Multi-view Clustering Analyses for District Heating Substations

Shahrooz Abghari

1 a

, Veselka Boeva

, Jens Brage

and H

akan Grahn

1 b

Department of Computer Science, Blekinge Institute of Technology, Karlskrona, Sweden

NODA Intelligent Systems AB, Karlshamn, Sweden

Keywords:

Data Mining, Multi-view Clustering, Multi-layer Clustering, Time Series, District Heating Substation.

Abstract:

In this study, we propose a multi-view clustering approach for mining and analysing multi-view network

datasets. The proposed approach is applied and evaluated on a real-world scenario for monitoring and

analysing district heating (DH) network conditions and identifying substations with sub-optimal behaviour.

Initially, geographical locations of the substations are used to build an approximate graph representation of

the DH network. Two different analyses can further be applied in this context: step-wise and parallel-wise

multi-view clustering. The step-wise analysis is meant to sequentially consider and analyse substations with

respect to a few different views. At each step, a new clustering solution is built on top of the one generated

by the previously considered view, which organizes the substations in a hierarchical structure that can be used

for multi-view comparisons. The parallel-wise analysis on the other hand, provides the opportunity to analyse

substations with regards to two different views in parallel. Such analysis is aimed to represent and identify

the relationships between substations by organizing them in a bipartite graph and analysing the substations’

distribution with respect to each view. The proposed data analysis and visualization approach arms domain

experts with means for analysing DH network performance. In addition, it will facilitate the identiﬁcation

of substations with deviating operational behaviour based on comparative analysis with their closely located

neighbours.

1 INTRODUCTION

District heating (DH) systems utilize hot water and

heat produced at a production unit for a number of

consumer units, i.e., buildings, in a limited geograph-

ical area through a distribution network. This part of

the system is referred to as the primary side. The con-

sumer unit itself consists of a heat exchanger, a circu-

lation network, and radiators for the rooms, which are

considered as the secondary side. The primary and

secondary sides are connected together through a sub-

station, which is responsible for adjusting the pressure

and the temperature of the supply water suitable for

the consumer unit.

In the DH domain, energy companies need to ad-

dress several conﬂicting goals such as satisfying con-

sumers’ heat demand including domestic hot water

(DHW) while minimizing production and distribu-

tion costs. Such complexity demands fault detection

and root cause analysis techniques for identiﬁcation

of deviating behaviours and faults. Undetected faults

https://orcid.org/0000-0002-3010-8798

https://orcid.org/0000-0001-9947-1088

can lead to underlying problems, which in return can

increase the maintenance cost and reduce the con-

sumers’ satisfaction. When it comes to monitoring

of a DH network there are different features and char-

acteristics that one needs to consider. Domain experts

often analyse substations individually or in a group

with regard to one speciﬁc feature or a combination

of features. While this provides useful information

for the whole network it does not take into account

the location of the substations along the distribution

network and their neighbouring substations automat-

ically. In other words, the operational behaviours of

the DH substations need to be assessed jointly with

surrounding substations within a limited geographi-

cal distance. Due to the nature of the data and the

fact that different data representations can be used, the

process of monitoring and identifying faults and de-

viating behaviours of the DH system and substations

can be treated as a multi-view data analysis problem.

Multi-view datasets consist of multiple data repre-

sentations or views, where each one may contain sev-

eral features (Deepak and Anna, 2019). Multi-view

learning is a semi-supervised approach with the goal

158

Abghari, S., Boeva, V., Brage, J. and Grahn, H.

Multi-view Clustering Analyses for District Heating Substations.

DOI: 10.5220/0009780001580168

In Proceedings of the 9th International Conference on Data Science, Technology and Applications (DATA 2020), pages 158-168

ISBN: 978-989-758-440-4

to obtain better performance by applying the relation-

ship between different views rather than one to facil-

itate the difﬁculty of a learning problem (Blum and

Mitchell, 1998; Ando and Zhang, 2007; Xu et al.,

2013). Due to availability of inexpensive unlabeled

data in many application domains, multi-view unsu-

pervised learning and speciﬁcally multi-view cluster-

ing (MVC) attract great attention (Deepak and Anna,

2019). The goal of multi-view clustering is to ﬁnd

groups of similar objects based on multiple data rep-

resentations.

We propose a multi-view clustering analysis ap-

proach for mining network datasets with multiple rep-

resentations. The proposed approach is used for mon-

itoring a DH network and identifying DH substa-

tions with sub-optimal operational behavior. We ini-

tially use geographical location of substations to di-

vide them into groups of similar substations based on

their distance and location. In that way, we are able

to: 1) group the substations (network nodes) based

on their location and distance, 2) build an approx-

imate graph representation of the DH network, and

3) order the substations using information about the

DH network structure and the average supply water

temperature for a speciﬁc period. Two different types

of analyses can then be applied in this scenario: i)

step-wise clustering to sequentially consider and anal-

yse substations with respect to a few different views;

ii) parallel-wise clustering to analyse substations with

regards to two different views in parallel.

2 RELATED WORK

MVC clustering algorithms have been proposed based

on different frameworks and approaches such as k-

means variants (Bickel and Scheffer, 2004; Cai et al.,

2013; Jiang et al., 2016), matrix factorization (Liu

et al., 2013; Zong et al., 2017), spectral methods (Ku-

mar and Daum

e, 2011; Wang et al., 2013) and

exemplar-based approaches (Meng et al., 2015; Wang

et al., 2015).

Bickel and Scheffers (Bickel and Scheffer, 2004)

proposed extensions to different partitioning and ag-

glomerative MVC algorithm. That study can proba-

bly be recognized as one of the earliest works where

an extension of k-means algorithm for two-view doc-

ument clustering is proposed. In another study (Cai

et al., 2013), the authors developed a large-scale

MVC algorithm based on k-means with a strategy for

weighting views. The proposed method is based on

the `

2,1

norm, where the `

norm is enforced on data

points to reduce the effect of outlier data and the `

norm is applied on the features. In a recent study,

Jiang et al. (Jiang et al., 2016) proposed an extension

of k-means with a strategy for weighting both views

and features. Each feature within each view is given

bi-level weights to express its importance both at the

feature level and the view level.

Liu et al. (Liu et al., 2013) proposed an MVC

algorithm based on joint non-negative matrix factor-

ization (NMF). The developed algorithm incorporates

separate matrix factorizations to achieve similar coef-

ﬁcient matrices and further meaningful and compara-

ble clustering solution across all views. In a recent

study, Zong et al. (Zong et al., 2017) proposed an ex-

tension of NMF for MVC that is based on manifold

regularization. The proposed framework maintains

the locally geometrical structure of multi-view data

by including consensus manifold and consensus coef-

ﬁcient matrix with multi-manifold regularization.

Kumar and Daum

e (Kumar and Daum

e, 2011)

proposed an MVC algorithm for two-view data by

combining co-training and spectral clustering. The

approach is based on learning the clustering in one

view to label the data and modify the similarity ma-

trix of the other view. The modiﬁcation of the sim-

ilarity matrices are performed using discriminative

eigenvectors. Wang et al. (Wang et al., 2013) pro-

posed a variant of spectral MVC method for situations

where there are disagreements between data views us-

ing Pareto optimization as a means of relaxation of the

agreement assumption.

Meng et al. (Meng et al., 2015) proposed an MVC

algorithm based on afﬁnity propagation (AP) for sci-

entiﬁc journal clustering where the similarity matrices

of the two views (text view and citations view) are in-

tegrated as a weighted average similarity matrix. In

another study, Wang et al. (Wang et al., 2015) pro-

posed a variant of AP where an MVC model consists

of two components for measuring 1) the within-view

clustering quality and 2) the explicit clustering con-

sistency across different views.

Fault detection and diagnosis (FDD) is an active

ﬁeld of research and has been studied in different

application domains. Isermann (Isermann, 1997; Is-

ermann, 2006) provided a general review for FDD.

Katipamula and Brambley (Katipamula and Bramb-

ley, 2005a; Katipamula and Brambley, 2005b) con-

ducted an extensive review in two parts on fault de-

tection and diagnosis for building systems. Xue et

al. (Xue et al., 2017) applied clustering analysis and

association rule mining to detect faults in substations.

Sandin et al. (Sandin et al., 2013) used probabilistic

methods and heuristics for automated detection and

ranking of faults in large-scale district energy sys-

tems. Calikus et al. (Calikus et al., 2019) proposed an

approach for automatically 1) discovering heat load

Multi-view Clustering Analyses for District Heating Substations

159

patterns in DH systems and 2) identifying buildings

with abnormal heat proﬁles and unsuitable control

strategies.

In contrast to the above mentioned methods, this

study proposes a multi-view data analysis approach

that can be applied for monitoring, evaluating and

visualizing the operational behaviour of DH substa-

tions. The geographical location data is used as a

backbone of the analysis and the operational perfor-

mance of the substations is further assessed in con-

junction with their neighbours.

3 PROBLEM FORMALIZATION

We have a network with N nodes, e.g., a DH network

linking a set of substations located in some geograph-

ical region. Assume that each network node, substa-

tion, i (i = 1,2,... , N) is monitored under n differ-

ent conditions (i.e., the measurements of n different

features are collected) for a given time period, e.g.,

m days. Each monitored condition j ( j = 1,2,...,n)

contains the measured levels of the corresponding fea-

ture for a period of m days in t different time points.

This leads to a set of n time series data matrices D

( j = 1,2, . . . , n), one per feature, for each network

node.

This multi-view data context can additionally be

complicated in the case of a real-world scenario such

as one related to a DH network. For example, the

operational behaviour of the substations varies dur-

ing heating and non-heating seasons which requires

separate analysis. Therefore, for each substation two

datasets are usually collected and available for further

analysis and comparison. Notice that in this study,

we are only interested in the operational behaviour of

the DH substations during heating season due to the

importance of space heating.

The main challenge in the above multi-view con-

text is how to use all available measurements about

the substations’ operational behaviour and perfor-

mance for better understanding and improved main-

tenance of the DH network. Exploiting the whole po-

tential of these real-world datasets is not trivial and it

requires suitable data analysis techniques to prevent

information loss.

4 METHODS

4.1 Clustering Analysis

In this study, we are interested in identifying homoge-

neous groups of substations by considering their loca-

tions and additionally analysing them with respect to

different views (features). Due to unavailability of the

labeled data, clustering analysis is applied to explore

hidden structures within the data. We apply two clus-

tering algorithms as follows:

1. Minimum Spanning Tree Clustering: We use Van-

derPlas’ (VanderPlas, 2016) Python implantation of

the minimum spanning tree (MST) clustering algo-

rithm for grouping substations based on their geo-

graphical location. The algorithm is based on con-

structing an approximate Euclidean minimum span-

ning tree (EMST), which considers only k nearest

neighbours of each point for building the minimum

spanning tree rather than the entire set of edges in a

complete graph.

2. Afﬁnity Propagation: We use the afﬁnity prop-

agation (AP) algorithm (Frey and Dueck, 2007) for

clustering the time series based on their similarities.

AP works based on the concept of message pass-

ing between data points to ﬁrst identify a suitable set

of exemplars and then to choose which data points

should pick which exemplars. One of the advantages

of AP, unlike other clustering algorithms, such as k-

means (MacQueen et al., 1967) which requires the

number of clusters as an input, is that it estimates the

optimal number of clusters from the data. In addition,

the chosen exemplars, the representative of the clus-

ters, are real data points which makes AP a suitable

clustering algorithm for this study.

4.2 Similarity Measures

We use different similarity measures, 1) to check the

similarity between daily time series proﬁles of each

feature, 2) to perform pairwise comparison between

exemplars of clustering solutions of different substa-

tions, and 3) to compute a similarity between two

clustering solutions by considering all pairs of mem-

bers. These similarity measures are as follows:

1. Dynamic Time Warping: Given two time series

Y = (y

,..., y

) and Y

= (y

,..., y

), the simi-

larity between Y and Y

can be measured using the dy-

namic time warping (DTW) algorithm. DTW is pro-

posed by Sakoe and Chiba (Sakoe and Chiba, 1978)

for spoken word detection with the focus of eliminat-

ing timing differences between two speech patterns.

In other words, DTW identiﬁes an optimal match-

ing between the given sequences by warping the time

axis. In order to align the time series Y and Y

length n and m respectively, a cost matrix, Q

n×m

computed. Each element, q

i j

, of Q

n×m

corresponds

to the distance (often Euclidean) between y

and y

the two series. Using the cost matrix, the DTW tries to

ﬁnd the best alignment path between these two time

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

160

series that is leading to minimum overall cost. The

best warping path should satisfy a different number of

conditions such as monotonicity, continuity, bound-

ary, warping window, and slope constraint.

2. Clustering Solution Similarities: Given two

clustering solutions C = {C

,. . . ,C

} and C

,. . . ,C

} of datasets X and X

, respectively,

the similarity, C

, between C and C

can be assessed

as follows (Abghari et al., 2019):

(C,C

) =

∑

i=1

(min

j=1

.d(c

))

∑

j=1

(min

i=1

.d(c

))

(1)

where c

and c

are exemplars of the clustering solu-

tion C

and C

, respectively. The weights w

and w

indicate the relative importance of clusters C

and C

compared to other clusters in the clustering solution

C and C

, respectively. For example, a weight w

a cluster C

can be calculated as the ratio of its cardi-

nality with respect to the size of X, i.e., w

= |C

|/|X|.

The C

has values in a range of [0,1]. Scores equal to

zero imply identical performance while scores close

to one show signiﬁcant dissimilarities.

3. Adjusted Rand Index: The quality of the results of

a clustering analysis can be validated by means of in-

ternal and external criteria. Internal criteria evaluate

the quality of the clustering solution produced by a

clustering algorithm that ﬁts the data in terms of, e.g.,

compactness and separation by using the inherent in-

formation of the data. External criteria on the other

hand, can be used for measuring the level of agree-

ments between the results of a clustering algorithm in

comparison with ground truth, the results of another

clustering algorithm on the same data, or same clus-

tering algorithm but by considering different views.

In this study, we apply a symmetric external val-

idation index for assessing the similarity (consen-

sus) between two clustering results generated on the

studied DH substations with respect to two differ-

ent views. The adjusted Rand index (ARI) (Hubert

and Arabie, 1985) is a correction of the Rand in-

dex (RI) (Rand, 1971) that measures the similarity

between two clustering solutions by considering the

level of agreements between the two groups. ARI is

computed as follows:

ARI =

RI − ExpectedRI

Max(RI) − ExpectedRI

(2)

ARI scores are bound between -1 and +1. A score less

than or equal to 0 represents random labelling and 1

stands for perfect match.

5 PROPOSED APPROACH

Geographical locations of N substations are initially

used for building an approximate graph representa-

tion of the DH network. We refer to the geograph-

ical location of the substations as the Location view

). This is performed by applying the MST cluster-

ing algorithm described in Section 4. The aim is to

connect substations based on their distance by build-

ing a minimum spanning tree and removing edges of

the tree with regard to a cut-off threshold. Therefore,

each cluster is represented by a tree that can be in-

terpreted as a representation of the DH network struc-

ture. In order to provide additional support for the do-

main experts, the graph representation can in turn be

used as a backbone for additional information about

the DH network, e.g., average yearly values and dif-

ferent forms of ranking.

On the foundation of the created grouping of the

substations we can perform further analysis by focus-

ing on a speciﬁc feature or subset of features and eval-

uate the substations’ operational behaviours in each

single location-based cluster. We study and evaluate

the following two scenarios:

1. Step-wise multi-view clustering (SW-MVC), we

can apply clustering analysis on substations that have

been grouped together at the previous step with re-

spect to a set of features, i.e., the substations can be

grouped by considering one feature at a time. This

scenario can be used when the domain experts are in-

terested in grouping similar substations based on their

performance with respect to one feature and then ﬁnd-

ing similar substations in each group by using another

feature and so on. Figure 1 shows how the results of

this analysis can be visualized based on the location

of the substations and two features.

…

: Location

: Feature 1

: Feature 2

Figure 1: SW-MVC analysis, each view represents the clus-

tering analysis based on one feature. Every analysing step is

based on the results obtained on the previously considered

view. Triangles represent substations.

As an example, consider two substations s

and s

in the cluster C

from v

, where the similarity of the

two substations can be analysed in terms of their op-

Multi-view Clustering Analyses for District Heating Substations

161

erational behaviour, ﬁrst based on Feature 1 (v

) and

then Feature 2 (v

). Here two scenarios can occur, ei-

ther s

and s

are grouped together with respect to v

since they performed similarly, or they are assigned

into different groups. In case of the ﬁrst scenario, af-

ter applying the second step of the analysis (i.e., using

) if s

and s

are in the same group this shows that

the operational behaviour of the two substations are

similar with respect to v

and v

. Otherwise, the two

substations are only similar with regards to v

and dis-

similar with regards to v

. In case of the second sce-

nario, the substations are dissimilar with respect to

both views. Nevertheless, in all cases the domain ex-

pert might be interested to further analyse groups of

substations with a smaller size.

2. Parallel-wise multi-view clustering (PW-MVC), in

this scenario a group of substations can be studied by

considering different features in parallel. For exam-

ple, the substations can be clustered separately with

respect to two different features (or subsets of fea-

tures). The produced clustering solution can further

be compared and analysed to ﬁnd out whether similar

substations, that have been grouped together based on

one feature, are still in the same group with respect to

the other feature. In addition, one can use a bipartite

graph, to present and visualize the relationships be-

tween a clustering solution based on one view and a

clustering solution produced on the other view. This

will provide domain experts more information by sup-

plying them with deeper insights about substations’

operational behaviours in different groups with re-

gards to two different views. For further analysis, one

can label clusters of each clustering solution with per-

formance indicators

and rank them from the highest

to the lowest performance.

The results of this pairwise comparison can be

used in conjunction with the SW-MVC analysis to

provide a better understanding of operational be-

haviour for each individual substation and the group

as a whole. Our initial assumption is that using the

SW-MVC analysis, one can construct a hierarchical

graph-model of a heating network for the area of

study. Those substations that are located in the same

cluster are assumed to share similar characteristics.

While the PW-MVC analysis focus is on identifying

a similar group of substations that are in the intersec-

tion of the two views.

The operational performance of a DH substation can be

evaluated with respect to different indicators, which are usu-

ally computed based on the quantitative relation between

the substation’s inputs and outputs.

6 EXPERIMENTS AND

EVALUATION

6.1 Dataset

The data used in this study is provided by an energy

company. The data consists of hourly average mea-

surements from 70 substations located in Southern

Sweden during 2015 to 2018. The dataset contains

eight features both from primary and secondary sides

of the DH network. The primary side data is always

available. The secondary side data on the other hand,

requires speciﬁc hardware to be extracted. Therefore,

in this study we mainly focus on primary side data to

analyse the operational behaviour of the DH substa-

tions.

Apart from these features there are two perfor-

mance indicators that are computed using both sides

of the DH network. The ﬁrst indicator is called the

least temperature difference (Frederiksen and Werner,

2013) which represents the difference between pri-

mary and secondary return temperatures. The least

temperature difference of a substation can be greater

than or equal to zero, though it can go below zero

due to usage of DHW. A lower value of this indicator

implies better performance. The second indicator is

referred to as substation effectiveness. It is the ratio

of the difference between primary supply and return

temperatures to the difference between primary sup-

ply temperature and the secondary return temperature.

The efﬁciency of a well-performing substation should

be close to one in a normal setting. However, due to

the affect of DHW generation on the primary return

temperature, it can represent values above one. Ta-

ble 1 shows the dataset features and the performance

indicators.

Table 1: Features included in the dataset.

No. Feature Notation Unit

1 T

Outdoor temperature °C

2 T

s,1

Primary supply temperature °C

3 T

r,1

Primary return temperature °C

4 ∆T

Primary delta temperature °C

5 G

Primary mass ﬂow rate l/h

6 Q

Primary heat kW

7 T

s,2

Secondary supply temperature °C

8 T

r,2

Secondary return temperature °C

Figure 2 shows the groups and graph network rep-

resentation produced by applying MST clustering on

the above mentioned 70 substations. The substations

are partitioned into nine clusters by applying the MST

clustering algorithm while the cut-off parameter is set

to 500 meters. That is substations with distance less

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

162

Figure 2: 70 substations located in Southern Sweden are grouped into nine clusters using the MST clustering algorithm. The

geographical location of the substations is referred to as the Location view (v

). Substations with distance less than 500 meters

from their closest neighbours are grouped together. The color of the substations represents the average T

s,1

in January 2018,

which for most substations is around 87 °C.

than 500 meters from their closest neighbour(s) are

grouped together. Five clusters represent as a tree,

i.e., edges of the tree represent the distance between

the substations (the tree nodes) and the remaining four

clusters are singletons. The substations’ colors repre-

sent their received average T

s,1

(°C) in January 2018.

In order to make the data ready for the experi-

ment, ﬁrst the duplicates are removed. Then we fo-

cused on extreme values which can appear as a result

of faults in measurement tools. We apply a Hampel

ﬁlter (Hampel, 1971) which is a median absolute de-

viation (MAD) based estimation to detect and smooth

out such extreme values. The ﬁlter is used with the

default parameters, i.e., the size of the window is set

to be seven and the threshold for extreme value detec-

tion is set to be three.

In the studied context, we have hourly measure-

ments data. This gives one time series every 24-hours

and in total 365 time series per year. Time series

with less than 24 measurement values are excluded.

Since we are expecting different behaviours from a

DH substation during heating and non-heating sea-

sons, the time series are divided into two groups with

respect to the outdoor temperature (T

). That is, if

the outdoor temperature is above a certain threshold,

threshold

, the DH substation behaviour can be cate-

gorized into the non-heating season otherwise to the

heating season. This threshold in Sweden can be set to

be T

threshold

= 10 °C. In order to assess each DH sub-

station’s operational behaviours during heating sea-

son and in comparison with other substations, the ex-

tracted time series are scaled with z-score normaliza-

tion. That is, each time series is scaled to have a mean

of zero and a standard deviation of one. Notice that in

the considered context the general shape of the time

series, rather than their amplitude, is important. Now

for every category, the time series related to one spe-

ciﬁc feature, i, can be compared in terms of similarity

with respect to a distance measure d(y

), where d

in this study is DTW . This leads to a similarity ma-

trix, SM

. In the next step, SM

is fed to a clustering

algorithm. Here we aim to group time series based

on their similarities into a number of clusters. Con-

sidering each feature as one view, we can analyse the

operational behaviour of a set of DH substations by

using the explained evaluation scenarios in Section 5.

6.2 Implementation and Availability

The proposed approach is implemented in Python ver-

sion 3.6. The afﬁnity propagation and the adjusted

Rand index are adopted from the scikit-learn mod-

ule (Pedregosa et al., 2011) and the MST clustering

algorithm is fetched from (VanderPlas, 2016). The

Multi-view Clustering Analyses for District Heating Substations

163

alignments between time series are identiﬁed using

dtwalign’s package

. The implemented code and the

experimental results are available at GitHub

7 RESULTS AND DISCUSSION

The initial view in our analyses is always the outcome

of the MST clustering, i.e., 70 substations in the stud-

ied area are grouped into nine clusters based on their

distances. We set the cut-off parameter to be 500 me-

ters which means any edges greater than 500 meters

are removed from the MST. The ﬁrst three clusters

(0, 1, and 2) include 15, 32, and 14 substations, re-

spectively (in total 61 of 70 substations). The remain-

ing substations are grouped into 6 clusters as follows:

cluster 3 contains 2 substations, clusters 4, 5, 7, and 8

are singletons and cluster 6 has 3 substations.

In the remainder of this study we only consider

and discuss the results produced on clusters 0, 1, and

2, since the majority of the substations are distributed

in these clusters. Each analysis can be performed

based on different combinations of the features in Ta-

ble 1. However, due to the page limit, we only report

the results of the analyses with respect to T

r,1

and

∆T

(the difference between T

s,1

and T

r,1

7.1 SW-MVC Analysis

Table 2 shows the results of SW-MVC analysis for

61 substations throughout the heating seasons from

2015 to 2018. For each MST cluster (v

), initially

substations are grouped based on T

r,1

) and then

for each created subgroup the clustering analysis is

performed using ∆T

). The information in Ta-

ble 2 can be used in three different ways: column-

wise, row-wise, or both. In the column-wise case, one

can see how the substations in each MST cluster are

grouped based on the other two views (i.e., v

and v

)

in different years. The row-wise analysis shows how

the substations in each MST cluster are grouped step-

wise based on ﬁrst v

and second v

. For example,

the domain experts might be interested in perform-

ing further analysis when the grouped substations in

are split into more subgroups based on v

. Num-

bers in bold in Table 2 represent the number of substa-

tions that are grouped into different clusters based on

as opposed to v

. By considering both cases one

can track the transition of the operational behaviour

of substations throughout the years.

https://github.com/statefb/dtwalign

https://github.com/shahrooz-abghari/MVC-DH-

Monitoring

Table 2: SW-MVC analysis based on T

r,1

and ∆T

from

2015 to 2018.

Year

: MST v

: T

r,1

: ∆T

Label #substations Label 0 1 2 Total

2015

5 0 5 5

10 1 1 9 10

1 32 0 11 12 9 32

5 0 3 2 5

9 1 6 3 9

2016

3 0 3 3

5 1 4 1 5

7 2 7 7

15 0 9 6 15

13 1 7 6 13

4 2 4 4

9 0 9 9

5 1 5 5

2017

8 0 8 8

1 1 1 1

5 2 5 5

1 3 1 1

11 0 8 3 11

4 1 1 3 4

16 2 8 8 16

1 3 1 1

2 0 2 2

9 1 9 9

3 2 3 3

2018

1 0 1 1

14 1 10 4 14

12 0 12 12

5 1 5 5

8 2 5 3 8

7 3 6 1 7

4 0 1 3 4

4 1 4 4

1 2 1 1

5 3 5 5

Note. Number of substations that are grouped into

different clusters based on v

as opposed to v

are

shown in bold.

Figure 3 depicts the SW-MVC analysis by consid-

ering T

r,1

as the ﬁrst view (squares) and ∆T

as the

second view (circles) in the period from 2015 to 2018.

Figure 4 represents the SW-MVC analysis speciﬁ-

cally for the MST cluster with label 0 during a period

covering 2017 and 2018. Notice, the color of squares

shows T

r,1

while the colored circles represent ∆T

These two features can be used as an assessment indi-

cator for the operational behaviour of the substations.

Technically, it is desired in a well performed substa-

tion that the T

r,1

has a lower value in comparison to

the T

s,1

. In other words, a greater delta means that

the substation is making more efﬁcient use of the sup-

plied heat for space heating.

Table 3 provides the statistics, the average values

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164

Figure 3: The results of SW-MVC analysis for the whole studied area contains 70 substations. Squares represent clusters of

substations based on the ﬁrst view, T

r,1

, and circles represent groups of substations with respect to the second view, ∆T

Note that the substations with similar colors in different MST clusters are not related.

of the actual measurements and their standard devia-

tions, regarding the DH substations that are discussed

in Figure 4. As one can see in 2017, substations

are grouped into 4 clusters, where clusters 0 (green

squares) and 2 (orange squares) contains the major-

ity of substations, 8 and 5, respectively. The two

other clusters include only one substation each (red

squares). The average T

r,1

for cluster 0 is approxi-

mately 43 °C and for cluster 2 is around 46 °C. Clus-

ters 1 and 3 both show the average T

r,1

of 48 °C. All

the grouped substations in the previous step stayed to-

gether based on the ∆T

, i.e., no new cluster is cre-

ated. The cluster with 8 substations (grey circles in-

side green squares) represents the ∆T

of 34.79 °C,

while the cluster with 5 substations (yellow circles in-

side orange squares) shows the ∆T

of 33.26 °C. The

other two clusters (yellow circles inside red squares)

show the same value, 34.25 °C for the ∆T

In 2018, the same number of substations, 15, are

grouped into only two clusters with an average T

r,1

of approximately 40 °C for cluster 0 (purple square)

and 44 °C for cluster 1 (orange squares). A majority

of the substations, 13 out of 14, are grouped in clus-

ter 1. This cluster is further divided into two clusters,

one with 10 DH substations (grey circles inside or-

ange squares) and the average ∆T

of approximately

34 °C and the other with 4 substations (orange cir-

cles inside orange squares) and the average ∆T

Figure 4: The results of SW-MVC analysis for the MST

cluster with label 0 and 15 substations in 2017 (top) and

2018 (bottom). Colored squares represent groups of substa-

tions based on v

: T

r,1

and colored circles represent groups

of substations based on v

: ∆T

Multi-view Clustering Analyses for District Heating Substations

165

Table 3: SW-MVC analysis for the MST cluster with label 0, 2017 to 2018.

Year

: MST v

: T

r,1

: ∆T

Label Label #substations Avg(°C) SD(°C) Label #substations Avg(°C) SD(°C)

2017 0

0 8 43.31 5.39 0 8 34.78 4.13

1 1 47.88 - 0 1 34.25 -

2 5 46.23 3.35 0 5 33.26 3.01

3 1 47.61 - 0 1 34.25 -

2018 0

0 1 39.91 - 0 1 31.01 -

1 14 44.31 4.66

0 10 33.71 4.11

1 4 36.63 2.10

Note. Avg: average, SD: standard deviation

approximately 37 °C. Cluster 0 represents one sub-

station (yellow circle inside purple square) with the

average ∆T

of 31.01 °C. In both years there are sub-

stations that show slightly different operational be-

haviour in comparison to their neighbouring substa-

tions, e.g., the red substations in 2017 and the purple

substation in 2018. The domain experts can investi-

gate the reasons why these substations performed dif-

ferently in comparison to the majority of substations.

In addition, it is important to mention that the order

in which views are used for the SW-MVC analysis af-

fects the results, which can be decided based on the

domain expert’s preferences.

7.2 PW-MVC Analysis

The aim of this analysis is to group the substations

based on two different views (i.e., v

and v

) and com-

pare the results of the clustering solution to ﬁnd out

which substations are similar based on both views.

Such analysis can provide useful information for the

domain experts while it applies for a period of time,

e.g., different years, where the transition of the opera-

tional behaviour of substations can be monitored. Ta-

ble 4 shows the distribution of the studied substations

based on PW-MVC analysis throughout the heating

season in the period from 2015 to 2018.

Figure 5 depicts the computed ARI scores for the

clustering solution based on T

r,1

and ∆T

of each

MST cluster in the period from 2015 to 2018. As

one can see, the ARI scores of the ﬁrst three clus-

ters are absolutely dissimilar in 2015, 2016, and 2018.

However, cluster 2 with 14 substations shows the ARI

score of 0.71 in 2017, which means the majority of

DH substations in this cluster performed similarly.

Other clusters, 3 to 8 represent the adjusted Rand in-

dex of 1 for all the years.

Figure 6 shows the PW-MVC analysis for MST

cluster with label 1 with respect to T

r,1

and ∆T

the period from 2015 to 2018. The substations with

similar cluster labels with regard to both features are

shown in red. The insight provided by Figure 6 can

Table 4: PW-MVC analysis is performed based on T

r,1

and

∆T

separately from 2015 to 2018.

Year

: MST v

: T

r,1

: ∆T

Label #substation 0 1 2 3 0 1 2

2015

0 15 5 10 5 10

1 32 32 11 12 9

2 14 5 9 6 5 3

Total 70 51 19 31 27 12

2016

0 15 3 5 7 6 9

1 32 15 13 4 32

2 14 9 5 10 4

Total 70 36 23 11 57 13

2017

0 15 8 1 5 1 15

1 32 11 4 16 1 10 22

2 14 2 9 3 10 4

Total 70 30 14 24 2 44 26

2018

0 15 1 14 9 6

1 32 12 5 8 7 26 6

2 14 4 4 1 5 4 10

Total 70 26 23 9 12 48 22

Figure 5: PW-MVC analysis, the computed ARI scores for

r,1

and ∆T

throughout 2015 to 2018.

be used for analysing the difference between oper-

ational behaviour of the red substations against the

greys within one speciﬁc year. In addition, the transi-

tion of the substations from one color group to another

can be tracked and further analysed.

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Figure 6: The results of PW-MVC analysis for the MST cluster with label 1 and 32 substations. Substations in red are those

that are grouped with the same label according to the both views, namely T

r,1

and ∆T

from 2015 to 2018.

8 CONCLUSIONS

We have proposed a multi-view clustering approach

for analysing datasets that consist of different data

representations. The proposed approach has been ap-

plied for monitoring and analysing operational be-

haviour of district heating substations. We have ini-

tially used the substations’ geographical information

to build an approximate graph representation of the

DH network. This graph structure has been used as a

backbone for further analysis of the network perfor-

mance.

In the above context, we have proposed and dis-

cussed two different types of analysis: 1) step-wise

multi-view clustering that sequentially considers and

analyses the operational behaviour of the DH substa-

tions with respect to different views and organizes the

substations into a hierarchical structure. That is, at

each step a new clustering solution is built on top of

the one generated in the previous step with respect to

the considered view. 2) parallel-wise multi-view clus-

tering that analyses substations with regards to two

different views in side by side. This enables the iden-

tiﬁcation of the relationships between neighbouring

substations by organizing them in a bipartite graph

and analysing their distribution with respect to the

two considered views. The proposed data analysis ap-

proach facilitates the visual analysis and inspections

of multi-view real-world datasets such as ones related

to the DH networks. For example, the proposed ap-

proach provides the opportunity to consider the DH

substations in close relation with their neighbours.

That is, those substations that demonstrate a deviat-

ing behavior from their neighbouring substations can

easily be identiﬁed for further investigation.

For future work, we are interested in expanding

our approach by adding a third scenario where the

clustering solution is the outcome of integration of

different views. We believe that the proposed ap-

proach provides a verity of analysis techniques to sup-

ply the domain experts with a complete picture about

the DH network operations. In addition, the proposed

approach can facilitate the identiﬁcation of substa-

tions with deviating behaviours and suggest initiation

of further inspections by domain experts.

Multi-view Clustering Analyses for District Heating Substations

167

ACKNOWLEDGEMENTS

This work is part of the research project “Scal-

able resource-efﬁcient systems for big data analyt-

ics“ funded by the Knowledge Foundation (grant:

20140032) in Sweden.

We would also like to thank Christian Johansson,

CEO of NODA Intelligent Systems, for his support

and valuable feedback.

REFERENCES

Abghari, S., Boeva, V., Brage, J., Johansson, C., Grahn,

H., and Lavesson, N. (2019). Higher order mining for

monitoring district heating substations. In 2019 IEEE

Int’l Conf. on Data Science and Advanced Analytics

(DSAA), pages 382–391. IEEE.

Ando, R. K. and Zhang, T. (2007). Two-view feature gener-

ation model for semi-supervised learning. In Proc. of

the 24th Int’l Conf. on Machine learning, pages 25–

32. ACM.

Bickel, S. and Scheffer, T. (2004). Multi-view clustering.

In ICDM, volume 4, pages 19–26.

Blum, A. and Mitchell, T. (1998). Combining labeled

and unlabeled data with co-training. In Proc. of the

eleventh annual Conf. on Computational learning the-

ory, pages 92–100.

Cai, X., Nie, F., and Huang, H. (2013). Multi-view k-

means clustering on big data. In Twenty-Third Int’l

Joint Conf. on artiﬁcial intelligence.

Calikus, E., Nowaczyk, S., Sant’Anna, A., Gadd, H., and

Werner, S. (2019). A data-driven approach for dis-

covery of heat load patterns in district heating. arXiv

preprint arXiv:1901.04863.

Deepak, P. and Anna, J.-L. (2019). Multi-View Clustering,

pages 27–53. Springer Int’l Publishing, Cham.

Frederiksen, S. and Werner, S. (2013). District Heating and

Cooling. Studentlitteratur AB.

Frey, B. J. and Dueck, D. (2007). Clustering by

passing messages between data points. Science,

315(5814):972–976.

Hampel, F. R. (1971). A general qualitative deﬁnition of

robustness. The Annals of Mathematical Statistics,

pages 1887–1896.

Hubert, L. and Arabie, P. (1985). Comparing partitions. J.

of classiﬁcation, 2(1):193–218.

Isermann, R. (1997). Supervision, fault-detection and fault-

diagnosis methods—an introduction. Control engi-

neering practice, 5(5):639–652.

Isermann, R. (2006). Fault-diagnosis systems: an introduc-

tion from fault detection to fault tolerance. Springer

Science & Business Media.

Jiang, B., Qiu, F., and Wang, L. (2016). Multi-view cluster-

ing via simultaneous weighting on views and features.

Applied Soft Computing, 47:304–315.

Katipamula, S. and Brambley, M. R. (2005a). Methods

for fault detection, diagnostics, and prognostics for

building systems-A review, part I. Hvac&R Research,

11(1):3–25.

Katipamula, S. and Brambley, M. R. (2005b). Methods for

fault detection, diagnostics, and prognostics for build-

ing systems-A review, part II. Hvac&R Research,

11(2):169–187.

Kumar, A. and Daum

e, H. (2011). A co-training approach

for multi-view spectral clustering. In Proc. of the 28th

Int’l Conf. on machine learning (ICML-11), pages

393–400.

Liu, J., Wang, C., Gao, J., and Han, J. (2013). Multi-view

clustering via joint nonnegative matrix factorization.

In Proc. of the 2013 SIAM Int’l Conf. on Data Mining,

pages 252–260. SIAM.

MacQueen, J. et al. (1967). Some methods for classiﬁcation

and analysis of multivariate observations. In Proc. of

the Fifth Berkeley Symp. on Mathematical Statistics

and Probability, volume 1, pages 281–297. Oakland,

CA, USA.

Meng, X., Liu, X., Tong, Y., Gl

anzel, W., and Tan, S.

(2015). Multi-view clustering with exemplars for sci-

entiﬁc mapping. Scientometrics, 105(3):1527–1552.

Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V.,

Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P.,

Weiss, R., Dubourg, V., Vanderplas, J., Passos, A.,

Cournapeau, D., Brucher, M., Perrot, M., and Duch-

esnay, E. (2011). Scikit-learn: Machine learning in

Python. J. of Machine Learning Research, 12:2825–

2830.

Rand, W. M. (1971). Objective criteria for the evaluation

of clustering methods. J. of the American Statistical

association, 66(336):846–850.

Sakoe, H. and Chiba, S. (1978). Dynamic programming

algorithm optimization for spoken word recognition.

IEEE Transactions on Acoustics, Speech, and Signal

Processing, 26(1):43–49.

Sandin, F., Gustafsson, J., and Delsing, J. (2013). Fault de-

tection with hourly district energy data: Probabilistic

methods and heuristics for automated detection and

ranking of anomalies. Svensk Fj

arrv

arme.

VanderPlas, J. (2016). mst clustering: Clustering via eu-

clidean minimum spanning trees. J. Open Source Soft-

ware, 1(1):12.

Wang, C.-D., Lai, J.-H., and Philip, S. Y. (2015). Multi-

view clustering based on belief propagation. IEEE

Transactions on Knowledge and Data Engineering,

28(4):1007–1021.

Wang, X., Qian, B., Ye, J., and Davidson, I. (2013). Multi-

objective multi-view spectral clustering via pareto op-

timization. In Proc. of the 2013 SIAM Int’l Conf. on

Data Mining, pages 234–242. SIAM.

Xu, C., Tao, D., and Xu, C. (2013). A survey on multi-view

learning. arXiv preprint arXiv:1304.5634.

Xue, P., Zhou, Z., Fang, X., Chen, X., Liu, L., Liu, Y., and

Liu, J. (2017). Fault detection and operation optimiza-

tion in district heating substations based on data min-

ing techniques. Applied Energy, 205:926–940.

Zong, L., Zhang, X., Zhao, L., Yu, H., and Zhao, Q. (2017).

Multi-view clustering via multi-manifold regularized

non-negative matrix factorization. Neural Networks,

88:74–89.

DATA 2020 - 9th International Conference on Data Science, Technology and Applications

168