![The schematic diagram of variable air volume AHU system [9,19,40].](https://www.researchgate.net/profile/Han-Zhang-212/publication/361229277/figure/fig1/AS:1165779009572865@1654954783856/The-schematic-diagram-of-variable-air-volume-AHU-system-9-19-40_Q320.jpg)
![The structure chart of the LSTM [36,37].](https://www.researchgate.net/profile/Han-Zhang-212/publication/361229277/figure/fig2/AS:1165779009568769@1654954783907/The-structure-chart-of-the-LSTM-36-37_Q320.jpg)
Content uploaded by Han Yuan Zhang
Author content
All content in this area was uploaded by Han Yuan Zhang on Jun 11, 2022
Content may be subject to copyright.
CORRECTED PROOF
Energy & Buildings xxx (xxxx) 112241
Contents lists available at ScienceDirect
Energy & Buildings
journal homepage: www.elsevier.com
Fault detection and diagnosis of the air handling unit via combining the
feature sparse representation based dynamic SFA and the LSTM network
Hanyuan Zhang a,⁎, Chengdong Li a, Qinglai Wei b,c,d, Yunchu Zhang a
aShandong Key Laboratory of Intelligent Buildings Technology, School of Information and Electrical Engineering, Shandong Jianzhu University, Jinan 250101, China
bThe State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China
cSchool of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing 100049, China
dInstitute of Systems Engineering, Macau University of Science and Technology, Macau 999078, China
ARTICLE INFO
Article history:
Received 14 March 2022
Received in revised form 12 May 2022
Accepted 5 June 2022
Keywords:
fault detection and diagnosis
air handling unit
feature sparse representation
slow feature analysis
long short-term memory (LSTM) network
ABSTRACT
In recent years, slow feature analysis (SFA) has been successfully employed to deal with the air handling unit
(AHU) system’s time-varying dynamic properties. However, since the derived slow features are the linear combi-
nations of all original variables, the conventional SFA based methods may suffer from poor model interpretabil-
ity and result in inaccurate fault detection and diagnosis (FDD) performance. To enhance the dynamic AHU sys-
tem FDD effectiveness, this paper presents a novel feature sparse representation based SFA algorithm through im-
posing the sparsity on the slow features, which is called the sparse three-way data based dynamic SFA (ST-
BDSFA). We make two contributions to propose the STBDSFA approach. One contribution is to infuse the feature
sparse representation technique into the constructed SFA based monitoring model by performing the sparse re-
striction on the loading vector, which can effectively eliminate the meaningless variables’coupling and select the
key variable responsible for the fault detection. Before building the STBDSFA algorithm, the other contribution is
to construct a new three-way data based dynamic SFA (TBDSFA) monitoring model to handle the AHU system’s
two-directional dynamic properties. In the established TBDSFA model, the auto-regressive moving average ex-
ogenous (ARMAE) model is first adopted to reveal the variables’auto-correlation relationships. Then, the multi-
way data analysis is applied to figure out the batch-wise dynamic property in multiple batch runs. After that, the
dynamic SFA model is further set up to sufficiently tackle the time-wise dynamic nature within a batch run. To ef-
fectively diagnose the pattern of detected fault, another innovative work is to develop a long short-term memory
(LSTM) based fault identification approach to classify the sparse slow features of fault snapshot dataset, owning
to the LSTM’s superiority of coping with the dynamic time-varying nature of the sparse slow features. By compar-
ing with some traditional and closely related FDD methods, the case study on the ASHRAE Research Project RP-
1312 experimental datasets are performed to verify the performance and effectiveness of the proposed FDD
scheme for the AHU system. To be specific, in comparison with the sparse principal component analysis, sparse
dynamic SFA, kernel locality preserving projection and sparse three-way data based SFA approaches, the devel-
oped STBDSFA based monitoring method not only reveals no fault detection delays to alert the eleven test faults,
but also achieves the highest average fault detection rates, i.e., 99.31 for the statistic and 99.77 for the SPE
statistic. Contrasted with the support vector machine, temporal convolutional network, convolutional neural net-
work and deep belief network based classification algorithms, the suggested STLSTM based fault identification
approach obtains the highest average fault diagnosis rate, i.e., 95.51 for the eleven fault snapshot datasets.
© 20XX
1. Introduction
Heating, ventilating, and air conditioning (HVAC) system contributes
to most of building energy consumption. As an important module of the
HVAC system, the faults occurring in the air handling unit (AHU) would
⁎Corresponding author.
E-mail address: zhanghanyuan18@sdjzu.edu.cn (H. Zhang).
reduce the indoor comfort as well as shorten the lives of equipments
and waste the building energy [1,2]. Therefore, to monitor and remove
the AHU system faults, it is very necessary to research an effective ap-
proach to perform the AHU system FDD [2,3].
With rich operating data recorded from the AHU system, the artifi-
cial intelligence (AI) based and the multivariate statistical analysis
(MSA) based FDD approaches have been becoming a research hotspot
recently [1,4,3]. The AI based methods have drawn more attentions be-
https://doi.org/10.1016/j.enbuild.2022.112241
0378-7788/© 20XX
Note: Low-resolution images were used to create this PDF. The original images will be used in the final composition.
CORRECTED PROOF
2H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Fig. 1. The schematic diagram of variable air volume AHU system [9,19,40].
Fig. 2. The structure chart of the LSTM [36,37].
cause they own the powerful capability to convert the original data
samples into a new feature space by the internal nonlinear mapping.
For instance, the deep neural network based FDD technique [5], the im-
proved Bayesian networks based fault identification algorithm [6], the
imbalanced data based support vector machine [7], and so forth. How-
ever, during the model training stage, most AI based methods use the
whole original variables as the model inputs. This defect always results
in an unsatisfying FDD effectiveness because some redundant variables
and process disturbances are usually included. On the contrary, the
MSA based methods have the superiority of eliminating noisy and re-
dundant variables using the dimension reduction techniques. By ex-
tracting the latent variables from the original process data, they can
capture the essential causes of process variations and directly construct
the visualized fault detection statistics. Thus, the MSA based methods
have been adopted to implement the AHU system FDD in recent years,
such as the ensemble empirical mode decomposition based PCA [8], an
improved feature filter based fault diagnosis algorithm [9], the
Bayesian network based nonlinear PCA [10], the unsupervised causal-
ity-based feature extraction method [11], the Riemannian metric based
locality preserving projection technique [12], and so on. Whereas, the
models of most abovementioned FDD approaches are built on the AHU
systems steady state operation with the time-invariant training dataset
[1,4]. In fact, the AHU system dynamically evolves from one operating
mode to another operating mode driven by the changes of outdoor and
indoor conditions [4,13]. Therefore, these FDD methods own the draw-
back to capturing the AHU system’s time-varying dynamic characteris-
tics.
As is known to all, including the AHU system, most of the modern
engineering systems are always characterized by the time-varying dy-
namics, due to the variations of operating conditions, control strategies
and external disturbances [2]. Hence, some data-driven based FDD and
prediction approaches have already been developed to deal with the en-
gineering systems’time-varying dynamic nature [2,3]. For example, in
the fields of mechanical fault diagnosis and life prediction, some data-
driven based methods, such as the statistical complexity measure based
unknown fault feature extraction approach [14], the common spatial
pattern based feature extraction algorithm [15], the uncertainty associ-
ated high-level methods fusion strategy [16] and a novel fuzzy-based
failure dynamic modeling method [17], have been suggested to per-
form the rolling bearing and gear FDD and remaining useful life predic-
tion tasks, by handling the vibration signals’time-varying dynamics.
Nevertheless, for the AHU system FDD, the AHU system’s time-varying
dynamic property have been rarely considered till now, which results in
a research gap of dynamically analyzing the system’s time-varying data
in the AHU FDD field.
Because of the variant outdoor and indoor environmental condi-
tions, the AHU system runs at different operating modes in a day to
maintain the indoor comfort. In the actual operation, the underlying
driving forces, i.e. the sequencing control logic, determines the switch
of these different operating modes. The AHU system is indeed charac-
terized by the two-directional time-varying dynamic properties: the
time-wise and the batch-wise dynamics [18,19]. In recent years, slow
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 3
Fig. 3. The flowchart of the developed STBDSFA algorithm.
feature analysis (SFA) based approaches have been employed to dispose
of the AHU system’s time-varying dynamic nature [19], because of its
capability to mine the slowly varying latent variables, which can catch
the underlying driving forces. For example, an enhanced kernel SFA
based AHU system FDD method was proposed and tested on the
ASHRAE Research Project RP-1312 in [19]. Considering the AHU sys-
tem’s time-varying dynamic nature, the improved SFA algorithm was
developed to detect the faults. Then, a new kernel discriminant SFA
model was built to nonlinearly diagnose the fault patterns through cal-
culating the similar degrees of the fault directions. Experimental results
indicated the significant performance improvements compared with
some related FDD approaches. The supervised learning based SFA
(SLSFA) framework was discussed to perform the time-varying dynamic
process FDD task [20]. By integrating the discriminant analysis tech-
nique into the SFA model, the SLSFA was presented to monitor the
process. To figure the tough issue of nonlinear fault variable diagnosis,
a novel nonlinear contribution plot was further developed after a fault
has been detected. Experimental results on the numerical and bench-
mark processes showed the superiority of the developed FDD method
contrasted with some traditional supervised learning methods. To sepa-
rately detect the faults from time-varying dynamic and steady-state op-
erating conditions, a concurrent monitoring strategy based SFA scheme
was presented in [21]. For this aim, multiple local SFA based models
were built to distinguish a steady derivation from the normal operating
condition. Then, a global SFA based model was constructed to detect
the dynamics anomalies according to the time variations. The feasibil-
ity and efficacy of the suggested scheme was verified on a typical multi-
phase dynamic system. Besides, a modified statistics SFA algorithm was
discussed to reveal the multiplicity information of the time-varying dy-
namic system [22]. Based on the statistics SFA (SSFA) model, the phase
recognition factor was first defined to automatically implement phase
division. After that, the data global structure analysis technique was
combined with the SSFA model, and different monitoring strategies
were proposed to detect the steady phases and transition period faults.
Compared with the traditional dynamic FDD methods, the SFA based
approaches are more competent and adequate to mine the time-varying
dynamic property, because they can characterize the underlying dri-
ving forces of the process [19,23,21]. However, the slow features ex-
tracted by these SFA based methods are the linear combinations of all
original variables, which can not reveal the meaningful variables for
the AHU system FDD.
By further analysis, the sparsity in the derived latent variables (the
slow features) has the critical significance to improve the fault detec-
tion effectiveness of the SFA based monitoring model [24,25]. When
the extracted slow features without sparsity (each slow feature is a lin-
ear combination of all original variables) are applied to the AHU system
FDD, they cannot uncover the significant correlations among original
variables. This drawback not only hinders the physical interpretation of
fault detection results but also fails to eliminate the original variables’
redundant coupling relationships. Therefore, to enhance the sensitivity
and detection ability for the fault, the sparse slow features are urgent to
be computed in the SFA based AHU system FDD methods. As a new rep-
resentation learning method, feature sparse representation technique
[26] has received considerable successes in the FDD field recently. For
the feature sparse representation, a sparsity criterion is imposed on the
established fault detection model, so that a group of significant vari-
ables with nonzero coefficients in the loading vectors are efficiently and
automatically chosen for the process monitoring. For instance, Liu et al.
[27] discussed a structured joint based sparse PCA model by involving
two regularization terms. For the fault detection, the norm was im-
posed to acquire the row-wise sparsity, while the graph Laplacian term
was introduced to incorporate structured variable correlation informa-
tion. Then, a two stage fault isolation strategy was developed to calcu-
late a score index for each variable. Compared with the PCA based
methods, two case studies demonstrated the presented method’s valid-
ity. Hu et al. [28] combined the sparsity with Fisher discriminant analy-
sis algorithm to perform fault diagnosis, by means of a proposed faulty
variable selection strategy. According to different natures of each fault
class, a probabilistic fault diagnosis strategy was put forward to recog-
nize the pattern of new fault sample. The experimental results on a ciga-
rette cut-made process proved the effectiveness of the constructed
method. To select the key fault variables, Yu et al. [29] suggested a
CORRECTED PROOF
4H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Fig. 4. Diagram of the data dynamic augmentation and three-way data analysis.
sparse exponential discriminant analysis based approach by integrating
the penalty of lasso or elastic net into the discriminant model. The opti-
mization problem was then solved with the help of a novel gradient di-
rection approach. In comparison with the related discriminant analysis
based method, the case studies on the two typical industrial processes
tested the fault diagnosis effect of the suggested method. Zhang et al.
[30] embedded a sparsity regularization layer into the convolutional
auto-encoder network to remove the extracted features’redundant in-
formation. Furthermore, a depthwise separable convolution block was
employed to reduce the computational cost. The experimental results
on an industrial benchmark and a real production process exhibited the
feasibility of the developed fault detection method in uncovering the
representative features. Inspired by the superiority and successful appli-
cation of the aforesaid feature sparse representation based FDD meth-
ods, in our work, the feature sparse restriction is imposed on the pro-
posed SFA based monitoring model to effectively preserve the fault dis-
criminative features and improve the AHU system FDD performance.
Before developing the feature sparse representation based SFA
method, we create an innovative way to build a new SFA based fault de-
tection model, called the three-way data based dynamic SFA (TBDSFA),
to display the variables’auto-correlation relationships and more effec-
tively handle the AHU system’s time-wise and batch-wise dynamics. To
be specific, the time-wise dynamics is related to the changes of operat-
ing modes in a batch run (one running day), while the batch-wise dy-
namics is characterized by the dynamic deviations in multiple batch
runs (different running days). In the established TBDSFA model, by
augmenting each input sample with its previous observations, the auto-
regressive moving average exogenous (ARMAE) model [31–33] is inte-
grated into the SFA to reveal the auto-correlation relationships of vari-
ables. On the other aspect, to adequately figure out the batch-wise dy-
namic variations and deviations, the multiway data analysis [19,34,35]
is fused into the suggested monitoring method to unfold the three-way
training dataset. Based on the unfolded two-way training matrix, the
dynamic SFA model is constructed to deal with the AHU system’s time-
wise dynamics more sufficiently. Based on the TBDSFA model, the fea-
ture sparse representation based SFA algorithm, referred to as the
sparse TBDSFA (STBDSFA), is constructed to detect the dynamic AHU
system fault.
After detecting a fault, the fault identification is needed to diagnose
the fault. However, the primary target of the constructed STBDSFA
model is to alert the fault rather than study how to identify fault pat-
tern. In this paper, the sparse slow features extracted from the devel-
oped STBDSFA is still dynamic time-varying, because of the STBDSFA’s
superiority of capturing the AHU system’s underlying driving forces.
During the fault diagnosis, the important issue is how to solve the time-
varying dynamic nature of the sparse slow features. The long short-term
memory (LSTM) network [36], as one time recurrent neural network,
implements excellently to handle the dynamic time-varying temporal
sequences. The idea of the LSTM is to establish a series of layers con-
nected with the feedback which uses the information from the previous
samples to calculate the output at the current time. In recent years, the
LSTM has been widely adopted to identify the fault pattern. For exam-
ple, a deep recurrent based LSTM was presented to dynamically esti-
mate the electrified vehicles’brake pressure in [37]. For this purpose, a
real time multivariate LSTM-RNN model was built using an established
speed estimation model. Experimental results on vehicle testing showed
good performance. In order to achieve higher fault diagnosis accuracy,
Han et al. [38] introduced an optimized memory-capable based LSTM
network to determine the optimal number of hidden layer nodes. The
experimental test proved the proposed LSTM network has much better
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 5
Fig. 5. The framework of the STLSTM based fault diagnosis.
performance than the conventional methods such as BP neural network,
original LSTM, and so on. In addition, Sun et al. [39] discussed the
Bayesian theory based LSTM to model the complex nonlinear dynamics,
which can yield the uncertainty estimates to simultaneously realize
fault detection, fault identification fault propagation analysis. Consid-
ering the LSTM’s above advantages, to diagnose the detected fault, we
employ the LSTM based method to classify the dynamic time-varying
sparse slow features of fault snapshot dataset derived by the STBDSFA
monitoring model.
As far as the authors know, the AHU system FDD based on the SFA
has arisen and attracted much attention recently [19]. To further en-
hance the FDD effectiveness by imposing the sparsity on the slow fea-
tures and tackling the two-directional time-varying dynamic properties,
we develop the STBDSFA and an improved LSTM based FDD scheme for
the AHU system in this paper. The main contributions of our work are
given as.
•A novel feature sparse representation based SFA algorithm,
termed as the STBDSFA, is developed to enhance the SFA based
model’s fault detection performance. By imposing the
penalization on the loading vectors’nonzero elements, the
presented STBDSFA algorithm can derive the sparse slow features
to reveal the meaningful variables for the AHU system fault
detection. Specifically, with the help of a lasso penalty, the
sparse restriction is first infused into the SFA based model to
shrink the coefficients of insignificant variables in the loading
vectors to be zero. Hence, the variables corresponding to nonzero
coefficients are automatically determined as key variables for
monitoring the AHU system. The optimization problem of the
STBDSFA is then worked out using the sparse generalized
eigenvalue (SGEV) technique. Finally, a deflation procedure is
adopted to sequentially generate the sparse loading vectors from
the training dataset.
•Before establishing the feature sparse representation based SFA
algorithm, a new three-way data based dynamic SFA (TBDSFA)
approach is constructed to efficiently figure out the batch-wise and
time-wise time-varying dynamics of the AHU system. In the
developed TBDSFA approach, the ARMAE model is first integrated
to uncover the variables’auto-correlation relationships, by
augmenting every sample with its previous samples within each
batch run dataset. Through variable-wisely unfolding the
augmented three-way normal matrix into a two-way training
dataset, the multiway data analysis is then applied to solve the
batch-wise dynamic nature in multiple batch runs. At last, the
dynamic SFA (DSFA) is further built to handle the time-wise
dynamic characteristics, by utilizing the two-way training dataset
as the model’s input.
•An innovative sparse three-way data LSTM (STLSTM) network is
suggested to effectively identify the pattern of detected fault,
which has the advantage to address the time-varying dynamics of
the sparse slow features. In the constructed STLSTM based fault
diagnosis method, the detected fault samples are first stored to
establish the fault snapshot dataset. The STBDSFA model is then
carried out on the fault snapshot dataset and multiple historical
fault datasets to extract the corresponding dynamic time-varying
sparse slow features. Next, the sparse slow features of historical
fault datasets are fed to the STLSTM network to train the
classification model. In the end, the trained STLSTM model is
employed to classify the sparse slow features of fault snapshot
dataset to diagnose the fault pattern.
•Detailed experiments of the developed AHU system FDD scheme
are implemented on the experimental datasets of the Research
Project RP-1312, and the comparisons with some related
approaches, such as the SPCA, SDSFA, KLPP and STBSFA based
fault detection approaches as well as the SVM, CNN, TCN and DBN
based fault diagnosis methods, are given out. In comparison with
the related methods, the outstanding fault detection effectiveness
of the constructed STBDSFA model is demonstrated. Besides, the
obtained experimental results also reveals the excellent fault
diagnosis performance of the suggested STLSTM classifier.
The rest parts of our work are arranged as follows. The AHU system,
the basic SFA, LSTM network, sparse generalized eigenvalue (SGEV)
technique and kernel density estimation (KDE) method are introduced
briefly in Section 2. In Section 3, the STBDSFA based monitoring
method is proposed. In Section 4, the STLSTM based fault identification
method is constructed. Section 5provides the proposed FDD scheme for
the AHU system. The case study on the RP-1312 are carried out in Sec-
tion 6. Finally, Section 7gives the conclusions.
2. Preliminaries
To promote the understanding of the proposed AHU system FDD
scheme, the operating mechanism of the AHU system is first intro-
duced, and then the basic SFA and LSTM methods are revealed and ex-
plained in this section. At last, the fundamental sparse generalized
eigenvalue (SGEV) technique and the kernel density estimation (KDE)
method are briefly introduced.
2.1. The AHU system
As an important module of the HVAC, the variable air volume AHU
system [40,41] shown in Fig. 1 is widely utilized in practice. As dis-
played in Fig. 1, the supply air is first heated, cooled and dehumidified
using the heating and cooling coils. Then, the supply air is imported to
the AHU terminals in multiple zones via the ductwork. Through the re-
turn duct, the air in multiple rooms is sent back. The supply and return
fans are applied to facilitate the air circulation in the ductworks. The
supply air static pressure controller (PC-1), the damper controller (DC-
1), the supply air temperature controller (TC-1) and the return air flow
rate controller (FC-1) are employed to keep the suitable supply air pres-
sure and temperature as well as the sufficient outdoor air ventilation.
In the actual operation, the time-wise dynamic behaviors exist in the
AHU system’s one batch run [19,18]. Besides, during several successive
days, the AHU system is also characterized by the batch-wise dynamics
among multiple batch runs [19]. According to the locations and causes,
the representative AHU system faults are the dampers stuck, the sensor
measurements deviation and the valves leakage, and so on. In order to
improve the AHU system FDD effect, the SFA based method has been
discussed to address the time-varying dynamics[19]. However, it is dif-
CORRECTED PROOF
6H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Fig. 6. The procedure of the STLSTM training and fault identification.
ficult to undertake the effective AHU system FDD, because in the con-
ventional SFA based models, the extracted slow features without spar-
sity are commonly applied to detection and diagnose the AHU system
fault. This shortcoming not only fails to reveal meaningful correlations
among variables, but also weakens the monitoring models’fault suscep-
tivity and detection ability [24,25,27].
2.2. The basic SFA
The principle of the SFA is to generate the outputs which possess
slow variations from the model inputs. For a input vector , the SFA
explores the conversion function vector to guarantee the output
vector with owns the slowest
variations. The objective function of the SFA is formulated as
(1)
s.t.
(2)
(3)
(4)
where denotes the j-th slow feature, represents the deriva-
tive of , and the operation is computed as .
The owns the slowest variation and the possesses the second
slowest variation, and so on.
Under the situation of the being a linear mapping, is for-
mulated as , where represents the loading vector. The
SFA optimization is then rewritten as
(5)
where and respectively indicate the covariances of the and
. The is approximately calculated as
[42,43]
2.3. The basic LSTM network
The long short-term memory (LSTM) is an enhanced version of the
recurrent neural network (RNN), which is widely used to handle the
temporal sequence data [36]. One outstanding merit of the RNN is that
the hidden layer’s neurons own a feedback mechanism to load the pre-
vious time series information in the current learning task. Whereas, af-
ter a long training time, the gradients of RNN nodes usually converge to
zero, which impedes the network learning procedure [37]. This phe-
nomenon is referred to as the ”vanishing gradient”problem. To over-
come this problem, the LSTM network has been studied and invented in
recent years.
The LSTM can learn the long-term dependencies because of its ca-
pacity to save the past information in the memory blocks [37,38]. That
is, the LSTM’s outputs rely on the past and present inputs. Therefore,
the LSTM is very suitable to model and classify the temporal sequences.
As revealed in Fig. 2, the LSTM network contains three periods respec-
tively corresponding to the input gate , the forget gate and the out-
put gate . The gates are all computed using the previous layer output
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 7
Fig. 7. The illustration of the suggested STBDSFA and STLSTM based FDD.
and the current layer input by utilizing a nonlinear sigmoid acti-
vation function. To selectively pass the information through, the sig-
moid activation function indicated by the symbol in Fig. 2 converts
the input into the range (0, 1), which is given as
(6)
The LSTM cell can be characterized by the Eqs. ()()()(7)–(9). The Eq.
(7) controls how much the previous information is dropped by the for-
get gate. As formulated in Eq. (8), the input gate selects the updated
new information. In Eq. (9), the candidate layer output is determined
by the output gate.
(7)
(8)
(9)
where and represent the weight vectors; and indi-
cate the bias vectors.
The candidate cell state is calculated using the tanh activation
function expressed by the symbol tanh in Fig. 2.
(10)
The denotes the LSTM’s internal memory cell state, which com-
bines the previous with the current candidate state .
(11)
where indicates the element-wise multiplication of two vectors.
In the output period, the current layer output is calculated as
(12)
where represents the tanh-activated cell state .
Based on the above calculation procedure, the LSTM network owns
the ability to decompose the new cell state into the retained historical
information and the selectable input information. This advantage of the
CORRECTED PROOF
8H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Table 1
The description of the sixteen original monitored variables [7,19,56–58].
Number Original variable Number Original variable
Supply air temperature Return air fan status
Supply air temperature set point Supply air fan speed control
signal
Outdoor air temperature Return air fan speed control
signal
Mixed air temperature Exhaust air damper control
signal
Return air temperature Outdoor air damper control
signal
Supply air duct static pressure Return air damper control
signal
Supply air duct static pressure set
point
Cooling coil valve control
signal
Supply air fan status Heating coil valve control
signal
LSTM not only eliminates the ”gradient vanishing”problem, but also
learns the long-term dependencies for the classification task.
2.4. The brief introduction of the SGEV technique and the KDE method
2.4.1. The fundamental SGEV technique
The SGEV technique developed in the references [44,45] offers a
general framework to solve the SGEV problem, which can find sparse
solutions to the generalized eigenvalue problem. The SGEV problem
can be approximately formulated as
(13)
where and Eq. (13) is a continuous optimiza-
tion problem.
Introducing the auxiliary variable , Eq. (13) is equivalently written
as
where the term is convex as , and
is jointly convex. So, Eq. (14) is a min-
imization of the difference of two convex functions over a convex set.
For the approximate SGEV given in Eq. (14). Let
(15)
where , so that Eq. (14) can be written as
and . Defining
(16)
The majorization function is then yielded
(17)
The majorization-minimization algorithm corresponding to this ma-
jorization function updates at iteration lby
(18)
Based on the Eq. (18), the following algorithm is given out
(19)
Finally, the SGEV algorithm is formulated in Eq. (20) according to
Eq. (19).
(20)
which is a sequence of quadratically constrained quadratic pro-
grams (QCQPs). For , the optimal solution of Eq. (20), i.e., ,
is unique, since the objective function is strictly convex. For , al-
though the objective function is no longer strictly convex, Eq. (20) also
has a unique optimal solution by solving a quadratic program. The
SGEV algorithm is detailed in Algorithm 1 [44,45].
Algorithm 1 The pseudo code of the sparse generalized eigenvalue algorithm
Require: , , and
1: Choose
2: Choose
3:
4: if then
5: if then
6: repeat
7:
8: if then
9:
10: else
11:
12: end if
13: until convergence
14: else
15: repeat
16:
17: if then
18:
19:
20:
21:
22: else
23:
24: end if
25: until convergence
26: end if
27: else
28: repeat
29:
30:
31: Image
32: until convergence
33:end if
34: return
2.4.2. The elementary KDE method
To inspect if a fault occurs, the control limits of the monitoring sta-
tistics constructed in the STBDSFA are required. As no prior knowledge
is available regarding the distribution of the process variables in the
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 9
Table 2
The introduction of the normal dataset [19,56,58].
Dataset Stored time Dataset Stored time
8/27/2008 9/5/2008
8/28/2008 2/11/2009
8/29/2008 5/6/2009
8/30/2008 5/7/2009
8/31/2008 5/8/2009
9/1/2008 5/15/2009
9/4/2008
AHU system, it is difficult to guarantee that the process variables con-
form to a specific distribution assumption. Hence, in this paper, the cor-
responding control limits of the built monitoring statistics are deter-
mined by applying the well-known KDE method [46,47], instead of di-
rectly calculating from a particular approximate distribution.
The KDE method is a nonparametric empirical density estimation
technique, which does not need any assumption of the process variables
distribution. Recently, data-driven KDE based method has become
more and more popular for control limit determination [46,47]. Specifi-
cally, a univariate kernel estimator with kernel ker in the KDE method
is defined by
(21)
where yis the data point under consideration, is an observation
value from the data set, his the smoothing parameter, nis the number
of observations. The ker denotes the kernel function which is selected as
Gaussian kernel in this work.
The corresponding control limits of established monitoring statistics
can be obtained using KDE method as follows. First, values of the moni-
toring statistics from three-way normal operating dataset are com-
puted. Then the univariate kernel density estimator is used to estimate
the density function of the corresponding monitoring statistics’normal
values. Lastly, the control limits of monitoring statistics are obtained by
calculating the points occupying the 99 area of density function in the
three-way normal operating dataset.
3. The developed STBDSFA based fault detection
Through the analysis of the AHU system’s operation, the AHU sys-
tem is always characterized by the time-wise and batch-wise dynamics
[19,18]. Although the SFA based approach has been utilized to figure
out the time-varying dynamics of the AHU system, the derived slow fea-
tures in the SFA based model are a linear combination of all original
variables instead of possessing the sparsity. This defect would reduce
the fault susceptibility and detection capability of the constructed AHU
system FDD models. In addition, to further enhance the FDD perfor-
mance, the variables’auto-correlations of the AHU system as well as its
time-wise and batch-wise dynamic properties should be well considered
and managed.
To cope with these problems, we discuss the STBDSFA algorithm
displayed in Fig. 3 to detect the AHU system fault, which involves three
modules. The first module employs the ARMAE model to augment each
batch run dataset for the purpose of capturing the variables’auto-
correlation relationships. The second module performs the multiway
data analysis on the multiple augmented normal batch run datasets to
solve the batch-wise dynamics, and establishes a DSFA model using the
unfolding two-way training matrix to sufficiently exploit the time-wise
dynamics. While the third one further constructs the STBDSFA model to
effectively and automatically choose the significant original variables
responsible for the fault detection, by integrating the feature sparse rep-
resentation technique into the established TBDSFA method. The three
modules of the proposed STBDSFA are introduced below in detail.
3.1. The ARMAE model for dynamically extending the datasets
The time-wise dynamic nature is characterized by the underlying
driving forces which are associated with the previous samples. To bet-
ter uncover the time-wise dynamic nature, the ARMAE model [31,32] is
utilized to handle the auto-correlation relationships existing in each
batch run. The goal of the ARMAE model is to augment every sample in
one batch run dataset with its previous observations. To be specific, the
augmented matrix of the i-th batch run dataset is constructed by
extending the current sample with its previous dsamples, which is
given as
(22)
where indicates the k-th sample, Kis the number of samples
and drepresents the time lags.
How to select the ideal value of the time lags dhas been still an open
problem [48,49]. If the time lags is too large, the augmented matrix
will become independent among different time instants. In the litera-
tures [48,49], the time lag is proved to be suitable to uncover the
variables’auto-correlation relationships. Hence, in this paper, the time
lag dis also empirically determined as 2 in the presented approach. Af-
ter all the multiple normal batch run datasets are augmented by the AR-
MAE model, the following multiway data analysis is carried out to solve
the batch-wise dynamic property.
3.2. The TBDSFA model construction
3.2.1. Eliminate the AHU system’s batch-wise dynamics
To figure out the batch-wise dynamic nature, the AHU system’s nor-
mal operating data is assumed to be stored for several adjacent days.
The acquired multiple batch datasets are expressed as , where
.Kand Jrespectively denote the numbers of samples and
variables in each batch run, and Iindicates the number of batch runs.
The Inormal batch datasets are augmented by the ARMAE
model to achieve the extended matrices ,
where denotes the numbers of augmented variables with the time lag
d.
All these extended matrices are then utilized to con-
struct a three-way matrix . As displayed in Fig. 4, with
the help of multiway data analysis [19,34,35], the three-way matrix is
further transformed into a two-way matrix to build the developed ST-
BDSFA fault detect model. The whole multiway data analysis procedure
is presented in detail below.
First of all, according to the time direction, the matrix
is splitted into the matrices with
. Then, the three-way training dataset is
unfolded into a two-way dataset by placing the Kmatrices
in the light of the batch-wise direction. The batch-wise dy-
namic variabilities in the matrix can be removed by the
batch-wise unfolding.
Secondly, the training dataset is normalized to acquire
the scaled dataset . To be specific, given the
, where indicates the i-th sam-
ple of the dataset , the sample is normalized as
CORRECTED PROOF
10 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Table 3
The basic information of the AHU system’s fault patterns [19,56,58].
Description Fault pattern Intensity Occurred time
Outdoor air damper stuck F1 Fully closed 5/7/2008
F2 40% open 5/8/2008
Outdoor air damper leaking F3 45% open 9/5/2007
F4 55% open 9/6/2007
Heating coil valve leaking F5 0.4 GPM 8/28/2007
F6 1.0 GPM 8/29/2007
F7 2.0 GPM 8/30/2007
Cooling coil valve stuck F8 Fully closed 5/6/2008
F9 Fully open 8/31/2007
F10 15% open 9/1/2007
F11 65% open 9/2/2007
(23)
where represents the mean operators and denotes the
standard deviation operators. By such normalized vectors, the normal-
ized matrix is established as .
In the last step, the normalized dataset is converted
into a novel dataset by utilizing a variable-wise unfolding
operation. This process is carried out by stacking the Knormalized ma-
trices according to the variable-wise direction.
3.2.2. Build the DSFA model to handle the time-wise dynamics
Due to the SFA’s excellent capability to reveal the underlying dri-
ving forces, we build the DSFA model using the unfolded two-way
dataset to adequately settle the AHU system’s time-wise
dynamics. However, the temporal series information in the training
dataset is destroyed when constructing the matrix with the
Ktime slice matrices . To cope with this problem, the pseudo
time series should be first established to set up the following sparse
DSFA model.
Let the represents the i-th sample in matrix .
According to the reference [19], its lnearest neighboring samples are
selected from the matrix through using the K-nearest neigh-
borhood (K-NN) criterion to construct the pseudo time series .
Based on the pseudo time series related to the sample , we
can reconstruct a pseudo time series of the dataset
as [19], where the
matrix indicates the pseudo time series of the matrix .
In our work, the -th and -th samples in the matrix
are respectively represented as the and
. Thus, the temporal variation information is thought to oc-
cur in the and . In the following subsection, the matrices
and are used to build the TBDSFA fault detec-
tion model.
Using the constructed matrix and the variable-wise un-
folding training matrix , the TBDSFA model is first estab-
lished by solving the following optimization problem.
(24)
where the vector indicates the loading vector, and represents
the i-th sample.
The temporal variation matrix of the matrix is cal-
culated as
(25)
where for .
Furthermore, we compute the temporal variation covariance matrix
and the covariance matrix . The objec-
tive function of the TBDSFA in Eq. (24) is reformulated as
The above optimization is resolved by implementing the eigenvalue
decomposition problem given as
(27)
The eigenvectors of the psmallest eigenvalues are collected to estab-
lish the loading matrix.
3.3. The STBDSFA model establishment
A drawback of the TBDSFA model is the loading vectors without the
sparsity. In other words, a majority of the elements in the loading vec-
tor are not zero. Thus, the loading vectors are associated with most
original variables. This drawback impedes the loading vectors’physical
interpretation and degrades the TBDSFA’s fault detection capability. To
eliminate the variables’meaningless coupling relationships and further
enhance the fault detection ability, in our work, the feature sparse rep-
resentation technique is incorporated into the TBDSFA model. Hence,
the STBDSFA algorothm is further presented to efficiently select the key
variables responsible for detecting the fault by inducing the sparse slow
features.
In our proposed STBDSFA algorithm, the penalization on the ele-
ments of loading vector is imposed on the TBDSFA’s objection function
to enforce the sparse solutions. In this way, the coefficients of insignifi-
cant variables are shrunk to be zero, and the variables with nonzero co-
efficients are determined as the significant variables. Hence, the devel-
oped STBDSFA algorithm can generate a series of sparse loading vectors
containing fewer nonzero elements, to explore the AHU system’s under-
lying driving forces.
3.3.1. Derive the STBDSFA model
In the STBDSFA, an additional cardinality constraint is imple-
mented on the optimization problem of the TBDSFA to compute the
sparse loading vectors. The STBDSFA’s objective function the is given
as
(28)
where , and indicates the relaxation fac-
tor to adjust the sparsity of the loading vector . Some elements of the
would shrink to be zero with large value of the .
The optimization formulated in Eq. (28) is discontinuous and non-
convex, thus the Eq. (28) is difficult to be resolved. To settle this prob-
lem, the relaxation technique discussed in the literatures [45,50] is uti-
lized to convert Eq. (28) into a convex optimization problem. The con-
straint is approximated to . Meanwhile, the is
approximately calculated as
(29)
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 11
Table 4
The profile of the first sixteen extracted SLVs using the STLSTM model.
Number Nonzero variables Number Nonzero variables
SLV1 , , , , , , SLV9 ,
SLV2 , , , , , SLV10 , , , ,
SLV3 , , , , , SLV11 , ,
SLV4 , , , , , SLV12 , , , , , ,
SLV5 , , , SLV13 , , , ,
SLV6 , , SLV14 , , , ,
SLV7 , , , , , SLV15 , , , , , ,
SLV8 , , SLV16 , , , , , , ,
where denotes the i-th element of the loading vector , and
indicates the absolute value of .
The problem in Eq. (28) is thus transformed into a continuous con-
vex optimization problem as follows
(30)
where .
Selecting a parameter satisfies
, Eq. (30) can be reformulated as
(31)
According to the literatures [45,50], the solution of Eq. (31) can be
computed by figuring out the following quadratically constrained qua-
dratic program (QCQP) problem.
(32)
where jrepresents the number of iterations.
The optimization problem given in Eq. (32) can be settled using the
sparse generalized eigenvalue (SGEV) algorithm [44,45]. To enhance
the sparsity of the loading vector, in our work, is approximately deter-
mined to be zero and is selected as the minimum of
to carry out the SGEV algorithm. According to the
principle suggested in references [44,45,50], the optimal sparse solu-
tion is selected from all the calculated sparse solutions by applying a set
of values to solve the Eq. (32).
Note that only the first sparse loading vector of the STBDSFA model
can be directly calculated by the SGEV algorithm. To achieve the other
sparse loading vectors, the matrix is deflated in our work based on the
previous sparse loading vectors. Thus, the orthogonalized projection
deflation technology [45,51] is employed in this paper.
(33)
where denote the sparse loading vector computed by
performing the SGEV method on the deflated matrices .
And represent the column vectors of the matrix with
. The eigenvectors of the TBDSFA model are determined as ini-
tial solutions to implement the deflation procedure and the aforemen-
tioned SGEV algorithm.
3.3.2. Monitor the new test sample
The developed STBDSFA model brings a set of sparse generalized
eigenvectors . These generalized eigenvectors
can characterize the varying trend of the sparse slow features. For the
AHU system, the slowly-varying sparse slow features catch the main
variation trends of the AHU system, while the rapidly-varying sparse
slow features capture the short-term system disturbances.
Retaining too many sparse slow features would focus needless atten-
tion on the short-term system disturbances, resulting in the STBDSFA’s
monitoring effectiveness deterioration. Therefore, to maintain the cru-
cial feature information of the AHU system, we only choose the first p
vital generalized eigenvectors to derive the slowest
varying sparse slow features. The parameter pis determined through
computing the cumulative slowness contribution rate (CSCR) index de-
fined in the literature [19]. In the developed STBDSFA model, the value
of pis determined in the light of the 95 CSCR.
Given the normalized test sample in a new batch run, its time
lagged vector is first constructed using the ARMAE model. Utilizing
the computed sparse slow features , the and SPE statistics
are then calculated to monitor the augmented test sample .
(34)
(35)
where . The statistic is built to quantify the AHU
system’s major variation trends; while the SPE statistic is established to
capture the short-term system disturbances.
To monitor the test sample , kernel density estimation (KDE) algo-
rithm [46,47] is applied to set up the control limits of the and SPE
statistics using the training data. With the help of the determined con-
trol limits, the STBDSFA can detect the AHU system faults.
4. The suggested STLSTM based fault diagnosis strategy
After the developed STBDSFA based method detected the AHU sys-
tem fault, the fault diagnosis procedure needs to be further investigated
to remove the fault effect. Due to the STBDSFA’s superiority of captur-
ing the AHU system’s underlying driving forces, the derived sparse slow
features is still dynamic time-varying in our work. Hence, in this sec-
tion, we present the STLSTM based fault diagnosis approach to effec-
tively deal with the time-varying dynamic property of the extracted
sparse slow features, recurring to the LSTM’s capability to learn the
long-term dependencies of the dynamic temporal sequences [36].
Suppose that fault samples are detected and collected, the fault
snapshot dataset is established utilizing
these fault samples. Then, the dataset is normalized according to the
training data. As shown in Fig. 5, the developed STBDSFA model is first
employed to extract the sparse slow features of both the dataset and
the classes historical fault datasets. Then, the sparse slow features of
historical fault datasets are fed to the STLSTM network to train the clas-
sification model. Finally, the trained STLSTM model is then employed
to classify the sparse slow features of fault snapshot dataset to identify
the fault pattern. Due to the STLSTM’s virtue to figure out the time-
varying dynamics of the sparse slow features, the pattern of detected
fault can be diagnosed more accurately and effectively.
The procedure of the suggested STLSTM based fault identification is
displayed in Fig. 6, which contains the STLSTM classifier training stage
and the STLSTM based fault diagnosis stage. In the STLSTM classifier
training stage, the digital labels of classes historical fault patterns are
first imported to the encoding operation unit to gain the correct label
vector. Next, the sparse slow features of classes historical fault
datasets are transformed into an estimated output vector through the
transformation of the STLSTM model and fully connected layer. Specifi-
CORRECTED PROOF
12 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Fig. 8. The monitoring charts for fault F5, (a) SPCA, (b) SDSFA, (c) KLPP, (d) STBSFA, (e) STBDSFA.
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 13
Fig. 9. The fault diagnosis rates of the STSVM, TCN, STCNN, STDBN and STL-
STM for the dataset .
cally, the fully connected layer is connected behind the STLSTM model
to generate the fully connected features which represent the global rep-
resentation of the extracted sparse slow features. The softmax cross en-
tropy is then utilized to compute the error (loss function) between the
correct label vector and the estimated output vector. Based on the ac-
quired error (loss function), the Adam optimizer is adopted to adjust
the model parameters until the classification accuracy reaches set
value. During the STLSTM based fault diagnosis stage, the sparse slow
features of fault snapshot dataset are first fed to the trained STLSTM
classifier to calculate an output vector. The softmax layer is then ap-
plied to convert the output vector into a probability vector. At last, the
pattern of detected fault is recognized by computing the largest elemen-
t’s index in the probability vector.
Below, the detailed implemented steps in the training stage of the
STLSTM classifier are given as.
(1) The encoding operation is carried out to encode the classes
historical fault patterns with the digital labels to obtain the correct
label vector . The different numbers denote the
multiple fault patterns.
(2) The sparse slow features of classes historical fault datasets are
calculated by the developed STBDSFA monitoring model, where
is the number of fault patterns. By employing the STLSTM model
and the fully connected layer, the entered sparse slow features are
converted into an estimated output vector . As illustrated
in Fig. 6, the fully connected layer derives the global representation
of the outcomes resulted from the STLSTM model by merging class-
distinctive local information in previous layers. This high-level fully
connected features are used as the final fault classification due to
the strong discrimination ability. Specifically, the relation between
the two successive layers is defined as [52,53]
(36)
where is the value of the g-th neuron at the l-th fully connected
layer and represents the activation function. denotes the
weight of the connection between the h-th neuron in the (l-1)-th
layer and the g-th neuron in the l-th layer, and indicates the bias
of the g-th neuron in the l-th layer.
(3) The softmax cross entropy loss function is computed to evaluate
the error between the output vector and the correct label vector ,
which is formulated as
(37)
(4) Based on the computed loss function, the Adam optimizer is
applied to adjust and update the parameters of the STLSTM network
and fully connected layer. The explanation of the Adam optimizer
can be referred to the literatures [18,54].
(5) The steps (1)–(4) are executed repeatedly until the number of
iterations or the classification accuracy comes up to the set value.
To identify the pattern of detected fault, as shown in Fig. 6, the com-
puted output vector is imported to the softmax layer, where the in-
volving Creal numbers is transformed into a new vector containing C
probability values. The i-th element of the is computed according to
Eq. (38). In order to determine the maximum probability of the output
vector belonging to a particular class in the correct label vector , the
largest element’s index in the should be figured out. The calculated
index is deemed to indicate a specific fault pattern of the AHU system.
(38)
It should be noted that the STLSTM network with some learnable
parameters would falls into the over-fitting problem, particularly when
the training dataset is relatively small. This limitation would reduce the
STLSTM’s generalization ability to classify a new test sample. There-
fore, we apply the dropout technique [39] to tackle the over-fitting is-
sue of the STLSTM network through dropping the neurons with a cer-
tain probability. The idea of the dropout technique is to follow the con-
cept of coadaptation avoidance among the hidden nodes, which sto-
chastically discards some hidden nodes in each iteration by building
multiple models and integrating them into a final model, when training
the STLSTM network [39,55]. The forward propagation of the dropout
is written as
(39)
where is the Bernoulli random vector with a certain probability.
The input node is dropped out, when the is equal to zero.
The fault diagnosis of the STLSTM is analogous to the STLSTM train-
ing stage. After the sparse slow features of the fault snapshot dataset is
extracted by the STBDSFA, these sparse slow features are fed to the
trained STLSTM classifier. Then, the fault pattern’s index is derived by
Eq. (38). Finally, the pattern of detected fault is diagnosed by identify-
ing the computed index.
5. The STBDSFA and STLSTM based FDD scheme
As displayed in Fig. 7, our developed AHU system FDD scheme is
composed of two portions: the STBDSFA based monitoring and the STL-
STM based fault identification. To be specific, the STBDSFA based mon-
itoring is further composed of the modeling stage and the detecting
stage. The STLSTM based fault identification includes the STLSTM clas-
sifier training phase and the fault snapshot dataset’s sparse slow feature
classifying phase. The details of our proposed AHU system FDD strategy
is formulated as follows.
CORRECTED PROOF
14 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Fig. 10. The monitoring charts for fault F8, (a) SPCA, (b) SDSFA, (c) KLPP, (d) STBSFA, (e) STBDSFA.
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 15
Fig. 11. The monitoring charts for fault F10, (a) SPCA, (b) SDSFA, (c) KLPP, (d) STBSFA, (e) STBDSFA.
CORRECTED PROOF
16 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Table 5
The comparison of FDTs (sample No.) for the five algorithms.
Fault Pattern SPCA SDSFA KLPP STBSFA STBDSFA
F1 155 121 121 121 121 121 121 121 121 121
F2 143 121 121 142 121 121 121 121 121 121
F3 121 121 124 128 121 242 121 121 121 121
F4 121 121 121 132 121 121 121 121 121 121
F5 130 127 241 124 123 241 121 121 121 121
F6 121 121 121 136 121 230 121 121 121 121
F7 121 121 121 143 121 140 127 122 121 121
F8 195 223 241 186 121 168 128 124 121 121
F9 135 121 121 121 121 123 121 121 121 121
F10 237 121 121 171 121 149 128 126 121 121
F11 121 132 121 133 133 121 121 121 121 121
5.1. The STBDSFA based monitoring
5.1.1. The modeling stage
(1) Augment the i-th batch run dataset to obtain the augmented
matrix for by the ARMAE model, which is designed
to reveal the time-wise dynamics more sufficiently.
(2) Establish the augmented three-way matrix , and the
batch-wise unfolding is applied to convert it into a normalized
dataset . The matrix is then rearranged
into a dataset according to the variable-wise direction.
(3) Reconstruct the pseudo time series dataset of training matrix
using the K-NN principle, and calculate the temporal
variation of the dataset according to Eq. (25).
(4) Build the TBDSFA model using the matrices and ,
then derive the STBDSFA algorithm by performing the feature
sparse representation technique on the TBDSFA model, as expressed
in Eq. (28).
(5) Convert the optimization problem of the STBDSFA into a
continuous convex optimization problem by the relaxation
technique, and iteratively calculate the sparse loading matrix
through using the SGEV algorithm and the deflation procedure.
(6) Figure out the training dataset’s and SPE monitoring indices and
adopt the KED algorithm to compute the corresponding control
limits.
5.1.2. The detecting stage
(1) Collect test data and augment it by the ARMAE model to
establish the extended vector .
(2) Normalize the augmented test sample using the training data,
then extract its sparse slow features .
(3) Figure out the and SPE monitoring indices of the augmented test
sample according to Eqs. (34) and (35).
(4) Compare the calculated statistics and SPE with their control
limits to alert the AHU system fault.
5.2. The STLSTM based fault identification
(1) Gather the detected fault samples to build fault snapshot dataset .
Then, extract the sparse slow features of the dataset and classes
historical fault datasets using the STBDSFA model.
(2) Train the STLSTM classifier using the sparse slow features of
historical fault datasets until the iterative optimization is
completed.
(3) Classify the sparse slow features of the dataset by employing the
established STLSTM classifier.
(4) Recognize the pattern of detected fault according to Eq. (38), by
calculating the largest element’s index in the probability vector .
6. The experiments and comparisons
6.1. Introduction of the experimental data
6.1.1. Description of the normal datasets
The developed STBDSFA and STLSTM based FDD scheme is exam-
ined on the AHU system datasets afforded by the ASHRAE research pro-
ject No. RP-1312 [56,57]. This project carried out a suite of on-site ex-
periments to emulate the time-varying dynamic natures of the AHU sys-
tem serving four zones. Thus, the ASHRAE project No. RP-1312 can of-
fer multiple fault patterns and normal operating experimental datasets.
More details about the project No. RP-1312 can be found in the litera-
ture [56,57]. The identical AHU-A and AHU-B systems are used in the
experiment. To produce a diversity of fault datasets, different fault pat-
terns are introduced to the AHU-A system. The AHU-B system operates
normally to achieve training datasets. The AHU-A and AHU-B systems
work for four zones, i.e., the zones A and B in the east, south, west and
inner. For a fair comparison, the zones A and B possess the same outside
thermal loadings. The AHU-A and AHU-B systems both run from 6:00 to
18:00 and stop working from 18:00 to 6:00 during a day. In spring,
summer and winter, the datasets of different faults and normal operat-
ing condition are gathered under the actual outdoor and indoor situa-
tions.
The sixteen crucial variables which contains the AHU system vari-
ables, set values, control signals and actuators’status, are chosen as the
monitored variables according to the references [7,19,56–58]. The de-
tailed description of these sixteen original monitored variables is given
out in Table 1. To handle the AHU system’s batch-wise dynamics during
a long-term operating time, the thirteen normal datasets
given in Table 2 are utilized to establish a three-way modeling matrix
[19,56,58]. These normal operating batches are collected from thirteen
different running days and the corresponding stored time is also re-
vealed in Table 2.
6.1.2. Description of the fault patterns
As displayed in Table 3, four typical types of the AHU system fault
are chosen to verify the FDD performance of the proposed STBDFSA
and STLSTM based method [19,56,58]. These four fault types involve
cooling coil valve stuck, heating coil valve leaking, and outdoor air
damper stuck or leaking. More specifically, each fault type also includes
multiple fault patterns according to its different fault intensities. For in-
stance, the fault type of heating coil valve leaking can be further di-
vided into three fault patterns in accordance with different fault intensi-
ties, i.e., 0.4 GPM, 1.0 GPM and 2.0 GPM. Therefore, in our work, a to-
tal of eleven fault patterns of the AHU system typical faults are adopted
to assess the effectiveness of the developed FDD scheme in this experi-
ment.
With the help of the AHU system’s operating mechanism and con-
trol strategies introduced in Section 2.1, the faulty and main affected
variables related to the abovementioned four fault types are further an-
alyzed. For the outdoor air damper stuck or leaking fault, the faulty
variable is the outdoor air damper control signal ( ); the main affected
variables are both the supply air temperature ( ), mixed air tempera-
ture ( ) and supply air duct static pressure ( ). The faulty variables of
the heating coil valve leaking and cooling coil valve stuck may be re-
spectively the heating coil valve control signal ( ) and the cooling coil
valve control signal ( ); the main affected variables are both the sup-
ply air temperature ( ) and return air temperature ( ).
In our work, we resample the normal and fault experimental data at
3 min interval to reduce the FDD methods’computation complexity.
Hence, there are a total of 480 samples in both the fault and normal
batch run datasets used in the simulation. To evaluate the fault detec-
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 17
Table 6
The comparison of the FDRs for the five algorithms.
Fault Pattern SPCA SDSFA KLPP STBSFA STBDSFA
F1 90.50% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
F2 74.02% 79.89% 93.58% 90.50% 92.46% 87.71% 100.00% 100.00% 100.00% 100.00%
F3 72.63% 78.77% 83.52% 87.43% 95.25% 63.69% 94.07% 97.46% 100.00% 100.00%
F4 74.86% 82.40% 81.00% 90.50% 97.21% 99.16% 97.46% 96.61% 100.00% 100.00%
F5 59.50% 72.07% 66.48% 81.23% 79.61% 66.48% 87.29% 89.83% 94.92% 97.46%
F6 70.67% 77.37% 81.84% 86.87% 98.06% 65.92% 88.14% 95.76% 97.46% 100.00%
F7 82.68% 84.92% 90.22% 91.90% 100.00% 92.18% 94.92% 99.15% 100.00% 100.00%
F8 63.69% 66.20% 66.48% 72.91% 86.87% 82.40% 83.05% 91.53% 100.00% 100.00%
F9 89.39% 100.00% 98.60% 100.00% 100.00% 99.44% 100.00% 100.00% 100.00% 100.00%
F10 62.85% 64.25% 67.04% 75.14% 91.39% 77.37% 89.83% 88.98% 100.00% 100.00%
F11 76.26% 87.71% 85.48% 92.18% 96.65% 100.00% 96.61% 100.00% 100.00% 100.00%
tion effect of the STBDSFA method, the test datasets are made up by
combining the first 120 normal samples with the first 360 fault samples
in related fault batch dataset. It should be noted that these normal sam-
ples and the corresponding fault samples are measured and stored in an
identical day. Therefore, the eleven fault patterns exhibited in Table 3
occur at the 121-th sample in the test datasets.
To diagnose the alarmed fault, a fault snapshot dataset is established
by utilizing the first 240 detected fault samples. And the remaining 240
data points in corresponding fault batch dataset are considered as his-
torical dataset. The sparse slow features of all the historical fault
datasets extracted by the STBDSFA are first utilized to train the sug-
gested STLSTM classifier, then the calculated sparse slow features of
fault snapshot dataset are fed to the trained classifier to diagnose the
fault pattern.
6.1.3. Analysis of the extracted sparse loading vectors
In the proposed STDBSFA algorithm, each data point is augmented
with its previous two samples. So the first sixteen sparse loading vectors
(SLVs) in the established sparse loading matrix are chosen to illustrate
the retained key variables with nonzero elements for the fault detection
and diagnosis. As displayed in Table 4, the selected key variables in
these sixteen SLVs cover all the sixteen original variables. Furthermore,
these SLVs explicitly exhibit the meaningful sparse characteristics
closely related to the control loops and process mechanisms. For in-
stance, according to the above analysis of faulty and main affected vari-
ables, the SLV1 and SLV4 both correctly represent the effect of the out-
door air damper control signal ( ) on the supply air temperature ( ),
mixed air temperature ( ) and supply air duct static pressure ( ). The
SLV3 and SLV16 indicate the influence of the heating coil valve control
signal ( ) on the supply air temperature ( ) and return air tempera-
ture ( ). The SLV7 and SLV10 reveal the effect of the cooling coil valve
control signal ( ) on the supply air temperature ( ) and return air
temperature ( ). The variable selection results shown in Table 4 prove
that the extracted SLVs can effectively choose the key variables with
significant physical meanings for detecting and diagnosing the AHU
system fault.
6.2. Comparative approaches and parameter setting
The effectiveness and feasibility of the developed STBDSFA ap-
proach is contrasted with the sparse PCA (SPCA), the sparse DSFA
(SDSFA), the kernel locality preserving projection (KLPP) [12,59]
which does not use the feature sparse representation technique and the
sparse three-way data based SFA (STBSFA) approaches. For the SPCA,
SDSFA and KLPP, training data is established using only one normal
dataset stored during a running day. For the STBDSFA, the time lags d
used in the ARMAE model is selected as 2 in line with the literatures
[48,49]. While the STBDSFA’s nearest neighbor number lis determined
as 3 by experience. For a fair comparison, the value of time lags din the
SDSFA is also chosen as 2, and the nearest neighbor number lin the ST-
BSFA is chosen as 3. For the KLPP, the Gaussian kernel function is uti-
lized and the kernel parameter is chosen to be 300. When establishing
the adjacency graph, the number of the nearest neighbors lis set to be
3. In the STBDSFA, STBSFA, KLPP, SDSFA and SPCA, the principal com-
ponents possessing contribution are kept. The control limits of
and SPE are determined by confidence level.
The fault identification effect of the suggested STLSTM classifier is
compared with some conventional classification algorithms, i.e., the
support vector machine (SVM), the temporal convolutional network
(TCN) [60,61], the convolutional neural network (CNN) and the deep
belief network (DBN). For the TCN based fault diagnosis approach, the
original eleven historical fault datasets without extracting
the sparse features are adopted to train the TCN classifier. While the
sparse slow features derived by the STBDSFA monitoring model are im-
ported to the SVM, CNN and DBN classifiers, respectively. And, these
modified classifiers are respectively called the STSVM, STCNN and
STDBN in this paper.
To be specific, a LSTM with two layers is employed in the STLSTM
classifier. During the STLSTM training, the batch size is selected as 128,
the numbers of epochs and hidden units are both chosen as 600, the
learning rate is determined as 0.001 by trial and error. In addition, the
Adam optimizer [18,54] is used to select the optimal parameters. The
STCNN consists of three convolution layers where the numbers of nodes
are respectively determined as 32, 64, 128, and the activation function
is used as Gaussian activation function. The structure of the STDBN con-
tains three hidden layers, the numbers of neurons in the 1st, 2nd and
3rd hidden layers are selected as 600, 400, 200, and the Gaussian func-
tion is also the activation function. To be fair, for the STCNN and
STDBN, the batch size and the epoch number are also chosen as 128 and
600 respectively, which are the same as that of the STLSTM. In addi-
tion, the learning rates of the STCNN and STDBN are optimized to be
both 0.0001 utilizing the same parameter determining method used in
the proposed STLSTM, i.e., by trial and error. In the TCN, the learning
rate is selected as 0.001 by trial and error, the batch size is set to be 64,
the number of hidden units is chosen as 500 and the expansion factor is
set as 2. In the STSVM classifier, the kernel function is determined as
Gaussian function and the kernel parameter is 1000, while the penalty
factor is chosen as 80.
6.3. The FDD assessment indices
To estimate the monitoring effect of the SPCA, SDSFA, KLPP, STB-
SFA and STBDSFA, the fault detection rate (FDR) and the fault detec-
tion time (FDT) are employed in our work. The FDR index is calculated
as the rate of detected fault data points over total real fault data points.
The fault is found after 3 continuous samples go beyond the control lim-
its of monitoring statistics. The FDT index is defined as the first number
of the earliest detected 3 continuous fault samples. Earlier FDT and
CORRECTED PROOF
18 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Fig. 12. The averaged FDRs of the five monitoring methods for faults F1 F11.
higher FDR manifest better monitoring effectiveness. Details about
these two performance indices can be found in the literatures [19,23,
62].
To assess the fault identification effect of the STSVM, TCN, STCNN,
STDBN and STLSTM classification methods, the fault diagnosis rate and
average fault diagnosis rate are applied in this paper. The fault diagno-
sis rate is calculated as the rate of correctly classified samples in one
fault pattern over the corresponding total real fault samples. The aver-
age fault diagnosis rate is the rate of all fault diagnosis rates’sum over
the number of all fault patterns. The higher value of fault diagnosis rate
implies better fault diagnosis performance for a certain fault pattern,
while the higher value of average fault diagnosis rate means more per-
suasive and much clearer identification result for the all fault patterns.
6.4. Comparison of the FDD results
6.4.1. The FDD results of the fault F5
The monitoring charts of the STBDSFA, STBSFA, KLPP, SDSFA and
SPCA for the fault F5 are shown in Fig. 8 The fault detection results of
the SPCA in Fig. 8(a) reveal the and SPE statistics respectively detect
the fault F5 at the 130-th and 127-th samples. Whereas, the SPCA’s
and SPE statistics both have the lowest FDRs because numerous fault
data points are below their corresponding control limits after the 130-
th and 127-th samples. According to Fig. 8(b), the SDSFA obtains an im-
proved monitoring results, where its statistic warns of the fault F5 at
the 241-th sample while its SPE statistic detects the fault F5 at the 124-
th sample. However, the SDSFA’sSPE statistic still possesses a relatively
large missing detected fault samples, because many fault data points go
down below the control limit after the 124-th data point. From Fig. 8
(c), it can be seen that the statistic of the KLPP detects the fault F5 at
the 123-th sample and the SPE statistic gives an alarm of the fault at the
241-th sample. But the KLPP’s statistic still results in a low FDR be-
cause many real fault samples are wrongly treated as the normal sam-
ples without giving fault alarms. From Fig. 8(d), the STBSFA’s statis-
tic gains much better fault detection effect in comparison with the
SPCA, SDSFA and KLPP, where fault F5 is alarmed at the 121-th data
point with fewer missing detected fault data points. Besides, for the ST-
BSFA, the SPE statistic also alerts the fault F5 at the 121-th data point
with only a few missing detected fault data points. In comparison with
the fault detection charts of the above four approaches, the proposed
STBDSFA displays much enhanced monitoring results. From Fig. 8(e),
its and SPE statistics find the fault F5 at the 121-th data point simul-
taneously with the least missing detected fault data points, which indi-
cates the STBDSFA achieves the earliest and the best fault detection ef-
fect. Therefore, the STBDSFA is the most valid method to detect the
fault F5 in the five monitoring methods.
The STSVM, TCN, STCNN, STDBN and STLSTM classifiers are ap-
plied to recognize the pattern of fault F5. After fault snapshot dataset
is constructed, the fault diagnosis rates of these four classification
methods are estimated. Specifically, the fault diagnosis rates of the
STSVM, TCN, STCNN, STDBN and STLSTM are respectively
and , which demonstrates that the
STLSTM achieves the highest fault diagnosis rate among the five ap-
proaches. For a more intuitionistic analysis, the five different fault diag-
nosis rates are further illustrated in Fig. 9. Therefore, we can make the
conclusion that the STLSTM is superior over the STDBN, STCNN, TCN
and STSVM to diagnose the fault F5.
6.4.2. The FDD results of the Fault F8
The fault F8 is also adopted to evaluate the monitoring performance
of the five algorithms. From Fig. 10(a), the and SPE statistics of the
SPCA respectively detect fault F8 at the 195-th and 223-th samples. Fur-
thermore, these two statistics both wrongly regard many real fault data
points as normal samples. Thus, the SPCA displays the worst fault de-
tection performance. On the contrary, the SDSFA obtains a reformative
monitoring results in Fig. 10(b). The SDSFA’s statistic indicates the
fault F8 at the 241-th data point with no missing detected fault data
points after the fault is detected, and the SDSFA’sSPE statistic goes be-
yond its control limit at the 186-th sample with not too many missing
detected fault data points after the 186-th sample. As plotted in Fig. 10
(c), the monitoring results of the KLPP are further improved. Specifi-
cally, the fault F8 is alerted by its and SPE statistics at the 121-th and
168-th samples respectively, and fewer fault samples fall below the cor-
responding control limits after the fault is detected. In comparison with
the SPCA, SDSFA and KLPP, the STBSFA owns much better fault detec-
tion results in Fig. 10(d), where the and SPE statistics respectively
alert the fault F8 at the 128-th and 124-th samples. These two statistics
achieve much higher FDRs because much fewer fault data points go
down below the control limits after the 128-th and 124-th samples. In
Fig. 10(e), the results of the STBDSFA reveal both the and SPE statis-
tics detect fault F8 at the 121-th data point with the highest FDRs, i.e.,
. Consequently, the STBDSFA is more effective to alarm the fault
F8 than the SPCA, SDSFA, KLPP and STBSFA.
After establishing the snapshot dataset of fault F8, the fault diag-
nosis rates of the STSVM, TCN, STCNN, STDBN and STLSTM for the
dataset are computed as and .
It can be seen that the STSVM obtains the worst fault diagnosis perfor-
mance, while the TCN and STCNN have approximately the same fault
diagnosis effectiveness which needs to be further improved. Although
the fault diagnosis rate of the STDBN is higher than these of the TCN
and STCNN, the STDBN’s fault recognition performance is no better
than that of the STLSTM. On the contrary, the STLSTM reveals the best
identification result for the dataset with the highest fault diagnosis
rate. This attributes to the STLSTM’s superior ability to mine the long-
term dependencies contained in the derived sparse slow features.
6.4.3. The FDD results of the Fault F10
Fig. 11 illustrates the monitoring results for the fault F10. As given
in Fig. 11(a), the SPCA gains the worst fault detection charts, where its
statistic alerts fault F10 at the 237-th sample while the SPE statistic
detects fault F10 at the 121-th sample. In addition, both the SPCA’s
and SPE statistics bring about much lower FDRs because a number of
fault data points are below the control limits after detecting the fault.
On the contrary, from Fig. 11(b), the and SPE statistics of the SDSFA
respectively alarm fault F10 at the 121-th and 171-th samples. How-
ever, they still result in unsatisfying FDRs because lots of real fault data
points are wrongly regarded as normal data points. According to Fig. 11
(c), the statistic of the KLPP brings an improved fault detection re-
sult, where the fault F10 is detected at the 121-th sample with much
fewer missing alarmed fault samples. However, the KLPP’sSPE statistic
also gains an unsatisfying monitoring result because the fault F10 is
alarmed at the 149-th sample and many real fault samples are wrongly
treated as the normal samples. Compared with the monitoring charts of
the SPCA, SDSFA and KLPP, Fig. 11(d) reveals an enhanced fault detec-
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 19
Fig. 13. The comparison of confusion matrices for the five classification methods, (a) STSVM, (b) TCN, (c) STCNN, (d) STDBN, (e) STLSTM.
tion effect of the STBSFA. From Fig. 11(d), the STBSFA’s and SPE sta-
tistics respectively go beyond the control limits after the 128-th and
126-th samples. Additionally, because of fewer missing detected fault
data points, the and SPE statistics display much higher FDRs. The ST-
BDSFA’s and SPE statistics shown in Fig. 11(e) warn of fault F10 at
the 121-th data point simultaneously without missing detected fault
data points, which acts the earliest and the most precisely to alert fault
F10 in the five monitoring methods. This case study also proves the ex-
cellent fault detection effect of the STBDSFA over the SPCA, SDSFA,
KLPP and STBSFA.
The calculated values of the fault diagnosis rates for the STSVM,
TCN, STCNN, STDBN and STLSTM in this case turn out to be
and , respectively. Obviously, the
five approaches exhibit different diagnosis performances to classify the
fault snapshot dataset . The STSVM, TCN and STDBN gain much
worse fault diagnosis results in comparison with the STCNN and STL-
CORRECTED PROOF
20 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
Table 7
The comparison of fault diagnosis rates for the five classification methods.
Fault snapshot dataset STSVM TCN STCNN STDBN STLSTM
76.67% 91.67% 91.25% 89.58% 97.92%
83.83% 86.25% 85.42% 87.50% 91.67%
63.75% 84.58% 78.75% 88.75% 93.75%
72.92% 78.75% 71.25% 86.25% 92.08%
79.33% 81.67% 93.33% 77.50% 97.72%
75.17% 87.50% 81.25% 85.00% 98.33%
72.08% 72.91% 83.33% 87.10% 95.42%
76.25% 83.75% 86.67% 90.42% 96.25%
83.16% 85.41% 89.58% 91.67% 98.75%
64.50% 76.67% 87.92% 76.25% 94.16%
81.67% 83.33% 79.58% 86.67% 94.58%
75.40% 82.96% 84.38% 86.06% 95.51%
STM. In contrast, the STCNN gives much better classification effective-
ness as the corresponding fault diagnosis rate is . On the whole,
the STLSTM achieve the best diagnosis performance owning to that its
fault diagnosis rate is computed to be . Again, this displays the
superiority of the developed STLSTM classifier over the STDBN,
STCNN, TCN and STSVM classifiers when recognizing the fault pat-
terns.
6.4.4. The FDD results of the faults F1 F11
The monitoring results of the SPCA, SDSFA, KLPP, STBSFA and ST-
BDSFA for the eleven faults are assessed in Table 5 and Table 6. Note
that all the test faults are step-changing faults. Hence, as revealed in
Table 5, the five methods possess the capability to alert the majority of
the test faults immediately after being introduced at the 121-th sample.
More specifically, the SPCA results in some fault detection delays for
faults F1, F2, F6 and F8 F11, while it has no fault detection delay to
alarm the rest faults. The SDSFA’s statistic owns the fault detection
delays only to warn of faults F3, F6 and F8, while its SPE statistic pos-
sesses the time delays to give an alarm of faults F2 F8, F10 and F11.
The statistic of KLPP only reveals the fault detection delays to detect
faults F5 and F11, whereas the KLPP’sSPE statistic displays the time de-
lays to warn of faults F3 and F5 F10. Instead, the STBSFA and ST-
BDSFA achieve almost no fault detection delays to alert these eleven
test faults, which displays much earlier fault detection effect compared
with the SPCA, SDSFA and KLPP.
Nevertheless, the five methods exhibit different values of the FDR
index listed in Table 6. Because of having no ability to tackle the AHU
system’s two-directional dynamic properties, the SPCA and SDSFA both
wrongly regard real fault data points as the normal ones after detecting
the faults. This results in lower FDRs for the and SPE statistics. Al-
though the statistic of KLPP achieves higher FDRs for the faults
F1 F11 in comparison with the SPCA and SDADF, the KLPP’sSPE sta-
tistic still brings about unsatisfactory FDRs for the eleven fault patterns.
Compared with the SPCA, SDSFA and KLPP, the STBSFA reveals a fur-
ther improved fault detection performance to warn of faults F1 F11,
due to its superiority of handling the AHU system’s two-directional dy-
namics. Specifically, the STBSFA’s and SPE statistics both acquire the
FDR values above for the eleven faults. Furthermore, the and
SPE statistics of the STBDSFA achieve almost FDRs to detect the
eleven test faults, which are the highest among the four monitoring
methods. This is because the STBDSFA can settle the two-directional
dynamics as well as perform the data dynamic expansion and the fea-
ture sparse representation of the AHU system. Moreover, the average
FDRs of these five monitoring algorithms for the eleven test faults are
visualized in Fig. 12, which also certifies the remarkable monitoring
performance of the STBDSFA over the SPCA, SDSFA, KLPP and STBSFA.
After the faults F1 F11 are detected, fault diagnosis rates of the
STSVM, TCN, STCNN, STDBN and STLSTM for the datasets
are computed and displayed in Fig. 13 and Table 7. To be specific, the
confusion matrices of the five classifiers are exhibited in Fig. 13. As
shown in Fig. 13, the figures in the deep orange squares indicate the
numbers of correctly classified data points, while the figures in the light
orange squares represent the numbers of wrongly classified data points.
From Fig. 13, for example, the figures in the light orange squares of the
seventh rows in Fig. 13(a), (b), (c) and (d) are much larger than those of
the seventh row in Fig. 13(e), which means that compared with the
STLSTM, more fault samples belonging to the fault pattern F7 are
wrongly recognized as the other fault patterns’samples by the STSVM,
TCN, STCNN and STDBN. Furthermore, there are more light orange
squares in Fig. 13(a), (b), (c) and (d) than those in Fig. 13(e), which il-
lustrates that the STSVM, TCN, STCNN and STDBN wrongly classify
more fault samples of the patterns F1 F11 into the other fault patterns
than the STLSTM.
To make a quantitative comparison, the fault diagnosis rates of the
STSVM, TCN, STCNN, STDBN and STLSTM for the datasets
are given out in Table 7. Besides, in order to make further comparative
analysis, the average fault diagnosis rates ( ) of the five classifica-
tion methods on the datasets are also listed in the last row of
Table 7. According to Table 7, the values of the STSVM, TCN,
STCNN, STDBN and STLSTM are and
, respectively. Thus, our proposed STLSTM classifier achieves the
highest average fault diagnosis rate for the snapshot datasets
among the five classification methods. In detail, compared with the
STSVM, TCN, STCNN and STDBN, the developed STLSTM also performs
much better to diagnose the patterns of the eleven detected faults. For
instance, the fault diagnosis rate of the snapshot dataset is
for the STLSTM, in contrast to only for the STDBN, for
the STCNN, for the TCN and for the STSVM. Similarly,
the fault diagnosis rate of the snapshot dataset is for the STL-
STM, in comparison with only for the STDBN, for the
STCNN, for the TCN and even for the STSVM. From Fig.
13 and Table 7, it can be concluded that the presented STLSTM ap-
proach is the most excellent for classifying the AHU system’s dynamic
time-varying sparse slow features. This is because the long-term depen-
dencies caught by the STLSTM take effect for the AHU system’s fault di-
agnosis task.
7. Conclusions
A new STBDSFA and STLSTM based scheme is proposed to accom-
plish the AHU system FDD. As far as the authors know, it is the first
time to integrate the feature sparse representation technique into the
SFA based method to monitor the AHU system. Specifically, in order to
enhance the SFA based monitoring model’s interpretability and im-
prove the fault detection ability, the sparse representation method is in-
fused into the SFA based model to propose the STBDSFA algorithm. In
the STBDSFA model, the penalization is imposed on the loading vec-
tor’s nonzero elements to shrink the coefficients of insignificant vari-
ables to be zero. In this way, the key variables responsible for the fault
detection are automatically and efficiently chosen. With the help of the
SGEV method and the deflation procedure, a series of sparse loading
vectors are sequentially figured out. Besides, the other two innovations
are formulated as follows. One innovation is to build the TBDSFA model
to cope with the AHU system’s two-directional dynamic properties, be-
fore constructing the feature sparse representation based SFA approach.
To be specific, the multiway data analysis is used to handle the batch-
wise dynamic property, and the dynamic SFA infused with the ARMAE
model is adopted to figure out the time-wise dynamic nature as well as
the auto-correlation relationships. The other innovation is to address
the troublesome issue of diagnosing the detected faults’patterns. After
the sparse slow features of the snapshot dataset and multiple historical
datasets are extracted by the STBDSFA monitoring model, a novel STL-
STM method is suggested to classify the sparse slow features, by consid-
ering the STLSTM’s capability to tackle the time-varying dynamic na-
ture of sparse slow features. The experiments and comparisons proves
CORRECTED PROOF
H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241 21
the outstanding monitoring effect of the developed STBDSFA algo-
rithm. In addition, the experimental results also validate the excellent
fault diagnosis effect of the presented STLSTM classifier to diagnose the
detected faults.
As aforesaid, the AHU system has different running modes due to
the time-wise dynamic behaviors. In the future, to enhance the AHU
system’s fault detection effectiveness, other modified SFA models are
suggested to be investigated to display and use the information of dif-
ferent running modes. Besides, to improve the fault identification capa-
bility, the LSTM based classification approaches need to be further
amended to better address the time-varying dynamic property of the ex-
tracted slow features.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influ-
ence the work reported in this paper.
Acknowledgments
This study is partly supported by the National Natural Science Foun-
dation of China (62003191, 62076150, 61903226, 62133008), the Key
Research and Development Plan of Shandong Province for Major Scien-
tific and Technological Innovation Project (2021CXGC011205), the
Taishan Scholar Project of Shandong Province (TSQN201812092), the
Natural Science Foundation of Shandong Province (ZR2020QF072), the
Key Research and Development Program of Shandong Province
(2019JZZY010115, 2019JZZY010120), the Youth Innovation Technol-
ogy Project of Higher School in Shandong Province (2019KJN005), and
the Doctoral Research Fund Project of Shandong Jianzhu University
(XNBS1821).
References
[1] Maryam Sadat Mirnaghi, Fariborz Haghighat, Fault detection and diagnosis of
large scale hvac systems in buildings using data-driven methods: A comprehensive
review, Energy Build. 229 (2020) 110492.
[2] Mojtaba Kordestani, Ali Akbar Safavi, Mehrdad Saif, Recent survey of large-scale
systems: architectures, controller strategies, and industrial applications, IEEE Syst.
J. 15 (4) (2021) 5440–5453.
[3] Mojtaba Kordestani, Mehrdad Saif, Marcos E. Orchard, Roozbeh Razavi-Far,
Khashayar Khorasani, Failure prognosis and applications a survey of recent
literature, IEEE Trans. Reliab. 70 (2) (2019) 728–748.
[4] A.P. Rogers, F. Guo, B.P. Rasmussen, A review of fault detection and diagnosis
methods for residential air conditioning systems, Build. Environ. 161 (2019)
106236.
[5] Kuei-Peng Lee, Bo-Huei Wu, Shi-Lin Peng, Deep learning based fault detection
and diagnosis of air-handling units, Build. Environ. 157 (2019) 24–33.
[6] Yang Zhao, Jin Wen Fu, Xuebin Yang Xiao, Shengwei Wang, Diagnostic bayesian
networks for diagnosing air handling units faults part i: Faults in dampers, fans,
filters and sensors, Appl. Therm. Eng. 111 (2017) 1272–1286.
[7] Ke Yan, Chaowen Zhong, Zhiwei Ji, Jing Huang, Semi-supervised learning for
early detection and diagnosis of various air handling unit faults, Energy Build. 181
(2018) 75–83.
[8] Guannan Li, Yunpeng Hu, An enhanced pca based chiller sensor fault detection
method using ensemble empirical mode decomposition based denoising, Energy
Build. 183 (2019) 311–324.
[9] Dan Li, Yuxun Zhou, Hu. Guoqiang, Costas J Spanos, Optimal sensor
configuration and feature selection for ahu fault detection and diagnosis, IEEE
Trans. Industr. Inf. 13 (3) (2016) 1369–1380.
[10] Zhanwei Wang, Lin Wang, Kunfeng Liang, Yingying Tan, Enhanced chiller fault
detection using bayesian network and principal component analysis, Appl. Therm.
Eng. 141 (2018) 898–905.
[11] Xu. Yubo, Jie Liu, High-speed train fault detection with unsupervised causality-
based feature extraction methods, Adv. Eng. Inform. 49 (2021) 101312.
[12] Muhammad Zohaib Hassan Shah, Lisheng Hu, Zahoor Ahmed, Modified lpp
based on riemannian metric for feature extraction and fault detection,
Measurement 193 (2022) 110923.
[13] Haitao Wang, Youming Chen, A robust fault detection and diagnosis strategy for
multiple faults of vav air handling units, Energy Build. 127 (2016) 442–451.
[14] Zhile Wang, Jianhua Yang, Guo Yu, Unknown fault feature extraction of rolling
bearings under variable speed conditions based on statistical complexity measures,
Mech. Syst. Signal Process. 172 (2022) 108964.
[15] Yunus Emre Karabacak, Nurhan Gürsel Özmen, Common spatial pattern-based
feature extraction and worm gear fault detection through vibration and acoustic
measurements, Measurement 187 (2022) 110366.
[16] Mojtaba Kordestani, Milad Rezamand, Marcos E. Orchard, Rupp Carriveau,
David Ting, Luis Rueda, Mehrdad Saif, New condition-based monitoring and fusion
approaches with a bounded uncertainty for bearing lifetime prediction, IEEE Sens.
J. 22 (9) (2022) 9078–9086.
[17] Milad Rezamand, Mojtaba Kordestani, Marcos Orchard, Rupp Carriveau, David
Ting, Mehrdad Saif, Condition monitoring and failure prognostic of wind turbine
blades, in: 2021 IEEE International Conference on Systems, Man, and Cybernetics
(SMC), 2021, pp. 1711–1718.
[18] Ding Li, Donghui Li, Chengdong Li, Lin Li, Long Gao, A novel data temporal
attention network based strategy for fault diagnosis of chiller sensors, Energy Build.
198 (2019) 377–394.
[19] Hanyuan Zhang, Chengdong Li, Ding Li, Yunchu Zhang, Wei Peng, Fault
detection and diagnosis of the air handling unit via an enhanced kernel slow
feature analysis approach considering the time-wise and batch-wise dynamics,
Energy Build. 253 (2021) 111467.
[20] Hanyuan Zhang, Xuemin Tian, Xiaogang Deng, Yuping Cao, Batch process fault
detection and identification based on discriminant global preserving kernel slow
feature analysis, ISA Trans. 79 (2018) 108–126.
[21] Shumei Zhang, Chunhui Zhao, Slow feature analysis based batch process
monitoring with comprehensive interpretation of operation condition deviation
and dynamic anomaly, IEEE Trans. Industr. Electron. 66 (5) (2018) 3773–3783.
[22] Hanyuan Zhang, Xuemin Tian, Xiaogang Deng, Yuping Cao, Multiphase batch
process with transitions monitoring based on global preserving statistics slow
feature analysis, Neurocomputing 293 (2018) 64–86.
[23] Chao Shang, Fan Yang, Biao Huang, Dexian Huang, Recursive slow feature
analysis for adaptive monitoring of industrial processes, IEEE Trans. Industr.
Electron. 65 (11) (2018) 8895–8905.
[24] Yun Kong, Zhaoye Qin, Tianyang Wang, Qinkai Han, Fulei Chu, An enhanced
sparse representation based intelligent recognition method for planet bearing fault
diagnosis in wind turbines, Renew. Energy 173 (2021) 987–1004.
[25] Jie Yang, Weimin Bao, Yanming Liu, Xiaoping Li, Junjie Wang, Yue Niu, Jin Li,
Joint pairwise graph embedded sparse deep belief network for fault diagnosis, Eng.
Appl. Artif. Intell. 99 (2021) 104149.
[26] Ioannis A. Kougioumtzoglou, Ioannis Petromichelakis, Apostolos F. Psaros,
Sparse representations and compressive sampling approaches in engineering
mechanics: A review of theoretical concepts and diverse applications, Probab. Eng.
Mech. 61 (2020) 103082.
[27] Yi Liu, Jiusun Zeng, Lei Xie, Shihua Luo, Su. Hongye, Structured joint sparse
principal component analysis for fault detection and isolation, IEEE Trans. Industr.
Inf. 15 (5) (2018) 2721–2731.
[28] Yunyun Hu, Yue Wang, Chunhui Zhao, A sparse fault degradation oriented fisher
discriminant analysis (fdfda) algorithm for faulty variable isolation and its
industrial application, Control Eng. Practice 90 (2019) 311–320.
[29] Wanke Yu, Chunhui Zhao, Sparse exponential discriminant analysis and its
application to fault diagnosis, IEEE Trans. Ind. Electron. 65 (7) (2018) 5931–5940.
[30] Chengyi Zhang, Yu. Jianbo, Lyujiangnan Ye, Sparsity and manifold regularized
convolutional auto-encoders based feature learning for fault detection of
multivariate processes, Control Eng. Practice 111 (2021) 104811.
[31] Lakhdar Aggoun, Yahya Chetouani, Fault detection strategy combining narmax
model and bhattacharyya distance for process monitoring, J. Franklin Inst. 358 (3)
(2021) 2212–2228.
[32] Liu Mei, Huaguan Li, Yunlai Zhou, Weilun Wang, Feng Xing, Substructural
damage detection in shear structures via armax model and optimal subpattern
assignment distance, Eng. Struct. 191 (2019) 625–639.
[33] M. Kordestani, M. Rezamand, M. Orchard, R. Carriveau, D.S.K. Ting, M. Saif,
Planetary gear faults detection in wind turbine gearbox based on a ten years
historical data from three wind farms, IFAC-PapersOnLine 53 (2) (2020)
10318–10323.
[34] Qingchao Jiang, Furong Gao, Xuefeng Yan, Hui Yi, Multiobjective two-
dimensional cca-based monitoring for successive batch processes with industrial
injection molding application, IEEE Trans. Industr. Electron. 66 (5) (2018)
3825–3834.
[35] Jinlin Zhu, Yuan Yao, Furong Gao, Multiphase two-dimensional time-slice
dynamic system for batch process monitoring, J. Process Control 85 (2020)
184–198.
[36] Somayeh Mirzaei, Jia-Lin Kang, Kuang-Yi Chu, A comparative study on long
short-term memory and gated recurrent unit neural networks in fault diagnosis for
chemical processes using visualization, J. Taiwan Inst. Chem. Eng. 130 (2021)
104028.
[37] Yang Xing, Chen Lv, Dynamic state estimation for the advanced brake system of
electric vehicles by using deep recurrent neural networks, IEEE Trans. Industr.
Electron. 67 (11) (2019) 9536–9547.
[38] Yongming Han, Ning Ding, Zhiqiang Geng, Zun Wang, Chong Chu, An optimized
long short-term memory network based fault diagnosis model for chemical
processes, J. Process Control 92 (2020) 161–168.
[39] Weike Sun, Antonio R.C. Paiva, Xu. Peng, Anantha Sundaram, Richard D. Braatz,
Fault detection and identification using bayesian recurrent neural networks,
Comput. Chem. Eng. 141 (2020) 106991.
[40] Suhrid Deshmukh, Stephen Samouhos, Leon Glicksman, Leslie Norford, Fault
detection in commercial building vav ahu: A case study of an academic building,
Energy Build. 201 (2019) 163–173.
[41] Cheng Fan, Xuyuan Liu, Peng Xue, Jiayuan Wang, Statistical characterization of
semi-supervised neural networks for fault detection and diagnosis of air handling
units, Energy Build. 234 (2021) 110733.
CORRECTED PROOF
22 H. Zhang et al. / Energy & Buildings xxx (xxxx) 112241
[42] Xinrui Gao, Yuri A.W. Shardt, Dynamic system modelling and process monitoring
based on long-term dependency slow feature analysis, J. Process Control 105
(2021) 27–47.
[43] YaPing Huang, JiaLi Zhao, YunHui Liu, SiWei Luo, Qi Zou, Mei Tian, Nonlinear
dimensionality reduction using a temporal coherence principle, Inf. Sci. 181 (16)
(2011) 3284–3307.
[44] Bharath K. Sriperumbudur, David A. Torres, Gert R.G. Lanckriet, A majorization-
minimization approach to the sparse generalized eigenvalue problem, Mach. Learn.
85 (1) (2011) 3–39.
[45] Shiyi Bao, Lijia Luo, Jianfeng Mao, Di Tang, Improved fault detection and
diagnosis using sparse global-local preserving projections, J. Process Control 47
(2016) 121–135.
[46] Jianhua Dai, Ye Liu, Jiaolong Chen, Feature selection via max-independent ratio
and min-redundant ratio based on adaptive weighted kernel density estimation, Inf.
Sci. 568 (2021) 86–112.
[47] Weichao Dong, Hexu Sun, Jianxin Tan, Zheng Li, Jingxuan Zhang, Huifang Yang,
Regional wind power probabilistic forecasting based on an improved kernel density
estimation, regular vine copulas, and ensemble learning, Energy 238 (2022)
122045.
[48] Kai Zhong, Dewei Ma, Min Han, Distributed dynamic process monitoring based
on dynamic slow feature analysis with minimal redundancy maximal relevance,
Control Eng. Practice 104 (2020) 104627.
[49] Jeremiah Corrigan, Jie Zhang, Integrating dynamic slow feature analysis with
neural networks for enhancing soft sensor performance, Comput. Chem. Eng. 139
(2020) 106842.
[50] Lijia Luo, Shiyi Bao, Jianfeng Mao, Di Tang, Fault detection and diagnosis based
on sparse pca and two-level contribution plots, Ind. Eng. Chem. Res. 56 (1) (2017)
225–240.
[51] Lester W. Mackey, Deflation methods for sparse pca, NIPS 21 (2008) 1017–1024.
[52] Jiaqing Gao, Hua Han, Zhengxiong Ren, Yuqiang Fan, Fault diagnosis for
building chillers based on data self-production and deep convolutional neural
network, J. Build. Eng. 34 (2021) 102043.
[53] Zufan Zhang, Zongming Lv, Chenquan Gan, Qingyi Zhu, Human action
recognition using convolutional lstm and fully-connected lstm with different
attentions, Neurocomputing 410 (2020) 304–316.
[54] Zihan Chang, Yang Zhang, Wenbo Chen, Electricity price prediction based on
hybrid model of adam optimized lstm neural network and wavelet transform,
Energy 187 (2019) 115804.
[55] E. Balaji, D. Brindha, Vinodh Kumar Elumalai, R. Vikrama, Automatic and non-
invasive parkinson’s disease diagnosis and severity rating using lstm network,
Appl. Soft Comput. 108 (2021) 107463.
[56] J. Wen, S. Li, Rp-1312 tools for evaluating fault detection and diagnostic
methods for air handling units (Tech. Rep.), ASHRAE, 2012.
[57] Price, B.A., Smith, T.F., 2003. Development and validation of optimal strategies
for building hvac systems. Dept. Mech. Eng., Univ. Iowa, Iowa City, IA, USA, Tech.
Rep. ME-TEF-03-001.
[58] J. Granderson, G.J. Lin, Inventory of data sets for afdd evaluation, Building
Technology and Urban Systems Division, Lawrence Berkeley National Laboratory,
2019.
[59] Yujie Zhou, Ke Xu, Fei He, Di He, Nonlinear fault detection for batch processes
via improved chordal kernel tensor locality preserving projections, Control Eng.
Practice 101 (2020) 104514–104528.
[60] Chengdong Li, Cunxiao Shen, Hanyuan Zhang, Hongchang Sun, Songping Meng,
A novel temporal convolutional network via enhancing feature extraction for the
chiller fault diagnosis, J. Build. Eng. 42 (2021) 103014.
[61] Weinan Li, Jin Gou, Zongwen Fan, Session-based recommendation with temporal
convolutional network to balance numerical gaps, Neurocomputing 493 (7) (2022)
166–175.
[62] Peipei Cai, Xiaogang Deng, Incipient fault detection for nonlinear processes
based on dynamic multi-block probability related kernel principal component
analysis, ISA Trans. 105 (2020) 210–220.
