Positioning Temperature Sensors in Confined Spaces Subject to Various Exogenous Impacts

The pressing need to minimise the impact of human economy activity on the natural environment has drawn significant attention over the last several decades. The growing world population and increasing pace of man-made pollution forces us to look for new solutions in terms of how energy is both generated and conserved. The building sector is the biggest consumer of energy. It accounts for 30–40% of worldwide consumption [1] and when the energy needed for construction and demolition is taken into account this number rises to about 50% [2]. In 2010 the energy delivered to buildings in absolute terms was 23.7 PWh and this is projected to rise to 38.4 PWh by 2040 [3].

In buildings, the majority of energy is consumed by various processes aimed at maintaining required parameters of the indoor environment. According to data summarised by [4], of its total energy consumption, the average office building in the UK dedicated: 55% to heating, ventilation and air-conditioning (HVAC) appliances; 17% to lighting; 10% to water heating; and only 5% each to food preparation, refrigeration and powering equipment; and a mere 4% remains for other processes. One of the most important parameters of the indoor environment, and the one which has the highest impact on the tenant’s wellbeing, is temperature [6]. Since buildings are not completely isolated from their surroundings, the indoor temperature will be influenced by external factors such as: irradiation, outdoor temperature, wind speed, humidity and atmospheric pressure. At the same time, the indoor environment will be created by internal sources of heat (humans, animals, and equipment), humidity (plants, water bodies), and air movement (various processes), etc.

In Fig. 1 we have represented the discrepancy between outdoor temperatures in Kraków (Poland) over 2016 and the desired indoor temperature in an office building. Depending on the building insulation, orientation and other factors over the whole year there will be periods which will require significant heating or cooling to be delivered to the building in order to create an acceptable indoor environment for efficient work. It is worth underlining that weather conditions not only impact the interior environment but also the building itself as a manmade structure prone to physical deterioration [7].

Figure 1.

All the above means that research into minimising energy consumption in buildings while maintaining the desired quality of indoor environment has become a very important and frequently investigated topic in scientific literature. The demand for such research is especially visible one considers the regular appearance of review papers (2010 – [7], 2013 – [8], 2017 – [9]) which aim to summarise and indicate past, present and future research directions. It is worth mentioning that analysis of thermal comfort is conducted for various spaces, such as: swimming pools [10]; sport halls [11]; classrooms [12]; or even churches [13].

The problem of optimal sensor placement is very often investigated in the literature. Various objective functions and constraints are considered. This problem exists in a multitude of disciplines, ranging from fault detection [14] to air contamination [15] or city water infrastructure [16]. Here we adapt this concept to find the optimal location for three temperature sensors (three sensors gives the minimal number of reference points which can be used to visualise the spatial distribution of investigated phenomena. A greater number of sensors can be used for more complex room layouts or perhaps when greater accuracy is required). Based on the proposed formula their readings will be used to estimate the temperature in the whole room. In the considered case, temperature sensors can be located only at the edges of the room. In reality this means walls. Some architectural designs would naturally allow them to be placed somewhere in the middle, such as on a pillar supporting the ceiling. The problem investigated here can be formulated as follows: select three locations (i=1,…,n) for temperature sensors for which, based on Eq. 1–4, the estimated temperature in points (j=1,…,m) will be closest to observed temperatures. In other words, find the minimal value of the objective function given in Eq. 4 .

For the purpose of the mathematical model we introduce an additional binary variable (x _i ) which will indicate whether a sensor has been installed in i or not. Please note that here the binary variable is subject to the constraint given in Eq. 2 which ensures that only three locations (C=3) will be selected. Such a formulation makes it possible to consider even more sensors. For example, in order to investigate the impact of their number on the accuracy of modelled temperature distribution. (1) x i = { 1 , if sensor has been installed in location i 0 , otherwise (2) ∑ i = 1 n x i ≤ C

In our research we have decided to estimate the temperature at points j based on three available temperature values from sensors located in certain locations i. However, we have decided that temperature will not be calculated based on the simple mean of three measurements. Instead we will use a weighted average, where weights are proportional to the distance between sensors in location i and point j. This will ensure that sensor located close to a given point has the greatest influence on determining the estimated temperature. Here those distances (d _ij ) have been gathered by on-site measurements. However, sometimes, on-site measurements of distance between given points may not have been performed. In such a situation one may use various distance metrics, such as Manhattan or Euclidian. In Eq. 3 only distances (Z _j ) between selected locations for sensors are considered and used in further calculations. (3) Z j = ∑ i = 1 n d i j x j j = 1 , … , m

After establishing which distances are taken into consideration, the formula given in Eq. 4 can be applied. Here the temperature ( t j ′ ) at point j is estimated based on readings from all three sensors. However, as noted earlier, the significance of a given sensor is strongly dependent on its distance from the investigated point j. (4) t j ′ = ∑ i = 1 n t i x i Z j d i j ∑ i = 1 n Z j d i j j = 1 , … , m

In order to assess the quality of the location of the three sensors suggested by the optimisation model we have used a MAPE (Mean Absolute Percentage Error) metric, which is common in modelling and forecasting. In the investigated problem this was integrated into the objective function given in Eq. 5. (5) min A = 1 m ∑ j = 1 m | t j ′ − t j t j ′ | ∗ 100 %

The whole procedure for estimating the temperature at a given point has been presented in Fig. 2. For example, the sum of the distance between sensors (i=1, 2 and 3) and point j=1 is 45 metres (Z ₁=45). Meanwhile, the temperature observed at point j=1 is 19°C. Based on Eq. 4 the temperature estimated at that point should be (Eq. 6) 19.23°C. Considering the proposed approach (inverse distance weighting) the meaning of sensors in Eq. 4 and 6 would be: i=2 (weight 3); i=1 (weight 4.5); i=1 (weight 2.25). This explicitly makes the assumption that things that are close to one another are more alike than those which are far apart. (6) t 1 ′ = ( 10 45 15 + 20 45 10 + 30 45 20 ) 9.75 → 19.23 ° C

Figure 2.

For the purpose of this research we have used thermal imaging from a Flir i7 infrared camera [14]. The whole procedure, as well as validation of this method accuracy in comparison to more traditional methods (for example, thermometers), has been described in [15]. The obtained results enabled us to claim that this method is sufficiently accurate to meet the required standards. Considering the layout of the selected lecture hall (at AGH University, Faculty of Management – Fig. 3) we evenly distributed 144 (i=1,..,144) measuring points (a matte black sheet of paper) from which 44 points located on the outline (j=1,…44) would be used as potential locations for sensors. As mentioned earlier the distances between individual points was calculated during on-site measurements and stored in a matrix (d _ij ).

Figure 3.

For further analysis we have assumed that the temperature distribution in this lecture hall will be investigated for four different scenarios. In those scenarios the lecture hall was subject to various exogenous impacts, such as: natural ventilation, air-conditioning, and intense heating. During investigation of natural ventilation only the window above the marked radiator was left open. As a benchmark we estimated the temperature distribution after an entire night with all HVAC appliances turned off apart from ventilation. It is important to add that measurements were performed when this space was not occupied. Three first scenarios: steady-state, intensive heating and natural ventilation were completed in February 2017, whereas that including air-conditioning was performed in May 2017. This was dictated by the fact that during the winter period it was impossible to turn on the air-conditioning due to the lack of a cooling medium. In Fig. 4, for clarity, we have presented the layout of the lecture hall and, in Fig. 5, the temperature distribution for all four scenarios.

Figure 4.

Figure 5.

The mathematical model presented in section 3 was implemented in MS Excel software and solved by means of the Solver add-in. Since the model formulation is nonlinear, an Evolutionary and a GRG (Generalised Reduced Gradient) method were used with default settings. More details about application of the MS Excel Solver can be found here [17]. For the considered possibility of installing 3 sensors in 44 potential locations this problem is not very complex from the computational perspective. For three sensors it is only possible to have 13,244 unique layout combinations – which implies that with the current speed of modern processors a brute-force method may be also applied and an optimal solution is guaranteed. However, if one would like to install, for example, 5 sensors, the number of unique layouts is greater than one million. In Fig. 6 we have presented the optimal locations of three sensors for each scenario separately.

Figure 6.

As can be observed on the layouts presented in Fig. 7 the optimal location for sensors is significantly different from the current position of the BMS sensor which controls the operation of the HVAC appliances. In the new locations suggested by the mathematical model the sensors tend to be closer to the sources of local “events” which impact the observed temperature. This is especially the case for natural ventilation and intensive heating. By applying the modified sensor positions and number (three sensors) it was possible to significantly reduce the temperature estimation errors in three scenarios, as shown in Table 1. Errors were not only reduced in terms of the MAPE criterion which is considered here; the maximal observed estimation errors were also smaller. The best results were observed in the heating scenario, where, by using the modified sensors layout, it was possible to reduce the observed MAPE criterion by 56% (in percentage points by almost 8%). Based on the results as well as on the illustration presented in Fig. 6 we can see that in this scenario people (students) working near a heat source will be operating in conditions which may reduce their thermal comfort. In the air-conditioning scenario, despite using more sensors, it wasn’t possible to significantly improve the temperature spatial distribution estimation errors. This results from the fact that the AC unit has a tendency to distribute cold air only across a limited space, and it takes time for it to reach more distant locations. What is more, in this scenario the possible sensor locations were not directly impacted by the AC’s operation. Here the perception of thermal comfort will be impacted not only by the cold air from the AC but also by the air speed – what is know as “cool breeze”. Accordingly to the questionnaire completed by students (as part of an OBMES [Office Building Energy Management System] project) this “cool breeze” causes a very unpleasant sensation which requires the exposed parts of body to be covered by clothing – even if the temperature observed in a given room is significantly above the thermal comfort threshold. It is important to underline that such remarks were made only by respondents under the direct impact of the AC’s operation.

Figure 7.

The quality of temperature estimation based on three sensors arranged in various layouts has been also presented by means of scatter plots in Figs 7–10. As can be observed, relatively promising values of the R2 criterion were only observed for the heating and natural ventilation scenario. As shown in Fig. 10 the suggested location of sensors wasn’t able to provide sufficient information to estimate the temperature distribution in this room. This is due to the fact that temperature sensors were situated far from the cold source and it usually takes time for cold air to reach them. And during this process people under the direct impact of the operating AC unit feel thermal discomfort from cold air and chill. Therefore, in such a situation it seems necessary to use an “artificial sensor” which will emulate the behaviour of the cold source and its impact on temperature distribution. We will investigate this issue in the next phases of our research.

Figure 8.

Figure 9.

Figure 10.

The research presented in this paper is part of a bigger project which aims to optimise the operation of Building Management Systems, and what is perhaps its most important part in particular, namely, the Heating Ventilation and Air Conditioning subsystems. The project’s objective is to minimise the volume of energy consumed for maintaining the required thermal conditions. The obtained results indicate that the usual use of a single temperature sensor to control the HVAC system is not enough and cannot depict the spatial variability of temperature and consequent differences in thermal comfort. What is more, process like heating, ventilation or air-conditioning differently impact the spatial distribution of observed temperature. In consequence, sensors optimally located for estimating temperature distribution during heating may not work properly for a cooling scenario. Therefore, the question remais open as to the optimal location of sensors which will enable the precise estimation of temperature distribution under various scenarios equally.

The conducted analysis revealed several interesting future research directions which should aim to answer the following questions: is investment in more sensors justified by the reduction in HVAC operation costs?; how does estimation accuracy increase with each additional sensor?; and how do other distance metrics (for example inverse distance-squared relationships) impact estimation precision?