[Objective] Carbon emissions from coal-fired power plants vary with combustion organization, flue-gas conditions, and operating-load disturbances. Conventional inventory methods cannot represent these hourly variations in a timely manner, whereas purely data-driven soft sensors may produce estimates that conflict with available process knowledge. This study aims to improve the accuracy and physical consistency of hourly carbon-emission soft sensing and to coordinate the selection of physical regularization and residual-correction parameters under a limited search budget. 

[Methods] The dataset contained 8,760 hourly records from a large coal-fired unit in eastern coastal China, covering 1 January to 31 December 2025. The records underwent temporal-consistency checks, screening of missing and physically invalid values, standardization using training-period statistics, and sliding-window construction. Data from January to October were used for training, November data were used for calibration, and 744 observations from December were reserved for testing. Flue-gas oxygen content, velocity, temperature, humidity, and pressure were used to construct proxy features for oxygen dilution, standard dry flue-gas flow, and combustion intensity. A long short-term memory (LSTM) network extracted current and historical operating information. Flow-direction, oxygen-direction, and temporal-smoothness regularization were incorporated into network training. Ridge regression then learned a mapping of the calibration residuals from 15 physics-related features, forming a physics-guided residual correction LSTM (PGRC-LSTM). An improved heart rate optimizer (IHRO) coupled heart-rate amplitude with fitness feedback, reused known fitness values, and performed a final boundary check in the favorable search interval. IHRO jointly selected one of three pretrained physical-weight models and the ridge regularization coefficient. All compared optimizers used the same candidate set, forward temporal-validation objective, search range, and budget of 90 evaluations. Twenty search seeds were used. Performance was evaluated using mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), and the coefficient of determination (R2). 

[Results] Under the fixed configurations used for basic-model screening, LSTM achieved the lowest mean MAPE of 6.9833% among the 11 compared models. Incorporating physical regularization reduced the mean MAPE to 6.7923%, and residual correction further reduced it to 5.5528%. The PGRC-LSTM therefore lowered MAPE by 1.4305 and 1.2395 percentage points relative to LSTM and the physics-guided regularized LSTM, respectively. In the conditional direction-consistency analysis, its average combined violation rate was 22.33%, compared with 24.91% for LSTM; this statistic was interpreted together with prediction error rather than as independent proof of engineering improvement. After optimization, IHRO-PGRC-LSTM obtained an MAE of 34,176.5604 kg, an RMSE of 46,830.4620 kg, a MAPE of 4.4631%, and an R2 of 0.9511 on the December test set. All four mean metrics were the best among the models listed in Table 5. The mean MAPE was 1.0897, 0.8342, and 0.3197 percentage points lower than those of unoptimized PGRC-LSTM, Bayesian optimization, and the baseline heart rate optimizer, respectively. Its MAPE standard deviation was higher than that of Bayesian optimization, indicating that the observed advantage primarily concerned mean accuracy rather than the lowest search variability. 

[Conclusion] The proposed method combines weak physical priors, residual correction, and constrained parameter search to improve hourly carbon-emission soft sensing when the required flue-gas inputs are synchronously available. Because online coal quality, total fuel and air flow, and flue-gas carbon dioxide concentration were unavailable, the imposed relationships are directional constraints rather than a strict carbon balance. The findings are limited to the historical unit data and search space examined. Independent-period, cross-season, and cross-unit validation is required before broader deployment.