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Production Scheduling Identification: An Inverse Optimization Approach for Industrial Load Modeling Using Smart Meter Data

Ruike Lyu, Hongye Guo, Qinghu Tang, Qixin Chen, Chongqing Kang

arXiv:2608.26485v1eess.SY

TL;DR

Accurate industrial load modeling is hindered when production parameters are private or unavailable, despite the need to use industrial flexibility in demand-side response. The paper proposes PSI, which infers parameters from smart-meter data using a modified STN and inverse optimization. On steel powder and cement plant data, PSI achieved modeling errors no greater than 8.5% and 5.2%, respectively, using 21 days of hourly data.

  • Problem

    Industrial load modeling often assumes access to sensitive production parameters that industrial users may not disclose.

  • Method

    PSI uses historical smart-meter data with a modified STN and an inverse-optimization algorithm to fit industrial load-model parameters.

  • Results

    5.2% and 8.5% nRMSEs for the two datasets were the lowest among the compared methods.

  • Takeaways & Limitations

    PSI demonstrates feasible industrial load parameter identification using only smart-meter data, including 21 days of hourly measurements.

  • Takeaways & Limitations

    The approach assumes electricity costs can be calculated from hourly consumption and prices, which may provide less information under long-term or slowly updated contracts.

Abstract

from arXiv · show

To cost-effectively manage the supply-demand balance of the power system, the flexibility of industrial users could be harnessed through demand-side response. To minimize the negative impact on the production of industrial users during demand-side response, general-purpose models such as the state-task network (STN) are widely used to model the energy-consuming constraints of industrial production processes. However, the required model parameters cannot be set because the required data are privately owned by industrial users and are not directly available, hindering the accurate modeling of industrial loads. In this paper, we propose production scheduling identification (PSI), an inverse-optimization-based approach for industrial load modeling under incomplete information. In PSI, industrial users' smart meter data are used to identify production scheduling parameters, thus addressing the problem of accurate load modeling when private data are unavailable. We implemented PSI with a modified STN and proposed a practical algorithm to obtain an effective solution. Numerical tests showed that PSI can identify the model parameters of a steel powder plant and a cement plant with acceptable accuracy, using only 21 days of hourly smart meter data. Compared with accurate models established with direct access to private data, the modeling error does not exceed 8.5% and 5.2%, respectively.

I. INTRODUCTION

Industrial load models require private production parameters that are often unavailable, limiting accurate grid-interaction planning. PSI addresses this gap by inferring parameters from smart-meter data through a modified STN and inverse optimization.

  • Motivation: Industrial load models define feasible production schedules that respect process constraints during grid interactions.The load model represents plant-level production scheduling at an hourly time scale.
  • Motivation: STN models represent materials as states and production processes as tasks, capturing energy-material relationships and production-stage constraints.
  • Research gap: Rated equipment power, interstage storage limits, and production objectives are sensitive parameters that may be inaccessible because of privacy concerns.
  • Proposed approach: PSI uses only industrial users’ smart-meter data to identify production scheduling parameters under incomplete information.
  • Proposed approach: The modified STN is linear, uses per-unit parameters, and permits task aggregation for computationally tractable identification.
  • Proposed approach: The practical algorithm measures fitting errors and iteratively uses data from multiple days to solve the inverse optimization problem.

II. CONCEPT AND FRAMEWORK OF PSI

PSI identifies industrial load-model parameters from historical smart-meter consumption, electricity prices, and public process knowledge when private facility data are inaccessible. It embeds production-process constraints in an inverse-optimization framework for applications such as demand-response estimation and baseline-load calculation.

  • Problem description: PSI identifies private industrial load-model parameters using historical smart-meter data, electricity prices, and public production-process knowledge.Facility-specific parameters include equipment ratings, storage capacity, and production goals, while unit energy and material consumption are treated as known prior knowledge.
  • Problem description: Industrial facility parameters are treated as inaccessible because they contain sensitive configuration, storage, and production-objective information.
  • PSI concept: PSI assumes smart-meter consumption reflects cost-minimizing production schedules under electricity prices and fits the parameters of the underlying optimization problem.
  • PSI concept: Unlike purely data-driven methods, PSI incorporates industrial production constraints, enabling physics-informed modeling from coarse-scale smart-meter data.
  • Framework: The PSI framework enhances the STN into an mSTN and uses a three-step inverse-optimization methodology for tractable identification.
  • Applications: Identified models can estimate industrial demand-response capacity, incentive-driven load changes, and baseline load for calculating reduced load and compensation.

III. MODIFIED STATE-TASK NETWORK MODEL

The modified state-task network (mSTN) changes the conventional STN to improve tractability under incomplete information while retaining production-scheduling constraints. It linearizes operation, unitizes task parameters, and permits task aggregation.

  • mSTN modifications: The mSTN linearizes discrete machine operation because short start-up times make within-interval operation nearly continuous and linear-program optimality conditions more tractable.
  • mSTN modifications: Per-unit task parameters address multivalue fitting because scaling certain state-related parameters can preserve the same electricity-consuming strategy.
  • mSTN modifications: Adjacent production tasks can be aggregated when the facility’s task count is unavailable, allowing PSI to determine an appropriate model scale.
  • Network representation: In the network, machines are tasks and feedstock, intermediate, and final products are states represented as nodes.
  • Mathematical formulation: Daily scheduling minimizes total energy cost subject to production targets, task power limits, buffer dynamics, initial states, and storage limits.
  • Mathematical formulation: mSTN scheduling variables describe task power, state quantities, and electricity consumption, while electricity prices and private facility parameters define the model inputs.

C. The dual problem of the mSTN

The mSTN’s linear-program structure permits a dual formulation used to construct optimality conditions for model identification.

  • Dual formulation: The mSTN dual problem is formulated through its Lagrangian dual function.
  • Dual formulation: Dual feasibility requires the coefficients in the Lagrangian dual function to equal 0.
  • Dual formulation: The dual variables are constrained to be nonnegative.

IV. MODEL IDENTIFICATION

Model identification uses historical prices and consumption to fit an mSTN load model that reproduces an industrial facility’s electricity-consumption strategy. The formulation treats facility parameters as decision variables and enforces optimality conditions through duality.

  • Identification formulation: Historical price and consumption data are used to identify a load model fitting the facility’s electricity-consumption strategy.
  • Identification formulation: The inverse-optimization objective measures average squared load-profile residuals.
  • Identification formulation: The formulation combines mSTN primal constraints with strong-duality and optimality conditions to represent observed scheduling solutions.
  • Identification variables: Facility model parameters θ are decision variables alongside primary scheduling variables and dual variables, while electricity prices are problem parameters.
  • Identification assumptions: The model assumes θ remains unchanged over short-to-medium periods such as a month; time-varying consumption strategies are left for future work.

B. Assumptions for Model Identification

The identification problem is made computationally feasible through assumptions that avoid using facility-private data. These assumptions constrain variables, production-state changes, maximum power, and available process knowledge.

  • The model-identification problem is a large-scale nonlinear optimization problem, so assumptions are introduced to improve computational feasibility without using private facility data.
  • Variables are bounded above by a large real number, such as 1e3, to avoid numerical problems.
  • After a production day, feedstock decreases, final product increases, and intermediate materials remain unchanged or increase.
  • Historical operation is assumed to include at least one period when all machines operate at rated power, using the entire dataset rather than one day.
  • The per-unit production-energy coefficients {g_i} are assumed known from process knowledge, while the total number of tasks remains unknown and is determined through model selection.

C. An Iterative Model Identification Algorithm

The iterative PSI algorithm addresses a difficult nonlinear identification problem by combining batchwise zero-order updates with model selection and practical stopping rules.

  • PSI uses zero-order stochastic gradient descent to solve the identification problem by estimating updates through suboptimal search rather than analytical gradients.
  • The algorithm trains each candidate task-count model on TRAIN and selects the best-performing model on the cross-validation set CV.
  • Each initialization step randomly selects a day and solves a batch problem to calculate θ^(k).
  • At each iteration, a randomly selected day yields an updated parameter θ^(k), which is averaged into θ using a fixed step size of 1/n.
  • Because the problem is nonconvex, feasible solutions found at a maximum iteration computation time may update parameters without requiring exact subproblem optima.

V. NUMERICAL RESULTS

The numerical study evaluates PSI on cement-plant and steel-powder-manufacturer datasets and compares it with mainstream machine-learning approaches.

  • PSI feasibility and performance are evaluated using datasets from a cement plant and a steel powder manufacturer.
  • The study uses mainstream machine-learning approaches as comparison methods because no prior data-driven industrial-load-modeling approach had been reported.
  • The comparison methods include multilayer perceptron, long short-term memory, and support vector regression approaches.

A. Dataset and Partitioning

The numerical tests generate price-responsive industrial consumption data, partition them for identification and evaluation, and assess PSI against accurate models and machine-learning baselines. PSI achieves its best steel-powder model-selection performance with four or five tasks and records nRMSEs of 5.2% and 8.5% on the two datasets.

  • Dataset and Partitioning: 41 days of 24-hour price-smart-meter pairs are generated using realistic prices and optimized industrial production schedules.
  • Dataset and Partitioning: The dataset uses the first 21 days for training, the middle 10 days for cross-validation, and the final 10 days for testing.
  • Dataset and Partitioning: PSI identifies an mSTN model from smart-meter data without requiring users’ energy-consumption parameters, and comparisons use an STN model with true parameters as the accurate model.
  • Model Selection: For the steel powder manufacturer, PSI cross-validation performance improves through four or five aggregated tasks and then degrades as task count increases.
  • Results and Comparison: 5.2% and 8.5% are PSI’s nRMSEs for the two datasets, the lowest among the compared methods.
  • Results and Comparison: PSI is more accurate than machine learning during high-electricity-price periods, when users avoid energy use to reduce costs.
  • Results and Comparison: Removing prior assumptions or reducing training samples to n = 10 or n = 5 reduces PSI performance, with prior assumption b having the largest impact.

VI. DISCUSSION

PSI requires assumptions about pricing, production objectives, and the suitability of the STN representation, and the authors note that practical deployment requires further improvement. The framework can be adapted to other load models, but doing so requires corresponding model and algorithm changes.

  • PSI requires further improvement before practical industry application because its assumptions may deviate from actual industrial conditions.The authors discuss alignment between PSI’s assumptions and real conditions as a direction for reducing reliance on those assumptions.
  • Industrial users may settle through power purchase agreements rather than directly facing fluctuating hourly electricity prices, reducing information available from historical data.Long-term contracts or time-of-use prices with long updating cycles can provide less variation for learning.
  • PSI assumes users minimize energy costs while meeting planned production targets, although users may also consider cost risk or deviations from production plans.These alternative objectives can differ from the basic scheduling objective used in PSI.
  • PSI assumes industrial energy-consumption mechanisms can be captured by STN, which omits parameter uncertainty and nonlinear production efficiency.These omitted features limit how fully the general-purpose STN can represent industrial loads.
  • Different load models can be incorporated into the PSI framework, with corresponding changes to model assumptions and other framework components.The authors propose constructing a model library when sufficient computational resources are available.

VII. CONCLUSION

The conclusion presents PSI as an inverse-optimization framework that fits modified-STN load-model parameters from historical smart-meter data under incomplete information. Tests on steel powder and cement plants achieved bounded modeling errors using 21 days of price-consumption data and outperformed pure data-driven approaches.

  • PSI fits modified-STN load-model parameters from historical smart-meter data when industrial-user information is incomplete.The framework models energy-consuming behavior under boundary conditions using a modified STN.
  • 8.5% and 5.25% were the respective maximum modeling errors for a steel powder manufacturer and cement plant using 21 days of price-consumption data.These errors were measured against verified accurate models obtained through direct access to private data.
  • PSI achieved significantly higher accuracy than mainstream pure data-driven approaches.
  • PSI facilitates effective load modeling with limited coarse-scale external meter data despite privacy barriers to obtaining industrial model parameters.This supports grid-load interaction work by load aggregators or virtual power plants that cannot directly access private parameters.
  • The modified STN state-change formulation represents intermediate-state changes using task energy consumption and production or consumption coefficients.For task i, energy consumption is represented as ∆E_i = P_i∆t; the associated coefficient describes production or consumption of an intermediate.
  • The mSTN uses per-unit production or consumption values after conversion rates are obtained, denoting these values by g_i when g_i = c_i.

APPENDIX B SUPPLEMENTARY DISCUSSION OF TASK AGGREGATION

The task-aggregation strategy reduces mSTN parameters and supports convergence by combining production links or tasks with similar throughput. The appendix also describes the aggregation procedure and compact pure-data-driven baselines used for comparison.

  • Task aggregation: Aggregating production links with similar throughput reduces mSTN parameters and is intended to enhance parameter-identification convergence.Links with similar throughput are likely to switch together under low or high electricity prices, making aggregation error intuitively small.
  • Task aggregation: A new task parameterized by g_i′ and P max_i′ can approximate adjacent tasks while neglecting the buffer between them.The same process can be generalized to multiple tasks.
  • Task aggregation: Similar task throughputs allow adjacent tasks to operate simultaneously at rated power during low-price periods without accumulating intermediates.This motivates aggregating adjacent tasks without substantial error when estimating machine parameters before facility construction.
  • Numerical implementation: The numerical test aggregates the 10-task model in the order (10, 3, 5, 9, 6, 2, 4, 8), using rated-power g_i values as prior knowledge.For the crusher task, the stated rated-power value is g = 15/20 = 0.75.
  • Data-driven baselines: The MLP baseline maps 24-hour energy prices to 24-hour electricity consumption through a 24-48-48-24 fully connected network.
  • Data-driven baselines: The LSTM baseline uses one LSTM layer, while the SVR baseline trains 24 independent models to predict each hourly consumption value.Both baselines use 24-hour energy prices as input.
  • Data-driven baselines: Few-parameter machine-learning baselines were selected because the scenario provides limited data and models are likely to underfit.Their listed hyperparameters resulted from extensive tuning.
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