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Energy and Buildings 95 (2015) 160–171

Contents lists available at ScienceDirect

Energy and Buildings

 j ournal homepage: www.elsevier .com/ locate /enbui ld

Energy audit of schools by means of cluster analysis

Rigoberto Arambula Lara a, Giovanni Pernigotto b, Francesca Cappellettic,∗,Piercarlo Romagnonic, Andrea Gasparella a

a Faculty of Science and Technology, Free University of Bozen-Bolzano, Italyb Department of Management andEngineering, University of Padova, Italyc Department of Designand Planning in Complex Environments, University IUAV of Venice, Italy

a r t i c l e i n f o

 Article history:Available online 26 March 2015

Keywords:

Energy retrofit

Energy audit

Schools refurbishment

Cluster analysis

Energy consumption

a b s t r a c t

More than 30 % of  the Italian schools have very low energy efficiency due to aging or poor quality of construction. The current European policy on energy saving, with the Commission Delegated Regulation

(EU) 244/2012, recommends a cost-optimal analysis of retrofit improvements, starting from some refer-

ence buildings. One relevant issue is the definition of aset of reference buildings effectively representative

of  the considered stock. A possible solution could be found using data mining techniques, such as the

K -means clustering method, which allows the division of  a large sample into more homogeneous and

small groups. This work adopts the cluster analysis to find out a few school buildings representative of 

a sample of  about 60 schools in the province of  Treviso, North-East of  Italy, thus reducing the number

of  buildings to be analyzed in detail to optimize the energy retrofit measures. Real consumption data

of  the scholastic year 2011–2012 were correlated to buildings characteristics through regression and

the parameters with the highest correlation with energy consumption levels used in cluster analysis to

group schools. This method has supported the definition of  representative architectural types and the

identification of  a small number of  parameters determinant to assess the energy consumption for air

heating and hot water production.

© 2015 Elsevier B.V. All rights reserved.

1. Introduction

According to the latest report about school buildings of the

Italian Association for the Environment Safeguard, in Italy 42 000

schools are currently in operation and about 60 % of them were

built before 1974[1]. Despite nearly50 % of the schools have under-

gone emergency repairs in thelast 5 years, more than 30 % requires

urgentmaintenance not only due to agingreasons butalso because

of the poor quality of the recent constructions.

The current interest in school buildings, not only in Italy but

also in Europe, is primarily related to two aspects: the high level

of energy consumption of this sector, and the inadequate level of 

comfort (both thermal and air quality). Numerous studies have

been carried out to determine both the real dimension of the prob-

lem and to propose technically and economicallyfeasible solutions,

while the governments have established tougher regulations and

standards that new and retrofitted constructions have to com-

ply with. The main problems in schools, as pointed out by many

∗ Corresponding author. Tel.: +39 0412571295.

E-mail address: francesca.cappelletti@iuav.it (F. Cappelletti).

authors, deal with not only the building envelope and system fea-

tures, but with the management as well.

Some years ago, Antoniniet al.[2] carriedout a surveyon a sam-

ple of 50 schools in the North-East Region of Veneto, Italy. Schools

were assessed as for the energy performance, through analytical

calculation methods, and for the environmental quality, through

experimental detections. It was found that schools in Veneto use

annually between 250kWh m−2 and 350 kWh m−2 (290kWh m−2

in average) including hot water for the gymnasiums and the can-

teens. About one third of this use is attributable to heat losses

through the building envelope. With respect to the heating sys-

tems, in addition to the oversized heat generators found in almost

all the buildings of the sample, the same analysis identified prob-

lems, only detectable through in situ measurements, related to an

incorrect positioning of the internal thermostatic probes or of the

heating elements or to a general bad management of the heating

system. Similarly, Filippín [3], starting from a study on 15 Argen-

tinian schools, reported a number of issues of management and

control related to the maintenance, the appropriate positioning of 

thermostats, the identification of critical areas, the monitoring of 

abnormal loads and the training of staff and students in the proper

use of the facilities.

http://dx.doi.org/10.1016/j.enbuild.2015.03.036

0378-7788/© 2015 Elsevier B.V. All rightsreserved.

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R. Arambula Lara et al./ Energy and Buildings 95 (2015) 160–171 161

Nomenclature

Symbols

 A area (m2)

C  referred to cluster or centroid

EP  normalized energy performance (Wh m−3 K−1 h−1)

F F -test statistic (–)H  capacity of the heating system (kW)

h hour (h)HDD heating degree days (Kd)

HDH  heating degree hours (Kh)K  number of partitions forK-means algorithm

Q  heating demand (Wh)

R2 index of determination (–)

S  dissipating surface (m2)

T  temperature (◦C)

U  thermal transmittance (W m−2 K−1)

V  conditioned volume (m3)

VIF  variance inflation factor (–)

Subscripts

0 initial

adj adjustedenv, gl referred to transparent envelope

env, o referred to opaque envelope

ext  external f  referred to floor

 f–g  floor in thermal contact with the ground

int  internal

k referred to the kth clusterocc  occupancy

r  referred to the roof 

vw vertical walls exposed to the external environment

win referred to the windows

Another important data collection, focused in particular onenergy consumption,installedpower and used fuel, was conducted

in the province of Perugia, Central Italy, by Desideri andProietti [4]

on 29 education institutes, distinguished by the type of construc-

tion. Thespecificelectricand heatconsumptions perunit of volume,

per class and student have been calculated. It was noted that the

energy needs for heating is about 80% of the total and, in the hypo-

thetical scenario in which all the examined buildings reduced the

consumption to the minimum detected in the sample, the achiev-

able savings in terms of energy per student and energy per cube

meter would be 47.6 % and 38 % respectively. Recently, consump-

tion data for space heating of a sample related to 120 school units

were collected in the province of Torino, North-West of Italy [5].

The schools were equipped with fuel meters, heat meters and cli-

matic probes allowing the derivation of a performance indicatorfor the heating consumption, to be used for an initial analysis of 

the building stock and a preliminary assessment of future bud-

get allocations for the Public Administration. Subsequently, using

techniques of multivariate statistical inference on a subgroup of 35

units within the monitored buildings, linear models were devel-

opedto correlatethe measuredconsumption withthermo-physical

and geometrical characteristics [5,6]. The difficulty of obtaining a

complete documentation describing the buildings led to focus on

the development of models based on a small number of indepen-

dent and easily detectable variables, such as the installed power

and the floor surface.

Recent studies focused on the identification of the most-

effectivemeasuresto be applied in thebuilding-system retrofitting.

Butala and Novak [7] indicated insulation and replacement of 

the windows as the most effective, cost-effective and necessary

interventions for 20 of 24 Slovenian buildings examined. For the

Greek climates characterized by a higher level of heating degree-

days, Dimoudi and Kostarela [8] quantified the effect of individual

retrofitting interventions to reduce both the needs for heating and

cooling in order to increase the indoor comfort of the occupants,

assessing the energy savings also in terms of decrease of the pollu-

tion agents in the considered environments. The potential savings,

after the improvement of the insulation level, were 28.7 % for heat-

ing, while more than 99 % in cooling, through the use of simple

ceiling fans and especially night ventilation.

When planning the retrofit or assessing the improvement

potential of a large stock of existing buildings, a large-scale assess-

ment of the consumption has to be carried out. In this framework

different auditing approaches can be adopted to find a bench-

mark to evaluate the energy performance of the building stock,

on one hand, and to assess the performance after retrofitting

interventions, on the other hand. Many studies on building stock

classification and benchmarking have been carried out, some of 

them concerning school buildings. A primary goal in these bench-

mark analyses was to define a stochastic model based on a few

variables, either a regressive model [6] or a targeted selection of 

statistically significant cases [9], suitable to firstly estimate the

margins for improvement.Hernandez et al. [10] proposed a methodto calculate the energy performance benchmark for a rating sys-

tem using a calculated energy performance indicator and grading

it according to standard EN 15217:2007 [11]. A group of primary

schools in Ireland was used as case-study and the main problem

they encountered was the lack of historical data, a problem that is

also present in Italy.

The European Commission is nowadays promoting the renova-

tion of existing buildings by the implementation of a cost-optimal

analysis of different retrofit improvements, starting from a refer-

ence building, whichhas to be representative of a buildingcategory

[12]. As it can be expected, defining the reference building in a

stock of existing ones implies the analysis of a large amount of 

information to find out how this set can be sub-grouped. In similar

cases when the building stock is very large, the application of sta-tistical techniques in order to group buildings with homogeneous

characteristics is necessary to focus the investigations on a small

number of representatives and possibly extend the results to the

others. Gaitani et al. [9], used clustering to identify a few repre-

sentative buildings on which to carry out detailed considerations

of retrofitting, thus anticipating the European directives. Analyz-

ing a sample of 1100 schools, i.e. 33 % of the Greek school building

stock, by means of algorithms of clustering and principal compo-

nents analysis, five typical buildings were selected and described

by seven characteristics,suchas theheated area, age of the building

andheatingsystem,envelope insulation, number of classrooms and

students and occupancy profile. In order to reduce the number of 

variables analyzed, the contribution of each to thefinal energyper-

formance was calculated individually. The application of clusteringin the analysis of existing buildings canbe found also in other stud-

ies. Indeed, clusteringanalysis is a powerful datamining technique,

used to find correlations and patterns,by whicha set of elements is

split into several homogeneous groups containing elements that

are much more similar to each other and significantly different

from those of any other group. Santamouris et al. [13], for exam-

ple, used clustering techniques to define energy classes based on

heating energyconsumption of a large sample of schools in Greece.

Some other authors applied the cluster analysis for the building

stock evaluation, not only to schools butalso to the householdmar-

ket [14]. In some studies the regression analysis associated to the

clustering has been used. This is the case of Filippín et al. [15], who

evaluated the historical heating natural gas consumption during

13 years in 72 apartments belonging to three different multifamily

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162 R. Arambula Lara et al./ Energyand Buildings 95 (2015) 160–171

buildings in La Pampa, Argentina. The stepwise regression method

was used to select the most representative variables that explain

the changes of heating and annual energy consumption during the

available period andthe clusteringto group the apartments accord-

ing to the variables explanatory of consumption, aiming to define

annual energy classes and their ranking.

Concerning the aim of the studies, in some cases the clustering

has been applied to classify the energy performance of buildings

[9,13,16], in other cases to define thebuildingtypologies in a build-

ing stock [17], sometimes to analyze the occupants’ behavior in

relation to the energy loads [18], or to check the necessary number

of typical load curves to represent the building behavior [19]. In

other contexts, clustering has been applied not on real buildings,

but on simulated ones. This is the case of Heidarinejad et al. [20],

who used cluster analysis to examine simulated energy consump-

tion of 134 U.S. LEED office buildings to classify buildings into high,

medium, and low energy use intensity clusters and to provide a

quantitative evaluation of the large difference in energy intensities

in high-performance office buildings.

In retrofitting, even if energy consumption data are often avail-

able, they have to be correlated with the buildings characteristics

in order to estimate the best actions to apply. While being a simple

task for a single building, this becomes quite difficult and time-

consuming when a large stock is considered. Moreover, even whenconsumptions are collected, information about buildings charac-

teristics is often very little.

The aim of this work is to explore a method for clustering a

large set of existing buildings in order to group them on the basis

of the characteristics that have the highest contribution on energy

consumption levels. The final aim is finding a few representative

schools, whichcould be monitored, modeled also by means of sim-

ulation calibration techniques, and analyzed, in order to evaluate

the impact of interventions, and to optimize, by a cost-optimal

approach, the list of the possible retrofit measures. Theset of build-

ings is composed of about 60 schools dated back from the 19th

century up to now and located in the province of Treviso, in the

North-East of Italy. For each school, energy use for heating and

for sanitary hot water was available for the period 2008–2013,together with a number of geometrical and thermo-physical data.

All this information has been elaborated with a precise approach

coherent with [21,22] in order to group them onthe base of similar

characteristics:

(1) Using regression analysis, the groupsof parameters ( predictors)

that are better correlated to the final energy consumption are

identified;

(2) The clustering method has been selected, adapted to the pecu-

liarities of the research objectives with the addition of some

rules and performed for clustering the sample of schools;

(3) Another regression has been performed to check the goodness

of the clustering and to validate the developed clusters;

(4) The obtained results have been studied, the centroids of eachcluster have been determined and identified as representative,

and other linear models have been developed to get an opti-

mized fitting to the data in each cluster.

Although the sample is composed of schools, this methodology

can be applied also to other buildings categories and could be of 

particular interest for historical buildings.

2. Statistical description of the school sample

 2.1. General description

A large database of information for 85 buildings that repre-

sent all of the Province-owned educational building stock has been

Fig. 1. Frequency distribution of buildings concerning the construction period.

analyzed. The province of Treviso is located in the Italian climatic

zone E (Cfa according to Köppen classification). Schools are situ-

ated in locations with a conventional number of Heating Degree

Days (HDD20) calculated with respect toa base temperature of20◦C

spanning from 2350 to 2700K d.For each building, data regarding geometry, thermal properties

of the building envelope and energy consumption of 5 years are

available. After a first control, some of the buildings have not been

considered for further analysis, because of missing or inconsistent

information. The final selected sample includes about 60 buildings,

all of which with consistent data availability. The age of these

buildingsis variable (Fig.1): many were built beforethe 1976, a few

of them in the recent past years. Fig. 1 shows the frequency distri-

bution of the buildings concerning the construction period. About

50 % of the schools were built before the publication of any energy

law, thefirst in Italy being the Lawnumber 373, in force since 1976.

A picture of the sample can be drawn by the descriptive statis-

tical analysis of the most common parameters. Three quarters of 

the schools have a gross heated volume lower than 20,000m3. Big-ger volumes actually occur when the same school occupies more

than one building (Fig. 2). In Fig. 3 the frequency distribution of 

the schools concerning the ratio between the dispersing area and

the heated volume (S /V ratio) shows that the sample is composed

mostly of quite compact buildings with a S /V ratio varying from 0.3

to 0.5. Finally, looking at the amount of the transparent surfaces,

67 % of the schools have a percentage of transparent envelope on

the total dispersing opaque envelope ( Aenv,  gl/ Aenv, op) in the range

6–12 % (Fig. 4a) and 78 % have a window to floor ratio ( Awin/ A f ) in

the range 10–20 % (Fig. 4b).

0

14

25

10

4  4 31 0

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

      <       4

       4   -        1        0

        1        0   -        2        0

        2        0   -        3        0

        3        0   -       4        0

       4        0   -       5        0

       5        0   -        6        0

        6        0   -       7        0

      >       7        0

   C  u  m  u   l  a   t   i  v  e   f  r  e  q  u  e  n  c  y

   F  r  e  q  u  e  n  c  y

Heated Volume [x1000 m³]

Fig. 2. Frequency of buildings by thegrossheated volume.

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R. Arambula Lara et al./ Energy and Buildings 95 (2015) 160–171 163

05

28

19

7

1 1   0 00

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

      <        0 .        2

        0 .

        2   -        0 .

        3

        0 .

        3   -        0 .

       4

        0 .

       4   -        0 .

       5

        0 .

       5   -        0 .

        6

        0 .

        6   -        0 .

       7

        0 .

       7   -        0 .

        8

        0 .

        8   -        0 .

        9

      >        0 .        9

   C  u  m

  u   l  a   t   i  v  e   f  r  e  q  u  e  n  c  y

   F  r  e  q  u  e  n  c  y

Dispersing Surface/ Volume ratio [m-1]

Fig. 3. Frequency of buildings by dispersing S/V ratio.

 2.2. Thermal resistance of building envelope

For each school of the sample, some more information about

the thermal transmittance of the structures is available. Datawere elaborated in order to obtain an area weighted average

value of thermal transmittance for the external opaque enve-

lope (vertical walls, floor, roof) and windows. In Fig. 5a and b

the frequency distribution of the average transmittance of the

03

710

14  16

9

1 00

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

      <       2

       2   -       4

       4   -       6

       6   -       8

       8   -       1       0

       1       0   -       1       2

       1       2   -       1       4

       1       4   -       1       6

      >       1       6

   C  u  m  u   l  a   t   i  v  e   f  r  e  q  u  e  n  c  y

   F  r  e  q  u  e

  n  c  y

Aenv, gl / Aenv, op [%]

03

16

29

11

0   1

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

      <       5

       5   -       1       0

       1       0   -       1       5

       1       5   -       2       0

       2       0   -       2       5

       2       5   -       3       0

      >       3       0

   C  u  m  u   l  a   t   i  v  e   f  r  e  q  u  e

  n  c  y

   F  r  e  q  u  e  n  c  y

Awin / Af [%]

(b)

(a)

Fig. 4. Frequency of buildings by the percentage of windows area over the opaque

envelope area (a) and over thetotal floor area (b).

48

  10

19

97

1

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

      <       0 .       5

       0 .

       5   -       0 .

       7

       0 .

       7   -       0 .

       9

       0 .

       9   -       1 .

       1

       1 .

       1   -       1 .

       3

       1 .

       3   -       1 .

       5

      >       1 .       5

   C  u  m  u   l  a   t   i  v  e   f  r  e  q  u  e  n  c  y

   F  r  e  q  u  e  n  c  y

Opaque Envelope Average U-value [W m-2 K-1]

0   1

24

8

17

8

1

0

0.2

0.4

0.6

0.8

1

0

10

20

30

40

50

      <       1 .

       5

       1 .

       5   -       2 .

       0

       2 .

       0   -       2 .

       5

       2 .

       5   -       3

       3   -       3 .

       5

       3 .

       5   -       4

      >       4

   C  u  m  u   l  a   t   i  v  e   f  r  e

  q  u  e  n  c  y

   F  r  e  q  u  e  n  c

  y

Windows average U-value [W m -2 K-1]

(a)

(b)

Fig. 5. Frequencyof buildings by thermal transmittance of opaque envelope(a) and

of windows (b).

opaque envelope and the average transmittance of the windows is

plotted.

 2.3. Occupancy schedule

In the schools dataset, the overall number of hours per year

in which the indoor temperature was maintained at the setpoint

is available for years from 2008 to 2013. This parameter changes

accordingto thetype of schooland to the extra-curricular activities

that take place in the same school building outside the teach-

ing timetable. Actually, the occupancy schedule is an information

more significant than the total amount of occupancy hours: thus,

using these schedules it is possible to calculate for each school thetotal occupancy hours during the corresponding heating period

(October 15th – April 15th). Consequently, the temperature dif-

ference between the indoor and outdoor temperature, which the

heating energy consumption strictly depends on, can be estimated

at each hour during the occupancy period. In the available dataset,

occupancy schedules of all the schools are available only for the

year 2011–2012. Since heating has to be provided during occu-

pancy hours in order to maintain the indoor air temperature at

the setpoint, it has been possible to calculate the specific Heat-

ing Degree Hours (HDH 20, occ ) for the scholastic year 2011–2012.

Meteorological data for the same period coming from 10 monitor-

ing stations administered by the regional environmental agency

(ARPA Veneto) in different locations of the province have been

used.

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164 R. Arambula Lara et al./ Energyand Buildings 95 (2015) 160–171

Fig. 6. Heating degreehours (HDH 20, occ ) foreach schoolduring heating period of thescholastic year 2011–2012.

As shown in Eq. (1), HDH 20, occ  are calculated as the sum of all

the hourly differences between the external air temperature and

a supposed internal setpoint temperature of 20◦C during every

occupancy hour of the heating period.

HDH 20,occ   = ˙ (T int  − T ext ) · h (1)

In Fig. 6 the results of this calculation are plotted, showing that therange of values present in this group of buildings is quite ample, as

it goes from around 8500 to 25 500K h.

 2.4. Heating systems and energy consumptions

As regards the heating system, most of the schools use nat-

ural gas boilers (87 %) while the others use heating oil fueled

boilers. Before 2012, all the boilers were traditional noncondens-

ing ones. During 2012, the traditional boiler has been replaced in

some schools with a condensing one. As for the value of the boiler

heating capacity: 16 % have boilers below 300 kW, 40 % between

300 kW and 600kW, 27 % between 600kW and 900 kW, and 17 %

more than 900kW. In order to compare the energy performance of 

schools, the energyconsumptionof each of them has been normal-ized with respect to its heated volume and heating degree hours.

Fig. 7 shows the trend of the energy index, which is very variable:

from 0.53 Wh m−3 K−1 h−1 to 8.41Wh m−3 K−1 h−1.

3. Method

As specified in the introduction, the first step regards the selec-

tion ofthe quantities todescribethesample andthen toperformthe

clustering. The annual energyconsumption of each building canbe

correlated to some of them, such as those describing the geometry,

the composition of the envelope, the characteristics of the condi-

tioning system, the control schedules and the weather conditions.

Theinfluence of each quantity on the heating demand is clearly dif-

ferent and the highest correlated parameters and variables can beused to characterize effectively the sample of buildings. Since the

aimof the work is to group schools with similar characteristics and

correlations between them and the energy consumption, we need

to define which variables are the most suitable to characterize the

heating demand and which ones can describe the properties of the

buildings’ set. As showed in previous paragraph, in order to make

the energy consumption for heating(Q ) easierto compare and, con-

sequently, the school easier to split into homogeneous groups, it

has been normalized coherently with the Italian National Guide-

lines for the Energy Labellingof Buildings [23] and EN 15217:2007.

Thus, Q have been divided by value of the conditioned volume (V )

and the heating degree hours (HDH 20, occ ) calculated according to

theapproach explainedbefore.In thisway, weather,size of building

and occupancy have been removed from the list of the descriptive

quantities and their effects directly accounted in the normalized

energy performance (EP ), expressed in [Wh m−3 K−1 h−1]. The list

of 12 candidate descriptive quantities includes:

•  The areaof thevertical walls exposed to theexternalenvironment

 Avw [m2];

•  The area of the roof  Ar  [m2];•  The area of the floor A f  [m

2];•   The area of thefloor inthermal contact with the ground A f–g  [m

2];•  The total area of the opaque envelope Aenv, o [m

2];•  T he total area of the transparent envelope Aenv, gl [m

2];•   The ratio between the windows area and the vertical walls area

 Awin/ Avw [−];•  The ratio between the windows area and the total floor area

 Awin/ A f  [−];•  The ratio betweenthe transparent envelopeand theopaque enve-

lope Aenv,  gl/ Aenv, op [−];•  The average thermal transmittance of the envelope U 

[W m−2 K−1];• The shape factor of the school, expressed in terms of ratio S /V 

[m−1

] between the dissipating surface S and the conditioned vol-ume V ;•  The capacity of the heating systemH [kW].

In orderto helpto findthe combinationsof thecandidate quanti-

tiesto define homogenous groups, we decidedto adopt themultiple

linear regression, since it offers the best advantages to perform

the following clustering [21]. Indeed, the selected quantities can

be employed both to identify the groups and to develop linear

predictive models for their elements. Other statistical techniques

could be implemented, such as multivariate ANOVA or the anal-

ysis of the correlations with Spearman’s index, depending on the

research objectives. In this case, considering the small size of the

sample and the aim of detecting linear relationships, only the prin-

cipal effects of the correlations between the candidate quantitiesand the EP have been investigated. The study of the interactions

and non-linear relationships will be considered in further devel-

opments with larger datasets, including more years of measured

data. Due to the chosen approach, EP can be treated as a response

quantity, which is, at most, a linear function of the 12 independent 

 predictors. Foreach oneof the12 descriptive quantities, the highest

value in the whole dataset is identified and used to normalize the

characteristics of each building. The predictors can be grouped in

4083 possible combinations starting from groups with 2 to groups

with12 predictors. The correspondinglinear models are elaborated

starting from the smallest groups. The adjusted index of determi-

nation (R2adj

) of each model is monitored because it is one of the

most meaningful indexes to consider in case of non-hierarchical

clustering [21]. The statistical significance of the model itself has

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R. Arambula Lara et al./ Energy and Buildings 95 (2015) 160–171 165

Fig. 7. Annual energy consumption per unit heated volume and per unit degreehour (scholastic year 2011–2012).

been controlled by means of the value of F -tests and the  p-values

and the multi-collinearity issues by means of thevariance inflation

 factor  (VIF ). Only models with significant  p-value with respect to

a significance level of 5 % or, at least, 10 % and, preferably, with

VIF values lower than 10 (i.e., without multi-collinearity issues) are

considered for the definition of the quantities for the clustering.

The calculation of the different models stops when, even includinglarger and larger groups,R2

adj  cannot be significantly improved. The

combinations of predictors with the highest R2adj

 are selected as set

of coordinates to define the“position” of each element in thesample

of schools.

The next two steps involve the clustering and its validation.

Among the different alternatives for the identification of clusters,

the K-means approach is one of the most popular techniques in

clustering and data mining. The technique is based on a simple

partitional algorithm that tries to find K non-overlapping clusters

[24,25]. By this method, K centroids are selected according to the

desired numberof clusters anddata pointsare assigned to theclos-

est centroid according to the squared Euclidean distances. Once the

clusters are defined, it is possible to validate them by checking if 

the combination of predictors with the highest R2adj with respect to

the whole dataset is the best for the cluster as well. If it is not, the

combination of predictors with the highest R2adj

 is found and used

as new coordinate system. If the cluster has an improved but still

poor R2adj

, if enough buildings are present (i.e., with more than 25

elements), itis possible torun thealgorithmagainin order to define

sub-clusters, but using the set of parameters with the highest R2adj

for the cluster to split.

Since the whole dataset for the clustering includes 58 elements,

we decided to imposeK I = 3 forthe first clustering andK II =2incase

of sub-clustering. A preliminary study has indicated that, for the

current dataset, usingK I = 3 in the first clustering is more effective

thanK I = 2. Onthe contrary, if K I = 4, some clusters aretoo small and,

so, the centroids have few representativeness. Furthermore, many

clusters obtained with K I =4 from this sample have the numberof buildings inadequate for the development of predictive linear

models according to the central limit theorem and so cannot be

validated. Clustering into 3 groups results to be a good compromise

between the sub-clustering levels and the adequacy of the cluster

size.

The K-means method is sensitive to the initial centroids C 0, k:

with the aim of increasing the robustness of the approach, an iter-

ative procedure has to be implemented, as well as an optimized

choice of thestarting conditions. As it is commonlydone inK-means

approaches, the initial virtual centroids are randomly generated

within the domain of the dataset. However, some constraints are

imposed in order to prevent cases in which C 0, k   are too far or too

close to each other andto partiallyavoid including very distant data

points in the same cluster. Specifically, for each initial centroid aninfluence regionis definedas percentage of the total size of thedata

cloud (30 % in this case). For sub-clustering, instead, since K II = 2

and the number of elements is lower, a different approach which

maximizes the distance between C 0, 1   and C 0, 2   is preferred. Con-

sequently, specific combinations of the values of the predictors are

used for the centroid initialization: the virtual centroidC 0, 1 hasthe

maximum value and C 0, 2   the minimum. After the creation of the

initial clusters, the centroids C 1, k  are calculated and the K-means

 Table 1

Results of the924 combinationswith 6 predictors: top-10 configurations selected for thefirst clustering and parameters considered foreach group.

ID 235 825 787 311 902 270 843 53 304 307

Predictors Avw   x x x x x x

 Ar    x

 A f  A f-g    x x x x x x x x

 Aenv, op   x x

 Aenv, gl   x x x x

 Awin/ Avw   x x x x x

 Awin/ A f    x

 Aenv, gl/ Aenv, op   x x x

U  x x x x x x x x x x

S/V  x x x x x x x x x x

H  x x x x x x x x x x

R2adj

  0.275 0.265 0.261 0.259 0.257 0.255 0.254 0.254 0.254 0.253

F value 4.59 4.43 4.35 4.32 4.28 4.24 4.23 4.23 4.23 4.22

 p-value

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166 R. Arambula Lara et al./ Energyand Buildings 95 (2015) 160–171

 Table 2

Results of thefirst clustering of the58 schools and thesubsequent secondregression in each cluster. In gray color theimproved regressions.

ID 235 825 787 311 902 270 843 53 304 307

C 1   R 2adj   0.399 0.400 0.542 0.802 0.355 0.585 0.461 0.543 0.695 0.486

F value   3.88 3.44 3.36 9.76 2.92 4.76 4.56 3.18 4.79 4.30

 p-value 0.01 0.02 0.08 10.In gray

color the improved regressions.

Cluster C1   C2   C3   C3.1   C3.2

ID Cluster Best model Cluster Best model Cluster Best model Cluster Best model Cluster Best model

304 162 304 629 304 403 403 396 403 457

Predictors

 Avw   x x x x x x x x x

 Ar    x x

 A f    x x x x x

 A f-g    x x x

 Aenv,op   x x x x x

 Aenv,gl   x

 Awin Avw   x x x x x x x x

 Awin A f    x x x x x x

 Aenv,gl Aenv,op   x x x x x

U  x x x x

S/V  x x x x x x x

H  x x x x x

R 2adj   0.695 0.988 0.565 0.676 0.058 0.486 0.891 0.989 0.162 0.369

F statistic  4.79 140.59 4.68 6.91 1.29 5.41 13.21 130.29 1.58 2.75

 p-value 0.08

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R. Arambula Lara et al./ Energy and Buildings 95 (2015) 160–171 167

IDs with the exception of ID-53 gave improvements considering

the first cluster, especially ID-311 and ID-304. Looking at the sec-

ond cluster, only ID-825, ID-902 and ID-304 gave a better fit of the

linear models to the data. Eventually, no ID led to improvements

for the last cluster: in many cases small groups were generated,

preventing the calculation of the regressions. As a whole, ID-304

brought the best results, since, for more than one cluster gen-

erated with its predictors, it allowed us to develop better linear

models.

In configuration ID-304, three main clusters are found:C 1 with

11 buildings, C 2   with 18 and C 3   with 29 (Table 2). With respect

to the value of the adjusted index of determination for model ID-

304 on the whole dataset, the indexes of C 1   (R2adj

  = 0.69) and C 2

(R2adj

  = 0.56) are increased while R2adj

 of  C 3   is still very low. These

results mean that the clustering gave an improvement for the clas-

sification and modeling of half of the buildings’ set, now split into

the homogeneous groups C 1 and C 2. The remaininghalf of the sam-

ple in C 3  needed some further investigation: since its number of 

elements was high enough, it has been determined to apply a sub-

clustering. For the sub-clusteringof C 3, the group of predictors with

the best R2adj

 is ID-403 and, thus, it has been used in the K-means

method. The groups resulting from this operation are clusters C 3.1

(with 10 schools)andC 3.2 (including 19 buildings). The linear modelID-403 has been assessed on these two sub-clusters but only the

adjusted index of determination of C 3.1   was improved. Since the

size of C 3.2 was not large enough, no additional sub-clustering has

been performed.

For each of the final clusters C 1, C 2, C 3.1 and C 3.2, we looked for

linear models with adjusted index of determination higher than

that of the linear models used for the selection of the predictors

to use in the clustering. When another combination of predictors

optimized R2adj

, it was selected to model that cluster. Four differ-

ent configurations were, in this way, selected: ID-162 is found to

be the best combination for C 1, ID-629 for C 2, ID-396 for C 3.1  and

ID-457 for C 3.2. The linear model ID-457 has still a low R2adj

 of 0.37.

That could indicate either a residual non-homogeneous group or

the impossibility of fitting well the data with a linear model orboth. Since C 3.2 is too small to be split into sub-clusters and to have

sufficient data to develop regressive models, a possible develop-

ment is to look for non-linear models to fit C 3.2  elements. For all

clusters, R2adj

 and  p-value have been improved. In Table 3 it is pos-

sible to see that the models developed for C 1   and C 2  are slightly

or not affected by multi-collinearity issues while the sub-clustersC 3.1   and C 3.2   are largely affected: this means that the errors on

the estimation of the regression coefficients for some predictors

can be high. Consequently, the models cannot be used for data-

extrapolation because the uncertainty on the relationship between

the normalized energy performance and some predictors can be

too large. Thus, these linear regressions can give robust results

only with the aim to study and model the buildings in the clus-

ters or groups of buildings withsimilar characteristics, without anypurpose of extensive generalization. In some cases in the litera-

ture, the Principal Component Regression (PCR) was used instead

of multivariate regression to overcome multi-collinearity effects

and it could be a possible further development to improve the

models. In [26], for example, in order to balance the efficiency and

accuracy of benchmarkingmodeling, a selectiveresidual-clustering

benchmarking method is proposed for building envelope energy

efficiency evaluation with multi or high dimensional data set. The

authors firstly calculated the variance inflation factor to determine

thedegree of multi-collinearity among explanatory variables. Then,

they applied either simple multivariate regression analysis for the

combinations of variables without multi-collinearity or principal

component regression to a sample of 480 buildings. Through PCR,

it has been noticed that main uncorrelated components can be

Fig. 8. Actual vs.Estimated energy consumption of schools in clusterC 1, C 2, C 3, C 3.1and C 3.2 . The dottedlines indicate a deviation of 20 %.

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168 R. Arambula Lara et al./ Energyand Buildings 95 (2015) 160–171

 Table 4

Building closest to the centroid of each cluster.

Predictors Units

CLUSTER 1 CLUSTER 2 CLUSTER 3.1 CLUSTER 3.2 CLUSTER 3CN 042-01 VB 049-01 CV 091-01 MB 083-02 CV 046-01

 Avw   m2 5111.66 2080.31 1773.71 2217.54 2178.25

 Ar    m2 5503.00 1720.60 1350.00 1612.00 1866.00

 A f-g    m2 5503.00 1762.00 1341.00 1612.00 1861.00

 Aenv, op   m2 10 328.23 2027.16 1881.26 3829.54 3983.55

 Aenv, gl   m2 1842.26 738.31 508.23 493.28 576.34

U  W m−2 K−1 0.82 0.95 1.07 1.27 1.09

S /V  – 0.33 0.38 0.38 0.41 0.42

 Awin/ Avw   – 0.36 0.35 0.29 0.22 0.26

 Awin/ A f    – 0.18 0.17 0.16 0.16 0.15

 A f    m2 10 185.00 4474.00 3205.00 3104.00 3739.00

H  kW 1420.00 378.00 644.00 822.00 283.00

 Aenv, gl/ Aenv, op   – 0.18 0.36 0.27 0.13 0.14

EP  Wh m−3 K−1 h−1 1.74 0.62 1.77 2.48 1.51

identified and more efficiently adopted to elaborate a regression

model.

As regards the comparison of the linear regression models to

the measured data, for each cluster, the normalized values of the

outputs (i.e., the normalized energy consumption) have been rep-

resented (Fig. 8). As it can be seen, the model fits very well the

measured data forC 1  and, consideringC 2, most of points are within

the error band of 20 %. Looking at the thirdgraph, it can be observed

that sub-clustering allowed to optimize the models fit to data, con-

firming what already observed about the indexes of determination.

In order to visualize all the parameters describing a particular

building at once, parallel coordinates plots were used. As seen in

Fig. 9, through this particular kind of representation it is possible

to visualizemultivariate data from each building using a number of parallel axes corresponding to the number of parameters. In each

parallel axis, the normalized value for one parameter is showed,

by means of a line crossing the axis at that particular level. This

way the comparison of the buildings characteristics can be done

referring to the spread of the variability of each parameter. In the

following analysis, the terms high, medium and low are relative to

the sample considered.

In the case of cluster C 1, buildings with roughly the biggest

external wall areas are included, as well as those with mediumval-

ues (0.4–0.6) for the S /V ratio. Even if this cluster seems the most

disperse with respect to parameters such as the area of opaque

envelope  Avw   (with values between 0.2 and 1) and the average

thermal transmittanceU (going from 0.3to 1),similarities between

buildings included in C 1  are found. For example, the S /V ratio andthe roof area  Ar , which have a smaller dispersion, range values

between 0.4 and 0.6 for the former and from 0.2 to 1 for the latter.

Energy consumption levels for all the buildings inside this group

are between 0.1 and 0.4, thus including some of the buildings with

lower energy demand per volume in the dataset.

With regard to buildings inside C 2, these are characterized by

small roof and ground floor area, with most of the schools within a

range from 0.1 to 0.2, finding medium to high values (0.3–0.7) for

the thermal transmittance and high values (between 0.4 and 1) for

the shape factor S /V . Nevertheless, a small dispersion is foundin the

majority of the parameters defining this group, exceptionmade for

the above-mentioned S /V and  Awin/ A f , both with values spanning

from medium to high, and Aenv,  gl/ Aenv, o, for which values from low

to medium (0.2–0.7) are found.

The buildings of the sample with the highest average thermal

transmittance of the envelope are found in cluster C 3  and, conse-

quently, someof theschools withthe highest levels of consumption

per volume belong to this last cluster. Nevertheless, almost all

of these schools have medium external wall areas (0.2–0.6) and

small roof and ground floor areas, with values ranging from

0.1 to 0.3. With respect to sub-clusters C 3.1   and C 3.2, the main

differences relate to Aenv,  gl/ Aenv, o, as well as to the reported energy

consumption levels. In the case of C 3.1, these quantities show an

ample range of values, going from 0.4 and 1 for the former and

from 0.2 to 1 for the latter. On the contrary, in C 3.2  the dispersion

is smaller and lower values for both parameters are found, being

 Aenv, gl/ Aenv, o from 0.2 to 0.4 and the energy demand in the range

between 0.1 and 0.3, that is the lowest for whole dataset. Despitegeneral similarities in C 3.1 and C 3.2  quantities, the two sub-clusters

have different energy consumption levels: buildings with medium

to high energy consumption are grouped in C 3.1, while schools

with low energy demand are included in C 3.2.

4.2. Reference buildings for each cluster 

Once the centroid coordinates for each cluster and sub-cluster

were defined, square Euclidean distances from every building to

theircorresponding centroid were calculated in order to determine

which building is the closest to the centroid inside each particular

group (Table 4). Being these, the school buildings with characteris-

tics that are more similar to the average values withintheir cluster,

these buildings are considered to be adequate reference buildingsfor the group they belong to.

With regard to cluster C 1, a distance of 0.1821 was measured

from the centroid to the school identified with the code CN 042-

01. This building belongs to a technical institute located in the

town of Conegliano. It has a total floor area of 10 185m2, being

the largest among reference buildings. Its average transmittance

(U =0.82Wm−2 K−1) is the lowest and, even if this is a relatively

compact building, it has the largest external wall surface. In con-

trast, school VB 049-01 is the second smallest reference building

and has also the lowest energy consumption per conditioned vol-

ume. It is situated in Valdobbiadene and is the closest to cluster C 2centroid with a total distance of 0.0924. The distance from cluster

C 3   centroid to the nearest school is 0.1016 for building CV 046-

01, which is located in Castelfranco Veneto. From the centroid of 

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R. Arambula Lara et al./ Energy and Buildings 95 (2015) 160–171 169

Fig. 9. Diagrams in parallel axis representing the coordinates (normalized parameters) of the schools belonging to each cluster and the centroid coordinates of clusters C 1,

C 2, C 3, C 3.1  and C 3.2. The gray dots in each graph x-axes indicates the parametersused for the optimized regression.

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170 R. Arambula Lara et al./ Energyand Buildings 95 (2015) 160–171

C 3.1, the smallest distance to a school is 0.1581, for building CV

091-01 (in Castelfranco Veneto) whereas school MB 083-02 (sit-

uated in Montebelluna) is the closest to C 3.2  centroid with a total

distance of 0.0707. With regard to their particular characteristics,

the latter is quite similar in geometry to C 3   reference building

whereas parameters with lower values are generally found in the

former.

5. Conclusion

In this work, we discussed the problem of classifying and mod-

eling the existing building stock and identifying a limited number

of representative buildings in order to develop strategies for an

extensive refurbishment according to the cost-optimal approach

suggested in the Commission Delegated Regulation [12]. We stud-

ied a sample of almost 60 schools in the province of Treviso, Italy,

and adopted a modified K-means approach as methodology for

their classification. Multiple Linear Regression techniques have been

used to drive and validate the cluster analysis, demonstrating some

potential to determine the classification and leading the clustering

strategy.

Themain aspects of the proposed method have been the follow-

ing ones:

(1) Starting from a multi-dimensional domaincomposed by all the

descriptive quantities of theschoolssample andcorrelated with

their normalized energy consumption, it has been possible to

reduce the variables from 12 to the main 6 ones.

(2) Once the best group of quantities has been defined, the cluster-

ing has been performed and the results validated by studying

the variation of the adjusted index of determination (R2adj

) and

other statistics, such as  p-values, F -values and variance infla-

tion factors, considered for diagnostic purposes. The method

has been iterated on more levels, until clustering was no more

meaningful and verifiable.

(3) The data in the clusters have been studied and described

with optimized regressions. For some clusters, such as C 1, C 2and C 3.1, a good data fitting has been achieved by the devel-

oped linear models: the adjusted index of determination is

almost 0.7 for C 2   and more than 0.9 for C 1   and C 3.1. How-

ever, some diagnostics revealed statistical multi-collinearity

issues, in particular for the sub-clusters and for the optimized

regressions. This underlined the impossibility of robust use of 

the found models for extrapolation and the necessity to fur-

ther investigate the data with alternative approaches, such

as the principal components regressions or some non-linear

methods. Otherwise, the dataset should be populated with

more points (not necessary with more buildings but with more

annual energy consumption data) to allow for more robust

findings.

(4) Even if there are some limitations in the extent of the mod-els outside the dataset, homogeneous groups and models have

been properly defined and, subsequently also their centroids.

The schools closest to those centroids have been determined.

Their floor area rangesfromaround 3100 m2 (C 3.2) tomore than

10 000m2 (C 1), with an installed heating capacity from less

than 400 kW (C 2) to more than1400kW (C 1), an average ther-

mal transmittance of the envelope from 0.82W m−2 K−1 (C 1)

to1.27 Wm−2 K−1 (C 3.2), a S /V ratio from 0.33 (C 1) to 0.41 (C 3.2)

anda ratio betweentransparentand opaque envelopefrom 0.13

(C 3.2) to 0.36 (C 2).

After the identification of the representative schools, it is now

possible to classify the schools according to a priority inter-

vention list, to apply the cost-optimal approach and to find

out the most convenient retrofit solutions. By means of the

regressive models, the achievable energy and economic sav-

ings can be estimated for the whole clusters. Moreover, some

comparison can be conducted starting on the data collected

during the next years of the survey, with the purpose of inves-

tigating the fault detection capabilities of the approach or of 

assessing the effects of some already implemented energy saving

measures.

 Acknowledgment

Theauthors would like to thank the Province of Treviso (Provin-

cia di Treviso) for making the schools database available for this

research.

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