# Sensitivity analysis of Gagge's Standard Effective Temperature for fan-first cooling control strategy in tropical mixed-mode buildings

Maxime Boulinguez, Aurélie Foucquier, Jean Castaing-Lasvignottes

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Maxime Boulinguez, Aurélie Foucquier, Jean Castaing-Lasvignottes. Sensitivity analysis of Gagge's Standard Effective Temperature for fan-first cooling control strategy in tropical mixed-mode buildings. Building and Environment, 2025, 272, pp.112667. 10.1016/j.buildenv.2025.112667. hal-04958917

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# Sensitivity analysis of Gagge's Standard Effective Temperature for fan-first cooling control strategy in tropical mixed-mode buildings

Maxime Boulinguez<sup>a,b,*</sup> <sup> ID </sup>, Aurélie Fouquier<sup>c</sup> <sup> ID </sup>, Jean Castaing-Lasvignottes<sup>a</sup> <sup> ID </sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Thermal comfort

Cooling

Control strategies

Mixed-mode

Tropical climate

Sensitivity analysis

Sobol

Shapley

## ABSTRACT

Mixed-mode (MM) cooling solutions integrate ceiling fans (CF) and traditional air conditioning (AC) simultaneously in a “Fan-first” scenario, enhancing comfort while reducing energy use. However, this dual approach complicates system sizing and consumption assessment due to the added layer of comfort control. To address this, we advocate using Gagge’s Standard Effective Temperature (SET) for MM cooling control in the tropics. We also evaluate its sensitivity to air velocity, clothing, and temperature, emphasising its applicability for control algorithms in Building Energy Models (BEM). BEM cooling algorithm should leverage fans’ air velocity and clothing flexibility as adaptive opportunities. This study first highlights the relevance of SET over alternative models found in the literature for tropical applications. Second, we use environmental datasets to assess air velocity and clothing contributions alongside temperature setpoints through two Sensitivity Analysis (SA) techniques. Traditional Sobol’s indices, assuming input independence, confirm that temperature is not the sole parameter influencing SET. More advanced Shapley Effect techniques, accommodating representative input distributions, reveal that air velocity and clothing outweigh temperature within an airspeed range of [0–2 m.s⁻¹]. This work provides new insights for designing actionable and adaptive control strategies for comfort in tropical indoor conditions to simulate and size MM-coupled cooling systems effectively. It also underscores the benefits of advanced SA techniques in evaluating comfort models.

## 1. Introduction

With growing demographics and the densification of urbanised territories, by 2030, nearly 60 % of the population will live in tropical cities and half of the population in the tropics by 2050 [1]. Adopting air conditioning as the only means of maintaining comfort conditions in tropical belt countries is the leading cause of the energy demand increase [2]. Comfort level, which tends to westernise, pushes designers to find alternatives to the exclusive use of mechanical compression cooling solutions to achieve comfort objectives. At the border between naturally ventilated buildings (NV) and fully air-conditioned buildings (AC), mixed-mode buildings (MM) offer the flexibility to target both comfort

and lower energy use in tropical climates. Such performance is achieved primarily through natural ventilation and ceiling fans, which are used most of the time and paired with AC systems when indoor conditions deteriorate (Fig. 1). Users are also encouraged to adapt their clothing accordingly. Malik et al. work [3] describes these so-called “Fan-First scenarios” regarding energy use performance. Ceiling fans permit first to delay the start of AC units and, secondly, to raise the setpoints. These shifts are possible because air velocity increases convection transfer on occupants’ skin, enhancing evapotranspiration heat loss. It thus leads to comfortable conditions at higher temperatures or relative humidity than with still air. The cooling effect of ceiling fans permits a rise by at least 2–3 °C typical setpoint [3–5], increasing it over the 24–26 °C/70 % RH range of a traditional Predicted Mean Vote (PMV) comfort zone.

**Abbreviations:** ANOVA, Analysis of variance; ASHRAE, American Society of Heating, Refrigerating and Air-conditioning Engineers; CF, Ceiling Fan; FAST, Fourier Amplitude Sensitivity Test; GSA, Global Sensitivity Analysis; HVAC, Heating Ventilation Air Conditioning; IPCC, Intergovernmental Panel on Climate Change; ISO, International Organization for Standardization; KNN, K-Nearest-Neighbours; LHS, Latin Hypercube Sampling; MEP, Mechanical, Electrical and Plumping; MM, Mixed-mode; NV, Natural Ventilation / Naturally Ventilated; NZEB, Net Zero Energy Building; OAT, One-factor-at-A-Time; PCD, Personal Comfort Device; PERENE, Reference framework for Bioclimatic design of buildings in La Réunion; PMV, Predicted Mean Vote; RBD, Random Balanced Design; RBD-FAST, Random Balance Design Fourier Amplitude Sensitivity Test; SA, Sensitivity Analysis; SET, Standard Effective Temperature; THM, Thermal Heat-balance Model.

* Corresponding author.

E-mail address: maxime.boulinguez@univ-reunion.fr (M. Boulinguez).

<footnote><sup>a</sup> PIMENT Laboratory University of La Réunion, 40 Av. De Soweto, Saint-Pierre 97410, La Réunion, France</footnote>

<footnote><sup>b</sup> Laboratoire d'Ecologie Urbaine Réunion, 137 rue Isautier, 97410 Saint Pierre, La Réunion, France</footnote>

<footnote><sup>c</sup> Grenoble Alpes University, CEA, Liten, Campus Ines, 73375 Le Bourget du Lac, France</footnote>
## Nomenclature

<table><tr><td>CLO</td><td>Clothing, clo</td></tr><tr><td>MET</td><td>Metabolism, met</td></tr><tr><td>RH</td><td>Relative Humidity, %</td></tr><tr><td>S<sub>i</sub></td><td>First order Sobol indice i, a.u.</td></tr><tr><td>S<sub>ij</sub></td><td>Second order Sobol indice ij, a.u.</td></tr><tr><td>T<sub>db</sub></td><td>Dry bulb temperature, °C</td></tr><tr><td>T<sub>g</sub></td><td>Globe temperature, °C</td></tr><tr><td>T<sub>r</sub></td><td>Mean radiant temperature, °C</td></tr><tr><td>T<sub>rm</sub></td><td>Running mean outdoor temperature, °C</td></tr><tr><td>V</td><td>Air velocity, m.s<sup>-1</sup></td></tr><tr><td>X<sub>i</sub></td><td>Input variable i, a.u.</td></tr><tr><td>φ<sub>i</sub></td><td>Shapley value of variable Xi, a.u.</td></tr></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Fan-First cooling strategy for Mixed-mode systems at isocomfort.</figcaption></figure>

## 1.1. Designing MM Fan-First coupled system challenges

This type of operation has been subject to numerous field survey research over the last few years. Although they highlighted the advantages of this type of design over traditional air-conditioned buildings in the tropics, neither of them described how designers should control the coupled CF and AC equipment nor simulate them in a Building Energy Model (BEM) control algorithm. Gokarakonda et al. [6] investigated MM cooling solutions in residential bedrooms in India. Their research aimed to analyse the impact of AC and fan usage behaviour on energy use conservation. Kent et al. [5], led a comfort and energy use field survey of over 35 occupants of a Net Zero Energy Building (NZEB) in Singapore for 11 weeks. They could measure the direct influence of desk and ceiling fan use by raising the thermostat setpoint by 2.5 °C up to 26.5 °C without changing the occupant's thermal satisfaction or air movement dissatisfaction. This study highlights an energy use reduction potential of 32 %. It also illustrates that the combined use of ceiling fans with AC could lead to under-sizing the chillers, pumps, and cooling towers due to higher nominal setpoints. Consequently, this can reduce the overall investment and operating costs. Hence, selecting the appropriate comfort model as an objective function is key to optimising MM-coupled system operation scenario.

The adaptive model [7], dedicated initially to NV buildings, appears as a good candidate for the MM building's comfort assessment, as shown

by Parkinson et al. [8]. Unfortunately, it cannot account for the demand for ceiling fans use energy. In a BEM simulation, the controlling algorithm simulates a cooling system. It is necessary in such cases to define rules to more efficiently target comfort, i.e. with less energy consumption. This adaptive model has been implemented in the Energy Plus Building simulation program in the native Energy Plus Building Management System and via a Python Plugin by Sanchez et al. [9,10]. Although it allows the designer to calculate an adaptive AC setpoint temperature, it does not offer the possibility to simulate ceiling fan use or clothing flexibility. Thus, it is only possible to assess AC energy use. Personal Comfort Devices (PCD), such as ceiling and desk fans, are not accounted for in the energy balance, making it challenging to optimise for the best operation scenario.

Thermal Heat-balance (THM) models are more adapted to simulate MM cooling control possibilities in BEM. THM depends on the four environmental parameters (air temperature (*t*), radiant temperature (*tr*), air velocity (*v*), relative humidity (*rh*) and individual parameters (clothing (*clo*), metabolism (*met*)). Unfortunately, the well-established ISO Predicted Mean Vote (PMV) model falls outside the typical tropical indoor condition range for what it considers its validity in terms of maximum temperature (30 °C) and maximum airspeed (1ms⁻¹) [11]. It is also prone to low accuracy when prediction falls outside the neutrality range. Gagge's two-node [12] model, which calculates the Standard Effective Temperature, disposes of a more refined thermoregulatory system suited for warm conditions [13]. According to Huang et al. [14], it is better to consider evapotranspiration heat loss in a warm environment and non-uniform air movement. Furthermore, it is the foundation of a few adaptive Thermal Heat-balance model alternatives, especially the SET-based adaptive model from Parkison et al.'s work [8], based on ASHRAE Global Comfort Database II [15].

The characteristics of Gagge's SET (Standard Effective Temperature) index mentioned earlier present a promising opportunity for a Fan-First control application. However, the variability of environmental conditions raises questions about the applicability of this index to the MM Fan-First strategy. The primary inquiry is how Gagge's SET index responds to the range of tropical indoor conditions. When using the SET, are air velocity and clothing good alternatives to AC temperature setpoint? How can we quantify the variation in input variables and their impact on the model's output in hot and humid conditions? It is crucial to understand how relative humidity, particularly at higher levels, affects evapotranspiration heat loss and consequently influences the selection of one control action over another. Conducting a Sensitivity analysis (SA) of the SET can help to address these important questions.

Sensitivity Analyses assess the relative contribution of input variables to a model output. It has been widely used in the building industry for several decades using BEM simulation programs (Trnsys, EnergyPlus, DOE2) or ISO13790:2008 simplified models, especially for studying the influence of building design components or climates on energy consumption, space conditioning loads and thermal comfort objectives [16]. De Wit et al. [17] studied the impact of material uncertainty on HVAC sizing and energy use to improve the decision-making process. Bomptemps et al. [18] used an association of SA techniques to study the reliability of dynamic simulation models and presented a framework for parameter selection for high CPU-demanding models. Breesch et al. [19, 20] used SA techniques to study different kinds of natural ventilation design with the objective of thermal comfort in buildings. Tian et al. [16] presented a tenth of other application examples of sensitivity analysis in building energy simulation in their review.

However, research on Thermal Heat Balance comfort models' sensitivity for coupled systems control has been limited, even more so in hot and humid conditions. Moutela et al. [21] demonstrated that metabolism and radiant temperature parameters significantly influence Fanger's model over the entire PMV validity range. At the same time, relative humidity and air speed only played a secondary role in the PMV deviation. Hasan et al. [22] concentrated solely on the sensitivity of PMV to variation in metabolism within the same domain as Moutela
et al. Finally, Ji et al. [23] explored the Gagge's model and its sensitivity to various thermoregulation sub-models. They compared them to measured physiological temperature to propose an updated two-node model. Still, they did not investigate the Gagge's SET sensitivity to its inputs.

## 1.2. Aim of this work, general methodology and expected outcomes

This research aims to advocate the selection of the Gagge's Standard Effective Temperature and evaluate its sensitivity to variations in air velocity, clothing insulation and AC temperature setpoint. The focus is on ensuring its suitability for a Fan-First Mixed-mode cooling system control, particularly in hot and humid indoor conditions.

The proposed steps of this work are outlined as follows:

1. Justify the selection of Gagge's SET as the most appropriate candidate for implementing a MM Fan-First control algorithm in the Building Energy Model (BEM) for a tropical office application. This comfort-based algorithm will manage three parameters simultaneously: dry bulb temperature ($T_{db}$), air velocity ($v$), and clothing (clo).

2. Describe the field measurements taken from an MM building, which will serve as the basis for the environmental ($T_{db}$, $T_r$, $rh$, $v$) and individual variables (met, clo) used in the sensitivity analysis variation datasets.

3. Conduct the sensitivity analysis of the SET using various techniques. This includes comparing the latest Shapley Effect technique with the traditional Sobol indices and quantifying the contribution of simulating air velocity control (ceiling fan or openings) and clothing (occupant adaptive opportunity) against controlling AC setpoint temperature solely.

This study represents a preliminary step in defining the methodology designers, and researchers can use when selecting a comfort model for an MM Fan-First control application. Additionally, it provides insights into how Gagge's SET index model behaves quantitatively during the hottest periods in a typical tropical climate. It also emphasises the advantages of utilising advanced SA techniques for assessing comfort models based on an actual case study.

# 2. Selection of the Gagge's Standard Effective Temperature for MM control application

## 2.1. Background in comfort modelling and selection criteria

Various models exist to predict users' comfort in buildings. However, it can be challenging to select one of them outside of a normative framework for the MM cooling solution [7]. Standards and literature contrast naturally ventilated (NV) buildings (field measurement-based empirical approach modelling) with air-conditioned (AC) buildings (analytical approach modelling, known as Thermal Heat-balance Model or THM). In NV buildings, occupants are considered fully adaptable, i.e., they maintain their comfort by operating openings, solar shading devices, and ceiling fans. In AC buildings, they are supposed to be passive users. This dichotomy is reflected in standards with the comfort models dedicated to NV and AC buildings, respectively, the Adaptive Model (AM) and THM Fanger's PMV (Predicted Mean Vote) [24,25]. From 2010, the ASHRAE 55 standard includes another THM model, the Gagge two-node model [26], to calculate a cooling effect (CE) [24] further entered as input to the Fanger model (named PMVCE). It aims to enhance air velocity's contribution over 1.0 m.s⁻¹, the normative limit of the PMV.

Building Energy Models (BEM) are widely used in the building industry to size AC systems appropriately. For coupled AC—Ceiling Fan systems, MEP engineers must simulate AC thermostat controls and a range of air velocities in BEM to estimate energy use and comfort

performance accurately. Additionally, incorporating clothing flexibility into BEM can enhance comfort and energy efficiency. For instance, reducing clothing insulation by 0.1 clo (20 % of a summer average of 0.5 clo per ASHRAE 55:2023) equates to a 0.6 °C decrease in operative temperature [27]. However, this raises challenges for comfort models to integrate all three controllable parameters—temperature, air velocity, and clothing—within the MM cooling strategy for use in BEM.

The case of mixed-mode cooling adds an extra degree of complexity to the model selection process. Whereas air conditioning is available in such buildings, they offer total flexibility to occupants to be active in their comfort, mainly through solar shading, ventilation openings, ceiling fan use and flexible dress code. Hence, adaptive opportunities should be considered.

Furthermore, our problem must be solved in a critical hot and humid condition. The selected model should be adapted and the most reliable in that field.

To select the most appropriate comfort model for simulating MM Fan-First coupled system operation in BEM, we defined the criteria of selection as follows:

(1) The comfort model chosen should give the ability to adjust variables such as the clothing (clothing flexibility), the airspeed (ceiling fans, ventilation openings) and the setpoint temperature of air conditioners

(2) Its ambience validity domain should contain hot summer tropical indoor conditions, especially for Temperature and Air velocity. ($24 > T_{db} > 34$ °C, $0 > v > 2.5$ m.s⁻¹)

(3) It should be the most reliable in warm and hot conditions for office applications

(4) It should be the most reliable in humid conditions where heat loss by perspiration is critical

## 2.2. Questioning original adaptive models for MM application

Research has highlighted the strong similarities between naturally ventilated and MM-cooled buildings regarding user adaptation and, therefore, the use of the adaptive model (AM: Adaptive Models) for MM buildings [8]. AM, specific to naturally ventilated buildings (NV), transcribes adaptability in a linear model, which takes the form $T_{comfort} = a \cdot T_m + b$, where $T_m$ is the running mean outdoor temperature of the previous days [7]. Behind the concept of adaptability, three main components are considered: behavioural, physiological and psychological [28]. Adaptive model simplicity allows rapid implementation in a BEM for controlling the air conditioning setpoint [9,10]; they are well suited to buildings cooled by natural ventilation or in high humidity conditions [25], which are typical of a tropical climate [14]; they are dedicated to building in which individuals are predisposed to a greater capacity to adapt to their environment [29] (as opposed to air-conditioned buildings).

Unfortunately, Zhang et al. [30] mention that it neglects the effect of air movement and other factors influencing thermal comfort even if new versions integrating additional environmental parameters, such as relative humidity [31], are being developed. Finally, none of these models allows air speed and clothing to be considered controllable parameters like the setpoint temperature, excluding them from our study.

## 2.3. Questioning original Thermal Heat-balance models and their adaptive alternatives for MM application

Alternative solutions exist to bridge the gap between Adaptive Models (AM) and Thermal Heat-balance (THM) Models. This combinatorial approach was introduced after the AM became part of the standard and is referred to as the *adaptive heat balance approach* (further named Adaptive THM). It seeks to enhance THM models by better considering the three physiological, psychological, and behavioural adaptation factors, thereby rationalising the adaptive concept within a thermo-
physiological framework. This is based on the observation that the Fanger and Gagge THM models have been the subject of numerous reliability studies. As an example, following Humphreys's research in 2002 [21], Cheung et al. [22] showed, based on the worldwide ASHRAE Global Comfort Database, an accuracy of PMV ISO of only 34 % for all cooling modes [15]. Adaptive THM, which expresses comfort as a function of six environmental and individual variables, alongside the concept of adaptability Comfort = f(T<sub>db</sub>, T<sub>r</sub>, RH, V, CLO, MET, *adaptation*), hold promise for comprehensive MM Fan-First cooling control.

Three primary Adaptive THM model types exist:

(1) **Output correction:** These models adjust THM outputs using a corrective factor k. Examples include Fanger and Toftum's ePMV [7,32,33], which use fixed k, or Yao et al.'s field-calibrated coefficients [32,35]. Zhang et al. proposed similar corrections for Gagge's SET [36]. The resulting model is of the following form: $THM_{corrected} = k \cdot THM$

(2) **Input modification:** Schweiker et al. developed the Adaptive Thermal Heat-Balance Model (ATHM) by modifying inputs like metabolic rate and clothing insulation. This method applies to both PMV and SET [37,38]

(3) **Regression-based model:** Parkinson et al. [8] extended the adaptive theory with a SET-based regression model ($SET_{comfort} = a \cdot T_{rm} + b$), highlighting that a third of the comfort effects must be attributed to the thermal balance influenced by the six individual and environmental parameters.

Despite these advancements, no consensus exists on the best model for MM-cooled buildings.

The MM control algorithm must assess comfort using a thermophysiological model (THM or Adaptive THM) to control multiple parameters. Currently, the available THM options are the Fanger one-node model (PMV indices) and the Gagge two-node model (SET indices), necessitating a choice between these two approaches. THMs evaluate the deviation from thermal neutrality. They carry out the thermal assessment of the occupant by establishing the difference between internal heat production and exchanges by conduction, convection, radiation, and respiration with the environment. The main advantage of Gagge's two-node model over Fanger's is the consideration of internal heat storage and transient thermoregulation systems. Two concentric cylindrical segments, called nodes, represent the skin and the human body's core. The exchanges occur between the ambience and the skin. The model evaluates the need to implement thermoregulation processes between the skin and the core body at each time step. Core and skin temperatures are then reassessed accordingly. Conversely, the more simplified Fanger's model only considers the exchanges between a single segment and node (the body) and the environment under steady-state conditions. [39]

The first aspect questioning the relevance of the PMV model is its validity domain. In a tropical climate, indoor ambient conditions which regularly reach $T_{db} > 30$ °C, $RH > 60 \%$, $0 \text{ m.s}^{-1} < V < 2.5 \text{ m.s}^{-1}$ go beyond the validity domain of Fanger's PMV ISO7730 (max $30 \circ C$, $1.0 \text{ m.s}^{-1}$) [11]. This domain is extended ($0.1 \text{ m.s}^{-1} < V < 2 \text{ m.s}^{-1}$) by ASHRAE 55:2010 and later with the introduction of the cooling effect (CE) determined from the Gagge's SET. The CE is subtracted from the dry bulb and radiant temperature in the PMV formulation to calculate the PMV<sub>CE</sub>. Tartarini et al. [40] found that neither PMV models (PMV or PMV<sub>CE</sub>) could accurately predict thermal sensations on the warm side, and they should only be used in neutral conditions, i.e. $|PMV| \le 1$. Aside from Fanger's model, the Gagge model has been the subject of multiple refinements since it was developed in 1971. Vellei et al. [28] mention the most up-to-date versions in their article and several research on the model's performance, namely good accuracy for neutral and hot or very hot conditions. In his book [24], Parsons mentions another advantage of Gagge's model over Fanger's steady-state equation. Thanks to its calculation procedure based on a time step of one minute, it is possible to

conveniently determine the physiological constants over the chosen duration, making the model more dynamic.

The other aspect that questions the relevance of PMV in these conditions is the cursory consideration of evapotranspiration, especially in the presence of high air velocity. Huang et al. [14] showed in their experiment for evaluating thermal sensation in the case of a warm environment and non-uniform air movement that the PMV model does not take into account the cooling effect of airflow when the skin wettedness is over the 'dry' threshold of 0.06. The model only integrates evapotranspiration through metabolic activity [41]. Thus, this latter remains constant at a fixed metabolism regardless of the environmental conditions. This hypothesis appears counterintuitive in hot and humid conditions where the influence of heat losses by evapotranspiration depends strongly on temperature, relative humidity and air movement. With the Fanger model, in the case of office activity such as typing, which remains relatively constant at around 1.1 MET, the environmental conditions, even extreme, will not influence evapotranspiration, one of the main driving forces behind regulating the human body in tropical conditions. Gagge's model integrates this phenomenon more completely. It depends on the skin's wettedness and the difference in partial vapor pressure between the skin and the environment. It further demonstrates that skin wettedness, defined as the ratio of evaporation on the surface to the maximum evaporation capacity in the environment, is a good indicator of discomfort and thermal stress well beyond the sweating rate proposed by Fanger. This highlights some limits of Fanger's PMV in these conditions since it does not account for evaporative heat loss enhancement by elevated air velocity, as Gagge's does.

## 2.4. Gagge's SET index selection

In this study, we selected the Gagge two-node model due to its ability to dynamically account for internal heat storage, transient thermoregulation processes, and the detailed physiological response to environmental conditions. Its Standard Effective Temperature (SET) index provides a robust framework for assessing thermal comfort in mixed-mode cooling applications under hot and humid tropical conditions. Furthermore, the SET index can be adapted within an adaptive heat balance framework to integrate better adaptive opportunities and controllable parameters: temperature, air velocity, and clothing insulation.

We conducted a sensitivity analysis to evaluate its response to variations in critical indoor conditions to ensure the suitability and robustness of the Gagge's SET index for MM Fan-First control applications. SA results quantify SET's response to airspeed and clothing modulation against other parameters, explicitly in tropical indoor conditions.

## 3. Data collection: Reunion MM building case study

A field study was conducted in a mixed-mode office building (Fig. 2) for seven weeks in the summers of 2023 and 2024 in Saint-Pierre, Reunion Island, to contextualise the analysis with actual field measurements of a tropical case study. Reunion Island is a French outermost territory close to Madagascar and Mauritius islands in the Indian Ocean. The building is located at the border between the Aw and Am climate subzones of the A-Tropical köppen climate group. The average annual daytime temperatures [7 a.m. to 6 p.m.] do not drop below 23 °C for Saint-Pierre. During the hot season, the daily temperature is expected to range from 24 to 34 °Celsius, with an average summer relative humidity of about 65-75 %. The building is an old concrete structure, a two-floor house of approximately 150 square meters located by the sea. It has been renovated, enlarged and converted into an office building for an architectural agency. It is typical of many offices in Reunion Island for small companies under 30 employees. The main blind façades are NE and NW-oriented, not shaded or insulated. The main opened façades are S-SW-oriented, facing a green courtyard. Most ground-floor ceilings are
<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Outdoor and indoor photographs of the case study building, “COArchitectes” Agency.</figcaption></figure>

shaded from the sun (great tree) or do not get direct sunlight as they are not in contact with the outdoors (first floor, rooftop veranda). Since specific spaces are single-oriented, they do not have natural cross ventilation for users’ comfort. These users have access to ceiling fans that can be combined with air conditioning for the hottest periods. There are four main spaces. A Mixed-Mode single office located on the first floor with double side openings, a Mixed Mode meeting room located on the ground floor with single side opening, a five desks NV fully cross-ventilated space (wood construction extension) and eight desks Mixed Mode office with single two side double door-windows openings (Fig. 2). All spaces have ceiling fans (brand: Hunter, model: Industrie II 132 cm). Mixed-mode spaces are also equipped with split air conditioner units (Fig. 3). Control of ceiling fans and AC thermostats is at the user’s discretion.

The ceiling fans offer three different rotation speeds. The office administrator limits the thermostat threshold for the AC unit at 26 °C per the local Reunion Island Standard for offices and commercial buildings

PERENE:2009 [42]. This setpoint is also recommended in Manzano-Aguagliaro et al. [43] review of bioclimatic strategies for thermal comfort. Several field studies were done in the summers of 2023 and 2024. Environmental sensors (Dragino LHT65) recording dry bulb temperature and relative humidity were installed on all workstations. Each space was also equipped with a comfort station, detailed in Table 1, measuring air temperature, globe temperature, relative humidity, and air velocity. It was placed in the centre of the room, away from direct solar radiation and architectural masks or internal partitions. Mean radiant temperature is calculated from globe temperature, air velocity and dry bulb temperature according to ISO 7726:2023 [44]. Clothing garments for top, bottom, shoes and extra layers were recorded for each vote over 1400 votes and translated to clothing values according to ISO 9920:2009.

The indoor conditions extrema are determined from data averaged over 5 min with a confidence of 6 standard deviations (99.7 % of the measurement points). Mean and standard deviation are also computed and mentioned in Table 3.

# 4. Sensitivity analysis of Gagge's Standard Effective Temperature

## 4.1. Global sensitivity analysis: Sobols's indices for independent variables

Various SA methods exist for independent variables. Their choice depends on the objectives set and the constraints of the models [45]. Screening techniques such as One-factor-at-A-Time (OAT) discretisation and the Morris method are widely used in engineering because they are straightforward and computationally efficient. They are often used if the input parameter variability is close to its nominal value [46]. Nevertheless, they are limited to qualitative analysis, cannot assess parameter interactions and combined effects and may not fully explore the input space, leading to overemphasising trends when extreme values dominate. We did not select this method because running the Gagge two-node model costs very little on a desktop computer, and only five parameters are being tested. Nevertheless, for further details on the technique, the reader should refer to Machard et al. [47,48] and Ioss and Lemaitre [45] and Tian et al. [16].

Global Sensitivity Analysis (GSA) based on the decomposition of the output variance, also called ANOVA (ANalysis Of VAriance), is more resource-intensive than the screening method. Still, they provide quantitative and valuable information. For example, Sobol's method can decompose all the output variance due to corresponding inputs and consider all interactions. The first-order Sobol indices $S_i$ can be used to assess the variability of the output (Y) of a function (f), knowing the

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Equipment location layout plan of the main floor of the study case building.</figcaption></figure>
<table><caption>**Table 1**
List of environmental equipment and sensor types for registering indoor climatic conditions.</caption><thead><tr><th>Supplier</th><th>Model</th><th>Probes</th><th>Measurements</th><th>Timestep</th></tr></thead><tbody><tr><td>DeltaOhm</td><td>HD32.1 – Thermal Microclimate Data Logger</td><td>Combined temperature and relative humidity PT 100 probe.<br>Globe temperature probe Ø 150mm</td><td>T<sub>db</sub>, T<sub>g</sub>, V, RH</td><td>15 s</td></tr><tr><td>Testo</td><td>400</td><td>Omnidirectional hot wire probe (0 °C...80 °C)<br>CO<sub>2</sub> probe with Temperature and relative humidity sensors<br>Globe temperature probe Ø 150 mm TC type K</td><td>T<sub>db</sub>, T<sub>g</sub>, V, RH</td><td>15 s</td></tr><tr><td>Kimo<br>Campbell Scientific + Testo</td><td>VT 110 / VT 115<br>CR1000</td><td>Hotwire thermo-anemometer<br>Hotwire thermo-anemometer<br>Combined temperature and relative humidity PT 100 probe.<br>Globe temperature probe Ø 152mm<br>Thermal anemometer</td><td>V, T<sub>db</sub><br>T<sub>db</sub>, T<sub>g</sub>, V, RH</td><td>15 s<br>1 s</td></tr></tbody></table>

variability of the input parameters ($X_i$), as shown in Eq. (1) representing the main effect. The second-order indices $S_{ij}$ can be used to assess the variability of the output (Y) of a function (f), knowing the variability of the input parameters ($X_i$) and ($X_j$), as shown in Eq. (2), representing the combined effects due to interactions. Finally, the sum of both represents the total effect as named total indices $S_{T_i}$ (Eq. (3)). It includes the first order. It fully accounts for the higher-order effects due to interactions between inputs. With Sobol's indices, it is then possible to measure the relative importance of the contribution of all input variables in terms of the main effect and interactions of inputs [49]. The main property of Sobol $S_i$ is its positiveness, and the sum of $S_i$ is equal to 1 when there are no interactions, which makes Sobol indices very easy to interpret.

$$S_i = \frac{\mathrm{Var}[E[Y|X_i]]}{\mathrm{Var}[Y]} \qquad (1)$$

$$S_{ij} = \frac{\mathrm{Var}[E[Y|X_i X_j] ]}{\mathrm{Var}[Y]} \qquad (2)$$

$$S_{Tj} = S_i + \sum_{i<j} S_{ij} + \sum_{j \neq i} \sum_{k \neq j} S_{ijk} + \dots \qquad (3)$$

Some methods exist to estimate those indices, such as Sobol and FAST (Fourier Amplitude Sensitivity Test). They all consider evaluating the model numerous times with different combinations of inputs; it becomes rapidly very computationally intensive. Typically, it could be counted on tens of thousands of evaluations. Monte Carlo sequences are well known but remain costly compared to their adaptation by Saltelli [45]. Multi-Fourier and coupled multi-Fourier transforms with random balance design (RBD) have been provided to lower the number of model evaluations [46]. These methods are well-suited to our case study, which involves rapid model evaluation and a small number of inputs, precisely five. Otherwise, they remain unstable and biased [45]. We used the GSA RBD-FAST method (Random Balance Design Fourier Amplitude Sensitivity Test [34,35]) associated with the method of generating samples by LHS permutation (Latin Hypercube Sampling) to determine the first-order Sobol indices. This method is as calculation time efficient as an OAT and has the advantage of quasi-random sampling across the whole range of variations. It is directly available in the SAlib Python library [50]. The Latin Hypercube sampling ensures a better homogeneous exploration of the variation domain of parameters than traditional

random sampling. The variation range is divided into N non-overlapping subranges with a probability of 1/N and is used for a random pickup process of values [19].

The variation range is based on actual building data, and the screening of the whole domain is bounded by the field study extrema in Table 2. The metabolism value is constant and equal to 1.1, the normative value of an office typing activity according to ISO7730. Unless metabolism is fixed at 1.1 MET, all variables are uniformly distributed. The choice of the uniform distribution is a strong hypothesis of variable independence. Furthermore, Haese et al. [51] described the detailed comfort field study in their article. Their analysis revealed a strong predominance of relative humidity regarding thermal sensations and user satisfaction. They [51] mentioned a shift at 68 % relative humidity for user satisfaction, independent of the other ambient conditions of the austral summer noted in the demonstrator building. Therefore, to evaluate small jumps in relative humidity, five series (i from 1 to 5) of sampling were executed using each time a different range of 10 relative humidity points, from 45 % to 95 %, called RH series (i) (see Table 2, note (1)). The LHS sampling technique mentioned above was used, and SET values were evaluated for all possible combinations. The five series of humidity ranges were plotted and compared.

## 4.2. The Shapley effects technique for dependent variables

All the methods mentioned above in Section 4.1 are well-suited for independent variables (uniform distribution with no correlation) but cannot handle correlated inputs. Nevertheless, in building engineering, Tian et al. [30] noted that it is common to work on building stock data that can have some form of correlation. Hence, the input space of parameter variation is not rectangular. For example, building age can be correlated to the building area of a city, or building geometry can be linked to a specific construction period or building stock [16]. As mentioned by Prieur et al. [52], “When the inputs are described by an empirical distribution on observational data, it is extremely rare that the variables are statistically independent”. Sobol’s indices lose their interpretability in the presence of correlated inputs [52], they can become negative, and their sum does not necessarily equal 1. The relative importance of correlated inputs can be assessed by different methods like LMG (Lindeman Merand and Gold), PMVD (Proportional Marginal Variance Decomposition) or conditional variable importance from

<table><caption>**Table 2**
Variation ranges bounds of uniform distribution for the independent variable framework based on actual data from summer 2023.</caption><thead><tr><th></th><th>Relative Humidity (%)<sup>1</sup></th><th>Dry Bulb Temperature (°C)</th><th>Mean radiant temperature (°C)</th><th>Air Velocity (ms<sup>-1</sup>)</th><th>Metabolism (met)</th><th>Clothing (clo)</th></tr></thead><tbody><tr><td>Lower bound (recorded min)</td><td>49</td><td>25.4</td><td>23.1</td><td>0.01</td><td>1.1<sup>2</sup></td><td>0.29</td></tr><tr><td>Upper bound (recorded max)</td><td>88</td><td>32.1</td><td>31.2</td><td>2.2</td><td></td><td>0.79</td></tr></tbody></table>

<footnote n="1">¹ Field measurement for determination of humidity ranges series for Sobol's method (1) [45: 55 %], (2) [55: 65 %], (3) [65: 75 %], (4) [75: 85 %], (5) [85 %: 95 %].</footnote>

<footnote n="2">² Typical fixed value for office activity sitting and typing as in ISO7730.</footnote>
random forest, which are recommended and described in Tian’s review of sensitivity analysis techniques for building energy analysis [16]. Those methods remain computationally intensive. To overcome this issue, Shapley values $\phi_i$ were introduced by Owen in 2014 [45] and are based on the coalitional game theory proposed by mathematician and economist Lloyd Shapley in his thesis back in 1951. It is recommended for dependent input variables. This concept aims to attribute the overall profit among players (or variables), given that they cooperate in a team (set of input variables). The marginal contribution of a player across all possible coalitions of players (also called permutations) defines the Shapley value. Shapley’s value has four axioms: (1) Efficiency – The sum of all effects represents 100 % of the gain, (2) Symmetry – If two players contribute the same way, they get the same gain, (3) Dummy – if a coalition has the same gain with and without player $X_i$, then $X_i$ gain should be zero, and (4) Additivity – When two models described by the prediction functions $f$ and $\tilde{f}$ are combined, the prediction should correspond to the sum of the contributions derived from $f$ and the contributions derived from $\tilde{f}$. It stands right in between Sobol’ $S_i$ and $S_{Ti}$, and their sums remain equal to 1 ($0 < \text{Si} < \phi_i < S_{Ti} < 1$). Contrary to Sobol’s total index $S_{Ti}$, which fully accounts for higher-order interaction (interaction of $X_{ij}$ is fully accounted in both $S_{Ti}$ and $S_{Tj}$), Shapley value $\phi_i$ accounts for the main effect and only the part of higher order contribution $X_{ij}$ of $X_i$ interacting with $X_j$ (with $j > i$). In other words, the interaction contribution is not counted twice like in the Sobol method. As mentioned by Prieur et al. [52], the first algorithms to calculate Shapley values were proposed in 2016 by Song et al. (Monte Carlo) and then in 2019 by Benoumechiara et al. (block bootstrap procedure). They are computationally intensive and impose an a priori knowledge of the conditional distribution. Broto et al. [53,54] introduced a subset

aggregation method based on KNN metamodel (K-Nearest-Neighbors) that can solve this issue and be used for any dataset when a sample of the inputs is available and their distributions are unknown, which is our case in this study. This method is available in R sensitivity package 1.30.0 under the shapleySubsetMc function [55].

The parameter variation ranges are based on the actual building data distribution recorded during the summer working hours of the MM office in 2023 and 2024. Fig. 4 shows the distribution of actual data in solid colours, red and blue. Based on these two datasets’ mean and standard deviation (Table 3), a normal distribution is fitted to better represent the on-site actual distributions. As mentioned above, distributions are non-rectangular and lose their independent feature. At this stage, it is impossible to subdivide by humidity ranges as in the previous section because fitted distributions are dependent on one another. As in the last section, the metabolism value will remain constant and equal to 1.1, the normative value of an office typing activity according to ISO7730. Clothing distribution will remain uniform between recorded extrema. Table 3 summarises all field measurements and distributions for the dependent framework.

# 5. Results and discussion

## 5.1. Contribution of parameters in the independent framework (Sobol indices)

Global sensitivity analyses are carried out across the entire variation domain for independent inputs and uniform distributions. After preliminary tests, it appeared that 10,000 draws were sufficient for converging the Sobol indices. SET is calculated using the two-node

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Field measured environmental variables and their fitted normal distribution LHS sampling based on the mean and standard deviation for summers 2023 and 2024.</figcaption></figure>
<table><caption>**Table 3**
Variation ranges and distribution types for the dependent variables framework based on actual data from summers 2023 and 2024.</caption><thead><tr><th>Dataset</th><th></th><th>Relative<br>Humidity (%)</th><th>Dry Bulb<br>Temperature (°C)</th><th>Mean radiant<br>temperature (°C)</th><th>Air Velocity<br>(ms<sup>-1</sup>)</th><th>Metabolism<br>(MET)</th><th>Clothing<br>(CLO)</th></tr></thead><tbody><tr><td rowspan="4">Summer 2023</td><td>min</td><td>49</td><td>25.4</td><td>23.1</td><td>0.01</td><td rowspan="4">1.1</td><td>0.29</td></tr><tr><td>max</td><td>88</td><td>32.1</td><td>31.2</td><td>2.2</td><td>0.79</td></tr><tr><td>mean</td><td>68.9</td><td>29.2</td><td>27.7</td><td>0.83</td><td></td></tr><tr><td>standard deviation</td><td>5.9</td><td>1.1</td><td>1.5</td><td>0.54</td><td></td></tr><tr><td rowspan="4">Summer 2024</td><td>min</td><td>51.7</td><td>23.2</td><td>21.1</td><td>0.01</td><td rowspan="4">1.1</td><td>0.29</td></tr><tr><td>max</td><td>87</td><td>31.8</td><td>31.5</td><td>0.96</td><td>0.79</td></tr><tr><td>mean</td><td>70.3</td><td>29.2</td><td>27.9</td><td>0.38</td><td></td></tr><tr><td>standard deviation</td><td>6.2</td><td>1.6</td><td>1.6</td><td>0.20</td><td></td></tr><tr><td></td><td>Distribution dependant<br>framework</td><td>Normal</td><td>Normal</td><td>Normal</td><td>Normal<br>truncated</td><td>Fixed</td><td>Uniform</td></tr></tbody></table>

model available in the *Pythermalcomfort v2.7* library [56] for each draw of the five RH series. Runtime is on the order of minutes on a desktop computer. For each series, the vertical extent of a violin diagram (Fig. 5) makes it possible to assess the dispersion of the output Y (SET index). The horizontal extent represents its probability density. The central bar corresponds to the interquartile range, representing 50 % of the data, centred on the median (white dot). The grey violins correspond to our reference case study in still air condition, whereas the blue ones represent our case study with real measured air velocities up to 2.2 ms⁻¹ (Summer 2023). Median lines function of RH ranges are plotted in blue (actual case with measured air velocity) or dashed grey (actual case in still air conditions). The comfort and acceptability categories of the SET are represented in colours from (A) to “hot, very unacceptable” in red) to (F) “slightly cold, slightly unacceptable” in light blue, according to Parsons’ correspondence table for SET [39]

Unsurprisingly, the probability of discomfort increases with relative humidity, as shown in Fig. 5. When in still air conditions, as shown in grey, for air velocities up to 0.15 ms⁻¹, the median of the SET increases by 1° to 2° SET for each series and by 5° SET from the RH series (1) to the RH series (5) as shown by the dashed line. The dispersion of the SET also doubles. In still air conditions, the SET index consistently trends toward uncomfortable levels, with 75 % of data points in very warm (Fig. 5 cat. C) and hot zones (Fig. 5 cat. B) at higher RH ranges. Conversely, the presence of higher air velocities (blue violin plots) mitigates potential discomfort, significantly shifting the SET distribution toward acceptable zones (Fig. 5 cat. E and D, blue violins). It is exacerbated at high RH, where having more than 10 % of points in the comfortable category (E) is difficult when in still air conditions (grey violins). With high air

velocities (blue violins), it is possible to lower the overall dispersion of the SET for each RH series with an average median of 5°SET under the one for still air conditions (blue line). Below 70 % RH, it is possible to have at least 50 % of the output in the comfortable and acceptable (E) to cool (F) categories. The remaining 50 % are mainly in the slightly warm category (D). Only from the RH series (3) with high air velocities can one observe more than 25 % of the points in an unacceptable situation (orange gradient cat. B and C) when all RH series with still air (grey violins) showed a significant amount of uncomfortable SET.

The partial variances (Table 4) represent the absolute contribution of the parameters $X_i$ on the output Y. They are expressed as a square of Y, i. e. SET². When comparing several sensitivity analyses, e.g. for several relative humidity ranges, it is necessary to validate that variance is not null, which could lead to a meaningless interpretation of first-order Sobol indices. All RH series show a total variance comprised between 4.8 and 13.4 SET². Fig. 6 presents the first-order sensitivity indices $S_i$ for each input parameter of each series. The Sobol index is the partial variance of Y knowing $X_i$, normalised by the total variance of Y (Eq. (1)). These Sobol indices represent a parameter’s relative contribution to the output’s variability. Within each series, we ensure that the sum of the Sobol indices is close to 1 and that the error bars are less than 5 % (Fig. 6). Then, we can confirm that there are no interactions of the input variables in the model evaluation. A parameter whose index is less than 0.1 can be considered to have a negligible influence on the output.

Fig. 6 shows that the relative contributions of the airspeed (v) and the clothing (clo) are of the same order of magnitude, whatever the relative humidity ranges. They are relatively constant at 0.25 from 45 % to 75 % RH and decrease slightly from 75 % RH. Their actual contribution (Table 4) increase with relative humidity to approximately double from the RH series (1) range to the RH series (5). In other words, the effect of a user action on one of these two parameters will have twice as much impact on the SET in conditions of high relative humidity than in dry conditions. At low relative humidity, the relative contribution (Fig. 6) of the two parameters (V) and (CLO) is slightly below (0.25) the contribution of temperature ($T_{db}$) (0.35). The more the relative humidity increases, the wider the gap becomes. It goes from 0.1-point difference in the RH series (1) to 0.15-point difference in the RH series (3) to finally reach 0.35-point difference for the series RH series (5). Thus, a user’s

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Range of variability of the SET index as a function of relative humidity ranges with hypothetical low air speeds (still air) 0.01 to 0.15 ms⁻¹ (grey violins) and with air speeds representative of the case study from 0.01 to 2.2 ms⁻¹ (blue violins) – (A) Hot, very uncomfortable, (B) Very Warm, very unacceptable, (C) Warm, uncomfortable, unacceptable, (D) Slightly warm, slightly unacceptable, (E) Comfortable, acceptable, (F) Slightly cold, slightly unacceptable.</figcaption></figure>

<table><caption>**Table 4**
Partial variance of the input parameters for a uniform distribution of independent inputs. Representation as a function of different relative humidity ranges for the tropical climate study case.</caption><thead><tr><th>RH series-<br>variables</th><th>(1)<br>[45 %–<br>55 %]</th><th>(2)<br>[55 %–<br>65 %]</th><th>(3)<br>[65 %–<br>75 %]</th><th>(4)<br>[75 %–<br>85 %]</th><th>(5)<br>[85 %–95 %]</th></tr></thead><tbody><tr><td><i>T</i><sub>db</sub></td><td>1.77</td><td>2.37</td><td>3.10</td><td>4.37</td><td>6.86</td></tr><tr><td><i>T</i><sub>r</sub></td><td>0.2</td><td>0.70</td><td>0.75</td><td>0.80</td><td>1.03</td></tr><tr><td><i>RH</i></td><td>0.01</td><td>0.02</td><td>0.02</td><td>0.06</td><td>0.10</td></tr><tr><td><i>V</i></td><td>1.33</td><td>1.54</td><td>1.89</td><td>2.10</td><td>2.65</td></tr><tr><td><i>CLO</i></td><td>1.46</td><td>1.60</td><td>1.96</td><td>2.23</td><td>2.73</td></tr><tr><td>Total</td><td>4.77</td><td>6.23</td><td>7.72</td><td>9.56</td><td>13.37</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Influence of control parameters on the SET. Representation of the first-order Sobol sensitivity indices of the input parameters for a uniform distribution of independent inputs. Representation as a function of different relative humidity ranges for the tropical climate study case.</figcaption></figure>

action on the clothing and the airspeed will have the same impact as acting on the temperature setpoint at low relative humidity. However, the effect will be at least twice as low from 75 % relative humidity. This is explained by the higher the relative humidity in the environment, the more difficult it is for evaporative exchanges to restore body temperature. At this stage, it becomes necessary to act on the setpoint temperature ($T_{db}$) of an air conditioner to complete the effects of air velocity (V) and clothing (CLO). Finally, the contribution of the average radiant temperature of the walls and windows remains marginal in our case study, for which the building is well protected from solar stress, as is a slight variation (10 % max) in relative humidity.

## 5.2. Contribution of parameters in the dependent framework (Shapley effects)

The distributions of variables are constructed by fitting normal distributions from experimental datasets for summer 2023 and 2024. It is first to be noticed that the measured indoor conditions (Fig. 4) are similar in 2023 and 2024 for relative humidity and mean radiant temperature. Nevertheless, air temperature presents a more flattened normal due to a concentration measurement between 26 and 27 °C, resulting from more intensive use of the AC unit in 2024. Recorded air velocities are very different from the two datasets, with a mean of 0.83 ±0.5 ms⁻¹ in 2023 divided by 2 in 2024 (0.38±0.2 ms⁻¹). For the rest of the analysis, we will mention low air velocity range of [0: 1ms⁻¹] for 2024 and high air velocity range [0: 2ms⁻¹] for 2023, as recorded extrema in Table 3. Recorded meteorological data from the closest station (Pierrefond airport, 5 km from the building site) can partially explain this discrepancy; it presents almost the same outdoor wind intensity distribution with a mean of around 3 m.s⁻¹ at 10 m high, but the wind directions differ. The southern façade of the building was far more exposed to strong winds (between 3 and 6 ms⁻¹ and over 6 ms⁻¹) in the summer of 2023 than in the summer of 2024. (30 % of S-facing winds in 2023 compared to 20 % in 2024, 24 % of W-SW winds in 2023 compared to 21 % in 2024). 2024's strong winds exclusively come from the E-SE direction, and a construction mask highly protects the building's opened façade. Hence, it provides less wind access for the S-SW single-oriented

opening spaces.

Fig. 7 presents the result of the Shapley values for both the dependent (dotted grey) and independent frameworks (coloured). The dotted barplot is the uniform distribution shown as a reference case from the independent framework Section 4.1. In a solid colour, the result of the Shapley method (Fig. 7) is similar to Sobol indices when compared with the average RH series (3) of [65 %: 75 %] (Fig. 6) because onsite measurement shows a normal distribution of RH centred on 70 % ± 6 %. Shapley effects are relevant to managing the independent framework. In this case of random draws in a rectangular space, the influence of temperature remains predominant (40 %), and airspeed and clothing make an equivalent contribution at ± 5 % for a total of 45 %. Those results align with Sobol's technique regarding trends, but one should remember that the absolute values of Sobol indices and Shapley values cannot be compared side by side.

According to the two datasets, Shapley values (Fig. 7) show different results. First, it is noted that, such as in the independent framework, one can reproduce the same trend concerning the influence of air velocity and clothing as for the independent framework. Their contribution to the SET remains equivalent at ± 5 %. It has been partially possible to reproduce the result sketched in Fig. 6 for the independent framework on two experimental datasets. Secondly, according to the average air velocity in the building, air temperature influence changes in front of clothing and air velocity. When mean air velocity is provided with a narrow range [0: 1 ms⁻¹], just like in the 2024 campaign (Table 3) (0.38 ±0.2 ms⁻¹), the influence of air temperature is 70 % more influential than when air velocity is available for an extended range [0: 2 ms⁻¹] ((0.83±0.54 ms⁻¹) (2023). It is worth noting that with a higher range, airspeed and clothing combined are more influential than dry bulb temperature itself. Fig. 7 illustrates this with a Shapley value of 0.37 ($T_{db}$) in 2024 when clo and v are respectively 0.2 and 0.27. Conservely, in 2023, when a higher range or air velocity is available, the Shapley value of $T_{db}$ drops to 0.22 when clo and v are 0.3 and 0.36, respectively. Combining these adaptive parameters can account for between 47 % and 66 % (Fig. 7) of the total output effect. Additionnaly, radiant temperature's contribution remains marginal in our two cases where it is close to air temperature, which is typical of NV buildings or conditioned
<figure><img src="image_10.png" /><figcaption>Fig. 7. Influence of control parameters on the SET. Representation of the Shapley values of the input parameters for a normal distribution of dependant inputs with uniform independent distribution as reference.</figcaption></figure>

buildings with high setpoints. Relative humidity, centred for both datasets around 70 ± 5 %, does not influence the SET. More significant variations are needed to assess its influence, just like in Fig. 6, between the RH series (1) and the RH series (5).

To balance parameter's effect on SET and energy consumption, it is necessary to simulate the building energy model (BEM) with a selection of cooling system type, size and power (ceiling fans and air conditioning units). Energy use inherently depends on those three cooling system characteristics and the coupled systems' operation in given climatic conditions. SET is defined in the BEM control algorithm as the objective function to find the most energy-efficient path to comfort. Notably, unlike temperature setpoint, air velocity and clothing control parameters operate at a very low cost (ceiling fan) or no cost (clothing flexibility). These factors account for more than 60 % of SET variance. This insight offers a new perspective on reducing energy use by optimizing system sizing in MM buildings based on SET control, knowing that a traditional ceiling fan's maximum power use is around 50 W, while in comparison, even a small AC unit's power is around 1500 W.

## 6. Conclusions and perspectives

This study initially allowed us to raise the question of the relevance of comfort tools for managing mixed cooling systems in tropical conditions. Several lessons can be learned from it.

First, we showed that the Gagge model presents an advantage over others when considering the tropical climate. We highlighted that the historical AM models are not suited to multiple parameter control in building simulation, such as coupled temperature, air velocity and clothing. It also demonstrates that the PMV has limitations, such as its validity range well below the actual tropical conditions. While the PMV airspeed range can be modified by the Cooling Effect (CE), this presents no significant improvement for the warmest part of the sensation scale. Our analysis demonstrated, on the contrary, that Gagge's SET ticks all the boxes of MM Fan-First cooling control in tropical climates. It better considers the phenomenon of evapotranspiration, is more reliable in warm conditions, and has a dynamic nature that could be easily coupled with the dynamic of Fan-First cascade cooling systems in the BEM control algorithm. The SET index is promising as it is the base of the recent "nudged" adaptive model of Parkinson's [8] for all kinds of buildings or

the ATHB framework of Schweiker [38].

The Sobol sensitivity analyses highlight critical insights into the dynamics of the SET index under various environmental and operational conditions. The studies confirm that the impact of parameters such as airspeed and clothing insulation is significant, regardless of the humidity range. Key findings reveal that at low to medium RH (45%–70%), airspeed and clothing collectively contribute about 50% of the total variability in SET, matching the influence of air temperature. With increasing RH, the combined contribution lowers, highlighting the diminishing effectiveness of the body's evaporative heat loss, resulting in a need to adjust the AC setpoint. Under still air conditions, the SET index often reaches uncomfortable levels, with 75% of data points falling in the very warm and hot categories at higher relative humidity (RH). However, increased air velocity reduces discomfort, shifting the SET distribution toward more comfortable zones.

Shapley effect analysis provides complementary results. When air velocities are limited [0: 1 m/s], Dry bulb temperature accounts for up to 35 % of the SET variance, and 47 % is accounted for clothing and air velocity. However, with a broader range of air velocities [0: 2 m/s], adaptive parameters (V and CLO) collectively influence the SET variability by 66 %, presenting a compelling case for their prioritisation in mixed-mode (MM) cooling strategies. In that case, temperature accounts for only 22 % of the total.

This study demonstrates that leveraging adaptive parameters like air velocity and clothing insulation using a SET-based control can substantially reduce energy use while maintaining comfort in MM buildings. Ceiling fans, with significantly lower power consumption (~50 W) compared to AC units (>1500 W), provide a sustainable pathway for optimising thermal comfort in tropical climates. The negligible influence of radiant temperature ($T_{db} \sim T_r$) and relative humidity variability within typical operating ranges (around 69% ± 5%) further simplifies MM control strategies.

This article presents a study with practical implications for selecting a comfort model for a MM Fan-First control objective. It introduces methods to investigate comfort models' behaviour, particularly emphasising the impact of the chosen distribution, variation range, and sensitivity analysis techniques. Its findings can have practical implications, as they help select SET and improve our understanding of its behaviour under the particular conditions of a tropical climate.
Nevertheless, defining the conditions that allow switching from one cooling mode to another remains an entire question. Indeed, as mentioned by Parkinson et al. [8], this remains a fertile area of research and will depend on many contextual factors for each building. The comparison between the model and actual data promises to be an essential second step for choosing the combined model based on SET. A technical solution must also be proposed to implement such a comfort-based control algorithm in BEM simulation tools to account for all three parameters and assess the energy use of both ceiling fans and air-conditioning units, thus linking SET and energy use.

Finally, this type of study could be an additional tool for comparing different comfort indices before implementing them in a control algorithm. This methodology could also be applied to various environmental conditions, such as the Continental and Mediterranean climates, which experience severe heatwaves during which fans could be beneficial in reducing energy use when coupled with AC.

## Disclosure instructions

While preparing this work, the authors used the free Grammarly app to check grammar correctness and improve readability. After using this tool, the authors reviewed and edited the content as needed and took full responsibility for the publication's content.

## CRediT authorship contribution statement

**Maxime Boulinguez:** Writing – original draft, Visualization, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.
**Aurélie Fouquier:** Writing – review & editing, Supervision, Resources, Methodology, Formal analysis, Conceptualization.
**Jean Castaing-Lasvignottes:** Writing – review & editing, Writing – original draft, Visualization, Supervision, Methodology, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgement

The authors thank Laboratoire d'Ecologie Urbaine Reunion which partially funded this study as part of the National Association for Research and Technology — CIFRE scheme under reference 2020-1566. We also want to acknowledge the technical support the ESIROI School of Engineering and CSTB PULSE Lab provided. We sincerely thank the “Co-architects” team members for letting us monitor their offices for two summers. We thank Thierry Mara for his scientific contribution in the field of sensitivity analysis, which helped us develop this study's methodology.

## Data availability

Data will be made available on request.

## References

[1] C.M. Rodriguez, M. D'Alessandro, Indoor thermal comfort review: the tropics as the next frontier, Urban Clim. 29 (Sep. 2019) 100488, https://doi.org/10.1016/j. uclim.2019.100488.

[2] S. Jim and Priyadarshi R Shukla, 'IPCC AR6 WGIII FullReport - Working Group III contribution to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change'. Accessed: Apr. 19, 2022. [Online]. Available: https://report.ipcc. ch/ar6wg3/pdf/IPCC_AR6_WGIII_FinalDraft_FullReport.pdf.

[3] A. Malik, et al., The potential for indoor fans to change air conditioning use while maintaining human thermal comfort during hot weather: an analysis of energy demand and associated greenhouse gas emissions, Lancet Planet. Health 6 (4) (Apr. 2022) e301-e309, https://doi.org/10.1016/S2542-5196(22)00042-0.

[4] E. Arens, H. Zhang, W. Pasut, Y. Zhai, T. Hoyt, and L. Huang, 'Air movement as an energy efficient means toward occupant comfort', p. 127.

[5] M.G. Kent, et al., Energy savings and thermal comfort in a zero energy office building with fans in Singapore, Build. Environ. 243 (Sep. 2023) 110674, https:// doi.org/10.1016/j.buildenv.2023.110674.

[6] S. Gokarakonda, C. Treeck, R. Rawal, Investigating optimum cooling set point temperature and air velocity for thermal comfort and energy conservation in mixed-mode buildings in India, Energies (Mar. 2022), https://doi.org/10.3390/ en15062259.

[7] R. de Dear, J. Xiong, J. Kim, B. Cao, A review of adaptive thermal comfort research since 1998, Energy Build. 214 (May 2020) 109893, https://doi.org/10.1016/j. enbuild.2020.109893.

[8] T. Parkinson, R. de Dear, G. Brager, Nudging the adaptive thermal comfort model, Energy Build. 206 (Jan. 2020) 109559, https://doi.org/10.1016/j. enbuild.2019.109559.

[9] D. Sánchez-García, D. Bienvenido-Huertas, C. Rubio-Bellido, Computational approach to extend the air-conditioning usage to adaptive comfort: adaptive-Comfort-control-Implementation script, *Autom. Constr.* 131 (Nov. 2021) 103900, https://doi.org/10.1016/j.autcon.2021.103900.

[10] D. Sánchez-García, J. Martínez-Crespo, U.R.-R. Hernando, C. Alonso, A detailed view of the Adaptive-Comfort-Control-Implementation Script (ACCIS): the capabilities of the automation system for adaptive setpoint temperatures in building energy models, Energy Build. 288 (Jun. 2023) 113019, https://doi.org/ 10.1016/j.enbuild.2023.113019.

[11] EN ISO 7730:2005 - Ergonomics of the thermal environment - analytical determination and interpretation of thermal comfort using calculation of the PMV and PPD indices and local thermal comfort criteria (ISO 7730:2005), 2006.

[12] A.P. Gagge, A.P. Fobelets, L.G. Berglund, A standard predictive index of HUMAN response to the thermal environment, Ashrae Trans 92 (part 2B) (1986) 23.

[13] M. Vellei and J.L. Dréau, 'On the prediction of dynamic thermal comfort under uniform environments', p. 17, 2020.

[14] L. Huang, E. Arens, H. Zhang, Y. Zhu, Applicability of whole-body heat balance models for evaluating thermal sensation under non-uniform air movement in warm environments, Build. Environ. 75 (May 2014) 108-113, https://doi.org/10.1016/j. buildenv.2014.01.020.

[15] V. Földváry Ličina, et al., Development of the ASHRAE Global Thermal Comfort Database II, Build. Environ. 142 (Sep. 2018) 502-512, https://doi.org/10.1016/j. buildenv.2018.06.022.

[16] W. Tian, A review of sensitivity analysis methods in building energy analysis, Renew. Sustain. Energy Rev. 20 (Apr. 2013) 411-419, https://doi.org/10.1016/j. rser.2012.12.014.

[17] S. de Wit, G. Augenbroe, Analysis of uncertainty in building design evaluations and its implications, Energy and Buildings 34 (9) (2002) 951-958, https://doi.org/ 10.1016/S0378-7788(02)00070-1.

[18] S. Bontemps, L. Mora, blatman geraud, Reliability of dynamic simulation models for building energy in the context of low-energy buildings, in: 13th Conference of International Building Performance Simulation Association, Chambéry, France, 2013.

[19] Breesch, Hilde and Arnold Janssens, Uncertainty and sensitivity analysis of the performances of natural night ventilation, 2004.

[20] H. Breesch, 'Natural night ventilation in office buildings'.

[21] R. Moutela, J.D. Carrilho, and M.G.D. Silva, 'Sensitivity of the Pmv index to the thermal comfort parameters', 2015, doi:10.13140/RG.2.1.3936.1447.

[22] M.H. Hasan, F. Alsaleem, M. Rafaie, Sensitivity study for the PMV thermal comfort model and the use of wearable devices biometric data for metabolic rate estimation, Build. Environ. 110 (Dec. 2016) 173-183, https://doi.org/10.1016/j. buildenv.2016.10.007.

[23] L. Ji, A. Laoudadi, C. Shu, L. Wang, M.A. Lacasse, Evaluation and improvement of the thermoregulatory system for the two-node bioheat model, Energy Build 249 (Oct. 2021) 111235, https://doi.org/10.1016/j.enbuild.2021.111235.

[24] ANSI/ASHRAE standard 55-2023 - Thermal environmental conditions for Human occupancy, 2023. Accessed: May 30, 2024. [Online]. Available: https://www.ashr ae.org/technical-resources/bookstore/standard-55-thermal-environmental-cond itions-for-human-occupancy.

[25] EN/TR 16798-2:2019 energy performance of buildings - ventilation for buildings - part 2: interpretation of the requirements in EN 16798-1 - indoor environmental input parameters for design and assessment of energy performance of buildings addressing indoor air quality, thermal environment, lighting and acoustics (Module M1-6), Aug. 2019.

[26] A.P. Gagge, Effect of energy conservation guidelines on comfort, acceptability and health. A Serie of Laboratory and Field Studies, John B. Pierce Foundation, 1976.

[27] S. Schiavon, K.H. Lee, Dynamic predictive clothing insulation models based on outdoor air and indoor operative temperatures, Build. Environ. 59 (Jan. 2013) 250-260, https://doi.org/10.1016/j.buildenv.2012.08.024.

[28] F.H. Mallick, 'Thermal comfort and building design in the tropical climates', p. 7, 1996.

[29] R.J. de Dear, G.S. Brager, Thermal comfort in naturally ventilated buildings: revisions to ASHRAE Standard 55, Energy Build 34 (6) (Jul. 2002) 549-561, https://doi.org/10.1016/S0378-7788(02)00005-1.

[30] S. Zhang, Z. Lin, Adaptive-rational thermal comfort model: adaptive predicted mean vote with variable adaptive coefficient, Indoor Air 30 (5) (Sep. 2020) 1052-1062, https://doi.org/10.1111/ina.12665.
[31] M. Vellei, M. Herrera, D. Fosas, S. Natarajan, The influence of relative humidity on adaptive thermal comfort, Build. Environ. 124 (Nov. 2017) 171–185, https://doi.org/10.1016/j.buildenv.2017.08.005.

[32] R. Yao, et al., Evolution and performance analysis of adaptive thermal comfort models – A comprehensive literature review, Build. Environ. 217 (Jun. 2022) 109020, https://doi.org/10.1016/j.buildenv.2022.109020.

[33] P. Ole Fanger, J. Toftum, Extension of the PMV model to non-air-conditioned buildings in warm climates, Energy Build 34 (6) (Jul. 2002) 533–536, https://doi.org/10.1016/S0378-7788(02)00003-8.

[34] J. Gao, Y. Wang, P. Wargocki, Comparative analysis of modified PMV models and SET models to predict human thermal sensation in naturally ventilated buildings, Build. Environ. 92 (Oct. 2015) 200–208, https://doi.org/10.1016/j.buildenv.2015.04.030.

[35] R. Yao, B. Li, J. Liu, A theoretical adaptive model of thermal comfort – Adaptive predicted mean vote (aPMV), Build. Environ. 44 (10) (Oct. 2009) 2089–2096, https://doi.org/10.1016/j.buildenv.2009.02.014.

[36] S. Zhang, Z. Lin, Standard effective temperature based adaptive-rational thermal comfort model, Appl. Energy 264 (Apr. 2020) 114723, https://doi.org/10.1016/j.apenergy.2020.114723.

[37] M. Schweiker, Combining adaptive and heat balance models for thermal sensation prediction: a new approach towards a theory and data-driven adaptive thermal heat balance model, Indoor Air 32 (3) (Mar. 2022), https://doi.org/10.1111/ina.13018.

[38] M. Schweiker, A. Wagner, A framework for an adaptive thermal heat balance model (ATHB), Build. Environ. 94 (Dec. 2015) 252–262, https://doi.org/10.1016/j.buildenv.2015.08.018.

[39] K.C. Parsons, Human thermal comfort. CRC Focus, CRC Press/Taylor & Francis Group, Boca Raton London New York, 2020.

[40] F. Tartarini, T. Cheung, S. Schiavon, E. Arens, The prediction accuracy of different predicted mean vote (PMV) model formulations, in: L.L. Wang, H. Ge, Z.J. Zhai, D. Qi, M. Ouf, C. Sun, D. Wang (Eds.), Proceedings of the 5th International Conference On Building Energy and Environment, Springer Nature, Singapore, 2023, pp. 2363–2370, https://doi.org/10.1007/978-981-19-9822-5_255.

[41] P.O. FANGER, Thermal Comfort Analysis and Applications in Environmental Engineering, Danish technical Press Copenhagen, 1970.

[42] 'PERENE Réunion, Mise à jour 2009. Règles de conception thermique et énergétique des bâtiments tertiaires et résidentiels adaptées aux zones climatiques de l'Ile de La Réunion'. 2009.

[43] F. Manzano-Agugliaro, F.G. Montoya, A. Sabio-Ortega, A. García-Cruz, Review of bioclimatic architecture strategies for achieving thermal comfort, Renew. Sustain. Energy Rev. 49 (Sep. 2015) 736–755, https://doi.org/10.1016/j.rser.2015.04.095.

[44] ISO 7726:2021 ergonomics of the thermal environment - instruments for measuring and monitoring physical quantities, 2023.

[45] B. Iooss and P. Lemaître, 'A review on global sensitivity analysis methods', Apr. 09, 2014, arXiv: arXiv:1404.2405. doi:10.48550/arXiv.1404.2405.

[46] J. Goffart, M. Woloszyn, EASI RBD-FAST: an efficient method of global sensitivity analysis for present and future challenges in building performance simulation, J. Build. Eng. 43 (Nov. 2021) 103129, https://doi.org/10.1016/j.jobe.2021.103129.

[47] A. Machard, C. Inard, J.-M. Alessandrini, C. Pelé, and J. Ribéron, 'Sensitivity analysis of the input parameters impacting summer thermal discomfort in buildings under future climate in Paris', presented at the IBPSA France 2020, Nov. 2020. Accessed: Jan. 22, 2024. [Online]. Available: https://cstb.hal.science/hal-04067828.

[48] A. Machard, 'Towards mitigation and adaptation to climate change : contribution to building design', phdthesis, Université de La Rochelle, 2021. Accessed: Mar. 11, 2024. [Online]. Available: https://theses.hal.science/tel-03675251.

[49] T.A. Mara, W.E. Becker, Polynomial chaos expansion for sensitivity analysis of model output with dependent inputs, Reliab. Eng. Syst. Saf. 214 (Oct. 2021) 107795, https://doi.org/10.1016/j.ress.2021.107795.

[50] J. Herman, W. Usher, SALib: an open-source Python library for Sensitivity Analysis, J. Open Source Softw. 2 (9) (Jan. 2017) 97, https://doi.org/10.21105/joss.00097.

[51] G. Haese, M. Boulinguez, P. Bernaud, A. Couzinet, An innovative approach to better understand hot discomfort, based on the measurement of global human responses, including physiological and sensory indicators - application to end users of mixed mode cooled buildings under tropical climate conditions. Ventilation, IEQ and Health in Sustainable Buildings, Aalborg University, Copenhagen, Denmark, 2023.

[52] B. Iooss and C. Prieur, 'Shapley effects for sensitivity analysis with correlated inputs: comparisons with Sobol' indices, numerical estimation and applications', Nov. 25, 2019, arXiv: arXiv:1707.01334. Accessed: Mar. 14, 2024. [Online]. Available: http://arxiv.org/abs/1707.01334.

[53] B. Broto, F. Bachoc, M. Depecker, Variance reduction for estimation of Shapley effects and adaptation to unknown input distribution, SIAMASA J. Uncertain. Quantif. 8 (2) (Jan. 2020) 693–716, https://doi.org/10.1137/18M1234631.

[54] M.B. Heredia, C. Prieur, N. Eckert, Global sensitivity analysis with aggregated Shapley effects, application to avalanche hazard assessment, Reliab. Eng. Syst. Saf. 222 (Jun. 2022) 108420, https://doi.org/10.1016/j.ress.2022.108420.

[55] B. Iooss, B. Broto, R sensitivity Package - Global sensitivity analysis of model outputs and importance measures, CRAN (2024).

[56] F. Tartarini, S. Schiavon, pythermalcomfort: a Python package for thermal comfort research, SoftwareX 12 (Jul. 2020) 100578, https://doi.org/10.1016/j.softx.2020.100578.