Background

This paper is an extension of work originally presented in proceedings of Frontiers of information technology (FIT’15) Conference 2015 (Habib and Zucker 2015). The energy systems of a typical contemporary building are usually complex and may contain several subsystems deployed independently of each other. In order to analyze various energy performance aspects of a given building, a lot of raw data is recorded during its monitoring Khan et al. (2011). The recorded data is studied at later stages in order to find interesting features, using a variety of visualization tools (Mourad and Bertrand-Krajewski 2002). The massive amount of recorded data makes any detailed performance analysis a formidable task. Moreover, there is a high chance of overlooking some important patterns in the data which, if noticed properly, may help identify faults that can compromise energy efficiency.

Patterns are regular, usually repetitive, sequences in a given data and may owe their existence to a specific event. A pattern is thus dependent on the characteristics of a system and may represent the underlying processes and structure of the system. Methods that can automatically identify interesting patterns from buildings’ data, help to get useful insights into the various parameters of energy usage as well as the source of faults in different components. In this context, data mining techniques like clustering are feasible tools to address these issues. The process of automatically finding the various patterns in the data can make the subsequent analysis easier, more feasible and lesser laborious (Miller et al. 2015; Iglesias and Kastner 2013; Narayanaswamy et al. 2014; Lin and Li 2009).

We aim to exploit machine learning for finding various patterns in energy related building data . The idea is to realize all this with minimum possible configuration changes and knowledge of the relevant field. More specifically, in order to automatically find different patterns in the adsorption chiller’s operation, in this article, we propose to use a bag of words representation (BoWR) with subsequent hierarchical clustering. The suggested method has been applied to the operation data of a water chiller and compared to another approach called dynamic time warping (DTW) using cophenetic correlation. The dynamic time warping (DTW) method uses a dynamic programming technique for defining the best alignment between the two time series data Keogh and Ratanamahatana (2004). Furthermore, the cophenetic correlation demonstrates that the cluster tree has a strong correlation with the distances between objects in the distance vector Lin and Li (2009). The On/Off state information required for the suggested technique is detected by using the k-means clustering algorithm. As we are taking the sensor readings that are placed outside the chiller, the sensors reading will reflect the behavior of the chiller during its operational cycle. Thus, the On (operational) states are of greater importance for assessing the performance of chillers and faults detection and diagnosis (FDD). Moreover, avoiding the Off cycle for finding different patterns will reduce the amount of data as well. The On (operational) cycles are discretized by using the symbolic aggregate approximation (SAX) method. These discretized values are called symbols or words. After transformation of the On cycles to words, a normalized histogram for each On cycle is created; called bag of words representation (BoWR). The normalized BoWR is used because the On (operational) cycle’s vary in there duration. The hierarchical clustering uses the normalized BoWR of the On cycles for finding the various operational patterns of the chiller. The details of the different clusters created by hierarchical clustering are also explained in detail.

The rest of the paper is arranged as follows. The next section discusses the state of the art methods available in the literature. In the subsequent section, the design of the demonstrated system is elaborated. This is followed by a section describing the methodology of the proposed solution for finding the patterns in the data. The penultimate section explains the different experiments and results, followed by a "Conclusion" section.

State of the art

This section discusses state of the art methods, from literature, proposed for finding operation patterns in different energy systems, in the context of buildings. The energy systems can be modeled using simulation tools.

Complex systems (CA)

“Complex systems consist of many interacting components and many hierarchical layers and, in general, it is impossible to reduce the overall behavior of the system to a set of properties characterizing the individual components. The interaction between components is able to produce properties at the collective level that are simply not present at the component considered individually” Avram and Rizescu (2014); Moffat (2010). The focus of complex systems is to study the system modules (subsystems), their interaction with each other, and how each module is contributing to the overall behavior of the system. Examples of complex systems are:

  • people creating social systems,

  • our nervous system, with brain spinal cord and neurons being the subsystems, and

  • a weather forecast system with factors like wind flow, pressure and temperature contributing in predictions.

  • cities can be considered as system and different aspect such as social physics, urban economics, transportation theory, regional science, and urban geography can be considered as subsystem (agents) for designing the cities Batty (2007).

The knowledge of complex systems can be used in all traditional disciplines of science, along with engineering and management. The main focus in the complex systems is on questions regarding parts (independent), overall behavior and relationships. The information provided with the complex systems, using sophisticated tools, is helpful to think analytically about these systems in detail and contributes in the modeling or simulation analysis of these systems.

Complex adaptive systems (CAS)

A Complex adaptive system (CAS) can be defined as an open system “with large variability and diversity of elements or agents, with dynamic interactions among them that create non-linear feedback systems” (Faucher 2010). Such systems are usually linked to the learning activities, in order to provide various features of CAS, like self-organization and unpredictability. They are also described as “special cases of complex systems, which can be called as ’complex macroscopic collection’ of relatively similar micro-structures that are partially connected. These macro-structures are formed to adapt the changes in the environment, and increase its survivability” (Kayman 2014).

The subsystems of a complex system are generally modeled as agents. The agents are usually goal-oriented, variable in number and, the condition of the environment can be affected by other agents. The three basic properties of the CAS are (Andrii 2014):

  1. 1.

    Adaptation: This characteristic of CAS is relevant to the adaptability of the system to changes in the environment.

  2. 2.

    Self-organizing: This characteristic is dependent on the structure of the system as well as its internal processes; the underlying question being how the larger dynamic system organizes itself in critical situations.

  3. 3.

    Emergence: This characteristic defines the qualitative change in the behavior of the system during a change in its observation scale. It is one of the common characteristic of CAS where the behavior of the system is more complex than the sum of the behaviors of the components of the system. The emergent property is lost when the system is decomposed into its component parts or when an elimination of some component occurs.

In order to study the complex systems, one has to take into account all; be it the components, their interaction or the overall behavior of the system. Still the emergent behavior cannot be discounted. One of the common methods used is to ignore some system details that mean to find a higher abstraction level of the system. In multidimensional scenarios, the space can be reduced by using mapping (generating a few equivalence classes). There are different factors usually ignored by CAS designers that can constrain the system and influence their long term performance, called as energy. The authors in Hadzikadic (2010) discusses the changes and influence due to the variation in available energy. Furthermore, adding the concepts of efficiency and resilience in complex adaptive systems can be beneficial in modeling (Korhonen and Snäkin 2015).

Buildings as CAS

Complex systems scale from large systems like ecosystem (Levin 1998; Grimm et al. 2005) or social ecological systems (Olsson et al. 2004) to smaller systems such as secure authentication systems (Habib et al. 2011) or buildings Oosterhuis (2012) and their energy system (Azar and Menassa 2010, 2011; Jensen et al. 2016). Limited area notwithstanding, the analysis of a building’s energy system is a complex task as it consists of several subsystems. In order to make a detailed analysis of the energy systems, the buildings are monitored using sensors. Nowadays, it is feasible to maintain a record of the historic operation data in the building. While there exist other domains that have considerably higher amounts of data, the operation data in buildings are specifically challenging, since there is commonly no appropriate underlying data model that can be generally applied to operation data; data is very specific to one building or component. Thus, data analytics methods have to supply a high degree of unsupervised automation in order to treat different types of data. Thus, today the main approach for data analysis is a simple visualization of the process parameters using time graphs as visualization tools. Such visualizations may further require manual followup performance analysis. Methods of analysis like these can be time intensive and there is always a chance to miss out some areas of interest that may eventually be of greater importance (Mourad and Bertrand-Krajewski 2002).

In order to make the buildings energy efficient, their prototype models are simulated for energy performance. For better designing and the ability to handle the dynamic nature of the building’s characteristics, each component of the building can be modeled as an active part; thus different components of the building will constitute a complex network (Oosterhuis 2012). There are many energy modeling methods that are generally used for predicting the buildings performance during the design phase. The actual energy consumption reading usually deviates from the predicted value during the modeling phase (Azar and Menassa 2010, 2011). Some of the reasons for this deviation are the dynamic parameters like occupant’s behavior, climate, and buildings properties (Azar and Menassa 2011). The agent based modeling can be used to handle such dynamic parameters. For example, the dynamic nature of occupants’ behavior can be correlated with the impact on energy consumption in commercial buildings (Azar and Menassa 2010, 2011) or in managing ventilation system in residential buildings (Jensen et al. 2016). There are several bottom up approaches put forward for the agent based modeling. The authors in Grimm et al. (2005) have proposed a framework using a pattern oriented approach for agent based modeling to handle the complexity and uncertainty problems.

Other methods for analysis of energy systems in buildings

The reasons for analyzing data from the energy subsystems are manifold and include such objectives like assessment of the overall system performance, comparison with other systems, calculation of operating costs, and prediction of energy consumption and faults etc. The International energy agency (IEA) has launched an implementing agreement (i.e., a technology initiative) called “IEA Solar Heating and Cooling Programme (SHC)”. Within this implementing agreement IEA SHC Task 38 “Solar Air-Conditioning and Refrigeration” was one of the research topics. The IEA SHC Task 38 (subtask A3a-B3b: “Monitoring procedure for solar cooling system”) defines a generic monitoring policy that provides information on sensor locations and naming for the evaluation of systems, evaluation of the system performance, and comparison of different energy systems (Napolitano et al. 2011). In the literature, one can find many methods for faults detection and diagnosis (FDD) in building components. One important area is concerned with the Heating, Ventilation and Air-Conditioning (HVAC) (Pietruschka et al. 2015; Isermann 2005; Fan and Qiao 2011; Katipamula and Brambley 2005; Capozzoli et al. 2015; Katipamula and Brambley 2005; Lee and Eun 2015; Narayanaswamy et al. 2014). Prior knowledge about the system can be useful in finding some of the simple undetected faults using first principles (i.e. energy balance, mass balance and other physical principles), but still there is a requirement for more sophisticated techniques to judge various aspects of a building’s energy performance. One known class of techniques that makes use of historic operation data describes the behavior of the system, characterized as black box models, which are fitted using the historical data (Katipamula and Brambley 2005, 2005). Faults can also be detected in buildings with machine learning algorithms using the information from the installed electricity consumption meters as shown in (Figueiredo et al. 2005), Domínguez et al. (2013). There are different parameters available that can be useful for the prediction of electricity consumption for each HVAC component; multivariate analysis can be used to calculate these parameters (Djuric and Novakovic 2012).

In order to detect various patterns in any energy system using data driven techniques, the focus is on extracting information from the recorded data using little to none domain expertise. There are several machine learning techniques that can be used for extracting information from the data, e.g. clustering can be used for finding similar daily performance patterns in the buildings (Miller et al. 2015; Seem 2005), detecting the abnormal performance from electricity consumption Seem (2007), and further enhancing the performance optimization algorithms (Kusiak and Song 2008). Moreover, at a larger scale, wavelet transformations and clustering can be used for the classification of electrical demand profiles of buildings (Florita et al. 2013).

The data is usually stored as a time series for later analysis. The time series data can be represented with different available techniques that can further help in finding the similarity between the data having same behavior. An example is the symbolic aggregate approximation (SAX), a category of Piecewise Aggregate Approximation (PAA), that can be used to improve the speed and usability of several analysis techniques Lin et al. (2007). The similarities between different time series data can even be calculated by simply using the Euclidean distance parameter, but the problem in this method is that even a slighter shift of data can lead to erroneous results (Lin and Li 2009). A comparison of time series data similarity algorithms (Euclidean, DTW, wavelets) is carried out in Lin and Li (2009) par rapport the method of bag of patterns using hierarchical clustering. The authors have concluded that the bag of patterns representation (BoPR) approach performed better for finding similarities in the time series data as compared to other methods. The use of bag of words model can be seen in various fields with classification (Anwar et al. 2015).

One of the well-known methods used for finding similar groups via data mining is clustering (Armano and Javarone 2013; Shah et al. (2015). The decision of the optimal number of clusters is an important issue in unsupervised methods, in general, and in hierarchical clustering, in particular. A clustering algorithm can give better results if the inter-cluster variations are minimum and intra-cluster variations are maximum (Tibshirani et al. 2001). Clustering algorithms can also be used for finding various energy states in the building, e.g., k-means clustering can be used to detect the state (On/Off) of machine, as data toggle between these two states (Habib et al. 2015; Zucker et al. 2015a, b). Another example of using clustering for finding system states can be found in Zucker et al. (2014), where the X-Means clustering algorithm is used for automatically detecting the system states (On/Off), in order to examine the operational data of adsorption.

Cluster evaluation methods

Cluster evaluation is usually carried out using graphical methods. One such way is to plot error measurement against the number of clusters. In this method the position, where the plot creates an “elbow” in the graph, can be taken as the number of clusters, since the “elbow” occurs at the point of sudden decrease in the error measurement (Ketchen and Shook 1996). There are other methods that can be used to find optimal number of clusters in the data, e.g. the Silhouettes criterion method Rousseeuw (1987), Davies-Bouldin’s criterion method Davies and Bouldin (1979) and Calinski-Harabasz criterion method Caliński and Harabasz (1974). Other than these techniques, a method from the literature is based on gap statistics analysis wherein the gap criterion finds the optimal number of clusters by estimating the “elbow” location as the number of clusters against the largest gap value (Tibshirani et al. 2001). The gap value can be defined as (Tibshirani et al. 2001)

$$\begin{aligned} Gap_n (k)=E_n \{log(W_k ) \}- log(W_k ), \end{aligned}$$
(1)

where \(E_n\) is the expected value, n is the size of the sample, k is the number of clusters that are being evaluated, and \(W_k\) is the dispersion measurement within the cluster and can be find as

$$\begin{aligned} W_K= \sum _{r=1}^{k}\dfrac{1}{2_{n_r}}D_r, \end{aligned}$$
(2)

where \(n_r\) represents the count of data points in the cluster r, and \(D_r\) denotes the sum of the pairwise distances for all data points in the cluster r.

Design of the demonstration system

This section discusses the architecture of the system that has been under observation for applying the proposed method. For this research, the data from selected solar adsorption chillers is used for the period of 16 months from January 2014 until April 2015. The monitoring policy with naming convention of IEA SHC Task 38 for solar and cooling had been followed (Napolitano et al. 2011). The design of the system is shown in Fig. 1, showing three different cycles in the system along with the installed sensors.

Fig. 1
figure 1

System architecture with the the required sensors

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The process involves the three main parts which can be summarized as follows:

  • The low temperature (LT)cycle is representing the part of the system that is handling the low temperature water produced by the chiller.

  • The medium temperature (MT) cycle represents the system portion where the unwanted heat of the system is transferred to the environment using cooling tower.

  • The high temperature (HT) cycle is showing the section of the system where heat is provided to produce cold water by the chiller.

The 18 different parameters of interest along with their description are given in Table 1.

Table 1 Parameters description
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In order to find patterns in the operational data (On cycles), different tests were performed in consultation with the experts in the field. There are additional features added for better results. The temperature difference between the return and supply temperature sensors of each of the cycle had been used as a feature that are given as,

$$\begin{aligned} \Delta Temp\_LT=|T\_LTre-T\_LTsu |\end{aligned}$$
(3a)
$$\begin{aligned} \Delta Temp\_HT=|T\_HTre-T\_HTsu |\end{aligned}$$
(3b)
$$\begin{aligned} \Delta Temp\_MT=|T\_MTre-T\_MTsu |\end{aligned}$$
(3c)

Therefore, the new set of features are added with other selected parameters for hierarchical clustering in next step. The following Table 2 shows the different features that have been used for the hierarchical clustering.

Table 2 Selected features for hierarchical clustering
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Methods

This section describes the methodology proposed in this paper. The first step followed in the analysis of data is always the preprocessing and finding outliers. The data used has already been processed; therefore it can be used without the preprocessing step.

The three methods used in this work were selected keeping in view their independence from two main factors, viz. configuration information and domain knowledge. The algorithms used in this research paper do not any require domain knowledge or configuration information, as illustrated in the Table 3.

Table 3 Selected algorithms analysis
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On state (operational) detection using K-means clustering

The distribution of the states for chiller vary a lot in the two states (On/Off), therefore the data can be classified in two clusters. It can be readily observed from Fig. 2 that the mean, minimum and maximum of the two clusters can be observed based on the difference between temperatures, flows, energy readings and pressures. For this purpose, the K-means clustering with two clusters and Euclidean distance setting is used to detect the On and Off state. After the detection of On/Off state, at each point of the time, the consecutive On states are marked as one On cycle. The same procedure is adopted for all consecutive Off states. The sensors are placed at the outer points of the solar cooling system, which means that during the On cycle, the data will be representing the system behavior; otherwise, during the non-operational period it will represent the behavior of the environment. Our interest lies in finding various patterns in the chiller’s data, therefore, only On cycles were considered for clustering.

Fig. 2
figure 2

Parallel coordinates showing the On/Off cycle mean,minimum and maximum patterns

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Symbolic aggregate approximation (SAX) transformation

After the detection of On/Off cycles, the data will be in the form defined by Eq. 4:

$$\begin{aligned} C_i=\big \{S_1,S_2,S_3,.......,S_N \big \}, \end{aligned}$$
(4)

where \(C_i\) is the ith cycle and \(S_t\) is the sensor value at time tick t. The data is normalized using Z-Scores which is given as:

$$\begin{aligned} Z(Data)=\dfrac{S_{t}-\mu }{\sigma }, \end{aligned}$$
(5)

where Z(Data) is the Z-score normalized form of the data, \(S_t\) is the sensor data at tth time tick, \(\mu\) represents the mean and \(\sigma\) is the standard deviation. After applying the Z-Score normalization, the data will be now in the form as defined in Eq. 6 while using the On/Off information from the k-means clustering algorithm,

$$\begin{aligned} Cycle_i= \{Z_1,Z_2,Z_3,......Z_{N_i}\}, \end{aligned}$$
(6)

where \(Cycle_i\) is the ith cycle of the data and in case if odd count of i is representing Off cycle then even count of i will be presenting the On cycle in data. \(Z_t\) is the normalized sensor value while \(N_i\) is the event count of \(Cycle_i\).

Fig. 3
figure 3

Example of SAX transformation of On duty with M = 4 and P = 5

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Each cycle data is first broken down into M non overlapping sub-sequences, in a uniform manner, just like the example illustrated in Fig. 3, wherein the partitions are represented by alphabets a, b, c and d. This process is called as chunking, and the period (x-axis) can be of different time length (P) depending on the application where it is used (Miller et al. 2015). The value of P is taken as 5 min in this research. The symbol of each data point is assigned according the breakpoints. The number of break points (M) taken for this research is 60. This transforms the data for each cycle to symbols. The SAX representation is specific for each a length of each cycle. In order to generalize the symbolic representation for each cycle with different lengths, the BoWR is used.

Bag of words representation (BoWR)

In order to represent the complete behavior of a cycle, with different parameters taken under consideration, each sensor’s data is converted to a BoWR of 60 characters and put together in a 540 character representation, as can be seen in Fig. 4.

Fig. 4
figure 4

540 character long representation of Cycle

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All the required parameters will be converted to z-score before transformation to SAX symbols. The value of M is taken as 60 for this research. \(BoWR_i\) pertains to BoWR of the ith On cycle containing all features (shown in Table 2). A pattern can be defined as:

$${\text{BoWR}}_{{\text{i}}}=\{ {\text{BoWR}}\_{\text{Sensor}}_{1}^{i} ,{\text{BoWR}}\_{\text{Sensor}}_{2}^{i} ,....,{\text{BoWR}}\_{\text{Sensor}}_{P}^{i} \} ,$$
(7)

The \(BoWR\_Sensor^i_{P}\) consists of a vocabulary set \(\{w_1,w_2,w_3,.....,w_M\}\) of sensor P. The associated histogram vector \(BoWR\_Sensor^i_{P}\) for \(i{\rm th}\) On cycle will be like the following:

$${\text{BoWR}}\_{{\text{Sensor}}_{{\text{P}}}^{{\text{i}}}} = \left( {{\text{V}}_{1}^{i} \;{\text{V}}_{2}^{i} \;{\text{V}}_{3}^{i} \ldots {\text{V}}_{M}^{i} } \right),$$
(8)

where P is representing the features (see Table 2) selected for finding out different patterns in the chiller’s data. \(V^i_j\) is the number of occurrences of \(w_j\) in the ith cycle, i.e.

$$\mathrm {V^i_j}= {\text{Count}}_i \rm(w_j),$$
(9)

where the subscript i in \(Count_i\) refers to the ith cycle.

In order to handle cycles of variable time lengths, a better idea is to normalize, i.e. use relative frequencies. With this in view, Eq. 9 can be modified as Eq. 10 below:

$$\text{V}^i_j= \dfrac{{\text{Count}}_i\rm(w_j)}{\text{N}_\text{i}},$$
(10)

where \(N_i\) is representing the number of time ticks in the ith cycle.

Hierarchical clustering

The hierarchical clustering technique groups the data over different scales by creating a cluster tree called dendrogram (Vesanto and Alhoniemi 2000). A dendogram shows a multilevel hierarchy of clusters, where the clusters (groups) at one level are joined together to constitute a cluster for the next level. This property of hierarchical clustering allows to decide the level of clustering that is the most appropriate for the task it is used for. The BoWR for each cycle is clustered using the hierarchical clustering technique. Figure 5 shows the dendogram of the \(BoWR_i\) given as input to the hierarchical clustering.

Fig. 5
figure 5

Dendogram of BoWR

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There are different techniques available to decide the best level or number of clusters for hierarchical clustering. One such technique is the gap method Tibshirani et al. (2001). A clustering algorithm gives better results when the intra-cluster difference is as small as possible while the inter-cluster difference is as high as possible.

Methodology overview

The steps involved, in the proposed method, are illustrated in Fig. 6. Below is a brief stepwise description of the method:

  • The first step is to find the On (operational) cycles in the data by using the k-means algorithm. The latter can be applied to any energy system because the two states are their in any energy dependent system and On duty cycle can be readily detectable.

  • The On cycles data are transformed to symbolic data with the SAX transformation method. This step also does not need any field knowledge and is applicable to almost all energy systems.

  • A BoWR was created for the symbols of each On cycle. This procedure does not need any field knowledge.

  • The BoWR are clustered by using the hierarchical clustering for finding various operation patterns of the chiller. This process does not need any field knowledge.

  • The gap statistics is used to find the optimal number of clusters in the data. This procedure does not need any field knowledge.

  • The cluster patterns can be further investigated using the average performance indicators of each cluster.

Fig. 6
figure 6

Methodology

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Experiments and results

The experiments had been performed on a data from water based chillers. Only On cycles were considered for clustering, as these are more appropriate for finding faults in the chiller. The hierarchical clustering was applied with dynamic time warping (DTW) and the proposed BoWR method. The comparison of the hierarchical clustering performance of the two methods was carried out with the help of cophenetic coefficients (Saraçli et al. 2013). The cophenetic correlation is technique that demonstrates the cluster tree strong correlation with the distances between objects in the distance vector. Table 4 lists the cophenetic coefficients with different hierarchical clustering methods Levin (2007) using the BoWR and DTW techniques. The BoWR has strong correlation with distance with all other objects in all the clustering methods. The best results for BoWR are attained with the Average method for hierarchical clustering.

Table 4 Cophenetic coefficients of dynamic time warping (DTW) and BoWR
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The first step performed for the BoWR, was to find the On cycles automatically. The results of the k-means clustering algorithm can be seen in Fig. 7. The last graph in Fig. 7 shows the On/Off cycles of the chiller. It can be observed from the behavior of the temperatures at the low temperature (LT), medium temperature (MT) and high temperature (HT) cycle are responding according to the detected On/Off state. It is clear from Fig. 7 that during the detected On cycle, the LT temperature decreases showing the cooling operation. At the same time, increase in the temperatures at HT and MT cycle of the chiller can be noticed. These simultaneous variation in temperature gives a clear signal that the chiller is in operational mode, which has also been detected by the proposed method of k-means clustering.

Fig. 7
figure 7

Demonstration of On/Off cycle detection with different temperature sensors using k-means clustering

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The selected features of the detected On cycles were converted to BoWR. The hierarchical clustering makes a clustering tree (dendogram) that gives the option to select the level (cutoff) for clustering. The gap statistics Tibshirani et al. (2001) had been used to find the optimal number of clusters, depending on the gap between different clusters. As it can be observed from Fig. 8, the gap statistics analysis gives the best gap distance with five clusters.

Fig. 8
figure 8

Gap distance of the BoWR data

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The cluster information of the five clusters are given in Table 5. The interesting pattern group is \(Cluster_1\) and \(Cluster_2\), as the average operation time in these clusters is greater than around 1 hour. For finding faults, \(Cluster_1\) patterns are more suitable since the average Coefficient of Performance (CoP) of cycles in this cluster is 0.16, in comparison to the average operational time of cycles that is around 68 hours, thus showing that the chiller’s performance is bad. A majority (\(98.75\,\%\)) of the On cycles lies in \(Cluster_1\). The latter represents the cycles with normal operational behavior, since its average CoP is 0.54 while the average operational time of the cycles in this cluster is around one hour. \(Cluster_3\), \(Cluster_4\) and \(Cluster_5\) are representing the cycles with shorter operational time, as the machine is in transient phase; thus the patterns in these clusters are different from a normal operational behavior of the chiller and are thus not plausible.

Table 5 Cluster information of the five clusters with hierarchical clustering
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For further investigation of the cycle behaviors in \(Cluster_1\), the graph in Fig. 9 had been drawn in order to show the behavior pattern of one of the On cycles in \(cluster_1\). The three graphs in Fig. 9 display the temperature difference (\(\Delta Temp\_LT,\ \Delta Temp\_HT,\ \Delta Temp\_MT\)),

Fig. 9
figure 9

On cycle pattern showing low performance (faulty) operation of chiller

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flows (\(Q7\_m3h,\ Q6a\_m3h,\ Q12\_m3h\)) and energy meter readings (\(Q7\_KW,\ Q6a\_KW,\ Q12\_KW\)) in the low, high and medium temperature cycles, respectively. The x-axis displays the time (in minutes) for the On cycle. In each graph, the values are represented with the intensity of the color given in the form of a vertical color code bar at the right hand side of each plot. It can be observed form Fig. 9 that at 30 min, the \(\Delta Temp\_MT\) becomes zero, showing that the cooling tower is not operating normally, thus causing no change in the temperature of MT cycle. It is also important to note that the flow variable (\(Q12\_m3h\)) for MT cycle is showing flow throughout the cycle. Due to this effect, the \(\Delta Temp\_LT\) has also started decreasing and at around 80 minute, the cooling has been stopped by the chiller. At the same time, the deriving heat (\(\Delta Temp\_HT\)) has been provided to the chiller but the chiller is not able to match the cooling load, thus showing lower coefficient of performance. This pattern of the chiller is giving a clue about the faults in the chiller that need to be diagnosed. Furthermore, to support the argument that \(Cluster_1\) is representing group of On cycles having bad performance, the histogram of CoP of On cycles grouped in \(Cluster_1\) are shown in Fig. 10. In order to find the CoP, the following equation is used (Napolitano et al. 2011).

$${\text{CoP}} = \frac{{{\text{Q}}7\_{\text{KW}}}}{{{\text{Q}}6{\text{a}}\_{\text{KW}}}}.$$
(11)

The histogram shows that \(80\,\%\) of the On cycles have CoP less than 0.2, thus representing low performance of the chiller.

Fig. 10
figure 10

Histogram of On cycle’s associated with cluster of faulty operation of chiller

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The same procedure had been adopted to see the behavior pattern of one of the On cycles in \(cluster_2\), as can be seen in Fig. 11. The x-axis displays the time, in minutes, for the On cycle.

Fig. 11
figure 11

On cycle pattern showing normal operation of chiller

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In each graph, the values are represented with the intensity of the color given in the attached color bar. It can be observed from Fig. 11 that the duration of the On cycle is 195 minutes. The temperature difference parameters show that there is cooling in the LT cycle, as \(\Delta Temp\_LT\) is representing it with time. The effect can be seen in the MT cycle as well. At the same time, HT cycle shows that the constant driving heat was provided to the chiller. It is also important to mention that the flow variable displays the flow in all the three cycle, whereas, same had been observed for the energy parameters. This behavior pattern shows the normal operation of the chiller. The histogram of CoP of On cycles, grouped in \(Cluster_2\), is shown in Fig. 12 in support of the argument that \(Cycle_2\) is representing the group of On cycles corresponding to the normal performance of the chiller. The histogram shows that the chiller is performing with CoP between 0.4 to 0.8, thus representing the normal behavior of the chiller.

Fig. 12
figure 12

Histogram of On cycle’s associated with cluster of normal operation of chiller

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Comparison of proposed method with CAS modeling

The main point in this research is to find various patterns in the operation of the energy system in buildings using minimum possible input from the engineers. For the analysis of the energy system, the data has been selected using IEA SHC Task 38. The issues that may surface, while modeling a current system using CAS, can be traced back to the complete knowledge of the system, its behaviors or states and the interaction of the subsystems; a problem of scale dealing with a very large state-space representation. Secondly, complex dynamic systems will require transitions between completely different behaviors in the form of what is called phase transitions. Hence a critical transition detection will require a detailed state-space model.

Conclusions

The main goal of this research work is to provide analysis algorithms that automatically find the various patterns in the energy system of a building using as little configuration or field knowledge as possible. A bag of word representation method with hierarchical clustering has been proposed to assess the performance of a building energy system. In the first phase, a k-means clustering algorithm is used to find the On (operational) cycles of the chiller. These On cycles are represented with symbols by using symbolic aggregate approximation (SAX) method. Furthermore, the symbolic representation is transformed to BoWR, which is provided to hierarchical clustering. The proposed method has been compared with dynamic time warping (DTW) method using cophenetic coefficients and it has been shown that the BoWR has produced better results as compared to DTW. The results of BoWR are further investigated and for finding the optimal number of clusters, gap statistics have been used. At the end, interesting patterns of each cluster are discussed in detail.

In future, the current research can be used in the field of automatic faults detection and diagnostics (FDD) in buildings, as the current research helps in finding the different performance patterns. This would help the experts in the field to look only for those areas where the performance is bad. Further research is needed in order to find intelligent ways of diagnosing the faults