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Nonlinear Observer-based Fault Diagnosis for a Multi-Zone Building⁎
10th IFAC Symposium on Fault Detection, 10th IFAC Symposium on Fault Detection, Supervision and Safetyon forFault Technical Processes 10th IFAC Symposium Symposium Detection, 10th IFAC Detection, Supervision and Safetyon forFault Technical Processes Available online at www.sciencedirect.com 10th IFACPoland, Symposium on Fault Detection, Warsaw, August 29-31, 2018 Supervision and for Technical Processes Supervision and Safety Safety for Technical Processes Warsaw, Poland, August 29-31, 2018 Supervision and Safety for Technical Processes Warsaw, Poland, August 29-31, 2018 Warsaw, Poland, August 29-31, 2018 Warsaw, Poland, August 29-31, 2018
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IFAC PapersOnLine 51-24 (2018) 544–549
Nonlinear Nonlinear Observer-based Observer-based Fault Fault Diagnosis Diagnosis Nonlinear Observer-based Fault for a Multi-Zone Building Nonlinear Observer-based Fault Diagnosis Diagnosis for a Multi-Zone Building for a Multi-Zone Building for a Multi-Zone Building Mona Subramaniam A, Tushar Jain
Mona Subramaniam A, Tushar Jain Mona Subramaniam Subramaniam A, A, Tushar Tushar Jain Jain Mona Mona Subramaniam A, Tushar Jain School of Computing and Electrical Engineering, School of Computing and Electrical Engineering, School of Computing and Electrical Engineering, Indian Institute of Technology Mandi, India175005 SchoolInstitute of Computing and Electrical Engineering, Indian of Technology Mandi, India175005 School of Computing and Electrical Engineering, Indian Institute of Technology Mandi, India(e-mail: [email protected], [email protected]). Indian Institute of Technology Mandi, India- 175005 175005 (e-mail: [email protected], [email protected]). Indian Institute of Technology Mandi, India175005 (e-mail: (e-mail: [email protected], [email protected], [email protected]). [email protected]). (e-mail: [email protected], [email protected]). Abstract: Abstract: Heating, Heating, Ventilation Ventilation and and Air Air Conditioning Conditioning (HVAC) (HVAC) systems systems are are installed installed in in comcomAbstract: Heating, Ventilation and Air Conditioning (HVAC) systems are installed in mercial buildings to meet the actual demand of heating and cooling at a given time such that Abstract: Heating, Ventilation and Air Conditioning (HVAC) systems are installed in comcommercial buildings to meet the actual demand of heating and cooling at a given time such that Abstract: Heating, Ventilation and Air the Conditioning (HVAC) systems installed in commercial buildings to meet the actual demand of heating and cooling at aa are given time such that the thermal comfort of occupants within rooms (or zones) can be ensured. This is achieved mercial buildings to meet the actual demand of heating and cooling at given time such that the thermal comfort of occupants within the rooms (or zones) can be at ensured. This issuch achieved mercial buildings tothe meet theatactual demand of heating and cooling a volume givenThis time that the thermal comfort of occupants within the rooms (or zones) can be ensured. This is achieved achieved by feeding air into zones a constant temperature using variable air (VAV) boxes, the thermal comfort of occupants within the rooms (or zones) can be ensured. is by feeding air into the at a constant temperature using variable air volume (VAV) boxes, the thermal comfort of zones occupants withinfor the (or zones) can The be ensured. This is achieved by air into the zones at aasolution constant temperature using variable air volume (VAV) boxes, which offers an energy-efficient aarooms multi-zone building. amount of supply air by feeding feeding air into the zones at constant variable volume which offersair aninto energy-efficient solution fortemperature multi-zoneusing building. Theair amount of(VAV) supplyboxes, air is is by feeding the zones at a constant temperature using variable air volume (VAV) boxes, which offers an energy-efficient solution for aa within multi-zone building. The amount of supply supply air is controlled with the help of dampers located the VAV boxes. Any occurrence of a fault which offers an energy-efficient solution for multi-zone building. The amount of air is controlled with the help of dampers located within the VAV boxes. Any occurrence of a fault which offers an energy-efficient solution for a within multi-zone building. The amount of and supply air is controlled with the help of dampers located within the VAV boxes. Any occurrence ofa aaloss fault in VAV dampers may cause undesirable results such as discomfort to occupants in controlled with the help of dampers located the VAV boxes. Any occurrence of fault in VAV dampers may cause undesirable results such as discomfort to occupants and aa aloss in controlled with the help of dampers located within the VAV boxes. Any occurrence of fault in VAV dampers may cause undesirable results such as discomfort to occupants and loss in energy efficiency. In this paper, we propose a nonlinear observer-based fault diagnosis method in VAVefficiency. dampers may cause undesirable results such as observer-based discomfort to occupants and amethod loss in energy In this paper, we propose a nonlinear fault diagnosis in VAV dampers may cause undesirable results such as observer-based discomfort to occupants and can amethod loss in energy efficiency. In this paper, we propose aa nonlinear fault diagnosis to extract a precise about the fault so that subsequently, the take energy efficiency. In information this paper, we propose nonlinear observer-based faultcontroller diagnosis method to extract a precise information about the fault so that subsequently, the controller can take energy efficiency. In this paper, we propose a nonlinear observer-based faultcontroller diagnosis method to extract aa precise information about the fault so that subsequently, the controller can take corrective actions ensuring the thermal comfort of the occupants. The novelty of this method lies to extract precise information about the fault so that subsequently, the can take corrective actions ensuring the thermal comfort of so thethat occupants. The novelty of this method lies to extract a precise information about the fault subsequently, the controller can take corrective actions ensuring the thermal comfort of The novelty of this this method method lies in its simple construction, which makes it applicable for diagnosing wide variety of actuator corrective actions ensuring the thermal comfort of the the occupants. occupants. The aanovelty of lies in its simple construction, which makes it applicable for diagnosing wide variety of actuator corrective actions ensuring the thermal comfort of the occupants. The novelty of this method lies in its simple construction, which makes it applicable for diagnosing a wide variety of actuator faults in HVAC systems. The effectiveness of the proposed method is successfully demonstrated in its simple construction, which makes it of applicable for diagnosing wide variety of actuator faults in HVAC systems. The effectiveness the proposed method is aasuccessfully demonstrated in its simple construction, which makes comprising it of applicable for diagnosing wide variety ofSIMBAD actuator faults in HVAC systems. The effectiveness the proposed method is successfully demonstrated on a case-study of a one-storey building of three zones, constructed using faults in HVAC systems. The effectiveness of the proposed method is successfully demonstrated on a case-study of a one-storey building comprising of three zones, isconstructed SIMBAD faults in HVAC systems. TheDevices). effectiveness of the proposed method successfullyusing demonstrated on of building of zones, using SIMBAD (SIMulator of Building And on aa case-study case-study of aa one-storey one-storey building comprising comprising of three three zones, constructed constructed using SIMBAD (SIMulator of Building And Devices). on a case-study of a one-storey building comprising of three zones, constructed using SIMBAD (SIMulator of And (SIMulator of Building Building And Devices). Devices). © 2018, IFACof(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. (SIMulator Building And Devices). Keywords: Keywords: Fault Fault Diagnosis, Diagnosis, HVAC, HVAC, Nonlinear Nonlinear models, models, Observers, Observers, Parameter Parameter identification, identification, Keywords: Fault Diagnosis, HVAC, Nonlinear models, Observers, Parameter identification, System identification, Thermal comfort, Multizone building, Energy efficiency. Keywords: Fault Diagnosis, HVAC, Nonlinear models, Observers, Parameter identification, System identification, Thermal comfort, Multizone building, Energy efficiency. Keywords: Fault Diagnosis, HVAC, Nonlinear models, Observers, Parameter identification, System Thermal comfort, Multizone building, Energy efficiency. System identification, identification, Thermal comfort, Multizone building, Energy efficiency. System identification, Thermal comfort, Multizone building, Energy efficiency. 1. required, 1. INTRODUCTION INTRODUCTION required, whose whose dynamics dynamics should should match match precisely precisely with with the the 1. required, whose whose dynamics should The match precisely with with dynamics of the real-life system. identification of the 1. INTRODUCTION INTRODUCTION required, dynamics should match precisely dynamics of the real-life system. The identification of the 1. INTRODUCTION should match precisely with the dynamics whose ofofthe the real-life system. system. The identification of parameters aa dynamics physics-based mathematical model poses HVAC (Heating, Ventilation and Air Conditioning) sys- required, dynamics of real-life The identification the parameters ofthe physics-based mathematical model of poses HVAC (Heating, Ventilation and Air Conditioning) sysdynamics of real-life system. The identification of the parameters of a physics-based mathematical model poses HVAC (Heating, Ventilation and Air Conditioning) syschallenges due factors strong between tems which are anVentilation integral part modern buildings conof ato physics-based mathematical model poses HVACwhich (Heating, andof Air Conditioning) sys- parameters due factors such such as as strong coupling coupling between tems are an integral part modern buildings conparameters of ato mathematical model poses HVAC (Heating, Ventilation andof Air Conditioning) sys- challenges challenges due to factors such as strong coupling between tems which are an integral part of modern buildings conzones, unpredictability outside temperature, internal sumes most of the energy resources. According to India challenges due tophysics-based factorsof such as strong coupling between tems which are an integral part of modern buildings conzones, unpredictability of outside temperature, internal sumes most of the energy resources. According to India due to factorswind, as strong coupling between tems which are anscenarios integral part of modern buildings con- challenges zones, solar unpredictability ofsuch outside temperature, internal sumes most of the energy resources. According to India gains, irradiation, humidity, etc. In this paper, energy securities 2047, IESS2047 (2015), HVAC zones, unpredictability of outside temperature, internal sumes most of the energy resources. According to India gains, solar irradiation, wind, humidity, etc. In thisinternal paper, energy securities scenarios 2047, IESS2047 (2015),toHVAC zones, unpredictability of outside temperature, sumes of the energy resources. According India solar wind, humidity, etc. In energy securities scenarios IESS2047 (2015), we present aairradiation, method that identifies the parameters of consumes about 55% of all 2047, the energy consumed byHVAC com- gains, gains, solar irradiation, wind, humidity, etc. In this this paper, paper, energy most securities scenarios 2047, IESS2047 (2015),by HVAC we present method that identifies the parameters of a a consumes about 55% of all the energy consumed comgains, solarmodel wind, humidity, etc. Indisturbance this paper, energy securities scenarios 2047, IESS2047 (2015),by we present aairradiation, method that identifies the parameters of consumes about 55% of all the energy consumed comnonlinear from the input-output and mercial sector buildings in India and thus provides aHVAC great we present method that identifies the parameters of a a consumes about 55% of all the energy consumed by comnonlinear model from the input-output and disturbance mercial sector buildings in India and thus provides a great present a method that identifies theThe parameters of a consumes about 55% allIndia the energy consumed by comnonlinear model fromSIMBAD the input-output input-output and disturbance mercial sector buildings in and thus provides aa great data collected from (2005). identification scope for research in of this field so that energy targets in we nonlinear model from the and disturbance mercial sector buildings in India and thus provides great data collected from SIMBAD (2005). The identification scope for research in this field so targets in modelfrom fromSIMBAD the input-output andproblem disturbance mercial sector buildings in India andthat thusenergy provides a great data collected collected from SIMBAD (2005). The The identification scope for research in this field that energy targets in process is as and India be by 2047. data (2005). identification scope could for research in this field so so that energy targets in nonlinear process is formulated formulated as an an optimization optimization problem and India could be met met in bythis 2047. data collected from SIMBAD (2005). The identification scope for research field so that energy targets in process is formulated as an optimization problem and India could be met by 2047. solved using nonlinear optimization function available in process is formulated as an optimization problem and India could be met by 2047. solved using nonlinear optimization function available in In a centralized HVAC systems, the thermal energy is ©formulated is as an optimization problem and India could be met by 2047. solved using nonlinear optimization function available MATLAB . In a centralized HVAC systems, the thermal energy is process © solved using in © . nonlinear optimization function available in MATLAB In aa centralized HVAC systems, the thermal energy is © supplied by a central supply subsystem, also known as In centralized HVAC systems, the thermal energy is solved using nonlinear optimization function available in MATLAB© .. supplied by a central supply subsystem, also known as MATLAB In a centralized HVAC systems, the thermal energy is © supplied by aaUnit central supply also known as The identified thermal model of the building can be used Air Handling (AHU), andsubsystem, is distributed throughout supplied by central supply subsystem, also known as MATLAB . identified thermal model of the building can be used Air Handling (AHU), andsubsystem, is distributed supplied by with aUnit central supply alsothroughout known as The The identified model ofmonitoring the can Air Handling Unit (AHU), is throughout for control and purposes. The the building the help ofand multiple end-use subsystems Theenergy-efficient identified thermal thermal model the building building can be be used used Air building Handling Unit (AHU), and is distributed distributed throughout for energy-efficient control andof monitoring purposes. The the with the help of multiple end-use subsystems The identified thermal model of the building can be used Air Handling Unit (AHU), and is distributed throughout for energy-efficient control and monitoring purposes. The the building with the help of multiple end-use subsystems energy-efficient operation of HVAC system is hampered consisting of variable air volumes (VAV) terminal boxes. for energy-efficient control and monitoring purposes. The the building with the help of multiple end-use subsystems energy-efficient operation of HVAC system is hampered consisting of variable air volumes (VAV) terminal boxes. for energy-efficient control and monitoring purposes. The the building with the help of multiple end-use subsystems energy-efficient operation of HVAC systemsuch is hampered hampered consisting of variable air volumes (VAV) terminal boxes. by the occurrence of faults or malfunctions as broken The AHU based on the requirement may appropriately energy-efficient operation of HVAC system is consisting of variable air volumes (VAV) terminal boxes. by the occurrence of faults of or HVAC malfunctions such as broken The AHU of based on the requirement mayterminal appropriately energy-efficient operation system is hampered consisting variable air volumes (VAV) boxes. by the occurrence of faults or malfunctions such as broken The on requirement may appropriately fan fault, air stuck fault, etc. process air based by filtering, heating, ventilating and by the occurrence of faultsfouling, or malfunctions such as broken The AHU AHU based on the thecooling, requirement mayventilating appropriately fan fault, air or or water-side water-side fouling, stuck damper damper fault, etc. process air by filtering, heating, and by the occurrence of faults or malfunctions such as broken The AHU based on thecooling, requirement appropriately fan fault, air or fouling, stuck damper fault, etc. process air by filtering, cooling, heating, ventilating and A by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air.may Following which fanreview fault, air or water-side water-side fouling, stuck damper fault, etc. process air by filtering, cooling, heating, ventilating and A review by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air. Following which fan fault, air or water-side fouling, stuck damper fault, etc. process air by filtering, cooling, heating, ventilating and A review by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air. Following which mary of automated fault diagnosis methods and their charVAV terminal boxes supplies the appropriate volumes of A review by Kim and Katipamula (2018) provides a sumadjusting the boxes humidity of supply air. Following which mary of automated fault diagnosis methods and their charVAV terminal supplies the appropriate volumes of A review by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air. Following which mary of of automated automated fault diagnosis methods and their charcharVAV boxes supplies the appropriate volumes of acterization based the likelihood of occurrences in processed air zone by adjusting damper fault their VAV terminal terminal boxeseach supplies the appropriate volumes of mary based on on thediagnosis likelihoodmethods of fault fault and occurrences in processed air into into each zone the by appropriate adjusting the the damper mary of components automated fault diagnosis methods and their charVAV terminal boxes supplies volumes of acterization acterization based on the likelihood of fault occurrences in processed air into each zone adjusting the damper various of a building. Shahnazari et al. (2018) position thereby achieving andby maintaining thermal acterization based on the likelihood of fault occurrences in processed air into each zone by adjusting the damper various components of a building. Shahnazari et al. (2018) position thereby achieving and maintaining the thermal acterization based on the likelihood of fault occurrences in processed air into each zone by adjusting the damper various components of a building. Shahnazari et al. (2018) position thereby achieving and maintaining thermal presents an integrated framework for fault detection and comfort of the user. various components of aframework building. Shahnazari et al. (2018) position thereby achieving and maintaining the thermal presents an integrated for fault detection and comfort of the user. various components of a building. Shahnazari et al. (2018) position thereby achieving and maintaining the thermal presents an integrated framework for fault detection and comfort of the user. isolation and fault-tolerant control of variable air volume presents an integrated framework for fault detection and comfort of the of user. andintegrated fault-tolerant controlfor of variable air volume The objective maintaining the user’s thermal comfort isolation presents an framework fault detection and comfort of the of user. isolation and fault-tolerant fault-tolerant control of variable variable air volume volume boxes. Thumati et al. (2011) presented an observer-based isolation and control of air The objective maintaining the user’s thermal comfort boxes. Thumati et al. (2011) presented an observer-based The objective of maintaining the user’s thermal comfort is achieved by ofinstalling an energy efficient control and boxes. The objective maintaining the user’s thermal comfort isolation and fault-tolerant control of variable air volume Thumati et al. (2011) presented an observer-based fault diagnosis systems, which online boxes. Thumatifor et HVAC al. (2011) presented anhas observer-based is achieved by installing an energy efficient control and The objective maintaining the user’s thermal comfort diagnosis for systems, which online fault fault is by an efficient control and monitoring system that a desired flow is achieved achieved by ofinstalling installing an energy energy efficient control and fault boxes. Thumati et HVAC al. Liang (2011) presented anhas observer-based fault diagnosis for HVAC systems, which has online fault learning capabilities. and Du. (2007) uses support monitoring system that provides provides a efficient desired volume volume flow fault diagnosis for HVAC systems, which has online fault is achieved by installing an energy control and learning capabilities. Liang and Du. (2007) uses support monitoring desired volume flow of processedsystem air to that each provides zone. To aadesign a model-based monitoring system that provides desired volume flow fault diagnosis for HVAC systems, which has online fault learning capabilities. Liang and Du. (2007) Weimer uses support support vector machines method for fault diagnosis. et al. learning capabilities. Liang and Du. (2007) uses of processed air to each zone. To design a model-based monitoring system that provides a desired volume flow vector machines method for fault diagnosis. Weimer et al. of processed air to each zone. To design aa model-based control and monitoring system, a mathematical model is learning of processed air to each zone. To design model-based capabilities. Liang and Du. (2007) uses support vector machines machines method forfault fault by diagnosis. Weimer etapal. (2012) diagnoses actuator using a two-tier vector method for fault diagnosis. Weimer et al. control and monitoring system, a mathematical model is of processed air to each zone. To design a model-based (2012) diagnoses actuator fault by using a two-tier apcontrol and monitoring system, a mathematical model is control andis monitoring system,byaMeitY, mathematical model is (2012) vector machines method for fault by diagnosis. etapal. diagnoses actuator fault using aaWeimer two-tier proach which includes a dynamic model based detector This work financially supported Govt. of India, under (2012) diagnoses actuator fault by using two-tier ap control andis monitoring system,byaMeitY, mathematical model is proach which includes a dynamic model based detector This work financially supported Govt. of India, under (2012) diagnoses actuator fault by using a two-tier ap theThis Visvesvaraya Ph.D Scheme and SERB DST under the grant proach which includes a dynamic model based detector work is financially supported by MeitY, Govt. of India, under and fast deciding steady state detector. The current trends proach includes dynamic model detector This work is financially supported MeitY, Govt.under of India, the Visvesvaraya Ph.D Scheme and by SERB - DST the under grant and fastwhich deciding steady aastate detector. Thebased current trends This proach includes dynamic model detector work is ECR/2016/001025. financially supported MeitY, Govt.under of India, agreement no. the Visvesvaraya Ph.D and SERB -- DST the grant and fast fastwhich deciding steady state state detector. Thebased current trends the Visvesvaraya Ph.D Scheme Scheme and by SERB DST under the under grant and deciding steady detector. The current trends agreement no. ECR/2016/001025. the Visvesvaraya Ph.D Scheme and SERB - DST under the grant agreement no. ECR/2016/001025. ECR/2016/001025. and fast deciding steady state detector. The current trends agreement no.
agreement no.2018, ECR/2016/001025. 2405-8963 © © IFAC (International Federation of Automatic Control) Copyright 2018 IFAC 544 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 544 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 544 Copyright © 2018 IFAC 544 10.1016/j.ifacol.2018.09.629 Copyright © 2018 IFAC 544
IFAC SAFEPROCESS 2018 Warsaw, Poland, August 29-31, 2018
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and challenges in fault detection and diagnosis (FDD) for smart buildings is discussed by Lazarova-Molnar et al. (2016).
Zone1
Tw5
N
Zone2 Tw2
E
Tw3
Zone3
Tw4 5.5 m
2. SYSTEM DESCRIPTION
Tw1 6.5 m
The main contributions of the paper are: first, we present a method to identify parameters of the physics-based nonlinear model of the building thermal dynamics. Secondly, we design a complete fault diagnosis system by using the identified nonlinear model for detecting and estimating stuck faults in the VAV dampers. The novelty of the FDD system lies in its simple construction, which makes it widely applicable for diagnosing a class of actuator faults in HVAC systems.
545
Tw6
9m
5m
Fig. 1. Three-zones in one-storey building scenario x˙ = Ax + Bu + Du (t)x + Ed y = Cx 3 ˙ i (t)Ai , where with Du (t) = i=1 m T x(t) = TzT TwT ˙2 m ˙ 3] ˙1 m u(t) = [m y(t) = Tz
2.1 Modelling of three-zones in a building The system considered is comprised of three-zones in a single-storey building Yam´e et al. (2015). The layout of the building is shown in the Fig. 1. For each zone i, (i = 1, 2, 3), denote the temperature of zone by Tzi , the mass flow rate at the output of the i-th VAV by m ˙ i , and the supply air temperature by Ts , and the walls temperature by Tw . There are a total of six walls, and their internal wall temperature is denoted by Twj , (j = 1, 2, . . . , 6). Three of them is a common walls between the zones, while the other interacts with the outside environment. Applying the first law of thermodynamics to each zone, the dynamics of the zones temperatures is described by dTz1 = k01 m˙ 1 (Ts − Tz1 ) + k11 (Tw1 − Tz1 ) dt +k12 (Tw4 − Tz1 ) + k13 (Tw5 − Tz1 ) +k14 (TO − Tz1 ) + k15 q1 dTz2 = k02 m˙ 2 (Ts − Tz2 ) + k21 (Tw2 − Tz2 ) dt (1) +k22 (Tw4 − Tz2 ) + k23 (Tw6 − Tz2 ) +k24 (TO − Tz2 ) + k25 q2 dTz3 = k03 m˙ 3 (Ts − Tz3 ) + k31 (Tw3 − Tz3 ) dt +k32 (Tw5 − Tz3 ) + k33 (Tw6 − Tz3 ) +k34 (TO − Tz3 ) + k35 q3 with the dynamics of the temperature of walls, given by dTw1 = k41 (Tz1 − Tw1 ) + k42 (TO − Tw1 ) dt dTw2 = k51 (Tz2 − Tw2 ) + k52 (TO − Tw2 ) dt dTw3 = k61 (Tz3 − Tw3 ) + k62 (TO − Tw3 ) dt (2) dTw4 = k71 (Tz1 − Tw4 ) + k72 (Tz2 − Tw4 ) dt dTw5 = k81 (Tz1 − Tw5 ) + k82 (Tz3 − Tw5 ) dt dTw6 = k91 (Tz2 − Tw6 ) + k92 (Tz3 − Tw6 ) dt where kmn , m = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), n = (1, 2, 3, 4, 5), are constant coefficients, TO is the outside air temperature, and qi is the heating load in the i-th zone. At a constant supply air temperature, equations (1), (2) can be lumped together into the following bilinear state-space form 545
(3)
T
T
d(t) = [q1 q2 q3 TO ] , T
T
Tz = [Tz1 Tz2 Tz3 ] , Tw = [Tw1 Tw2 Tw3 Tw4 Tw5 Tw6 ] . The parameters A, Ai , B, E in (3) are matrices of appropriate dimension and C = [I3 03×6 ], where I• is an identity matrix of dimension • and 0• denotes a zero matrix of dimension •. The state of the bilinear system (3) can also be represented as x˙ = Ax + Bu + A(u ⊗ x) + Ed (4) where the symbol ⊗ denotes the Kronecker product and A = [A1 A2 A3 ] is the bilinearity matrix. 2.2 Fault modelling Under the event of actuator faults, the mass flow rate at the output of VAVs, denoted by uf (t), is given by uf (t) = u(t) + fa (t), (5) T
T
˙ 2,f m ˙ 3,f ] , fa = [f1 f2 f3 ] . Here, ˙ 1,f m where uf = [m we consider stuck fault in VAV dampers. That is, for the i-th VAV, its output signal m ˙ i,f is a constant. Under actuator fault scenario, the state-space system is given by x˙ = Ax + Bu + Du (t)x + Ed + F (t)fa , (6) where F = [F1 (t) 06×3 ], with F1 = diag(k01 (Ts − Tz1 ), k02 (Ts −Tz2 ), k03 (Ts −Tz3 )) and 06×3 denotes the zero matrix, is a time-varying matrix of appropriate dimension. The problem we aim to solve is to detect the occurrence of actuator fault and estimate precisely its magnitude for control and monitoring purposes. 3. MODEL IDENTIFICATION SIMBAD supplies a black-box model under the Simulink environment, which is illustrated in Fig. 2. For the development of a model-based fault diagnosis system, we present a Airßow Setpoint
Heating load
Flow controller Flow sensor
Damper & Actuator
Zone dynamics
VAV box
Fig. 2. Structure of the system in SIMBAD
Zones Temp.
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method for identifying the parameters of the model (3). All variables in (3) are measured from SIMBAD, except the walls temperature. Using the input-output and measured disturbance data, parameters of the model are estimated using a nonlinear optimization algorithm. To proceed with the identification, we discretized the system (3) using a forward differentiation rule with a sample time of h x(k + 1) = [Ah + I]x(k) + [Bu(k) + Du (k)x(k) + Ed(k)]h. (7) ˜ = Bh, D ˜ u (k) = Du (k)h, Denoting A˜ = Ah + I , B ˜ = Eh, (7) can be rewritten as E ˜ ˜ ˜ ˜ u (k)x(k) + Ed(k) x(k + 1) = Ax(k) + Bu(k) +D (8) y(k) = Cx(k) Let us denote the estimated variables by T˜z and T˜w . At k = 0, we use the available data and the initial condition of walls temperature Tw (0) to get T˜z (1) ˜ Tz (0) + Bu(0) ˜ ˜ ˜ u (0) Tz (0) + Ed(0) = A + D Tw (0) Tw (0) T˜w (1) (9) for k ≥ 1, the estimate is computed as T˜z (k + 1) ˜ Tz (k) + Bu(k) ˜ = A T˜w (k) T˜w (k + 1) (10) ˜ ˜ u (k) Tz (k) + Ed(k) +D T˜w (k) y˜ = T˜z The objective function y − y˜ 22 is minimized using the nonlinear least square optimization method available in MATLAB to obtain the discrete-time state equation pa˜ B, ˜ D ˜ u and E. ˜ Subsequently, the continuous rameters A, ˜ time parameters are computed by A = h1 (A˜ − I), B = h1 B, 1 ˜ 1 ˜ Du = h Du and E = h E. The identified parameters are given in the Table 1. Table 1. Identified Parameter values Parameter k01 k12 k14 k02 k22 k24 k03 k32 k34 k41 k51 k61 k71 k81 k91
Value 2.0427 × 10−04 5.3023 × 10−05 3.6321 × 10−06 2.3754 × 10−04 1.2005 × 10−04 2.5815 × 10−06 1.8038 × 10−04 9.3119 × 10−05 2.1423 × 10−06 1.7207 × 10−05 9.4988 × 10−07 1.8324 × 10−05 2.7660 × 10−14 2.8143 × 10−14 1.7356 × 10−05
Parameter k11 k13 k15 k21 k23 k25 k31 k33 k35 k42 k52 k62 k72 k82 k92
Value 1.5534 × 10−04 4.4707 × 10−05 1.8629 × 10−07 1.1537 × 10−05 1.2700 × 10−04 2.2873 × 10−07 1.3383 × 10−04 2.8062 × 10−14 1.6678 × 10−07 9.3684 × 10−07 1.9566 × 10−05 1.8321 × 10−06 9.9811 × 10−05 9.7686 × 10−05 2.7487 × 10−06
Fig. 3. Block structure of the observer ˆ + Ed x ˆ˙ = Aˆ x + Bu + Du x − (C T LCAC T + C T LCDu C T )(y − yˆ) yˆ = C x ˆ
(11)
with L ∈ R3×3 denoting the gain matrix.
The block diagram of the observer is illustrated in Fig. 3. Here the objective is to synthesize the matrix L of the observer such that the error, defined by, e = x − x ˆ tends to 0 asymptotically. To prove this, first we compute the derivative of the error given by e˙ = x˙ − x ˆ˙ . Substituting x˙ ˙ and x ˆ from (3) and (11) respectively, yields e˙ = (A + C T LCAC T C)e + (Du + C T LCDu C T C)e, which can be written as e˙ = (A¯ + H(t))e,
(12)
where A¯ = (A + C T LCAC T C) and H(t) = (Du + C T LCDu C T C). The following result would be helpful to establish the stability of the error dynamics and subsequently compute the L matrix. Lemma 1. Chen (1995) Assume that all eigenvalues of A¯ have negative real parts, then for any given positive definite symmetric matrix N , the Lyapunov equation A¯T M + M A¯ = −N
has a unique symmetric solution M which is positive definite. n Definition 2. For any vector x ∈ Rn , x 2 = ( i=1 x2i )1/2 1/2 and for any matrix A ∈ Rm×n , A 2 = λmax (AT A). Theorem 3. Assume that all eigenvalues of A¯ have negative real parts, there exists a positive definite symmetric matrix N such that H(t) satisfies H(t) 2 <
λmin (N ) 2λmax (M )
where M is the solution of the Lyapunov equation, then the system e˙ = (A¯ + H(t))e
4. NONLINEAR FAULT DIAGNOSIS In this section, we describe the nonlinear observer-based fault diagnosis system for HVAC . Let the state vector of the observer is denoted by x ˆ. We propose the dynamics of the observer as 546
is asymptotically stable. Proof. We use V (e) = eT M e as a Lyapunov function candidate. Consider g(e) = H(t)e, the derivative of V (x) along the trajectories of the system is given by
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¯ + g(e)) + eT (A¯T + g T (e))M e V˙ (x) = eT M (Ae T = −e (M A¯ + A¯T M )e + 2eT M g(e)
= −eT N e + 2eT M g(e) The first term on the right-hand side is negative definite, while the second term is (in general) indefinite. Since g(e) is continuous, it satisfies g(e) 2 → 0 as e 2 → 0 e 2 Therefore, for any γ = H(t) 2 , there exists r > 0 such that g(e) 2 < γ e 2 , ∀ e 2 < r Hence, V˙ (e) < −eT N e + 2 M 2 H(t) 2 e 22 , ∀ e 2 < r But eT N e ≥ λmin (N ) e 22 where λmin (•) denotes the minimum eigenvalue of a matrix. Note that λmin (N ) is real and positive since N is symmetric and positive definite. Thus, V˙ (e) < −[λmin (N ) − 2 M 2 H(t) 2 ] e 22 , ∀ e 2 < r Choosing λmin (N ) λmin (N ) H(t) 2 < = 2 M 2 2λmax (M ) ensures that V˙ (e) is negative definite. Hence, the origin is asymptotically stable. ¯ C, ¯ where The matrix A¯ can be written as A¯ = A − L ¯ = C T L and C¯ = −CAC T C and now L ¯ is the matrix L to be computed which is in fact equivalent to computing ¯ is observable. the matrix L. Note that the pair (A, C) ¯ by selecting We can arbitrarily place the eigenvalues of A, the L matrix, on the left-hand side of the complex plane. The damper position of a VAV box is limited to operate into the range [0, 1], where ‘0’ denotes fully close and ‘1’ denotes fully open. Depending on the damper pressure, the flow meter imposes a limitation on the amount of airflow rate, u(t), i.e. umin ≤ u(t) ≤ umax with umin and umax denoting the minimum value and the maximum value of the airflow rate respectively. Consequently, we choose an appropriate L matrix such that the matrix A¯ is Hurwitz and the upper bound on the Euclidean norm of H(t) is satisfied. Whenever an actuator fault occurs, the error dynamics is given as e˙ = (A¯ + H(t))e + F (t)fa . For fault detection, we compute the residual vector by ri (t) = yi (t) − yˆi (t) (13) where ri (t) denotes the residual of the i-th zone. The detection and isolation logic under a fault-free damper or a stuck in damper for the i-th VAV is then given by ri (t) < th, no actuator fault (14) ri (t) ≥ th, fault occurred in the i-th actuator where th is a given threshold value. To achieve the complete fault diagnosis for HVAC systems, an estimation of the stuck is required. Let the matrices in the system equa A11 A12 B1 tion be partitioned such that, A = ,B= , A A B2 21 22 E1 D11 (t) D12 (t) . For the considered and E = Du = E2 D21 (t) D22 (t) 547
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three zone system, we have B2 = D12 (t) = D21 (t) = D22 (t) = 0. Taking the derivative of the output variable in (3) yields y˙ = C x˙ B1 D11 (t) 0 Tz A11 A12 Tz + u+ =C 0 0 Tw 0 A21 A22 Tw E F + 1 d + 1 fa E2 0 T T˙z = [A11 A12 ] z + B1 u + D11 (t)Tz + E1 d + F1 fa Tw (15) Note that we do not have a direct measurement of the temperature of the walls. From the experimental analysis, it is observed that the effect of the fault on the dynamics of wall temperatures is very slow and the deviation is also of very low magnitude. Therefore, we use the wall temperatures from the nonlinear observer for the estimation of the fault. The fault estimation filter is expressed as Tz fˆa (t) = F1−1 T˙z − [A11 A12 ] ˆ + B1 u Tw (16) +D11 (t)Tz + E1 d
where fˆa (t) denotes the estimation of the fault. Here, the matrix F1 is always nonsingular. The estimation of the fault using (16) is triggered whenever it is detected by the fault detection logic (14). 5. SIMULATION RESULTS
To set-up our case-study for simulation purposes, we used SIMBAD (SIMulator of Building And Devices), which is a Matlab/Simulink toolbox developed by CSTB SIMBAD (2005) primarily for building simulation. This simulator enables a powerful and in-depth implementation of any real-life building with all physical phenomena of zones and large HVAC library database El Khoury et al. (2005). 5.1 Three-zones building setup Using SIMBAD simulator, we build a one-story building consisting of three zones to demonstrate the effectiveness of the developed fault diagnosis system. The considered case-study is illustrated in Fig. 1, where the net area of the building is 168 square meters. The building is divided into three zones: Zone-1 and Zone-3 have two windows while Zone-1 has one, where it is assumed that the windows blinds are always closed. The internal gains within this building are defined by the occupancy characteristics which consists of the occupancy type and a number of occupants. The type of occupancy is selected as “Offices” where the working time of occupants is modeled from 08h00 to 18h00 with a lunch break of two hours starting at 12h00. We considered six occupants in Zone-1, four occupants in Zone-2, while five occupants in Zone-3. We also modeled constant interzonal mass flows between each zones. The ventilation system in each zone has two air-flow paths inside the zone, where one path allows the exterior air-flow while the other air-flow path is controlled by the VAV box to maintain the thermal comfort of occupants.
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The VAV terminal box consists of the grid model, the 3way valve model and the detailed dynamic heating coil (the reheat coil), where the VAV damper process is composed of a damper with its driving motor and a flow sensor to measure the supply air-flow rate. The motorized damper model takes into account the hysteresis of the shaft and the type of blades to determine the air flow rate through the damper. The position of the damper is used to control the air-flow rate and its input-output relationship is generally nonlinear. In the considered scenario, the minimum and maximum airflow rate are umin = 0.49 × 10−3 kg/s and umax = 0.3163kg/s.
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Fig. 4. Outside temperature, Supply temperature and Internal gains data used for model identification and validation A nonlinear model is identified using the procedure as described in section 3 with temperatures of walls and zones initialized to T˜w (0) = T˜z (0) = 20°C. The output of the identified system is compared with the output of the benchmark model by supplying both the system and the model with the same input and disturbance signals. Fig. 5 illustrates the comparison between the response of the system and the identified model. It is observed that the identified model closely matches with the actual system. The identified model is validated by comparing the physical system with the model for a different set of seeded random signals as input flow for seven days. The outside temperature, supply temperature and the internal gains are shown in Fig. 4. Corresponding to the stimulus given to model and system, the zone temperature response obtained is illustrated in the Fig. 6. It is observed that the model identified closely matches the actual system for a different set of signals as well. A slight mismatch occurs after the sixth day, which is possibly due to a sudden change in the outside air temperature. However, the dynamics of the identified model is similar to the benchmark model. 548
Fig. 6. Validation of the identified model using a seven days data with a different set of input signals 5.3 Results of fault diagnosis The observer is designed by choosing the matrix L = 60 × I3 such that condition of Theorem 3 is satisfied. We considered three faults scenario, where a damper stuck fault of different magnitude occurs in all zones at the end of the day 1. The damper of zone 1 gets stuck at a position of 0.6, while the damper of zone 2 and zone 3 get stuck at a position of 0.5 and 0.65 respectively. Due to these faults the airflow into the zone 1 is now constant at 0.0417kg/s, airflow into zone 2 at 0.0238kg/s and airflow into zone 3 at 0.0545kg/s. The air temperature of the outside environment, the supply air temperature and the internal gains are shown in Fig. 4. The airflow set-point to the VAV box (see Fig. 2) is kept at a constant level of 0.16kg/s, but due to faults the actuator is unable to supply the desired flow rate. Consequently, there is a drop in the temperature of the zones as illustrated in Fig. 7, which affects the thermal comfort of the users. It is worth observing that the observer successfully detects the faults, where the threshold value is th = 0.0145. The residual vector is shown in Fig. 8. Subsequently, using (16), the fault vector fˆa is estimated. The estimate of airflows delivered by VAVs is given by u ˆf (t) = u(t) + fˆa (t), which
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systems. A real-life case-study of the three-zones onestorey building scenario is implemented in the SIMBAD simulator, which shows that the proposed fault diagnosis strategy has a great potential in practical applications for energy efficient fault-tolerant control (FTC) in building automation systems. A preliminary integrated FDD and FTC approach for HVAC systems has been presented in A and Jain (2018).
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6. CONCLUSION In this paper, we presented a nonlinear observer-based fault diagnosis method for detecting and estimating stuck faults in VAV dampers. Due to the occurrence of these faults, the thermal comfort is heavily affected as well as deteriorating the energy-efficient operation of HVAC 549
A, M.S. and Jain, T. (2018). Fault tolerant economic model predictive control for energy efficiency in a multizone building. In 2nd IEEE Conference on Control Technology and Applications, August 21-24. Copenhagen, Denmark. Chen, C. (1995). Linear system theory and design. Oxford University Press, Inc. El Khoury, Z., Riederer, P., Couillaud, N., Simon, J., and Raguin, M. (2005). A multizone building model for MATLAB/Simulink Environment. In Ninth International IBPSA Conference, 525–532. Montr´eal, Canada. IESS2047 (2015). Commercial lighting and appliances documentation. Technical report, India Energy Security Scenarios 2047, Govt. of India, http://indiaenergy.gov.in/iess/docs/CommercialLighting˙Appliances-documentation.pdf. Kim, W. and Katipamula, S. (2018). A review of fault detection and diagnostics methods for building systems. Science and Technology for the Built Environment, 24(1), 3–21. Lazarova-Molnar, S., Shaker, H.R., Mohamed, N., and Jørgensen, B.N. (2016). Fault detection and diagnosis for smart buildings: State of the art, trends and challenges. In 2016 3rd MEC International Conference on Big Data and Smart City (ICBDSC), 1–7. Muscat, Oman. Liang, J. and Du., R. (2007). Model-based fault detection and diagnosis of hvac systems using support vector machine method. International Journal of Refrigeration, 30, 1104–1114. Shahnazari, H., Mhaskar, P., House, J.M., and Salsbury, T.I. (2018). Heating, ventilation and air conditioning systems: Fault detection and isolation and safe parking. Computers and Chemical Engineering, 108, 139–151. SIMBAD (2005). Centre scientifique et technique du batiment, france. SIMBAD building and HVAC toolbox. Thumati, B.T., Feinstein, M.A., Fonda, J.W., Turnbull, A., Weaver, F.J., Calkins, M.E., and Jagannathan, S. (2011). An online model-based fault diagnosis scheme for hvac systems. In IEEE International Conference on Control Applications (CCA), 70–75. Denver, CO, USA. Weimer, J., Ahmadi, S.A., Araujo, J., Mele, F.M., Papale, D., Shames, I., Sandberg, H., and Johansson, K.H. (2012). Active actuator fault detection and diagnostics in hvac systems. In Proceedings of the Fourth ACM Workshop on Embedded Sensing Systems for EnergyEfficiency in Buildings, BuildSys ’12, 107–114. ACM, Toronto, Ontario, Canada. Yam´e, J., Jain, T., and Sauter, D. (2015). An online controller redesign based fault-tolerant strategy for thermal comfort in a multi-zone building. In IEEE Conference on Control Applications, 1901–1906. Sydney, NSW, Australia.
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Nonlinear Nonlinear Observer-based Observer-based Fault Fault Diagnosis Diagnosis Nonlinear Observer-based Fault for a Multi-Zone Building Nonlinear Observer-based Fault Diagnosis Diagnosis for a Multi-Zone Building for a Multi-Zone Building for a Multi-Zone Building Mona Subramaniam A, Tushar Jain
Mona Subramaniam A, Tushar Jain Mona Subramaniam Subramaniam A, A, Tushar Tushar Jain Jain Mona Mona Subramaniam A, Tushar Jain School of Computing and Electrical Engineering, School of Computing and Electrical Engineering, School of Computing and Electrical Engineering, Indian Institute of Technology Mandi, India175005 SchoolInstitute of Computing and Electrical Engineering, Indian of Technology Mandi, India175005 School of Computing and Electrical Engineering, Indian Institute of Technology Mandi, India(e-mail: [email protected], [email protected]). Indian Institute of Technology Mandi, India- 175005 175005 (e-mail: [email protected], [email protected]). Indian Institute of Technology Mandi, India175005 (e-mail: (e-mail: [email protected], [email protected], [email protected]). [email protected]). (e-mail: [email protected], [email protected]). Abstract: Abstract: Heating, Heating, Ventilation Ventilation and and Air Air Conditioning Conditioning (HVAC) (HVAC) systems systems are are installed installed in in comcomAbstract: Heating, Ventilation and Air Conditioning (HVAC) systems are installed in mercial buildings to meet the actual demand of heating and cooling at a given time such that Abstract: Heating, Ventilation and Air Conditioning (HVAC) systems are installed in comcommercial buildings to meet the actual demand of heating and cooling at a given time such that Abstract: Heating, Ventilation and Air the Conditioning (HVAC) systems installed in commercial buildings to meet the actual demand of heating and cooling at aa are given time such that the thermal comfort of occupants within rooms (or zones) can be ensured. This is achieved mercial buildings to meet the actual demand of heating and cooling at given time such that the thermal comfort of occupants within the rooms (or zones) can be at ensured. This issuch achieved mercial buildings tothe meet theatactual demand of heating and cooling a volume givenThis time that the thermal comfort of occupants within the rooms (or zones) can be ensured. This is achieved achieved by feeding air into zones a constant temperature using variable air (VAV) boxes, the thermal comfort of occupants within the rooms (or zones) can be ensured. is by feeding air into the at a constant temperature using variable air volume (VAV) boxes, the thermal comfort of zones occupants withinfor the (or zones) can The be ensured. This is achieved by air into the zones at aasolution constant temperature using variable air volume (VAV) boxes, which offers an energy-efficient aarooms multi-zone building. amount of supply air by feeding feeding air into the zones at constant variable volume which offersair aninto energy-efficient solution fortemperature multi-zoneusing building. Theair amount of(VAV) supplyboxes, air is is by feeding the zones at a constant temperature using variable air volume (VAV) boxes, which offers an energy-efficient solution for aa within multi-zone building. The amount of supply supply air is controlled with the help of dampers located the VAV boxes. Any occurrence of a fault which offers an energy-efficient solution for multi-zone building. The amount of air is controlled with the help of dampers located within the VAV boxes. Any occurrence of a fault which offers an energy-efficient solution for a within multi-zone building. The amount of and supply air is controlled with the help of dampers located within the VAV boxes. Any occurrence ofa aaloss fault in VAV dampers may cause undesirable results such as discomfort to occupants in controlled with the help of dampers located the VAV boxes. Any occurrence of fault in VAV dampers may cause undesirable results such as discomfort to occupants and aa aloss in controlled with the help of dampers located within the VAV boxes. Any occurrence of fault in VAV dampers may cause undesirable results such as discomfort to occupants and loss in energy efficiency. In this paper, we propose a nonlinear observer-based fault diagnosis method in VAVefficiency. dampers may cause undesirable results such as observer-based discomfort to occupants and amethod loss in energy In this paper, we propose a nonlinear fault diagnosis in VAV dampers may cause undesirable results such as observer-based discomfort to occupants and can amethod loss in energy efficiency. In this paper, we propose aa nonlinear fault diagnosis to extract a precise about the fault so that subsequently, the take energy efficiency. In information this paper, we propose nonlinear observer-based faultcontroller diagnosis method to extract a precise information about the fault so that subsequently, the controller can take energy efficiency. In this paper, we propose a nonlinear observer-based faultcontroller diagnosis method to extract aa precise information about the fault so that subsequently, the controller can take corrective actions ensuring the thermal comfort of the occupants. The novelty of this method lies to extract precise information about the fault so that subsequently, the can take corrective actions ensuring the thermal comfort of so thethat occupants. The novelty of this method lies to extract a precise information about the fault subsequently, the controller can take corrective actions ensuring the thermal comfort of The novelty of this this method method lies in its simple construction, which makes it applicable for diagnosing wide variety of actuator corrective actions ensuring the thermal comfort of the the occupants. occupants. The aanovelty of lies in its simple construction, which makes it applicable for diagnosing wide variety of actuator corrective actions ensuring the thermal comfort of the occupants. The novelty of this method lies in its simple construction, which makes it applicable for diagnosing a wide variety of actuator faults in HVAC systems. The effectiveness of the proposed method is successfully demonstrated in its simple construction, which makes it of applicable for diagnosing wide variety of actuator faults in HVAC systems. The effectiveness the proposed method is aasuccessfully demonstrated in its simple construction, which makes comprising it of applicable for diagnosing wide variety ofSIMBAD actuator faults in HVAC systems. The effectiveness the proposed method is successfully demonstrated on a case-study of a one-storey building of three zones, constructed using faults in HVAC systems. The effectiveness of the proposed method is successfully demonstrated on a case-study of a one-storey building comprising of three zones, isconstructed SIMBAD faults in HVAC systems. TheDevices). effectiveness of the proposed method successfullyusing demonstrated on of building of zones, using SIMBAD (SIMulator of Building And on aa case-study case-study of aa one-storey one-storey building comprising comprising of three three zones, constructed constructed using SIMBAD (SIMulator of Building And Devices). on a case-study of a one-storey building comprising of three zones, constructed using SIMBAD (SIMulator of And (SIMulator of Building Building And Devices). Devices). © 2018, IFACof(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. (SIMulator Building And Devices). Keywords: Keywords: Fault Fault Diagnosis, Diagnosis, HVAC, HVAC, Nonlinear Nonlinear models, models, Observers, Observers, Parameter Parameter identification, identification, Keywords: Fault Diagnosis, HVAC, Nonlinear models, Observers, Parameter identification, System identification, Thermal comfort, Multizone building, Energy efficiency. Keywords: Fault Diagnosis, HVAC, Nonlinear models, Observers, Parameter identification, System identification, Thermal comfort, Multizone building, Energy efficiency. Keywords: Fault Diagnosis, HVAC, Nonlinear models, Observers, Parameter identification, System Thermal comfort, Multizone building, Energy efficiency. System identification, identification, Thermal comfort, Multizone building, Energy efficiency. System identification, Thermal comfort, Multizone building, Energy efficiency. 1. required, 1. INTRODUCTION INTRODUCTION required, whose whose dynamics dynamics should should match match precisely precisely with with the the 1. required, whose whose dynamics should The match precisely with with dynamics of the real-life system. identification of the 1. INTRODUCTION INTRODUCTION required, dynamics should match precisely dynamics of the real-life system. The identification of the 1. INTRODUCTION should match precisely with the dynamics whose ofofthe the real-life system. system. The identification of parameters aa dynamics physics-based mathematical model poses HVAC (Heating, Ventilation and Air Conditioning) sys- required, dynamics of real-life The identification the parameters ofthe physics-based mathematical model of poses HVAC (Heating, Ventilation and Air Conditioning) sysdynamics of real-life system. The identification of the parameters of a physics-based mathematical model poses HVAC (Heating, Ventilation and Air Conditioning) syschallenges due factors strong between tems which are anVentilation integral part modern buildings conof ato physics-based mathematical model poses HVACwhich (Heating, andof Air Conditioning) sys- parameters due factors such such as as strong coupling coupling between tems are an integral part modern buildings conparameters of ato mathematical model poses HVAC (Heating, Ventilation andof Air Conditioning) sys- challenges challenges due to factors such as strong coupling between tems which are an integral part of modern buildings conzones, unpredictability outside temperature, internal sumes most of the energy resources. According to India challenges due tophysics-based factorsof such as strong coupling between tems which are an integral part of modern buildings conzones, unpredictability of outside temperature, internal sumes most of the energy resources. According to India due to factorswind, as strong coupling between tems which are anscenarios integral part of modern buildings con- challenges zones, solar unpredictability ofsuch outside temperature, internal sumes most of the energy resources. According to India gains, irradiation, humidity, etc. In this paper, energy securities 2047, IESS2047 (2015), HVAC zones, unpredictability of outside temperature, internal sumes most of the energy resources. According to India gains, solar irradiation, wind, humidity, etc. In thisinternal paper, energy securities scenarios 2047, IESS2047 (2015),toHVAC zones, unpredictability of outside temperature, sumes of the energy resources. According India solar wind, humidity, etc. In energy securities scenarios IESS2047 (2015), we present aairradiation, method that identifies the parameters of consumes about 55% of all 2047, the energy consumed byHVAC com- gains, gains, solar irradiation, wind, humidity, etc. In this this paper, paper, energy most securities scenarios 2047, IESS2047 (2015),by HVAC we present method that identifies the parameters of a a consumes about 55% of all the energy consumed comgains, solarmodel wind, humidity, etc. Indisturbance this paper, energy securities scenarios 2047, IESS2047 (2015),by we present aairradiation, method that identifies the parameters of consumes about 55% of all the energy consumed comnonlinear from the input-output and mercial sector buildings in India and thus provides aHVAC great we present method that identifies the parameters of a a consumes about 55% of all the energy consumed by comnonlinear model from the input-output and disturbance mercial sector buildings in India and thus provides a great present a method that identifies theThe parameters of a consumes about 55% allIndia the energy consumed by comnonlinear model fromSIMBAD the input-output input-output and disturbance mercial sector buildings in and thus provides aa great data collected from (2005). identification scope for research in of this field so that energy targets in we nonlinear model from the and disturbance mercial sector buildings in India and thus provides great data collected from SIMBAD (2005). The identification scope for research in this field so targets in modelfrom fromSIMBAD the input-output andproblem disturbance mercial sector buildings in India andthat thusenergy provides a great data collected collected from SIMBAD (2005). The The identification scope for research in this field that energy targets in process is as and India be by 2047. data (2005). identification scope could for research in this field so so that energy targets in nonlinear process is formulated formulated as an an optimization optimization problem and India could be met met in bythis 2047. data collected from SIMBAD (2005). The identification scope for research field so that energy targets in process is formulated as an optimization problem and India could be met by 2047. solved using nonlinear optimization function available in process is formulated as an optimization problem and India could be met by 2047. solved using nonlinear optimization function available in In a centralized HVAC systems, the thermal energy is ©formulated is as an optimization problem and India could be met by 2047. solved using nonlinear optimization function available MATLAB . In a centralized HVAC systems, the thermal energy is process © solved using in © . nonlinear optimization function available in MATLAB In aa centralized HVAC systems, the thermal energy is © supplied by a central supply subsystem, also known as In centralized HVAC systems, the thermal energy is solved using nonlinear optimization function available in MATLAB© .. supplied by a central supply subsystem, also known as MATLAB In a centralized HVAC systems, the thermal energy is © supplied by aaUnit central supply also known as The identified thermal model of the building can be used Air Handling (AHU), andsubsystem, is distributed throughout supplied by central supply subsystem, also known as MATLAB . identified thermal model of the building can be used Air Handling (AHU), andsubsystem, is distributed supplied by with aUnit central supply alsothroughout known as The The identified model ofmonitoring the can Air Handling Unit (AHU), is throughout for control and purposes. The the building the help ofand multiple end-use subsystems Theenergy-efficient identified thermal thermal model the building building can be be used used Air building Handling Unit (AHU), and is distributed distributed throughout for energy-efficient control andof monitoring purposes. The the with the help of multiple end-use subsystems The identified thermal model of the building can be used Air Handling Unit (AHU), and is distributed throughout for energy-efficient control and monitoring purposes. The the building with the help of multiple end-use subsystems energy-efficient operation of HVAC system is hampered consisting of variable air volumes (VAV) terminal boxes. for energy-efficient control and monitoring purposes. The the building with the help of multiple end-use subsystems energy-efficient operation of HVAC system is hampered consisting of variable air volumes (VAV) terminal boxes. for energy-efficient control and monitoring purposes. The the building with the help of multiple end-use subsystems energy-efficient operation of HVAC systemsuch is hampered hampered consisting of variable air volumes (VAV) terminal boxes. by the occurrence of faults or malfunctions as broken The AHU based on the requirement may appropriately energy-efficient operation of HVAC system is consisting of variable air volumes (VAV) terminal boxes. by the occurrence of faults of or HVAC malfunctions such as broken The AHU of based on the requirement mayterminal appropriately energy-efficient operation system is hampered consisting variable air volumes (VAV) boxes. by the occurrence of faults or malfunctions such as broken The on requirement may appropriately fan fault, air stuck fault, etc. process air based by filtering, heating, ventilating and by the occurrence of faultsfouling, or malfunctions such as broken The AHU AHU based on the thecooling, requirement mayventilating appropriately fan fault, air or or water-side water-side fouling, stuck damper damper fault, etc. process air by filtering, heating, and by the occurrence of faults or malfunctions such as broken The AHU based on thecooling, requirement appropriately fan fault, air or fouling, stuck damper fault, etc. process air by filtering, cooling, heating, ventilating and A by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air.may Following which fanreview fault, air or water-side water-side fouling, stuck damper fault, etc. process air by filtering, cooling, heating, ventilating and A review by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air. Following which fan fault, air or water-side fouling, stuck damper fault, etc. process air by filtering, cooling, heating, ventilating and A review by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air. Following which mary of automated fault diagnosis methods and their charVAV terminal boxes supplies the appropriate volumes of A review by Kim and Katipamula (2018) provides a sumadjusting the boxes humidity of supply air. Following which mary of automated fault diagnosis methods and their charVAV terminal supplies the appropriate volumes of A review by Kim and Katipamula (2018) provides a sumadjusting the humidity of supply air. Following which mary of of automated automated fault diagnosis methods and their charcharVAV boxes supplies the appropriate volumes of acterization based the likelihood of occurrences in processed air zone by adjusting damper fault their VAV terminal terminal boxeseach supplies the appropriate volumes of mary based on on thediagnosis likelihoodmethods of fault fault and occurrences in processed air into into each zone the by appropriate adjusting the the damper mary of components automated fault diagnosis methods and their charVAV terminal boxes supplies volumes of acterization acterization based on the likelihood of fault occurrences in processed air into each zone adjusting the damper various of a building. Shahnazari et al. (2018) position thereby achieving andby maintaining thermal acterization based on the likelihood of fault occurrences in processed air into each zone by adjusting the damper various components of a building. Shahnazari et al. (2018) position thereby achieving and maintaining the thermal acterization based on the likelihood of fault occurrences in processed air into each zone by adjusting the damper various components of a building. Shahnazari et al. (2018) position thereby achieving and maintaining thermal presents an integrated framework for fault detection and comfort of the user. various components of aframework building. Shahnazari et al. (2018) position thereby achieving and maintaining the thermal presents an integrated for fault detection and comfort of the user. various components of a building. Shahnazari et al. (2018) position thereby achieving and maintaining the thermal presents an integrated framework for fault detection and comfort of the user. isolation and fault-tolerant control of variable air volume presents an integrated framework for fault detection and comfort of the of user. andintegrated fault-tolerant controlfor of variable air volume The objective maintaining the user’s thermal comfort isolation presents an framework fault detection and comfort of the of user. isolation and fault-tolerant fault-tolerant control of variable variable air volume volume boxes. Thumati et al. (2011) presented an observer-based isolation and control of air The objective maintaining the user’s thermal comfort boxes. Thumati et al. (2011) presented an observer-based The objective of maintaining the user’s thermal comfort is achieved by ofinstalling an energy efficient control and boxes. The objective maintaining the user’s thermal comfort isolation and fault-tolerant control of variable air volume Thumati et al. (2011) presented an observer-based fault diagnosis systems, which online boxes. Thumatifor et HVAC al. (2011) presented anhas observer-based is achieved by installing an energy efficient control and The objective maintaining the user’s thermal comfort diagnosis for systems, which online fault fault is by an efficient control and monitoring system that a desired flow is achieved achieved by ofinstalling installing an energy energy efficient control and fault boxes. Thumati et HVAC al. Liang (2011) presented anhas observer-based fault diagnosis for HVAC systems, which has online fault learning capabilities. and Du. (2007) uses support monitoring system that provides provides a efficient desired volume volume flow fault diagnosis for HVAC systems, which has online fault is achieved by installing an energy control and learning capabilities. Liang and Du. (2007) uses support monitoring desired volume flow of processedsystem air to that each provides zone. To aadesign a model-based monitoring system that provides desired volume flow fault diagnosis for HVAC systems, which has online fault learning capabilities. Liang and Du. (2007) Weimer uses support support vector machines method for fault diagnosis. et al. learning capabilities. Liang and Du. (2007) uses of processed air to each zone. To design a model-based monitoring system that provides a desired volume flow vector machines method for fault diagnosis. Weimer et al. of processed air to each zone. To design aa model-based control and monitoring system, a mathematical model is learning of processed air to each zone. To design model-based capabilities. Liang and Du. (2007) uses support vector machines machines method forfault fault by diagnosis. Weimer etapal. (2012) diagnoses actuator using a two-tier vector method for fault diagnosis. Weimer et al. control and monitoring system, a mathematical model is of processed air to each zone. To design a model-based (2012) diagnoses actuator fault by using a two-tier apcontrol and monitoring system, a mathematical model is control andis monitoring system,byaMeitY, mathematical model is (2012) vector machines method for fault by diagnosis. etapal. diagnoses actuator fault using aaWeimer two-tier proach which includes a dynamic model based detector This work financially supported Govt. of India, under (2012) diagnoses actuator fault by using two-tier ap control andis monitoring system,byaMeitY, mathematical model is proach which includes a dynamic model based detector This work financially supported Govt. of India, under (2012) diagnoses actuator fault by using a two-tier ap theThis Visvesvaraya Ph.D Scheme and SERB DST under the grant proach which includes a dynamic model based detector work is financially supported by MeitY, Govt. of India, under and fast deciding steady state detector. The current trends proach includes dynamic model detector This work is financially supported MeitY, Govt.under of India, the Visvesvaraya Ph.D Scheme and by SERB - DST the under grant and fastwhich deciding steady aastate detector. Thebased current trends This proach includes dynamic model detector work is ECR/2016/001025. financially supported MeitY, Govt.under of India, agreement no. the Visvesvaraya Ph.D and SERB -- DST the grant and fast fastwhich deciding steady state state detector. Thebased current trends the Visvesvaraya Ph.D Scheme Scheme and by SERB DST under the under grant and deciding steady detector. The current trends agreement no. ECR/2016/001025. the Visvesvaraya Ph.D Scheme and SERB - DST under the grant agreement no. ECR/2016/001025. ECR/2016/001025. and fast deciding steady state detector. The current trends agreement no.
agreement no.2018, ECR/2016/001025. 2405-8963 © © IFAC (International Federation of Automatic Control) Copyright 2018 IFAC 544 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 544 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 544 Copyright © 2018 IFAC 544 10.1016/j.ifacol.2018.09.629 Copyright © 2018 IFAC 544
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and challenges in fault detection and diagnosis (FDD) for smart buildings is discussed by Lazarova-Molnar et al. (2016).
Zone1
Tw5
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2. SYSTEM DESCRIPTION
Tw1 6.5 m
The main contributions of the paper are: first, we present a method to identify parameters of the physics-based nonlinear model of the building thermal dynamics. Secondly, we design a complete fault diagnosis system by using the identified nonlinear model for detecting and estimating stuck faults in the VAV dampers. The novelty of the FDD system lies in its simple construction, which makes it widely applicable for diagnosing a class of actuator faults in HVAC systems.
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Fig. 1. Three-zones in one-storey building scenario x˙ = Ax + Bu + Du (t)x + Ed y = Cx 3 ˙ i (t)Ai , where with Du (t) = i=1 m T x(t) = TzT TwT ˙2 m ˙ 3] ˙1 m u(t) = [m y(t) = Tz
2.1 Modelling of three-zones in a building The system considered is comprised of three-zones in a single-storey building Yam´e et al. (2015). The layout of the building is shown in the Fig. 1. For each zone i, (i = 1, 2, 3), denote the temperature of zone by Tzi , the mass flow rate at the output of the i-th VAV by m ˙ i , and the supply air temperature by Ts , and the walls temperature by Tw . There are a total of six walls, and their internal wall temperature is denoted by Twj , (j = 1, 2, . . . , 6). Three of them is a common walls between the zones, while the other interacts with the outside environment. Applying the first law of thermodynamics to each zone, the dynamics of the zones temperatures is described by dTz1 = k01 m˙ 1 (Ts − Tz1 ) + k11 (Tw1 − Tz1 ) dt +k12 (Tw4 − Tz1 ) + k13 (Tw5 − Tz1 ) +k14 (TO − Tz1 ) + k15 q1 dTz2 = k02 m˙ 2 (Ts − Tz2 ) + k21 (Tw2 − Tz2 ) dt (1) +k22 (Tw4 − Tz2 ) + k23 (Tw6 − Tz2 ) +k24 (TO − Tz2 ) + k25 q2 dTz3 = k03 m˙ 3 (Ts − Tz3 ) + k31 (Tw3 − Tz3 ) dt +k32 (Tw5 − Tz3 ) + k33 (Tw6 − Tz3 ) +k34 (TO − Tz3 ) + k35 q3 with the dynamics of the temperature of walls, given by dTw1 = k41 (Tz1 − Tw1 ) + k42 (TO − Tw1 ) dt dTw2 = k51 (Tz2 − Tw2 ) + k52 (TO − Tw2 ) dt dTw3 = k61 (Tz3 − Tw3 ) + k62 (TO − Tw3 ) dt (2) dTw4 = k71 (Tz1 − Tw4 ) + k72 (Tz2 − Tw4 ) dt dTw5 = k81 (Tz1 − Tw5 ) + k82 (Tz3 − Tw5 ) dt dTw6 = k91 (Tz2 − Tw6 ) + k92 (Tz3 − Tw6 ) dt where kmn , m = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), n = (1, 2, 3, 4, 5), are constant coefficients, TO is the outside air temperature, and qi is the heating load in the i-th zone. At a constant supply air temperature, equations (1), (2) can be lumped together into the following bilinear state-space form 545
(3)
T
T
d(t) = [q1 q2 q3 TO ] , T
T
Tz = [Tz1 Tz2 Tz3 ] , Tw = [Tw1 Tw2 Tw3 Tw4 Tw5 Tw6 ] . The parameters A, Ai , B, E in (3) are matrices of appropriate dimension and C = [I3 03×6 ], where I• is an identity matrix of dimension • and 0• denotes a zero matrix of dimension •. The state of the bilinear system (3) can also be represented as x˙ = Ax + Bu + A(u ⊗ x) + Ed (4) where the symbol ⊗ denotes the Kronecker product and A = [A1 A2 A3 ] is the bilinearity matrix. 2.2 Fault modelling Under the event of actuator faults, the mass flow rate at the output of VAVs, denoted by uf (t), is given by uf (t) = u(t) + fa (t), (5) T
T
˙ 2,f m ˙ 3,f ] , fa = [f1 f2 f3 ] . Here, ˙ 1,f m where uf = [m we consider stuck fault in VAV dampers. That is, for the i-th VAV, its output signal m ˙ i,f is a constant. Under actuator fault scenario, the state-space system is given by x˙ = Ax + Bu + Du (t)x + Ed + F (t)fa , (6) where F = [F1 (t) 06×3 ], with F1 = diag(k01 (Ts − Tz1 ), k02 (Ts −Tz2 ), k03 (Ts −Tz3 )) and 06×3 denotes the zero matrix, is a time-varying matrix of appropriate dimension. The problem we aim to solve is to detect the occurrence of actuator fault and estimate precisely its magnitude for control and monitoring purposes. 3. MODEL IDENTIFICATION SIMBAD supplies a black-box model under the Simulink environment, which is illustrated in Fig. 2. For the development of a model-based fault diagnosis system, we present a Airßow Setpoint
Heating load
Flow controller Flow sensor
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Fig. 2. Structure of the system in SIMBAD
Zones Temp.
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method for identifying the parameters of the model (3). All variables in (3) are measured from SIMBAD, except the walls temperature. Using the input-output and measured disturbance data, parameters of the model are estimated using a nonlinear optimization algorithm. To proceed with the identification, we discretized the system (3) using a forward differentiation rule with a sample time of h x(k + 1) = [Ah + I]x(k) + [Bu(k) + Du (k)x(k) + Ed(k)]h. (7) ˜ = Bh, D ˜ u (k) = Du (k)h, Denoting A˜ = Ah + I , B ˜ = Eh, (7) can be rewritten as E ˜ ˜ ˜ ˜ u (k)x(k) + Ed(k) x(k + 1) = Ax(k) + Bu(k) +D (8) y(k) = Cx(k) Let us denote the estimated variables by T˜z and T˜w . At k = 0, we use the available data and the initial condition of walls temperature Tw (0) to get T˜z (1) ˜ Tz (0) + Bu(0) ˜ ˜ ˜ u (0) Tz (0) + Ed(0) = A + D Tw (0) Tw (0) T˜w (1) (9) for k ≥ 1, the estimate is computed as T˜z (k + 1) ˜ Tz (k) + Bu(k) ˜ = A T˜w (k) T˜w (k + 1) (10) ˜ ˜ u (k) Tz (k) + Ed(k) +D T˜w (k) y˜ = T˜z The objective function y − y˜ 22 is minimized using the nonlinear least square optimization method available in MATLAB to obtain the discrete-time state equation pa˜ B, ˜ D ˜ u and E. ˜ Subsequently, the continuous rameters A, ˜ time parameters are computed by A = h1 (A˜ − I), B = h1 B, 1 ˜ 1 ˜ Du = h Du and E = h E. The identified parameters are given in the Table 1. Table 1. Identified Parameter values Parameter k01 k12 k14 k02 k22 k24 k03 k32 k34 k41 k51 k61 k71 k81 k91
Value 2.0427 × 10−04 5.3023 × 10−05 3.6321 × 10−06 2.3754 × 10−04 1.2005 × 10−04 2.5815 × 10−06 1.8038 × 10−04 9.3119 × 10−05 2.1423 × 10−06 1.7207 × 10−05 9.4988 × 10−07 1.8324 × 10−05 2.7660 × 10−14 2.8143 × 10−14 1.7356 × 10−05
Parameter k11 k13 k15 k21 k23 k25 k31 k33 k35 k42 k52 k62 k72 k82 k92
Value 1.5534 × 10−04 4.4707 × 10−05 1.8629 × 10−07 1.1537 × 10−05 1.2700 × 10−04 2.2873 × 10−07 1.3383 × 10−04 2.8062 × 10−14 1.6678 × 10−07 9.3684 × 10−07 1.9566 × 10−05 1.8321 × 10−06 9.9811 × 10−05 9.7686 × 10−05 2.7487 × 10−06
Fig. 3. Block structure of the observer ˆ + Ed x ˆ˙ = Aˆ x + Bu + Du x − (C T LCAC T + C T LCDu C T )(y − yˆ) yˆ = C x ˆ
(11)
with L ∈ R3×3 denoting the gain matrix.
The block diagram of the observer is illustrated in Fig. 3. Here the objective is to synthesize the matrix L of the observer such that the error, defined by, e = x − x ˆ tends to 0 asymptotically. To prove this, first we compute the derivative of the error given by e˙ = x˙ − x ˆ˙ . Substituting x˙ ˙ and x ˆ from (3) and (11) respectively, yields e˙ = (A + C T LCAC T C)e + (Du + C T LCDu C T C)e, which can be written as e˙ = (A¯ + H(t))e,
(12)
where A¯ = (A + C T LCAC T C) and H(t) = (Du + C T LCDu C T C). The following result would be helpful to establish the stability of the error dynamics and subsequently compute the L matrix. Lemma 1. Chen (1995) Assume that all eigenvalues of A¯ have negative real parts, then for any given positive definite symmetric matrix N , the Lyapunov equation A¯T M + M A¯ = −N
has a unique symmetric solution M which is positive definite. n Definition 2. For any vector x ∈ Rn , x 2 = ( i=1 x2i )1/2 1/2 and for any matrix A ∈ Rm×n , A 2 = λmax (AT A). Theorem 3. Assume that all eigenvalues of A¯ have negative real parts, there exists a positive definite symmetric matrix N such that H(t) satisfies H(t) 2 <
λmin (N ) 2λmax (M )
where M is the solution of the Lyapunov equation, then the system e˙ = (A¯ + H(t))e
4. NONLINEAR FAULT DIAGNOSIS In this section, we describe the nonlinear observer-based fault diagnosis system for HVAC . Let the state vector of the observer is denoted by x ˆ. We propose the dynamics of the observer as 546
is asymptotically stable. Proof. We use V (e) = eT M e as a Lyapunov function candidate. Consider g(e) = H(t)e, the derivative of V (x) along the trajectories of the system is given by
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¯ + g(e)) + eT (A¯T + g T (e))M e V˙ (x) = eT M (Ae T = −e (M A¯ + A¯T M )e + 2eT M g(e)
= −eT N e + 2eT M g(e) The first term on the right-hand side is negative definite, while the second term is (in general) indefinite. Since g(e) is continuous, it satisfies g(e) 2 → 0 as e 2 → 0 e 2 Therefore, for any γ = H(t) 2 , there exists r > 0 such that g(e) 2 < γ e 2 , ∀ e 2 < r Hence, V˙ (e) < −eT N e + 2 M 2 H(t) 2 e 22 , ∀ e 2 < r But eT N e ≥ λmin (N ) e 22 where λmin (•) denotes the minimum eigenvalue of a matrix. Note that λmin (N ) is real and positive since N is symmetric and positive definite. Thus, V˙ (e) < −[λmin (N ) − 2 M 2 H(t) 2 ] e 22 , ∀ e 2 < r Choosing λmin (N ) λmin (N ) H(t) 2 < = 2 M 2 2λmax (M ) ensures that V˙ (e) is negative definite. Hence, the origin is asymptotically stable. ¯ C, ¯ where The matrix A¯ can be written as A¯ = A − L ¯ = C T L and C¯ = −CAC T C and now L ¯ is the matrix L to be computed which is in fact equivalent to computing ¯ is observable. the matrix L. Note that the pair (A, C) ¯ by selecting We can arbitrarily place the eigenvalues of A, the L matrix, on the left-hand side of the complex plane. The damper position of a VAV box is limited to operate into the range [0, 1], where ‘0’ denotes fully close and ‘1’ denotes fully open. Depending on the damper pressure, the flow meter imposes a limitation on the amount of airflow rate, u(t), i.e. umin ≤ u(t) ≤ umax with umin and umax denoting the minimum value and the maximum value of the airflow rate respectively. Consequently, we choose an appropriate L matrix such that the matrix A¯ is Hurwitz and the upper bound on the Euclidean norm of H(t) is satisfied. Whenever an actuator fault occurs, the error dynamics is given as e˙ = (A¯ + H(t))e + F (t)fa . For fault detection, we compute the residual vector by ri (t) = yi (t) − yˆi (t) (13) where ri (t) denotes the residual of the i-th zone. The detection and isolation logic under a fault-free damper or a stuck in damper for the i-th VAV is then given by ri (t) < th, no actuator fault (14) ri (t) ≥ th, fault occurred in the i-th actuator where th is a given threshold value. To achieve the complete fault diagnosis for HVAC systems, an estimation of the stuck is required. Let the matrices in the system equa A11 A12 B1 tion be partitioned such that, A = ,B= , A A B2 21 22 E1 D11 (t) D12 (t) . For the considered and E = Du = E2 D21 (t) D22 (t) 547
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three zone system, we have B2 = D12 (t) = D21 (t) = D22 (t) = 0. Taking the derivative of the output variable in (3) yields y˙ = C x˙ B1 D11 (t) 0 Tz A11 A12 Tz + u+ =C 0 0 Tw 0 A21 A22 Tw E F + 1 d + 1 fa E2 0 T T˙z = [A11 A12 ] z + B1 u + D11 (t)Tz + E1 d + F1 fa Tw (15) Note that we do not have a direct measurement of the temperature of the walls. From the experimental analysis, it is observed that the effect of the fault on the dynamics of wall temperatures is very slow and the deviation is also of very low magnitude. Therefore, we use the wall temperatures from the nonlinear observer for the estimation of the fault. The fault estimation filter is expressed as Tz fˆa (t) = F1−1 T˙z − [A11 A12 ] ˆ + B1 u Tw (16) +D11 (t)Tz + E1 d
where fˆa (t) denotes the estimation of the fault. Here, the matrix F1 is always nonsingular. The estimation of the fault using (16) is triggered whenever it is detected by the fault detection logic (14). 5. SIMULATION RESULTS
To set-up our case-study for simulation purposes, we used SIMBAD (SIMulator of Building And Devices), which is a Matlab/Simulink toolbox developed by CSTB SIMBAD (2005) primarily for building simulation. This simulator enables a powerful and in-depth implementation of any real-life building with all physical phenomena of zones and large HVAC library database El Khoury et al. (2005). 5.1 Three-zones building setup Using SIMBAD simulator, we build a one-story building consisting of three zones to demonstrate the effectiveness of the developed fault diagnosis system. The considered case-study is illustrated in Fig. 1, where the net area of the building is 168 square meters. The building is divided into three zones: Zone-1 and Zone-3 have two windows while Zone-1 has one, where it is assumed that the windows blinds are always closed. The internal gains within this building are defined by the occupancy characteristics which consists of the occupancy type and a number of occupants. The type of occupancy is selected as “Offices” where the working time of occupants is modeled from 08h00 to 18h00 with a lunch break of two hours starting at 12h00. We considered six occupants in Zone-1, four occupants in Zone-2, while five occupants in Zone-3. We also modeled constant interzonal mass flows between each zones. The ventilation system in each zone has two air-flow paths inside the zone, where one path allows the exterior air-flow while the other air-flow path is controlled by the VAV box to maintain the thermal comfort of occupants.
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Tz1 (o C)
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The VAV terminal box consists of the grid model, the 3way valve model and the detailed dynamic heating coil (the reheat coil), where the VAV damper process is composed of a damper with its driving motor and a flow sensor to measure the supply air-flow rate. The motorized damper model takes into account the hysteresis of the shaft and the type of blades to determine the air flow rate through the damper. The position of the damper is used to control the air-flow rate and its input-output relationship is generally nonlinear. In the considered scenario, the minimum and maximum airflow rate are umin = 0.49 × 10−3 kg/s and umax = 0.3163kg/s.
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A dataset is formed using the data collected from SIMBAD for system identification. The supply temperature has an average temperature of 35°C. Internal gains are generated using the occupancy information of each zone. The airflow rates into each zone are considered as a seeded random signal (different seed for different zones). Fig. 4 shows the outside temperature TO , the supply temperature Ts and the internal gains [q1 , q2 , q3 ] for seven days out of which the data for five days is used for identification and the rest is used for validation.
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Fig. 4. Outside temperature, Supply temperature and Internal gains data used for model identification and validation A nonlinear model is identified using the procedure as described in section 3 with temperatures of walls and zones initialized to T˜w (0) = T˜z (0) = 20°C. The output of the identified system is compared with the output of the benchmark model by supplying both the system and the model with the same input and disturbance signals. Fig. 5 illustrates the comparison between the response of the system and the identified model. It is observed that the identified model closely matches with the actual system. The identified model is validated by comparing the physical system with the model for a different set of seeded random signals as input flow for seven days. The outside temperature, supply temperature and the internal gains are shown in Fig. 4. Corresponding to the stimulus given to model and system, the zone temperature response obtained is illustrated in the Fig. 6. It is observed that the model identified closely matches the actual system for a different set of signals as well. A slight mismatch occurs after the sixth day, which is possibly due to a sudden change in the outside air temperature. However, the dynamics of the identified model is similar to the benchmark model. 548
Fig. 6. Validation of the identified model using a seven days data with a different set of input signals 5.3 Results of fault diagnosis The observer is designed by choosing the matrix L = 60 × I3 such that condition of Theorem 3 is satisfied. We considered three faults scenario, where a damper stuck fault of different magnitude occurs in all zones at the end of the day 1. The damper of zone 1 gets stuck at a position of 0.6, while the damper of zone 2 and zone 3 get stuck at a position of 0.5 and 0.65 respectively. Due to these faults the airflow into the zone 1 is now constant at 0.0417kg/s, airflow into zone 2 at 0.0238kg/s and airflow into zone 3 at 0.0545kg/s. The air temperature of the outside environment, the supply air temperature and the internal gains are shown in Fig. 4. The airflow set-point to the VAV box (see Fig. 2) is kept at a constant level of 0.16kg/s, but due to faults the actuator is unable to supply the desired flow rate. Consequently, there is a drop in the temperature of the zones as illustrated in Fig. 7, which affects the thermal comfort of the users. It is worth observing that the observer successfully detects the faults, where the threshold value is th = 0.0145. The residual vector is shown in Fig. 8. Subsequently, using (16), the fault vector fˆa is estimated. The estimate of airflows delivered by VAVs is given by u ˆf (t) = u(t) + fˆa (t), which
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systems. A real-life case-study of the three-zones onestorey building scenario is implemented in the SIMBAD simulator, which shows that the proposed fault diagnosis strategy has a great potential in practical applications for energy efficient fault-tolerant control (FTC) in building automation systems. A preliminary integrated FDD and FTC approach for HVAC systems has been presented in A and Jain (2018).
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REFERENCES
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Fig. 7. Zone temperature under actuator faults is illustrated in Fig. 9. Note that when there is no fault in VAV dampers, then u ˆf (t) = u(t). Clearly, the fault estimation filter is able to estimate precisely the magnitude of the fault. 0.02 0 -0.02 0.02
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Fig. 9. Estimate of the airflow at the output of VAV dampers
6. CONCLUSION In this paper, we presented a nonlinear observer-based fault diagnosis method for detecting and estimating stuck faults in VAV dampers. Due to the occurrence of these faults, the thermal comfort is heavily affected as well as deteriorating the energy-efficient operation of HVAC 549
A, M.S. and Jain, T. (2018). Fault tolerant economic model predictive control for energy efficiency in a multizone building. In 2nd IEEE Conference on Control Technology and Applications, August 21-24. Copenhagen, Denmark. Chen, C. (1995). Linear system theory and design. Oxford University Press, Inc. El Khoury, Z., Riederer, P., Couillaud, N., Simon, J., and Raguin, M. (2005). A multizone building model for MATLAB/Simulink Environment. In Ninth International IBPSA Conference, 525–532. Montr´eal, Canada. IESS2047 (2015). Commercial lighting and appliances documentation. Technical report, India Energy Security Scenarios 2047, Govt. of India, http://indiaenergy.gov.in/iess/docs/CommercialLighting˙Appliances-documentation.pdf. Kim, W. and Katipamula, S. (2018). A review of fault detection and diagnostics methods for building systems. Science and Technology for the Built Environment, 24(1), 3–21. Lazarova-Molnar, S., Shaker, H.R., Mohamed, N., and Jørgensen, B.N. (2016). Fault detection and diagnosis for smart buildings: State of the art, trends and challenges. In 2016 3rd MEC International Conference on Big Data and Smart City (ICBDSC), 1–7. Muscat, Oman. Liang, J. and Du., R. (2007). Model-based fault detection and diagnosis of hvac systems using support vector machine method. International Journal of Refrigeration, 30, 1104–1114. Shahnazari, H., Mhaskar, P., House, J.M., and Salsbury, T.I. (2018). Heating, ventilation and air conditioning systems: Fault detection and isolation and safe parking. Computers and Chemical Engineering, 108, 139–151. SIMBAD (2005). Centre scientifique et technique du batiment, france. SIMBAD building and HVAC toolbox. Thumati, B.T., Feinstein, M.A., Fonda, J.W., Turnbull, A., Weaver, F.J., Calkins, M.E., and Jagannathan, S. (2011). An online model-based fault diagnosis scheme for hvac systems. In IEEE International Conference on Control Applications (CCA), 70–75. Denver, CO, USA. Weimer, J., Ahmadi, S.A., Araujo, J., Mele, F.M., Papale, D., Shames, I., Sandberg, H., and Johansson, K.H. (2012). Active actuator fault detection and diagnostics in hvac systems. In Proceedings of the Fourth ACM Workshop on Embedded Sensing Systems for EnergyEfficiency in Buildings, BuildSys ’12, 107–114. ACM, Toronto, Ontario, Canada. Yam´e, J., Jain, T., and Sauter, D. (2015). An online controller redesign based fault-tolerant strategy for thermal comfort in a multi-zone building. In IEEE Conference on Control Applications, 1901–1906. Sydney, NSW, Australia.