Detection and diagnosis for sensor fault in HVAC systems

Transcription

1 Energy Conversion and Management 48 (27) Detection and diagnosis for sensor fault in HVAC systems Zhimin Du *, Xinqiao Jin School of Mechanical Engineering, Shanghai Jiao Tong University, 23 Shanghai, PR China Received 7 August 25; received in revised form 2 March 2; accepted 27 September 2 Available online December 2 Abstract Principal component analysis (PCA) is presented to detect single sensor faults in heating, ventilation and air conditioning (HVAC) systems. Three PCA models, based on energy balance and air side and water side flow pressure balances, respectively, are set up to detect whether there is any abnormality occurring in the systems. With any fault discovered, a joint angle plot, which compares the new fault vector with known ones in the library, can be used to isolate the faulty sensor indeed. As pre-diagnosis, some expert rules are used to improve the diagnosing process. With the strategy of joint angle plot combined with expert rules, the single sensor fault can be timely isolated on line. Ó 2 Elsevier Ltd. All rights reserved. Keywords: Principal component analysis; Joint angle plot; Fault detection and diagnosis; HVAC systems. Introduction Recently, to satisfy the increasing demands of indoor air quality, heating, ventilation and air conditioning (HVAC) systems have become more and more complex. There are two key factors to ensure good performance of the systems, one is a suitable control strategy and the other is reliable measurement. Actually, there are many control loops with sensors in HVAC systems. Typical local controls associated with monitors in the systems are air handling unit supply air temperature control with temperature sensor, supply air static pressure control with pressure sensor, outdoor air flow control with flow rate sensor, variable speed pump (in the second chilled water loop) control with pressure sensor. Obviously, the normal or even optimal operation of the systems strongly relies on the capacity of the controls mentioned above. On the other hand, these controls strongly rely on reliable measurements. Unfortunately, drifting or biasing of the sensors is usually * Corresponding author. Tel.: ; fax: addresses: duzhimin@situ.edu.cn (Z. Du), xqjin@sjtu.edu.cn (X. Jin). inevitable after the systems are used for a relatively long term. So, the sensor with a fault may lead to the real operation of the systems being misrepresented. Furthermore, it may mislead the controls to execute a wrong/ unsuitable operation. As a result, inaccurate measurements may lead to worsened indoor air quality or waste of energy consumption, although there are suitable control strategies. Thus, to find a suitable method to detect, diagnose or even tolerate the faults occurring in the systems is a significant target. The studies of fault detection and diagnosis in HVAC systems started in the 98s [,2] and are paid more attention recently. Annex25 [3], Annex34 [4] and many other studies [5 ] concerned all kinds of faults. There have been two main methods in the past to detect or diagnose faults. One is based on the model of the systems and the other is based on knowledge. The model based method obtains the normal values of the parameters through the model of the systems first. Then, 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 must be obtained. Stylianou [] used a first order model to detect faults of temperature sensors by comparing the 9-894/$ - see front matter Ó 2 Elsevier Ltd. All rights reserved. doi:./j.enconman

2 94 Z. Du, X. Jin / Energy Conversion and Management 48 (27) Nomenclature x measure vector L loading matrix R model projection matrix T temperature ( C) M flow rate (kg/s) H enthalpy (kj/kg) h relative humidity (%) q heat exchange (kj/kg) P pressure DP pressure difference C control signal f fault vector F faults matrix PCA principal component analysis PCS principal component subspace RS residue subspace SPE square prediction error variable air volume x * real value of measure vector Greek symbols d a threshold for SPE h angle between two vectors lying in PCS c angle between two vectors lying in RS Subscripts and superscripts ˆ modelled part of vector ~ un-modelled part of vector sup supply air fre outdoor air rtn return air rec recycle air exh exhaust air ws supply water wr return water st supply air static pressure w chilled water passing by air handling unit ch chiller bp bypass n number of measurement samples r number of faults in F actual temperature decay with the model output using hypothesis testing. Wang [2] developed a law based sensor fault diagnosis strategy, which took all the commonly used temperature and flow rate sensors in a chilling plant into account at the same time. Moreover, Howell [3] and Haves [4] also paid attention to this method. The knowledge based approaches, including expert systems [5], neural networks [] and fuzzy theory [7], have been widely used to detect and diagnose faults recently. Lee [7] investigated fault diagnosis in a simulated air handling unit using a two stage artificial neural network. In addition, Wang and Chen [8] developed a strategy based on a neural network model to diagnose measurement faults of the outdoor air and supply air flow rate sensors. For some simple systems, it is easy to detect faults using such experiential knowledge. However, it is challenging to apply a knowledge based approach to large scale systems such as the HVAC systems because it is too complicated to develop the detailed expert system. Because of the difficulty in building pure mathematical models, the statistical method of principal component analysis (PCA) is developed in this paper. To improve the intelligence of the statistical method, some essential basis of the mathematical model of the systems are employed. These essentials include energy balance, air side flow pressure balance and water side flow pressure balance, which describe the relations of mass conservation, heat conservation and pressure balance among the variables. So, three PCA models, based on one energy balance and two flow pressure balances, are built to detect single sensor faults with fixed bias occurring in HVAC systems. Furthermore, the joint angle plot combined with some expert rules is used to diagnose the fault source. 2. Fault detection and diagnosis methodology 2.. Geometric interpretation of PCA [ 22] According to principal component analysis, a measurement vector x, which describes a running condition of the systems, can be decomposed into two parts (Fig. ), x ¼ ^x þ ~x ðþ where ^x ¼ LL T x ¼ Rx ð2þ is the modelled part that represents the projection on the principal component subspace (PCS), and ~x ¼ðI RÞx ¼ erx ð3þ Residual Subspace x~ ^x x Fig.. Decomposition of measurement vector. Principal Component Subspace

3 Z. Du, X. Jin / Energy Conversion and Management 48 (27) is the un-modelled part on the residual subspace (RS). So, the PCA model divides the measuring space into two orthogonal subspaces: PCS and RS. PCS refers to the condition that normal data variation occurs, while RS refers to the condition that abnormal variation or noises may occur. Under normal operation, most of the projection of x is on the PCS, while the projection on the RS is very little. When a sensor is biased, however, the projection on the RS will 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 abnormality is the squared prediction error (SPE), SPEðxÞ ¼k~xk 2 ¼ x T ði RÞx ð4þ The system is considered abnormal if SPEðxÞ > d a ð5þ where d a denotes a confidence limit or threshold for the SPE Joint angle plot [23] to diagnose Fault library Steady state information via plant test or historical data is available, so the fault signatures can be obtained. If a fault occurs, the measurement vector x can be represented as x ¼ x þ f ðþ Here, x * denotes the real value of the measurement for normal operation of the system, f denotes a fault vector. Then, with the PCA method, the vector f can be decomposed into two components; one lying in the PCS that can be denoted as ^f and the other lying in the RS denoted as f ~. After being normalized, they can be expressed as ^f ¼ Rf krf k ~f ¼ e Rf kerfk So, the fault signature library of all known faults can be obtained and expressed as follows: bf ¼½^f ; ^f 2 ;...; ^f r Š; ef ¼½ f ~ ; f ~ 2 ;...; f ~ r Š ð8þ Then, the two fault signature matrices include the information of all known faults of the systems in both the PCS and RS Fault diagnosis based on joint angle plot When a new sensor x measurement is obtained, it can be decomposed and then be normalized as follows: ^x ¼ Rx krxk ~x ¼ e Rx kerxk Then, the angles between the known fault signatures ðbf ; ef Þ and the new measurement vector ð^x; ~xþ are used to diagnose (Fig. 2) [23] the fault. The cosine value between the new measurement and one of the known fault signatures gives the relative measure of collinearity between them: ð7þ ð9þ x 2 ^ f cos h i ¼ ^x T^f i cos c i ¼ ~x T f ~ i ð ^x T^f ðþ i ; ~x T f ~ i ; i ¼ rþ When the cosine value is close to or, it means that the new measurement vector is nearly collinear with the fault direction. Therefore, once a fault is discovered, it can be isolated as the one whose cosine values are both close to or through a joint angle plot. 3. System description and models 3.. System description f Principal Component x * x ^ ~ x~x~ ~ f The typical HVAC systems in Fig. 3 can be partitioned into water side and air side. In the water side, the supply chilled water coming from the chiller is circulated to the air handling unit by the 2nd pump, and the return chilled water passing by the air handling unit is circulated back to the chiller by the st pump. Actually, the air handling unit is the place of heat and humidity exchange between the air and the water. In the air side, on the other hand, the supply air, which is a mixture of outdoor air and recycled air, is circulated to the air handling unit by the supply fan and will exchange heat and humidity with the chilled water. After being cooled (in summer condition) by the chilled water, the supply air is circulated to the variable air volume () terminals to meet the indoor requirement. Finally, with the return fan, the return air is divided into two parts: the exhaust air and the recycled air. The former is discharged into the outside space and the latter is reemployed in another air circle. In the systems, five main control loops are included. As to the water side, the P pressure controller adjusts the bypass valve, while the P 2 pressure controller supervises the 2nd pump to maintain the pressure difference between the supply and return chilled water. As to the air side, first is the T sup controller to ensure the proper supply air temperature through adjusting the chilled water valve located at the inlet of the air handling unit. Second, the supply fan is moderated to control the supply air static pressure. Last, the outdoor, recycle and exhaust air dampers are adjusted by the M fre controller to maintain the outdoor air x (New measure) Known Fault Fig. 2. Scheme of joint angle plot. x

4 9 Z. Du, X. Jin / Energy Conversion and Management 48 (27) TPTP Controller TP2 Controller Water Pipe Pump Water valve Air Duct Fan Air Damper F Flow Rate Sensor Chiller st Pump 2 nd Pump Return chilled water Supply chilled water Controller P T Pressure Sensor Temperature erature Sensor Outdoor Air F Supply Air F Supply Fan T P Controller Recycle Air Controller Air handling unit Exhaust Air F Return Air Return Fan Fig. 3. HVAC systems. requirement. To ensure these controllers working normally, the accuracy of the concerned sensors are significant of course. So, single sensor faults are the focus in this paper PCA model I based on energy balance The energy balance shown in Fig. 4 is an important balance to describe the relations among the variables of HVAC systems [24]. The energy balance represents not only the air mixing process but also the heat exchange process in the air handling unit between the air and the water. On one hand, the air mixing process can be described as Eqs. () (3). Among them, Eq. () reflects the quality balance in the mixing process: M exh ¼ M rtn M rec ¼ M rtn ðm sup M fre Þ T exh ¼ T rtn h exh ¼ h rtn ðþ ð2þ ð3þ where M refers to flow rate, T refers to temperature, h refers to humidity and the subscripts fre, sup, rtn, rec and exh refer to fresh (outdoor), supply, return, recycle and exhaust air, respectively. On the other hand, the heat exchange process between the air and the water can be expressed as Eq. (4), where q Chiller air,water heat losses or gains of air ducts to the environment are neglected: H fre þ H rtn ¼ H sup þ H exh þ q air;water ð4þ Therefore, the energy balance of HVAC systems that includes both the air mixing process and the heat exchange processes can be described as the relations above. In Eq. (4), the character H refers to enthalpy that can be represented by temperature, humidity and flow rate, while q air,water, is the heat exchange rate between the air and the water that is related to T ws (supply chilled water temperature), T wr (return chilled water temperature) and M w (chilled water flow rate passing the air handling unit). So, Eqs. () (4), referring to the energy balance of HVAC systems can be described as certain functions of T fre, T sup, T rtn, M fre, M sup, M rtn, h fre, h rtn, T ws, T wr and M w. In other words, these measurement variables can be combined together but have some connections through the energy balance of the systems. Therefore, the PCA model I based on the energy balance is set up using these measurement variables, and the corresponding matrix is denoted as 2 T fre T sup T rtn T ws T wr M fre M sup M rtn M 3 w h fre h rtn T 2 fre T 2 sup I ¼ T 2 rtn T 2 ws T 2 wr M 2 fre M 2 sup M 2 rtn M 2 w h2 fre h 2 rtn T n fre T n sup T n rtn T n ws T n wr M n fre M n sup M n rtn M n w hn fre h n rtn 3.3. PCA model II and III based on flow pressure balance n H fre H exh Air handling unit Fig. 4. Energy balance. H sup H rtn Besides the energy balance, the flow pressure balances are the other key relations in HVAC systems. The flow pressure balance, which describes the relations of flow balance and pressure balance among the variables, includes not only the mass flow but also the mass conservation

5 Z. Du, X. Jin / Energy Conversion and Management 48 (27) equations. Since the whole systems can be partitioned into two subsystems, the flow pressure balance can also be divided into two aspects: the air side and the water side Model II based on air side flow pressure balance The air side flow pressure balance is illustrated in Fig. 5. The flow resistances of the terminals are considered to be variable that depend on the positions of the dampers. Since the outdoor, recycle and exhaust air dampers are changeable according to the action of the outdoor air flow controller, the resistances of these three dampers are variable. The flow resistances of the supply and return air ducts are constant, and the resistance of the air handling unit, including the filter and cooling coil, is also considered to be constant. The resistance to air leakage through the building envelope to the outside is thought to be constant too. So with these flow resistances, the corresponding pressure differences between inlet and outlet of these facilities can be denoted as DP, DP AHU, DP fan,sup, DP fan,rtn, DP duct,sup, DP duct,rtn and DP rec. Therefore, the air side flow-pressure balance that is based on air mass conservation and pressure balance can be shown as Eqs. (5) (7). Among them, Eq. (5) describes air mass conservation, while Eqs. () and (7) represent the pressure balances: M rec ¼ M rtn M exh ¼ M sup M fre DP fan;sup DP AHU DP duct; sup DP ¼ P zone P mix DP fan;rtn DP duct;rtn DP rec ¼ P zone P mix ð5þ ðþ ð7þ Considering Eqs. (5) (7), the flow pressure balance model concerns the variables of M fre, M sup, M rtn, P st and C fan,sup and C fan,rtn (the control signals for the supply and return fans). So, these six measurement variables are used to construct the PCA model II based on the air side flow pressure balance, and the corresponding matrix is denoted as 2 3 M fre M sup M rtn P st C fan;sup C fan;rtn M 2 fre M 2 sup M 2 rtn P 2 st C 2 fan;sup C 2 fan;rtn II ¼ M n fre M n sup M n rtn P n st C n fan;sup C n fan;rtn Model III based on water side flow pressure balance The water side flow pressure balance is shown in Fig.. The flow resistances of all the water pipes are considered to be constant, and the resistance of the chiller is constant too. Since the water valve of the air handling unit varies according to the action of the optimal controller, its resistance is variable, and the resistance of the bypass valve is also n P atm = P mix Air handling unit M fre Outdoor air M sup ΔPP damper AHU Supply fan + ΔP P fan, sup Supply air duct ΔP P duct,sup M rec Recycle air damper ΔP P rec terminal ΔP P M exh P atm = Exhaust air damper M rtn ΔP P fan, rtn + ΔP P duct,rtn Return fan Return air duct P zone P atm = Building construction Fixed flow resistance Variable flow resistance Fig. 5. Air side flow pressure balance. Pipe P mix M w 2 nd pump Pipe4 Water valve of air handling unit Chiller st pump M ch Δ P chiller Δ Δ Δ P pipe M bp P pump2 P pipe4 First Chilled Bypass-valve Second Chilled Water Loop Water Loop Δ P valve Δ P pipe 3 Pipe 3 ΔPP pump ΔPP pipe2 P mix2 Pipe2 Fixed flow resistance Variable flow resistance Fig.. Water side flow pressure balance.

6 98 Z. Du, X. Jin / Energy Conversion and Management 48 (27) variable because the P controller moderates it. With these resistances, the pressure differences between the inlet and outlet of these facilities can be denoted as DP chiller, DP pump, DP pump2, DP pipe, DP pipe2, DP pipe3, DP pipe4 and DP valve. So, the pressure balance can be illustrated as Eqs. (8) and (9), and then, they can be simplified as Eq. (2). The flow balance describing the mass conservation can be expressed as Eq. (2): P mix P mix2 ¼ DP pump DP chiller DP pipe DP pipe2 ¼ P ð8þ P mix2 P mix ¼ DP pump2 DP pipe4 DP valve DP pipe3 ¼ DP pump2 P 2 ð9þ P 2 P ¼ DP pump2 ð2þ M ch ¼ M bp þ M w ð2þ Obviously, the water side flow pressure balance includes M w, M ch, P, P 2 and C pump,2nd (control signal for 2nd pump) according to Eqs. (2) and (2). In a similar way, the matrix of PCA model III based on the water side flow pressure balance can be described as 2 3 M w M ch P P 2 C pump;2nd M 2 w M 2 ch P 2 P 2 2 C 2 pump;2nd III ¼ M n w M n ch P n P n 2 C n pump;2nd 4. Fault detection and diagnosis strategy The logic diagram of the fault detection and diagnosis strategy is illustrated in Fig. 7. First of all, the three PCA models based on energy balance and two flow pressure balances are set up through the following steps: () After being divided into three groups according to the three balances, the historical normal operation data should be scaled to zero mean and unit variance. (2) With the number of principal components optimized, the correlation matrix of models I, II and III can be obtained under normal operation. With the eigenvalues and eigenvectors calculated, PCA models I, II and III can be set up by partitioning the measurements into two orthogonal subspaces: PCS and RS. n5 Normal Historical Data Scaled to zero mean New Measure Data and unit variance Energy, Air-side Eigenvalue & Eigenvector & Water-side PCA Model I,II,III Balances PCS RS Fault Detection ( x ) > δ a SPE > Y I &N II &Y III Y I &Y II &N III Y I &N II &N III N I &N II &Y III N I &Y II &N III M w M fre, M sup, M rtn T fre, T sup, T rtn,t ws, P, P 2, M ch T wr,h fre, Pre-diagnosis Fault Diagnosis h rtn Joint angle plot N I &N II &N III P st Conclusions Fig. 7. Fault detection and diagnosis strategy logic. Secondly, with the three detection models, new measurements can be examined whether there is any abnormality in the systems through comparing the SPE values of the new measures with the threshold. Thirdly, expert rules are used to pre-diagnose the fault source. Since single sensor faults are considered in this paper, and the three detection models have common sensors, the six rules shown in Table can be summarized to improve the diagnosing process. At last, a joint angle plot can be used to diagnose further to confirm the fault source. 5. Validation results Sensor fault detection and diagnosis using principal component analysis and joint angle plot are tested. They are examined using the simulator of HVAC systems that has already been developed [25]. The running time of the HVAC systems is from 7:45 AM to 8: PM. In the simulation, the fault generator has been incorporated, and it can generate a fixed bias for Table Expert rules for pre-diagnosis No. Rules SPE(x) >d a? Possible fault source Model I Model II Model III Y I &N II &N III Yes No No T fre, T sup, T rtn, T ws, T wr, h fre, h rtn 2 Y I &Y II &N III Yes Yes No M fre, M sup, M rtn 3 Y I &N II &Y III Yes No Yes M w 4 N I &Y II &N III No Yes No P st 5 N I &N II &Y III No No Yes M ch, P, P 2 N I &N II &N III No No No No fault

7 Z. Du, X. Jin / Energy Conversion and Management 48 (27) the sensors. A fixed bias refers to the difference between the measurement value and its real value being constant. It should be noted again that only single sensor faults are considered in the paper. 5.. Fault detection The detection effect of PCA model I is shown in (a) and (b) of Fig. 8, which concern several cases of single sensor faults, including positive and negative biases. When the T sup sensor is biased with 8% at 2:3 PM, for example, the judging index, SPE, greatly exceeded its threshold (3.2989), indicating the occurrence of a certain fault. Similar analysis in the cases of T rtn, M fre and M w with faults shows their SPE values all exceeded the thresholds after the faults occurred, indicating the occurrence of abnormality. In addition, the detection capacity of PCA model II is illustrated in Fig. 9. As for the sensors concerned in model II (M fre, M sup and P st ), these abnormalities can also be timely discovered because the SPE values went beyond the threshold after the faults occurred. Finally, as for PCA model III, three fault cases (P, P 2 and M w ) are tested. Fig. shows that all of them can be detected quickly by comparing their SPE values with the thresholds. SPE SPE : AM 2: PM : PM 2: PM 3: PM 4: PM Pst sensor biased with 2% at 2:3PM Mfre sensor biased with -5% at 2:3PM Msup sensor biased with % at 2:3PM Threshold(.874) Fault occurred (2:3PM-4PM) Fig. 9. Detection for single sensor fault using PCA model II. P sensor biased with 2% at 2:3PM P2 sensor biased with % at 2:3PM Mw sensor biased with % at 2:3PM Threshold(.22) Fault occurred (2:3PM-4PM) a Tsup sensor biased with 8% at 2:3PM Trtn sensor biased with 7% at 2:3PM Threshold(3.2989) 9: AM : AM : AM 2: PM : PM 2: PM 3: PM 4: PM Fig.. Detection for single sensor fault using PCA model III. SPE Fault diagnosis 5 (9AM-2:3PM) Fault occurred (2:3PM-4PM) Off line diagnosis Fig. is the joint angle plot to isolate the fault source when the T sup sensor is biased with 8% at 2:3 PM. All the 9: AM : AM : AM 2: PM : PM 2: PM 3: PM 4: PM diagnosing area b SPE Mfre sensor biased with 5% at 2PM Fault occurred Mw sensor biased with -% at 2:3PM (2:3PM-4PM) Threshold(3.2989) Fault occurred (2PM-4PM) RS Direction Bias(Tsup) Bias(Mfre) Bias(Msup) Bias(Mrtn) Bias(Mw) Bias(Tfre) Bias(Trtn) Bias(Twr) Bias(Tws) Bias(hfre) Bias(hrtn) 9: AM : AM : AM 2: PM : PM 2: PM 3: PM 4: PM Fig. 8. Detection for single sensor fault using PCA model I: (a) temperature sensor fault and (b) flow rate sensor fault PCS Direction Fig.. Joint angle plot to diagnose T sup sensor fault.

8 7 Z. Du, X. Jin / Energy Conversion and Management 48 (27) possible sources (sensors concerned in PCA model I) are validated. The angle components in both the PCS and RS between this new fault direction and the directions of the faults in the library are shown. Note that the cosines of the angles between these new fault vectors and the T sup fault vectors in the library go to the (, ) corner of the plot, suggesting essentially perfect collinearity with a T sup fault in both subspaces. So, it is unambiguous that the T sup sensor biased in deed. On the other hand, one is likely to be misled if the angle in only one of the subspaces, but not in both, was examined. For example, in the PCS direction, the new fault has a vector that is also collinear with that of a T ws sensor bias. However, the T ws bias does not align with the new fault vector in the RS direction. This confirms that both the PCS and RS need to be jointly examined for effective diagnosis. In fact, the method to diagnose the faulty sensor is to examine whose values are located in the (, ) or (, ) corner of the plot. Similar analysis of the M fre sensor fault shows that the joint angle plot shown in Fig. 2 can also isolate this fault source accurately because the cosines of the two angles between the new fault vectors and the M fre sensor fault vectors in the library go to the (,) corner of the plot. In addition, after a fault (P st sensor biased with 2% at 2:3 PM) is detected by PCA model II (Fig. 9), the joint angle plot shown in Fig. 3 can confirm the fault source to be P st sensor successfully. At last, after the fault (P 2 sensor biased with % at 2:3 PM) is detected by PCA model III (Fig. ), the joint angle plot shown as Fig. 4 can isolate that fault source too. RS Direction RS Direction diagnosing area.5 Bias(Pst) Bias(Mfre) Bias(Mrtn) PCS Direction Fig. 3. Joint angle plot to diagnose P st sensor fault. diagnosing area Bias(Msup).5 Bias(P2) Bias(Mch) Bias(Mw) On line diagnosis Obviously, if the real fault source can be isolated quickly through on line diagnosis, some remediation can be taken, such as replacing the biased sensor or adjusting the control strategy accordingly to avoid undesirable conditions that PCS Direction Bias(P) RS Direction diagnosing area PCS Direction Fig. 2. Joint angle plot to diagnose M fre sensor fault. Bias(Mfre) Bias(Mrtn) Bias(Msup) Bias(Mw) Bias(Tfre) Bias(Trtn) Bias(Tsup) Bias(Twr) Bias(Tws) Bias(hfre) Bias(hrtn) Fig. 4. Joint angle plot to diagnose P 2 sensor fault. may worsen indoor air quality or waste energy consumption. The strategy combining joint angle plots with some expert rules in Table can be used to isolate the fault source on line. When the T sup sensor is biased, since only PCA model I discovered the fault but PCA models II and III did not, the temperature or humidity sensor can be focused according to Rule (Y I &N II &N III ) in Table. Furthermore, Fig. 5 illustrates that the cosines between the new fault and the T sup fault in both subspaces are close to gradually after 2:3 PM, indicating that it is the T sup sensor that is the source. It took about min on line to isolate the fault source. When the M fre sensor is biased, the possible fault source can be M fre, M sup or M rtn according to Rule 2 (Y I &Y II &- N III ), but through the joint angle plot shown in Fig.,

9 Z. Du, X. Jin / Energy Conversion and Management 48 (27) Cosine of the angles Faulty operation cos( θ ): PCS Direction cos( γ ): RS Direction Cosine of the angle Faulty operation cos( θ ): PCS Direction cos( γ ): RS Direction - 2: PM 2:5 PM 2:3 PM 2:45 PM : PM :5 PM :3 PM Fig. 5. On line joint angle plot of T sup. - 2: PM 2:3 PM : PM :3 PM Fig. 8. On line joint angle plot of P 2. Cosine of the angles :3 PM :45 PM 2: PM 2:5 PM 2:3 PM Faulty operation Fig.. On line joint angle plot of M fre. cos( θ ): PCS Direction cos( γ ): RS Direction the M fre sensor can be confirmed. It took about three minutes on line to isolate this fault source. When the P st sensor is biased, according to Rule 4 (N I &Y II &N III ), the possible fault source may be P st, and the joint angle plot shown in Fig. 7 further confirmed this. It took about fifteen minutes on line to isolate this source. When the P 2 sensor is biased, with Rule 5 (N I &N II &Y III ), the possible fault source may be M ch, P or P 2, but through Cosine of the angle Faulty operation cos( θ ): PCS Direction cos( γ ): RS Direction - 2: PM 2:3 PM : PM :3 PM Fig. 7. On line joint angle plot of P st. the joint angle plot shown in Fig. 8, the P 2 sensor can be confirmed to be the source indeed. It took about twenty minutes on line to isolate this source.. Conclusions Principal component analysis and joint angle plots based on three balances are presented to detect and diagnose the fixed bias of single sensors in HVAC systems. In order to improve the detection, three PCA models, based on the energy balance and the air side and water side flow pressure balances, are built in this paper. These balances can indirectly reflect the mathematical models of the systems because they extract basic but essential information of the systems, so they can improve the statistical model greatly. It is promising to use a statistical method combined with some essential physical models to diagnose in the future. It is because accurate mathematical models are still difficult or complicated to build for HVAC systems. On the other hand, the pure statistical model lacks the necessary physical information after all. Although the contribution plot has been widely used to isolate a fault source [24], its diagnosing ability in HVAC systems is still limited. Since the effect of some faults may propagate to other variables in the control loop due to feedback action, the real faulty sensor may be hidden and be difficult to isolate, while the joint angle plot presented in this paper extracts sensor fault signatures that are the vectors of movement of the fault in the PCS and RS. The directions of these vectors are compared with the corresponding vector directions of known faults in the fault library. Moreover, some expert rules are integrated with the joint angle plots to improve the on line diagnosing process. References [] Usoro PB, Schick IC, Negahdaripour S. An innovation-based methodology for HVAC system fault detection. J Dyn Syst Meas Control 985;7: [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 989;95(2).

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