Application of a Model Based on Rough Set Theory (RST) for Estimating the Temperature of Brine from Vertical Ground Heat Exchangers (VGHE) Operated with a Heat Pump—A Case Study

Abstract

This work presents the results of a study that used a model based on rough set theory (RST) to assess the brine temperature of vertical ground heat exchangers (VGHEs) to feed heat pumps (HP). The purpose of this research was to replace costly brine temperature measurements with a more efficient approach. The object of this study was a public utility building located in Poland in a temperate continental climate. The building is equipped with a heating system using a brine–water HP installation with a total capacity of 234.4 kW, where the lower heat source consists of 52 vertical ground probes with a total length of 5200 m. The research was conducted during the heating season of 2018/2019. Based on the data, the heat energy production was determined, and the efficiency of the system was assessed. To predict the brine temperature from the lower heat source, a model based on RST was applied, which allows for the analysis of general, uncertain, and imprecise data. Weather data, such as air temperature, solar radiation intensity, degree days of the heating season, and thermal energy consumption in the building, were used for the analysis. The constructed model was tested on a test dataset. This model achieved good results with a Mean Absolute Percentage Error (MAPE) of 12.2%, a Coefficient of Variation Root Mean Square Error (CV RMSE) of 14.76%, a Mean Bias Error (MBE) of −1.3%, and an R-squared (R²) value of 0.98, indicating its usefulness in estimating brine temperature. These studies suggest that the described method can be useful in other buildings with HP systems and may contribute to improving the efficiency and safety of these systems.

1. Introduction

Thermal energy stored in the ground can be freely used to heat or cool buildings. The ground is characterized by significant thermal capacity and temperature stability, especially in deeper layers [1]. During the heat flow in the ground, conduction plays the most significant role in heat exchange. Evaporation, condensation, freezing, and thawing of water have less importance. The ground temperature changes with depth, with more significant variations observed in near-surface layers, where heat accumulates from solar radiation, and temperature variability depends on the seasons. Weather factors, such as atmospheric pressure, solar radiation, humidity, air velocity, precipitation, and the type of ground cover, influence temperature at these layers. When the depth reaches a certain magnitude, typically above 13 to 15 m [1,2], heat primarily originates from the Earth’s interior, known as the geothermal heat of the ground. The temperature steadily increases with depth unless there are disrupting factors, such as vertical ground probes.

Heating systems that operate in conjunction with heat pumps (HPs) and employ vertical ground heat exchangers (VGHE) harness the thermal energy stored in the subterranean aqueous environment. Alterations in the temperature of the near-surface layers due to climatic oscillations significantly influence the magnitude of heat flux [3]. The proportion of geothermal heat’s contribution to the overall energy budget of the subsurface experiences substantial variation contingent upon factors including depth and subsurface composition. Within contemporary scientific discourse, there is a prevalent discourse pertaining to the analytical or numerical optimization of the operation of VGHE, which serves as lower-temperature heat sources for HPs across diverse configurations, sizes, geological contexts, and climatic conditions. The overarching objective is to minimize perturbations in subsurface temperature and enhance the efficiency of HP systems [4]. The prediction of subsurface temperature values holds considerable importance across multiple energy applications, including the utilization of the subsurface as a heat reservoir for HP, energy storage, and building thermal regulation. The assessment of subsurface temperature during the operation of HP systems is a pivotal step in the design and monitoring of geothermal HP installations; this facilitates the optimal utilization of available thermal resources and ensures the operational efficiency of the system. Various methodologies are employed for subsurface temperature assessment during HP operation:

• Field measurements: One of the simplest methods involves precise measurement of ground temperature; this can be performed using thermocouples or temperature sensors placed in boreholes at various depths in the ground. These measurements are accurate but can be costly and time-consuming [2,5,6,7,8,9];
• Computer simulations: These allow for modeling the behavior of the HP system under specific ground conditions; this enables the prediction of how ground temperature will change throughout the year [10,11,12]. However, there is no universal formula available thus far [13,14,15,16,17,18,19]. Real measurement results are required for simulation calculations to validate the assumed model;
• System performance monitoring: Another method is to monitor the performance of the HP itself. By tracking its operation and energy consumption, one can assess how efficiently it utilizes the available lower heat source temperature [2,20,21,22,23,24,25,26,27].

Ultimately, the accuracy of ground temperature assessment depends on the available resources and the complexity of the project. It is also essential to consider seasonal variations in ground temperature, as this can have a significant impact on the performance of a geothermal HP system. Therefore, regular monitoring and assessment of ground temperature are crucial to ensure the efficiency and durability of such a system.

In computer simulations of VGHEs using software programs, it is important to consider that the average monthly temperature at the inlet and outlet of the brine exchanger should not be lower than −1.5 °C. Exceeding the maximum unit thermal output (qv) of a VGHE or the maximum unit heat extraction from the ground can lead to excessive cooling of the ground and an increased frequency of freezing and unfreezing cycles of the filling material [28]; this can result in the leakage of the filling material around the VGHE pipe and, in some cases, even damage to the pipe. It also reduces the energy efficiency of the HP. Additionally, excessive load on the VGHE can lead to a long-term reduction in brine temperature in subsequent heating seasons due to inadequate ground regeneration [2]. Therefore, it is essential to forecast brine temperatures during the heating season to ensure the efficiency and durability of such a system under variable building heating loads. The authors have noted a lack of research in the literature that considers changes in brine temperature with varying building heating demands in real-world conditions [2,20], for which obtaining reliable and accurate data without precise and costly measurements can be challenging. In this study, the authors aimed to assess the suitability of a selected model, based on rough set theory (RST), for estimating changes in brine temperature using readily available weather data, such as daily average outdoor air temperature, solar radiation intensity, degree days of the heating season, and thermal energy consumption in a sample public utility building. Such an approach is novel in this type of research and can serve as an alternative to precise and costly measurements of brine temperature, which can indirectly reflect the average ground temperature affected by the operation of an HP during the heating season.

To evaluate the quality of the adopted predictive model, the following indicators were used: R², MBE, CV RMSE, and MAPE, which are recognized as statistical calibration standards accepted by ASHRAE [29]; this will offer assistance in answering the question of whether the strategy based on RST is reasonable for evaluating changes in brine temperature amid the operation of HPs within the warming season and what level of precision can be accomplished with it.

2. Materials and Methods

2.1. Subject of the Research

The investigation was carried out within a temperate continental climate region (Dfb) [30], situated in northeastern Poland, designated as the IV climatic zone [31], with a calculated external temperature of −22 °C. Per ASHRAE 169-2021 standards, this locale corresponds to thermal climate zone 6A [32]. The region experiences 3071 heating degree days (HDD) over a standard heating season, with a base temperature of +18.3 °C [33]. The subject building, spanning 7187 m² in floor area and 22,091 m³ in volume, was erected in the early 1990s, featuring a single-story configuration with a basement and serving as housing for laboratories and lecture halls. To enhance heating system efficiency, a series of modernization efforts were undertaken, encompassing upgrades to the central heating system, ventilation system, and heat source, alongside thermal improvements to the building envelope. For heating purposes, two brine–water HPs, each rated at 117.2 kW, were installed within the building. These HPs employ a cascade configuration, drawing thermal energy from a lower heat source, which consists of 52 VGHEs, each reaching a depth of 100 m. Each individual vertical heat exchanger is designed to provide 3.54 kW, culminating in a collective capacity of 182 kW and spanning a total length of 5200 m [2,20]. The operation of these HPs is orchestrated through a parallel bivalent system with a thermal node rated at 125 kW. This cascade HP system, with a combined capacity of 234.4 kW, comprises a primary MASTER HP and a subordinate SLAVE one. Control of both HPs is centralized under the MASTER HP, whereas the SLAVE one is devoid of an autonomous HP controller, relying on the MASTER HP for operation guidance. Both devices are equipped with dual compressors to expedite the cycle duration of the equipment. When the heating capacity required exceeds the initial-stage HP’s capacity, the HP controller activates the second-stage unit.

Figure 1 provides a graphical representation of the cascade configuration featuring two MASTER and SLAVE HPs, each boasting a heating capacity of 117.4 kW.

To achieve the optimum cycle length of HPs and the associated better annual operation rate, two buffer tanks with a capacity of 1000 dm³ each were installed (Figure 2).

They provide hydraulic decoupling of volume flows in HP circuits and heating circuits, thus ensuring a more balanced operation of the devices when their heating capacities do not match the instantaneous demand. They also maintain the necessary minimum flow rate for the HPs. From the buffer tanks (Figure 2), thermal energy is transported to distribution units that distribute heat to individual heating circuits with different technological parameters, depending on their needs (50/40 °C, 40/30 °C, and 45/35 °C). If the temperature in the buffer tank falls below the setpoint value on the controller, the HPs will be activated. In the event that the HPs are unable to maintain and provide the required temperature on the supply line to the distribution units, supplementary heating of the installation water will be carried out by the peak source, a thermal node with a capacity of 125 kW operating in a parallel bivalent system with the HPs (Figure 1).

The compact thermal node working in conjunction with the HPs is shown in Figure 3.

The system is controlled using a freely programmable HP controller following a heating curve inclined towards a value of 1.0. Setting the curve at this slope allows for achieving a heating system temperature of +57 °C when the external temperature is −20 °C. At higher external temperatures, the heating system supply temperatures also change according to the heating curve, becoming progressively lower. The controller provides for the cooperation of a cascade of HPs with a compact heat substation, except that the priority of operation is given to the HPs, and when these at certain times are insufficient, the heating medium will then be reheated by means of the district heating system.

2.2. Monitoring the System of an HP Installation

Monitoring of the heat source’s operation is carried out using a supervisory control and visualization system, which records and balances essential system operation parameters, including:

• Measurement of brine temperature at the inlet and outlet of vertical probes (Figure 4);
• Measurement of external temperature;
• Measurement of solar radiation intensity;
• Measurement of energy accumulated in a given day and cumulatively since the HP installation was started;
• Measurement of temperature and the amount of energy transferred from individual circuits of the installation;
• Measurement of electrical energy consumption;
• Measurement of energy generated by the HPs on the upper source side;
• Measurement of energy supplied to the HP by the lower source.

Measurements were made at a frequency of every 5 min. The method of metering the ground vertical probes, where, among other things, brine temperatures were recorded, is shown in Figure 4. The individual metering of the probes makes it possible to control the performance of the probe and, in subsequent years of use, evaluate the correctness of its operation, which is very important during use.

All data is collected and archived and can be displayed numerically. Based on the obtained data for the selected heating season of 2018/2019, the production of thermal energy for the building’s needs was determined.

During the analyzed period, based on readings from heat meters, the HPs produced 446.55 MWh of thermal energy for heating purposes, covering 100% of the heat demand (without the need to activate the thermal node). The heat loss from the two 1000 dm³ buffer tanks during the analyzed period was 2.28 MWh. The average annual coefficient of performance (COP), based on the energy balance of the HPs in the 2018/2019 heating season, was 3.95.

The generation of thermal energy in individual months varied significantly, closely related to external temperatures ranging from −12.3 °C to +17.9 °C, with an average of +3.5 °C. The temperature of the lower source (brine at the inlet) during the heating season averaged +2.28 °C, with an average of +0.81 °C from late November to the end of the heating season. Solar radiation intensity varied from 0.2 to 253 W/m², with an average value of 63 W/m², as shown in Figure 5.

The recorded daily energy consumption for heating the building during the analyzed heating season ranged from 779 to 3839 kWh, with an average value of 1996 kWh. It was directly correlated with external air temperature values (Figure 5). Therefore, additional calculations were made for the heating degree day values, assuming a base temperature of +18.3 °C, as illustrated in Figure 6.

2.3. Method of Predicting the Supply Temperature of the Lower Heat Source

The input data, based on actual measurements, were included in the set of features characterizing the studied object. The features describing each day are in quantitative form, such as daily average external air temperature, daily average solar radiation intensity, daily HDD, daily energy demand of the building, and the brine supply temperature from the lower heat source. A qualitative parameter is the months of the heating season. The adopted database was randomly divided into two subsets in a ratio of 80/20. The analysis was conducted on 251 days of the heating season, from which a training set (information system) consisting of 188 records and a test set consisting of 63 records were selected.

Randomly selected records from the training set were compiled into a decision table, which included the data attributes (conditional attributes) adopted for analysis, denoted by symbols c1–c5. The brine supply temperature from the lower heat source, which is the vertical heat exchangers, is a decision attribute denoted by the symbol d.

Table 1. Information system (decision table).

Object Number	Condition Attributes					Decision Attribute
	c1	c2	c3	c4	c5	d
1	5.75	1	13.18	12.6	1595	1.82
6	−2.64	1	47.12	20.9	2598	1.03
23	1.79	1	41.7	16.5	2082	0.67
38	3.27	2	18.4	15.0	1872	0.83
63	3.43	9	5.91	14.9	1826	1.14

For the previously mentioned attributes, domains were determined according to the following assumptions: c₁—daily average external air temperature [°C]; c₂—the month number of the heating season; (1—January, 2—February, 3—March, 4—April, 5—May; 6—September; 7—October; 8—November; 9—December); c₃—daily average solar radiation intensity [W/m²]; c₄—daily HDD for a base temperature of +18.3 °C [°C·d]; c₅—daily energy demand of the building [kWh]; d—brine supply temperature from the lower heat source [°C].

contains selected sample objects (from the subset of 188 records) that make up the information system and illustrates how they are described. The adopted data and their corresponding symbols are also shown in Figure 5 and Figure 6.

The variables in Table 1 serve as conditional attributes, forming the basis for a predictive model aimed at estimating the supply temperature of brine from a lower heat source. This model relies on RST [34], a tool designed to describe uncertain and imprecise knowledge while also modeling decision systems [35]. The selected conditional attributes encompass a variety of data feature encoding methods, spanning both quantitative and qualitative formats. The integration of the Valued Tolerance Relation (VTR) proves to be advantageous in this context. The incorporation of the Valued Tolerance Relation (VTR) within RST facilitates the determination of upper and lower approximations for sets with varying degrees of indiscernibility; this allows for the comparison of two data sets and yields results within the range of 0 to 1, which indicates the degree of indiscernibility. This range is derived from membership assumptions grounded in fuzzy set theory. A result closer to 1 indicates greater similarity (indiscernibility) among objects concerning the analyzed feature, while a result closer to 0 signifies a higher level of distinctiveness [36,37,38]. In the applied predictive model, decisions are made based on the following relationship: specific decisions are reached when certain conditions are met.

For example, for object 1 (Table 1), if (c₁ = 5.75), (c₂ = 1), (c₃ = 13.18), (c₄ = 12.6), and (c₅ = 1595), then (d = 1.82).

In the method presented, only two primary coefficients—quality and accuracy of approximation—are employed to control the significance of conditional attributes in relation to the decision attribute and the generated decision rules. These coefficients are simple to apply and interpret. The overall process of constructing the model using RST is depicted in Figure 7. A comprehensive explanation of the predictive model, utilizing both quantitative and qualitative variables, can be found in the referenced papers [39,40].

3. Research Results and Analysis

After selecting representative decision rules (Figure 7), it was possible to determine the values of the brine temperature from the lower heat source. A test set was used for this purpose, consisting of data randomly selected from a pool of 251 days in the heating season. This set amounted to 63 instances, for which conditional attributes were denoted as c₁–c₅. In the next stage, by applying the Valued Tolerance Relation (VTR), membership levels to decision rules were determined, allowing for the selection of the appropriate brine temperature value from the lower heat source. The obtained results are presented in Figure 8, where the brine supply temperatures from the lower heat source, calculated using the predictive model (RST), are compared with the actual values obtained from measurements.

Analyzing the results presented in Figure 8, it can be observed that the calculations obtained from the model differ from the actual data in the range of 0.02 to 1.55 °C, with an average value of 0.22 °C. The confidence interval for the examined group of objects varies within the range of 0.15 to 0.29 °C.

In the subsequent phase of this study, the calculation of metrics was undertaken to assess the performance of the predictive model that had been constructed. The quality assessment of the developed models was based on CV RMSE, MBE, and R², which are accepted as statistical calibration standards by the ASHRAE Guideline [29]. Another metric frequently used in the literature, MAPE, was used to assess the quality [29,38].

Evaluation metrics were calculated using Formulas (1)–(4).

$$MAPE=\frac{1}{n_g}\sum _{m=1}^{n_g}\left|\frac{t_i-t_i^P}{t_i}\right|·100\% m=\mathrm{1,2},3\dots ,n_g$$

(1)

$$CV RMSE=\frac{\sqrt{\sum _{m=1}^{n_g}\frac{{\left(t_i-t_i^P\right)}^2}{t_i}}}{\frac{1}{n_g} \sum _{m=1}^{n_g}{t}_i}·100\% m=\mathrm{1,2},3\dots ,n_g$$

(2)

$$MBE=\frac{\sum _{m=1}^{n_g}\left(t_i-t_i^P\right)}{\sum _{m=1}^{n_g}{t}_i}·100\% m=\mathrm{1,2},3\dots ,n_g$$

(3)

$$R^2={\left(\frac{n_g·\sum _{m=1}^{n_g}{t}_i·t_i^P-\sum _{m=1}^{n_g}{t}_i·\sum _{m=1}^{n_g}{t}_i^P}{\sqrt{\left(n_g·\sum _{m=1}^{n_g}{t}_i^2-{\left(\sum _{m=1}^{n_g}{t}_i\right)}^2\right)·\left(n_g·\sum _{m=1}^{n_g}{t}_i^{P 2}-{\left(\sum _{m=1}^{n_g}{t}_i^P\right)}^2\right)}}\right)}^2$$

(4)

where: “t_i—is the actual value (quantity) in the facility i, and t^p_i—is the forecast value (quantity) in the facility i. The difference between t_i and t^p_i is divided by the actual value t_i and m is the index of number of test object, n_g is the number of objects (m = 1, 2, 3, …, n_g)”.

According to the ASHRAE Guideline [29] criteria, for the model to be considered well-calibrated, the value of the evaluation indices should not exceed:

• MBE index: from −5% to +5%;
• CV RMSE: index 15%.

In any case, the value of the coefficient of determination should be R² ≥ 0.75.

The calculated indicators for assessing the quality and accuracy of the model are summarized in

Table 2. Model quality characteristics.

Assessment Indicator	Results
MAPE (%)	12.2
CV RMSE (%)	14.76
MBE (%)	−1.3
R2	0.98

.

The results presented in Table 2 show that the model estimates the brine temperature from the lower heat source to be of acceptable quality. The obtained error values fall within the permissible limits set by the ASHRAE Guideline; this provides a basis for asserting that the applied approximation method allows for achieving good results in estimating the brine temperature values from the lower heat source, reflecting the temperature of the ground subjected to the operation of the HP during the heating season.

4. Conclusions and Perspectives

Based on 251 days of HP operation during the 2018/2019 heating season in the analyzed public utility building, a set of quantitative and qualitative variables characterizing the building’s heating demand was extracted. These variables were used to assess the applicability of a model using RST to estimate the brine temperature from the lower heat source under the influence of HP operation during the heating season. The results obtained were compared with actual values and then evaluated according to the model quality assessment standards proposed by ASHRAE. This analysis led to the following conclusions:

• The model’s calculations exhibit a mean deviation of 0.22 °C from the actual data, with a confidence interval for the examined group of objects ranging from 0.15 to 0.29 °C;
• The quality assessment metrics for the applied model are as follows: MBE = −1.3%, CV RMSE = 14.76%, and R² = 0.98. Consequently, it can be concluded that the method outlined in this article delivers satisfactory results in estimating the brine temperature of the lower heat source, closely approximating the temperature of the ground influenced by the HP’s operation;
• The presented tool proves invaluable for swiftly analyzing the determination of the lower heat source’s operating temperature during the heating season, thus mitigating the risks of ground freezing or HP damage in cases of excessively low brine temperatures;
• The outcomes of this case study suggest the potential utilization of a rough set theory-based model in other buildings reliant on HP-based heating systems. Therefore, future research endeavors aim to extend this study to other facilities, with the goal of verifying whether the described method yields comparable results to those obtained for the analyzed public utility building;
• As part of further research, the authors want to use other methods, such as artificial neural networks or the Takagi–Sugeno type fuzzy model, to predict brine temperature;
• Access to information about the brine temperature can offer insights into the load placed on the lower heat source, as an excessive load has the potential to significantly decrease the brine temperature in the boreholes, thereby influencing system performance.