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Continuous-Time Aggregation of Massive Flexible HVAC Loads Considering Uncertainty for Reserve Provision in Power System Dispatch

Jingguan Liu, Xiaomeng Ai, Jiakun Fang, Shichang Cui, Shengshi Wang, Wei Yao, Jinyu Wen

arXiv:2609.02408v1eess.SY

TL;DR

Massive HVAC scheduling must capture intra-hour flexibility while handling computational complexity, heterogeneous feasible regions, and outdoor-temperature uncertainty. The paper develops a continuous-time aggregation and reformulation framework integrated with hierarchical reserve dispatch, with case studies demonstrating aggregation accuracy, flexibility utilization, and uncertainty handling.

  • Problem

    Massive HVAC scheduling requires accurate aggregation of intra-hour flexibility despite computational complexity, neglected thermal dynamics, and outdoor-temperature uncertainty.

  • Method

    The paper combines a continuous-time HVAC aggregation model, affine transformation, distributionally robust uncertainty treatment, tractable reformulations, and hierarchical reserve dispatch.

  • Results

    Case studies demonstrate effectiveness and scalability in aggregation accuracy, intra-hour flexibility utilization, and reliable handling of outdoor-temperature uncertainty.

  • Takeaways & Limitations

    The framework supports scheduling massive HVACs for reserve provision while balancing aggregation accuracy, intra-hour flexibility utilization, computational tractability, and uncertainty risk.

Abstract

from arXiv · show

Heating, ventilation, and air conditioning (HVAC) loads, with their rapid response capabilities, can provide considerable intra-hour flexibility on the demand side for reserve provision in order to follow the fast variations of renewables. However, scheduling massive HVACs is challenging due to computation complexity and the uncertainty of outdoor temperature. In this paper, we first introduce a novel continuous-time (CT) aggregation model to reveal the potential intra-hour flexibility of HVACs. For accurate aggregation, a new affine transformation is designed to handle the heterogeneity in high-dimensional feasible region. Further, for reliable aggregation in practical environment, the outdoor temperature uncertainty is constructed by distributionally robust chance constrains and integrated into the aggregation model. Secondly, for the tractable calculation of the proposed CT aggregation model, a cascade of tailored reformulation techniques is proposed, including the Bernstein polynomial spline, polytope projection, and linearization transformation. Thirdly, a customized hierarchical dispatch framework is proposed via incorporating the proposed CT aggregation model into reserve provision in power system dispatch, so as to efficiently schedule massive HVACs to cope with the renewable uncertainty. Case studies verify the effectiveness and scalability of the proposed CT aggregation model in aggregation accuracy, intra-hour flexibility utilization, and uncertainty handling.

I. INTRODUCTION

The paper addresses the difficulty of accurately and reliably aggregating massive HVAC flexibility in continuous time under heterogeneous parameters and outdoor-temperature uncertainty. It proposes a CT aggregation model, tractable reformulations, and a hierarchical dispatch framework for reserve provision.

  • Motivation: Massive HVACs can provide intra-hour reserve flexibility because of their thermal inertia and rapid response capacities.Aggregating individual HVACs is necessary because each unit has limited capacity.
  • Research gaps: Exact aggregation is computationally intractable because heterogeneous HVAC feasible regions must be combined through their Minkowski sum.Existing approximations therefore trade computational efficiency against aggregation accuracy or disaggregation feasibility.
  • Research gaps: Discrete-time aggregation neglects intra-hour thermal dynamics, potentially underestimating flexibility needed for increasingly frequent intra-hour ramping events.The paper motivates extending polytope-based aggregation to a continuous-time formulation.
  • Research gaps: Ignoring outdoor-temperature uncertainty can miscalculate aggregated flexibility and cannot guarantee disaggregation feasibility in practice.The paper treats this uncertainty as part of reliable aggregation.
  • Proposed approach: The proposed framework combines CT aggregation with uncertainty modeling, tractable reformulations, and hierarchical dispatch for reserve provision under renewable uncertainty.The dispatch formulation schedules aggregated HVAC reserve alongside thermal-unit decisions and intra-day deployment.

II. CT AGGREGATION MODEL OF MASSIVE FLEXIBLE HVACS

The CT aggregation model represents individual HVAC thermal dynamics and operating limits as continuous-time feasible regions, then uses affine transformation to aggregate heterogeneous units. Indoor comfort and HVAC power bounds remain explicit constraints.

  • Affine aggregation: The model uses affine transformation to map a base feasible set into an approximation of each heterogeneous HVAC feasible region before aggregation.The paper’s notation distinguishes the base set, affine-transformed base set, and aggregated feasible sets.
  • Individual CT model: The model begins with a second-order equivalent thermal parameter model to represent each HVAC’s continuous-time thermal dynamics.The individual model is the basis for constructing each HVAC’s feasible region.
  • Operational constraints: Indoor temperature is constrained within the set-point tolerance to preserve user comfort.The bound is expressed around the indoor temperature set-point.
  • Operational constraints: HVAC cooling power is bounded by minimum and maximum power limits.These limits constrain the feasible operating trajectories of each unit.
  • Individual CT model: Each HVAC feasible region is expressed as an H-representation of a continuous-time polytope.The formulation is built from the individual HVAC dynamic and operational constraints.

B. Aggregation Model Based on Affine Transformation

The paper develops an affine-transformation inner approximation for aggregating heterogeneous HVAC feasible regions, extends it to outdoor-temperature uncertainty, and reformulates the continuous-time model for tractable dispatch. The resulting framework supports reserve provision by incorporating aggregated HVAC flexibility into hierarchical power-system dispatch.

  • Motivation and approximation: The exact Minkowski sum of heterogeneous HVAC feasible regions is computationally intractable, motivating an inner-approximation model that preserves disaggregation feasibility.Inner approximation avoids constraint violations during disaggregation, unlike outer approximation, while polytope models balance accuracy and computational efficiency.
  • Affine transformation: A base polytope is transformed for each HVAC using affine matrices and translation vectors, then aggregated through the approximate Minkowski sum of the transformed sets.The affine transformation is constrained to remain within each HVAC’s feasible polytope before aggregation.
  • Geometric adaptability: General affine transformations can reproduce the exact diamond in the two-dimensional example, whereas uniform or diagonal scaling produces less accurate square or rectangular approximations.The comparison illustrates the greater geometric adaptability of general affine transformations in high-dimensional feasible regions.
  • Uncertainty handling: Outdoor-temperature uncertainty is modeled through distributionally robust chance constraints using an allowable violation probability and an ambiguity set defined by empirical mean and covariance.The uncertainty is integrated because outdoor temperature affects the HVAC feasible region and ignoring it can prevent guaranteed disaggregation feasibility in practice.
  • Reformulation: Bernstein polynomial splines, polytope projection, and linearization transform the continuous-time aggregation problem into a tractable linear programming model.The spline representation maps continuous-time functions into finite-dimensional algebraic variables, while the final model is solvable by commercial solvers.
  • Power-system dispatch: The hierarchical dispatch framework formulates aggregated HVAC reserve scheduling as a two-stage continuous-time stochastic optimization problem that becomes a tractable mixed-integer linear program after reformulation.The first stage schedules reserve capacity and unit commitment; the second deploys thermal-unit and HVAC reserves under renewable-uncertainty scenarios.

C. Overall Hierarchical Dispatch Framework

The framework integrates aggregated HVAC flexibility into day-ahead reserve provision through four sequential processes, enabling parallel aggregation and response without iterative disaggregation optimization.

  • Framework overview: The day-ahead framework determines HVAC reserve provision and thermal-unit commitment through four sequential processes.It follows a hierarchical dispatch structure for power-system reserve provision.
  • Parameter Collection: HVAC aggregators collect managed-device parameters and construct the base set for aggregation.
  • Parallel Aggregation: Aggregators independently calculate affine aggregation matrices and submit the resulting dispatchable HVAC region to the system operator.
  • Dispatch Distribution: The system operator centrally computes dispatch results and distributes them to HVAC aggregators.
  • Parallel Response: Aggregators respond in parallel and allocate dispatch results to their HVACs without solving disaggregation optimization problems.Equation (23) directly disaggregates signals while maintaining feasible HVAC power.
  • Framework properties: The four processes are non-iterative and parallel-enabled, giving the framework low implementation barriers.

V. CASE STUDIES

Case studies use modified 6-bus and IEEE 118-bus systems to evaluate the aggregation model, with 5-minute forecast and dispatch data and heterogeneous HVAC parameters.

  • Study design: Case studies on modified 6-bus and IEEE 118-bus systems evaluate the proposed model’s effectiveness and scalability.Simulations use MATLAB 2021a and GUROBI 11.0.0.
  • 6-bus system: The 6-bus test system contains three thermal units, a 110MW wind farm, and 210MW rated total load.
  • 6-bus system: The 6-bus dispatch horizon spans 24 hours with 1-hour periods, while Fig. 5 provides 5-minute forecast curves and corresponding DT/CT profiles.
  • HVAC configuration: The 6-bus system includes 40 heterogeneous HVACs at bus 3 within one aggregator.The risk level is 1% and reserve deployment and capacity prices are 13$/MWh and 4$/MWh, respectively.

A. Validation of Aggregated Model for HVACs

The validation compares CT HVAC aggregation approaches against a direct-control benchmark, emphasizing the trade-off between flexibility, aggregation accuracy, disaggregation feasibility, and computational burden.

  • Benchmark: Case 4 is the benchmark because its comprehensive individual-HVAC model guarantees optimality but imposes heavy computational burden.The benchmark is not considered practical for HVAC scheduling.
  • Aggregation accuracy: Case 3 identifies the highest HVAC reserve capacity but has a 28% infeasible disaggregation rate because its outer approximation ignores parameter heterogeneity.Its larger-than-exact feasible region cannot guarantee disaggregation feasibility.
  • Disaggregation feasibility: Case 5 achieves 100% disaggregation feasibility while strictly respecting HVAC operational constraints, although its aggregation is conservative.Feasibility is guaranteed by containment constraint (17).
  • Flexibility representation: The proposed affine transformation retains more HVAC flexibility and produces larger adjustable capacity bounds than the other inner-approximation models.It adapts the base-set structure to the high-dimensional HVAC feasible region.
  • Sensitivity analysis: When heterogeneity exceeds 1.4, the proposed model’s deviation exceeds 5%, representing a tractability trade-off despite its lowest deviation overall.For homogeneous parameters, Case 2 and the proposed model identify optimal HVAC reserve capacity.
  • Computational performance: The proposed aggregation sharply reduces dispatch scale while obtaining a near-optimal solution, with slightly higher CPU time paid for greater aggregation accuracy.The authors characterize the model as accurate with relatively low computational complexity.

B. Advantage of CT Aggregation

The CT aggregation model captures intra-hour HVAC flexibility more effectively than the traditional DT version, improving reserve utilization and overall dispatch economy despite higher computation time.

  • Dispatch framework: The two-stage stochastic dispatch fixes day-ahead HVAC reserve and commitment decisions before scenario-based intra-day regulation deploys thermal and HVAC reserves.Real-time economic dispatch is simulated at 5-minute granularity after day-ahead optimization.
  • Day-ahead scheduling: The CT model prepares more HVAC reserve capacity and thermal generation initially, resulting in a slightly higher first-stage total cost than Case 5a.Case 5a denotes traditional DT aggregation using the proposed affine approximation.
  • Real-time operation: Ignoring intra-hour flexibility in Case 5a causes reserve shortages, more expensive thermal regulation, wind curtailment, and load shedding during real-time operation.Load shedding occurs from 13:20–13:50, while wind curtailment occurs during several fast net-load variation intervals.
  • Computational trade-off: The proposed model requires more CPU time than traditional DT aggregation because it contains more variables, reflecting its use of intra-hour flexibility.The additional computation is explicitly described as the cost of utilizing intra-hour flexibility.

C. Impact of Outdoor Temperature Uncertainty

The uncertainty analysis shows that ignoring outdoor-temperature uncertainty overestimates usable HVAC flexibility, whereas the proposed model supports reliable aggregation and tunes the economy–risk balance.

  • Practical reliability: Ignoring outdoor-temperature uncertainty produces more cost-effective reserve on paper but may yield dispatch results that are inapplicable under uncertain conditions.The uncertainty-free case identifies larger HVAC flexibility and lower total I+II cost.
  • Scenario testing: Monte Carlo testing evaluates one HVAC scheduling plan across 1000 actual outdoor-temperature scenarios using forecasted and realistic indoor-temperature trajectories.The forecast trajectory is shown as a solid blue line and scenario variations as gray lines.
  • Risk sensitivity: Increasing the risk level changes the feasible-region size and reduces system costs as the system obtains more aggregation flexibility.The feasible-region size is described as proportional to the total trace of Γ_aff,BB.
  • Overall outcome: The proposed model efficiently handles outdoor-temperature uncertainty while providing ε_m as a controllable measure for balancing economy and risk attitude.The authors present this as a regulation measure for practical applications.

D. Scalable Test

The 118-bus test demonstrates that the proposed aggregation remains computationally scalable while identifying more HVAC flexibility and achieving the best reported economic outcome among compared aggregation models.

  • Test system: The scalability test uses an IEEE 118-bus system with 54 thermal units, 4405 MW maximum load, and wind capacity scaled to 1600 MW.The system provides a larger dispatch setting for testing computation and flexibility identification.
  • Computational scalability: The benchmark cannot calculate an optimal solution within one day, whereas the proposed model solves in a reasonable CPU time using a much smaller optimization problem.Reported CPU times include 526.3 s for the proposed case and more than one day for the benchmark.
  • Flexibility identification: The proposed model identifies substantially more HVAC reserve than Case 1 and Case 2, whose capacities equal only 33.6% and 80.0% of Case 5, respectively.The comparison concerns total HVAC reserve capacity.
  • Economic outcome: Case 5 achieves the lowest expected wind-curtailment cost and the best economy among the compared aggregation cases.The reported expected wind-curtailment costs are 19.7, 15.3, and 6.0 k$ for Cases 1, 2, and 5, respectively.
  • Problem size: The proposed model uses 519,742 variables, compared with 4,225,512 for the direct-control benchmark.The reduced optimization scale supports the reported computational advantage.

VI. CONCLUSIONS

The paper develops a continuous-time aggregation and dispatch framework for uncertain massive HVAC flexibility, then evaluates its accuracy, utilization, scalability, and uncertainty handling. Future work targets greater parameter heterogeneity and multiple source uncertainties.

  • The proposed framework combines continuous-time HVAC aggregation, outdoor-temperature uncertainty modeling, linear reformulation, and hierarchical dispatch for reserve provision.
  • Case studies demonstrate effectiveness and scalability in aggregation accuracy, intra-hour flexibility utilization, and outdoor-temperature uncertainty handling.The studies use modified 6-bus and IEEE 118-bus systems.
  • Future work will address greater parameter heterogeneity to improve aggregation accuracy and extend the model to multiple source uncertainties.

A. Proof of Inner-approximation

The proof constructs an affine-approximation aggregated set from a base set and expresses the relevant sets using polytope H-representations and containment relationships. Auxiliary matrices support the resulting formulation.

  • The affine-approximation aggregated set is constructed from a base set and affine transformation variables.
  • The set is expressed in polytope H-representation using coefficient matrices and an inequality description.
  • The proof uses a containment relationship between the affine approximation and the relevant set.
  • Auxiliary matrix variables are introduced in the formulation, and combining intermediate relations yields the stated constraints.

C. Proof of the Reformulation of DRCCs

The reformulation proof applies a Bonferroni approximation to distribute constraint violation probability and converts the resulting uncertain constraints into an equivalent deterministic form. The supplied passages also reference wind-power scenarios and continuous-time dispatch.

  • The Bonferroni approximation assigns each constraint row a fixed violation probability ε_m.
  • The reformulation separates constraints without uncertain parameters from those requiring an equivalent deterministic representation.
  • The deterministic form uses coefficient matrices, affine terms, auxiliary variables, and row-wise inequalities.
  • The per-row violation probability is defined using ε_m = ε_Ldim ⁄ Ldim, where Ldim is the number of elements in M_u.
  • The supplied figure text identifies stochastic scenarios of wind power and wind power curves in continuous-time dispatch.
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