Unmasking the Thermal Behavior of Single-Zone Multi-Room Houses: An Empirical Study

In single-zone multi-room houses (SZMRHs), temperature controls rely on a single probe near the thermostat. This practice often results in temperature discrepancies that cause both thermal discomfort and energy waste. Automatic vent registers, among other similar solutions, have faced adoption barriers due to installation, cost, and maintenance constraints. Utilizing per-room sensors with smart thermostats (STs) to control based on average temperature has gained acceptance by major ST manufacturers and demonstrated initial potential to diminish thermal discomfort. This paper empirically characterizes temperature discrepancies in SZMRHs and studies their effects, particularly with respect to thermal comfort and demand response (DR). Our aim is to leverage room-level data sourced from 1000 houses across the United States and two real-world testbeds to identify the shortcomings of SZMRHs and to diagnose the deficiencies via parameter identification. We discover that comfortable DR durations (CDRDs) for rooms are typically 70% longer or 40% shorter than for the room with the thermostat. When averaging, rooms at the control temperature’s bounds are typically deviated around -3°F to 2.5°F from the average. Moreover, in 95% of houses, we identified one or two rooms experiencing notably higher solar gains compared to the rest of the rooms, while 85% and 70% of houses demonstrated lower heat input and poor insulation, respectively. This study serves as a benchmark for assessing thermal comfort and DR services in the existing housing stock, while also highlighting room-level retrofitting needs. Our approach sets the stage for more granular, precise control strategies of SZMRHs.


INTRODUCTION
Commercial HVAC systems enable zonal controls, facilitating temperature optimization in each respective zone using a variety of control solutions.However, in residential settings, the heating and cooling mechanisms are typically dictated by a single temperature probe that oversees the entire building's thermal regulation due to the cost of multi-zone systems [12].This invariably culminates in variances in room temperatures, adversely impacting occupant thermal comfort [17,21] and the quality of other services provided by these buildings such as demand response (DR).This issue is particularly accentuated in multi-floor dwellings, a prevalent residential layout in the United States.
Although few studies have empirically identified temperature variations [17,22], a comprehensive analysis, especially one involving a larger sample size, has yet to be undertaken.Yet, despite not having sufficient evidence of how severe the deviations are, research has proposed numerous solutions, such as automatic vent registers [12,20,22,23], remotely controlled space heaters [14], and averaging strategies [17].Companies like Keen Home [7] and Alea Labs [15] have produced smart vents, but adoption is limited due to the need for complex installation processes and waste production [6].On the other hand, averaging strategies offered by major Smart Thermostat (ST) companies, such as ecobee [4] and Nest [19], have a higher likelihood of adoption.This is predominantly because STs already have a substantial market presence in the US, with an adoption rate of 11% [13].However, the effectiveness of these averaging methodologies has not been thoroughly investigated.Additionally, leveraging the data from STs, especially with the advent of multisensor setups, offers an opportunity to shed light on the root causes of such discrepancies.
In this study, we analyze multi-room sensing data derived from a large sample of STs in the United States and two single-family residential test beds.The focus of these inquiries is to evaluate the extent of temperature variations, gauge the efficacy of commercially available thermostats, and diagnose the limitations contributing to these deviations.Specifically, the main contributions of this paper are: (1) assessing temperature deviations during operational conditions and DR events, (2) examining the benefits and limitations of averaging methodologies through an experimental study and a large dataset encompassing approximately 900 houses, (3) identifying the underlying causes of thermal variations by performing a thermal characterization of up to 100 houses, equipped with five additional sensors.Our findings provide a valuable benchmark to judge the quality of services such as thermal comfort and DR provided by the existing housing stock.They also shed light on the need for room-level retrofitting and thus enhancing energy efficiency.Ultimately, our empirical analysis lays for more granular, precise control strategies for residential HVAC systems.
The remainder of the paper is structured as follows: Section 2 reviews related literature, followed by an explanation of our methodologies and datasets in Section 3. We then identify temperature variations in Section 6, evaluate averaging techniques in Section 5 and diagnose potential factors leading to these variations (Section 6).The paper concludes with a discussion in Section 7.

LITERATURE REVIEW
We divide the literature review into two primary areas of focus.First, we consider studies that delineate the challenges of single-zone multi-room houses (SZMRHs) and/or propose enhancements.Our examination of the initial set of literature reveals a lack of studies that utilize a large sample size to empirically analyze the intensity of these variations and identify their root causes.That is why, we conducted a second literature search where we investigated the use of ecobee datasets to study the thermal attributes of buildings.We believe that the methods used in this area of research could be deployed in a room-level granularity to identify the reasons for deviations considering existing housing stock.Though there is an intersection of these two literature areas, to the best of our knowledge there are no studies identifying room-level thermodynamic models of SZMRHs.

Single-Zone Multi-Room Houses
Studies conducted on typical Canadian homes revealed that temperatures fell within a comfortable range (±1 degree Celsius from thermostat reading) about 40% of the time, indicating a substantial proportion of time spent in uncomfortable thermal conditions [22].However, this may be underestimating the problem given that it disregards the possible deviations that might occur from the setpoint by the thermostat reading itself.
Some studies [12,20,22,23] and commercial ventures [7,15] have proposed multi-zone control enabled by automatic vent registers as a solution.However, these techniques may present substantial risks, such as potential damage to ducts and motors due to overpressure accumulation [12,23].Additionally, closing registers have been shown to cause more leaks in the ducts, which reduces the efficiency of the system [24].Moreover, these techniques do not scale efficiently to multi-story buildings [14].
Further investigations have assessed the efficacy of supplemental equipment, like register booster fans and space heaters, in tandem with multi-nodal sensing controllers [14].Register booster fans were found to be ineffective in rooms with inadequate insulation or a significant distance from the furnace.In contrast, although space heaters were proven to be useful, they pose challenges in terms of installation costs and aesthetic integration.
The potential utility of sensor networks for enhanced control of SZMRHs was initially explored by [17], who conducted a simulation analysis of various strategies based on either averaging room temperatures or prioritizing the worst-performing room.Additionally, several studies have revealed a prevalent trend among occupants of multi-story homes to condition their entire dwelling to maximize comfort in one or two rooms, leading to substantial energy waste [14,20].This demonstrates that a more homogeneous temperature distribution among the house would also result in improved energy savings, as shown in [17].

Building Thermal Characterization
Numerous studies have worked on ecobee's Donate Your Data (DYD) [5] datasets for building thermal parameter estimation.In [11], an ecobee dataset, comprising 10,000 houses with STs, was utilized to predict thermal time constants (TTCs).Their findings revealed notable disparities in TTCs between the summer and winter months.Thermal building parameters, as well as TTCs, were identified by utilizing winter months' data from 4,000 houses in Ontario and New York by [1].The work presented in [8] utilized data from 1,000 houses within the ecobee dataset to train various data-driven models, including a grey box model featuring indoor temperature, outdoor temperature, HVAC runtime, and solar irradiance.Nevertheless, no analysis relating to the thermal parameters of the buildings was conducted.Among the studies reviewed, [25] bear the closest resemblance to our research in terms of TTC identification.They conducted a dual parameter identification analysis during heating and cooling seasons to infer TTCs and equivalent temperatures due to solar and internal heat gains.As the primary aim of this study was to create a simulation environment for control algorithms, the detailed calculations were not further elaborated upon.
Given the shortcomings of current methodologies for improving comfort in SZMRHs, there is an imperative need for the implementation of more straightforward, scalable, and safe strategies to enhance comfort in SZMRHs.Preliminary successes of averaging methodologies have been documented in a limited number of simulation studies [17], and these techniques have been adopted by ST enterprises.Nonetheless, several important knowledge gaps remain: the severity of the temperature variations, the efficiency of these averaging strategies, and the underlying reasons for the limitations that cannot be addressed merely by averaging.

METHODOLOGY
In this section, we first delve into the metrics adopted for quantifying the successful operation of buildings, providing a comprehensive understanding of the measures we utilize.Following this, we outline the parameter identification methodology applied to extract the thermal parameters of rooms.This procedure is essential for diagnosing limitations inherent in the existing housing stock.Lastly, we present an overview of the datasets utilized throughout the paper, illustrating the breadth and depth of empirical evidence underpinning our study.

Comfortable Operation Metric
Various metrics have been deployed to evaluate the comfortable operation of buildings.Percentage indices have been used to evaluate the frequency of temperature deviations outside the comfort range [21,22], while risk indices measure human thermal comfort perception [3].Cumulative indices are the only kind that considers both frequency and the magnitude of the deviation together.However, most of them require complex parameters and detailed knowledge about occupant characteristics.For instance,   is an asymmetric discomfort index that sums over occupied hours of overheating [2], but its use often demands occupancy data, which many residences lack.Degree-hours criterion defined by ISO 7730 [9] is easy to apply and does not require extensive knowledge like others, but it is a cumulative value making it harder to compare between buildings.Unmet load hours, though widely recognized, is sensitive to sample size and indiscriminately penalizes both minor and significant setpoint deviations.The closest metric to our work was introduced in a very recent study by [22].Their method computes the relative frequency (RF) for each deviation interval (e.g., 2 to 3°F) by dividing the number of times the room was in that interval by the length of the whole dataset.Although this is a useful method that demonstrates how often a value was in a certain deviation interval from the thermostat reading, it fails to capture the general behavior of the room in two ways: (1) it does not consider discomfort as a deviation from user-requested setpoint, (2) it can only focus on a certain interval, thus does not produce generalizable results.These shortcomings suggest the need for a metric that can be used on a room-level resolution and consider both the magnitude and the frequency of the deviation from the setpoint, while still being limited to a certain scale (i.e., 0-1).In response to this methodological gap, we will be using a new metric: the Comfortable Operation Index (COI).This index, as shown below, measures how successful the HVAC is at conditioning each room, considering a specific temperature setpoint and deadband.() is the RF of the given temperature deviation interval  from the temperature setpoint.The set {−, . . ., } represents the range of temperature deviations defined as within the comfort zone.The set {−, . . .,  } is the temperature deviation limits of the analysis range (decided by the analyst).Using the maximum absolute deviation in the dataset as  makes the denominator 1, converting the function to approximate a probability density function, as applied in this study.However, analysts might prefer limiting the range to different percentiles.For results consistent with ours,  should equate the dataset's maximum deviation (denominator = 1) and  should be set at 2°F.Utilizing alternative  and  values confines comparisons to specific rooms or houses within a given study.In essence, COI represents the portion of the RF curve's area within the comfort range, compared to the total area for the analysis range (See Figure 1).
The COI has an inherent limitation: it requires a singular setpoint and a deadband, making it less applicable to HVAC systems operating with dual setpoints for heating and cooling.While the COI can adapt by taking the average of both setpoints in 'auto' mode, this approach becomes misleading if users exclusively employ either heating or cooling modes.To address this, we introduce the Comfortable Cooling Index (CCI) for analyzing cooling operations.Unlike COI, CCI defines the comfort zone as {−, . . ., 0} and calculates deviation relative to the cooling setpoint.

Parameter Identification
3.2.1 Free floating periods.Free Floating Periods (FFPs) have been successfully used to estimate grey box modeling parameters from ST data [1,11,25].They can be described as periods where no additional heating or cooling occurs.Constraints defined for extracting FFPs are different for heating and cooling seasons.For heating season, FFPs were constrained to occur between 10 p.m. and 7 a.m., given that the internal and solar gain during this period is negligible [25].Additionally, there needed to be a temperature difference of 2°F for the sensor of focus, the FFP had to last at least an hour, and the outdoor temperature had to be lower than the indoor temperature.For cooling season, in contrast, solar gain is not negligible.Hence, we considered the period where solar gain was high to infer the largest solar gain, akin to the approach taken by [25].In this case, the FFP was expected to happen between 10 a.m. and 5 p.m. Similarly, there had to be a temperature difference of 2°F for the sensor of focus, the FFP needed to last at least an hour, and the outdoor temperature had to be higher than the indoor temperature during free floating.Following [1,25], we are going to use a thermal Resistance-Capacitance () modeling approach for each sensor using FFPs.The structure of the model can be found as follows: where   () is the indoor air temperature of the sensor of focus at time ,   is the mean outdoor temperature,  is the thermal resistance of the room times heat input (in this case, solar and internal gain). is the thermal resistance times capacitance (also referred to as TTC).For the analysis of the heating season, we made the assumption of no internal or solar heat gains during the FFP, leading to the exclusion of the  term and inference of only the  value.However, in the cooling season analysis, both the  and  terms were inferred.The scipy.optimize library and its curve_fit function were employed for parameter identification.After identification, outliers were eliminated by using two standard deviation range from mean and error filtering of Root Mean Square Error (RMSE) values that are larger than 1°F.

Balance Point. The balance point method (introduced by [1])
focuses on the relation between the energy usage of a building and outdoor temperature.Certain constraints are applied to extract suitable periods of data for the deployment of this method.First, a selection criterion is applied to isolate the nights when the heating system was operational and no cooling events occurred.The focus on nighttime data serves to minimize the influence of additional heat gains, such as solar or internal gains.The specified night period extends from 10 PM to 7 AM.As some residences employ a twostage heating system, the run times of both the first and second stages are combined into a single metric to maintain consistency across different households.Few households in the sample have second-stage heating, which further justifies this unification.
Subsequently, the refined dataset is subjected to the balance point method.The structure of the model is as follows (details of the model can be found in [1]): where  ℎ, is the heating duty cycle of the night,   is the average indoor air temperature of the sensor of focus at night,   is the average outdoor temperature at night,  is the thermal resistance of the room times heating power. itself is not easy to interpret, but it can be used to make comparisons between rooms to decide if certain rooms are achieving less heat input from the HVAC system.The scipy.stats library and its linregress function were employed for parameter identification.After identification, a two-step filtering process was undertaken.Firstly, R-values below 0.7 were excluded, and subsequently, values deviating beyond two standard deviations from the mean were removed.In this test bed, an ecobee thermostat was installed in a four-story, ten-room house located in Pittsburgh, PA.Data was accumulated over a 31-day period.The preprocessing included two steps: (1) considering the instances when the thermostat setpoint was 78°F; and (2) extracting a period of time where the average outdoor temperature was almost equal for both averaging and thermostat-based control (i.e., 72.5°F).This preprocessing led to a remaining operational period of 264 hours.It can be seen that data collection did not take place during a significantly hot period with a low setpoint; therefore, temperature variations might be more pronounced in the hotter summer months.

ecobee DYD dataset.
This study utilizes the ecobee DYD dataset, which comprises data from 1,000 houses across the United States for the year 2017.Comprehensive details of this dataset are elaborated in [18].The analysis undertaken is divided into heating and cooling seasons due to two primary reasons: first, ecobee maintains separate cooling and heating setpoints rather than a single one; and second, RC values were observed to significantly vary between winter and summer months [11].We also used different heating and cooling season definitions for thermal comfort and parameter identification analysis.
For thermal comfort analysis, this study focuses on households possessing at least one additional sensor beyond the thermostat reading, yielding a total of 887 and 845 households included in the cooling and heating season analyses, respectively.The counts vary over the year as some households acquire additional sensors subsequently.The heating and cooling seasons for the thermal comfort analysis are defined separately for each household by identifying the longest period during which only heating or cooling was active.
In parameter identification analysis, we used houses with five additional sensors which resulted in up to 100 houses.Earlier parameter identification studies have predominantly focused on either the winter or summer months.However, our preliminary investigation suggests a marked improvement in the identification percentage of parameters when the entire heating or cooling season is taken into account.For instance, we define a heating season for each house as the period that commences with the first activation of the heating system during the nine-month window (i.e., excluding summer months) and concludes with the final deactivation of the heating system within the same period.The same methodology was conducted for cooling season.Our study revealed a substantial increase in the number of heating and cooling days compared to the conventional 90-day periods.

CHARACTERIZING TEMPERATURE VARIATIONS
In this section, we assess the extent of temperature deviations during regular operation and DR events.Our analysis provides an understanding of discomfort in relation to setpoints, a factor previously neglected in research.Furthermore, we explore the often disregarded thermal comfort implications of DR events [16].

Variations with an Existing Thermostat
Figure 1 uses data from Test bed 1 and the COI (explained in Section 3.1).On the vertical axis, RF values are displayed, ranging from 0, signifying that a particular deviation is never experienced, to 1, indicating that only that specific deviation is experienced.One should note that optimal operation of the thermostat would result in the peak location of the thermostat being located at 0°F deviation.However, Figure 1 shows that the thermostat probe's measurement is persistently divergent by 3°F, yielding a sustained fluctuation.Low COI values (shown in the legend) underscore the criticality of the situation for each individual room.We observe that the thermostat is not even the most comfortable room while the severity increases as we move to rooms located on the upper floors.While the deviation of the thermostat could be attributed to faulty installation of the mercury thermometer, the main reasons behind deviations in other rooms cannot be explained easily and need indepth examination.For comparison, it is reasonable to assume that in a single-zone, single-room system, the thermostat line would reflect the entirety of the system given that control in Test bed 1 is solely dictated by the thermostat temperature.The outcomes of this preliminary investigation not only emphasize the significance of assessing variations from the setpoint rather than the thermostat reading, but they also illustrate the gravity of scenarios wherein rooms frequently deviate from the comfort range.

Demand Response Effects
On June 19, 2023, we replicated a DR event at 12:00pm on our entirely unoccupied Test bed 2, during which the average outdoor temperature was 85°F. Figure 2 The distinct time intervals seen in each room's temperature rise highlight the unique thermal behavior of individual spaces.For instance, rooms 6, 7, and 8 reach their peak temperatures close to midnight, which may be a consequence of their specific orientation.Conversely, Room 2 experiences its peak temperature relatively early, likely attributable to its extensive window area.Room 1, on the other hand, displays a very small increase in temperature over a long period, which can be explained by its superior insulation.
In order to validate that these discrepancies are not isolated to a single household but persist across larger samples, we expanded our analysis to residences fitted with five additional sensors during the cooling season (Section 3.3.3).We identified the FFPs, which are the periods between 12 pm and 5 pm when the outdoor temperature exceeds the indoor temperature for at least an hour.These intervals reflect typical DR events as the HVAC system remains off for substantial durations.However, note that DR events happening during heatwaves could create larger discrepancies than we identified here.In each dwelling, the CDRD and temperature deviation from the initial temperature is computed.Table 1 shows the statistical analysis results considering all houses with a valid FFP.Fast-Reacting Rooms (FSRs) and Slow-Reacting Rooms (SRRs) are defined as rooms that achieve their CDRDs the quickest or slowest compared to other rooms in the same household, respectively.We define the comfort gap as a pair of numbers showing the differences between the maximum and minimum (a) CDRDs and (b) temperature deviations within a house.These results indicate that, on average, the comfort gap among rooms in the same house is 52 minutes and 2.37°F.Moreover, CDRDs can be 70% longer or 40% shorter than the duration of the room with the thermostat, on average.Further, distinct rooms can deviate from the thermostat reading by an average of 48% more or 34% less.The considerably large standard deviations underscore the substantial variability and distinctive thermal behavior of each house.
The analysis in this section highlights considerable variations in the performance of thermostats during operational and DR periods with significant discrepancies from setpoint values across different

EVALUATING THE EFFICACY OF AVERAGING
Our second contribution is a critical evaluation of the effectiveness of averaging methodologies, conducted on our designated Test bed 2 and ecobee dataset by utilizing the methodology explained in Section 3.1.While the efficacy of averaging has been investigated through simulations in [17], to the best of our knowledge, no research has conducted experimental studies or utilized large datasets.Figure 3 displays the RF of temperature deviations from the cooling setpoint for averaging and thermostat-based control in Test bed 2. Thermostat-based control, as depicted by the CCI values in the legend, impairs cooling comfort, with conditions worsening as one moves to higher floors (given in ascending order).Nevertheless, the zero-deviation positioning of the thermostat peak indicates that the HVAC system meets its goal.Compared to thermostat-based control, averaging resulted in higher comfort with 45% improvement on average (excluding the basement).Yet, it is not without limitations.First, the thermostat temperature was -6°F deviated from the setpoint, resulting in overcooling most of the time.It can also be seen that certain rooms are still less comfortable.The second room, although located in close proximity to the HVAC system, possessed numerous windows, which makes it prone to external affects.In upper-floor rooms, pinpointing whether the deviations on these floors arise from duct air leakage, insulation inefficiencies, or solar gain necessitates a thorough analysis.For instance, this phenomenon has been observed before in a single household and is attributed to insufficient fan power and/or duct leaks by gathering measurements with air velocity sensors [14].We will try to identify such deficiencies using the ecobee dataset in Section 6.
The ecobee dataset explained in Section 3.3.3 is also leveraged to understand the efficacy of averaging.The configuration of sensors, as shown in Figure 4, provides insight into the operation of houses with additional sensors (counts of which are given in the legend).During the cooling season, thermostat temperatures tend to register on the lower end of the spectrum in comparison to other sensors.This observation could be attributed to common practices of thermostat installation, such as closer proximity to air vents or the air conditioning unit than the upper floors, contributing to lower readings in the computation of average indoor temperature.Conversely, additional sensors generally record higher temperatures, possibly due to being situated further from the air conditioning unit, thereby receiving less cooled air, or due to exposure to higher levels of solar radiation during the day.Analyzing the heating season, a lesser degree of deviation is noticeable compared to the cooling season.For all households, the median temperature aligns with zero deviation from the indoor heating setpoint, indicating that all rooms achieve this reference temperature half the time.Yet, additional sensors typically report similar or lower temperatures than the thermostat reading, aligning with our previous assumption of thermostats being situated closer to the HVAC system.From this analysis, we infer that temperature variations pose a less significant issue during the heating season compared to the cooling season.Potential explanations for this phenomenon include the superior performance of HVAC systems in heating mode, stack effect causing hot air to rise to the upper floors, and the contribution of internal and solar heat gains in maintaining room temperatures.A limitation of the preceding methodology is that the aggregated temperature deviations might mask the unique differences among individual houses.Therefore, we need to conduct an additional analysis focusing on the hottest and coldest rooms within each house.For this particular analysis, we restrict our study to houses equipped with five additional sensors to attain a greater degree of granularity in temperature variations.Our analysis is based on the CCI values of the thermostat, the room with the lowest performance, and the room with the highest performance within each house.As illustrated in Table 2, on average, in each house, one room performs 15% worse than the room where the thermostat is located, while another room is 7% more comfortable.Table 3 presents a statistical analysis of temperature deviations from both the setpoint and the average temperature (i.e., control temperature) for the coldest and hottest rooms.Deviation from the setpoint reveals the challenge the HVAC system encounters to keep the rooms closer to the setpoint.However, it does not explain what the target of the control mechanism was for the extreme rooms.On the other hand, deviation from the control temperature signifies that even at the targeted temperature, rooms are anticipated to exhibit a certain degree of deviation.Consequently, even with a perfectly functioning HVAC system, it is expected that the coldest and hottest rooms will a deviation range of approximately -3°F to 2.5°F from the control temperature across both seasons.

DIAGNOSING THE LIMITATIONS
As the last contribution, we diagnose the reasons behind these deviations using a parameter identification methodology on up to 100 houses, each equipped with five additional sensors.

Building Thermal Parameter Identification
6.1.1Thermal Time Constant () Identification.Utilizing the FFPs according to Section 3.2.1,we inferred the  values for both cooling and heating seasons for each sensor-house pair.The subsequent process of outlier removal allowed us to infer suitable FFPs for 78.9% and 87.6% of the house-sensor pairs in the heating and cooling seasons, respectively.The increased success rate of thermal parameter identification during the cooling season could be attributed to the added parameter , which introduces further flexibility to the process.The  values we obtained show a good alignment with those reported in previous research [1,11,25].This high rate of identification in our study is likely a result of our methodological choice to extract cooling and heating data over a nine-month period (Section 3.3.3).To display the benefit of considering a longer period, we replicated the cooling season analysis using a three-month summer period and found that utilizing a longer period improved the total duration of FFPs by 84%, which in turn resulted in a 21% improvement in the percentage of identified parameters.The distribution of  values for heating and cooling seasons were displayed in Figure 5.Each subfigure consists of two plots: the upper plot shows the collective distribution of  values, while the lower plot presents the variances in room behavior within individual houses, color-coded by state.For instance, Figure 5(a) reveals a unimodal distribution of  values, ranging from 1 to 27 hours for the cooling season.Contrarily, Figure 5(b) showcases a bimodal distribution for the heating season with a larger discrepancy in  values ranging from 1 to 157 hours.The corresponding boxplot highlights intra-house differences as large as 20 and 143 hours among rooms for cooling and heating seasons, respectively.Given that  values are solely dependent on the physical structure of the building, such variations between the  values for summer and winter are unexpected.Yet, a similar discrepancy was observed in a prior study, which was attributed to the disparity in behavioral changes of occupants, such as a tendency to leave windows open more frequently during summer [11].Additionally,  value for the room where the thermostat is located at is marked with light blue for each house.If the room where the thermostat is located were to accurately represent the thermal behavior of the rest of the house, the marker would be positioned in the middle of each boxplot with a notably small range.However, the plot clearly shows that this is often not the case.
In the heating season, there's a 30% chance that another room in the same house will have an  value twice as high as the room with the lowest  value.This probability slightly decreases to 24% during the cooling season.Furthermore, our analysis revealed that, more than 50% of the time, a difference of 4 hours and 54 hours is observed in the  values among rooms for the cooling and heating seasons, respectively.6.1.2 Identification.The identification of thermal  values during the cooling season concurrently produces  values.The distribution of  values can be seen in Figure 6.Upon conducting outlier and error filtering, 87.2% of the  values were successfully inferred.Our study reveals differences as significant as 3.2°F among rooms within the same house (see Figure 6).According to our findings, there is a 38% probability that one room will endure twice the solar heat gain compared to another room in the same house.Furthermore, there is a 25% likelihood that one room will experience three times the solar heat gain of its counterparts within the same dwelling.Despite lacking a concrete physical interpretation, these  values provide insightful data for retrofitting decisions, especially for identifying rooms that are most susceptible to high  solar irradiance.Ultimately, the varied positions of the markers suggest that rooms with thermostats may experience notably less or more solar gain compared to other rooms within the same household.This wide range of outcomes underscores that thermostat measurements may inaccurately estimate the degree of solar gain across the household, with no discernible bias toward either lower or upper limits.4, was utilized to infer the  values.This resulted in the  value distribution ranging from 26 to 204°F, presented in the top plot of Figure 7 with a successful identification rate of 55.9%.The boxplot in Figure 7 illuminates the considerable variability in the  values across different sensors within the same house, reflecting the operational characteristics of the HVAC systems.The variation spans a broad spectrum: while some houses exhibit marginal differences, others reach to deviations of approximately 80°F.The analysis suggests that in a quarter of the instances, a room within a house receives 20% more heating input compared to the room with the least heating input.This disparity underscores the room-level variations in heating distribution, potentially signifying areas of improvement in HVAC system operation.Subsequent decisions regarding retrofitting could be informed by these insights, such as the need to enhance duct sealing to mitigate air leakage or to clean ducts to facilitate better airflow.It should be noted that since there are no studies that utilized the ST data for such purposes, there is no accepted metric for us to use.However, our approach is akin to the smart meter data based inefficient house detection methodology conducted by [10].While this limited approach does not conclusively indicate such inefficiencies, it does help identify rooms with a high risk of such deficiencies.
Our observations indicate that a maximum of two rooms within a house exhibit such deficiencies.Although the actual number may be higher, this is a plausible estimate considering the constraints of having only six sensor readings from each house.Table 4 presents the number of houses exhibiting these deficiencies in just one room, two rooms, and the total.The percentages have been calculated based on the total data available for each instance separately, which explains why the denominator varies for each row.Overall, 96% of the houses exhibit High Solar Gain, with 78% in one room and 18% in two rooms.Low Heating Input emerges as the second most common deficiency, occurring in 85% of the houses overall, with 70% of the instances being confined to just one room.Low Heating Input is recorded in 78% of the houses, 69% of which observe it in only one room.Lastly, 70% of the houses are at risk of Poor Insulation, with 15% experiencing it in two rooms.In our final analysis, we examined the assumption that thermostat placements -ideally unaffected by vents or solar radiationaccurately represent overall house conditions.We computed the differences in thermal parameters across rooms from the room thermostat is located, with the statistical analysis results presented in Table 5.While the relatively low mean/median values might seem to initially endorse this assumption, the high standard deviations indicate significant variation in each home's thermostat-related behavior.This prevents us from drawing robust conclusions regarding the thermal conditions experienced by thermostats due to their physical locations.However, we can assert that they do not adequately represent the rest of the house.

DISCUSSION AND CONCLUSION
This study offers a comprehensive exploration of the thermal behavior variability in SZMRHs using data from two residential houses and a publicly available dataset from ecobee.
Our initial analysis identifies significant temperature discrepancies across rooms, often overlooked by traditional thermostats.We displayed a consistent 3°F discrepancy between the thermostat's reading and the setpoint, with comfort indices falling below 50% within rooms.Further, we noted a significant disparity during DR events, where the duration of comfort was typically 70% longer or 40% shorter compared to the thermostat-controlled room.
The subsequent analysis underscores that while averaging may mitigate some discrepancies, substantial deviations outside comfort bounds persist.In our test bed, averaging techniques demonstrated an average of 45% improvement in cooling operations.Despite this improvement, the thermostat reading still exhibited an average deviation of -6°F.On a larger scale, we found that individual rooms display an expected deviation ranging from -3°F to 2.5°F from this average (i.e., control temperature).This considerable discrepancy underscores that, even with averaging, the targeted temperatures for rooms are typically outside of the comfort bounds (±2°F).
Our detailed parameter identification study further illuminates the potential root causes of these deviations.In 95% of houses, we found rooms with High Solar Gain, 85% with Low Heating Input, and 70% with Poor Insulation.Moreover, we observe that common assumptions about thermostat placement are often incorrect, as they are typically located in rooms that do not accurately reflect the whole house's thermal conditions.
In conclusion, despite moderate improvements with averaging techniques, significant temperature deviations in SZMRHs still exist, affecting thermal comfort during regular operations and DR events.The methodology used in this study can benefit future HVAC designs by enabling ST companies to leverage these algorithms for room deficiency detection, enhancing energy efficiency retrofit cost estimates, and guiding optimal thermostat placement for accurate representation of a home's thermodynamic characteristics.Some limitations of this study are: 1) the dataset, which could have influenced our findings due to constraints such as low temperature resolution or the limited number of houses equipped with sensors, 2) the metrics we employed to identify deficiencies have not been validated.Despite these potential variances stemming from limitations, the findings remain illustrative.While subsequent studies might develop refined metrics, leveraging building thermal parameters for more nuanced room-level deficiency evaluations, our analysis still offers substantial insights.Future work could include an analysis on estimating overriding behavior during DR events by utilizing room-level occupancy and temperature data.Moreover, existing model predictive control methodologies could be extended to room-level granularity using the methods we deployed.

21 Figure 1 :
Figure 1: Relative frequencies of temperature deviations from setpoint.Grey area indicates comfort zone and legend displays COI values.
(a) shows the Comfortable Demand Response Duration (CDRD) (defined as the duration of time it takes for a room's temperature to rise by 2°F in the absence of cooling) for each room colored by the final temperature reached, providing a detailed view of how a DR event can differently affect comfort levels in various rooms.For example, Room 1 could maintain a comfortable temperature throughout an 8-hour DR event, whereas Room 2 could only do so for around 15 minutes.Moreover, Figure 2(b) illustrates the temperature increase over a 12-hour span, beginning at 12:00 p.m. Rooms are categorized and color-coded based on their location within the structure, arranged in ascending order.

Figure 2 :
Figure 2: Temperature variations during the DR event.

Figure 3 :
Figure 3: Relative frequencies of temperature deviations from the cooling setpoint.Grey area indicates comfort zone and legend displays CCI values.

Figure 4 :
Figure 4: Distribution of temperature deviations from setpoints in houses with varied sensor counts.

Figure 5 :
Figure 5: This histogram depicts the collective distribution of  values (top), accompanied by boxplots for individual room distributions (bottom) for cooling and heating seasons.Markers indicated in light blue represent the  values for the room where the thermostat is located.

Figure 6 :
Figure 6: This histogram depicts the collective distribution of  values (top), accompanied by boxplots for individual room distributions (bottom).Markers indicated in light blue represent the  values for the room where the thermostat is located.
3.3.1 Test bed 1.The first test bed of this study (from here on named as Test bed 1) was a two-story, six-room house in Pittsburgh, PA, outfitted with a conventional mercury thermometer and a central HVAC system.During the data collection period, both heating and cooling were active, reflecting seasonal transitions.The thermostat and Room 1 are located on the first floor, while the other rooms are on the second floor.Temperature probes, consistent with ASHRAE guidelines, were placed 130-160 cm above the floor.Although data was gathered over 17 days in operational conditions, only 200 hours were usable for analysis due to sensor disconnections.

Table 1 :
Statistical analysis of the CDRDs and temperature deviations of rooms during DR events rooms in a house.It becomes evident that a single thermostat does not capture the temperature complexity of a SZMRH effectively.

Table 2 :
CCI values for houses with 5 additional sensors

Table 3 :
Summary of room deviations from setpoint and average temperature After obtaining the , , and  values for each house, we carried out an independent analysis to identify homes with rooms that are at a high risk of having Poor Insulation, Low Solar Gain, High Solar Gain, and Low Heating Input.All comparisons were made based on the performance of other rooms within the same house.In order to identify rooms with deficiencies, we set certain assumptions.Rooms with  values one standard deviation below or above their mean are expected to face Low Solar Gain and High Solar Gain, respectively.Rooms exhibiting a  value one standard deviation below the mean of the house are assumed to have Low Heating Input.Poor Insulation is first detected by analyzing  data for both heating and cooling seasons by incorporating values that sit below one standard deviation from their mean.We then observed fewer houses with multiple rooms encountering Poor Insulation in the cooling season, whereas Poor Insulation from the heating season is more prevalent in houses but affects fewer rooms.This observation aligns with our initial assumption of lower  values in the cooling season due to behaviors such as occupants leaving windows open more frequently.As a result, to accurately represent the actual physical attributes of the house, and not merely occupant behavior-related deficiencies, we define Poor Insulation using  values from the heating season.
Figure 7: This histogram depicts the collective distribution of  values (top), accompanied by boxplots for individual room distributions (bottom).Markers indicated in light blue represents the  values for the room where the thermostat is located.6.2 Interpretation of the Parameters

Table 4 :
Number of houses with deficiencies

Table 5 :
Statistical Analysis of the thermal parameter difference of rooms from the thermostat