Analyzing and modeling thermal complaints in a commercial building in France

In this section, the thermal comfort standards were assessed according to the ISO Standard 7730. This standard suggests the Fanger’s model (Fanger 1982) in which the PMV and PPD indices are used for assessing and predicting thermal comfort in indoor environments including buildings (Katafygiotou and Serghides 2014; Soutullo et al. 2014; Trebilcock et al. 2014; Dudzińska and Kotowicz 2015; Gilani et al. 2015; Natarajan et al. 2015; Cheung et al. 2019). The CBE Thermal Comfort tool (Hoyt et al. 2017) was used to calculate the PMV and PPD indices, in which the input parameters were indoor air temperature (C), RH (%), mean radiant temperature (C), air velocity (m/s), clothing insulation value (clo), and metabolic rate of the users (met). Indoor air temperature and RH ratios corresponding to the complaints in relation to HVAC system were obtained from the ambient sensor data. Mean radiant temperatures were calculated using the formula (Equation 1) proposed by Nagano and Mochida (2004). (1) Tr=0.99×Ta−0.01,R2=0.99 {T_{\rm{r}}} = 0.99 \times {T_{\rm{a}}} - 0.01,{R^2} = 0.99 where T_r represents the mean radiant temperature and T_a represents the indoor air temperature. Air velocity was assumed to be 0.15 m/s, which is below the maximum allowable air velocity in offices according to ISO7730 Standard. Metabolic rates and clothing insulation values of occupants were calculated by using the corresponding tables in ISO7730 Standard. Subsequently, the metabolic rate was determined as 1.2 met, which corresponds to office sedentary activities. The checklist of clothing in the ISO7730 Standard was used to obtain the clothing insulation (clo) values, which were determined according to the most likely garments to be worn. Subsequently, the clo values were determined as 0.57 and 1.1 clo for cooling and heating seasons, respectively.

Moreover, the operative temperature was calculated to check the compatibility of this parameter with the recommended values in the ISO7730 Standard. Equation 2, which is given in the ASHRAE55-2010, is used to calculate the operative temperatures. (2) To=A×Ta+(1−A)×Tr {T_{\rm{o}}} = A \times {T_{\rm{a}}} + \left( {1 - A} \right) \times {T_{\rm{r}}} where T_o represents operative temperature and A is weighting factor that depends on air velocity (v_r) and was determined as 0.5 according to the ASHRAE55-2010.

Regression models investigate the relationship between independent variables and an outcome (dependent) variable. If the dependent variable is a categorical variable, discriminant analysis, probit analysis, logarithmic linear regression, and logistic regression can be used for data analysis. In particular, logistic regression is widely used since it enables to use continuous explanatory variables and it is easier to handle more than two explanatory variables simultaneously (Fagerland and Hosmer 2012).

Logistic regression models can be considered as a type of multiple regression models. Logistic regression models are not linear, and thus they are more complex compared with the multiple regression models (Tabachnick and Fidell 2007). Logistic regression function can be formalized as follows: (3) π(x)=eβ0+β1⋅X1+eβ0+β1⋅X \pi \left( x \right) = {{{e^{{\beta _0} + {\beta _1} \cdot X}}} \over {1 + {e^{{\beta _0} + {\beta _1} \cdot X}}}} in which π(x) = E(y/x) value is a conditional probability, which must be transformed to linearize β₀ + β₁ · X in the model. This transformation is a logit transformation and is expressed as follows: (4) g(x)=ln(πx1−πx)=β0+β1⋅X g\left( x \right) = \ln \left( {{{{\pi _x}} \over {1 - {\pi _x}}}} \right) = {\beta _0} + {\beta _1} \cdot X

Transformation variable g(x) is linear to the parameter in the model and is continuous (Hosmer and Lemeshow 2005).

If the number of independent variables is more than 1, then logistic regression model is generalized and called multivariate logistic regression (Chen and Dey 2003). Considering that there are P sets of independent variables, the logit of the multivariate logistic regression model is obtained from the following equation: (5) g(x)=β0+β1⋅X1+β2⋅X2+⋯+βp⋅Xp g\left( x \right) = {\beta _0} + {\beta _1} \cdot {X_1} + {\beta _2} \cdot {X_2} + \cdots + {\beta _p} \cdot {X_p}

In such cases, the multivariate logistic regression function is formulized as follows: (6) π(x)=eβ0+β1⋅X1+β2⋅X2+⋯+βp⋅Xp1+eβ0+β1⋅X1+β2⋅X2+⋯+βp⋅Xp \pi \left( x \right) = {{{e^{{\beta _0} + {\beta _1} \cdot {X_1} + {\beta _2} \cdot {X_2} + \cdots + {\beta _p} \cdot {X_p}}}} \over {1 + {e^{{\beta _0} + {\beta _1} \cdot {X_1} + {\beta _2} \cdot {X_2} + \cdots + {\beta _p} \cdot {X_p}}}}}

In this study, a multivariate logistic regression model was developed since the number of independent variables in the data set is more than 1. The independent variables were identified as the operative temperature (°C), RH (%), and seasons. Thermal complaints were identified as the dependent variable. A multivariate logistic regression model was established to understand the relationship between the independent variables and the dependent variable. The analysis was performed by using the SPSS program.

This section presents the assessment of indoor environmental conditions against ISO 7730 Standard per heating and cooling seasons. Operative temperature (°C) and RH (%) were taken into account in the analysis. The maximum and minimum allowable operative temperatures recommended by ISO 7730 are 20 and 24°C for heating season, respectively. Figure 1 shows the distribution of operative temperatures corresponding to the date of the complaints as well as the recommended maximum and minimum allowable values for the heating season. It should be noted that each dot in the following figures represents the measurement of indoor environmental parameter that corresponds to an occupant's thermal complaint.

Fig. 1

The complaints were collected between January 2017 and May 2018, and thus the period covers two heating seasons starting from October until the end of March. A total of 24 and 5 thermal complaints out of 29 were collected in the heating seasons of 2017 and 2018, respectively. As can be seen from the figure, all operative temperatures were within the recommended values. The maximum operative temperature was observed on October 9, 2017 with a value of 23.9°C, which is almost at the maximum allowable operative temperature recommended by ISO 7730 Standard. The minimum operative temperature was observed on November 28, 2017 with a value of 20.8°C. Figure 2 shows the distribution of RH measurements as well as the recommended maximum and minimum allowable values for the heating season. It should be noted that the minimum and maximum allowable RH ratios are 30 and 70% for the heating season, respectively.

Fig. 2

As can be seen, 2 out of 29 measurements were below the recommended minimum RH ratios. The minimum RH was observed on January 26, 2018 with a value of 19.3% whereas the maximum RH was observed on October 9, 2017 with a value of 49.1%.

Regarding the cooling season, ISO 7730 Standard recommends 23 and 26°C as the maximum and minimum allowable operative temperatures, respectively. The complaints were collected between January 2017 and May 2018, and thus the period covers two cooling seasons starting from April until the end of September. It should be noted that the experimental campaign covered only the months of April and May in the cooling season of 2018. A total of 11 and 13 thermal complaints out of 24 were collected in the cooling seasons of 2017 and 2018, respectively. Figure 3 shows the distribution of operative temperatures as well as the recommended maximum and minimum allowable values for the cooling season.

Fig. 3

As can be seen from the figure, all the operative temperatures were within the recommended values. The maximum operative temperature was observed on August 28, 2017 with a value of 25.9°C, which is close to the maximum allowable operative temperature. The minimum operative temperature was observed on May 25, 2018 with a value of 23.2°C. Figure 4 shows the distribution of RH measurements as well as the recommended maximum and minimum allowable values for the cooling season. It should be noted that the minimum and maximum allowable RH ratios are 30 and 70% for the cooling season, respectively.

Fig. 4

As can be seen from the figure, all RH ratios were within the recommended values. The maximum humidity was observed on August 28, 2017 with a value of 52.3%. The minimum humidity was observed on April 11, 2018 with a value of 38.0%.

This section investigates the compatibility of PMV and PPD values with ISO 7730 Standard. The maximum and minimum allowable PMV values are −0.5 and +0.5 for both heating and cooling seasons. Figure 5 shows the distribution of PMV values as well as the recommended maximum and minimum allowable values for both seasons. It should be noted that each dot in Figures 5 and 6 represents the PMV values calculated according to the measurement of indoor environmental parameter that corresponds to an occupant's thermal complaint.

Fig. 5

The results showed that 96% of the PMV values comply with ISO 7730 Standard, which means that 96% of indoor environmental conditions are satisfactory for the occupants. In addition, the allowable PPD value is less than 10% for both heating and cooling seasons per ISO 7730 Standard. Figure 6 shows the distribution of PPD values and the recommended allowable values for both seasons.

Fig. 6

The PPD values suggested that there are two incidents in which occupants were not satisfied with the indoor conditions. These incidents were observed in the heating season. However, there are 53 thermal complaints filed in these particular conditions. Figure 7 presents the distribution of thermal complaints filed by the occupants. The results showed that occupants have both warm and cold complaints regardless of the season. Therefore, a predefined comfort set point do not ensure occupant thermal satisfaction in the built environment.

Fig. 7

In this analysis, thermal complaint is the binary dependent variable with categories, namely too warm and too cold. The season, operative temperature, and RH values are independent variables.

For the binary dependent variable, a total of 28 and 25 responses were obtained for “too warm” and “too cold”, respectively. The base category for thermal complaints is “too warm” (Table 2).

Table 3 shows the descriptive statistics for the numerical independent variables, which are operative temperature and RH.

For the season, which is a categorical independent variable, the performance is evaluated with respect to a base category. The base category for season is “cooling” and Table 4 presents the frequency of season with respect to thermal complaints.

The first model, a null model (the intercept-only model), was without any predictors. The overall accuracy of the first prediction model was 52.8% as shown in Table 5.

In the second model, the predictors were included in the model (Step 1) and chi-squared goodness-of-fit test was conducted. Table 6 gives the overall test for the model which includes the predictors. The results showed that all p values are <0.05, and thus the prediction model based on predictors is statistically significant.

Table 7 presents the −2 log likelihood, Cox & Snell R², and Nagelkerke R² values. R² values in this model can be interpreted similar to the R² values obtained in the regression analysis. Nagelkerke R² value is a standardized value. It should be noted that Nagelkerke R² value is always more than Cox & Snell R² value. The results showed that 85.4% of the change in the dependent variable can be explained by the developed model.

The results of the second model are given Table 8. The model correctly predicted 26 out of 28 thermal complaints that said “too warm.” Besides, it correctly predicted 22 out of 25 thermal complaints that said “too cold.” Overall, the accuracy of the model is 90.6%.

The statistical results of the variables are given Table 9. The p values indicate whether or not the independent variables are significant. Since the p value of RH is more than 0.05, it can be concluded that RH has no effect on the thermal complaints. On the other hand, the operative temperature and the season have a significant effect on the thermal complaints because the p values of these variables are <0.05.

Since the B value of operative temperature is negative, the probability of a thermal complaint to be “too warm” increases if the operative temperatures increase. In addition, since the B value of the season is negative, the probability of a thermal complaint to be “too warm” increases in the cooling season compared with the heating season.