## 2 Methods
### \(\sum IAQ\) Model Scheme Morphology  
According to the new proposal, for models with air quality sub-indices \(IAQ(P_{j})\) (developed with the standard EN 16798-1:2019 as a reference for IEQ model creation [22; 33] and with assumptionsdescribed in Reference [34]), in the case where indoor air has many pollutants, \(P_{1},\ldots j\), the combined \(\Sigma\)IAQ\({}_{\rm index}\) equation is:  
\[\Sigma IAQ_{index}=W_{PI}\cdot IAQ(P_{1})_{\rm index}+W_{P2}IAQ(P_{2})_{\rm index },\ldots,W_{Pj}IAQ(P_{j})_{\rm index} \tag{8}\]  
where the \(W_{P1},\ldots\ p_{j}\) weighting system for IAQ components is created on the basis of the arithmetic mean and the concept of \"excess concentration\" is introduced only for groups of pollutants with similar concentration values. There is a difference in concentration \(\Delta c_{j}\) between the observed concentration of pollutant \(c_{j}\) and the reference concentration \(c_{ref}\) (\(c_{ELV}\) or \(c_{LCI}\)), which is below the current concentration in contaminated rooms. Thus, the excess concentration is:  
\[\Delta c_{j}=c_{j}-c_{ref} \tag{9}\]  
The weights \(W_{1,\ldots\ j}\) for all three IAQ models are determined on the basis of arithmetic means or by adjusting all the values of \(\Delta c_{j}\) in a given model using Equation (10):  
\[W_{j}(IAQ_{comfort/health})=\frac{\Delta c_{j}}{\sum\limits_{j=1\ldots 7} \Delta c_{1\ldots 7}} \tag{10}\]  
where the sum of the adjusted weights \(W_{j}\) of all ventilated pollutants described with sub-indices should be unity. The weight values for a given IAQ model, (e.g., \(IAQ_{comfort}\)) may be different, but the sum of the sub-index weights must be \(\leq\)1.0. The values of the reference concentrations are the concentration levels that are acceptable or recommended as limit values for various pollutants \(P_{j}\). In the case of the \(\sum IAQ_{quality}\) submodel as part of the \(IEQ_{index}\) model, weights should be used for the \(VOC_{odorous}\) (HCHO and TVOC) reference threshold concentrations of odors. The weights in the weighting system should be adjusted to unity according to the Equation (11):  
\[W_{HCHO}=\frac{\Delta c_{j}}{\sum\limits_{j=2,3}\Delta c_{2,3}}=\frac{(c_{j}- c_{ref})}{(c_{TVOC}-c_{ref})+(c_{HCHO}-c_{ref})} \tag{11}\]  
There are, however, non-typical cases in which the scales have different values. This is the case for formaldehyde, the concentration of which is many times lower in the building than the threshold level \(c_{th}\). According to the WHO [7], the admissible value \(c_{ref}\) is also higher than the concentration in the building. In this case, the authors recommend taking the reference value as zero. Then, the weight \(W_{HCHO}\) described by Equation (11) (for two pollutants), would not be negative (\(c_{HCHO}-c_{ref}\)). The ASHRAE Guideline 10 (2011) [3] recommends that the IEQ model (and appropriate weights, \(W_{i}\)) should contain synergy effects of environmental parameters included in the subcomponents and their sensory perceptions. Figure 5 shows the extended IAQ\({}_{\rm index}\) model with its sub-indices treated as components of the IEQ\({}_{\rm index}\), but also with sub-indices of the \(IAQ_{comfort/health}\) type, i.e., pollutants that do not belong to the IEQ model but are important to health and the energy balance of a building with a mechanical ventilation system. The experimental dependencies of the percentage of persons dissatisfied, \(\%PD\), and the values of the concentrations of pollutants, \(c_{j}\), sensed in indoor air in the appropriate ranges are of fundamental significance in the sub-indices relevant to the IEQ model [43]. From the dependencies, expressed as the curves for \(PD\)(CO\({}_{2}\)) or \(PD\)(VOC\({}_{\rm odorous}\)), the equations of the models are derived Equations (12)-(14):  
\[\Sigma\rm IAQ_{quality}=W_{1}\cdot IAQ(CO_{2})+W_{2}\cdot IAQ(TVOC)+W_{3}\cdot IAQ (HCHO) \tag{12}\]  
\[