A numerical study reports lower error scores for two data-driven reductions than for SAL in all three simulated industrial-load cases, and than for OVB in the two cases where OVB was available. The reductions also used fewer listed variables.
The work targets the approximation of complex mixed-integer industrial-load constraints for economic dispatch or market clearing. Its data-driven model was trained on optimal energy-usage data from the original constraints.
A smaller representation of a complex problem
The reduced representation treats an industrial plant as an adjustable load fleet made up of multiple adjustable loads. Linear constraints describe the fleet's equipment composition, power ratings and production goals.
To fit the representation, the study uses inverse optimization on the discrepancy between optimal energy consumption under the original constraints and the reduced constraints. Bilinear complementary-slackness conditions are transformed using the Fortuny-Amat transformation, and the transformed conditions are used to form a mixed-integer linear programming problem solvable by commercial solvers.
Training used zeroth-order stochastic gradient descent, or ZOSGD. It processed data in batches and updated parameters iteratively; the paper states that its computational complexity is independent of the number of days in the dataset.
Three cases, one simulated test period
The evaluation covered three industrial-load datasets: cement, steel powder and steelmaking. The first two were STN-based and the third was RTN-based.
All three were numerical simulations using PJM hourly market prices from July 2022. July 1 through 21 supplied training data, while July 22 through 31 supplied test data. The optimization ran with Gurobi V11.0.0 and YALMIP in MATLAB on a workstation with a 3.7 GHz CPU and 128 GB of RAM.
The reference methods were SAL, which uses maximum and minimum values from historical load curves, and OVB, an analytical inner-approximation method. OVB was not usable for the steelmaking case because linear relaxation methods were absent.
The reported scores differed by plant
For cement, reported nRMSE was 25.4% with OVB and 10.2% with SAL. The two data-driven variants recorded 8.9% for D3R-1 and 9.5% for D3R-2, putting both below the two comparison scores.
For steel powder, reported nRMSE was 36.7% with OVB and 23.0% with SAL. D3R-1 recorded 17.2% and D3R-2 recorded 10.3%, again below the reported comparison scores.
For steelmaking, OVB was listed as not available. SAL's nRMSE was 8.3%, compared with 3.6% for D3R-1 and 4.2% for D3R-2.
The original formulations listed 196 continuous and 288 integer variables for cement, 490 and 720 for steel powder, and 0 continuous and 10,208 integer variables for steelmaking. D3R-1 used 24 continuous variables and no integer variables in each case, while D3R-2 used 48 continuous variables and no integer variables in each case. The count separates continuous variables from those constrained to integer values.
Across the evaluated cases, the paper reports a reduction error ranging from 4% to 10% and says that D3R outperformed analytical methods in accuracy.
What the comparison does not establish
The OVB comparison also had an important qualification. It converged for cement after 767 seconds but did not converge for steel powder within 24 hours; the study used values from unconverted states for that case.
The evidence covered only three simulated industrial-load cases and PJM July 2022 price inputs. No real-world deployment or direct market-clearing evaluation was reported, so the figures do not establish how the reductions would perform in industrial operations or actual market clearing.
The report also gives no confidence intervals, hypothesis tests, formal uncertainty estimates or run-to-run variability. The percentages therefore compare these model settings; they do not show how stable the differences would be across repeated runs.
The document is labeled arXiv:2608.24390v1 and dated Aug. 25, 2026. It says the work was supported in part by the National Natural Science Foundation of China under Grants 52107102 and 52321004.
Paper data and sources
Original title: Data-Driven Dimension Reduction for Industrial Load Modeling Using Inverse Optimization
Authors: Ruike Lyu, Hongye Guo, Goran Strbac, Chongqing Kang
Journal/Repository: arXiv
Status: Peer-reviewed
First online: 2026-08-25
DOI: 10.1109/tsg.2025.3545339
Original paper · Full text