for each earlier period, the LP (21) is solved to find an inner affine approximation of states guaranteeing feasibility of current and future HVAC operation. At runtime, each building solves LP (27) based on current temperature and exogenous state to determine minimum and maximum feasible power for that period. These intervals are summed across buildings to obtain aggregate power limits reported to the grid operator. If a regulation signal in this range is received, it is disaggregated by computing a convex combination of the extreme feasible control profiles of each building, ensuring recursive feasibility is preserved for future steps.

Technical innovations

• Adapting backward reachable set theory to multi-zone HVAC fleet aggregation to encode full-horizon thermal and comfort constraints into per-period reachable state sets under worst-case exogenous uncertainty.
• Developing a tailored affine inner approximation method of high-dimensional backward reachable sets as polytopes, enabling scalable linear programming computation.
• Employing explicit lifted polyhedral reformulation and linear containment conditions (Prop. 1) to efficiently compute inner approximations without explicit polytope projection.
• Introducing a scalable, time-causal real-time aggregation and disaggregation policy where aggregate flexibility is represented as closed-form Minkowski sums of building-level power intervals.
• Providing a closed-form convex combination disaggregation strategy (Prop. 2) guaranteeing recursive feasibility while matching any requested aggregate regulation signal within the flexibility bounds.

Baselines vs proposed

• Direct summation methods [14-16]: Less accurate, outer approximations, lack recursive feasibility vs proposed: guarantees recursive feasibility with inner approximations.
• Boundary optimization methods [18-22]: Limited scalability due to large centralized optimization vs proposed: parallel per-building LPs enabling scalable computation.
• Geometric transformation methods [5,9,10,23]: Assume full-horizon information violating time causality vs proposed: time-causal aggregation under sequential information revelation.

Figures from the paper

Figures are reproduced from the source paper for academic discussion. Original copyright: the paper authors. See arXiv:2605.13836.

Fig 2: Illustration of the real-time coordination process.

Limitations

• Exact reachable set computation is intractable; inner approximations may be conservative limiting flexibility characterization accuracy.
• The method requires precise building thermal models and exogenous uncertainty sets, which may be challenging to obtain or vary in practice.
• No adversarial attack or deliberate manipulation scenarios considered; robustness beyond bounded uncertainty sets is not evaluated.
• The paper lacks extensive comparative empirical evaluation on large-scale real building datasets or real deployment measurements.
• Closed-form disaggregation assumes convex feasible sets; nonlinearities or actuator constraints beyond power limits are not modeled.

Open questions / follow-ons

• How to reduce the conservatism of the affine inner approximation to achieve tighter aggregate flexibility estimates without sacrificing tractability?
• Can the method be extended to incorporate nonlinear thermal dynamics and more complex HVAC control constraints?
• How to adaptively learn or update exogenous uncertainty sets and building parameters online to improve model accuracy?
• What is the performance and robustness under real-world noisy measurements, communication delays, or partial information about building states?

Why it matters for bot defense

For bot-defense and CAPTCHA practitioners interested in distributed resource aggregation or online decision-making under uncertainty, this work illustrates how to tackle sequential, high-dimensional constraint coupling efficiently via offline reachable set computation and time-causal policies. The key insight is transforming a complex multi-period coupling into tractable per-period constraints using backward reachability and suitable approximations—preserving recursive feasibility in real time. Similarly, CAPTCHA systems aggregating responses under uncertainty or sequential interaction could benefit from reachable set-based formulations to guarantee solvability while managing complexity and time-causal information structures. The paper also underscores the value of parallelizable offline computation to enable scalable real-time operation, a foundational pattern in bot defense for coordinating large distributed systems reliably.

Cite

@article{arxiv2605_13836,
  title={ Reachable-Set Decomposition for Real-Time Aggregation of Multi-Zone HVAC Fleets },
  author={ Jingguan Liu and Xiaomeng Ai and Cong Chen and Shaoze Li and Shichang Cui and Jiakun Fang and Jinyu Wen },
  journal={arXiv preprint arXiv:2605.13836},
  year={ 2026 },
  url={https://arxiv.org/abs/2605.13836}
}

Read the full paper

• arXiv:2605.13836 (abstract)
• arXiv:2605.13836 (PDF)