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Leveraging Time-Causal State Variable Aggregation for Real-Time Schedule of Massive Air Conditioners

Jingguan Liu, Xiaomeng Ai, Shichang Cui, Xizhen Xue, Shengshi Wang, Jiakun Fang, Jinyu Wen, Yang Shi

arXiv:2609.02410v1eess.SY

TL;DR

Massive AC scheduling must address strong temporal coupling and time-causal uncertainty while operating efficiently in active distribution networks. The paper proposes TCA-ADP, which combines time-causal state aggregation with offline value-function learning and online disaggregation; case studies report favorable accuracy, uncertainty handling, and scalability. The approach balances economic performance with tractability for massive AC scheduling.

Abstract

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Air conditioner (AC) loads offer promising flexibility for active distribution networks to manage uncertainties, such as those in renewable energy generation, electricity prices, and load demand. However, real-time scheduling of ACs is challenging due to their massive temporal coupling constraints and time-causal uncertainties. To address this, a novel time-causal aggregation-based approximate dynamic programming (TCA-ADP) algorithm is proposed for efficient scheduling. The time-causality requirements for aggregating state variables are first analyzed to align with the real-time sequential decision-making process. Subsequently, an enhanced aggregation model is developed to ensure both high accuracy and adherence to time causality. The aggregation process is further reformulated as a linear program to optimize aggregation parameters and enable tractable computation. Accordingly, the TCA-ADP leverages aggregated state variables to approximate the value function as a new way, balancing computational efficiency and economy against the large value function space of massive ACs. By training the value function offline using historical data, the TCA-ADP efficiently achieves near-optimal real-time scheduling of massive ACs through parallel and closed-form disaggregation. Case studies demonstrate the effectiveness and scalability of the TCA-ADP, highlighting its aggregation accuracy, uncertainty handling, and the trade-off between economy and tractability.

A. Abbreviations:

The paper frames massive ACs as flexible resources for active distribution networks, while addressing uncertainty and real-time scheduling constraints through TCA-ADP.

  • AC loads provide demand-side flexibility for managing renewable intermittency and real-time net-load changes in active distribution networks.Source: §I · p. 2
  • Real-time scheduling must handle strong inter-period coupling in massive AC populations while decisions remain dependent only on currently observed information.Source: §I · p. 2
  • Existing uncertainty-aware scheduling methods may be limited online because they require global uncertainty information and can violate time causality.Source: §I · p. 2
  • ADP uses offline historical-data learning and sequential single-period optimization to obtain near-optimal real-time scheduling with trained value functions.Source: §I · p. 2
  • TCA-ADP introduces time-causal aggregation, an enhanced aggregation model, and linear-program parameter optimization for efficient massive-AC scheduling.Source: §I · pp. 2–3
  • The paper reports a trade-off between aggregation accuracy, uncertainty handling, economy, and tractability, with applicability also noted for other large-scale flexible resources.Source: §I · p. 3

III. TCA-ADP SCHEDULING ALGORITHM

The scheduling problem is reformulated as a sequential MDP whose states, decisions, uncertainty observations, and transitions support real-time optimization. ADP then replaces the intractable expectation calculation with a post-decision value-function approximation.

  • The TCA-ADP algorithm comprises MDP reformulation, time-causal aggregation, parameter determination, value-function fitting, and practical application.Source: §III · p. 3

MDP Reformulation

Post-Decision Value Function

  • Classical dynamic programming is computationally impractical in stochastic environments because evaluating all uncertainty scenarios creates a curse of dimensionality.Source: A. MDP Reformulation Under Bellman… · p. 4
  • The post-decision state represents the system after current decisions but before next-period uncertainty, allowing ADP to avoid optimization over high-dimensional random factors.Source: A. MDP Reformulation Under Bellman… · p. 4

MDP Reformulation

B. Time-Causal State Variable Aggregation Model

The paper aggregates massive AC flexibility sets to approximate value functions while addressing the large state-variable space and preserving real-time feasibility.

1) Time-Causality Requirement:

Time-causal aggregation aligns approximate AC flexibility sets with sequential decisions by preserving the temporal structure of individual constraints. A diagonal affine transformation improves geometric adaptability without breaking that structure.

Time-Causality Requirement

  • A time-causal set has constraints in period t involving only current-period and next-period variables, matching an MDP-style sequential process.Source: 1) Time-Causality Requirement · pp. 4–5
  • Individual AC sets and their exact aggregate are naturally time-causal, so the approximate aggregate must preserve time causality for real-time implementation.Source: 1) Time-Causality Requirement · p. 5

Aggregation Model

  • The proposed model replaces uniform scaling with a general affine transformation to improve geometric adaptability while retaining inner-approximation feasibility.Source: 2) Aggregation Model · p. 5
  • A diagonal affine matrix scales dimensions independently, and its parameters are determined so the transformed base set closely approximates each original AC set.Source: 2) Aggregation Model · p. 5
  • The aggregated variable shares the dimension of the individual AC variable, while the base-set construction uses predefined parameters averaged across ACs at a node.Source: 2) Aggregation Model · p. 5

3) An Illustrative Toy Example:

The toy example compares uniform scaling, diagonal affine transformation, and dense affine transformation for approximating a three-period time-causal set. The proposed diagonal method balances accuracy with causality.

2) Objective Function Reformulation:

The aggregation parameters are optimized by maximizing an inner approximation and then reformulating the nonlinear problem as a tractable linear program. Projection removes redundant variables and resolves dimension loss.

Objective Function Reformulation

D. Value Function Fitting Around Aggregated State Variables

The method condenses massive AC state variables into aggregated variables and fits their value function with convex piecewise-linear functions. Offline-trained segment slopes enable a linear-program reformulation of the Bellman decision step.

Aggregation

  • Aggregated state variables condense massive AC states into two sets, enabling efficient value-function fitting and reformulation of scheduling decisions.The aggregated state variables are used to replace the original massive-AC constraints and decision representation.Source: D. Value Function Fitting Around… · p. 7

Value-function fitting

  • Convex piecewise-linear functions fit the value function with monotonically increasing slopes, combining fitting accuracy with low computational burden.Each aggregated state variable is divided into N_S segments, with segment slopes r_a and r_m constrained to preserve convexity.Source: D. Value Function Fitting Around… · p. 7
  • Segment quantities v_a and v_m are optimized, while slopes r_a and r_m are predetermined and the post-decision state accumulates quantities across segments.Later segments receive no value until earlier segments are fully filled, consistent with convexity.Source: D. Value Function Fitting Around… · p. 7

Linear reformulation

  • After selecting segment slopes, the Bellman equation is recast as a linear program whose solution provides near-optimal decision variables.The linear program treats the aggregated post-decision state and segment quantities as decision variables.Source: D. Value Function Fitting Around… · p. 7

Offline training

  • Historical-data training updates segment slopes through sample slope estimation and leveling, maintaining convexity during offline iterations.Each iteration generates a Monte Carlo training scenario, solves the decision problem, updates slopes, and advances the system state.Source: D. Value Function Fitting Around… · p. 7

E. Practical Application Framework

The practical framework separates aggregation and value-function fitting offline from real-time scheduling online. Online decisions use trained value functions and are disaggregated to individual ACs through parallel, closed-form calculations.

Process 1: Offline State Variable Aggregation

  • Offline aggregation computes the approximate aggregate AC set, with parallel calculation of aggregation parameters providing scalability for massive AC populations.The aggregation parameters of different ACs are orthogonal, allowing parallel computation.Source: E. Practical Application Framework · p. 8

Process 2: Offline Value Function Fitting

  • Offline value-function fitting uses historical data and Algorithm 1 to train the value functions of aggregated state variables.The fitted functions are subsequently used in the online scheduling process.Source: E. Practical Application Framework · p. 8

Process 3: Online Real-time Application

  • In each online period, the algorithm observes current information, solves the aggregate scheduling problem with trained value functions, updates the state, and disaggregates decisions.This sequence follows the real-time application procedure period by period.Source: E. Practical Application Framework · p. 8

Online disaggregation

  • Individual AC power profiles are obtained directly as explicit functions of the optimal aggregate power profile, avoiding an additional optimization problem.The closed-form disaggregation is performed after the aggregate scheduling problem is solved.Source: E. Practical Application Framework · p. 8
  • Parallel and closed-form disaggregation supports scalable, fast, and feasible power allocation while inner approximation guarantees individual AC operational constraints.The case studies use modified 8-bus and IEEE 123-bus systems to evaluate effectiveness and scalability.Source: E. Practical Application Framework · p. 8, §IV · p. 8

A. Validation of TCA-ADP Algorithm

Case studies compare TCA-ADP with hindsight, uniform-scaling, unaggregated, myopic, and MPC approaches. Results show accurate aggregation, feasible disaggregation, economically responsive scheduling, and a cost–participation trade-off for AC flexibility.

Training performance

  • Case 1 reaches a 3.03% solution gap after aggregation, while unaggregated Case 1b reaches 0.76% but requires 182.45 seconds per iteration and converges slowly.Aggregated Cases 1 and 1a converge after 500 iterations with each iteration taking only a few seconds.Source: 1) Real-time Scheduling Results of ADN · p. 9

Real-time scheduling

  • TCA-ADP shifts AC demand toward low-price periods and reduces demand during high-price periods, coordinating AC flexibility with generation and transactions.This price-responsive behavior is reported in the Case 1 ADN scheduling results.Source: 1) Real-time Scheduling Results of ADN · p. 9

Real-time disaggregation

Sensitivity analysis

  • As AC-response price increases, ADN demand for response energy decreases and real-time scheduling cost rises; at 1 $/kWh, AC response is no longer used.Lower prices make AC flexibility more cost-effective, while excessively low prices could reduce user participation.Source: 1) Real-time Scheduling Results of ADN · p. 10

Aggregation accuracy

  • TCA-ADP produces near-optimal scheduling outcomes close to Benchmark and consistently identifies more response energy than uniform scaling.The authors summarize these results as favorable aggregation accuracy and comparable economic scheduling results.Source: 1) Real-time Scheduling Results of ADN · p. 10

1) Deterministic Scenario:

The TCA-ADP algorithm achieves near-optimal scheduling under uncertainty while remaining computationally tractable and scalable for massive AC populations. Case studies show that aggregation improves convergence and preserves online efficiency, with a solution gap around 4%.

Deterministic scheduling results

  • TCA-ADP achieves a global near-optimal solution with the lowest 3.03% solution gap, compared with 19.62%, 15.12%, and 10.13% for Myopic and MPC cases.Source: 1) Deterministic Scenario · p. 10
  • Myopic scheduling exhausts stored thermal energy during lower-price periods, leaving little flexibility when prices rise and producing the poorest economic performance.Source: 1) Deterministic Scenario · p. 10
  • MPC improves over Myopic scheduling by anticipating near-future price increases and storing thermal energy in advance.Source: 1) Deterministic Scenario · p. 10

Uncertainty handling

  • TCA-ADP embeds uncertainty information in trained value-function slopes, enabling global near-optimal real-time scheduling without exact future predictions.Source: 1) Deterministic Scenario · p. 10
  • After 1,200 training iterations, TCA-ADP approaches a 3% solution gap and achieves an average 3.5% gap across 200 real-time testing scenarios.Source: 2) Uncertain Scenarios · p. 11

Scalability and tractability

  • In the 123-bus system with 1,000 ACs, TCA-ADP reaches around 4% after 1,200 iterations and requires only 6.8 s per training iteration versus about 9,000 s for unaggregated ADP.Aggregation reduces the optimization count from 96,000 to 288 problems per iteration.Source: C. Scalability Test · p. 11
  • TCA-ADP maintains the smallest testing gap with average online CPU time of 0.52 s per scenario and nearly constant iteration time as AC scale increases.Source: C. Scalability Test · pp. 11–12

Trade-off

APPENDIX

The appendix recasts individual AC thermal dynamics and temperature constraints into equivalent flexibility-set representations. These transformations provide the H-representation parameters used in the aggregation formulation.

D. Derivation of Disaggregation Results

The disaggregation derivation represents aggregate AC power as a sum of transformed individual profiles and then recovers individually feasible powers through an explicit inverse transformation.

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