A Neural Network for Monitoring and Characterization of Buildings with Environmental Quality Management, Part 1: Verification under Steady State Conditions

Abstract

Introducing integrated, automatic control to buildings operating with the environmental quality management (EQM) system, we found that existing energy models are not suitable for use in integrated control systems as they poorly represent the real time, interacting, and transient effects that occur under field conditions. We needed another high-precision estimator for energy efficiency and indoor environment and to this end we examined artificial neural networks (ANNs). This paper presents a road map for design and evaluation of ANN-based estimators of the given performance aspect in a complex interacting environment. It demonstrates that in creating a precise representation of a mathematical relationship one must evaluate the stability and fitness under randomly changing initial conditions. It also shows that ANN systems designed in this manner may have a high precision in characterizing the response of the building exposed to the variable outdoor climatic conditions. The absolute value of the relative errors ($MaxARE$) being less than 1.4% for each stage of the ANN development proves that our objective of monitoring and EQM characterization can be reached.

1. Introduction

A published review [1] and papers [2,3,4] introduced the concept of environmental quality management (EQM) including a feasibility of application of the selected statistical methods [5] or artificial neural networks [6] to control heating systems. This paper is a first attempt to address the full system of monitoring and management of environmental quality in buildings from the control point of view.

During the review of current technology [7,8,9] the shortcomings of the current energy models became clear. Amongst many reasons one can highlight the uncertainty in measurement of the global temperature that is key to operating temperature [10], the effect of interactions that is not reflected in the energy models and costs [11,12], difficulties with applying multi-criterial analysis [13], and interaction of building with a city [14].

Nevertheless, the problem is two-fold: (1) most of the currently used energy models solve a system of simultaneous heat, air, and water transfer equations that are partial and hysteretic, second-order and non-linear, with the simplified linearization process and experimental determination of the cross-linking effects (so called Onsager’s effects); and (2) the environment is transient with repeatable daily cycling conditions—the resulting cumulative errors are unspecified. Thus, one may consider the currently used energy models as less suitable for building an automatic control system.

To achieve a high precision level of the energy prediction in the smart buildings, the authors decided to change the paradigm of energy evaluation. We decided to combine the performance monitoring with the post-occupancy optimization of energy use. The process is further complicated by use of the operational room temperature to control the system that is calculated on the basis of the measured walls, fraction of windows, and outdoor temperature. In turn, the operational room temperature is transient and must vary within the adaptable climate range.

To proceed towards the ultimate objectives of monitoring and EQM characterization, we reexamined a previous case [9] and posed general questions about ruggedness and fit of the artificial neural network (ANN) model to the mathematical relation of several inputs to one output. In doing so, we found that despite the many papers published on ANNs, several technical questions related to the resulting precision of the networks, such as effect of initial conditions, weights and bias in the ANN, effect of missing data, shifts in building response related to the outdoor conditions, and outliers caused by misfunction of existing controllers, are not addressed. We decided to divide the whole field into three or four papers.

A review of data-driven building energy prediction [15] lists 153 articles focusing on data source, feature types, model utilization and prediction outputs, use of data-driven models for multi-step prediction, as well as the time dependence of energy related data. The authors highlight that hybrid models (gray box) should also be supported by a statistical analysis to improve their accuracy. In this review, the machine learning methods and the deep learning methods are discussed in a particularly detailed manner.

As EQM technology integrates energy and indoor environment, we decided to review some examples of application of ANN to thermal comfort. One reason for using ANN is to approximate the nonlinear variables and get the input-output mapping of Heating, Ventilation, Air Conditioning (HVAC) control system [16]. One study [17] compares six different algorithms and using box diagrams to present accuracy of the solution, identifies the best combination. Another [18] takes into account seven outdoor and indoor climate and four other variables to evaluate individual thermal comfort. So far, we have dealt with ANNs having one hidden layer [19,20], but in deep learning models we observe a recent trend to use more complex structures [21,22].

Thus, the short review of the literature indicates that a combination of statistical methods in the preliminary stage and classic, one hidden layer type of ANN with a focus on the highest attainable precision, could be the recommended approach.

This paper, being the first part of the series, deals with the full range of considerations as an element of the next generation methodology providing the monitoring and characterization of EQM technology. Instead of the air temperature that in current engineering approaches is considered as measured quantity affected by the energy balance in the room and the convective cooling of the measured property, we will use the operational air temperature that is precisely calculated from the energy balance. As the results of these two concepts are close under moderate summer conditions, we selected the summer period for input of the measured data and with 20 different input data may compare these calculations with the previous case [9]. The methodology applied here is based on that of Latka and Matysek [23] which is based on the equation $y=f\left(\mathit{X}\right)$ and represents a mathematical model [24]. The estimator $y=f\left(\mathit{X}\right)$ is part of a larger complex to monitor and control the indoor environment in a building with EQM system, as shown in Figure 1.

In this paper we analyze a simplified case with constant room temperature as such a case was previously considered (see [9]). This paper seeks to create a highly-efficient, smart building control system that can be applied to any type of facility regardless of its functionality. This approach is one of the novelties in the subject matter described, but it also has certain consequences. One of them, as already mentioned, relates to including the weather predictive capabilities in the model, another is to calculate the transient operational room temperature that must vary within a prescribed time function and stay within the adaptable climate range. Another novelty is integrating energy with the indoor environment in the management system.

It is worth highlighting that using artificial intelligence to solve the control issue with a new energy evaluation paradigm leads to extraordinary convergence of the model results with the measured results. However, to achieve this convergence, it is necessary to perform the required analytical work. The demonstrated procedure is another novelty in the area of environmental quality management.

The solution presented in the article is a continuation of many years of research, which was the basis for the creation of many articles. In previous studies [25,26,27,28,29,30], the experimental base was discussed in detail along with the possibilities it creates for conducting “in situ” research in the field of automatic process control. The issues concerning the work of individual technical systems (HVAC, lighting, blinds, etc.) in terms of reducing energy consumption through the implementation of specially developed control algorithms are presented elsewhere (see [6,26,31,32,33]), as are issues related to providing comfort in an energy-efficient building (see [10,29,30,34,35]). Following the development of technology, conducting an analysis of scientific research results, we designed a new solution that is the subject of this article. To develop this solution, we used measurements published in different sources [10,26,27,28,29,30,31,32,33,34].

2. Methods

2.1. Requirements for Building Automatics Control System (BACS)

With time, as the number of energy sources and storage capabilities grew (e.g., solar thermal, solar photovoltaic, water-based heat pump, water buffer or hot water tank, air-earth heat exchanger, air-water preheat coils), the role of heating ventilation and air conditioning (HVAC) interactions became apparent. Furthermore, post-construction requirements of a good indoor climate that were parallel to energy efficiency, imposed the optimization of HVAC under occupancy stage and thereby created a new set of problems and opportunities [36,37,38]. For instance, some inaccessible valves that previously were set during construction needed accessibility to permit adjustments, thus, new controls to expand the adjustment ranges had to be added. Yet the biggest impact was the requirement for additional monitoring information of temperatures or intensity of the flowing media as the control system needed to develop predicting capabilities [29].

Field monitoring is necessary for two different reasons: (a) safety and security, and (b) information to predict settings of these operational controls that are affected by thermal impedance of the dynamically operated building. Incidentally, we need both the history of these settings and the weather as well as the forecast of weather. Moreover, we need a model of the time response of the building. As we discussed elsewhere [1,2,3,4,5,6,7,8,9], the EQM technology does not rely on the currently existing energy models but develops a neural network-based model (see later text). Since having a well-designed BACS is the key to optimizing the energy performance of the building, one may ask the question—what are the requirements for building an automatics control system?

In the European Union (EU) the main driver for the upgrading energy efficiency is the Energy Performance in Buildings Directive (EPBD) from 2019 which highlights the importance of building automation and monitoring for non-residential buildings. Furthermore, the EU proposed a systematic auditing methodology and established energy performance certificates (EPCs). Such a certificate is issued by an energy expert, and provides information about details of the energy systems, energy consumption, and potential retrofitting measures. The mandatory certificate is implemented throughout all of the EU and helps to increase the energy efficiency of buildings. A study of voluntary European certification conducted in 2014 found that emerging, comprehensive energy certification was the eu.bac audit. In principle, the features and functionalities of building automation and control systems rely on guidelines set by the European standard, EN15232 “Energy performance of buildings—Impact of Building Automation, Controls and Building Management”.

While the eu.bac tool is useful for acquiring insight as part of the building automation system’s commissioning process, the tool is very detailed and needs a large amount of data to define the system. In most cases such data are missing. Time and resources to acquire the data limit the potential users of the tool, especially for retrofit projects. Thus, one may consider this standard as an excellent checklist to be supported with climate consideration (not addressed in the EN 15232 standard). Amongst many papers dealing with auditing and evaluation of BAC systems [39,40,41] one finds that large project savings on improvements to BAC systems are comparable to those of retrofitting building enclosures [39]. Yet, this not an issue of choice between different approaches; both are complementary elements of the same process and both are subjects to the economic analysis of cost–benefit relations.

In the previous series of papers, we addressed the issues related to the next generation of construction technology from a building science point of view [1,2,3,4,6,7,8]. This paper, as well as two more papers to follow, will address the issues related to the building automation, controls, and building management systems that as we know [25,39,40] may have an impact on both energy efficiency and building science. This paper employs two types of verification: (1) traditional verification of ANNs that includes a validation stage after training and before testing, and (2) comparison of the measured and ANN calculated results.

2.2. Finding the Best Number of Neurons in the Hidden Layer

We used a process discussed previously [42], as “$y$” represents the room air operational temperature, and $\mathit{X}$ is a vector that includes 20 parameters described in the later text of the paper that affect the value $y$ in one or another manner (Figure 2). Two two-layered ANNs, P1 and P2 after training and validation have neurons in the second (hidden) layer, s^{1}, varying from 1 to 50. Weight and bias were randomly assigned, and calculation was repeated five times (called approach and indexed as one to five). The results are shown below using so-called boxplots [43].

The criterion for the choice of the ANN was selected as the minimum value of the maximum absolute value of the relative error [44,45] calculated for a given approach. An additional condition about resilience when changing initial weight and bias for neurons was needed to ensure repeatability of results in the tested problem [46,47,48,49].

$$MainCrit=\min \left[{\left(MaxAR{E}_{\mathrm{TEST}}\right)}_{s^{\left\{1\right\}},approch}\right]$$

(1)

where $MainCrit$ is main criterion for choosing the best neural network structure$; s^{\left\{1\right\}}$ is the number of neurons in the hidden layer$; MaxAR{E}_{\mathrm{TEST}}$ is maximum absolute relative error obtained for the testing stage:

$$MaxAR{E}_{\mathrm{TEST}}=max\left(\left|\frac{y_{iTest}-y_{ANNiTest}}{y_{iTest}}\right|\right)$$

(2)

where $y_{iTest}$ is a target for the network in testing stage$; y_{ANNiTest}$ is an output for the network in the testing stage.

The selection of minimum value of the maximum absolute value ($\left|y_{ANNi}\right|$) ensures that each relative error will be between zero and the maximum absolute relative error, as long as we do not use the ANN outside of the training range [50].

While the information about the maximum absolute value of the error is in the mathematical model [51] under testing, the expectation is that the calculated ANN results for conditions not used in training are not farther apart from the target values.

For the sake of clarity, we distinguish between input preparation and other elements of ANN’s architecture.

2.3. Data Preparation

Measurements were performed for 20 min each starting on 17 July and ending on 6 August for a total of 21 days in the summer (the operational temperature, see definitions, is season dependent but in this project, we are not concerned with seasonal differences). The 20 physical parameters that may affect the value of operative temperature in the tested room are as follows:

$$\mathrm{Vector}{\mathit{X}}_i,\mathrm{where}i=1\mathrm{to}20$$

(3)

• Measured time in decimal notation every 20 min
• Degree of opening valve for interior earth–air heat exchanger (%)
• Degree of opening valve for exterior earth–air heat exchanger (%)
• Exterior air temperature (mean of both earth–air heat exchangers)
• Relative humidity of exterior air measured on the inlet to earth–air heat exchangers (EAHXs)
• Irradiation, W/m², measured on north elevation
• Irradiation, W/m², measured on east elevation
• Irradiation, W/m², measured on south elevation
• Exterior air temperature (measured on the roof of the building)
• Temperature of the cooling water in the tank
• Temperature of the cooling water on return from the tested room
• Efficiency of the cooling exchanger (%)
• Steering of the floor cooling valve (%)
• Temperature in the adjacent room on side 1 (it is used instead of wall surface temperature and we are dealing with a steady state evaluation)
• Temperature in the adjacent room on side 2 (it is used instead of wall surface temperature)
• Temperature in the room below (it is used instead of floor surface temperature)
• Temperature in the room above (it is used instead of ceiling surface temperature)
• Steering of the cooling valve for the floor system
• Efficiency of ventilator in climate-convector (%)
• Angle of setting in the solar shutters

$$\mathrm{Target}:y_i=\mathrm{temperature}\mathrm{in}\mathrm{the}\mathrm{tested}\mathrm{room},°\mathrm{C}$$

(4)

The above measures constituted the vector ${\mathit{X}}_i$ and together with the room temperature (targets) $y_i$ were used for teaching, validation, and testing of the ANN. There were 1409 sets and 60% were used for training$\left({\mathit{X}}_{i\mathrm{Tr}},y_{i\mathrm{Tr}}\right)$ and 20% each for validation $\left({\mathit{X}}_{i\mathrm{Val}},y_{i\mathrm{Val}}\right)$ and testing $\left({\mathit{X}}_{i\mathrm{Test}},y_{i\mathrm{Test}}\right)$. An algorithm for ascribing data sets was written as a loop so that they moved to the next iteration after coming to the starting point. The first three sets were ascribed to the training, number four was ascribed to validation, and number 5 to testing. In this manner, for each approach (see Figure 2) we reduced random effects of the comparison and network validation. The data for validation were taken from measurements from which the estimator $y=f\left(\mathit{X}\right)$ was determined [40]. Since the data included changes in temperature, humidity, weather, and other factors, and the validation was performed on independent data, uncertainty of the estimator in the validation stage was compared with training and testing stages. The $MSE$ (6) was chosen as the uncertainty indicator. The process follows the description from previous studies [52,53].

2.4. Pre- and Post-Processing, Learning Parameters, and General Equation of the ANN’s Architecture for a Given Approach

We decided that all input data would be normalized during pre-processing and denormalized during post-processing using the “mapminmax” function (i.e., a linear transformation into the interval of given boundaries) [52,53].

$$Va{l}^{\prime }=\frac{Va{l}_{org}-Va{l}_{min}}{Va{l}_{max}-Va{l}_{min}}·\left({Va{l}^{\prime }}_{max}-{Va{l}^{\prime }}_{min}\right)+{Va{l}^{\prime }}_{min}$$

(5)

where $Va{l}_{org}$ is the original value; $Va{l}^{\prime }$ is the transformed value; $Va{l}_{max}$ and $Va{l}_{min}$ are the original interval boundaries; ${Va{l}^{\prime }}_{max}$, ${Va{l}^{\prime }}_{min}$ are the desired range boundaries, from −1 to 1.

The research uses the Levenberg–Marquardt teaching algorithm [53,54]. This algorithm has shown satisfactory performance in preliminary studies [9]. As a performance function the mean squared error (MSE) was chosen:

$$MSE=\frac{{{\displaystyle \sum }}_{i=1}^n{\left(y_i-y_{\mathrm{ANN}i}\right)}^2}{n}=\frac{{{\displaystyle \sum }}_{i=1}^n{\left(e_i\right)}^2}{n}$$

(6)

$\mathrm{where} n$ is the number of experimental sets for each learning stage (training $\left({\mathit{X}}_{i\mathrm{Tr}},y_{i\mathrm{Tr}}\right)$, validation $\left({\mathit{X}}_{i\mathrm{Val}},y_{i\mathrm{Val}}\right)$, tests $\left({\mathit{X}}_{i\mathrm{Test}},y_{i\mathrm{Test}}\right)$);$y_i$ is a target for the network$; y_{\mathrm{ANN}i}$ is an output of the network respective to the i-th target.

Implicitly, the error was also defined (5):

$$e_i=y_i-y_{\mathrm{ANN}i}$$

(7)

Other parameters of the learning process are shown in

Table 1. Learning parameters for each approach.

Learning Parameter	Value
performance function goal	0
minimum performance gradient	10−10
maximum validation failures	12
maximum number of epochs to train	100,000
learning rate	0.01
momentum	0.9

.

To identify the best possible relationship $y=f\left(\mathit{X}\right)$ we selected the general network structure as feedforward [53,54] with one hidden layer (denoted with {1}) and one output layer (denoted with {2}) (Figure 3).

To restrict the scope of analysis the same activation function was selected. The chosen function was tansig, a hyperbolic tangent sigmoid [53,55], mathematically equivalent to tanh (8). Similarly, for the output layer a linear function of activation was used—purelin (12)—as recommended [50] for cases of non-linear functions.

The choices of the general network architecture and activation functions [9] were confirmed in the preliminary analysis.

$${\mathit{f}}^{\left\{1\right\}}{}_{activ}(\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}})=\frac{e^{2·\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}}-1}{e^{2·\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}}+1}=tanh(\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}})$$

(8)

where$\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}$—argument of the transfer function was:

$$\mathit{a}\mathit{r}{\mathit{g}}^{\left\{1\right\}}=W^{\left\{1\right\}}·\mathit{X}+B^{\left\{1\right\}}$$

(9)

where $\mathit{X}$ is the input column vector; $B^{\left\{1\right\}}$ is the column vector of biases for the hidden layer; and $W^{\left\{1\right\}}$ is the matrix of weights of input arguments for the hidden layer:

$$B^{\left\{1\right\}}=\left[\begin{array}{c}{\mathrm{b}}_{1^{\left\{1\right\}}}\\ {\mathrm{b}}_{2^{\left\{1\right\}}}\\ ⋮\\ {\mathrm{b}}_{\mathrm{p}-1^{\left\{1\right\}}}\\ {\mathrm{b}}_{{\mathrm{p}}^{\left\{1\right\}}}\\ {\mathrm{b}}_{\mathrm{p}+1^{\left\{1\right\}}}\\ ⋮\\ {\mathrm{b}}_{{\mathrm{s}}^{\left\{1\right\}}}\end{array}\right],W^{\left\{1\right\}}=\left[\begin{array}{ccccccccc}{\mathrm{w}}_{1,1}^{\left\{1\right\}}& {\mathrm{w}}_{1,2}^{\left\{1\right\}}& {\mathrm{w}}_{1,3}^{\left\{1\right\}}& & {\mathrm{w}}_{1,t-1}^{\left\{1\right\}}& {\mathrm{w}}_{1,t}^{\left\{1\right\}}& {\mathrm{w}}_{1,t+1}^{\left\{1\right\}}& & {\mathrm{w}}_{1,20}^{\left\{1\right\}}\\ {\mathrm{w}}_{2,1}^{\left\{1\right\}}& {\mathrm{w}}_{2,2}^{\left\{1\right\}}& {\mathrm{w}}_{2,3}^{\left\{1\right\}}& & {\mathrm{w}}_{2,t-1}^{\left\{1\right\}}& {\mathrm{w}}_{2,t}^{\left\{1\right\}}& {\mathrm{w}}_{2,t+1}^{\left\{1\right\}}& & {\mathrm{w}}_{2,20}^{\left\{1\right\}}\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮& & ⋮\\ {\mathrm{w}}_{\mathrm{p}-1,1}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}-1,2}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}-1,3}^{\left\{1\right\}}& \dots & {\mathrm{w}}_{\mathrm{p}-1,t-1}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}-1,t}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}-1,t+1}^{\left\{1\right\}}& \dots & {\mathrm{w}}_{\mathrm{p}-1,20}^{\left\{1\right\}}\\ {\mathrm{w}}_{\mathrm{p},1}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p},2}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p},3}^{\left\{1\right\}}& & {\mathrm{w}}_{\mathrm{p},t-1}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p},t}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p},t+1}^{\left\{1\right\}}& & {\mathrm{w}}_{\mathrm{p},20}^{\left\{1\right\}}\\ {\mathrm{w}}_{\mathrm{p}+1,1}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}+1,2}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}+1,3}^{\left\{1\right\}}& & {\mathrm{w}}_{\mathrm{p}+1,t-1}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}+1,t}^{\left\{1\right\}}& {\mathrm{w}}_{\mathrm{p}+1,t+1}^{\left\{1\right\}}& & {\mathrm{w}}_{\mathrm{p}+1,20}^{\left\{1\right\}}\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮& & ⋮\\ {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},1}^{\left\{1\right\}}& {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},2}^{\left\{1\right\}}& {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},3}^{\left\{1\right\}}& & {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},t-1}^{\left\{1\right\}}& {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},t}^{\left\{1\right\}}& {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},t+1}^{\left\{1\right\}}& & {\mathrm{w}}_{{\mathrm{s}}^{\left\{1\right\}},20}^{\left\{1\right\}}\end{array}\right]$$

(10)

Looking at Equation (8), it should be noted that it is simultaneously the column vector of the hidden layer {1} outputs ${\mathit{Y}}^{\left\{1\right\}}$. This vector takes the following form:

$${\mathit{Y}}^{\left\{1\right\}}=\left[\begin{array}{c}{y}_{1^{\left\{1\right\}}}\\ y_{2^{\left\{1\right\}}}\\ ⋮\\ y_{\mathrm{p}-1^{\left\{1\right\}}}\\ y_{{\mathrm{p}}^{\left\{1\right\}}}\\ y_{\mathrm{p}+1^{\left\{1\right\}}}\\ ⋮\\ y_{{\mathrm{s}}^{\left\{1\right\}}}\end{array}\right]$$

(11)

In Figure 3 we notice that the output vector ${\mathit{Y}}^{\left\{1\right\}}$ of the hidden layer {1} enters the output layer {2} as the input. Taking this into consideration as well as the fact that in the output layer the transfer function was purelin (12) [53,55] and that the number of neurons was constant and equal to $s^{\left\{2\right\}}=1^{\left\{2\right\}}$ (in accordance with the single variable output [49]), the following equations for the layer {2} can be written:

$$f^{\left\{2\right\}}{}_{activ}\left(ar{g}^{\left\{2\right\}}\right)=a·ar{g}^{\left\{2\right\}}$$

(12)

where$a=1$ is the directional coefficient$; ar{g}^{\left\{2\right\}}$ is the purelin transfer function argument:

$$ar{g}^{\left\{2\right\}}=W^{\left\{2\right\}}·{\mathit{Y}}^{\left\{1\right\}}+{\mathrm{b}}_1^{\left\{2\right\}}$$

(13)

where $W^{\left\{2\right\}}$ (14) isthe row vector of weights of input arguments ${\mathit{Y}}^{\left\{1\right\}}$ for the output layer; ${\mathrm{b}}_{1^{\left\{2\right\}}}$ is the bias (scalar) for the output layer.

$$W^{\left\{2\right\}}=\left[\begin{array}{cccccccc}{\mathrm{w}}_{1,1}^{\left\{2\right\}}& {\mathrm{w}}_{1,2}^{\left\{2\right\}}& \dots & {\mathrm{w}}_{1,j-1}^{\left\{2\right\}}& {\mathrm{w}}_{1,j}^{\left\{2\right\}}& {\mathrm{w}}_{1,j+1}^{\left\{2\right\}}& \dots & {\mathrm{w}}_{1,s^{\left\{1\right\}}}^{\left\{2\right\}}\end{array}\right]$$

(14)

The above analysis of components of the general structure of the ANN (Figure 3) permits formulating the equation $\mathrm{y}=\mathrm{f}(\mathrm{X}$) as agglomeration of Equations (3) though (14) in the form of Equation (15):

$$y={\mathrm{denorm}}_{\mathrm{mapminmax}}\left(f^{\left\{2\right\}}{}_{\mathrm{activ}}\left(W^{\left\{2\right\}}·\left({\mathit{f}}^{\left\{1\right\}}{}_{activ}\left(W^{\left\{\mathbf{1}\right\}}·{\mathrm{norm}}_{\mathrm{mapminmax}}\left(\mathit{X}\right)+{\mathit{B}}^{\left\{\mathbf{1}\right\}}\right)\right)+{\mathrm{b}}_1^{\left\{2\right\}}\right)\right)$$

(15)

where${\mathrm{norm}}_{\mathrm{mapminmax}}$ is the data preprocessing operation$; {\mathrm{denorm}}_{\mathrm{mapminmax}}$ is the data postprocessing operation.

3. Results

3.1. Robustness Study of the Examined Neural Network Structures

As mentioned before, we checked 50 cases of ANNs with five created approaches, where weight and bias were randomly ascribed to permit assessment of the ANN’s robustness. If the ANN structure, despite changes in the initial weight and bias, maintains consistency in the basic qualifiers such as $MARE$ (mean absolute relative error) and $\mathrm{R}$ (Pearson’s coefficient), one can assume that the structure is not sensitive to the initial conditions [55]. Additionally, one verifies if the initial data affect the stability of the system [55]. Therefore, if the ANN’s qualifiers such as $MARE$, $R$, $MSE$ or others are satisfactory, the repeatability of the ANN is confirmed [24].

Figure 4 shows a boxplot for values of MARE (Equation (16)) shown against the number of neutrons in the hidden layer:

$$MARE=\frac{1}{n} \cdot {\displaystyle {\sum }_{i=1}^n\left|\frac{y_i-y_{\mathrm{ANN}i}}{y_i}\right|}=\frac{1}{n} \cdot {\displaystyle {\sum }_{i=1}^n\left|\frac{e_i}{y_i}\right|}$$

(16)

Figure 5 and Figure 6 show Pearson’s correlation coefficient R [52] and the mean squared error (MSE) to illustrate how these three qualities bring forth similar conclusions.

All three figures show that the ANNs with more than 32 neurons in the hidden layer are sensitive to the initial conditions. They also prove that structures with less than five neurons in the hidden layer show insufficient accuracy fit to the functional relationship. Figure 4 eliminates structures for $s^{\left\{1\right\}}=6,7,11,12,16,18,19,23,31$. Figure 5 eliminates $s^{\left\{1\right\}}=9,12,16,18,20,23,28,31$ and Figure 6, for $s^{\left\{1\right\}}=6,9,11,12,16,18,20,21,22,23,28,30$ because they had too wide of an interquartile range (IQR) (i.e., differences were too large versus the median) [24]. Effectively, only the structures $s^{\left\{1\right\}}=8,10,13,14,15,17,24,25,26,27,29$ can be considered as suitable.

3.2. Overfitting and Underfitting Study of the Examined Neural Network Structures

The next item in the analysis in the stage of testing was the degree of ANN fitting (i.e., the relation between ANN model prediction and the target values) [49]. In this case, as all the y values had a magnitude of 10², a summed square of residuals ($\mathrm{SSE}$) was selected to measure the discrepancy between the data and an estimation model (see Equation (17)):

$$SSE={\displaystyle {\sum }_{i=1}^n{(y_i-y_{ANNi})}^2}={\displaystyle {\sum }_{i=1}^n{(e_i)}^2}$$

(17)

This qualifier is used for$e_i>1$ as the power increases the difference between target and model prediction. Figure 7 shows boxplot versus the number of neurons in the hidden layer.

The results presented in Figure 7 show that for most structures, underfitting or overfitting phenomenon occurs. It can also be noticed that the overfitting has significant impact for almost every structure starting from $s^{\left\{1\right\}}=15$.

Finally, after analyzing the results from Figure 7, it is possible to indicate only a few structures for which the influence of the phenomenon of underfitting or overfitting is acceptable or has negligible significance. Such structures correspond to $s^{\left\{1\right\}}=8,13,14,20$.

3.3. Identification of the Best Possible Mathematical Relationship of $y=f\left(\mathit{X}\right)$

So far, results in Section 3.1 (robustness) and Section 3.2 (over and underfitting) indicate that structures with ${\mathrm{s}}^{\left\{1\right\}}=8,13,14$ and their approaches are suitable to represent the phenomenon described by the equation $y=f\left(\mathit{X}\right)$. Figure 8 presents a boxplot illustrating the results of the maximum absolute relative error calculated for the network testing stage (2), constituting the basis for the main criterion for choosing the best possible relationship $y=f\left(\mathit{X}\right)$ (1).

This figure presents the results obtained for all analyzed structures to prove that the structure indicated as the best in the following part of the publication (Section 3.3.1) met the main selection criterion (1) for the full analyzed range of $s^{\left\{1\right\}}$ ($1\le s^{\left\{1\right\}}\le 50$).

Table 2. Maximum absolute relative error obtained for the test stages for neural network structures with $s^{\left\{1\right\}}=8,13,\mathrm{and}14$ neurons.

	$\mathit{M}\mathit{a}\mathit{x}\mathit{A}\mathit{R}{\mathit{E}}_{\mathit{T}\mathit{E}\mathit{S}\mathit{T}}$
${\mathit{s}}^{\left\{1\right\}}$	8	13	14
$\mathrm{approach}1$	0.0164	0.0105	0.0228
$\mathrm{approach}2$	0.0192	0.0300	0.0280
$\mathrm{approach}3$	0.0224	0.0242	0.0191
$\mathrm{approach}4$	0.0401	0.0145	0.0290
$\mathrm{approach}5$	0.0159	0.0194	0.0194

shows the maximum absolute relative error obtained for the test stages for the three selected structures with $s^{\left\{1\right\}}$= 8, 13, and 14 neurons.

Results shown in Figure 8 and Table 2 for the main criterion for the selection of the best identification of the equation $y=f\left(\mathit{X}\right)$, as was defined in Equation (1), were fulfilled in approach 1, by the structure with $s^{\left\{1\right\}}=13$. It is marked with bold numbers. Furthermore, looking at Figure 7, the smallest change in the SSE value was also seen for the structure $s^{\left\{1\right\}}=13$. This means that overfitting or underfitting had the smallest significance for this structure.

Effectively, we can state that the structure $s^{\left\{1\right\}}=13$, using $\mathrm{approach}1$ is the best approximation of the relation $y=f\left(\mathit{X}\right)$ for input described in Section 2.2.

3.3.1. The Best $y=f\left(\mathit{X}\right)$ Relation Obtained

The learning process for the selected, best neural network is shown in Figure 9 and Figure 10. The figures present the calculated value of the network performance (Equation (6)) and other learning parameters (gradient, momentum, validation checks). Both of these graphs were depicted according to learning epochs.

In Figure 9, it can be seen that the ANN obtained the best results for the 8th epoch, therefore this epoch represents the final results of the learning process of the network. This figure also shows that the ANN learning process for the training stage had an uninterrupted downward trend until its completion. In addition, we notice that for the validation stage from the beginning of the ANN learning process up to the 8th epoch, there was a continuous improvement in the value of the performance (Equation (6)). This fact also confirms the course of validation checks shown in Figure 10.

Figure 11 shows a histogram of residuals of the network, $e_i$, (Equation (7)), in each of the ANN learning stages (training, validation, test). The x-axis presents the values of the residuals ascribed for the given bin. The y-axis presents the number of occurrences of $e_i,$ covering the range of a given bin. In the figure discussed, it can be seen that the histogram is characterized by a Gaussian distribution and that the data assigned to a specific learning stage have been distributed in an even manner. The discussed figure from the point of view of the quality of the modeled relation $y=f\left(\mathit{X}\right)$ is purely illustrative, because the values of $e_{i }$ (errors) on it are not related to the reference output values of $y_{i }$ which this relationship should seek. Therefore, Figure 12 shows a histogram of relative errors calculated with Equation (18), in the form of 50 bins. In turn, Figure 13 presents a graph showing for which of the assigned measurement samples the relative error occurred. In addition to the analysis of residuals and relative errors in the input data, representation of Pearson’s correlation coefficient, R, was calculated [48] to establish the degree of the ANN model fit. To demonstrate this we used, R², as the qualifier [48]. Figure 14 shows four graphs: training, validation, testing, and all data in relation between $y_{ANNi}$ (output) and $y_i$ (target). The first three relate to specific stages of the ANN learning process, the last shows them all.

$$RE=\frac{y_i-y_{ANNi}}{y_i}$$

(18)

Results shown in Figure 12 and Figure 13 indicate that the relative error for the representation $y=f\left(\mathit{X}\right)$ is less than 1.4% (i.e., a very good result).

Table 3. Coefficient of determination obtained for the best analyzed neural network.

R2	Value
Training stage	0.99818
Validation stage	0.99718
Testing stage	0.99740
All data	0.99782

shows the coefficient of determination, R², indicating a very strong correlation [54] between the results of ANN calculation $y_{ANNi}$ and the values $y_i$ to which the relationship $y=f\left(\mathit{X}\right)$ should converge. As the difference between the ideal fit and the studied case is less than 0.3%, one may conclude that the input data, architecture of the ANN, and methodology of its teaching were correct.

To present the results of the described mapping quality, Figure 15 presents an example of the value of the function $y=f\left(\mathit{X}\right)$ drawn for arguments $x_1$ and $x_2$ (3). This function was drawn on the background of $y_i$ (target), absolute errors ($e_i),$ and relative errors (RE) made by the functional mapping $y=f\left(\mathit{X}\right)$. In this figure, it can be seen that the function maps all reference output values $y_i$ with proper accuracy (RE < 1.4%). It can also be seen that the abrupt change in the value of $x_2$ does not cause discrepancies in the values of $e_i$ or RE, which indicates the compliance of the results described in this chapter.

The number of input arguments ($x_u$ gdzie $u=1,2,\dots ,20$) and the length of the results, implied that instead of presenting the functional mapping relative to the other arguments $x_u$ and $x_p$ (where $u\ne p=1,2,\dots ,20$), the equation $y=f\left(\mathit{X}\right)$ will be presented.

The equation $y=f\left(\mathit{X}\right)$ for the best analyzed neural network is presented here as Equation (1) with two decimal places (low precision) to show the significance of a given parameter for the resulting value. Yet, Appendix A (available online) shows the actual values in the neural network as obtained from the training process. Equation (19) and those in Appendix A are a specific form of the ANN described by Equation (15) and elements of matrix are defined and identified by numbers.

$$y=denor{m}_{mapminmax}\left(f^{\left\{2\right\}}{}_{activ}\left(W_{1\mathrm{x}13}^{\left\{\mathbf{2}\right\}}·\left({\mathit{f}}^{\left\{1\right\}}{}_{activ}\left(W_{13\mathrm{x}20}^{\left\{\mathbf{1}\right\}}·nor{m}_{mapminmax}\left(\mathit{X}\right)+B_{13\mathrm{x}1}^{\left\{\mathbf{1}\right\}}\right)\right)+{\mathrm{b}}_1^{\left\{2\right\}}\right)\right)$$

(19)

More details of the content of the elements of Equation (19) are given in Appendix A in Equation (A1).

4. Discussion

This paper is a part of tools developed for environmental quality management (EQM) system [1,2,3,4,5,6,7,8,9] by two groups: (a) ANN group with US team (MAE Dept., Clarkson University, Potsdam, NY, USA, RD services Cookeville, TN, USA and DFI Enterprise in Morrisville, NY, USA) and Poland team (Lbooking Inc and Cracow University of Technology, Cracow, Poland) and (b) energy modeling group with Canada (ETS, Montreal, QC, Canada) and Saudi Arabia (JUC, Jubal, Saudi Arabia). A cornerstone of this technology is integration of mechanical and structural elements of buildings and use of the integrated control systems for steering of these mechanical devices to satisfy requirements of energy efficiency and indoor environment. The EQM building operates in transient conditions and while all current energy and hygrothermal models are parametric in that they allow comparisons of results related to a given change in the input data, they are not suitable to address transient and cyclic short-term events as they do not account for energy related to air and moisture movements and moisture hysteresis in porous materials.

Therefore, we intend to change the paradigm for steering and control of the indoor environment and energy in the EQM-based residential buildings either new or retrofitted to monitoring to develop an EQM solution for a given building. In the feasibility study [9] we used an ANN to a selected case study. This paper follows the case study and examines the same data set as a component of a potential system of monitoring and EQM solution. This paper presents a full range of design and testing of a ANN that aims at a high precision of estimation of the air temperature in the room located in the experimental building equipped with an earth–air heat exchanger (EAHX) for preconditioning of air during the summer period [2,31]. The case study deals with a model of temperature control and uses 20 input parameters that describe all possible effects, namely temperatures of all adjacent rooms, outdoor climatic conditions including solar radiation, performance of EAHX and water chiller.

The anticipated estimator was termed as $y=f\left(\mathit{X}\right)$ as it was supposed to represent a mathematical model. The input data to the vector $\mathit{X}$ were selected in a three-week summer period with variable outdoor conditions (more than 1400 data sets), but similar enough to expect a pattern characteristic for the given building. Indeed, the results exceeded the expectations. The absolute value of the relative errors ($MaxARE$) was for this estimator less than 1.4% for each stage of the ANN development.

The estimator (Equation (19)) was established after evaluation of 250 variants for the basic equation $y=f\left(\mathit{X}\right)$. This evaluation process included a search for the best structure of ANNs (see Figure 2) and satisfying diverse performance criteria that also included ruggedness (Section 3.1) as well as overfitting and underfitting (Section 3.2). These two elements of analysis showed large effect of initial conditions leaving only three cases out of 50 initially considered cases (or 15 out of 250). This comparison implies that 94% of cases did not have the required stability (Section 3.1) or fitness to represent the equation of $y=f\left(\mathit{X}\right)$.

It must also be highlighted that the above presented estimator was verified in this paper as a self-standing item. Yet the whole project involves an integrated control system and the energy estimator will be expanded to include exchange of air and there also will be another estimator addressing the indoor environment characterization for a given climatic season.

It is worth mentioning that the above presented model was successfully validated for the case of its individual performance. Yet the whole project involves an integrated control system and the energy estimator will be expanded in a few different aspects, particularly with respect to missing data. One should remember that this estimator will be validated twice: first, during the work in the “Measurement data reconstruction and temperature estimation module” (Figure 1), and secondly, for the whole control system (Figure 1). To achieve this goal, measurements will be performed that will correlate with the period of time (climatic season), for which the estimator was designed. Afterwards, the appropriate indicators will be selected. All these validations will be conducted in accordance with previously described procedures [49,56,57,58].

Nevertheless, while recording measurements for ${\mathit{X}}_i$ samples often have part of input data missing or distorted. Such cases are commonly encountered when studying buildings with working control systems. Therefore, in the future, the estimator for the real objects will use a control system element called “Data analysis/reconstruction” (Figure 1). A project of such an element will be presented in the course of further work. This element will function on the basis of an autoencoder or Hopfield’s net [59].

The above presented estimator allows for an initial handling of data so that they will not disturb measured results or negatively affect the process of analysis. Using Hopfield’s net reduces the negative effects of such, as loss of measurements leads to empty fields in the data or failure in control of some parameters or even non-optimal settings used for some devices or sub-systems. Thanks to these measures the estimator will provide stability and efficiency to the operation of the BAC system.

Moreover, the estimator (Figure 2) is built as a modular and elastic tool that employs several different statistical methods. One can continue to modify and expand the scope of the operation of this tool without any collision with the control processes. The statistical analysis complements the limitations in the inherent scope of ANNs because it gives us the capability to determine which factors have a critical influence on the process, to identify statistical outliers or even find a malfunctional device. In turn, one can make a decision on what will be considered as input to the ANN that is being created.

Perhaps, one should explain that our team participates in three interdisciplinary, partly sponsored industrial projects in which buildings with very different thermal mass (a historic building functioning as a hotel, a modern office building, and a houseboat) were retrofitted. With such different thermal impedance of each building, we were looking for a modular and universal solution to control heating/cooling systems. The control system must work with a high precision under adaptable indoor climate conditions (linear transition of air temperature) for 12 cm thick walls on the boat and for more than a meter of masonry walls in the historic building.

The model described in the paper has the following features:

(a)

statistical analysis of data;

(b)

increased performance as higher precision means larger savings of energy;

(c)

the best model of the analyzed room.

The search for a high precision system is justified by the volume of buildings being retrofitted. The difference between 5% and 15% errors in the energy balance means the difference between a country being within international agreements on carbon gas emission or failing it on energy, carbon emissions, and socio-economic progress.

This paper is presented as a road map for the design and evaluation of ANN-based estimators of the given performance aspect in a complex interacting environment. It demonstrates that to create a model for a precise representation of a mathematical relationship one must evaluate stability and fitness under randomly changing initial conditions. It also shows that a properly designed ANN system may have a very high precision in characterizing the underlying dependence as in the analyzed case where the response factor of the building was exposed to the variable outdoor climatic conditions. The absolute value of the relative errors ($MaxARE$) was for this estimator less than 1.4% for each stage of the ANN development which proves that our objective of monitoring and EQM characterization can be reached. A similar procedure was used in another case study [57].

5. Future Research Program

In this paper, we used a term “operational temperature” to describe the indoor climate effect because we needed to add convective and conductive effects to the radiative heat exchange (particularly as we will primarily use the radiative heating systems) and this is done under the term of operational air temperature. To this end we need to complete procedures of handling data prior to application to the ANN and development of indexes to be used in the ANN characterization of indoor environment.

As this paper dealt with verification under steady state conditions, we could use air temperatures for adjacent rooms, but for the transient conditions one needs to use the wall surface temperatures as well. Thus, in addition to characterizing the effect of air exchange, the next estimator must deal with an operational temperature calculated from the balance of energy in the room surroundings. Furthermore, as our focus is on the interaction between different factors affecting the transient response of the building, we will use a modular statistical software [5] to evaluate the consistency of the input data set and remove statistical outliers.