Study on Sensor Fault-Tolerant Control for Central Air-Conditioning Systems Using Bayesian Inference with Data Increments

A lack of available information on heating, ventilation, and air-conditioning (HVAC) systems can affect the performance of data-driven fault-tolerant control (FTC) models. This study proposed an in situ selective incremental calibration (ISIC) strategy. Faults were introduced into the indoor air (Ttz1) thermostat and supply air temperature (Tsa) and chilled water supply air temperature (Tchws) sensors of a central air-conditioning system. The changes in the system performance after FTC were evaluated. Then, we considered the effects of the data quality, data volume, and variable number on the FTC results. For the Ttz1 thermostat and Tsa sensor, the system energy consumption was reduced by 2.98% and 3.72% with ISIC, respectively, and the predicted percentage dissatisfaction was reduced by 0.67% and 0.63%, respectively. Better FTC results were obtained using ISIC when the Ttz1 thermostat had low noise, a 7-day data volume, or sufficient variables and when the Tsa and Tchws sensors had low noise, a 14-day data volume, or limited variables.


Introduction 1.Background
Statistics show that building energy consumption has exceeded one-third of the total global energy consumption [1][2][3], and 60% is consumed by building energy systems (for ventilation, heating, and cooling).With the development of the economy and the improvement in living standards, people have a higher and higher demand for indoor air environments [4,5].The application of HVAC systems in buildings is also becoming increasingly widespread [6][7][8].Many data-driven techniques have been proposed in order to improve the performance of HVAC systems [9].These techniques and applications include system fault detection and diagnosis [10][11][12][13], and building energy consumption prediction [14,15], control, and optimization [16,17].The key to these applications and technologies lies in the building sensing environment.The within-control-loop sensors in building systems have a huge impact on energy consumption [18], control efficiency, indoor environmental quality, and comfort [19,20].Sensor faults can negatively impact data-driven applications [21] and big data analytics [22], which, in turn, diminishes the value of smart buildings.Regardless of how advanced data-driven applications are, the sensing environment remains a core prerequisite for realizing smart buildings [23,24].

Research and Challenges in Data Incremental Learning for FTC
In real-world scenarios, long-term continuous data collection is important for FTC models before and after a model's implementation.For every data-driven FTC method, a certain amount of historical data is required for training and validation.Although building energy consumers and managers want to implement FTC method modeling as soon as possible, the available historical data may not always be sufficient.Data incremental learning (DIL) [37][38][39] utilizes continuously collected data to extend the knowledge of an existing model (i.e., either by updating or re-training the data-driven model [40][41][42]).It can learn dynamic features to adapt to the trends of the new input time series data.This makes DIL another potential solution to the shortage of available data for FTC tasks.However, DIL in the field of FTC has undergone limited investigation.The quantity and quality Sensors 2024, 24, 1150 3 of 22 of available modeling data are important factors that affect the accuracy of FTC models.FTC models using the DIL strategy can be updated or retrained using newly collected data from an HVAC system.By learning from recent data, the adaptability and calibration accuracy of the FTC model can be improved.Many data-driven sensor calibration methods have been investigated for different amounts of training data [43].Ng et al. [40] proposed an enhanced self-organizing incremental neural network for evaluating the potential of incremental learning.In a machine-learning-based strategy for building energy prediction, Singh et al. [41] reduced the prediction error via data incremental learning and enrichment.
The application of DIL in FTC tasks for HVAC systems is relatively understudied.There is a lack of sufficient investigations on the influence of DIL on FTC models' performance in new building energy system scenarios with increasing amounts of data.In addition, it is unclear whether DIL is applicable for improving FTC performance.Answers to these questions would substantially contribute to the selection of appropriate FTC solutions for real-world applications and improve performance.

Research Objectives
To address the current research gap, this study proposed an in situ selective incremental calibration strategy.Multiple linear regression-Bayesian inference and principal component analysis were used to realize in situ sensor calibration and data filtering, respectively.The proposed data incremental strategy not only used historical datasets to update the calibration model but also retrained the model using all the data, including the newly generated data from the building energy system.Based on the EnergyPlus-Python co-simulation testbed, an FTC study was carried out by selecting three target sensors in a central air-conditioning (CAC) system.Finally, the impacts of the data quality (under noise and steady-state conditions), data volume, and number of variables were evaluated.The research objectives of this study can be summarized as follows: (1) The first objective was to propose an FTC strategy for the in situ selective incremental calibration of HVAC system sensors based on the multiple linear regression-Bayesian inference (MLR-BI) and principal component analysis (PCA) methods.(2) The second objective was the quantification of the FTC results for three target sensors in the central air-conditioning system, including the target variables, energy consumption, and thermal comfort.(3) The last objective was to explore the impacts of several influencing factors (i.e., the data quality, data volume, and number of variables) on the FTC results and determine the appropriate FTC strategies for the target sensors.

Principle of In Situ Selective Incremental Calibration
This study proposed an ISIC strategy.The strategy consists of multiple linear regression-Bayesian inference and principal component analysis.These two methods are used to realize in situ sensor calibration and data filtering, respectively.Principal component analysis can effectively filter the data samples for calibration and reduce the influence of outliers on the calibration results.Then, multiple linear regression-Bayesian inference employs these filtered data samples for in situ sensor calibration, providing more accurate calibrated data.Ultimately, these calibrated data are fed into the CAC system for FTC.These two methods are presented in Sections 2.1 and 2.2.

Fault Calibration Using Multiple Linear Regression-Bayesian Inference
In the sensor calibration, Bayesian inference is applied to calculate the offset constants and unknown parameters in the model to minimize the distance function [44].As shown in Equations ( 1)-( 3), the prior probability function of the error x is denoted as π(x), and the distance function D(x) is constructed to derive the likelihood function (P(Y b |x)).The normalization function (P(Y b )) of the fault values is obtained by sampling with the Markov chain Monte Carlo algorithm [45].Finally, the posterior distribution (P(x|Y b )) and the error x are determined.In addition, σ denotes the standard deviation of the prior distribution.
As shown in Equation ( 4), multiple linear regression (MLR) is used in this study to construct the system term [46,47] and incorporate the sensor-level term to improve the calibration accuracy [29].f (V ′ c ) in Equation ( 6) represents the regression function of the system-level model containing the target sensor to be calibrated.f V p in Equation ( 5) is the benchmark of the system and represents all variable information in the system except the target sensor to be calibrated.The compensation function for the target sensor is defined as Equation (7). where is the physical sensor other than the target sensor to be calibrated, α 0 and β 0 are the constant terms of the MLR model, and α 1 − α p and β 1 − β c are the coefficients corresponding to each of the above variables.V B,c is the benchmark of the target sensor to be calibrated.V f is the fault data.

Data Filtering Using Principal Component Analysis
Principal component analysis (PCA) is a commonly used method for sensor fault detection and process-monitoring methods in industry areas [48].For a given multivariate dataset X, PCA transforms a set of correlated original variables into a new set of mutually uncorrelated or orthogonal forms.In HVAC systems, individual sensors are often multidimensional and interrelated.Therefore, the original variables can be represented with fewer principal components due to the redundancy between the variables.During the data transformation process, each data sample is decomposed into two subspaces, the principal component space and the residual subspace [49].For any new test data sample z, the original data are divided into two parts, as shown in Equation ( 8): where ẑ and ∼ z denote the projections in the principal component subspace and residual subspace, respectively.Typically, ∼ z is used to reflect anomalous deviations from the original statistical correlations between variable measurements.As shown in Equation (9), the Q statistic is built in the residual subspace to detect anomalous deviations, such as sensor bias.
where Q a is the computational threshold for the Q statistic Q z used to detect sensor faults.In this study, the principal element selection method of cumulative contribution was used to select the number of principal components and thus determine the threshold Q a .This method defines the number of principal elements as k when the cumulative contribution of the kth principal element is greater than or equal to some empirical value.A cumulative contribution of variation greater than or equal to 85% is used as the basis for judging the selection of the number of principal elements [50].The cumulative contribution is shown in Equation (10), and Q a can be found using Equation (11).
where c α is the normal deviation corresponding to the upper (1 , and λ i is the ith eigenvalue of the covariance array.If the Q statistic is greater than its corresponding threshold Q a , there is a sensor fault.If the Q statistic does not exceed Q a , there is no sensor fault.

Process of ISIC Fault-Tolerant Control Strategy
In this study, an FTC strategy for in situ selective incremental calibration was proposed based on an in situ calibration (IC) strategy (Figure 2a). Figure 2b shows the in situ selective incremental calibration process in detail.At  , the calibrated data  , are obtained from the fault dataset ( ) and the historical dataset via MLR-BI calculation.The sample  can be obtained by inputting the calibrated data into the HVAC system.At the moment  , the sample  is first filtered with the threshold  calculated using

Process of ISIC Fault-Tolerant Control Strategy
In this study, an FTC strategy for in situ selective incremental calibration was proposed based on an in situ calibration (IC) strategy (Figure 2a). Figure 2b shows the in situ selective incremental calibration process in detail.At τ 1 , the calibrated data T c,1 are obtained from the fault dataset (τ 1 ) and the historical dataset via MLR-BI calculation.The sample X τ 1 can be obtained by inputting the calibrated data into the HVAC system.At the moment τ 2 , the sample X τ 1 is first filtered with the threshold Q a calculated using PCA.If X τ 1 is less than or equal to Q a , then X τ 1 and the historical dataset constitute the incremental dataset (τ 2 ).The incremental dataset (τ 2 ) and the fault dataset (τ 2 ) are passed through MLR-BI to obtain the calibrated data T c,2 .These data are carried over to the HVAC system to obtain the sample X τ 2 , and so on, until the end of the simulation.The incremental dataset size is shown in Equation ( 12): where τ denotes the time of calibration; ∆τ denotes the time step; X denotes the dataset, which is of size m × n; and Q a is the threshold calculated via PCA from the historical dataset.

Process of ISIC Fault-Tolerant Control Strategy
In this study, an FTC strategy for in situ selective incremental calibration was proposed based on an in situ calibration (IC) strategy (Figure 2a). Figure 2b shows the in situ selective incremental calibration process in detail.At  , the calibrated data  , are obtained from the fault dataset ( ) and the historical dataset via MLR-BI calculation.The sample  can be obtained by inputting the calibrated data into the HVAC system.At the moment  , the sample  is first filtered with the threshold  calculated using PCA.If  is less than or equal to  , then  and the historical dataset constitute the incremental dataset ( ).The incremental dataset ( ) and the fault dataset ( ) are passed through MLR-BI to obtain the calibrated data  , .These data are carried over to the HVAC system to obtain the sample  , and so on, until the end of the simulation.The incremental dataset size is shown in Equation ( 12): where  denotes the time of calibration; ∆ denotes the time step;  denotes the dataset, which is of size  × ; and  is the threshold calculated via PCA from the historical dataset.

Thermal Comfort Metrics
In this study, the predicted mean valuation (PMV) and predicted percentage dissatisfaction (PPD) were selected to evaluate indoor thermal comfort.The PMV model was proposed by Fanger in the late 1960s [51].The PPD represents the percentage of the population that is dissatisfied with the thermal environment.As shown in Table 1, the PMV represents the average of hot and cold sensations in the majority of people in the same environment, and the metric utilizes a 7-level scale.The target variable data under FTC conditions were compared with the target variable data under normal conditions to calculate the mean absolute error (MAE), as shown in Equation (13).The smaller the MAE, the higher the accuracy of the FTC data.In addition, this paper used the calibration accuracy (ξ ca ) to measure the results of the calibration method, as shown in Equation (14).
where ŷi is the target variable data for the FTC, y i is the target variable data of the CAC system under normal conditions, and ŷca,i is the calibrated data of the target variable.
As shown in Equations ( 15) and ( 16), the RE f was used to quantify the degree of energy consumption deviation from the fault energy consumption after FTC.The RE n was used to quantify the degree of energy consumption close to normal levels after FTC.
where E f , E n , and E FTC are the total energy consumption for the CAC system's operation under fault, normal, and FTC conditions, respectively.

Analysis of the Influencing Factors of MLR-BI-Based FTC Strategies
This study considered two FTC strategies, in situ calibration and in situ selective incremental calibration, applied in a real operating environment and evaluated in terms of the following aspects: (1) Data quality: The results of the FTC of the two strategies were evaluated using steadystate (10:30-17:30) versus non-steady-state data and different standard deviations of noise (SDN) (0.01, 0.05, 0.1, 0.15, 0.5, 1, 1.5, and 2.0), respectively.(2) Data volume: Different time periods of 1 day, 7 days, 14 days, 1 month, 2 months, and 3 months were selected as the original training set for MLR-BI.(3) The number of variables: Regression models for the target sensors in different variable scenarios were constructed as shown in Figure 3.

Central Air-Conditioning System and Target Building
Figure 4 shows the schematic diagram of the target CAC system, which was a CAC system consisting of an independent fresh air system (air-handling units) and an indoor terminal (a fan coil).The system design was based on the official EnergyPlus example (5ZoneFanCoilDOASCool) [52].The target building was located in Chicago, IL, USA, and was a single-story office building.This building had a core thermal zone in the middle and four thermal zones around it.The building had a floor area of 465 m 2 , a height of 3.0 m, and a return air chamber at the top (with a height of 0.6 m).Each zone had a return air chamber and windows, and the cooling (heat) loads of the five thermal zones were shared by an independent fresh air system and five indoor terminals.

Operation Schedule and Fault Settings
Table 2 depicts the operating schedule of the target system.The CAC system was switched on at 07:00, and a bias fault was introduced into the target sensor at 12:00.The The independent variables are the chilled water return temperature (T chw,r ), the cooling water supply temperature (T cw,s ), the cooling water return temperature (T cw,r ), the chilled water mass flow rate (M chw ), the cooling water mass flow rate (M cw ), and the chiller's energy consumption (E chiller ).

Case Study 4.1. Central Air-Conditioning System and Target Building
Figure 4 shows the schematic diagram of the target CAC system, which was a CAC system consisting of an independent fresh air system (air-handling units) and an indoor terminal (a fan coil).The system design was based on the official EnergyPlus example (5ZoneFanCoilDOASCool) [52].The target building was located in Chicago, IL, USA, and was a single-story office building.This building had a core thermal zone in the middle and four thermal zones around it.The building had a floor area of 465 m 2 , a height of 3.0 m, and a return air chamber at the top (with a height of 0.6 m).Each zone had a return air chamber and windows, and the cooling (heat) loads of the five thermal zones were shared by an independent fresh air system and five indoor terminals.

Central Air-Conditioning System and Target Building
Figure 4 shows the schematic diagram of the target CAC system, which was a CAC system consisting of an independent fresh air system (air-handling units) and an indoor terminal (a fan coil).The system design was based on the official EnergyPlus example (5ZoneFanCoilDOASCool) [52].The target building was located in Chicago, IL, USA, and was a single-story office building.This building had a core thermal zone in the middle and four thermal zones around it.The building had a floor area of 465 m 2 , a height of 3.0 m, and a return air chamber at the top (with a height of 0.6 m).Each zone had a return air chamber and windows, and the cooling (heat) loads of the five thermal zones were shared by an independent fresh air system and five indoor terminals.

Operation Schedule and Fault Settings
Table 2 depicts the operating schedule of the target system.The CAC system was switched on at 07:00, and a bias fault was introduced into the target sensor at 12:00.The

Operation Schedule and Fault Settings
Table 2 depicts the operating schedule of the target system.The CAC system was switched on at 07:00, and a bias fault was introduced into the target sensor at 12:00.The FTC method using data increments was used to calibrate the system starting at 12:30, which, Sensors 2024, 24, 1150 9 of 22 in turn, achieved FTC.In addition, the system operated from 07:00 to 18:00 on weekdays.The system simulation model data output time step was 30 min.In this study, three within-control-loop sensors were selected, which were the thermostat of hot zone 1, the supply air temperature of the air-handling unit, and the chilled water supply temperature sensor of the CAC mainframe.Table 3 describes the setpoints of the target sensors and the magnitudes of the faults introduced.

PCA Filtering Fault-Tolerant Data Results
In this section, the T tz1 thermostat is used as an example to show the PCA filtering data.The training set was data from July during the operation of the CAC system.The confidence level was 0.05. Figure 5 shows the T tz1 thermostat filtering data using the PCA fault detection algorithm in August.The number of principal components was three and the cumulative contribution was 91.12%.The threshold was 1.64.Samples with Q-statistics exceeding the threshold were rejected and did not participate in the subsequent in situ incremental calibration.
Sensors 2024, 24, x FOR PEER REVIEW 9 of 22 FTC method using data increments was used to calibrate the system starting at 12:30, which, in turn, achieved FTC.In addition, the system operated from 07:00 to 18:00 on weekdays.The system simulation model data output time step was 30 min.In this study, three within-control-loop sensors were selected, which were the thermostat of hot zone 1, the supply air temperature of the air-handling unit, and the chilled water supply temperature sensor of the CAC mainframe.Table 3 describes the setpoints of the target sensors and the magnitudes of the faults introduced.

PCA Filtering Fault-Tolerant Data Results
In this section, the  thermostat is used as an example to show the PCA filtering data.The training set was data from July during the operation of the CAC system.The confidence level was 0.05. Figure 5 shows the  thermostat filtering data using the PCA fault detection algorithm in August.The number of principal components was three and the cumulative contribution was 91.12%.The threshold was 1.64.Samples with Q-statistics exceeding the threshold were rejected and did not participate in the subsequent in situ incremental calibration.Note: The cumulative contribution was 91.12% and greater than 85%.Then the number of principal components was three.

Comparison of the Accuracy between MLR-BI Calibration and PCA Reconstruction
Via the ISIC strategy, the PCA reconstruction method was selected in this section for a comparison study with the MLR-BI calibration method.Figure 6 shows the calibration accuracy of the target sensors for both the steady-state and non-steady-state data cases.In most situations, the calibration accuracy of the MLR-BI method was better than that of Note: The cumulative contribution was 91.12% and greater than 85%.Then the number of principal components was three.

Comparison of the Accuracy between MLR-BI Calibration and PCA Reconstruction
Via the ISIC strategy, the PCA reconstruction method was selected in this section for a comparison study with the MLR-BI calibration method.Figure 6 shows the calibration accuracy of the target sensors for both the steady-state and non-steady-state data cases.In most situations, the calibration accuracy of the MLR-BI method was better than that of PCA reconstruction.For the T tz1 thermostat and T sa sensor, the calibration accuracy using Sensors 2024, 24, 1150 10 of 22 MLR-BI was greater than that using PCA reconstruction in the steady-state case.For the T tz1 thermostat and T sa sensor, the MLR-BI calibration accuracies were 97.4% and 96.6%, respectively.For the T tz1 thermostat, the calibration accuracy using MLR-BI was greater than that using PCA reconstruction in the non-steady-state case.The T tz1 thermostat was calibrated to 97.9% and 94.4% accuracies using MLR-BI and PCA, respectively.The T chws sensor was calibrated to almost the same accuracy using both methods.
Sensors 2024, 24, x FOR PEER REVIEW 10 of 22 PCA reconstruction.For the  thermostat and  sensor, the calibration accuracy using MLR-BI was greater than that using PCA reconstruction in the steady-state case.For the  thermostat and  sensor, the MLR-BI calibration accuracies were 97.4% and 96.6%, respectively.For the  thermostat, the calibration accuracy using MLR-BI was greater than that using PCA reconstruction in the non-steady-state case.The  thermostat was calibrated to 97.9% and 94.4% accuracies using MLR-BI and PCA, respectively.The  sensor was calibrated to almost the same accuracy using both methods.

FTC Results for a Typical Day
Figure 7 shows the changes in the temperature and energy consumption of the CAC system on a typical day (1 August) under different conditions.After the fault was introduced into the sensor, the measurement data showed a fault deviation.This resulted in a deviation in the original set temperature from the normal temperature.When the measurement data were calibrated, the set temperature was allowed to return to its original state.At 12:00 p.m., a +2 °C bias fault was introduced into the  thermostat.The  dropped from 26.00 °C to 24.86 °C.At 13:00 p.m., either the IC or ISIC method was used to bring the temperature back up to 25.64 °C, and the energy consumption decreased from 14.88 M to 14.24 MJ.When a +2 °C bias fault was introduced into the  sensor, the  decreased from 14.00 °C to 12 °C, and the energy consumption increased from 15.03 MJ to 15.20 MJ.When FTC was performed at 13:00, the  and energy consumption were 13.90 °C and 14.09 MJ, respectively.At 13:00, the  increased from 5.00 °C to 7.00 °C, and the energy consumption was reduced from 15.60 MJ to 13.87 MJ.    7 shows the changes in the temperature and energy consumption of the CAC system on a typical day (1 August) under different conditions.After the fault was introduced into the sensor, the measurement data showed a fault deviation.This resulted in a deviation in the original set temperature from the normal temperature.When the measurement data were calibrated, the set temperature was allowed to return to its original state.At 12:00 p.m., a +2 • C bias fault was introduced into the T tz1 thermostat.The T tz1 dropped from 26.00 • C to 24.86 • C. At 13:00 p.m., either the IC or ISIC method was used to bring the temperature back up to 25.64 • C, and the energy consumption decreased from 14.88 M to 14.24 MJ.When a +2 • C bias fault was introduced into the T sa sensor, the T sa decreased from 14.00 • C to 12 • C, and the energy consumption increased from 15.03 MJ to 15.20 MJ.When FTC was performed at 13:00, the T sa and energy consumption were 13.90 • C and 14.09 MJ, respectively.At 13:00, the T chws increased from 5.00 • C to 7.00 • C, and the energy consumption was reduced from 15.60 MJ to 13.87 MJ.
Sensors 2024, 24, x FOR PEER REVIEW 10 of 22 PCA reconstruction.For the  thermostat and  sensor, the calibration accuracy using MLR-BI was greater than that using PCA reconstruction in the steady-state case.For the  thermostat and  sensor, the MLR-BI calibration accuracies were 97.4% and 96.6%, respectively.For the  thermostat, the calibration accuracy using MLR-BI was greater than that using PCA reconstruction in the non-steady-state case.The  thermostat was calibrated to 97.9% and 94.4% accuracies using MLR-BI and PCA, respectively.The  sensor was calibrated to almost the same accuracy using both methods.

FTC Results for a Typical Day
Figure 7 shows the changes in the temperature and energy consumption of the CAC system on a typical day (1 August) under different conditions.After the fault was introduced into the sensor, the measurement data showed a fault deviation.This resulted in a deviation in the original set temperature from the normal temperature.When the measurement data were calibrated, the set temperature was allowed to return to its original state.At 12:00 p.m., a +2 °C bias fault was introduced into the  thermostat.The  dropped from 26.00 °C to 24.86 °C.At 13:00 p.m., either the IC or ISIC method was used to bring the temperature back up to 25.64 °C, and the energy consumption decreased from 14.88 M to 14.24 MJ.When a +2 °C bias fault was introduced into the  sensor, the  decreased from 14.00 °C to 12 °C, and the energy consumption increased from 15.03 MJ to 15.20 MJ.When FTC was performed at 13:00, the  and energy consumption were 13.90 °C and 14.09 MJ, respectively.At 13:00, the  increased from 5.00 °C to 7.00 °C, and the energy consumption was reduced from 15.60 MJ to 13.87 MJ.  ; and (f) system energy consumption of the  sensor.

Thermal Comfort in August
Figure 8 depicts the average PPD and PMV in August in thermal zone 1.In the three sensors, the  thermostat fault had a greater impact on indoor thermal comfort.After FTC with IC and ISIC, the PPD decreased from 10.13% to 9.45% and 9.46%, respectively.The PMV increased from −0.38 to −0.07 and −0.06, respectively.Indoor dissatisfaction was reduced, and thermal sensation was more moderate.For the  sensor after FTC with IC and ISIC, the PPD decreased from 10.20% to 9.44% and 9.57%, respectively.The PMV increased from −0.12 to −0.05 and −0.06, respectively.The thermal environment in thermal zone 1 was restored to normal levels.

Thermal Comfort in August
Figure 8 depicts the average PPD and PMV in August in thermal zone 1.In the three sensors, the T tz1 thermostat fault had a greater impact on indoor thermal comfort.After FTC with IC and ISIC, the PPD decreased from 10.13% to 9.45% and 9.46%, respectively.The PMV increased from −0.38 to −0.07 and −0.06, respectively.Indoor dissatisfaction was reduced, and thermal sensation was more moderate.For the T sa sensor after FTC with IC and ISIC, the PPD decreased from 10.20% to 9.44% and 9.57%, respectively.The PMV increased from −0.12 to −0.05 and −0.06, respectively.The thermal environment in thermal zone 1 was restored to normal levels.

Thermal Comfort in August
Figure 8 depicts the average PPD and PMV in August in thermal zone 1.In the three sensors, the  thermostat fault had a greater impact on indoor thermal comfort.After FTC with IC and ISIC, the PPD decreased from 10.13% to 9.45% and 9.46%, respectively.The PMV increased from −0.38 to −0.07 and −0.06, respectively.Indoor dissatisfaction was reduced, and thermal sensation was more moderate.For the  sensor after FTC with IC and ISIC, the PPD decreased from 10.20% to 9.44% and 9.57%, respectively.The PMV increased from −0.12 to −0.05 and −0.06, respectively.The thermal environment in thermal zone 1 was restored to normal levels.For the target sensors, Figures 9 and 10 show the FTC results in August under different standard deviation of noise (SDN) conditions, respectively.Oriented to different SDNs, both the IC and ISIC strategies could maintain a good calibration accuracy.Using these two strategies, the MAE of the T tz1 thermostat varied less, at approximately 0.17 • C. The RE f and RE n were approximately 3.00% and 0.20%, respectively.In this case, the FTC results with ISIC were better than those with IC.When the SDN was from 0.01 to 0.15, the MAE of T sa was small and stable, at less than 0.02 • C. The RE f was higher than 4.60%, and the RE n was lower than 0.05%.When the SDN was between 0.01 and 0.15, the FTC results with IC and ISIC were almost the same.Except for the T chws sensor at SDN = 0.1, the FTC results for both strategies were better at SDN.At other standard deviations of noise, the MAE was lower than 0.1 • C, the RE f was close to 5.00%, and the RE n was lower than 0.05%.At SDN = 2, the RE f and RE n of ISIC were 4.8% and 0.1%, respectively, which are better than the FTC results with IC.The ISIC strategy excluded the calibrated data that exceeded the threshold during the data increment process, which improved the calibration accuracy and made the FTC results better.

Data Quality (a) Data Noise
For the target sensors, Figures 9 and 10 show the FTC results in August under different standard deviation of noise (SDN) conditions, respectively.Oriented to different SDNs, both the IC and ISIC strategies could maintain a good calibration accuracy.Using these two strategies, the MAE of the  thermostat varied less, at approximately 0.17 °C.The  and  were approximately 3.00% and 0.20%, respectively.In this case, the FTC results with ISIC were better than those with IC.When the SDN was from 0.01 to 0.15, the MAE of  was small and stable, at less than 0.02 °C.The  was higher than 4.60%, and the  was lower than 0.05%.When the SDN was between 0.01 and 0.15, the FTC results with IC and ISIC were almost the same.Except for the  sensor at SDN = 0.1, the FTC results for both strategies were better at SDN.At other standard deviations of noise, the MAE was lower than 0.1 °C, the  was close to 5.00%, and the  was lower than 0.05%.At SDN = 2, the  and  of ISIC were 4.8% and 0.1%, respectively, which are better than the FTC results with IC.The ISIC strategy excluded the calibrated data that exceeded the threshold during the data increment process, which improved the calibration accuracy and made the FTC results better.4 and 5 show the FTC results for the target sensors in a steady state and nonsteady state.For the  thermostat, the FTC results were better with the IC strategy using steady-state data and the ISIC strategy using non-steady-state data.The FTC results with ISIC were better than those with IC when non-steady-state data were used.For the  thermostat, the MAE,  , and  were 0.170 °C, 2.98%, and 0.20% for the FTC with ISIC using non-steady-state data, respectively.When steady-state data were used, the FTC results with IC were better than the results with ISIC for the  thermostat.In this situation, the MAE,  , and  were 0.168 °C, 3.39%, and 0.22%, respectively.For the  sensor, the FTC results with IC were better and independent of the steady-state and nonsteady-state data.In the case of the ISIC strategy, the FTC results were better using steadystate data for the  sensor.The MAE,  , and  were 0.105 °C, 4.41%, and 0.26%, respectively.The calibration accuracy of the  sensor was almost the same with the IC and ISIC strategies whether steady-state data or non-steady-state data were used.4 and 5 show the FTC results for the target sensors in a steady state and nonsteady state.For the T tz1 thermostat, the FTC results were better with the IC strategy using steady-state data and the ISIC strategy using non-steady-state data.The FTC results with ISIC were better than those with IC when non-steady-state data were used.For the T tz1 thermostat, the MAE, RE f , and RE n were 0.170 • C, 2.98%, and 0.20% for the FTC with ISIC using non-steady-state data, respectively.When steady-state data were used, the FTC results with IC were better than the results with ISIC for the T tz1 thermostat.In this situation, the MAE, RE f , and RE n were 0.168 • C, 3.39%, and 0.22%, respectively.For the T sa sensor, the FTC results with IC were better and independent of the steady-state and non-steady-state data.In the case of the ISIC strategy, the FTC results were better using steady-state data for the T sa sensor.The MAE, RE f , and RE n were 0.105 • C, 4.41%, and 0.26%, respectively.The calibration accuracy of the T chws sensor was almost the same with the IC and ISIC strategies whether steady-state data or non-steady-state data were used.

Data Volume
For the target sensors, Figures 11 and 12 show the FTC results with IC and ISIC for different data volumes.The FTC results for a 1-day volume were better when the T tz1 thermostat adopted the IC strategy.In this situation, the MAE, RE f , and RE n were 0.15 • C, 3.30%, and 0.13%, respectively.When the data volume was in the range of 7 days to 3 months, the FTC results of IC and ISIC were almost the same and deteriorated with the increase in the data volume.When 1-month data were selected for the T sa sensor, the FTC results with IC were better, with the MAE, RE f , and RE n being 0.07 • C, 4.49%, and 0.18%, respectively.When other data amounts were selected, the FTC results with IC and ISIC were almost the same.For the T chws sensor, the calibration accuracies of IC and ISIC were close to each other when the amount of data was small.When a 3-month volume was chosen, the MAE, RE f , and RE n of IC were 0.20 • C, 4.51%, and 0.04%, respectively.The results of IC were better than those of ISIC.Low outside temperatures exist in May and June.This could result in the possible shutdown of the CAC system.The data quality was not good in May and June.This also caused poor FTC results for the target sensor using 2-month (June-July) and 3-month (May-July) data.
Sensors 2024, 24, x FOR PEER REVIEW 15 of 22 thermostat adopted the IC strategy.In this situation, the MAE,  , and  were 0.15 °C, 3.30%, and 0.13%, respectively.When the data volume was in the range of 7 days to 3 months, the FTC results of IC and ISIC were almost the same and deteriorated with the increase in the data volume.When 1-month data were selected for the  sensor, the FTC results with IC were better, with the MAE,  , and  being 0.07 °C, 4.49%, and 0.18%, respectively.When other data amounts were selected, the FTC results with IC and ISIC were almost the same.For the  sensor, the calibration accuracies of IC and ISIC were close to each other when the amount of data was small.When a 3-month volume was chosen, the MAE,  , and  of IC were 0.20 °C, 4.51%, and 0.04%, respectively.The results of IC were better than those of ISIC.Low outside temperatures exist in May and June.This could result in the possible shutdown of the CAC system.The data quality was not good in May and June.This also caused poor FTC results for the target sensor using 2-month (June-July) and 3-month (May-July) data.

Variable Number
Figures 13 and 14 show the FTC results with IC and ISIC in different variable scenarios for the target sensors.The FTC results of the  sensor using IC and ISIC were relatively similar.Both strategies achieved better FTC results in variable scenario A, and the MAE,  , and  were 0.17 °C, 2.98%, and 0.20% when ISIC was used, respectively.In different variable scenarios, the FTC results of the  sensor were better in variable scenario F. The FTC results with ISIC were slightly better than those with IC, and the MAE and  were 0.17 °C and 4.65%, respectively.For different variable scenarios, the FTC results of the  sensor with IC and ISIC were basically the same, and the  was approximately 4.87%.

Variable Number
Figures 13 and 14 show the FTC results with IC and ISIC in different variable scenarios for the target sensors.The FTC results of the T tz1 sensor using IC and ISIC were relatively similar.Both strategies achieved better FTC results in variable scenario A, and the MAE, RE f , and RE n were 0.17 • C, 2.98%, and 0.20% when ISIC was used, respectively.In different variable scenarios, the FTC results of the T sa sensor were better in variable scenario F. The FTC results with ISIC were slightly better than those with IC, and the MAE and RE f were 0.17 • C and 4.65%, respectively.For different variable scenarios, the FTC results of the T chws sensor with IC and ISIC were basically the same, and the RE f was approximately 4.87%.

Conclusions
This paper proposed an in situ selective incremental calibration strategy.This strategy was used to address the problem of the lack of available information about HVAC systems affecting the performance of data-driven fault-tolerant control models.Using the EnergyPlus-Python co-simulation testbed, the central air-conditioning system of a singlestory office building was simulated, and faults were introduced into the T tz1 thermostat, T sa sensor, and T chws sensor.This study quantified the changes in target variables, energy consumption, and thermal comfort before and after fault-tolerant control.The effects of the data quality, data volume, and number of variables on the fault-tolerant control results were evaluated.The main conclusions are as follows: (1) The fault-tolerant control strategy using data increments can lead to good faulttolerant control results for a central air-conditioning system.Compared with the sensor fault operation, the fault-tolerant control strategy reduced the total energy consumption by 2.98%, 3.72%, and 4.87% for the T tz1 thermostat and the faulty T sa and T chws sensors, respectively.For the T tz1 thermostat and faulty T sa , the predicted percentage dissatisfaction was reduced by 0.67% and 0.63%, respectively.The system energy consumption and indoor thermal comfort were close to normal levels after fault-tolerant control.(2) For the T tz1 thermostat and T sa sensor, better fault-tolerant control results were obtained by using in situ selective incremental calibration when the standard deviation of noise was small.When non-steady-state data were used, better results were obtained by using in situ selective incremental calibration for the T tz1 thermostat.For the T chws sensor, the data quality had less influence on the fault-tolerant control results.(3) Compared with in situ calibration, the T tz1 thermostat obtained good fault-tolerant control results with the in situ selective incremental calibration strategy with a 7-day data volume and sufficiently variable scenarios.The T sa and T chws sensors obtained better fault-tolerant control results with the in situ selective incremental calibration strategy with a 14-day data volume and variable scenarios with limited information.

Figure 1 22 Figure 1 .
Figure 1 illustrates the research framework of this study, including three main steps: (1) The FTC strategy for in situ selective incremental calibration (ISIC) was proposed.MLR-BI and PCA were used to realize fault calibration and data filtering, respectively.A brief flow of the ISIC strategy is shown in Appendix A. (2) The fault modeling and FTC of the indoor air thermostat (T tz1 ), supply air temperature (T sa ), and chilled water supply temperature (T chws ) in the CAC system using the EnergyPlus-Python co-simulation testbed was carried out to demonstrate the variations in target variables and energy consumption on a typical day and the changes in thermal comfort in August.(3) The effects of the data quality, data volume, and number of variables on the FTC results were evaluated.FOR PEER REVIEW 6 of 22

Figure 1 .
Figure 1.Research framework of this study.

Figure 2 .
Figure 2. Flow chart of in situ calibration and in situ selective incremental calibration.

Figure 2 .
Figure 2. Flow chart of in situ calibration and in situ selective incremental calibration.

Figure 3 .
Figure 3. Target sensor variable scenario settings.Note: Taking the variable scenario G as an example, the dependent variable involved in the MLR-BI modeling is the chilled water supply temperature.The independent variables are the chilled water return temperature ( , ), the cooling water supply temperature ( , ), the cooling water return temperature ( , ), the chilled water mass flow rate ( ), the cooling water mass flow rate ( ), and the chiller's energy consumption ( ).

Figure 3 .
Figure 3. Target sensor variable scenario settings.Note: Taking the variable scenario G as an example, the dependent variable involved in the MLR-BI modeling is the chilled water supply temperature.The independent variables are the chilled water return temperature (T chw,r ), the cooling water supply temperature (T cw,s ), the cooling water return temperature (T cw,r ), the chilled water mass flow rate (M chw ), the cooling water mass flow rate (M cw ), and the chiller's energy consumption (E chiller ).

22 Figure 3 .
Figure 3. Target sensor variable scenario settings.Note: Taking the variable scenario G as an example, the dependent variable involved in the MLR-BI modeling is the chilled water supply temperature.The independent variables are the chilled water return temperature ( , ), the cooling water supply temperature ( , ), the cooling water return temperature ( , ), the chilled water mass flow rate ( ), the cooling water mass flow rate ( ), and the chiller's energy consumption ( ).

Figure 5 .
Figure 5. Demonstration of PCA filtering of fault-tolerant data using  thermostat as an example.Note: The cumulative contribution was 91.12% and greater than 85%.Then the number of principal components was three.

Figure 5 .
Figure 5. Demonstration of PCA filtering of fault-tolerant data using T tz1 thermostat as an example.Note: The cumulative contribution was 91.12% and greater than 85%.Then the number of principal components was three.

Figure 6 .
Figure 6.Comparison of accuracy using MLR-BI calibration and PCA reconstruction.(a) Steady-state data.(b) Non-steady-state data.

Figure
Figure7shows the changes in the temperature and energy consumption of the CAC system on a typical day (1 August) under different conditions.After the fault was introduced into the sensor, the measurement data showed a fault deviation.This resulted in a deviation in the original set temperature from the normal temperature.When the measurement data were calibrated, the set temperature was allowed to return to its original state.At 12:00 p.m., a +2 • C bias fault was introduced into the T tz1 thermostat.The T tz1 dropped from 26.00 • C to 24.86 • C. At 13:00 p.m., either the IC or ISIC method was used to bring the temperature back up to 25.64 • C, and the energy consumption decreased from 14.88 M to 14.24 MJ.When a +2 • C bias fault was introduced into the T sa sensor, the T sa decreased from 14.00 • C to 12 • C, and the energy consumption increased from 15.03 MJ to 15.20 MJ.When FTC was performed at 13:00, the T sa and energy consumption were 13.90 • C and 14.09 MJ, respectively.At 13:00, the T chws increased from 5.00 • C to 7.00 • C, and the energy consumption was reduced from 15.60 MJ to 13.87 MJ.

Figure 7 .
Figure 7. Changes in target variables and system energy consumption under three operating conditions on 1 August: (a)  ; (b) system energy consumption for  thermostat; (c)  ; (d) system energy consumption for the  sensor; (e) ; and (f) system energy consumption of the  sensor.

Figure 8 .
Figure 8. Changes in thermal comfort before and after FTC in thermal zone 1 in August: (a) PPD; (b) PMV.

Figure 7 .
Figure 7. Changes in target variables and system energy consumption under three operating conditions on 1 August: (a) T tz1 ; (b) system energy consumption for T tz1 thermostat; (c) T sa ; (d) system energy consumption for the T sa sensor; (e) T chws ; and (f) system energy consumption of the T chws sensor.

Sensors 2024 , 22 Figure 7 .
Figure 7. Changes in target variables and system energy consumption under three operating conditions on 1 August: (a)  ; (b) system energy consumption for  thermostat; (c)  ; (d) system energy consumption for the  sensor; (e) ; and (f) system energy consumption of the  sensor.

Figure 8 .
Figure 8. Changes in thermal comfort before and after FTC in thermal zone 1 in August: (a) PPD; (b) PMV.

Figure 8 .
Figure 8. Changes in thermal comfort before and after FTC in thermal zone 1 in August: (a) PPD; (b) PMV.

Figure 9 .
Figure 9. MAEs of target sensors for IC and ISIC FTC with different SDNs: (a)  thermostat; (b)  sensor; and (c)  sensor.

Figure 10 .
Figure 10.Relative errors of August energy consumption of target sensors for IC and ISIC FTC with different SDNs: (a)  thermostat; (b)  sensor; and (c)  sensor.

Figure 10 .
Figure 10.Relative errors of August energy consumption of target sensors for IC and ISIC FTC with different SDNs: (a) T tz1 thermostat; (b) T sa sensor; and (c) T chws sensor.(b) Steady State and Non-steady State Tables4 and 5show the FTC results for the target sensors in a steady state and nonsteady state.For the T tz1 thermostat, the FTC results were better with the IC strategy using steady-state data and the ISIC strategy using non-steady-state data.The FTC results with ISIC were better than those with IC when non-steady-state data were used.For the T tz1 thermostat, the MAE, RE f , and RE n were 0.170 • C, 2.98%, and 0.20% for the FTC with ISIC using non-steady-state data, respectively.When steady-state data were used, the FTC results with IC were better than the results with ISIC for the T tz1 thermostat.In this situation, the MAE, RE f , and RE n were 0.168 • C, 3.39%, and 0.22%, respectively.For the T sa sensor, the FTC results with IC were better and independent of the steady-state and non-steady-state data.In the case of the ISIC strategy, the FTC results were better using steady-state data for the T sa sensor.The MAE, RE f , and RE n were 0.105 • C, 4.41%, and 0.26%, respectively.The calibration accuracy of the T chws sensor was almost the same with the IC and ISIC strategies whether steady-state data or non-steady-state data were used.

Figure 11 . 11 .
Figure 11.MAEs of target sensors for IC and ISIC with different data volumes: (a)  thermostat; (b)  sensor; and (c)  sensor.11.MAEs of target sensors for IC and ISIC with different data volumes: (a) T tz1 thermostat; (b) T sa sensor; and (c) T chws sensor.

Figure 12 .
Figure 12.Relative errors of August energy consumption of target sensors for IC and ISIC with different data volumes: (a)  thermostat; (b)  sensor; and (c)  sensor.

Figure 12 .
Figure 12.Relative errors of August energy consumption of target sensors for IC and ISIC with different data volumes: (a) T tz1 thermostat; (b) T sa sensor; and (c) T chws sensor.

Figure 13 .
Figure 13.MAEs of target sensors for IC and ISIC in different variable scenarios: (a)  thermostat; (b)  sensor; and (c)  sensor.

Figure 14 .
Figure 14.Relative errors of energy consumption in August of target sensors for IC and ISIC in different variable scenarios: (a)  thermostat; (b)  sensor; and (c)  sensor.

Figure 13 . 22 Figure 13 .
Figure 13.MAEs of target sensors for IC and ISIC in different variable scenarios: (a) T tz1 thermostat; (b) T sa sensor; and (c) T chws sensor.

Figure 14 .
Figure 14.Relative errors of energy consumption in August of target sensors for IC and ISIC in different variable scenarios: (a)  thermostat; (b)  sensor; and (c)  sensor.

Figure 14 .
Figure 14.Relative errors of energy consumption in August of target sensors for IC and ISIC in different variable scenarios: (a) T tz1 thermostat; (b) T sa sensor; and (c) T chws sensor.

Table 2 .
CAC system operation schedule.

Table 3 .
Target sensor setpoints and fault settings.

Table 2 .
CAC system operation schedule.

Table 3 .
Target sensor setpoints and fault settings.

Table 4 .
MAEs of target sensors during IC and ISIC in steady state and non-steady state.

Table 5 .
Relative errors of August energy consumption of target sensors during IC and ISIC in steady state and non-steady state.
For the target sensors, Figures11 and 12show the FTC results with IC and ISIC for

Table 4 .
MAEs of target sensors during IC and ISIC in steady state and non-steady state.

Table 5 .
Relative errors of August energy consumption of target sensors during IC and ISIC in steady state and non-steady state.
Target sensors to be calibratedV 1 − V pPhysical sensors other than the target sensor to be calibrated x Constant terms of the MLR modelsα 1 − α p ,β 1 − β cCoefficients corresponding to each of the above variables c α Normal deviation corresponding to the upper (1 − α) percentile Figure A1.Brief flowchart of FTC strategy for in situ selective incremental calibration. c