Research on Model Calibration Method of Chiller Plants Based on Error Reverse Correction with Limited Data

Traditional model calibration methods are supported by sufficient data. However, sufficient data cannot always be satisfied in reality. To solve this problem, a model calibration method based on error reverse correction was investigated. Firest, a traditional calibration method was used for preliminary model calibration to obtain simulation data and errors. Then, the sources of the simulation errors were analyzed to determine the distribution characteristics of the corresponding operating conditions of the model. Finally, the performance of the model was reversely corrected by adding a correction term to the original model. The proposed calibration method was tested on a real chiller plant. The results showed that the proposed method improved prediction accuracy by 2.61%. The maximum mean bias error ( MBE ) for monthly chiller energy consumption was 2.66% with the proposed method, while it was 10.42% with the traditional method. Overall, in scenarios with limited data, the proposed calibration method can effectively improve the accuracy of simulation results.


Introduction
The HVAC systems account for 40~60% of energy consumption in buildings (Xiao, 2022).As an important link in the central air conditioning system, chiller plant energy consumption occupies a major part of the total energy consumption (Wang, 2018;Chen, 2022).Therefore, reducing the operating energy consumed by chiller plant equipment has become an important way to save energy (Shi, 2021).In the analysis of chiller plant energy efficiency, simulation has been widely used, and a number of simulation software have been developed, including EnergyPlus, TRNSYS, and Modelica, etc. Simulation models can be used to evaluate system performance, identify opera-tional problems, propose improvement solutions, and verify the effectiveness of energy saving measures (Karami, 2018;Fan, 2021).In order to improve the reliability of simulation results, we need to establish a baseline model that can accurately predict the operating conditions and energy con-sumption of the system.Calibration is one of the most important steps (Fu, 2019).The calibration process aims at closing, or at least reducing, the performance gap.It consists of tuning the different unknown input parameters within a defined range in order to match the simulated and measured values (Guyot, 2020).Tüysüz (2020) divided the calibration process into four main stages: modeling, measuring the calibration data, making improvements for calibration, and acquiring results.In the preliminary stage, the model input data required for calibration and the calibration target data are collected.Then, under the condition that the model inputs match the actual inputs, the selected calibration parameters are then tuned so that the simulated outputs match the observed outputs.Therefore, high quality data is an important basis for calibration.When collecting data for calibration, two dimensions must be considered: the attribute dimension (i.e., the parameters used for calibration, such as temperature, flow rate, and power) and the time dimension (i.e., the time resolution for monitoring various data, such as daily, monthly, hourly, etc.).The time dimension of the data required for calibration depends on the size of the study population and the use of the simulation.The calibration process can be performed based on yearly, monthly, daily, or hourly data.Chong (2021) observed a clear trend in increasing temporal data resolution as researchers moved from urban scale to building to component/system energy simulations.The calibration process is typically based on hourly data in simulation studies aimed at system energy optimiza-tion.Lee (2012) proposed a simulation-optimization approach for the energy efficiency of a chilled water system, and the accuracy of the simulation system was verified using hourly data.Similarly, Li (2020) used simulations to find the optimal operation strategy for a data center cooling system with a water-side economizer by selecting practical weather data, cooling load, and equipment control signals as inputs.Simulation results were compared with hourly measured data to verify the accuracy of the model predictions.Given the hourly variability in energy consumption and equipment operating conditions, many researchers have agreed that chiller plant equipment models should be calibrated based on hourly or even smaller time scales data (Li, 2018).In the above, we discussed the requirements for the time dimension of the data for chiller plant model calibration, and we will now introduce the characteristics of the Proceedings of the 18th IBPSA Conference Shanghai, China, Sept. 4-6, 2023 https://doi.org/10.26868/25222708.2023.1630attribute dimension.The type of measured data required for model calibration is related to the model characteristics.Martin (2019) used the airflow rate and the mixed air temperature to calibrate a variable air volume fan model in EnergyPlus to accurately predict the fan power and supply air temperature.Huang (2017) developed a simulation model for the primary loop of the studied chiller plant and calibrated the chiller models using one week of hourly measured data.The temperatures of the condenser and chilled water entering the chillers were selected as input variables during the calibration process.As a result, the relative error in chiller power was less than 5% by adjusting the chiller performance curves.Yin (2016) selected the chiller, cooling tower, and fan power as target variables for the calibration of HVAC systems.In summary, model calibration for HVAC equipment places more emphasis on the identification of performance values/curves.In addition, the type of data used for calibration needs to match the characteristics of the simulation model, and there are differences in the type of data required by different simulation models.Monfet (2013) calibrated the EnergyPlus model of a central cooling plant and analyzed the differences in model inputs and performance values/curves of key equipment models in EnergyPlus and TRNSYS.Fu (2019) modeled the cooling and control systems of an actual data center using Modelica and detailed the model inputs, calibration parameters, and calibration targets when calibrating critical equipmen.Table 1 shows a summary of the literature reviewed.In order to obtain a baseline model that can accurately predict the actual operating condition of the system, most current studies often select the operating power, medium temperature, and other state parameters as calibration targets and perform hourly calibration.At the same time, almost all calibration studies are based on a complete type of hourly measured data, but this is not always the case.One of the longstanding challenges of energy and buildings research has been the issue of data access (Summerfield, 2012).The available measured data for model calibration varies in quality, quantity, and frequency from one chiller plant to another (Martin, 2019).In reality, the lack of time dimension of data collection for chiller plant operation is objective, and there are a variety of limited data scenarios; e.g., the lack of hourly measured data for equipment power consumption available only as a monthly cumulative value (Cacabelos, 2015) and the lack of itemized equipment power consumption (Hinkelman, 2022), etc.This chiller plant is often in operation for a long period of time, so the need for energy saving renovation is more urgent.However, little attention has been paid to model calibration based on limited data.The calibration process encounters difficulties when the measured data condi-tions do not match the data requirements of the simulation tools commonly used for model calibration.This study focuses on the calibration method of chiller plants based on limited data.

Research Object
Figure 1 depicts the system diagram of a water-cooled chiller system, which is used to supply chilled water to the air conditioning system.The chiller system consists of nine chillers, nine condenser pumps, nine chilled pumps, and eleven cooling towers distributed in two separate loops (Loop A and Loop B).When calibrating the chiller plant simulation system, it is common practice to individually calibrate the equipment in the system.As the main components of the chiller plant, the chillers consume about 35~40% of the energy consumption of the air conditioning system (Liu, 2017).Therefore, the chillers are selected as the study object to introduce the calibration method.Some studies employed standard least-squares linear regression techniques to calibrate chiller models (Hydeman, 2002).Other key studies proposed an optimization-based calibration approach (Fu, 2019;Huang, 2017).We summarize the ideal data conditions required to calibrate the DOE-2 chiller model based on the above studies.Table 2 shows both the ideal dataset and the limited dataset dealt with in this paper.It can be seen that there is no meas-ured data for the cooling water supply temperature in the limited dataset.Furthermore, only monthly statistics for equipment energy consumption are available.It is impossible to evaluate the hourly accuracy of chiller performance under the limited data conditions.To improve the prediction accuracy of the model with limited data, this paper proposes a model calibration method based on error reverse correction.

Modeling
The chillers are modeled using the Modelica language.The Modelica Buildings library contains components for HVAC system modeling, equipment performance libraries, etc., that can be directly used for modeling or improved based on existing models (Wetter, 2014).We chose to use an improved DOE-2 electrical chiller model, and the model path in the Building library was "Building.Fluid.Chillers.ElectricReformulatedEIR".The chiller model consists of three performance curves: "CAPFT" -a curve that represents available cooling capacity as a function of condenser leaving and evaporator leaving fluid temperature; "EIRFT" -a curve Proceedings of the 18th IBPSA Conference Shanghai, China, Sept. 4-6, 2023 3704 https://doi.org/10.26868/25222708.2023.1630that represents the full load efficiency as a function of condenser leaving and evaporator leaving fluid temperature; and "EIRFPLR" -a curve that represents the efficiency as a function of evaporator leaving fluid temperature and the partload ratio.Detailed curves are described in Equations ( 1)-( 3), and the chiller power calculation method is described in Equation ( 4).CAPFT = a 1 + T chw,out (a 2 + a 3 T chw,out ) + T cw,out (a 4 + a 5 T cw,out ) + a 6 T chw,out T cw,out (1) (3) where Tchw,out is the chilled water supply temperature, °C; Tcw,out is the cooling water supply temperature, °C; PLR is the part load ratio; Pref is the nominal power of the chiller, kW; P is the power of the chiller, kW; and a, b, c are the coefficients of the performance curve.

Optimization-Based Calibration
This stage aims to match the simulation results with the observed outputs by adjusting the calibration parameters.We used the GenOpt optimization engine and employed particle swarm optimization (PSO) to complete the calibration parameters update.Normalized mean bias error (NMBE) can be used as an indicator of all biases in the simulation predictions, which can be expressed by Equation ( 5).
where   and   are the measured and simulated values, respectively; m ̅ is the mean of the measured values; n is the number of data points ( n monthly = 12, n daily = 365, n hourly = 8760).

Error-Based Reverse Correction
The performance of the chiller depends on the operating conditions of the unit, and this is also true for the simulation model.After determining the performance curve coefficients, the power of the DOE-2 chiller model is directly related to Tchw,ou, Tcw,out, and PLR.This paper defined the state parameter that affects the model performance as the performance-related state parameter.By mining the relationship between simulation errors and the distribution of the performance-related state parameters, it is possible to determine the range of model operating condition distributions that cause errors.Based on this, model performance can be corrected within the determined range of the operating condition to effectively improve model accuracy.In Figure 2, the flow chart of error-based reverse correction is presented.After completing the optimization-based calibration based on the monthly measured data of chillers energy consumption, the hourly simulated data of performance-related state parameters and the monthly simulated data of chillers energy consumption can be obtained.The mean bias error (MBE) was used to calculate the error between the simulation and reality, determining the degree of proximity for each month.The formula is as follows: where   and   are the measured and simulated values, respectively;  is monthly interval.Then, the performance-related state parameter that can reflect the difference in the distribution of the device operating conditions under the baseline dataset and the required-corrected dataset is selected as the key state parameter (Parm_key).The interval in which the key state parameter is centrally distributed in the baseline dataset is defined as the baseline interval (I1).In the undercorrected dataset, the part of the key state parameter that is centrally distributed and different from the baseline interval is defined as the under-corrected interval (I2).Similarly, in the over-corrected dataset, the part of the key state parameter that is centrally distributed and different from the baseline interval is defined as the over-corrected interval (I3).Finally, the model performance is reversely corrected by adding a correction term α to the original model.As showen in Equation( 7).
The correction term α is a piecewise function with the key state parameter as the independent variable, which is assigned a value greater than 1 in the under-correction interval to amplify the model performance, and a value less than 1 in the over-correction interval to retrace the model performance, and 1 in the other intervals to Proceedings of the 18th IBPSA Conference Shanghai, China, Sept. 4-6, 2023 3705 https://doi.org/10.26868/25222708.2023.1630maintain the original level of performance correction.In addition, the value of the correction term can be determined by optimization.The objective of the optimization is to minimize the difference between the model output and the corresponding measurement.The difference is defined by NMBE.The formulation of the optimization problem is shown in Equation (8).
Calibration Results Evaluation Some criteria are selected to evaluate the calibrated model.NMBE and CV(RMSE) are the most commonly used (Ruiz, 2017).NMBE can be calculated by Equation ( 5), and CV(RMSE) can be calculated by Equation ( 9).
Furthermore, the acceptable criteria defined by ASHRAE, IPMVP, and FEMP are shown in Table 3.The calibrated model is then approaching reality and can be used to evaluate the effectiveness of various energy conservation measures be-fore applying them to an actual chiller plant.

Case Case Description
In this case, a chiller plant located in Xiamen City, China, was chosen as the research object.The configuration of the chiller plant is shown in Figure 1.The chiller plant has two separate cooling loops (Loop A and Loop B).Due to the high energy consumption, we focused on the model calibration for the chillers.The nominal data for the chillers are listed in Table 4.There are five identical chillers in Loop A and four in Loop B. Therefore, two models need to be calibrated; denoted as Chiller_A and Chiller_B, respectively.We determined the number of units, the connection type, and the value of the nominal parameters in combination with the design information.The operating data of the chillers in 2021 were monitored, as shown in Table 5.The input data required for chiller simulation, such as the return temperature and the flow rate of chilled water, the return temperature and the flow rate of cooling water, etc., are monitored each hour.
However, only monthly monitoring data is available for the energy consumption of the chillers.

Calibration Scheme
In order to compare the effectiveness of the calibration method proposed in this study, we created three calibration schemes: 1. Init: Calibrating the chiller model using the optimization-based calibration method.2. Opt: On the basis of Init, calibrating the chiller model based on error reverse correction, and the value of the correction terms are determined by optimization.

Init: Optimization-Based Calibration
In Init, we performed only the traditional optimizationbased calibration.Figure 3 shows the chiller calibration system model in Modelica.The measured data of chilled water supply temperature were selected as the set point in the simulation.The calibration target was the total energy consumption of the chiller.When calibrating two types of chillers (Chiller_A and Chiller_B), the number of correction coefficients to be optimized is significant, and it is difficult to determine a suitable range for finding the optimum.Therefore, we applied an exhaustive optimization method to select the best performance coefficients from the reference samples of chillers provided by the Building library.The chiller energy consumption NMBE monthly was used as a screening index.Finally, the nominal COP of Chiller_A and Chiller_B were calibrated considering the possible performance degradation of the chillers.Min(NMBE)was chosen as the optimization objective, and the optimization process was completed using GenOpt with PSO.Finally, the values of the correction terms for the different over-correction intervals and under-correction intervals were determined.GenOpt was used to determine the values of correction terms.In the proposed approach, the particle swarm optimization was selected.

Results
Init: Optimization-Based Calibration Table 6 shows the optimization results of the calibration parameters in Init.As can be seen from Table 6, the nominal COP of Chiller_A fell from 5.32 to 5.21.Therefore, it is necessary to consider the performance degradation that occurs during the operation of the equipment.7 shows the classification results of the baselined dataset, over-corrected dataset, and under-corrected dataset.The monthly MBE within the range of −3% to 3% was considered to be an acceptable error level, and the measured data were divided into three groups according to the calibration results of Init.The data in March and April belonged to the over-corrected dataset.The data in February, November, and December belonged to the under-corrected dataset.The data in the remaining months belonged to the baseline dataset.We chose PLR together with Tchw,out as key state parameters.By the same method, the distribution ranges of overcorrected intervals and under-corrected intervals for the two loop chillers were determined.On this basis, the values of the correction terms were determined in next step.Table 8 shows the results of the determined interval ranges, where α A,i , α B,i are the correction term set for Chiller_A and Chiller_B, respectively.On this basis, the values of the correction terms were determined in next step.Table 9 shows the values of the correction terms in Opt. Figure 6 shows the monthly measured and simulated

Discussion
As can be seen from the above results, the calibration accuracy of the model can be improved by a calibration method based on error reverse correction under limited data conditions.To achieve this goal, we improved the traditional calibration method.By analysing the causes and characteristics of calibration errors, the chiller models were reverse corrected to improve the calibration accuracy.
Table 10 summarizes the NMBE, and CV(RMSE) values for the calibrated model in two schemes, it could be seen that the accuracy of the model calibration was significantly improved after applying the error-based reverse correction.As can be seen in Figure 7, the MBE values for the same month belonging to the baseline dataset were almost unchanged in the three schemes.In addition, the MBE values in the over-corrected dataset and the undercorrected dataset were significantly reduced by the errorbased reverse correction.After optimizing the original model by adding correction terms, this did not affect the prediction accuracy of the model in the baseline dataset.
This confirms that the interval classification results shown in Table 8 are accurate.

Conclusions
In this study, a model calibration method based on error reverse correction was proposed to improve the model prediction accuracy when only the monthly measured energy consumption data were available.The proposed method was tested in a chiller plant in Xiamen, China, via simulation.Based on the results of the test case studies, conclusions could be drawn as follows: 1.When only the monthly measured energy consumption data are available, the calibration results with the optimization-based method can meet the acceptable criteria, but the accuracy of the chiller energy consumption in different months greatly varies.
In Init, the CV(RMSE) was 3.96%, which met the acceptable criteria.However, the MBE values in July, April, and December were −0.02%, −8.14%, and 10.42%, respectively.2. Based on the simulation data and the error information obtained from the preliminary calibration, it can be found that there is a relationship between the model simulation accuracy and the model operating conditions.By analyzing the distribution characteristics of the key state parameters in the different months, it is possible to determine the range of operating conditions where errors occur.This is the basis for the reverse correction of model performance.
3. Improving the model by adding correction terms can significantly improve the predictive accuracy of the model.In Opt, the CV(RMSE) was reduced from 3.96% to 1.35%.The maximum MBE was reduced from 10.42% to 2.66%.This study was an initial attempt to calibrate the chiller model with limited data.With the proposed calibration method, accurate simulation results were obtained even under limited data conditions.The calibrated model could be used to support the simulation verification of energysaving measures.This helps facilitate the use of simulation in engineering cases.

Figure 1 :
Figure 1: The schematic diagram of a water-cooled chiller system.The DOE-2 model is the most commonly used chiller model.The calibration methods of the DOE-2 chiller model have been demonstrated in many field studies.Some studies employed standard least-squares linear regression techniques to calibrate chiller models(Hydeman, 2002).Other key studies proposed an optimization-based calibration approach(Fu, 2019;Huang, 2017).We summarize the ideal data conditions required to calibrate the DOE-2 chiller model based on the above studies.Table2shows both the ideal dataset and the limited dataset dealt with in this paper.It can be seen that there is no meas-ured data for the cooling water supply temperature in the limited dataset.Furthermore, only monthly statistics for equipment energy consumption are available.It is impossible to evaluate the hourly accuracy of chiller performance under the limited data conditions.To improve the prediction accuracy of the model with limited data, this paper proposes a model calibration method based on error reverse correction.

Figure 2 :
Figure 2: Schematic illustrating the process of errorbased reverse correction.The simulation dataset is divided into the baseline dataset (data in M1 and M4) and the required-corrected dataset (data in M2 and M3) according to the simulation error level.The required-corrected dataset include the undercorrected dataset (data in M2), with a large negative bias, and the over-corrected dataset (data in M3), with a large positive bias.Then, the performance-related state parameter that can reflect the difference in the distribution of the device operating conditions under the baseline dataset and the required-corrected dataset is selected as the key state parameter (Parm_key).The interval in which the key state parameter is centrally distributed in the baseline dataset is defined as the baseline interval (I1).In the undercorrected dataset, the part of the key state parameter that is centrally distributed and different from the baseline interval is defined as the under-corrected interval (I2).Similarly, in the over-corrected dataset, the part of the key state parameter that is centrally distributed and different from the baseline interval is defined as the over-corrected interval (I3).Finally, the model performance is reversely corrected by adding a correction term α to the original model.As showen in Equation(7).P=P ref •CAPFT•EIRFT•EIRFPLR•α(Parm_key)

Figure 3 :
Figure 3: Chiller calibration system model in Modelica.Opt: Error-Based Reverse Correction Opt was performed based on Init.Since the performancerelated state parameters of the chiller are Tchw,ou, Tcw,out, and PLR, it was first necessary to filter out the appropriate key state parameter based on the distribution range of the parameters.Then, the ranges of over-correction intervals and under-correction intervals were determined based on

Figure 4
Figure4shows the monthly measured and simulated chiller energy consumption.After completing the optimization-based calibration, the accuracy of chiller energy consumption significantly varied from month to month.The monthly MBE reached −8.14% in April and 10.42% in December.The prediction accuracy of the calibrated model was not satisfactory.

Figure 4 :
Figure 4: Energy consumption of the chiller.Opt: Error-Based Reverse Correction Table7shows the classification results of the baselined dataset, over-corrected dataset, and under-corrected dataset.The monthly MBE within the range of −3% to 3% was considered to be an acceptable error level, and the measured data were divided into three groups according to the calibration results of Init.The data in March and April belonged to the over-corrected dataset.The data in February, November, and December belonged to the under-corrected dataset.The data in the remaining months belonged to the baseline dataset.Table7: The classification results of the baselined dataset, over-corrected dataset, and under-corrected dataset.
Figure 5 shows the distribution of key state parameters for the chillers in Loop A in the baseline dataset and in April.In the dot-density map, the darker the color, the more chiller operating points are distributed.It is obvious that the two were different, and the operating conditions of the chillers in April were concentrated in the blue rectangle: PLR ∈ (0.56,0.83) and T chw,out ∈ (7.4,7.7)(a) Dot-density map of Tchw,out versus PLR in the baseline dataset for chillers in Loop A; (b) dotdensity map of Tchw,out versus PLR in April for chillers in Loop A.

Figure 7 :
Figure 7: The MBE values in two schemes.

Table 3 :
Acceptance criteria for the calibration process.

Table 4 :
Technical parameters of the chillers in the cooling system.

Table 6 :
Optimization results of calibration parameters in Init.

Table 7 :
The classification results of the baselined dataset, over-corrected dataset, and under-corrected dataset.
Figure 6: Energy consumption of the chiller in Opt.

Table 10 :
Monthly NMBE, and CVRMSE, valuesfor the calibrated model in three schemes.