## 2 Overview of the IMBPC HVAC Solution Hardware and Software
### Software
The value of the PMV index is a seven-point thermal sensation scale, between \(-3\) (cold) and \(+3\) (hot), 0 being neutral [18]. The classical way in which the PMV index (denoted here as \(\Theta\)) can be computed was detailed in [19] and is dependent on six variables: metabolic rate (\(M_{r}\)), clothing insulation (\(I_{cl}\)), inside air temperature (\(TA_{i}\)), inside air relative humidity (\(RH_{i}\)), air velocity (\(V_{ai}\)), and mean radiant temperature (\(\overline{T}_{r}\)). As the standard computation of PMV involves the solution of a nonlinear equation, with a high, non-constant, computation time, we have proposed the use of a static RBFNN to approximate the PMV, for specified values of \(M_{r}\), \(I_{cl}\), and \(V_{ai}\), denoted as a context vector in [20], that achieves a constant computation time at a desired accuracy/speed-up tradeoff. This way, assuming that the RBFNN \(\Theta\) models are designed off-line for the context vectors adequate to a specific installation (using tabulated values for \(M_{r}\)--according to the use of each room, for \(I_{cl}\)--according to the season and the geographical location, and for \(V_{ai}\), obtained from the equipment's specifications), the PMV can be estimated as:  
\[\hat{\Theta}[k]=f_{\Theta}\big{(}TA_{i}[k],RH_{i}[k],\overline{T}_{r}[k]\big{)}, \tag{2}\]  
where \(f_{\Theta}(. )\) denotes the static RBFNN PMV model, for the specific context \(C=\{M_{r},I_{cl},V_{a_{i}}\}\). The forecasts of \(\Theta\) (denoted as \(\hat{\Theta}[k+j]\)), over \(PH\), can also be obtained by the \(f_{\Theta}(. )\) model, provided the forecasts of \(TA_{i}\), \(RH_{i}\) and \(\overline{T}_{r}\) over \(PH\) are available, i.e.,  
\[\hat{\Theta}[k+j]=f_{\Theta}\big{(}\hat{TA}_{i}[k+j],\hat{RH}_{i}[k+j],\hat{ \overline{T}}_{r}[k+j]\big{)},j\in\{1\ldots PH\}. \tag{3}\]  
The measurements of \(TA_{i}\) and \(RH_{i}\) are easily performed by the SPWS devices, discussed later on. Measurement of \(\overline{T}_{r}\) is not so simple. The mean radiant temperature can be defined as the uniform temperature of an imaginary enclosure in which radiant heat transfer from the human body equals the radiant heat transfer in the actual non-uniform enclosure [18]. It can be estimated using different methods [18]:* From the plane radiant temperature in six opposite directions, weighted according to the projected area factors for a person;
* Or, using a black globe thermometer, which was the method used in a previous work [8]. As the latter solution is completely unaesthetic, the former approach is used here, which means that the temperature of the enclosures needs to be measured by the SPWS devices. Regarding the forecasts of the three variables, they are obtained by dynamic RBNNs, implementing Nonlinear Auto Regressive models with eXogeneous inputs (NARX). Specifically,  
\[\hat{T}A_{i}[k]=f_{TA_{i}}\Bigg{(} \begin{array}{c}\{TA_{i}[k]\},\;\{RH_{i}[k ]\},\;\{TA[k]\},\\ \{SR[k]\},\;\{TR[k]\},\;\{M[k]\}\end{array} \Bigg{)} \tag{4}\]  
\[\hat{R}H_{i}[k]=f_{HR}\Bigg{(} \begin{array}{c}\{RH_{i}[k]\},\;\{TA_{i}[k]\}, \;\{RH[k]\},\\ \{SR[k]\},\;\{TR[k]\}\end{array} \Bigg{)} \tag{5}\]  
\[\hat{\overline{T}}_{r}[k]=f_{\overline{T}_{r}}\big{(}\{\overline{T}_{r}[k]\},\;\{TA_{i}[k]\},\;\{TR[k]\},\;\{M[k]\}\big{)}, \tag{6}\]  
where \(\{v[k]\}\) denotes a set of delayed values of a generic variable \(v\) at instant \(k\), whose number and delays need to be determined, \(M[k]\) denotes measured values of the movement detection signal in each room, which must be obtained by the SPWS devices, and \(TR[k]\) denotes the reference temperature to be applied to the HVAC. The inclusion of movement detection in \(f_{TA_{i}}(. )\) and \(f_{\overline{T}_{r}}(. )\), models the influence of occupancy in these two models (for more details please see [14]).