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Tolerant control for multiple faults of sensors in VAV systems
Energy Conversion and Management 48 (2007) 764–777 www.elsevier.com/locate/enconman
Tolerant control for multiple faults of sensors in VAV systems Zhimin Du *, Xinqiao Jin School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 27 February 2006; accepted 10 September 2006 Available online 14 November 2006
Abstract Principal component analysis, joint angle plots and reconstruction schemes are presented in this paper to detect, isolate and evaluate multiple sensors faults occurring in variable air volume (VAV) systems. Multi-level principal component analysis models, including system level and local level, are built to detect multiple faults occurring in VAV systems. As the initial detection, a system level model is used to discover the abnormalities in view of the whole systems. Two local level models are used to further confirm the occurrence of the faults. Moreover, with the multiple faults separated into different locations by the two local level detection models, joint angle plots are used, respectively, to isolate the faults one by one. Finally, the reconstruction scheme is used to estimate the magnitude of the bias to recover from the faulty operation. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Tolerant control; Multiple faults; Sensor; Variable air volume; Detection; Isolation; Reconstruction
1. Introduction Recently, to satisfy the increasing demands on indoor comfort and environment, heating, ventilation and air conditioning systems have become more and more complex. Inadequate control of systems may result in poor indoor environment and more energy consumption. There are two key factors to ensure good performance of VAV systems: one is suitable control strategies and the other is reliable measurements. There are many control loops with sensors in VAV systems. Typical local controls associated with sensors are air handling unit supply air temperature control with temperature sensor, outdoor air control with flow rate sensor, static pressure control with pressure sensor etc. The normal or even optimal operation of the systems strongly relies on the capacity of these controls, while these controls strongly rely on the reliable measurements of their sensors. Unfortunately, biasing of the sensors is usually inevitable after VAV systems are used for a relatively long term. Therefore, *
Corresponding author. Tel.: +86 21 62933351; fax: +86 21 62932601. E-mail addresses: [email protected] (Z. Du), [email protected] (X. Jin). 0196-8904/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.09.007
the sensor with a fault may lead to the real operation condition of the systems being misrepresented. Furthermore, it may mislead the controls so as to execute a wrong/unsuitable operation. As a result, inaccurate measurements may lead to worsened indoor air quality or wasted energy consumption, although there are suitable control strategies. Finding a suitable method to detect, diagnose or even tolerate the fault occurring in the VAV systems is a significant target. Studies of fault detection and diagnosis in air conditioning systems started in the 1980s [1,2] and is being paid more attention recently. Annex25 [3], Annex34 [4] and many more studies [5–12] are concerned about all kinds of faults in air conditioning systems. There have been two main methods in the past to detect or to diagnose the sensor faults. One is based on the model of the systems, and the other is based on knowledge. The model based method is to obtain normal values of the parameters through the models of the systems first. Whether or not the fault occurs can be judged by comparing the actual value with the normal one. The premise of this method is that an accurate mathematical model can be obtained. Stylianou and Nikanour [13] used a first order model to detect faults of temperature sensors by comparing
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777
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Nomenclature x P R T M H h q C f F PCA SPE PCS RS x*
measure vector loading matrix model projection matrix temperature (°C) flow rate (kg/s) enthalpy (kJ/kg) relative humidity (%) heat exchange rate (kJ/kg) control signal a fault vector faults matrix principal component analysis square prediction error principal component subspace residue subspace real or estimated value of a measure vector
Greek symbols da threshold for SPE the actual temperature decay with the model output using hypothesis testing. Wang and Wang [14] developed a law based sensor fault diagnosis strategy, which took the commonly used temperature and flow rate sensors in the chilling plant into account at the same time. Other researchers, such as Howell and Maddison [15] and Haves et al. [16], paid attention to this method as well. The knowledge based approaches, such as expert systems [17], neural net [18] and fuzzy theory [19], are also widely used to detect and diagnose faults. Lee et al. [20] investigated the fault diagnosis in a simulated air handling unit using a two stage artificial neural network. After that, Wang and Chen [21] developed a strategy based on a neural network model to diagnose measurement faults of the flow rate sensors of outdoor air and supply air. For some simple systems, it is easy to detect and diagnose faults using such experiential knowledge. However, it is challenging to apply the knowledge based approach to large scale systems such as VAV systems because it is difficult to develop the complicated expert system. Recently, Wang and Xiao [22] presented a principal component analysis (PCA) method to detect a single sensor fault occurring in an air handling unit. He used contribution plots plus some simple rules to isolate the single fault source. In addition, he also developed a tolerant control strategy based on principal component analysis, which can be used to detect and diagnose single sensor faults in central chilling systems [23]. However, three more aspects are still worthy of further investigation. First, multiple faults of sensors should be investigated. After all, multiple faults are much more popular than single faults in VAV systems. In addition, it is necessary to seek a suitable diagnosis method or strategy, as a contribution plot is too weak
h c b
angle between two vectors lying in PCS angle between two vectors lying in RS magnitude of bias
Subscripts and superscripts ^ modelled part of vector projected on PCS ~ un-modelled part of vector projected on RS sup supply air fre outdoor air rtn return air rec recycle air exh exhaust air w water ws supply water wr return water set set point n number of measurement samples r number of faults in F
to isolate complex faults with propagation characteristics in VAV systems. At last, it is promising to develop a robust scheme not only to isolate multiple faults simultaneously but also reconstruct them one by one. Multi-level principal component analysis, joint angle plots and reconstruction schemes are presented in this paper to detect, isolate and evaluate multiple sensors faults occurring in VAV systems. First of all, multi-level principal component analysis models, including system level and local level, are built to detect multiple faults occurring in VAV systems. As the initial detection, a system level model is used to discover the abnormalities in view of the whole systems. The local level models are used to further confirm the occurrence of the faults. Moreover, with multiple faults separated into different locations by local level detection models, joint angle plots are used, respectively, to isolate the faults one by one. Finally, the reconstruction scheme can be used to estimate the magnitude of the bias so as to recover the operation of the systems. 2. System description and faults focused 2.1. System description Typical VAV systems are shown in Fig. 1, which are composed of air handling units, supply fan, return fan, air ducts, air dampers, VAV terminals, controllers and sensors. The supply air, which is a mixture of outdoor air and recycle air, is circulated to air handling units by the variable speed supply fan and will exchange heat with the chilled water. After being cooled (in summer condition) by the chilled water, it will be circulated to the VAV terminals to meet the indoor requirement. Finally, with the var-
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Controlle r Controller
Supply Fan Outdoor Air
F
VAV
T
F Supply Air
VAV
VAV
VAV
VAV
VAV
VAV
F
Fow Rate Sensor
P
Pressure Sensor
T
Temperature Sensor
P
Air handling unit
Recycle Air
Controller Exhaust Air
VAV Controller Controller F Return Air
Return Fan
Fig. 1. Schematic diagram of VAV systems.
iable speed return fan, the return air may be divided into exhaust air and recycle air, which are discharged into the outside space and reemployed to another air circle again, respectively. 2.2. Sensor fault cases Since supply air temperature and outdoor air flow rate are the key variables that affect indoor air quality and energy consumption, the focus is on both the supply air temperature control loop and the outdoor air flow rate control loop, and their commonly occurring faults related to the two locations are concerned. In this paper, multiple faults can occur in one control loop not simultaneously but orderly. As for different control loops, multiple faults can occur simultaneously. Besides the normal operation of the systems, the following 10 fault cases are considered in this paper: four cases for single faults and six cases for multiple faults. Case Case Case Case Case
1: 2: 3: 4: 5:
Case 6: Case 7: Case 8: Case 9: Case 10:
Tsup sensor biased with 8% at 12PM; Mfre sensor biased with 20% at 12PM; Twr sensor biased with 7% at 12PM; Msup sensor biased with 20% at 12PM; Tsup and Mfre sensors biased with 8% and 20% at 12:30PM, respectively; Tsup and Mrtn sensors biased with 10% and 20% at 10AM, respectively; Twr and Msup sensors biased with 7% and 20% at 12:30PM, respectively; Tsup and Mfre sensors drifted with 0.01 °C/min and 0.0012 kg/s/min at 10AM, respectively; Tsup and Msup sensors drifted with 0.01 °C/ min and 0.004 kg/s/min at 11AM, respectively; Tsup and Mfre sensors complete failure occurring at 12PM and 1PM, respectively.
3. Multiple faults tolerant control methodology 3.1. Principal component analysis to detect multiple faults 3.1.1. Overview of principal component analysis [24–27] According to the principal component analysis method, a measurement vector x that describes a running condition of VAV systems can be decomposed into two orthogonal parts (Fig. 2), x ¼ ^x þ ~x
ð1Þ
where ^x ¼ PP T x ¼ Rx
ð2Þ
is the modelled part that represents the projection on the principal component subspace (PCS) and e ~x ¼ ðI RÞx ¼ Rx
ð3Þ
is the un-modelled part on the residual subspace (RS). So, the principal component analysis model divides the measurement space into two orthogonal subspaces: principal component subspace and residue subspace. The principal component subspace refers to the condition in which normal data variation occurs, while the residue subspace
Residual Subspace
x
x~ x^
Principal Component Subspace
Fig. 2. Decomposition of measurement vector.
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refers to the condition in which abnormal variation and noise may occur. Under normal running conditions, most of the projection of x is on the principal component subspace, while the projection on the residue subspace is very little. When faults occur, however, the projection on the residue subspace can be greatly increased. As a result, the magnitude of ~x reaches unusual values compared to those obtained during normal conditions. A typical statistic for detecting abnormal conditions is the squared prediction error (SPE), e SPEðxÞ ¼ k~xk ¼ xT ðI RÞx ¼ xT Rx 2
ð4Þ
The system is considered abnormal if ð5Þ
SPEðxÞ > da
where da denotes a confidence limit or threshold for the SPE. 3.1.2. Multi-level detection models: system level and local level Multiple detection models, including system level and local level are to be developed to detect the multiple faults occurring in VAV systems. As initial detection, the system level model is used to discover faults or abnormalities in view of the whole system, while the two local level models are used not only to confirm the occurrence of faults (further detection), but also to pre-isolate the faults sources and divide them into different locations so as to improve the isolating process. 3.1.2.1. System level detection model. As the crucial balance in VAV systems, the energy balance, as shown in Fig. 3, combines each part of the systems and describes the relations among the variables [22]. The energy balance repre2
T 1sup
6 2 6 T sup I¼6 6 4 T nsup
Chiller
qair,water Hfre Air handling unit
On the other hand, the heat exchange process between the air side and the water side can be expressed as Eq. (9), where heat losses or gains of air ducts to the environment are neglected H fre þ H rtn ¼ H sup þ H exh þ qair;water
M 1fre
M 1sup
M 1rtn
T 1fre
T 1rtn
h1fre
M 2w
T 2ws
T 2wr
M 2fre
M 2sup
M 2rtn
T 2fre
T 2rtn
h2fre
M nw
T nws
T nwr
M nfre
M nsup
M nrtn
T nfre
T nrtn
hnfre
T exh ¼ T rtn hexh ¼ hrtn
ð7Þ ð8Þ
where M refers to flow rate, T refers to temperature, h refers to relative humidity and the subscripts fre, sup, rtn, rec and exh refer to fresh (outdoor), supply, return, recycle and exhaust air, respectively.
ð9Þ
Therefore, the energy balance of the VAV systems that includes both an air mixing process and heat exchange process can be described as the relations above. In Eq. (9), character H refers to enthalpy that can be represented by temperature, humidity and flow rate, while qair,water, the heat exchange rate between the air side and the water side, is related to Tws (supply water temperature), Twr (return water temperature) and Mw (water flow rate). So, Eqs. (6)–(9) referring to the energy balance of the VAV systems, can be described as certain functions of Tsup, Tws, Twr, Mw, Mfre, Msup, Mrtn, Tfre, Trtn, hfre and hrtn. In other words, these measurement variables can be combined together and have some relations through the energy balance of the systems. Therefore, the system level PCA model I can be set up using these eleven measurement variables, and the corresponding matrix is denoted as
T 1wr
ð6Þ
Hrtn
Fig. 3. Schematic diagram of energy balance.
T 1ws
M exh ¼ M rtn M rec ¼ M rtn ðM sup M fre Þ
Hsup
Hexh
M 1w
sents not only the air mixing process but the heat exchange process in the air handling unit between the air side and the water side. On one hand, the air mixing process can be described as Eqs. (6)–(8). Among them, Eq. (6) reflects the quality balance in the mixing process,
767
h1rtn
3
7 h2rtn 7 7 7 5 hnrtn n11
3.1.2.2. Local level detection models. As mentioned above, the focus is on two crucial control loops. One is the supply air temperature control loop as shown in Fig. 4, and the other is the outdoor air flow rate control loop as shown in Fig. 5. The local level detection models to be built are based on these two control loops. In the supply air temperature control loop (Fig. 4), Tsup is maintained at the set point (Tsup,set) to control the air handling unit, typically through adjusting the water valve. The main relevant variables in this subsystem include Tsup, Tsup,set, Cw (water valve control signal), Mw, Tws and Twr. These variables have strong relationships among each other because of this control loop. So, the first local level detection model can be built as
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Tws
Mw
Fault
Controller
Fault
Sensors Sensor
Fault
Tsup,set
Twr Fault
Cw
Water Valve
Setpoint
Sensor Air handling unit
Tsup Sensor
Fault Fig. 4. Supply air temperature control loop.
Msup Fault
Fault
Mfre,set
Controller
Cfre
Actuator
Setpoint
Mrtn Sensors Sensor
Fault
Air Dampers
Mfre Sensor
Fault Fig. 5. Outdoor air flow rate control loop.
2
T 1sup
6 2 6 T sup A¼6 6 4 T nsup
M 1w
T 1ws
T 1wr
T 1sup;set
C 1w
M 2w
T 2ws
T 2wr
T 2sup;set
C 2w
M nw
T nws
T nwr
T nsup;set
C nw
3
can be achieved and the multiple faults can be separated into different locations.
7 7 7 7 5
3.2. Joint angle plots to isolate multiple faults n6
Similarly, in the outdoor air flow rate control loop (Fig. 5), Mfre is maintained at the set point (Mfre,set) through adjusting the air dampers to meet the demand of indoor air quality. Five main variables, Mfre, Mfre,set, Cfre (outdoor air damper control signal), Msup and Mrtn, can be combined together in this loop. So, the second local level detection model can be described as 3 2 1 M fre M 1sup M 1rtn M 1fre;set C 1fre 7 6 2 6 M fre M 2sup M 2rtn M 2fre;set C 2fre 7 7 6 B¼6 7 5 4 n n n n n M fre M sup M rtn M fre;set C fre n5 In addition, models A and B can be considered to be independent in the detection process since they describe different control loops, and a fault occurring in one loop will not severely interfere and destroy the correlation of variables of the other model but only that of variables of its own, which will be validated later in the tests. So, with the assurance of independent detecting of the two local level models, the aim to pre-isolate the faults
Although the contribution plot has been widely used to isolate the fault source in many applications, its capacity in isolating sensor faults in VAV systems is weak. VAV and its control systems are complicated. The contribution plots come from an underlying correlation model that does not provide a causal relationship among the sensors/variables. They only show which group of variables of the systems is highly correlated with the fault. They do not provide direct fault isolation. For those complex sensor faults occurring whose effects may propagate to other parts of the systems and cause symptoms, the symptoms of a sensor fault may be hidden due to the feedback of the symptoms. For example, suppose the supply air temperature sensor has a positive bias, the control signals will be decreased because of the feedback control. Then, by the actions of the control loop, the fault effect of the Tsup sensor is propagated to other relevant variables, such as Cw, Twr and Mw, and the contributions of those variables will inevitably increase. So, after being detected, it is difficult to isolate the fault accurately in the contribution plots.
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Because of the weak isolating capacity of contribution plots, joint angle plots [28] are used to solve the problem. 3.2.1. Fault knowledge Steady state information via VAV plant test or historical data is available, so fault knowledge can be obtained. When a fault occurs in VAV systems, the measurement vector x can be represented using a fault vector, f (x = x* + f). Here, x* denotes the vector for normal operation condition just prior to a fault. Then the vector (f) can be decomposed using the above mentioned principal component analysis model into two components: one is lying in the principal component subspace that can be denoted as f^ , and the other is lying in the residue subspace, denoted as f~ . After being normalized, they can be described as Rf f^ ¼ ; kRf k
e Rf f~ ¼ e k Rf k
ð10Þ
Then, the fault knowledge library of all known fault signature vectors can be expressed as follows: h i Fb ¼ f^ 1 ; f^ 2 ; . . . ; f^ r ; ð11Þ Fe ¼ f~ 1 ; f~ 2 ; . . . ; f~ r So the two fault signature matrices that include all the known fault information of the systems lie in both the principal component subspace and the residue subspace. 3.2.2. Fault isolation using joint angle plots When a new sensor measurement is obtained, it can be decomposed and then normalized as follows: ^x ¼
Rx ; kRxk
~x ¼
e Rx e k Rxk
ð12Þ
Then, angle analysis between the known fault signatures ð Fb ; Fe Þ of the systems and the new measurement vector signature ð^x; ~xÞ are used for the fault isolation (Fig. 6) [28]. The cosine value between the new measure and one of the known fault signatures gives the relative measure of collinearity between them cos hi ¼ ^xT f^ i ; cos ci ¼ ~xT f~ i ð1 6 ^xT f^ i ; ~xT f~ i 6 1; i ¼ 1 rÞ
x* f^1
x
(New measure)
~ f1 f1
3.3. Fault reconstruction Fault reconstruction is used to provide the best estimate of the real value of x, using a compensatory reconstruction method. The single fault condition is discussed first. Since the normalized fault vector fi has been known through isolation, suppose the magnitude of the bias is b, then the estimated vector can be expressed as Eq. (14) x ¼ x b fi
ð14Þ
As a result, after the fault is removed from the measurement value, the new SPE will be 2 e SPEðx Þ ¼ Rx ð15Þ So, the value of SPE after compensating would be reduced greatly only if the supposition is relatively correct. On the contrary, if the supposition about the direction of the fault (negative bias or positive bias) and its value is incorrect, the value of SPE will never be reduced. For example, if the bias of one sensor is negative while it is supposed to be positive, the value of SPE would never be greatly reduced through false compensation. As a result, the problem of reconstruction can be changed into one optimal problem that finds an optimal b to minimize the value of SPE, which is described by Eq. (16) minðSPEÞ !
dðSPEÞ ¼0 db
ð16Þ
As to multiple faults, the reconstruction can be decomposed into two local reconstructions, since multiple faults can be separated into two different locations through models A and B. The corresponding faults can be estimated one by one through solving the optimal problem mentioned above.
The logic diagram of multiple faults tolerant control is illustrated in Fig. 7. First of all, multi-level (system level and local level) principal component analysis models are set up through the following steps:
~ x
x^
When the two cosine values are both close to 1 or 1, it means that the new measurement vector is nearly collinear to the fault direction in both the principal component subspace and the residue subspace. Therefore, once the SPE plot detects a fault, the fault can be isolated as the one whose cosine values mentioned above are both close to 1 or 1 through joint angle plots.
4. Multiple faults tolerant control logic
Principal Component
x2
ð13Þ
769
x1
Known Fault
Fig. 6. Schematic diagram of joint angle plot.
(1) After being divided into three groups according to matrices I, A and B, the historical normal operation data should be scaled to zero mean and unit variance.
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Normal historical data New measure
Scaled to zero mean and unit variance
Model I
Eigenvalue & Eigenvector
Fault Detection System-Level Detection:
N
SPE> δ α
PCS
Y Local-Level Detection:
Model A
Model B
SPE> δ α
SPE>δ α
YA / NA
Fault Isolation YA &NB Pre-isolation:
Isolation:
YB&NA
Tsup & Mfre control loop
Joint angle plots based on Model A
Fault Reconstruction d ( SPE ) =0 dβ A A
XA − β AfiA
XA , new Tsup Controller
Principal co mponent analysis Model I, A and B
YB/NB
YA &YB
Tsup control loop
RS
Mfre control loop
NA &NB Tfre, Trtn , h fre, h rtn
Joint angle plots based on Model B
d ( SPE ) =0 dβB B
XB − β BfiB XB , new Mfre Controller
Fig. 7. Logic diagram of multiple faults tolerant control.
(2) With the number of principal components optimized, the correlation matrix of models I, A and B can be obtained under normal operation. With the eigenvalues and eigenvectors calculated, principal component analysis models I, A and B can be set up by partitioning the measurements into two orthogonal subspaces: the principal component subspace and the residue subspace. Secondly, faults detection, including system level and local level is performed. On one hand, system level detection examines whether there is any abnormality by comparing the value of SPE of the new measurement vector with the threshold of model I. It can be considered as the initial detection to discover any abnormal operation of the whole systems. On the other hand, the two local level models (A and B) are used to confirm further the occurrence of the faults discovered by system level detection through similar comparisons, respectively. Thirdly, multiple faults isolation is performed. Multiple faults can be pre-isolated into different locations
(Tsup or Mfre control loop) through local level detection. With the pre-isolation, joint angle plots based on relevant local models are used to isolate the faults sources one by one. At last, with confirmation of the faulty sensors, fault reconstruction to estimate the value of the bias is conducted according to different local models. As a result, the corrected values of the sensors are transferred into the corresponding controllers. 5. Validation Tolerant control for multiple faults in VAV systems using multiple principal component analysis, joint angle plots and reconstruction schemes are tested and validated. They are performed using the developed simulator [29] of VAV systems. Typical weather data from Shanghai Weather Data Record is pre-processed and selected to build the training model. In the tests, the air conditioning systems worked between 7:45AM to 8:00PM.
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In the simulation, the fault generator has been incorporated, and it can generate three types of faults of sensors: fixed, drifting and complete failure. 5.1. Fault detection 5.1.1. Single sensor fault Detection of single sensor bias is tested first in this section. The detection results of the fixed bias of single sensors, including Tsup, Mfre, Twr and Msup (Cases 1–4), are shown in Fig. 8. Under normal operation, all the SPE values are less than the thresholds of both system level and local level models, respectively. When the Tsup sensor is biased with 8% at 12PM, the SPE value exceeds the threshold of system level model I as shown in Fig. 8a, indicating the abnormality of the systems. Furthermore, detection of local level model A can be used to confirm the occurrence of the fault in the supply air temperature control loop by comparing the SPE value with its threshold in Fig. 8b. On the other hand, detection of model B can be used to confirm no fault occurred in the outdoor air flow rate control loop by similar comparison. So, with further detection of the two local level models, it can be pre-isolated that
771
some fault occurred only in the supply air temperature control loop. When the Mfre sensor is biased with 20% at 12PM, first of all, the SPE value goes beyond the threshold of model I as shown in Fig. 8a, indicating that the systems are running abnormally. Then, detection of model A illustrates that no fault occurred in the supply air temperature control loop because its SPE value is always less than its threshold, while detection of model B indicates the occurrence of a certain fault in the outdoor air control loop by comparing the SPE value with its threshold. Moreover, similar to the analysis of the fixed bias of the Twr and Msup sensors, they can also be detected well through multi-level detection (Fig. 8). 5.1.2. Multiple sensors faults As to multiple sensors faults with fixed biases, Cases 5–7 are tested. When the Tsup and Mfre sensors are biased with 8% and 20% at 12:30PM simultaneously, system level detection based on model I indicates the abnormality of the systems (Fig. 9a), while the two local level SPE plots (Figs. 9b and c) based on detection models A and B indicate the occurrence of faults in both the supply air temperature and outdoor air control loops, respectively.
Fig. 8. SPE plots for single sensor faults. (a) System level detection (model I based); (b) local level detection (model A based); (c) local level detection (model B based).
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Fig. 9. SPE plots for multiple sensors faults: fixed biases. (a) System level detection (model I based); (b) local level detection (model A based); (c) local level detection (model B based).
When the Tsup sensor is biased with 10% and the Mrtn sensor is biased with 20% at 10AM, first of all, the system level model discovers the abnormality shown in Fig. 9a in view of the whole system. Then, the multiple faults are further confirmed and separated into different control loops through the two local level detection plots as shown in Figs. 9b and c. When the Twr and Msup sensors are biased with 7% and 20% at 12:30PM simultaneously, it can be discovered through the system level SPE plot in Fig. 10a. Also, the two local level detections in Figs. 10b and c can confirm the faults occurred in both the supply air temperature and outdoor air control loops. In addition, the multiple faults of drifting and complete failure are also tested and the results are shown in Fig. 10. For Cases 8 and 9, drifting faults can be discovered well when the magnitude of the faults increase to some extent, and for Case 10, complete failure can also be detected in timely manner. 5.2. Multiple faults isolation For Case 5, since they are separated into two local control loops, joint angle plots can be used to isolate the faults sources among all of the sensors concerned in models A and B. On one hand, for model A, the angle components in both the principal component subspace and residue sub-
space between this local fault direction and the directions of the faults in the library are shown in Fig. 11a. Note that the cosines of the angles between this new local fault vector and the Tsup sensor fault vector in the library go to the (1, 1) corner of the plot, suggesting essentially perfect collinearity with a Tsup fault in both subspaces. So, it is unambiguous that the Tsup sensor is biased in the supply air temperature control loop indeed. Actually, the method to diagnose the fault source according to a local level model is to examine whose values locate in the corner of (1, 1) or (1, 1) in the corresponding joint angle plot. On the other hand, similar analysis is made according to model B. Since the cosines of the angles between this new local fault vector and the Mfre sensor fault vector in the library go to the (1, 1) corner of the plot in Fig. 11b, indicating perfect collinearity with a Mfre fault in both the principal component subspace and residue subspace directions, it is unambiguous that the Mfre sensor is biased in the outdoor air control loop. In sum, the sources are the Tsup and Mfre sensors together. As to Case 6, the two local joint angle plots in Fig. 12 illustrate that the sources are the Tsup and Mrtn sensors because their values locate in the corner of (1, 1) and (1, 1), respectively. As to Case 7 finally, the two joint angle plots (Fig. 13) based on models A and B show that Twr and Msup sensors are the real faults sources.
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Fig. 10. SPE plots for multiple sensors faults: drifting and complete failure. (a) System level detection (model I based); (b) local level detection (model A based); (c) local level detection (model B based).
Isolation area 1
0.5
0.5
Tsup Mw
0
Tws Twr
-0.5
RS Direction
RS Direction
Isolation area 1
Mfre 0
Mrtn Msup
-0.5
-1
-1
-1
-0.5
0 PCS Direction
0.5
1
-1
-0.5
0 0.5 PCS Direction
1
Fig. 11. Joint angle plots to isolate Tsup Mfre fixed faults (Case 5). (a) Joint angle plot based on model A; (b) joint angle plot based on model B.
5.3. Multiple faults reconstruction On line isolation for Cases 5–7 are illustrated in Figs. 14–16. As to the Tsup Mfre faults occurring at 12:30PM simultaneously (Case 5), the Tsup bias can be isolated on line at about 12:45PM, while it cost more time to isolate the Mfre bias that can be identified at 2PM (Fig. 14). As to the Tsup Mrtn faults occurring at 10AM (Case 6), the
Tsup and Mrtn biases can be isolated successfully at 10:08AM and 10:10AM (Fig. 15). In addition, several minutes after the Twr Msup faults occurred (Case 7), they can be isolated at 12:35PM (Fig. 16). After the multiple faults are isolated one by one on line, the reconstruction module can estimate the magnitudes of the sensor biases that are illustrated in Table 1. The relative errors between the estimated and real values are less than
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Isolation area 1
0.5
0.5
RS Direction RS Direction
1
RS Direction
Tsup Mw
0
Tws Twr
-0.5
Mfre Mrtn
0
Msup
-0.5
-1
-1
-1
-0.5
Isolation area
0 PCS Direction
0.5
1
-1
-0.5
0 PCS Direction
0.5
1
Fig. 12. Joint angle plots to isolate Tsup Mrtn fixed faults (Case 6). (a) Joint angle plot based on model A; (b) joint angle plot based on model B.
1
0.5
0.5 RS Direction
RS Direction
Isolation area 1
Tsup Mw Tws Twr
0
Msup
-0.5
-0.5
-1
-1
-1
-0.5
0 PCS Direction
0.5
Mfre Mrtn
0
-1
1
-0.5
0
Isolation area PCS Direction
0.5
1
Fig. 13. Joint angle plots to isolate Twr Msup fixed faults (Case 7). (a) Joint angle plot based on model A; (b) joint angle plot based on model B.
1
1
0.5
0.5
0
Cosine of the angles
Cosine of the angles
cos(θ): PCS Direction
cos(θ): PCS Direction cos(ν): RS Direction
0
-0.5
-0.5
-1 12:15 PM
cos(ν): RS Direction
12:30 PM
12:45 PM
1:00 PM
1:15 PM
Time
-1 12:00 PM
12:30 PM
1:00 PM
1:30 PM
2:00 PM
2:30 PM
Time
Fig. 14. On line isolation for Tsup Mfre fixed faults (Case 5). (a) On line joint angle plot for Tsup; (b) on line joint angle plot for Mfre.
thirty percent. Besides the estimated value of the biases, the detection time and isolation time are also shown in Table 1. Finally, the reconstruction results for Case 6 are illustrated in Fig. 17. Compared with faulty operation
(10AM–12PM), the SPE values decrease greatly to be less than the thresholds of models A and B after the faults are reconstructed (12–4PM). Similar analysis for Case 7 in Fig. 18, compared with faulty operation (12:30–
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777 1
1 cos(θ): PCS Direction
cos(θ): PCS Direction
cos(ν): RS Direction
cos(ν): RS Direction 0.5 Cosine of the angles
Cosine of the angle
0.5
0
0
-0.5
-0.5
-1 9:30 AM
775
9:45 AM
10:00 AM
10:15 AM
-1 9:30 AM
10:30 AM
9:45 AM
10:00 AM
10:15 AM
10:30 AM
Time
Time
Fig. 15. On line isolation for Tsup Mrtn fixed faults (Case 6). (a) On line joint angle plot for Tsup; (b) on line joint angle plot for Mrtn.
1
1 cos(θ): PCS Direction
cos(θ): PCS Direction
cos(ν): RS Direction
cos(ν): RS Direction 0.5 Cosine of the angles
Cosine of the angles
0.5
0
-0.5
-1 12:00 PM
0
-0.5
12:15 PM
12:30 PM
12:45 PM
1:00 PM
-1 12:00 PM
12:15 PM
Time
12:30 PM
12:45 PM
1:00 PM
Time
Fig. 16. On line isolation for Twr Msup fixed faults (Case 7). (a) On line joint angle plot for Twr; (b) on line joint angle plot for Msup.
Table 1 Multiple faults tolerant control results Case
Faults
Occurred time
Real bias
Detection time
Isolation time
Estimate bias
Relative error (%)
5
Tsup Mfre Tsup Mrtn
12:30PM
1.0 °C 0.2 kg/s 1.5 °C 2.0 kg/s
12:35PM 12:35PM 10:05AM 10:05AM
12:45PM 2PM 10:08AM 10:10AM
0.78 °C 0.26 kg/s 1.67 °C 1.78 kg/s
22 30 12 11
Twr Msup
12:30PM
1.0 °C 2.0 kg/s
12:33PM 12:33PM
12:35PM 12:35PM
1.19 °C 1.93 kg/s
6 7
10AM
19 3.5
Fig. 17. SPE plots before and after reconstruction (Case 6). (a) Local level (model A based); (b) local level (model B based).
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Fig. 18. SPE plots before and after reconstruction (Case 7). (a) Local level (model A based); (b) local level (model B based).
1:30PM), shows that most of the SPE values are less than the corresponding thresholds when the faults are recovered (1:30–4PM). 6. Conclusions Tolerant control methods for multiple faults of sensors in VAV systems using multi-level principal component analysis, joint angle plots and reconstruction schemes are developed in this paper. Multi-level principal component analysis models, which include models I, A and B, are set up. As system level detection, model I is used to detect the abnormal conditions initially in view of the whole system. After the initial detection, local level models A and B are used to confirm further the occurrence of the faults. Moreover, they can also be used to pre-isolate the faults and separate the sources into the two special control loops. With the multiple faults separated into two locations, joint angle plots according to the corresponding local models (A and B) are used to isolate the faults sources. Actually, the approach of joint angle plots extracts fault signatures that are the vectors of movement of the fault in the principal component subspace and the residue subspace. The directions of these vectors are compared with the corresponding vector directions of known faults in a fault library. The isolation is based on comparison of the joint angles between the vectors of the current fault and those of the known ones. Finally, the reconstruction scheme can be used to estimate the magnitude of the bias so as to recover the operation of the systems. References [1] Usoro PB, Schick IC, Negahdaripour S. An innovation-based methodology for HVAC system fault detection. J Dyn Syst Meas Control 1985;107:284–5. [2] Anderson D, Graves L, Reinert W, Kreider JF, Dow J, Wubbena H. A quasi-real-time expert system for commercial building HVAC diagnostics. ASHRAE Trans 1989;95(2). [3] Hyvarnen J et al. IEA ANNEX 25, building optimization and fault diagnosis source book. Paris: International Energy Agency; 1995.
[4] Dexter AL, Pakanen J. Demonstrating automated fault detection and diagnosis methods in real buildings. VTT Building Technology, Finland (ISBN 951-38-5726-3), ANNEX 34 Final Report, 2001, IEA. [5] Piette MA, Kinney SK, Philip H. Analysis of an information monitoring and diagnostic system to improve building operation. Energy Build 2001;33(8):783–91. [6] Comstock MC, Braun JE. Development of analysis tools for the evaluation of fault detection and diagnostics in chillers. Report #HL99-20. Purdue University, Ray W. Herrick Laboratories, West Lafayette, IN, 1999. [7] Peitsman H, Bakker VE. Application of black-box models to HVAC systems for fault detection. ASHRAE Trans 1996;102(2):628–40. [8] Rossi TM, Braun JE. A statistical, rule-based fault detection and diagnostic method for vapor compression air conditioners. Int J Heat Ventilat Air Condition Refrig Res 1997;3(1):19–37. [9] Yoshida H, Iwami T, Yuzawa H, Suzuki M. Typical faults of airconditioning systems and fault detection by ARX model and extended Kalman filter. ASHRAE Trans 1996;102(1):557–64. [10] Lee WY, Park C, Kelly GE. Fault detection in an air-handling unit using residual and recursive parameter identification methods. ASHRAE Trans 1996;102(2):528–39. [11] Ngo D, Dexter AL. A robust model-based approach to diagnosing faults in air-handling units. ASHRAE Trans 1999;105(1):1078–86. [12] House JM, Vaezi-Nejad H, Whitcomb JM. An expert rules set for fault detection in air handling units/discussion. ASHRAE Trans 2001;107(1):858–71. [13] Stylianou M, Nikanour D. Performance monitoring, fault detection, and diagnosis of reciprocating chillers. ASHRAE Trans 1996;102(1):615–27. [14] Wang SW, Wang JB. Robust sensor fault diagnosis and validation in HVAC systems. Trans Inst Meas Control 2002;24(3):231–62. [15] Howell J, Maddison EJ. Fault detection in HVAC plants based on constraint suspension. Build Serv Eng Res Technol 1995;16(4):207–13. [16] Haves P, Salsbury TI, Wright JA. Condition monitoring in HVAC subsystem using first principles models. ASHRAE Trans 1996;102(1):519–27. [17] Tzafestas S. Second generation expert systems: requirements, architectures and prospects. In: IFAC/IMACS symposium on fault detection, supervision and safety for technical process, Baden-Baden, 1991. [18] Hemmelblau DM, Use of artificial neural networks to monitor faults and for troubleshooting in the process industries. In: IFAC symposium on on-line fault detection and supervision in the chemical process industries, Newark, 1992. [19] Dexter AL, Ngo D. Fault diagnosis in HVAC systems: a multi-step fuzzy model-based approach. Int J HVAC&R Res 2001;7(1):83–102. [20] Lee WY, House JM, Shin DR. Fault diagnosis and temperature sensor recovery for an air-handling unit. ASHRAE Trans 1997;103(1):621–33.
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777 [21] Wang SW, Chen YM. Fault-tolerant control for outdoor ventilation air flow rate in building based on neural network. Build Environ 2002;37(7):691–704. [22] Wang SW, Xiao F. AHU sensor fault diagnosis using principal component analysis method. Energy Build 2004;36:147–60. [23] Wang SW, Cui JT. Sensor-fault detection, diagnosis and estimation for centrifugal chiller systems using principal-component analysis method. Appl Energy 2005;82:197–213. [24] Jackson JE, Mudholkar GS. Control procedures for residuals associated with principal components analysis. Technometrics 1979;21:341–9. [25] Edward J. User’s guide to principal components. Wiley; 1991.
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[26] Jolliffe IT. Principal component analysis. NewYork: Springer-Verlag; 1986. [27] Dunia Ricardo, Joe Qin S. Joint diagnosis of process and sensor faults using principal component analysis. Control Eng Practice 1998;6:457–69. [28] Yoon Seongkyu, MacGregor John F. Fault diagnosis with multivariate statistic models. Part I: Using steady state fault signatures. J Process Control 2001;11(1):387–400. [29] Jin XQ. Study on simulation of VAV air-conditioning system and online optimal control. Shanghai: Shanghai Jiaotong University; 1999.
Tolerant control for multiple faults of sensors in VAV systems Zhimin Du *, Xinqiao Jin School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200030, PR China Received 27 February 2006; accepted 10 September 2006 Available online 14 November 2006
Abstract Principal component analysis, joint angle plots and reconstruction schemes are presented in this paper to detect, isolate and evaluate multiple sensors faults occurring in variable air volume (VAV) systems. Multi-level principal component analysis models, including system level and local level, are built to detect multiple faults occurring in VAV systems. As the initial detection, a system level model is used to discover the abnormalities in view of the whole systems. Two local level models are used to further confirm the occurrence of the faults. Moreover, with the multiple faults separated into different locations by the two local level detection models, joint angle plots are used, respectively, to isolate the faults one by one. Finally, the reconstruction scheme is used to estimate the magnitude of the bias to recover from the faulty operation. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Tolerant control; Multiple faults; Sensor; Variable air volume; Detection; Isolation; Reconstruction
1. Introduction Recently, to satisfy the increasing demands on indoor comfort and environment, heating, ventilation and air conditioning systems have become more and more complex. Inadequate control of systems may result in poor indoor environment and more energy consumption. There are two key factors to ensure good performance of VAV systems: one is suitable control strategies and the other is reliable measurements. There are many control loops with sensors in VAV systems. Typical local controls associated with sensors are air handling unit supply air temperature control with temperature sensor, outdoor air control with flow rate sensor, static pressure control with pressure sensor etc. The normal or even optimal operation of the systems strongly relies on the capacity of these controls, while these controls strongly rely on the reliable measurements of their sensors. Unfortunately, biasing of the sensors is usually inevitable after VAV systems are used for a relatively long term. Therefore, *
Corresponding author. Tel.: +86 21 62933351; fax: +86 21 62932601. E-mail addresses: [email protected] (Z. Du), [email protected] (X. Jin). 0196-8904/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2006.09.007
the sensor with a fault may lead to the real operation condition of the systems being misrepresented. Furthermore, it may mislead the controls so as to execute a wrong/unsuitable operation. As a result, inaccurate measurements may lead to worsened indoor air quality or wasted energy consumption, although there are suitable control strategies. Finding a suitable method to detect, diagnose or even tolerate the fault occurring in the VAV systems is a significant target. Studies of fault detection and diagnosis in air conditioning systems started in the 1980s [1,2] and is being paid more attention recently. Annex25 [3], Annex34 [4] and many more studies [5–12] are concerned about all kinds of faults in air conditioning systems. There have been two main methods in the past to detect or to diagnose the sensor faults. One is based on the model of the systems, and the other is based on knowledge. The model based method is to obtain normal values of the parameters through the models of the systems first. Whether or not the fault occurs can be judged by comparing the actual value with the normal one. The premise of this method is that an accurate mathematical model can be obtained. Stylianou and Nikanour [13] used a first order model to detect faults of temperature sensors by comparing
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777
765
Nomenclature x P R T M H h q C f F PCA SPE PCS RS x*
measure vector loading matrix model projection matrix temperature (°C) flow rate (kg/s) enthalpy (kJ/kg) relative humidity (%) heat exchange rate (kJ/kg) control signal a fault vector faults matrix principal component analysis square prediction error principal component subspace residue subspace real or estimated value of a measure vector
Greek symbols da threshold for SPE the actual temperature decay with the model output using hypothesis testing. Wang and Wang [14] developed a law based sensor fault diagnosis strategy, which took the commonly used temperature and flow rate sensors in the chilling plant into account at the same time. Other researchers, such as Howell and Maddison [15] and Haves et al. [16], paid attention to this method as well. The knowledge based approaches, such as expert systems [17], neural net [18] and fuzzy theory [19], are also widely used to detect and diagnose faults. Lee et al. [20] investigated the fault diagnosis in a simulated air handling unit using a two stage artificial neural network. After that, Wang and Chen [21] developed a strategy based on a neural network model to diagnose measurement faults of the flow rate sensors of outdoor air and supply air. For some simple systems, it is easy to detect and diagnose faults using such experiential knowledge. However, it is challenging to apply the knowledge based approach to large scale systems such as VAV systems because it is difficult to develop the complicated expert system. Recently, Wang and Xiao [22] presented a principal component analysis (PCA) method to detect a single sensor fault occurring in an air handling unit. He used contribution plots plus some simple rules to isolate the single fault source. In addition, he also developed a tolerant control strategy based on principal component analysis, which can be used to detect and diagnose single sensor faults in central chilling systems [23]. However, three more aspects are still worthy of further investigation. First, multiple faults of sensors should be investigated. After all, multiple faults are much more popular than single faults in VAV systems. In addition, it is necessary to seek a suitable diagnosis method or strategy, as a contribution plot is too weak
h c b
angle between two vectors lying in PCS angle between two vectors lying in RS magnitude of bias
Subscripts and superscripts ^ modelled part of vector projected on PCS ~ un-modelled part of vector projected on RS sup supply air fre outdoor air rtn return air rec recycle air exh exhaust air w water ws supply water wr return water set set point n number of measurement samples r number of faults in F
to isolate complex faults with propagation characteristics in VAV systems. At last, it is promising to develop a robust scheme not only to isolate multiple faults simultaneously but also reconstruct them one by one. Multi-level principal component analysis, joint angle plots and reconstruction schemes are presented in this paper to detect, isolate and evaluate multiple sensors faults occurring in VAV systems. First of all, multi-level principal component analysis models, including system level and local level, are built to detect multiple faults occurring in VAV systems. As the initial detection, a system level model is used to discover the abnormalities in view of the whole systems. The local level models are used to further confirm the occurrence of the faults. Moreover, with multiple faults separated into different locations by local level detection models, joint angle plots are used, respectively, to isolate the faults one by one. Finally, the reconstruction scheme can be used to estimate the magnitude of the bias so as to recover the operation of the systems. 2. System description and faults focused 2.1. System description Typical VAV systems are shown in Fig. 1, which are composed of air handling units, supply fan, return fan, air ducts, air dampers, VAV terminals, controllers and sensors. The supply air, which is a mixture of outdoor air and recycle air, is circulated to air handling units by the variable speed supply fan and will exchange heat with the chilled water. After being cooled (in summer condition) by the chilled water, it will be circulated to the VAV terminals to meet the indoor requirement. Finally, with the var-
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Controlle r Controller
Supply Fan Outdoor Air
F
VAV
T
F Supply Air
VAV
VAV
VAV
VAV
VAV
VAV
F
Fow Rate Sensor
P
Pressure Sensor
T
Temperature Sensor
P
Air handling unit
Recycle Air
Controller Exhaust Air
VAV Controller Controller F Return Air
Return Fan
Fig. 1. Schematic diagram of VAV systems.
iable speed return fan, the return air may be divided into exhaust air and recycle air, which are discharged into the outside space and reemployed to another air circle again, respectively. 2.2. Sensor fault cases Since supply air temperature and outdoor air flow rate are the key variables that affect indoor air quality and energy consumption, the focus is on both the supply air temperature control loop and the outdoor air flow rate control loop, and their commonly occurring faults related to the two locations are concerned. In this paper, multiple faults can occur in one control loop not simultaneously but orderly. As for different control loops, multiple faults can occur simultaneously. Besides the normal operation of the systems, the following 10 fault cases are considered in this paper: four cases for single faults and six cases for multiple faults. Case Case Case Case Case
1: 2: 3: 4: 5:
Case 6: Case 7: Case 8: Case 9: Case 10:
Tsup sensor biased with 8% at 12PM; Mfre sensor biased with 20% at 12PM; Twr sensor biased with 7% at 12PM; Msup sensor biased with 20% at 12PM; Tsup and Mfre sensors biased with 8% and 20% at 12:30PM, respectively; Tsup and Mrtn sensors biased with 10% and 20% at 10AM, respectively; Twr and Msup sensors biased with 7% and 20% at 12:30PM, respectively; Tsup and Mfre sensors drifted with 0.01 °C/min and 0.0012 kg/s/min at 10AM, respectively; Tsup and Msup sensors drifted with 0.01 °C/ min and 0.004 kg/s/min at 11AM, respectively; Tsup and Mfre sensors complete failure occurring at 12PM and 1PM, respectively.
3. Multiple faults tolerant control methodology 3.1. Principal component analysis to detect multiple faults 3.1.1. Overview of principal component analysis [24–27] According to the principal component analysis method, a measurement vector x that describes a running condition of VAV systems can be decomposed into two orthogonal parts (Fig. 2), x ¼ ^x þ ~x
ð1Þ
where ^x ¼ PP T x ¼ Rx
ð2Þ
is the modelled part that represents the projection on the principal component subspace (PCS) and e ~x ¼ ðI RÞx ¼ Rx
ð3Þ
is the un-modelled part on the residual subspace (RS). So, the principal component analysis model divides the measurement space into two orthogonal subspaces: principal component subspace and residue subspace. The principal component subspace refers to the condition in which normal data variation occurs, while the residue subspace
Residual Subspace
x
x~ x^
Principal Component Subspace
Fig. 2. Decomposition of measurement vector.
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777
refers to the condition in which abnormal variation and noise may occur. Under normal running conditions, most of the projection of x is on the principal component subspace, while the projection on the residue subspace is very little. When faults occur, however, the projection on the residue subspace can be greatly increased. As a result, the magnitude of ~x reaches unusual values compared to those obtained during normal conditions. A typical statistic for detecting abnormal conditions is the squared prediction error (SPE), e SPEðxÞ ¼ k~xk ¼ xT ðI RÞx ¼ xT Rx 2
ð4Þ
The system is considered abnormal if ð5Þ
SPEðxÞ > da
where da denotes a confidence limit or threshold for the SPE. 3.1.2. Multi-level detection models: system level and local level Multiple detection models, including system level and local level are to be developed to detect the multiple faults occurring in VAV systems. As initial detection, the system level model is used to discover faults or abnormalities in view of the whole system, while the two local level models are used not only to confirm the occurrence of faults (further detection), but also to pre-isolate the faults sources and divide them into different locations so as to improve the isolating process. 3.1.2.1. System level detection model. As the crucial balance in VAV systems, the energy balance, as shown in Fig. 3, combines each part of the systems and describes the relations among the variables [22]. The energy balance repre2
T 1sup
6 2 6 T sup I¼6 6 4 T nsup
Chiller
qair,water Hfre Air handling unit
On the other hand, the heat exchange process between the air side and the water side can be expressed as Eq. (9), where heat losses or gains of air ducts to the environment are neglected H fre þ H rtn ¼ H sup þ H exh þ qair;water
M 1fre
M 1sup
M 1rtn
T 1fre
T 1rtn
h1fre
M 2w
T 2ws
T 2wr
M 2fre
M 2sup
M 2rtn
T 2fre
T 2rtn
h2fre
M nw
T nws
T nwr
M nfre
M nsup
M nrtn
T nfre
T nrtn
hnfre
T exh ¼ T rtn hexh ¼ hrtn
ð7Þ ð8Þ
where M refers to flow rate, T refers to temperature, h refers to relative humidity and the subscripts fre, sup, rtn, rec and exh refer to fresh (outdoor), supply, return, recycle and exhaust air, respectively.
ð9Þ
Therefore, the energy balance of the VAV systems that includes both an air mixing process and heat exchange process can be described as the relations above. In Eq. (9), character H refers to enthalpy that can be represented by temperature, humidity and flow rate, while qair,water, the heat exchange rate between the air side and the water side, is related to Tws (supply water temperature), Twr (return water temperature) and Mw (water flow rate). So, Eqs. (6)–(9) referring to the energy balance of the VAV systems, can be described as certain functions of Tsup, Tws, Twr, Mw, Mfre, Msup, Mrtn, Tfre, Trtn, hfre and hrtn. In other words, these measurement variables can be combined together and have some relations through the energy balance of the systems. Therefore, the system level PCA model I can be set up using these eleven measurement variables, and the corresponding matrix is denoted as
T 1wr
ð6Þ
Hrtn
Fig. 3. Schematic diagram of energy balance.
T 1ws
M exh ¼ M rtn M rec ¼ M rtn ðM sup M fre Þ
Hsup
Hexh
M 1w
sents not only the air mixing process but the heat exchange process in the air handling unit between the air side and the water side. On one hand, the air mixing process can be described as Eqs. (6)–(8). Among them, Eq. (6) reflects the quality balance in the mixing process,
767
h1rtn
3
7 h2rtn 7 7 7 5 hnrtn n11
3.1.2.2. Local level detection models. As mentioned above, the focus is on two crucial control loops. One is the supply air temperature control loop as shown in Fig. 4, and the other is the outdoor air flow rate control loop as shown in Fig. 5. The local level detection models to be built are based on these two control loops. In the supply air temperature control loop (Fig. 4), Tsup is maintained at the set point (Tsup,set) to control the air handling unit, typically through adjusting the water valve. The main relevant variables in this subsystem include Tsup, Tsup,set, Cw (water valve control signal), Mw, Tws and Twr. These variables have strong relationships among each other because of this control loop. So, the first local level detection model can be built as
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Tws
Mw
Fault
Controller
Fault
Sensors Sensor
Fault
Tsup,set
Twr Fault
Cw
Water Valve
Setpoint
Sensor Air handling unit
Tsup Sensor
Fault Fig. 4. Supply air temperature control loop.
Msup Fault
Fault
Mfre,set
Controller
Cfre
Actuator
Setpoint
Mrtn Sensors Sensor
Fault
Air Dampers
Mfre Sensor
Fault Fig. 5. Outdoor air flow rate control loop.
2
T 1sup
6 2 6 T sup A¼6 6 4 T nsup
M 1w
T 1ws
T 1wr
T 1sup;set
C 1w
M 2w
T 2ws
T 2wr
T 2sup;set
C 2w
M nw
T nws
T nwr
T nsup;set
C nw
3
can be achieved and the multiple faults can be separated into different locations.
7 7 7 7 5
3.2. Joint angle plots to isolate multiple faults n6
Similarly, in the outdoor air flow rate control loop (Fig. 5), Mfre is maintained at the set point (Mfre,set) through adjusting the air dampers to meet the demand of indoor air quality. Five main variables, Mfre, Mfre,set, Cfre (outdoor air damper control signal), Msup and Mrtn, can be combined together in this loop. So, the second local level detection model can be described as 3 2 1 M fre M 1sup M 1rtn M 1fre;set C 1fre 7 6 2 6 M fre M 2sup M 2rtn M 2fre;set C 2fre 7 7 6 B¼6 7 5 4 n n n n n M fre M sup M rtn M fre;set C fre n5 In addition, models A and B can be considered to be independent in the detection process since they describe different control loops, and a fault occurring in one loop will not severely interfere and destroy the correlation of variables of the other model but only that of variables of its own, which will be validated later in the tests. So, with the assurance of independent detecting of the two local level models, the aim to pre-isolate the faults
Although the contribution plot has been widely used to isolate the fault source in many applications, its capacity in isolating sensor faults in VAV systems is weak. VAV and its control systems are complicated. The contribution plots come from an underlying correlation model that does not provide a causal relationship among the sensors/variables. They only show which group of variables of the systems is highly correlated with the fault. They do not provide direct fault isolation. For those complex sensor faults occurring whose effects may propagate to other parts of the systems and cause symptoms, the symptoms of a sensor fault may be hidden due to the feedback of the symptoms. For example, suppose the supply air temperature sensor has a positive bias, the control signals will be decreased because of the feedback control. Then, by the actions of the control loop, the fault effect of the Tsup sensor is propagated to other relevant variables, such as Cw, Twr and Mw, and the contributions of those variables will inevitably increase. So, after being detected, it is difficult to isolate the fault accurately in the contribution plots.
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777
Because of the weak isolating capacity of contribution plots, joint angle plots [28] are used to solve the problem. 3.2.1. Fault knowledge Steady state information via VAV plant test or historical data is available, so fault knowledge can be obtained. When a fault occurs in VAV systems, the measurement vector x can be represented using a fault vector, f (x = x* + f). Here, x* denotes the vector for normal operation condition just prior to a fault. Then the vector (f) can be decomposed using the above mentioned principal component analysis model into two components: one is lying in the principal component subspace that can be denoted as f^ , and the other is lying in the residue subspace, denoted as f~ . After being normalized, they can be described as Rf f^ ¼ ; kRf k
e Rf f~ ¼ e k Rf k
ð10Þ
Then, the fault knowledge library of all known fault signature vectors can be expressed as follows: h i Fb ¼ f^ 1 ; f^ 2 ; . . . ; f^ r ; ð11Þ Fe ¼ f~ 1 ; f~ 2 ; . . . ; f~ r So the two fault signature matrices that include all the known fault information of the systems lie in both the principal component subspace and the residue subspace. 3.2.2. Fault isolation using joint angle plots When a new sensor measurement is obtained, it can be decomposed and then normalized as follows: ^x ¼
Rx ; kRxk
~x ¼
e Rx e k Rxk
ð12Þ
Then, angle analysis between the known fault signatures ð Fb ; Fe Þ of the systems and the new measurement vector signature ð^x; ~xÞ are used for the fault isolation (Fig. 6) [28]. The cosine value between the new measure and one of the known fault signatures gives the relative measure of collinearity between them cos hi ¼ ^xT f^ i ; cos ci ¼ ~xT f~ i ð1 6 ^xT f^ i ; ~xT f~ i 6 1; i ¼ 1 rÞ
x* f^1
x
(New measure)
~ f1 f1
3.3. Fault reconstruction Fault reconstruction is used to provide the best estimate of the real value of x, using a compensatory reconstruction method. The single fault condition is discussed first. Since the normalized fault vector fi has been known through isolation, suppose the magnitude of the bias is b, then the estimated vector can be expressed as Eq. (14) x ¼ x b fi
ð14Þ
As a result, after the fault is removed from the measurement value, the new SPE will be 2 e SPEðx Þ ¼ Rx ð15Þ So, the value of SPE after compensating would be reduced greatly only if the supposition is relatively correct. On the contrary, if the supposition about the direction of the fault (negative bias or positive bias) and its value is incorrect, the value of SPE will never be reduced. For example, if the bias of one sensor is negative while it is supposed to be positive, the value of SPE would never be greatly reduced through false compensation. As a result, the problem of reconstruction can be changed into one optimal problem that finds an optimal b to minimize the value of SPE, which is described by Eq. (16) minðSPEÞ !
dðSPEÞ ¼0 db
ð16Þ
As to multiple faults, the reconstruction can be decomposed into two local reconstructions, since multiple faults can be separated into two different locations through models A and B. The corresponding faults can be estimated one by one through solving the optimal problem mentioned above.
The logic diagram of multiple faults tolerant control is illustrated in Fig. 7. First of all, multi-level (system level and local level) principal component analysis models are set up through the following steps:
~ x
x^
When the two cosine values are both close to 1 or 1, it means that the new measurement vector is nearly collinear to the fault direction in both the principal component subspace and the residue subspace. Therefore, once the SPE plot detects a fault, the fault can be isolated as the one whose cosine values mentioned above are both close to 1 or 1 through joint angle plots.
4. Multiple faults tolerant control logic
Principal Component
x2
ð13Þ
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x1
Known Fault
Fig. 6. Schematic diagram of joint angle plot.
(1) After being divided into three groups according to matrices I, A and B, the historical normal operation data should be scaled to zero mean and unit variance.
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Normal historical data New measure
Scaled to zero mean and unit variance
Model I
Eigenvalue & Eigenvector
Fault Detection System-Level Detection:
N
SPE> δ α
PCS
Y Local-Level Detection:
Model A
Model B
SPE> δ α
SPE>δ α
YA / NA
Fault Isolation YA &NB Pre-isolation:
Isolation:
YB&NA
Tsup & Mfre control loop
Joint angle plots based on Model A
Fault Reconstruction d ( SPE ) =0 dβ A A
XA − β AfiA
XA , new Tsup Controller
Principal co mponent analysis Model I, A and B
YB/NB
YA &YB
Tsup control loop
RS
Mfre control loop
NA &NB Tfre, Trtn , h fre, h rtn
Joint angle plots based on Model B
d ( SPE ) =0 dβB B
XB − β BfiB XB , new Mfre Controller
Fig. 7. Logic diagram of multiple faults tolerant control.
(2) With the number of principal components optimized, the correlation matrix of models I, A and B can be obtained under normal operation. With the eigenvalues and eigenvectors calculated, principal component analysis models I, A and B can be set up by partitioning the measurements into two orthogonal subspaces: the principal component subspace and the residue subspace. Secondly, faults detection, including system level and local level is performed. On one hand, system level detection examines whether there is any abnormality by comparing the value of SPE of the new measurement vector with the threshold of model I. It can be considered as the initial detection to discover any abnormal operation of the whole systems. On the other hand, the two local level models (A and B) are used to confirm further the occurrence of the faults discovered by system level detection through similar comparisons, respectively. Thirdly, multiple faults isolation is performed. Multiple faults can be pre-isolated into different locations
(Tsup or Mfre control loop) through local level detection. With the pre-isolation, joint angle plots based on relevant local models are used to isolate the faults sources one by one. At last, with confirmation of the faulty sensors, fault reconstruction to estimate the value of the bias is conducted according to different local models. As a result, the corrected values of the sensors are transferred into the corresponding controllers. 5. Validation Tolerant control for multiple faults in VAV systems using multiple principal component analysis, joint angle plots and reconstruction schemes are tested and validated. They are performed using the developed simulator [29] of VAV systems. Typical weather data from Shanghai Weather Data Record is pre-processed and selected to build the training model. In the tests, the air conditioning systems worked between 7:45AM to 8:00PM.
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In the simulation, the fault generator has been incorporated, and it can generate three types of faults of sensors: fixed, drifting and complete failure. 5.1. Fault detection 5.1.1. Single sensor fault Detection of single sensor bias is tested first in this section. The detection results of the fixed bias of single sensors, including Tsup, Mfre, Twr and Msup (Cases 1–4), are shown in Fig. 8. Under normal operation, all the SPE values are less than the thresholds of both system level and local level models, respectively. When the Tsup sensor is biased with 8% at 12PM, the SPE value exceeds the threshold of system level model I as shown in Fig. 8a, indicating the abnormality of the systems. Furthermore, detection of local level model A can be used to confirm the occurrence of the fault in the supply air temperature control loop by comparing the SPE value with its threshold in Fig. 8b. On the other hand, detection of model B can be used to confirm no fault occurred in the outdoor air flow rate control loop by similar comparison. So, with further detection of the two local level models, it can be pre-isolated that
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some fault occurred only in the supply air temperature control loop. When the Mfre sensor is biased with 20% at 12PM, first of all, the SPE value goes beyond the threshold of model I as shown in Fig. 8a, indicating that the systems are running abnormally. Then, detection of model A illustrates that no fault occurred in the supply air temperature control loop because its SPE value is always less than its threshold, while detection of model B indicates the occurrence of a certain fault in the outdoor air control loop by comparing the SPE value with its threshold. Moreover, similar to the analysis of the fixed bias of the Twr and Msup sensors, they can also be detected well through multi-level detection (Fig. 8). 5.1.2. Multiple sensors faults As to multiple sensors faults with fixed biases, Cases 5–7 are tested. When the Tsup and Mfre sensors are biased with 8% and 20% at 12:30PM simultaneously, system level detection based on model I indicates the abnormality of the systems (Fig. 9a), while the two local level SPE plots (Figs. 9b and c) based on detection models A and B indicate the occurrence of faults in both the supply air temperature and outdoor air control loops, respectively.
Fig. 8. SPE plots for single sensor faults. (a) System level detection (model I based); (b) local level detection (model A based); (c) local level detection (model B based).
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Fig. 9. SPE plots for multiple sensors faults: fixed biases. (a) System level detection (model I based); (b) local level detection (model A based); (c) local level detection (model B based).
When the Tsup sensor is biased with 10% and the Mrtn sensor is biased with 20% at 10AM, first of all, the system level model discovers the abnormality shown in Fig. 9a in view of the whole system. Then, the multiple faults are further confirmed and separated into different control loops through the two local level detection plots as shown in Figs. 9b and c. When the Twr and Msup sensors are biased with 7% and 20% at 12:30PM simultaneously, it can be discovered through the system level SPE plot in Fig. 10a. Also, the two local level detections in Figs. 10b and c can confirm the faults occurred in both the supply air temperature and outdoor air control loops. In addition, the multiple faults of drifting and complete failure are also tested and the results are shown in Fig. 10. For Cases 8 and 9, drifting faults can be discovered well when the magnitude of the faults increase to some extent, and for Case 10, complete failure can also be detected in timely manner. 5.2. Multiple faults isolation For Case 5, since they are separated into two local control loops, joint angle plots can be used to isolate the faults sources among all of the sensors concerned in models A and B. On one hand, for model A, the angle components in both the principal component subspace and residue sub-
space between this local fault direction and the directions of the faults in the library are shown in Fig. 11a. Note that the cosines of the angles between this new local fault vector and the Tsup sensor fault vector in the library go to the (1, 1) corner of the plot, suggesting essentially perfect collinearity with a Tsup fault in both subspaces. So, it is unambiguous that the Tsup sensor is biased in the supply air temperature control loop indeed. Actually, the method to diagnose the fault source according to a local level model is to examine whose values locate in the corner of (1, 1) or (1, 1) in the corresponding joint angle plot. On the other hand, similar analysis is made according to model B. Since the cosines of the angles between this new local fault vector and the Mfre sensor fault vector in the library go to the (1, 1) corner of the plot in Fig. 11b, indicating perfect collinearity with a Mfre fault in both the principal component subspace and residue subspace directions, it is unambiguous that the Mfre sensor is biased in the outdoor air control loop. In sum, the sources are the Tsup and Mfre sensors together. As to Case 6, the two local joint angle plots in Fig. 12 illustrate that the sources are the Tsup and Mrtn sensors because their values locate in the corner of (1, 1) and (1, 1), respectively. As to Case 7 finally, the two joint angle plots (Fig. 13) based on models A and B show that Twr and Msup sensors are the real faults sources.
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Fig. 10. SPE plots for multiple sensors faults: drifting and complete failure. (a) System level detection (model I based); (b) local level detection (model A based); (c) local level detection (model B based).
Isolation area 1
0.5
0.5
Tsup Mw
0
Tws Twr
-0.5
RS Direction
RS Direction
Isolation area 1
Mfre 0
Mrtn Msup
-0.5
-1
-1
-1
-0.5
0 PCS Direction
0.5
1
-1
-0.5
0 0.5 PCS Direction
1
Fig. 11. Joint angle plots to isolate Tsup Mfre fixed faults (Case 5). (a) Joint angle plot based on model A; (b) joint angle plot based on model B.
5.3. Multiple faults reconstruction On line isolation for Cases 5–7 are illustrated in Figs. 14–16. As to the Tsup Mfre faults occurring at 12:30PM simultaneously (Case 5), the Tsup bias can be isolated on line at about 12:45PM, while it cost more time to isolate the Mfre bias that can be identified at 2PM (Fig. 14). As to the Tsup Mrtn faults occurring at 10AM (Case 6), the
Tsup and Mrtn biases can be isolated successfully at 10:08AM and 10:10AM (Fig. 15). In addition, several minutes after the Twr Msup faults occurred (Case 7), they can be isolated at 12:35PM (Fig. 16). After the multiple faults are isolated one by one on line, the reconstruction module can estimate the magnitudes of the sensor biases that are illustrated in Table 1. The relative errors between the estimated and real values are less than
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Isolation area 1
0.5
0.5
RS Direction RS Direction
1
RS Direction
Tsup Mw
0
Tws Twr
-0.5
Mfre Mrtn
0
Msup
-0.5
-1
-1
-1
-0.5
Isolation area
0 PCS Direction
0.5
1
-1
-0.5
0 PCS Direction
0.5
1
Fig. 12. Joint angle plots to isolate Tsup Mrtn fixed faults (Case 6). (a) Joint angle plot based on model A; (b) joint angle plot based on model B.
1
0.5
0.5 RS Direction
RS Direction
Isolation area 1
Tsup Mw Tws Twr
0
Msup
-0.5
-0.5
-1
-1
-1
-0.5
0 PCS Direction
0.5
Mfre Mrtn
0
-1
1
-0.5
0
Isolation area PCS Direction
0.5
1
Fig. 13. Joint angle plots to isolate Twr Msup fixed faults (Case 7). (a) Joint angle plot based on model A; (b) joint angle plot based on model B.
1
1
0.5
0.5
0
Cosine of the angles
Cosine of the angles
cos(θ): PCS Direction
cos(θ): PCS Direction cos(ν): RS Direction
0
-0.5
-0.5
-1 12:15 PM
cos(ν): RS Direction
12:30 PM
12:45 PM
1:00 PM
1:15 PM
Time
-1 12:00 PM
12:30 PM
1:00 PM
1:30 PM
2:00 PM
2:30 PM
Time
Fig. 14. On line isolation for Tsup Mfre fixed faults (Case 5). (a) On line joint angle plot for Tsup; (b) on line joint angle plot for Mfre.
thirty percent. Besides the estimated value of the biases, the detection time and isolation time are also shown in Table 1. Finally, the reconstruction results for Case 6 are illustrated in Fig. 17. Compared with faulty operation
(10AM–12PM), the SPE values decrease greatly to be less than the thresholds of models A and B after the faults are reconstructed (12–4PM). Similar analysis for Case 7 in Fig. 18, compared with faulty operation (12:30–
Z. Du, X. Jin / Energy Conversion and Management 48 (2007) 764–777 1
1 cos(θ): PCS Direction
cos(θ): PCS Direction
cos(ν): RS Direction
cos(ν): RS Direction 0.5 Cosine of the angles
Cosine of the angle
0.5
0
0
-0.5
-0.5
-1 9:30 AM
775
9:45 AM
10:00 AM
10:15 AM
-1 9:30 AM
10:30 AM
9:45 AM
10:00 AM
10:15 AM
10:30 AM
Time
Time
Fig. 15. On line isolation for Tsup Mrtn fixed faults (Case 6). (a) On line joint angle plot for Tsup; (b) on line joint angle plot for Mrtn.
1
1 cos(θ): PCS Direction
cos(θ): PCS Direction
cos(ν): RS Direction
cos(ν): RS Direction 0.5 Cosine of the angles
Cosine of the angles
0.5
0
-0.5
-1 12:00 PM
0
-0.5
12:15 PM
12:30 PM
12:45 PM
1:00 PM
-1 12:00 PM
12:15 PM
Time
12:30 PM
12:45 PM
1:00 PM
Time
Fig. 16. On line isolation for Twr Msup fixed faults (Case 7). (a) On line joint angle plot for Twr; (b) on line joint angle plot for Msup.
Table 1 Multiple faults tolerant control results Case
Faults
Occurred time
Real bias
Detection time
Isolation time
Estimate bias
Relative error (%)
5
Tsup Mfre Tsup Mrtn
12:30PM
1.0 °C 0.2 kg/s 1.5 °C 2.0 kg/s
12:35PM 12:35PM 10:05AM 10:05AM
12:45PM 2PM 10:08AM 10:10AM
0.78 °C 0.26 kg/s 1.67 °C 1.78 kg/s
22 30 12 11
Twr Msup
12:30PM
1.0 °C 2.0 kg/s
12:33PM 12:33PM
12:35PM 12:35PM
1.19 °C 1.93 kg/s
6 7
10AM
19 3.5
Fig. 17. SPE plots before and after reconstruction (Case 6). (a) Local level (model A based); (b) local level (model B based).
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Fig. 18. SPE plots before and after reconstruction (Case 7). (a) Local level (model A based); (b) local level (model B based).
1:30PM), shows that most of the SPE values are less than the corresponding thresholds when the faults are recovered (1:30–4PM). 6. Conclusions Tolerant control methods for multiple faults of sensors in VAV systems using multi-level principal component analysis, joint angle plots and reconstruction schemes are developed in this paper. Multi-level principal component analysis models, which include models I, A and B, are set up. As system level detection, model I is used to detect the abnormal conditions initially in view of the whole system. After the initial detection, local level models A and B are used to confirm further the occurrence of the faults. Moreover, they can also be used to pre-isolate the faults and separate the sources into the two special control loops. With the multiple faults separated into two locations, joint angle plots according to the corresponding local models (A and B) are used to isolate the faults sources. Actually, the approach of joint angle plots extracts fault signatures that are the vectors of movement of the fault in the principal component subspace and the residue subspace. The directions of these vectors are compared with the corresponding vector directions of known faults in a fault library. The isolation is based on comparison of the joint angles between the vectors of the current fault and those of the known ones. Finally, the reconstruction scheme can be used to estimate the magnitude of the bias so as to recover the operation of the systems. References [1] Usoro PB, Schick IC, Negahdaripour S. An innovation-based methodology for HVAC system fault detection. J Dyn Syst Meas Control 1985;107:284–5. [2] Anderson D, Graves L, Reinert W, Kreider JF, Dow J, Wubbena H. A quasi-real-time expert system for commercial building HVAC diagnostics. ASHRAE Trans 1989;95(2). [3] Hyvarnen J et al. IEA ANNEX 25, building optimization and fault diagnosis source book. Paris: International Energy Agency; 1995.
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