Two-stage Residual Learning for Modelling Strength-Informed Optimisation of Thermal Systems

Models

• ANN denotes artificial neural network, while TSRL denotes two-stage residual learning.MLP is multilayer perceptron and SiLU is sigmoid linear unit.Source: Nomenclature · p. 2

Reliability

• GMR and WMR denote good and weak modelled regions, respectively.The residual score s_i is the predicted residual magnitude for observation i.Source: Nomenclature · p. 2

Metrics

• RMSE, MAE, MSE, ROC, and AUC name the reported error and classification metrics.They denote root mean square error, mean absolute error, mean squared error, receiver operating characteristic, and area under the ROC curve.Source: Nomenclature · p. 2

Outputs

• THR is turbine heat rate, while TE is thermal efficiency.The refinery nomenclature includes AGO for atmospheric gas oil and PHGOT for performance-heater gas outlet temperature.Source: Nomenclature · p. 2

Data and optimisation

• N_ref is the number of refinery modelling observations, defined as 817.CO2 denotes the refinery optimisation ceiling, and x_target denotes the prescribed scaled input target.Source: Nomenclature · p. 2

1 Introduction

• Surrogate-based optimisation reduces computational cost by replacing repeated detailed-model calls with a quickly evaluated approximation.Thermal-system operation still requires decisions to satisfy equipment, material, energy, and safety constraints.Source: §1 · p. 2
• The study uses ANN surrogates because their differentiable form suits optimisation, but split-level RMSE, MAE, and R2 do not locate errors in the input–output mapping.Separate data are needed for parameter estimation, model selection, and final evaluation to avoid optimistic error estimates from reusing selection data.Source: §1 · p. 3
• Domain restrictions retain input combinations represented in data, but satisfying them does not establish local prediction accuracy.Prior approaches include validity domains, data-domain constraints, spatial areas of applicability, and Mahalanobis constraints.Source: §1 · p. 3
• Prior uncertainty approaches use repeated training or stochastic forward passes, and their spread measures disagreement rather than a threshold calibrated against deployed-model errors.The cited gas-turbine uncertainty study did not carry its uncertainty into surrogate-based operational optimisation.Source: §1 · p. 3
• Earlier residual learners improved prediction accuracy, while a pointwise-error predictor returned continuous uncertainty without locked thresholds or constrained-programme solution labels.Residual learning had not used correction magnitude as a reliability signal in the cited studies.Source: §1 · p. 3
• The paper addresses the lack of point-specific modelling-strength measures for refinery and gas-turbine operating points before they reach operators.It calibrates residual scores against observed base errors and assesses domain-constrained solutions without adding an uncertainty model to the optimisation formulation.Source: §1 · p. 3
• TSRL fits a base ANN, then trains a residual ANN on signed base-model errors and uses predicted residual magnitude to label solutions GMR or WMR.Validation data link residual scores to observed base errors to select percentile thresholds by maximising F1; the thresholds are locked before testing and solution scoring.Source: §1 · p. 3
• The framework is examined on 817 simulated crude-distillation observations and operating records from a 395 MW industrial gas turbine.The refinery inputs span crude blends and utility conditions; turbine measurements include operating dynamics and sensor noise.Source: §1 · p. 3

2 Methods

• The base ANN supplies the optimisation response, while a residual ANN learns signed errors; its corrected prediction adds the residual and its magnitude scores reliability at unknown operating points.The base network is fixed before residual learning, and only the base ANN enters the optimisation problem.Source: §2 · p. 4

2.3 Residual-score calibration and reliability classification

• Percentile thresholds for observed absolute base error and residual score are cross-matched on validation data, and the pair with the highest F1 is locked before testing.The high-error reference class is defined relative to validation percentiles, although an accepted process-error limit can replace that percentile definition.Source: §2 · p. 4, §2.3 · p. 7

2.1 Data partition, scaling, and evaluation

• Validation observations are split 70/15/15 for training, validation, and testing, and inputs are min–max scaled to [0, 1] using shared limits across splits.The validation split guides model selection and threshold calibration; the test split is evaluated once after modelling choices are fixed.Source: §2.1 · p. 4

2.2 Base and residual surrogate models

• Separate one-hidden-layer MLPs with SiLU activations model each output; the base network is fixed before a second network learns its signed residuals.Hyperparameters are tuned independently, and the selected output has the largest positive validation RMSE improvement among outputs whose training RMSE also decreases.Source: §2.2 · pp. 5–6

2.4 Constrained optimisation and local assessment

• Domain constraints include a covariance-aware Mahalanobis-distance limit; after IPOPT returns a locally optimal solution, its residual score is compared with locked Ts to assign GMR or WMR.The residual assessment does not change the optimisation problem, which retains the base surrogate in its objective and constraints.Source: §2.4 · p. 7

3 Results and Discussion

• Performance and residual-score screening precede GMR or WMR labelling of computed optimisation solutions.The refinery case is considered before the gas turbine study.Source: §3 · p. 8
• The refinery simulation’s controlled input variation provides a direct test of approximation error.Source: §3 · p. 8
• The gas-turbine study applies TSRL to sensor-based industrial records with measurement variation and a less uniform operating history.Source: §3 · p. 8

3.1.1 Rigorous process model

• The refinery surrogate ground truth is a validated steady-state Aspen HYSYS model of a 291.7 t/h crude distillation unit.Its flowsheet includes crude preheat, a fired heater, an atmospheric column, side strippers, and pump-around circuits.Source: §3.1.1 · p. 8

3.1.2 Crude blend input space and sampling design

• The four crude-blend fractions define a unit-sum feasible space, sampled by projecting Latin-hypercube designs onto the simplex.The projection mildly concentrates samples toward the simplex centroid but retains space-filling coverage and permits near-vertex blends.Source: §3.1.2 · p. 8

3.1.3 Automated simulation campaign

• A Python–HYSYS interface writes blend fractions through a flowsheet spreadsheet and reads outputs after solver re-convergence.The spreadsheet interface decouples sampling from individual unit operations, and each sample uses a hold–write–release cycle.Source: §3.1.3 · p. 8

3.1.4 Data description

• The dataset contains 817 observations, with seven crude-flow and utility inputs and seven product and emissions outputs.Crude rates span near zero to about 56–65 kg/s, while reboiler duty varies by about 3.5% of its mean; the gas stream is mostly zero or near zero with a few larger observations.Source: §3.1.4 · p. 9

3.1.5 Surrogate performance and residual screening

• Fuel Oil test RMSE falls from 10.3 to 4.7 g/s with residual correction.The comparison is reported in Table 3.Source: Table 3 · p. 11, §3.1.5 · p. 11
• The locked 11.92 g/s residual-score threshold detects 10 of 11 high-error Fuel Oil test observations.The validation ROC AUC is 0.99; the locked threshold yields test precision 0.83, recall 0.91, and F1 0.87.Source: §3.1.5 · p. 12, Figure 4 · p. 12

3.1.6 Optimisation Cases

• All CO2-sweep solutions are labelled WMR; their Fuel Oil residual scores rise from 13.1 g/s at an 8,198.8 kg/h CO2 limit to 101.5 g/s at 8,446.8 kg/h.Each score exceeds the locked 11.92 g/s threshold.Source: §3.1.6 · p. 13, Table 4 · p. 14
• Arab Heavy targets down to 19.48 kg/s remain GMR, while lower targets become WMR as Arab Medium and Kuwait replace more feed.As the target falls from 51.73 to 0.12 kg/s, predicted Kerosene rises from about 12.2 to 13.1 kg/s and Fuel Oil falls from 46.55 to 44.22 kg/s.Source: §3.1.6 · p. 14, Table 5 · p. 15

3.2.1 System and data description

• The case uses operating records from a 395 MW gas turbine, with Power ranging from 185.7 to 395.7 MW across represented ramping operation.Inputs describe the air path, fuel path, ambient conditions, and performance; the dataset statistics are reported in Table 6.Source: §3.2.1 · p. 15, Table 6 · p. 15

3.2.2 Base and corrected surrogate performance

• THR test RMSE fell from 151 to 105 kJ/kWh, while Power test RMSE rose from 1.02 to 1.08 MW and TE test RMSE stayed at 0.46 percentage points.THR validation RMSE fell from 158 to 99 kJ/kWh, a 37% improvement; THR test R2 rose from 0.94 to 0.97.Source: §3.2.2 · pp. 15–16, Table 7 · p. 16, Figure 7 · p. 16

3.2.3 THR residual reliability diagnostic

• Validation AUC was 0.78; applying locked thresholds gave test precision 0.60, recall 0.55, and F1 0.57.Predicted and true signed residuals had a test Spearman correlation of 0.72 and residual RMSE of 105 kJ/kWh; the validation residual-score threshold was 161.32 kJ/kWh.Source: §3.2.3 · pp. 16–17, Figure 8 · p. 17

3.2.4 Thermal-efficiency optimisation across fuel-flow targets

• At scaled GFFR target 0.8, predicted TE reached the observed upper bound of 43.05% and was the only GMR; TE fell at target 1.0.The five nominal scaled targets ranged from 0.2 to 1.0; the optimisation constrained base outputs to observed ranges and Mahalanobis distance to its 95th percentile.Source: §3.2.4 · pp. 17–18, Table 8 · p. 18

• TSRL was applied to a simulated crude distillation unit and a 395 MW industrial gas turbine to label the modelling strength of computed optimal solutions.Source: §4 · p. 18
• Residual correction improved Fuel Oil and THR predictions, but screening was more effective for Fuel Oil; all eight refinery CO2-sweep solutions were WMR despite satisfying the Mahalanobis constraint.For the gas turbine, scaled GFFR target 0.8 gave the highest predicted TE and the only GMR among five targets.Source: §4 · p. 18
• Domain constraints kept solution searches close to data envelopes, while residual scores distinguished GMR from WMR within those data-driven domains.Source: §4 · p. 18

• Thresholds calibrated on one simulated refinery dataset and one gas-turbine unit may not transfer across operating regimes; labels remain relative to validation-error percentiles.Transfer to another unit requires checking score calibration against its operating data and, where available, an engineering error limit.Source: Limitations and future work · p. 18
• Screened optimisation solutions still require independent checks through simulator reruns or controlled plant trials.Source: Limitations and future work · p. 19