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Privacy-Preserving Coordinated Operation of Multi-Player Industrial Network Using Secure Aggregation

Akshdeep Singh Ahluwalia, Zachary Wilson, Jeffrey E. Arbogast, Can Li

arXiv:2609.16402v1cs.CE

TL;DR

Coordinated demand response among independent industrial stakeholders can use network flexibility, but centralized scheduling exposes proprietary models and distributed exchanges can still reveal private parameters. This paper combines secure aggregation with ICC–ADMM and two-phase revenue sharing; in a 31-day simulation it reduces network cost by 19.77% versus decentralized operation and comes within 3.08% of the centralized benchmark.

  • Problem

    Coordinating industrial demand response requires exchanging information that may reveal proprietary plant models and parameters, while limited disclosure through distributed optimization alone provides no formal privacy guarantee.

    Source: §1 · p. 3, §4.5 · p. 28

  • Method

    The framework combines ICC–ADMM with Diffie–Hellman neighbor masking for secure aggregation and redistributes realized savings through a two-phase revenue-sharing mechanism.

    Source: §1 · p. 5, §7 · p. 45

  • Results

    19.77% lower total network cost than decentralized operation and a full-month cost within 3.08% of the centralized social-welfare benchmark were achieved in a 31-day rolling-horizon simulation.

    Source: §7 · p. 46

  • Takeaways & Limitations

    Revenue sharing ensured net cost reductions for all plants relative to their status quo in the simulated network.

    Source: §7 · p. 46

    Under all-but-one collusion, revealed iterates combined with side information can support inference of a remaining plant’s private objective parameters.

    Source: §6 · p. 45, §7 · p. 46

Abstract

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Electrified chemical industries with operational flexibility can reduce operating costs by shifting production and distribution decisions in response to time-varying electricity prices. However, chemical plants operate within process networks where coordinated demand response can exploit flexibility across multiple stakeholders. Centralized coordination requires access to stakeholders' local scheduling models and proprietary operational data, often incompatible with data-privacy requirements. Distributed optimization with an independent central coordinator (ICC) avoids direct model sharing, but iterative exchange of coupling variables can still reveal private model parameters. We propose a privacy-preserving distributed coordination framework for coordinated demand response in industrial networks. The framework integrates secure aggregation with an ICC-based alternating direction method of multipliers (ADMM) algorithm, so plant-level messages are numerically masked and become useful to the ICC only after aggregation. We test the framework on a multi-plant industrial gas network in which three air-separation units jointly schedule production and shipments to shared customer regions. To support stable participation, we incorporate a two-phase revenue-sharing mechanism that reallocates savings so every plant improves relative to its decentralized status quo. In a 31-day rolling-horizon simulation with synthetic data representing heterogeneous electricity prices and demand, the coordinated policy reduces total network cost by 19.77% relative to decentralized operation and achieves a full-month cost within 3.08% of a centralized social-welfare-maximization benchmark. We further quantify a conservative worst-case collusion mode, showing how unmasked iterates and auxiliary information can expose private objective parameters.

1. Introduction

Coordinated demand response can exploit flexibility across connected plants, but independent stakeholders need both privacy for their local models and incentives to share coordination gains. The paper combines secure aggregation with ICC–ADMM coordination and a second-phase revenue-sharing mechanism, then evaluates the approach on an industrial gas network and under worst-case collusion.

Motivation and privacy gap

  • ADMM avoids sharing complete plant models, but communicated iterates can still reveal private feasible regions and cost parameters.Prior work demonstrated parameter inference under adversarial attacks, a risk not previously assessed in coordinated demand response studies.Source: §1 · p. 4

Privacy-aware coordination

  • Plants numerically mask shipping estimates so masks cancel upon aggregation, allowing the ICC to recover only quantities needed for ADMM updates.The proposed scheme uses Diffie–Hellman-based neighbor masking.Source: §1 · pp. 4–5

Revenue sharing

  • Realized coordination savings are redistributed in a second phase using Nash bargaining.This follows secure-aggregation-based ADMM coordination in the first phase.Source: §1 · p. 5
  • After secure-aggregation-based ADMM achieves coordination in the first phase, realized savings are redistributed in a second phase using Nash bargaining to provide all participants sufficient incentive to cooperate.Source: §1 · pp. 4–5

Case study

  • The industrial-scale case study jointly optimizes production and shipping over a 31-day horizon while preserving privacy during coordination.It uses mixed-integer linear programming scheduling models for a multi-plant industrial gas network.Source: §1 · p. 5

Collusion analysis

  • The privacy analysis evaluates an extreme, practically unlikely case in which all but one stakeholder collude against the remaining participants.This stress test lies outside the stated protection regime, which requires at least two plants to remain honest.Source: §1 · p. 5

2. Related Works

Prior work studies demand response within individual plants and coordinated production across industrial supply chains, while distributed methods enable cooperation among independent stakeholders. The reviewed literature leaves unresolved whether exchanged coordination iterates protect proprietary plant information.

Industrial demand response

  • Air-separation, chlor-alkali, and steel studies model demand response through storage, switching, or operating-mode choices, with process-specific constraints on flexibility.Examples include temperature-limited response speed in membrane cells and trade-offs between low-price production, energy efficiency, and electrode consumption.Source: §2 · pp. 6–7

Multi-plant coordination

  • Multi-plant industrial-gas planning coordinates production, inventories, and routing under time-varying electricity costs or customer-service requirements.The cited frameworks mainly concern integrated supply-chain planning rather than explicit electricity-market demand response.Source: §2 · p. 7

Independent stakeholders

  • Centralized scheduling for independent stakeholders can require disclosure of proprietary models and data while creating difficult large-scale optimization problems.These confidentiality and computational concerns motivate distributed architectures.Source: §2 · p. 8

Distributed coordination

  • Distributed ADMM studies coordinate stakeholders through selected coupling variables rather than complete process models, including networks coupled through material flows.Earlier work also considered customer compensation so participants improve relative to their status quo.Source: §2 · p. 8

Privacy gap

  • Limited model disclosure does not itself guarantee privacy because exchanged ADMM iterates may encode private constraints, operating limits, and cost parameters.Prior studies showed that suitable side information and adversarial settings can enable parameter inference.Source: §2 · pp. 8–9

Research contribution

  • The paper addresses the unresolved exposure of sensitive plant data during coordinated demand response with a secure-aggregation-based approach.The reviewed literature presents coordination as economically compelling but requiring potentially revealing information exchange.Source: §2 · p. 9

3. Background

The ICC needs aggregate plant messages for coordination, but individual schedules may reveal commercially sensitive operational information. The section introduces secure aggregation, compares its privacy and communication trade-offs, and explains its fit with ADMM.

3. Background

  • The ICC needs the aggregate demand vector for ADMM updates, while individual messages can expose plant schedules, operating limits, outages, ramping, and flexibility.Secure aggregation lets the coordinator recover the aggregate without directly observing individual plant vectors.Source: §3 · pp. 9–10

3.1. Secure aggregation methods

  • Secure aggregation includes neighbor masking, homomorphic encryption, and secret sharing; the paper adopts masking because it is suited to repeated coordination.Neighbor masks cancel under summation, leaving the desired aggregate while concealing individual messages.Source: §3.1 · pp. 10–11

3.2. Threats models

  • The threat models include honest-but-curious participants or coordinator, colluding entities, and passive outside eavesdroppers.The honest-but-curious parties follow the protocol but may log intermediate information; outside eavesdroppers do not alter messages.Source: §3.2 · pp. 14–15

3.3. Secure aggregation against threat models

  • Neighbor masking and Shamir tolerate collusion by up to |P| −2 plants, while conventional Paillier relies on a trusted decryptor; threshold Paillier with τ = |P| removes that single-decryptor bottleneck.All three primitives protect against honest-but-curious behavior and passive eavesdropping when implemented correctly.Source: §3.3 · pp. 16–17

3.4. Secure aggregation for iterative coordination

  • Neighbor masking requires one online stage and |P| transmissions per aggregation step; Paillier and coordinator-mediated Shamir require three stages.The comparison excludes one-time setup costs; the paper selects masking to minimize repeated online interaction.Source: §3.4 · pp. 17–18, Table 2 · p. 18

3.5. Intuition behind using neighbor masking in ADMM

  • ADMM exchanges shipment iterates and uses aggregated quantities for updates, but repeated messages may leak private plant parameters; neighbor masking preserves the aggregate while concealing individual iterates.The paper treats the |P| −1 collusion case as a conservative limitation benchmark, beyond the stated protection for up to |P| −2 colluding plants.Source: §3.5 · p. 19

4. Methods

The methods formulate plant-level and network-wide scheduling and use ICC-based ADMM with neighbor masking to select feasible, status-quo-improving coordinated schedules while keeping plant decisions local and limiting the ICC to aggregate quantities. A transfer mechanism allocates coordination savings to support plant participation.

4.1. Problem Setting

  • Because products are substitutable across plants, coordination can reallocate production and shipments toward plants’ lower-cost or lower-carbon electricity hours while meeting customer demand.The framework assumes common product portfolios and customer regions, with electricity prices treated as exogenous.Source: §4.1 · p. 20

4.2. Status quo optimization

  • Each plant independently solves a mixed-integer status-quo scheduling problem, with feasibility determined by plant constraints and initial operating state, while electricity prices affect cost but not feasibility.Plant-side constraints include production rates, capacities, material balances, bounds, and inventory dynamics.Source: §4.2 · p. 21

4.3. Social welfare optimization

  • The social-welfare formulation coordinates plants to satisfy total network demand, but is solved distributively because centralized optimization would require sharing plant models.The paper uses ICC-governed ADMM with neighbor masking instead.Source: §4.3 · pp. 21–22

4.4. Distributed optimization using secure aggregation

  • Neighbor masking lets the ICC recover shipment aggregates while concealing individual plant vectors, and local copies of its dual variable keep coordinator operations purely aggregative.Pairwise masks cancel under summation; the ICC observes only masked vectors and the aggregates required by the algorithm.Source: §4.4 · pp. 23–24, §4.5 · p. 28

4.5. Step-by-step description of Algorithm 1

  • Plants solve local scheduling subproblems, while the ICC aggregates masked messages and selects only feasible network schedules that improve on the status quo.Candidate offset heuristics correct aggregate shipment mismatch, but plants must check each candidate’s local feasibility and cost; production schedules and terminal states remain local.Source: §4.5 · pp. 24, 27, 28
  • The ICC evaluates an upper-bound heuristic only when every plant confirms its assigned shipment is feasible, then retains the best feasible candidate improving on status-quo cost.Plants commit the locally stored candidate associated with the selected heuristic when the incumbent is updated.Source: §4.5 · p. 29
  • Plants adjust ADMM penalties by residual balancing and rescale dual variables after penalty changes; termination requires residual tolerances and an incumbent cost below status quo.Penalty updates occur periodically, increasing or decreasing the penalty according to the relative primal and dual residuals.Source: §4.5 · pp. 30–31
  • Neighbor masking reveals only aggregate quantities needed for ADMM and heuristic construction, protecting against an honest-but-curious ICC, eavesdroppers, and collusion by up to |P|−2 plants.The stated guarantee holds while at least two plants do not collude.Source: §4.5 · p. 31

4.6. Revenue sharing and fair payoff allocation

  • When coordination produces positive total savings, Nash bargaining yields a budget-balanced, individually rational transfer with equal surplus for each plant.The two-phase mechanism quantifies total savings and distributes them through a closed-form transfer rule.Source: §4.6 · pp. 32–33

5. Case study and results

A three-plant industrial gas network is simulated over 31 rolling-horizon days to benchmark coordinated scheduling against decentralized and centralized operation. Coordination reduces full-month network cost, while revenue sharing, plant-level cost shifts, and day-wise schedule gaps characterize its benefits and performance.

5.1. Simulation setup

  • Three air-separation units jointly schedule production and shipments to three customer regions in a 31-day synthetic-data simulation with heterogeneous electricity prices and rolling seven-day forecasts.Each day, only the first 24 hours of the week-ahead schedule are implemented.Source: §5 · p. 33, §5.1 · pp. 34–35

5.2. Simulation benchmarking results

  • 19.77% lower total network cost than the status quo, with full-month cost within 3.08% of the centralized social-welfare trajectory.The comparison comes from the 31-day closed-loop rolling-horizon simulation.Source: §5.2 · p. 36, Figure 4 · p. 36
  • After revenue sharing, all plants reduce costs versus their status quo: ASU1 by 23.93%, ASU2 by 36.86%, and ASU3 by 12.07%.Before sharing, the UB schedule raises ASU2 costs by 102.43% while reducing ASU3 costs by 65.87%.Source: §5.2 · pp. 37–38, Table 3 · p. 37
  • 28.77% lower network operating cost accompanies essentially unchanged shipping cost (-0.49%), while the plant-level decomposition shows ASU3 operating cost down 69.34% and ASU2 up 137.17%.The cost shifts indicate production reallocation away from the high-price ASU3 toward underutilized ASU2.Source: Table 4 · p. 37, §5.2 · p. 38

5.3. ICC-ADMM algorithm performance results

  • The day-wise state-matched UB gap ranges from 4.55% to 19.33%, averaging 8.87%, but later-horizon suboptimality is partly replaced by daily re-optimization.The state-matched gap compares each feasible week-ahead schedule with a centralized optimum initialized from the same plant states; it differs from full-month unrealized savings.Source: §5.3 · p. 39, Table 5 · p. 39
  • On day 4, the smallest state-matched gap was 4.55%; the ICC began observing nontrivial feasible upper bounds around iteration 16.Figure 5 illustrates representative convergence for that day.Source: §5.3 · p. 39, Figure 5 · p. 40

6. Adversarial attack under worst-case collusion

The section tests how an extreme collusion regime can use leaked ADMM iterates and side information to recover private plant cost parameters. Recovery improves primarily with observations across different days, with approximately a month of exposure required for high accuracy.

Threat model

  • When the ICC colludes with all but one plant, colluders can reconstruct the remaining plant’s masks and recover its unmasked iterates.This is a conservative stress test outside the stated protection regime, which requires at least two non-colluding plants.Source: §6 · p. 40

Adversarial recovery model

  • The adversary fits private cost coefficients by minimizing disagreement between leaked ADMM behavior and trajectories induced by candidate costs under its approximate local model.The method is structured as empirical risk minimization because the adversary lacks the target’s full feasible region and operational trajectories.Source: §6 · pp. 41–42

Adversary side information

  • The attack assumes revealed mode schedules, day-boundary inventories, hourly production bounds, and cost ratios to make recovery identifiable and tractable.Without structural constraints such as known cost ratios, the ERM can admit solutions that under-estimate fixed costs.Source: §6 · pp. 42–43

Adversarial attack setup

  • The leakage experiment samples up to five iterations from each day’s final ten ADMM iterations and evaluates 100 feasible randomized realizations per setting.It varies both the number of attacked days and the number of leaked iterates per day.Source: Figure 6 · p. 41, §6 · pp. 43–44

Attack outcome and interpretation

  • Approximately one month of exposure is required for high-accuracy cost recovery across the tested rates of one to five leaked iterates per day.Revealing iterates across distinct days improves recovery more than adding correlated iterates within a day.Source: §6 · pp. 44–45

7. Conclusion

The paper combines privacy-aware ICC–ADMM coordination with revenue sharing and evaluates the approach in a three-plant industrial gas network. Its conclusion reports lower network cost than decentralized operation, near-centralized monthly cost, and residual privacy risk under extreme collusion; Appendix A describes two alternative secure-aggregation schemes.

7. Conclusion

  • Diffie–Hellman neighbor masking lets the ICC recover aggregate quantities for ADMM updates and upper-bound construction while concealing individual plant messages.Because convergence guarantees do not generally hold for the resulting nonconvex mixed-integer subproblems, the framework uses secure-aggregation-compatible upper-bound heuristics.Source: §7 · p. 45
  • The 31-day, three-plant simulation reduced total network cost by 19.77% versus decentralized operation and placed full-month cost within 3.08% of the centralized social-welfare benchmark.The day-wise state-matched upper-bound gap averaged 8.87%, and revenue sharing ensured net cost reductions for all plants.Source: §7 · p. 46
  • If all but one plant collude, they can reconstruct the remaining plant’s unmasked iterates and combine them with side information to infer private objective parameters.The conclusion identifies differential privacy, broader heuristics, and larger-network evaluation as directions for future work.Source: §7 · p. 46

Appendix A. Homomorphic Encryption and Secret Sharing for Secure Aggregation

  • Appendix A presents Paillier encryption and Shamir secret sharing as alternatives that let the ICC recover aggregate messages without directly observing individual plant contributions.The approaches differ in computation, communication, trust assumptions, and collusion robustness.Source: Appendix A · p. 47
  • Threshold Paillier distributes decryption across authorized parties but requires |P|+2τ online transmissions, while the Shamir workflow requires |P|(|P|−1) share-forwarding transmissions plus |P| aggregate-share uploads.Both protocols add communication structure around recovery of the aggregate rather than individual plant messages.Source: Appendix A · pp. 50, 52

Appendix B. ADMM Subproblems and Residuals

Appendix B spells out the plant and coordinator ADMM updates and residuals, while Appendix C describes heuristics for allocating aggregate mismatch to form feasible upper-bound candidates. The coordinator’s update depends on aggregate shipments, and the heuristic variants use splits that remain compatible with aggregate-only information.

Appendix B. ADMM Subproblems and Residuals

  • Appendix B specifies the plant and coordinator subproblems for the ADMM iterations and the residual expressions used for stopping and hyperparameter tuning.The plant and coordinator perform separate updates, followed by scaled dual-variable updates.Source: Appendix B · p. 53, Appendix B.1 · p. 53

Appendix B.1. Scaled augmented Lagrangian and ADMM updates

  • Each plant solves a mixed-integer quadratic x-update locally, while the ICC updates shipment copies and the scaled dual variables.The local plant problem is solved with an off-the-shelf MIP solver.Source: Appendix B.1 · p. 53

Appendix B.2. Closed-form coordinator update

  • The closed-form coordinator update applies a common shift to each plant’s shipment copy, which the ICC computes from the aggregate shipment quantity.The resulting coordinator minimizer is unique for the strictly convex quadratic objective.Source: Appendix B.2 · pp. 54–55

Appendix B.3. Two-block form and residuals

  • The primal residual tracks constraint mismatch, while the dual residual tracks changes in local shipment copies.The dual residual approaches zero as shipment copies and their aggregate stabilize.Source: Appendix B.3 · p. 56

Appendix C. Upper-bound heuristics for allocating the aggregate mismatch

  • Upper-bound heuristics allocate the aggregate network-balance mismatch to plant-side candidate shipment targets, which are then certified by plant-side subproblems.The appendix also describes an instance-dependent heuristic used in the experiments.Source: Appendix C · p. 56

Appendix C.1. Extending AssignToTwoPlants to AssignTomPlants

  • AssignToTwoPlants generalizes to allocating mismatch across a selected subset of plants, with equal, single-plant, and two-plant splits as special cases.Candidate subsets may be restricted using engineering considerations while retaining the ICC’s aggregate-only requirement.Source: Appendix C.1 · pp. 56–57

Appendix C.2. Non-uniform split ratios and capacity-guided splits

  • Non-uniform splits can weight mismatch by plant attributes or electricity prices; the implemented price-weighted rule assigns more adjustment to plants with lower prices.Capacity-guided weights are another option, using exogenous capacity information without requiring individual shipment vectors at the ICC.Source: Appendix C.2 · pp. 57–58

Appendix D.1. Feasibility of budget-balanced transfers

Appendix D characterizes when budget-balanced transfers can make coordinated costs acceptable to every plant, then gives the Nash-bargaining transfer rule. Strict aggregate savings permit strict improvement for every plant.

Appendix D.1. Feasibility of budget-balanced transfers

  • Budget-balanced transfers can leave every plant no worse than its status quo when total coordinated savings cover the total deficits; strict surplus permits strict improvement for all.The feasibility argument partitions plants into those with savings and those with deficits, then reallocates surplus to cover deficits.Source: Appendix D.1 · pp. 59–60

Appendix D.2. Nash bargaining solution under budget balance

  • With positive total savings, the Nash-bargaining solution equalizes each plant’s post-transfer surplus relative to its status quo.The transfers are budget-balanced and maximize the product of plant surpluses.Source: Appendix D.2 · pp. 60–61

Appendix E. ADMM implementation details

The appendix reports the ADMM settings used across simulation days and describes targeted changes on days with instability and poor convergence.

Appendix E. ADMM implementation details

  • The algorithm uses a maximum of 30 iterations and identical primal and dual stopping tolerances of 0.01 across all simulation days.Source: Appendix E · p. 61
  • Penalty updates occur every iteration, while the initial penalty varies across days from 1.68 to 2.43, with mean 1.96 and standard deviation 0.195.The authors describe the initial penalty as tightly concentrated around 2.Source: Appendix E · p. 61
  • For most days, the ADMM configuration uses β = 0.75, τ = 1.5, θ = 10, ρ0 = 1.9, ρf = 1, tolerances of 0.01, and Kmax = 30.Source: Appendix E · p. 61
  • On days 23 and 31, the authors use aggressive penalty growth and lower primal-update damping in response to unusually high UB gaps of approximately 18%–20%.The intervention aims to tighten primal feasibility and increase the influence of the latest primal update.Source: Appendix E · pp. 61–62

Appendix F.1. Day-wise state-matched UB gap

Figure F.7 reports a day-wise gap between ADMM-derived feasible upper bounds and centralized social-welfare optima, matched on the ADMM-realized plant states.

Appendix F.1. Day-wise state-matched UB gap

  • The day-wise UB gap compares each ADMM-derived best feasible upper bound with a centralized social-welfare optimum initialized from the same ADMM-realized plant states.Source: Appendix F.1 · p. 62, Figure F.7 · p. 62
  • Matching the initialization states makes the comparison conditional on the plant states realized by the ADMM-UB trajectory for that day.Source: Appendix F.1 · p. 62
  • The figure presents the gap by simulation day, with summary statistics noted in its legend and in main-text Table 5.Source: Appendix F.1 · p. 62

Appendix F.2. Heuristic upper-bound evolution on day 4

The day-4 comparison finds the best feasible upper bound among the heuristics and describes how its mismatch allocation relates to plant prices and operating constraints.

Appendix F.2. Heuristic upper-bound evolution on day 4

  • $124,213.75 is the day-4 network cost of the best feasible upper bound, obtained by epsilon_split_1_2 at iteration 28.This heuristic allocates the aggregate mismatch across ASU1 and ASU2, excluding ASU3.Source: Appendix F.2 · p. 63, Figure F.8 · p. 63
  • The ASU1–ASU2 split is consistent with ASU3 having a substantially higher daily average electricity price on day 4.Source: Appendix F.2 · p. 63
  • Heuristics involving ASU3, all-plant splits, and electricity-weighted splits remain dominated on day 4, while epsilon_split_P1 and epsilon_electricity_split yield no feasible upper bounds over the displayed iterations.Source: Appendix F.2 · p. 63

Appendix F.3. Performance of heuristics on the case study

The case-study results compare heuristic upper bounds across the rolling-horizon simulation; the appendices also describe an adversarial reconstruction model and the ICC–ADMM implementation.

Appendix F.3. Performance of heuristics on the case study

  • ϵ split_1_2 wins on 19 of 31 days and has the lowest overall average gap among the listed heuristics, at 8.23%.The authors relate this performance to lower average electricity prices for ASU1 and ASU2 and ASU1’s largest capacity in the engineered case study.Source: Appendix F.3 · p. 64
  • Feasibility does not guarantee frequent wins: ϵ split_3 is feasible on 27 days but wins twice and has a 13.41% average gap.ϵ split_2 is feasible on 26 days and wins 6, making it a strong baseline among single-plant instance-independent rules.Source: Appendix F.3 · pp. 64–65
  • ϵ electricity_split wins on two days but is feasible on only six, so price-aware allocation can help when shipment targets fit plant constraints and inventories but can be brittle when feasibility is tight.The authors recommend maintaining a portfolio combining broadly feasible instance-independent rules with targeted instance-dependent rules.Source: Appendix F.3 · p. 65

Appendix G. Adversarial empirical risk minimization model

  • The adversarial convex ERM fits a cost-consistent latent plant trajectory to revealed iterates, ADMM center terms, inventory accounting, and capacity bounds to estimate private cost coefficients.Aggregate side-information ratio constraints improve identifiability of fixed-cost components.Source: Appendix G · pp. 65, 67, 68

Appendix H. ICC–ADMM algorithms

  • The ICC–ADMM implementation masks plant shipment copies before sending them to the coordinator, while masks cancel when messages are summed to form the aggregate mismatch.The coordinator also checks global feasibility and whether a candidate improves on network status quo.Source: Appendix H · pp. 69, 70, 72

Appendix I. Air separation unit model

The appendix formulates weekly ASU scheduling as an hourly MILP for plants producing liquid nitrogen, oxygen, and argon. It represents operating modes, production, electricity use, and liquefier transitions while meeting daily targets.

Appendix I. Air separation unit model

  • The model tracks operating mode, product quantities, market electricity purchases, and stored liquid inventory for plants producing LIN, LOX, and LAR.Plants differ in production capacity, storage capacity, and operating characteristics.Source: Appendix I · pp. 75–76
  • Each ASU chooses hourly operating modes and production rates to minimize electricity cost while meeting prescribed production targets.The model covers OFF, Liquid Start-up, and Liquid Production modes.Source: Appendix I · p. 75
  • The Liquid Production feasible region is approximated by the convex hull of historical production data, while liquefier start-up and shut-down requirements constrain mode transitions.The model also identifies minimum up-time and downtime constraints as possible limits on excessive switching.Source: Appendix I · pp. 75–76
  • The weekly planning horizon has 168 hourly decision periods, with daily product targets and historical periods used to enforce mode-transition constraints.The model uses hourly electricity prices, with day-ahead prices treated as known and later prices represented by forecasts.Source: Appendix I · pp. 75–77

Appendix I.1. Mass balance

The appendix specifies ASU scheduling constraints for inventory, energy, production, and liquefier operation. It also states boundary conditions, cost assumptions, and the simulated study's scope.

Appendix I.1. Mass balance

  • Inventory evolves as production minus dispatch, is limited by plant- and product-specific storage capacity, and daily targets require dispatch at each day's end.Third-party product purchases are excluded because assigned production targets are assumed feasible within plant capacity.Source: Appendix I.1 · pp. 79–80, Appendix I.6 · p. 85

Appendix I.2. Energy balance

  • Electricity purchased equals electricity consumed because the modeled ASUs have no auxiliary power source or in-house energy production.Consumption depends on operating mode and, in Liquid Production mode, production quantity.Source: Appendix I.2 · p. 80

Appendix I.3. Internal operations

  • Only Liquid Production permits nonzero output, represented within a convex hull of historical production samples; mode-dependent consumption combines fixed and production-dependent electricity requirements.The OFF mode has no electricity consumption, while Liquid Start-up has a fixed requirement but no production.Source: Appendix I.3 · pp. 81–82

Appendix I.4. Mode-transition constraints

  • Liquefier mode changes follow the state graph and are constrained by minimum OFF time, start-up duration, and minimum ON time.The transition constraints also prohibit transitions not allowed by the graph.Source: Appendix I.4 · pp. 82–84

Appendix I.5. Boundary conditions

  • Scheduling uses initial inventory and operating mode, a prescribed minimum terminal inventory, and recent operating history for transition-related lag constraints.The historical horizon extends only as far back as the largest transition-related lag required.Source: Appendix I.5 · pp. 84–85

Appendix I.6. Objective function

  • The static status-quo model minimizes electricity operating cost and omits shipping costs included in the main text's coordinated demand-response case study.Fleet sizing and vehicle routing are not modeled explicitly because transportation capacity is assumed sufficient.Source: Appendix I.6 · p. 85

Disclaimer

  • The case-study results are based on synthetic data and a simulated industrial gas supply chain, not an actual Air Liquide supply chain.Source: Disclaimer · p. 86
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