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Coupling-Aware Aggregation of Multi-Zone HVAC Loads under Uncertainty: A Two-level Framework

Jingguan Liu, Han Jiang, Xiaomeng Ai, Shengshi Wang, Xizhen Xue, Shichang Cui, Jinming Hou, Jiakun Fang, Jinyu Wen

arXiv:2609.03253v1eess.SY

TL;DR

Multi-zone HVAC coupling and uncertainty propagation make reliable aggregate-flexibility quantification difficult, while existing approaches face dimensional and scalability limitations. The paper proposes a two-level framework that derives building-level coupled expressions and uses matrix-transformed inner approximations with linear-program parameter selection; simulations report accurate, reliable aggregation with low computational complexity.

  • Problem

    Multi-zone coupling propagates uncertainty across zones and creates dimensional mismatches that limit existing aggregation methods, undermining reliable aggregate-flexibility quantification.

  • Method

    The framework reduces coupled building dynamics to analytical power-interface expressions, then constructs different-dimensional polytope inner approximations and optimizes aggregate parameters.

  • Results

    Case 1 achieves the lowest average UPR of 3.28%, while the proposed method achieves an average UPR of approximately 5% when α=1.4 under parameter heterogeneity.

  • Takeaways & Limitations

    The framework provides accurate and reliable aggregate flexibility estimates while balancing robustness and conservatism under uncertainty.

Abstract

from arXiv · show

Aggregating building heating, ventilation, and air-conditioning (HVAC) loads unlocks substantial demand-side flexibility for power systems. Yet multi-zone coupling creates intricate interdependencies and uncertainty propagation, complicating the quantification of aggregate flexibility. To address this issue, this paper proposes a coupling-aware two-level aggregation framework. At the building level, tailored Gaussian elimination and coordinate transformation techniques are employed to recast the high-dimensional thermal dynamics as an equivalent lower-dimensional analytical expression. This expression streamlines the subsequent aggregator-level stage by (i) clarifying the propagation of zone-level uncertainties to the building-level interface, (ii) decoupling intra-building multi-zone coupling from inter-building aggregation, and (iii) providing full-dimensional building-level flexibility sets that enable tractable reformulation. At the aggregator level, existing geometric aggregation approaches are generalized by a newly developed matrix-transformation technique. This technique effectively constructs inner approximations between polytopes of different dimensions, producing closed-form images of high-dimensional multi-zone HVAC flexibility in power subspace. The resulting inner approximation is then recast as a customized separatable linear program that efficiently determines the optimal aggregate parameters. Case studies validate the effectiveness of our framework, highlighting its accuracy, reliability, and scalability.

A. Abbreviation

The paper models multi-zone HVAC flexibility under thermal coupling and uncertainty, then develops analytical and geometric tools for reliable aggregation.

  • HVAC loads provide demand-side flexibility, but individually scheduling many loads becomes computationally prohibitive at scale.
  • Multi-zone coupling creates dimensional mismatch, uncertainty propagation, and computational difficulty for existing aggregation approaches.
  • The building-level method uses Gaussian elimination and coordinate transformation to reduce high-dimensional coupled dynamics to a lower-dimensional power-interface expression.
  • The framework extends polytope-based aggregation through matrix transformations that construct inner approximations between polytopes of different dimensions.

B. Aggregate Flexibility Set

The aggregate flexibility set is the Minkowski sum of building and zone HVAC flexibility sets, but coupling makes exact computation difficult. The framework addresses this through building-level decomposition, matrix-transformed inner approximations, optimization, and disaggregation.

  • B. Aggregate Flexibility Set: The aggregate flexibility set is represented as the Minkowski sum of HVAC power flexibility across the controlled buildings and zones.
  • B. Aggregate Flexibility Set: Multi-zone coupling introduces interdependencies among zone powers and propagates uncertainties across zones, complicating aggregate flexibility computation.
  • B. Aggregate Flexibility Set: The framework decomposes aggregation into building and aggregator levels because coupling occurs within buildings while different buildings remain independent.
  • B. Aggregate Flexibility Set: At the building level, an analytical expression captures intra-building coupling and uncertainty propagation at the building power interface.
  • B. Aggregate Flexibility Set: At the aggregator level, matrix transformations construct inner approximations in power subspace and enable closed-form Minkowski-sum calculation.
  • B. Aggregate Flexibility Set: A linear program determines aggregate parameters, followed by a fast disaggregation strategy for dispatching aggregate power to individual HVAC schedules.

B. Building-Level Analytical Expression

The building-level formulation reduces high-dimensional multi-zone thermal dynamics to a concise, full-dimensional polytope at the power interface. Gaussian elimination and coordinate transformation clarify uncertainty propagation, remove equality constraints, and decouple building-level coupling from later aggregation.

  • Building-level projection: The framework projects high-dimensional building thermal dynamics onto an equivalent reduced subspace at the building-level power interface.The projection is illustrated in Fig. 4 and forms the basis for the lower-dimensional analytical expression.
  • Dimensional reformulation: Step 1 lifts the building formulation into a high-dimensional space before eliminating temperature states and redundant variables.The lifting procedure enables reformulation of the aggregate set as a decoupled Minkowski sum of building-level powers.
  • Benefits: The transformed formulation provides a lower-dimensional analytical expression while preserving a full-dimensional geometry suitable for downstream aggregation.This addresses the difficulty caused by equality constraints, redundant variables, and uncertainty-dependent coupling in the original set.
  • Dimensional reformulation: Gaussian elimination removes temperature-state equality constraints by substituting the thermal dynamics into the building flexibility set.This produces a medium-dimensional polytope with coefficient matrices derived from the original and reformulated dynamics.
  • Full-dimensional representation: A tailored coordinate transformation removes power-coupling equalities and converts the reduced representation into a full-dimensional building-level polytope.The resulting representation retains the building-level flexibility while eliminating redundant variables and equality constraints.
  • Benefits: The compact expression clarifies how zone-level uncertainties propagate to the building-level power interface.It also confines multi-zone coupling to the building-level formulation, reducing the burden of subsequent inter-building Minkowski-sum calculations.

1) Physical Insight:

The building-level representation exposes how uncertainty affects the flexibility set through both the original multi-zone coupling and the coordinate transformation. This links the geometry of the transformed set to uncertainty propagation at the building interface.

  • Physical insight: The transformed set reflects uncertainty through the interaction of inherent multi-zone coupling and the applied coordinate transformation.The resulting geometry therefore captures how uncertain parameters influence the building-level flexibility representation.

2) Mathematical Tractability:

At the aggregator level, matrix transformation enables inner approximation between polytopes with different dimensions while keeping the result in power space. The affine representation supports closed-form aggregation and tractable parameter determination under uncertainty.

  • Motivation: The original building-level set is non-full-dimensional because equality constraints and redundant variables hinder downstream aggregation.The transformed low-dimensional set is full-dimensional in the new coordinates and supports efficient aggregate-parameter calculation.
  • Motivation: Existing polytope methods require matching dimensions, but multi-zone coupling creates dimensional mismatches and can make exact projection NP-hard.These restrictions make traditional approaches unsuitable for directly aggregating coupled HVAC flexibility sets.
  • Matrix transformation: The proposed matrix-transformation technique reformulates different-dimensional inner approximation as an affine polytope containment problem, avoiding exact projections.This is the central mechanism illustrated by Fig. 5.
  • Affine approximation: Affine mapping produces a closed-form Minkowski sum and offers greater transformation flexibility for aggregation than alternative inner-approximation techniques using the same base set.The affine-transformed base set is parameterized by aggregate variables determined later in the framework.
  • Uncertainty handling: Distributionally robust chance constraints characterize the inner approximation under uncertainty with allowable violation probability 𝜀.The ambiguity set is built from the empirical mean vector and covariance matrix of the uncertain parameters.
  • Matrix transformation: The transformation matrix flexibly matches polytope dimensions and ensures the approximation lies purely in power space, where it remains summable.Setting the matrix to the identity recovers existing approaches, so the method generalizes them to multi-zone-coupled loads.

D. Aggregate Parameter Determination

The framework converts uncertain polytope inclusion into a tractable optimization that selects aggregate parameters for a largest-volume inner approximation. Inter-building decoupling further enables parallel solution across buildings.

  • Tractable Reformulation: The inner-approximation condition is recast as linear constraints, enabling tractable determination of aggregate parameters under uncertainty.The resulting parameters preserve the inner approximation's accuracy and reliability.
  • Robustness Selection: The robustness parameter ε trades robustness against conservatism: smaller ε improves uncertainty protection but tightens the approximation.The paper states that ε should be selected to safeguard against observed uncertainty without unduly restricting flexibility.
  • Robustness Selection: The data-driven rule selects ε* as the largest risk level guaranteeing the stated condition for historical samples.This rule is based on historical samples for each building.
  • Optimization Problem: The parameter-selection program maximizes tr(Γagg), making the approximate flexibility set the largest-volume subset of the aggregate set.The volume of the approximate set scales with the trace of Γagg.
  • Computational Complexity: Inter-building decoupling decomposes the global problem into NB independent linear programs that can be solved in parallel and recombined into the aggregate solution.This block-separable structure accelerates computation for large-scale implementation.

E. Fast and Feasible Disaggregation Strategy

The proposed two-level disaggregation first allocates scheduled aggregate power across buildings, then distributes each building's allocation across zones. The procedure is designed for computational efficiency, feasible operation, and consistent zone comfort.

  • Computational Efficiency: Building-level allocation uses closed-form expressions, while zone-level problems can run independently and in parallel to support large-system computation.The strategy avoids optimization whose size grows with building count at the building-allocation stage.
  • Building-Level Disaggregation: Any scheduled aggregate power profile in the approximate set can be split into building-level allocations that remain feasible and sum to the scheduled profile.Building-level allocations are computed using the closed-form disaggregation method.
  • Zone-Level Disaggregation: Building allocations are distributed among zones by independently solving smaller optimization problems for zone power and indoor temperatures.The formulation includes normalized temperature constraints and HVAC dynamics and limits.
  • Zone-Level Disaggregation: Normalized temperatures map each zone to [-1,1], while the objective minimizes deviations among zones to promote consistent comfort.The normalization enables direct comparison across zones with distinct thermal dynamics.
  • Operational Feasibility: The zonal disaggregation problem always has a feasible solution, guaranteeing feasibility of the resulting zonal schedules.This guarantee follows from the stated feasibility of problems (24.a)-(24.e).
  • Scope Boundary: The framework's current scope excludes heterogeneous building response patterns and inter-building fairness considerations.The authors identify these as possible directions for future work.

IV. CASE STUDIES

Case studies show that the proposed two-level aggregation captures more HVAC flexibility than competing approximations while preserving feasible schedules and near-benchmark economic performance. Its accuracy remains strong as parameter heterogeneity increases.

  • Illustrative Example: In the illustrative example, the proposed affine-transformation method recovers 89.5% of the exact polytope volume, compared with 61.7% for homothet, 39.4% for virtual-battery, and 14.4% for hyper-box approximations.The competing methods lose coverage by neglecting intertemporal coupling, using predefined storage models, or relying on limited geometric transformations.
  • Simulation Setup: The aggregator-level evaluation uses 100 buildings containing 2 to 8 zones, while excluding the virtual-battery model because it failed to converge within 24 hours.Experiments use 31 days of hourly Spanish day-ahead electricity prices from July 1–31, 2024.
  • Large-Scale Aggregation: The proposed framework achieves the lowest average UPR, 3.28%, with the smallest fluctuations across 31-day scheduling scenarios.Cases 2 and 3 produce more conservative approximations and underestimate available flexibility.
  • Large-Scale Aggregation: Case 1 nearly overlaps with the centralized benchmark in a representative day while outperforming hyper-box and homothet aggregation approaches.The schedule charges thermal storage during low-price periods and discharges it during peak prices.
  • Large-Scale Aggregation: The proposed disaggregation keeps all zone power and temperature profiles within limits, confirming feasible schedules and satisfied thermal-comfort constraints.The profiles are evaluated using normalized power and temperature representations.
  • Heterogeneity Analysis: At heterogeneity α=1.4, the proposed method achieves an average UPR of approximately 5%, while traditional homothet approaches become increasingly conservative.The proposed generalized affine transformations adapt to high-dimensional thermal coupling more effectively than scaling and translation.

C. Effectiveness of Aggregation Reliability

Under forecast uncertainty, the proposed risk-aware aggregation reduces flexibility utilization to improve reliability, with substantially fewer temperature violations than uncertainty-ignorant scheduling. The framework also remains computationally practical for larger zone counts.

  • Impact of Uncertainty: The uncertainty-aware strategy reduces response energy by 12.8%, from 14.248 MWh to 12.428 MWh, while increasing scheduling cost by 4.5%.The reduction reserves HVAC flexibility to hedge against forecast errors.
  • Out-of-Sample Reliability: The uncertainty-aware schedule lowers the out-of-sample violation rate from 98.85% to 1.85% across 2,000 test scenarios.The uncertainty-ignorant schedule reaches temperature boundaries and incurs severe constraint violations under forecast errors.
  • Risk Attitude: Lower allowable violation probabilities reduce response energy while increasing scheduling cost at an accelerating rate.More risk-averse settings require the aggregator to reserve more flexibility against uncertainty.
  • Computational Efficiency: The inner-approximation computation remains under 4 minutes for a 14-zone building, despite runtime increasing with zone count.The dominant inner-approximation step is a linear program suitable for solvers such as CPLEX or Gurobi, while parallel aggregation keeps runtime largely independent of building count.
  • Conclusion: The framework balances robustness and conservatism while delivering accurate flexibility estimates with low computational complexity.The conclusion presents this balance as a central outcome of the uncertainty-aware aggregation approach.
  • Limitations and Future Work: Future work will address heterogeneous response willingness, inter-building fairness, ultra-large buildings, and individual HVAC parameter-estimation errors.These issues define the stated scope boundaries for extending the framework to additional real-world complexities.

VI. APPENDIX

The appendix establishes the dimensionality reduction, polytope-containment conditions, and robustness-selection rule underlying the aggregation framework. It also shows that overly conservative constraint satisfaction can restrict flexibility.

  • Coordinate transformation: Gaussian-Jordan elimination identifies the transformation matrices needed to relate the reduced auxiliary variables to the building-level representation.The construction uses basis selection and matrix blocks to derive the required relationships.
  • Dimensionality relation: The coordinate transformation reduces the auxiliary-variable dimension from N_n^I N_T to N_n^I N_T−N_T.The resulting dimensional difference is established after transforming the original HVAC representation.
  • Polytope containment: Lemma 1 guarantees affine inner containment between polytopes of different dimensions through auxiliary variables and linear matrix inequalities.The lemma extends Farkas’-lemma-based containment conditions to transformed inner and outer polytopes.
  • Robustness selection: Choosing ε=ε* provides robustness for every ξ_n∈X_n under the bound established in (28.c).The result follows by combining formulation (20) with the uncertainty bound in (28.b).
  • Robustness selection: When the constraints in (28.c) are strictly over-satisfied, the resulting approximation becomes excessively conservative.This identifies the practical cost of selecting robustness margins that are too restrictive.

D. Proof of Proposition 4

The proposition’s proof constructs aggregate variables from the transformed building-level representation while preserving aggregate power balance. A dimensional-mismatch example compares the proposed inner approximation with fixed-dimension alternatives.

  • Proof of Proposition 4: The constructed aggregate point satisfies the aggregate power-balance equation through the inverse aggregate transformation.The proof explicitly derives the equality ensuring aggregate power balance.
  • Proof of Proposition 4: Membership in the aggregate base polytope implies the existence of a base point whose transformed image equals the target aggregate point.This establishes the required representation before mapping back to the building-level sets.
  • Dimensional-mismatch case: In a two-zone, two-period HVAC example, the full-dimensional polytope contains two aggregate-power and two auxiliary variables before projection.The example ignores uncertainty to isolate the dimensional-mismatch issue.
  • Dimensional-mismatch case: Both fixed- and differing-dimension approaches guarantee inner approximations, but the fixed-dimension result is not guaranteed to remain strictly within the P1-P2 subspace.The proposed comparison concerns projection of a three-dimensional polytope onto the two-dimensional aggregate-power subspace.
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