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Robust sensor fault diagnosis and validation in HVAC systems Shengwei Wang and Jin-Bo Wang Transactions of the Institute of Measurement and Control 2002; 24; 231 DOI: 10.1191/0142331202tm030oa The online version of this article can be found at: http://tim.sagepub.com/cgi/content/abstract/24/3/231

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Transactions of the Institute of Measurement and Control 24,3 (2002) pp. 231–262

Robust sensor fault diagnosis and validation in HVAC systems Shengwei Wang and Jin-Bo Wang Department of Building Services Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong

A robust fault diagnosis and validation strategy for temperature sensors and ow meters in central chilling plant is developed, which is based on the Žrst law of thermodynamics. The strategy evaluates soft sensor faults (biases) by examining and minimizing the sum of the squares of concerned mass or steady state energy balance residuals represented by the corrected measurement over a period. It considers systematically all the concerned energy balances and obtains the best estimates of the sensor biases by minimizing the sum of the mean squares of normalized residuals of all energy balances involved. The genetic algorithm technique is employed to determine the global minimal solution to the multimodal objective function, which can be difŽcult to achieve by traditional gradient-direc ted search methods. Performance of an advanced robust fault detection, diagnosis and evaluation (FDD&E ) scheme is compared with that of a sequential scheme, which was reported earlier, in simulation tests. The robust scheme is superior to the sequential scheme in robustness to abrupt sensor faults, such as biases, etc. The robust scheme is applied to a central chilling plant in an existing commercial building, providing satisŽed bias estimates. As a basic method, the sensor FDD&E strategy is of practical value in heating, ventilation and air-conditioning (HVAC) systems as well as in systems where the measurements of liquid ow variables are essential to control and performance monitoring.

Key words: bias evaluation; building; chilling plant; fault detection and diagnosis; HVAC; sensor fault; sensor validation; soft fault.

Address for correspondence: Shenwei Wang, Department of Building Services Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong. E-mail: beswwangKpolyu.edu.hk

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Nomenclature a b cp df f I(j) Ibp1 Ibp2 J, k, l M M(j) Mb Mbp Mbp1 Mbp2 N Nop Npt r Sr Srsq T Tsb Trb Trch Tref Ts Tr Trch x d d r

n

CoefŽcient Constant SpeciŽc heat capacity at constant pressure (kJ/kg°C) Relative difference of the best Žtness values from two consecutive GA runs Fitness value or Žtness function Index denoting operation status of jth chiller (or ow) (0,1) Index denoting positive bypass ow direction (0,1) Index denoting negative bypass ow direction (0,1) Serial number Flow meter or ow rate (l/s) Chilled water ow meter or the ow rate of jth chiller (l/s) Building chilled water ow meter or the ow rate (l/s) Flow rate or ow meter of bypass (l/s) Flow rate or ow meter in positive bypass ow direction (l/s) Flow rate or ow meter in negative bypass ow direction (l/s) Total chiller number in the plant Total number of chillers operating simultaneously Total number of sample points balance residual sum of residuals sum of the squares of the balance residuals Temperature sensor or temperature (°C) Building supply temperature sensors or the temperature (°C) Building return temperature sensor or the temperature (°C) Chiller common return temperature sensor or the temperature (°C) Reference temperature Chiller supply temperature sensors or the temperature (°C) Chiller chilled water return temperature sensor or the temperature (°C) Chiller common return temperature sensor or the temperature (°C) Any meter or process variable Constant additive sensor bias Effect of sensor biases on balance residual White random noise

Superscripts ˆ· : [i] igen 0

Sample value Normalized variable Sampling instant Number of generation Initial estimate Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Subscripts A B best C x 1.

Related to control volume A Related to control volume B Best value Related to control volume C Related to any meter or the process variable Introduction

Research on techniques of automated fault detection and diagnosis (FDD) in building heating, ventilation and air-conditioning (HVAC) systems has become very active and extensive in recent years (Usoro et al., 1985; Liu and Kelly, 1989; Pape et al., 1991; Norford and Little, 1993; Hyvarinen, 1995; Dexter, 1996). An advanced FDD scheme aims at assisting system operators to monitor the current state of the system through inferring the nature and extent of any fault using the sensor and control signals which are accessible in building management systems. Application of such FDD techniques could lead to improved occupant comfort, reduction of energy consumption, prompt and economic equipment maintenance, and longer equipment life. Among the various faults in HVAC systems, complete and partial equipment failures (hard and soft faults) and hard sensor faults have been dealt with extensively (Lee et al., 1996a, 1996b; Yoshida et al., 1996; Dexter and Benouarets, 1996; Haves et al., 1996; Jiang et al., 1995; Stylianou and Nikanpour, 1996; Peitsman and Bakker, 1996; Tsutsui and Kamimura, 1996). Usually, these types of faults affect directly the availability and quality of system services and energy use efŽciency, and therefore became the Žrst FDD target problems. Soft sensor faults, such as biases or drifts, are among the typical faults commonly found in HVAC systems. Awareness has been growing that sensor errors are one of the main obstacles to improving the performance of building control schemes and successfully applying fault detection techniques in automated HVAC commissioning and performance monitoring systems (Dexter, 1999). Development of automated FDD techniques for soft sensor faults is of practical value. A biased sensor provides deceptive information to control and monitoring systems and operators. The effects could be more energy consumption (Kao and Pierce, 1983), failure in applications of advanced control, optimization and system/component FDD techniques (Usoro et al., 1985; Stylianou and Nikanpour, 1996), and unreliable results in system/component performance assessments. As a matter of fact, any action or decision made based on biased sensor signals could be erroneous. This can be particularly serious in HVAC systems, since the temperature differentials are usually small, and biases of even moderate magnitudes could result in drastic errors in control, FDD and performance monitoring schemes. Generally, there are three kinds of approaches for sensor FDD: model-based, knowledge-based and measurement aberration detection (Henry and Clarke, 1993). The model-based approach employs either dynamic or static system/ component models. Implicit analytical redundancy in the dynamic relationships Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Fault diagnosis and validation in HVAC systems

between measurements and system inputs is exploited. Residuals are generated and examined by the decision-making process, resulting in fault decisions (Isermann, 1984; Baseeville, 1988; Frank, 1990; Patton and Chen, 1991). In the knowledge-based method, qualitative models of the process are built and manipulated using heuristic reasoning. Techniques used include expert systems (Tzafestas, 1991), neural nets (Hemmelblau, 1992; Lee et al., 1997) and fuzzy logic (Vachekov and Matsuyama, 1992). The measurement aberration detection method examines the output of a sensor independently for the indications of faults, assuming that the ‘true’ measurement signal has certain time and/or frequency domain properties and that a ‘fault’ may occur randomly at any time (Yung and Clarke, 1989). The model-based method is the one of the methods used in most modern FDD schemes. This method is powerful in the detection and diagnosis of abrupt changes. However, it has some problems in application. One is that it is in general difŽcult to develop a model that is robust to process disturbances and modiŽcations to the plant (Henry and Clarke, 1993). This is true for the highly nonlinear and dynamic HVAC systems. The other is that, in dealing with the nonabrupt type soft sensor faults (slowly developing biases), model-based methods may encounter problems in distinguishing sensor biases from system or component performance degradations since both faults evolve simultaneously. The SEVA (sensor validation) approach avoids the problem of process model validity and some other problems with the model-based method (Henry and Clarke, 1993). A SEVA sensor (or smart sensor) is designed to have a built-in microprocessor. The fault detection and diagnosis of the sensor is dealt with internally by evaluating the steady state and dynamic characteristics of the elements of the sensor (Yung and Clarke, 1997; Clarke and Fraher, 1996). However, this approach does not make use of knowledge of the systems, which can be valuable and, usually, not difŽcult to obtain. In the Želd of HVAC, Usoro et al. (1985) studied the detection and diagnosis of an abrupt bias in a room temperature sensor using a model-based method. Stylianou and Nikanpour (1996) also used a model-based method to detect soft sensor faults, aiming at making sure the measurements were reliable when monitoring the performance of a laboratory chiller. Lee et al. (1997) investigated the detection and automatic recovery of a faulty supply air temperature sensor in an air-handling unit (AHU). Recently, Wang and Wang (1999) reported a law-based strategy for the fault detection, diagnosis and evaluation (FDD&E) of nonabrupt biases of the temperature sensors and ow meters in a central chilling plant and presented dynamic simulation test results. The law-based sensor FDD&E strategy is based on the fundamental mass and (steady state) energy conservation (balance) relationships. Those relationships are easy to build and their validity is absolute and independent of plant performance degradations and change in working conditions. The strategy, therefore, resolves the two problems with the commonly used model-based sensor FDD methods mentioned above. Sensor bias values are estimated basically by minimizing the sum of the squares of the corrected residuals of each of the involved balances, sequentially. However, robustness is, sometimes, a problem of the FDD&E strategy, particularly in dealing with the measurements containing abrupt biases. Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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This paper presents an improved scheme of the sensor FDD&E and the application to an existing building HVAC system. The robust scheme considers systematically all the concerned energy balances and estimates the average values of the sensor biases by minimizing the sum of the mean squares of normalized residuals of all energy balances involved. The genetic algorithm (GA) technique is employed to determine the global minimal solution to the multimodal objective function, which can be difŽcult to achieve by gradient-directed search methods. Simulation tests are conducted to compare the performances of the improved robust FDD&E scheme and the existing scheme. Results of Želd application are presented and analysed. 2.

Chilling plant and monitoring sensors

Figure 1 shows the schematic of a typical primary–secondary chilling plant commonly used in large HVAC systems. The chilled water system is separated into primary and secondary loops by a decoupling bypass line. In the primary loop, multiple chillers are employed. Each chiller has an associated constant primary pump. Usually, but not always, the secondary chilled water pumps are variable speed, which are usually controlled to maintain a constant differential pressure across the building supply and return near chilling plants or the (most) remote

Figure 1

Schematic of a primary–secondary chilling system

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Fault diagnosis and validation in HVAC systems

AHU. Two-way control valves are used to modulate the chilled water ow through individual AHU terminals. Variation of total building cooling load is reected in the ow rate and temperature difference in the secondary loop. Bypass chilled water ows in either the ‘positive’ or the ‘negative’ direction, depending on the difference between the circulation ow rates in the primary and the secondary loops. As building cooling load varies, the number of chillers (and primary pumps) required may vary also, resulting in the cycling on or off of different chillers at different times. The sensors to be examined are the temperature sensors and ow meters (Figure 1), which include the building supply ow meter (Mb), building supply and return temperature sensors (Tsb, Trb), chilled water ow meter and supply and return temperature sensors associated to each chiller (M(j), Ts(j), Trch) and bypass ow meter (Mbp). Because the bypass ow meter measures the ows in both directions (positive and negative) and it is possible that the meter has different characteristics for the two directions, the meter is treated as two different meters. Mbp1 and Mbp2 are used to denote the positive and negative direction ows respectively. A ow switch is usually used in association with each individual chiller to detect the corresponding chilled water ow condition. When a primary pump is activated, the corresponding ow switch sends out a signal, indicating that the corresponding chiller can be started. I(j) is used to denote this signal. If the chilled water is owing through the evaporator of the corresponding chiller I(j) = 1, else I(j) = 0. There also exists a ow switch in the bypass line to indicate ow direction. Ibp1 and Ibp2 are used to denote the bypass ow direction signals. When the ow direction is positive, Ibp1 = 1 and Ibp2 = 0. When the ow direction is negative, Ibp1 = 0 and Ibp2 = 1. 3.

Sequential FDD&E scheme

The sequential FDD&E scheme refers to the scheme presented by Wang and Wang (1999). A brief review of the scheme is presented in this section. The scheme evaluates the biases of the ow meters and temperature sensors in the chilling plant by estimating their values, which allows the locations and degrees of the sensor faults be identiŽed simultaneously. For the sensor installation condition speciŽed in Figure 1, the sequential FDD&E scheme consists of four properly conŽgured estimators (Estimators 1, 2, 3 and 4), as illustrated by Figure 2. Each estimator is composed of a steady state detector, a preprocessor and a bias evaluator, as shown in Figure 3. The steady state detector Žltrates the measurement data which are collected in the periods when the concerned balance is in transients. The preprocessor manipulates the measurement data in steady state into coefŽcients or constants. The evaluator solves the estimation equations for the estimates of the concerned sensor biases. The chiller return temperature sensor (Trch) is chosen as the common temperature reference point whose health state is assumed good, i.e., d Trch = 0. Estimator 1 estimates the individual chiller ow meter biases ( d M(j)) and the sum of the biases of the building supply ow meter and the by-pass meters ( d Mb+d Mbp1 , d Mb+d Mbp2 ). Using the outputs of Estimator 1 ( d M(j)) as parameters, Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

Wang and Wang

Figure 2

Sequential FDD&E scheme for the chilling plant

Figure 3

Structure of a bias estimator

237

Estimator 2 determines the relative bias of the individual chiller supply temperature sensor with respect to the building supply temperature sensor ( d Ts(j) - d Tsb). Estimator 3 estimates the bias of the building return temperature sensor ( d Trb), which is actually relative with respect to the common reference (the chiller return) temperature sensor. Using the outputs from the preceding three estimators as known parameters, Estimator 4 estimates the biases of the building supply ow meter and temperature sensor ( d Mb, d Tsb). Finally, the bypass ow meter biases ( d Mbp1 , d Mbp2 ) and individual chiller supply temperature sensor biases ( d Ts(j)) are determined. The four estimators were developed to minimize, respectively, the sum of the Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Fault diagnosis and validation in HVAC systems

squares of corrected residuals of the ow balance, the physical redundancy (control volume C, when bypass chilled water ows in the negative direction), the control volume B energy (heat) balance (when the bypass chilled water ows in the positive direction) and the control volume A heat balance, as represented by Equation (1), where r[ i] denotes a corrected balance residual represented by Equation (2), which is calculated from the data resulting from eliminating sensor biases ( d x) from the raw measurements (xˆ[ i] ). r(·) denotes a (steady state) conservation relationship as the function of the involved measurements, whose value should be zero if the measurement data were true variables (x), i.e., r(x) = 0. minimize

|O d x

r

(r[ i] )2 ; [r[ i] = r[Mi] , r[Ci], r[Bi] , r[Ai] ]

(1)

i

= r(xˆ[ i] - d x)

[ i]

(2)

Equation (3) represents a raw residual (rˆ[ i] ) which is calculated from the raw data. The relationship between a corrected and the corresponding raw residual can be represented by Equation (4), where d [ri] is the effect of the sensor biases on the balance residual. Its detailed expression, as a function of measurements and sensor biases, can be deduced from Equation (5) for a speciŽc balance relationship. rˆ[ i] = r(xˆ[ i] ) r[ i] = rˆ[ i] - d d

[ i] r

(3) [ i] r

(4)

= r(xˆ[ i] ) - r(xˆ[ i] - d x)

(5)

Equations (6) to (9) are the four raw balance residuals used in the development of the four estimators mentioned above.

O N

i] [ i] i] [ i] ˆ [bi] + M ˆ [bp1 ˆ [bp2 rˆ[Mi] = M Ibp1 + M Ibp2 -

O

ˆ [ i](j)I[ i](j)] [M

(6)

j=1

N

rˆ[Bi] = r cpw

[ i] [ i] [Ibp1 I (j)M[ i](j)(Tˆ[si] (j) - Tˆ[sbi] )]

(7)

j=1

i] i] )I[bp2 rˆ[Ci] = (Tˆ[rbi] - Tˆ[rch

H

[ i] ˆ [bi](Tˆrb - Tˆ[sbi] ) rˆ[Ai] = r cpw M

O N

i] ˆ [ i] (j)I[ i] (j)(Tˆ[rch [M - Tˆs[ i](j))]

j=1

J

(8) (9)

Equation (10) represents the normal equation in general form corresponding to the problem represented by Equation (1). Estimators 1 to 4 are derived by applying this equation to each of the residuals deŽned above. The normal equations Žnally obtained for the four estimators are represented by Equations (11) to (14). Success in obtaining a set of unique solution to the above four equations, i.e., the estimates of the sensor biases, relies on the combinations of the operating chillers in different periods when the measurement data are collected. Such chiller combination information is identiŽable through analysing the coefŽcients (matrix). Detailed procedures of the derivation of these equations, the formulae for calculatDownloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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239

ing the coefŽcients and constants, and the techniques of solving and identifying unique solution sets can be found in Wang and Wang (1999). ­

FO i

(rˆ[ i] - d

aM.1,2

¼ aM.1,j+ 2

¼ aM.1,N+ 2

aM.2,1

aM.2,2

¼ aM.2,j+ 2

¼ aM.2,N+ 2

aM.3,1

aM.3,2

¼ aM.3,j+ 2

¼ aM.3,N+ 2

¼

¼

ê

ù

¼

ë aM.N+ 2 ,1 aM.N+2 ,2

aM.N+ 2,j+ 2 é

ê ê

ê aB.l,1

ë

¼

d

M

d

¼ ¼

aB.N,1 ¼ aB.N,j ¼ aB.N,N

-d

+d +d

Mbp1ù Mbp2

úê úê

=

Trch

O

û ë

ù

úê ú úê ú bM.2

ú = ê ¼

ú

(11)

bM.j+ 2

(j)

M

¼

ë bM.N+ 2 û

û

(N)

M

ù

Ts

Tsb

¼

é

ù

ú ê ú ú ê ú bB.1 ¼

(j) - d

Ts

Tsb

ú = ê bB.l ú

(12)

¼

¼ (N) - d

d

é bM.1

bM.3

(1)

(1) - d

d

¼ aB.l,j ¼ aB.l,N ú ê d

Trb

Mb

(10)

ê¼

ù é

¼ ¼

¼ ¼

Mb

d

aM.N+ 2 ,N+2 û ëd

¼ ¼

d

5

úê úê

=0

¼

aB.1,1 ¼ aB.1,j ¼ aB.1,N ¼

G

éd

ú

aM.l+ 2,1 aM.l+ 2 ,2 ¼ aM.l+ 2,j+ 2 ¼ aM.l+ 2 ,N+2 ¼

)

­ ( d x)

é aM.1,1

ê ê

[ i] 2 r

Ts

û

Tsb

ë

bB.N

û

[i] [(Tˆ[rbi] - Tˆ[i] rch)Ibp2 ]

i

O

(13) I[i] bp2

i

+ aA.1,2 d

aA.1,1 d

Mb

aA.2,1 d

Mb + aA.2,2 d

Tsb

+ aA.1,3 d

Tsb + aA.2,3 d

+ aA.1,4 d

2 Tsb

+ aA.1,5 d

2 Mbd Tsb

= bA.1

Tsb + aA.2,4 d

2 Mb

+ aA.2,5 d

Tsbd

2 Mb

= bA.2

Mbd Tsb Mbd

(14)

The sequential FDD&E scheme was validated using dynamic simulation data in the condition that the sensor biases were Žxed during the time the evaluation was performed and that white random noises were added to the measurements in addition to the Žxed biases. In practice, however, abrupt bias may occur to one or more sensors. As will be demonstrated by simulation test, such abrupt sensor biases could lead to a noticeable decrease of estimation accuracy. The following robust FDD&E scheme is developed to settle the problem. Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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4.

Fault diagnosis and validation in HVAC systems

Robust FDD&E scheme

Figure 4 illustrates the overall structure of the robust FDD&E scheme developed. It employs the sequential FDD&E scheme and a robust GA Estimator. The GA Estimator is designed to estimate the biases of the building supply ow meter and temperature sensor ( d Mb, d Tsb), and the biases of the chiller supply temperature sensors ( d Ts(j)). The biases of the chiller ow meters and the building return temperature sensor ( d M(j), d Trb,) are estimated by the sequential scheme. The GA Estimator estimates d Mb, d Tsh and d Ts(j) by minimizing the sum of the mean squares of the normalized corrected residuals of the control volume A and control volume B heat balances, as represented by Equation (15). · · Srsq.A Srsq.B minimizeu d x + ; [ d x = d Mb, d Tsb, d Ts(j), j = 1, ¼, N] (15) nA nB

S

D

where · Srsq.A =

O O

(r·[Ai] )2

(16)

(r·[Bi] )2

(17)

i

· Srsq.B =

i

Figure 4 Robust scheme for sensor fault detection, diagnosis and estimation Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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241

The normalized corrected residuals (r·[Ai], r·[Bi] ) are deŽned as represented by Equations (18) and (19), which are against the number of the chillers operating at the instant [i]. When the chillers are different in cooling capacity, the number of a larger chiller is deŽned as the ratio of its nominal capacity to that of the smallest chiller. r[Ai] rˆ[Ai] d r[Ai] r·[Ai] = [ i] = [ i] - [ i] Nop Nop Nop

(18)

r[Bi] rˆ[Bi] d r[Bi] r·[Bi] = [ i] = [ i] - [ i] Nop Nop Nop

(19)

In concept, the minimization problem (Equation 15) might be reduced to solving the corresponding normal equations in the sensor biases, as was done in the sequential FDD&E scheme. However, the resulting normal equations would be a system of polynomials in the bias estimates, which has a total number of N+2 equations and their highest order is 3. It could be quite difŽcult to use traditional gradient-directed search methods to Žnd the global minimum solutions. As an alternative, GA is employed to cope with the problem of Equation (15). In the robust scheme, the sequential scheme is employed not only to estimate the biases of the individual chiller and bypass ow meters and the building return temperature sensor, but also to assist the GA Estimator in two ways. First, the sequential scheme is used to check whether a unique set of estimates can be obtained based on collected measurement data, since, generally, a GA scheme alone does not have the ability to do so. Secondly, the sequential scheme is used to produce a set of initial estimates, which are then used to determine and narrow the search space of the GA Estimator. 5.

Genetic algorithms

GA is an advanced search and optimization technique. It has been developed to imitate the evolutionary principle of natural genetics. GA was invented by Holland (1992) and further developed by his students and colleagues in the 1960s and the 1970s. Goldberg (1989), Davis (1991) and Mitchell (1996) provided comprehensive overviews and introductions to GA. Deb (1996) compared the GA search method with traditional methods (the direct exhaustive search method and the gradient-directed search method ) for function optimization. One of the main advantages of GA is that it is generally robust in Žnding global optimal solutions, particularly in multimodal and multiobjective optimization problems (Deb, 1996). Extensive research on the theoretical fundamentals and applications of GA is still going on, aimed at better computation efŽciency, improved robustness, and so on (Salomon, 1998). Generally, GA uses three operators (selection, crossover and mutation) to imitate the natural evolution processes. The working procedures of a simple GA using binary coding are summarized as follows. 1) Initialization: to create an initial population of bit-strings (chromosomes ), randomly, which represent a population of trial solution candidates. Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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2) Evaluation: to calculate the Žtness corresponding to each bit-string (each trial solution candidate) in the population. The Žtness function is designed such that the better the value of the optimization objective function, the larger the Žtness. 3) GA operations: to repeat the following operations until a new population of bitstrings is formed. Selection (or reproduction): to select a pair of good bit-strings in the current population (parent population). The probability of a string in the parent population being selected (to mate with other string) is an increasing function of its Žtness. Thus, a trial solution candidate that produced better values of objective function is more likely to be selected than those that produced inferior values. Crossover: to exchange (crossover) randomly chosen portion(s) of the two strings, with certain probability (crossover probability or crossover rate), to form two offspring. Crossover is the dominating operation in GA. Mutation: to mutate the offspring bit-strings at each locus with a certain probability (the mutation probability or mutation rate). Mutation rate is usually small. 4) Replacement: to replace the parent population with the obtained offspring bitstrings. 5) Evaluation: the same as step 2. 6) Termination: to stop after a prespeciŽed number of generations (one loop from step 3 to step 5 is called a generation) or when a criteria that determines the convergence is satisŽed. Otherwise, go to step 3. The procedures described above form a run. They are the basis for most applications of GA. There are a number of details to Žll in, such as the size of the population and the probabilities of crossover and mutation. 6.

GA Estimator

Figure 5 shows schematically the owchart of the developed GA Estimator. It starts with the initial bias estimates and steady state measurement data generated in the sequential FDD&E scheme. The component with greyed background represents the procedures of a GA run. Multiple runs are allowed. Equation (20) represents the Žtness function (f), which is the reciprocal of the objective function of the original minimization problem (Equation 15). Given a set of d Ts(j), d Mb and d Tsb, the sums of the squares of the corrected heat · normalized · balance residuals of the control volumes A and B (Srsq.A, Srsq.B) can be calculated using the formulae given in Appendix A. The bias estimates of other sensors from the sequential FDD&E scheme are used as known parameters. · · Srsq.A Srsq.B + (20) f = f( d Mb, d Tsb, STs(j)) = 1 nA nB

YS

D

The variables constituting the GA search space are the relative biases of the individual supply temperature sensors with respect to the building supply temperature sensor ( d Ts(j) - d Tsb). The remaining two estimates ( d Mb, d Tsb) corresponding Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

Wang and Wang

Figure 5

243

Flowchart of the GA Estimator

to each of the GA trial sets are determined internally by employing the Estimator 4. Such a treatment is adopted to increase the convergence speed. At the beginning of each run, the search interval for each variable is set. The initial estimate of the relative temperature sensor bias ( d 0Ts(j) - d 0Tsb) is used as the centre of the searching space, and a default radius is used to limit the range over which the variable is to be sought. Two kinds of initializations are involved at the beginning of a run in the GA Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Estimator. The Žrst is the initialization that produces the initial population to start a GA run. This is one of the GA procedures as described in the preceding section. The second is the initialization that sets or resets the intervals of the search variables and the seed of the random number generator. For different runs, the seed is changed. Termination of a GA run is decided if the number of the current generation is equal to a prespeciŽed maximum number. At least two runs of the GA processes are necessary when running the GA Estimator. The criterion to stop the GA Estimator is based on the comparison of the best Žtness values of two consecutive runs. If the relative difference between the two maximum Žtness values (df) is less that a threshold value ( e f, e.g., equals to 0.001), then, the GA Estimator is stopped. A GA driver developed by Carroll (1999) is used. Relevant adaptations of the source code were made for implementing the multiple runs and the termination rules, resetting the control parameters, etc. Table 1 shows the control parameters used in the GA Estimator. Detailed explanations and discussions about the determinations of these control parameters can be found in Carroll (1996). 7.

Simulation tests

Many tests using simulation and Želd data were conducted to validate and compare the performance of the sequential and the robust FDD&E schemes. Two simulation tests using measurement data in different conditions are presented. The dynamic simulation program for a chilling system developed by Wang (1998) uses TRANSYS as a platform to generate the measurement data. The chilling system simulated is the same as the real system described in next section. In the simulation, four identical duty chillers are used, and the sensors to be examined are as shown in Figure 1. Fixed sensor biases are introduced throughout the simulation. Random noises are added to sensor outputs at each simulation step. Descriptions of the relevant system operations are available in Wang and Wang (1999). Table 1

Control parameters of the GA Estimator

Selection Crossover Mutation Maximum generation of a GA run Population size Length of chromosome Interval of search space Threshold value

Tournament scheme and elitism Uniform, with probability of 0.7; one child from a pair of parents for next generation Both creep and jump mutations enabled (creep mutation rate is 0.01; jump mutation rate is 0.02) 100 30 8 bits for each of the searching variables From -2 to 2°C for each variable, with the initial estimate as the centre e f = 0.001

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7.1

245

Test 1

In Test 1, the measurement data are from the simulation of 4 days operation directly. The introduced sensor biases are given in Table 2. No bias is introduced to the common reference temperature sensor (chiller return temperature sensor). Estimates of the sensor biases produced by the sequential and the robust schemes are also given in the table. The results show that there are no differences in the bias estimates of the individual chiller ow meters and the building return temperature sensor ( d M(j), d Trb) from the two schemes. This is expected because Estimators 1 and 3 are not affected by the use of the GA Estimator. Minor differences exist between the bias estimates of the building and bypass ow meters ( d Mb, d Mbp1 , d Mbp2 ) and the building and the chiller supply temperature sensors ( d Tsb, d Ts(j)). Those differences are negligible compared to the true bias values; these indicate that the robust FDD&E scheme performs as well as the sequential scheme in estimating the sensor biases in usual conditions. 7.2

Test 2

Test 2 still uses the measurement data of Test 1, but an abrupt bias is introduced to the building supply temperature sensor (Tsb). The introduced sensor biases, the time averages and the estimates produced by the two schemes are given in Table 3. The abrupt bias has a small magnitude (0.3°C) and lasts only a short time (approximately 40 min). It was expected that the estimates from the sequential scheme would not be inuenced by the abrupt bias and would be close to the time averages, since the Table 2

The introduced and estimated sensor biases – Test 1 ( d

Sensor

Mb Mbp1 Mbp2 M(1) M(2) M(3) M(4) Tsb Trb Ts(1) Ts(2) Ts(3) Ts(4)

Introduced bias (l/s) (l/s) (l/s) (l/s) (l/s) (l/s) (l/s) (°C) (°C) (°C) (°C) (°C) (°C)

-14.04 -1.54 5.72 12.24 2.28 -13.82 -2.17 -0.653 0.414 1.580 -0.034 -0.099 0.650

Trch

= 0)

Estimates Robust scheme Test 1

Sequential scheme Test 1

-12.14 -3.91 3.69 11.96 2.19 -13.87 -2.17 -0.729 0.417 1.465 -0.124 -0.190 0.563

-13.67 -2.37 5.18 11.96 2.19 -13.87 -2.17 -0.740 0.417 1.515 -0.130 -0.188 0.563

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Table 3

Test condition and the result – Test 2 ( d

Sensor

Mb Ts b Mbp1 Mbp2 Ts(1) Ts(2) Ts(3) Ts(4)

Trch

= 0)

Introduced bias

(l/s) (°C) (l/s) (l/s) (°C) (°C) (°C) (°C)

Estimates

Fixed

Abrupt/ [occurring time (h)]

Time average

Robust scheme

Sequential scheme

-14.041 -0.653 -1.535 5.721 1.582 -0.034 -0.099 0.650

– 0.3/[19:06–19:46] – – – – – –

-14.041 -0.651 -1.535 5.721 1.582 -0.034 -0.099 0.650

-11.818 -0.763 -2.625 3.099 1.450 -0.092 -0.257 0.497

-7.035 -0.944 -5.407 0.317 1.077 0.124 -0.630 0.130

magnitude is small and the lasting time is quite short. However, the actual estimates are quite different from the time averages. The abrupt bias of the building supply temperature sensor affected not only the bias estimate of this sensor itself, but also the bias estimates of the chiller supply temperature sensors and the building and by-pass ow meters. The largest error of the temperature sensor bias estimates is about 0.54°C ( d Ts(3)). On the other hand, the result produced by the robust scheme is much better. The building ow meter bias estimate becomes closer to the introduced value (from -7.035 l/s of the sequential scheme to -11.808 l/s of the robust scheme, the true value is -14.041 l/s). The bias estimate of the building supply temperature sensor is -0.763°C, which is much closer to the true value of -0.651°C than the sequential scheme estimate of -0.994°C. The largest estimation error of the temperature sensor biases is reduced to 0.16°C ( d Ts(3)). Actually, all the accuracy of the bias estimates are improved by the robust scheme. The result produced by the sequential FDD&E scheme in this test reveals that the sequential scheme is not robust in certain conditions. The reason is that the introduced abrupt bias to the building supply temperature sensor (Trb) happened to be in a period which constitutes one combination of the simultaneously operating chillers. The misleading information contained in the measurement data from the period affected the output of Estimator 2, i.e., d Ts(j) - d Tsb, which further inuenced the output of Estimator 4. The sum of the squares of the balance residuals of the control volume B (Srsq.B) was indeed minimized by the estimates produced by the sequential scheme. At the same time, the sum of the squares of the balance residuals of the control volume A (Srsq.A) was also minimized, when the bias estimates of the building ow meter and the building supply temperature sensor were considered alone. However, the individual chiller supply temperature sensor biases also affects the sum of the squares of the heat balance residuals of the control volume A. The estimates of the individual chiller supply temperature sensors could be adjusted to further minimize Srsq.A, while Srsq.B is not affected signiŽcantly. The minimizDownloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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247

ation object function of the GA Estimator considered the two heat balances systematically, which resulted in better performance of the robust FDD&E scheme. 8.

Site study and validation in existing buildings

The existing chilling system (Figure 6) serves a 46 stories high commercial ofŽce building with a usable area of about 74 000 m2 . Indirect seawater-cooled chillers supply chilled water to terminal AHUs. The chilling system operates 24 h a day, including public holidays. The chilling plant is located in the basement. Five identical centrifugal chillers are used, four for normal duty and one for stand-by. Each chiller has a design cooling capacity of 3100 kW. Associated to each chiller are a primary chilled water pump, a cooling water pump and a seawater heat exchanger. Four pumps are used in the secondary chilled water loop, three for duty and one for stand-by. All these pumps are constant speed pumps. The two seawater pumps are variable speed pumps. Each AHU is equipped with a PID controller, which controls the supply air temperature by adjusting the chilled water ow through the cooling coil. A twoway modulating valve is used. The frequency of the AC power to the seawater pumps and the sequencing of the chillers and the secondary pumps are supervised manually. The sequencing of the secondary pumps is automatically controlled according to the differential pressure of the secondary chilled water loop and the bypass ow rate. In the chilled water networks, the building management system (BMS) monitor-

Figure 6

Schematic of an existing chilling system

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Fault diagnosis and validation in HVAC systems

ing sensors in the chilling plant include the chilled water temperate sensors installed for the building supply (Tsb), individual chiller supply and return (Ts(j), Tr(j)). Chilled water ow meters are available for the building total supply (Mb), the bypass (Mbp) and individual chillers (M(j)). The building return temperature sensor (Trb) is not installed. There are also some other monitoring points, as shown in Figure 6, which are not covered in the current study. 8.1

Revision to estimators and scheme

Three major revisions are made to the estimators and scheme described previously, since the Želd sensor installation condition is different from that speciŽed in Figure 1. Figure 7 shows the sequential scheme, which serves as a part of the robust scheme employed in the Želd study. Due to the absence of the building return temperature sensor (Trb), Estimator 3 (see Figure 2) is not used. An estimator (Estimator 8) is added to estimate the biases of the chiller return temperature sensors ( d Tr(j)), which is described in Appendix B. Estimator 4 is revised, wherein the control volume A heat balance i] residual is calculated using Equation (21). Tˆ[rch is an artiŽcial chiller return temi] perature measurement calculated by Equation (22). I[bp2 is included in Equation [ i] (21), meaning that rˆA can be calculated only for the measurements when the

Figure 7

Sequential scheme estimators for the real chilling plant

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249

bypass chilled water ows in the negative direction. Equation (21) is also used when calculating the Žtness values in the GA Estimator. rˆ

[ i] A

=r c I

[ i] pw bp2

O

H

O N

i] ˆ [bi](Tˆ[rch - Tˆ[sbi] ) M

ˆ [ i](j)I[ i](j) (Tˆ[ri](j) - Tˆ[si](j))] [M

j=1

J

(21)

N

[ i] = Tˆrch

j=1

I[ i](j) (Tˆ[ri](j) - d

O

(j))

Tr

(22)

N

[ i]

I (j)

j=1

8.2

Data collection and sensor fault

The measurement data from the monitoring sensors are recorded in the BMS, which are then retrieved from the central computer station of the BMS. Data from several intermittent days within 1 month are collected. The measurement sampling interval is 5 min. The BMS is set not to update the sensor outputs if the change of a new reading is within a given threshold value. Sensor faults (biases) are introduced to three of the chilled water temperature sensors (Tsb, Ts(2), and Ts(3)) through changing the deŽnitions of the relevant temperature sensors in the BMS outstations. The values of the introduced biases are given in Table 4. Prior to introducing these faults, checks and calibration of the temperature sensors in the chilling plant are conducted. No special instruments or procedures are used to check the ow meters. However, records of the original commissioning data of the chilled and cooling water ows of the individual chillers are available. 8.3

Results and discussions

Table 5 presents the output of the robust FDD&E scheme based on the collected measurement data. The biases introduced to the three temperature sensors (Tsb, Ts(2), Ts(3)) are successfully diagnosed. The largest error of the three estimates is 0.25°C ( d Tsb). The common reference used in the estimation is the artiŽcial chiller return temperature sensor (Trch). The use of this artiŽcial temperature sensor as the common reference assumes implicitly that the sum (or the mean) of the biases of the indiTable 4

Biases manually introduced to three of the BMS temperature sensors

Temperature sensor

Tsb

Ts(2)

Ts(3)

Fault – bias (°C)

1.5

-1.0

1.5

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Fault diagnosis and validation in HVAC systems

Table 5

Bias estimates of the sensor in an existing chilling plant M(1) 3.2

M(2) 17.9

M(3) 17.7

M(4) 6.8

M(5) 17.0

Sensor Bias Estimate (°C)

Ts(1) -0.08

Ts(2) 1.10

Ts(3) -1.47

Ts(4) 0.14

Ts(5) 0.27

Sensor Bias Estimate (°C)

Tr(1) -0.24

Tr(2) -0.11

Tr(3) 0.04

Tr(4) 0.24

Tr(5) 0.08

-0.16

-1.21

1.51

0.10

-0.19

d

Sensor Bias Estimate (L/s)

Tr

(j) - d

Ts

(j) (°C)

Mb -4.9

Mbp1 -2.2

Mbp2 16.8

Tsb 1.75

vidual chiller return temperature sensors ( d Tr(j)) are zero (see Appendix B). Since the temperature sensors were checked and adjusted before the data collection, the real bias values of the Žve individual chiller return temperature sensors (Tr(j)) can be regarded as randomly distributed around zero. In such a circumstance, the implicit assumption becomes, approximately, the real situation. The estimates of the individual chiller return temperature sensor biases ( d Tr(j)) obtained are around zero with limited errors, as can be seen in Table 5. The resulting bias estimates of the differential temperature sensors for the Žve chillers ( d r(j) - d s(j), j = 1, ¼, 5) are -0.16, -1.21, 1.51, 0.10, -0.19°C, respectively. The two temperature sensor biases introduced ( d Ts(2) and d Ts(3)) inuenced the corresponding differential temperature measurements (Tr(2) - Ts(2), Tr(3) - Ts(3)). Their effects ( d r(2) - d s(2), d r(3) - d s(3)) were correctly diagnosed. The estimated biases of the bypass chilled water ow meter (negative direction, d Mbp2 ), the chilled water ow meters of chillers 2, 3 and 5 ( d M(2), d M(3), d M(5)) turn out not to be low (see Table 5). For chillers 2, 3 and 5, the biases account for more than 10% of the measured values. The remaining four chilled water ow meter biases are negligible, considering the measurement uncertainties of the ow meters. The original commissioning data of chiller chilled water ow rates (130 l/s) provided by the chiller supplier were used as a reference for comparison, since there is no other simple method to compare accurately the ow meter bias estimation results with the actual unknown condition. Another reason is that the chilling system is required to operate continuously and interruption is prohibited. As shown in Figure 8(A), the measured (raw) chilled water ow rates of different chillers are different from one another, even when the chillers involved are operating simultaneously. The largest difference is about 20 l/s. This contradicts the fact that the chillers, and also the primary pumps, are identical, which means that the chilled water ow rates of the different chillers should be approximately the same. The contradiction disappears in the corrected chiller ow rates, as can be seen in Figure 8(B), though each corrected chiller ow rate still varies a little when the number of the simultaneously operating chillers (and secondary pumps) is different. All the corrected chiller chilled water ow rates are almost the same and are close to the commissioning data. Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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251

Figure 8 (A) The raw and (B) the corrected chiller chilled water ow rates

8.4

Sensitivity of balance residual to sensor bias

8.4.1 Flow balance residual: The raw ow balance residual indicates the existence of the ow meters biases as shown in Figure 9(A). In principle, the balance residuals should be randomly around zero, but the raw residual does not exhibit this characteristic: it deviates signiŽcantly from zero. This indicates that some, if not all, of the ow meters are biased. Site technicians are very cautious of chilled water leakage in the chilling plant. No report was found during the period of data collection. It has also been carefully Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Fault diagnosis and validation in HVAC systems

Figure 9 (A) The raw and (B) corrected chilled water ow balance residuals

checked if chilled water owed through any of the evaporators of idle chillers, by tracing the individual supply and return temperatures of the idle chillers. The rising rate of each of these temperature measurements was found to be normal after the chillers were shut down. This indicates that there was no bypass ow through the evaporators. There cannot be any other reason why some of the chilled water disappeared or was generated. Therefore, violation of the law of mass conservation by the raw ow measurements presents a clear evidence that the ow meters must have problems. Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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253

On the other hand, as shown in Figure 9(B), the corrected ow balance residual varies randomly around zero. For the raw residual, the mean and the standard deviation are -9.01 and 21.4 l/s, respectively; for the corrected residuals, they are 0.04 and 7.45 l/s. 8.4.2 Heat balance residuals: Similarly to the ow balance residual, the heat balance residuals also have the ability to detect the existence of the biases of involved sensors. The raw and the corresponding corrected heat balance residuals of the control volume B and A are shown in Figures 10 and 11, respectively. It can be seen that the raw heat balance residuals of the two control volumes deviate from zero severely. Obviously, the reason is that three large temperature sensor biases were introduced. After the raw measurements had been corrected with the obtained bias estimates, the large biases in the balance residuals diminished. 8.4.3 Physical redundancy residuals: Figure 12 shows the redundancy residuals of some of the redundant (chiller return) temperature sensors (Tr(1) - Tr(4), Tr(2) - Tr(4), Tr(3) - Tr(4)). The samples presented in the Žgure are from two different periods. The Žrst is the period during which the data were collected and used for the FDD&E purposes. The second is a short period about 1 month later. The later data were collected speciŽcally to examine whether the temperature sensor biases changed. As can be seen in the three Žgures, small magnitude biases of the sensors existed during the Žrst period. The relative bias of the return temperature sensor of chiller 4 with respect to the return temperature sensor of chiller 1 is relatively larger, about 0.5°C (Figure 12A). This was diagnosed by the obtained estimates (see Table 5). The other two are smaller. The corrected redundancy residuals are

Figure 10 Comparison of the control volume B heat balance residuals (normalized) Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Fault diagnosis and validation in HVAC systems

Figure 11 Comparison of the control volume A heat balance residuals (normalized)

around zero. Figures 12(B) and (C) show that the biases of the three temperature sensors (Tr(2), Tr(3), Tr(4)) changed between the Žrst and second periods. Chiller 1 and chiller 4 did not operate simultaneously in the second period, and the redundancy residual of the return temperature sensor of these two chillers did not exist. 9.

Conclusion and discussion

The strategy, based directly on ow and heat balance constraints in a steadystate operation, can detect and identify simultaneously nonabrupt additive sensor biases. The balance residuals are sensitive indicators of the existence of ow meter and temperature sensor biases. Analysis of the residuals under various operating conditions of the chilling plant and minimization of the sum of the squares of the corrected balance residuals make it possible to locate biased sensors and to estimate the magnitudes of the biases. The sequential sensor FDD&E scheme, which estimates sensor biases by minimizing the sum of balance residual squares sequentially, is not sufŽciently robust and is sensitive to certain measurement disturbances, such as abrupt biases of relatively short duration. The improved robust sensor FDD&E scheme estimates the sensor biases by considering the heat balances involved for different control volumes systematically, in addition to the sequential scheme. Field and simulation tests show that the robustness of the robust scheme is increased signiŽcantly, which can meet the requirements of reliability and accuracy of a sensor FDD&E strategy in HVAC practice. The robust scheme uses a GA technique as the optimizing (minimization) tool, which is simple to implement and robust in obtaining a global optimum. In HVAC systems, as well as in other systems, the measurements of liquid ow variables are essential to control and performance monitoring. There always exists Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

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Figure 12

Redundancy residuals of chiller return temperatures

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some type, or certain degree, of physical and/or analytical redundancies in the measurements, such as the cases presented in this paper. As a basic approach, the sensor FDD&E strategy presented in this paper is of practical value. Since it is based on the most fundamental principles, such a strategy is robust to measurement disturbances and to the uncertainty of system/component performance degradations. Although the FDD&E scheme is robust to the disturbances of abrupt sensor biases, it is not designed to detect and estimate them. In order to make the BMS capable of dealing with all possible sensor errors that may occur in practical systems, it would be desirable to incorporate certain methods for detecting and evaluating abrupt biases in the sensor FDD&E scheme for commercial applications. Acknowledgements The paper presents the contribution of the authors in the research project IEA Annex 34 (Application of Fault Detection and Diagnosis in Real Buildings). The work presented in the paper was supported by a grant from the Research Grants Council of the Hong Kong SAR (Project No. PolyU 5016/99E). References Baseeville, M. 1988: Detecting changes in signals and systems – a survey. Automatica 24, 309–26. Carroll, D.L. 1996: Genetic algorithms and optimising chemical oxygen–iodine lasers. In Wilson, H. Batra, R., Bert, C., Davis, A., Schapery, R., Stewart, D. and Swinson, F., editors. Developments in theoretical and applied mechanics, Vol. XVIII. Tuscaloosa, AL: School of Engineering, The University of Alabama, 411–24. Carroll, D.L. 1999: FORTRAN Genetic algorithm (GA) driver. Version 1.7.0. http://www. staff.uniuc.edu/|carroll/ga.html. Clarke, D.W. and Fraher, P.M.A. 1996: Modelbased validation of a DOX sensor. Control Engineering Practice 4, 1313–20. Davis, L. 1991: Handbook of genetic algorithms. New York: Van Nostrand Reinhold. Deb, K. 1996: Genetic algorithms for function optimisation. In Ferrera, F. and Verdegay, J.L., editors. Genetic algorithms and soft computing. Heidelberg: Physica-Verlag, 3–29. Dexter, A.L. 1996: Computer-aided evaluation of HVAC system performance: the practical application of fault detection and diagnosis techniques in real buildings. Proposal for A New

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Appendix A: Generalized formula for calculating the sum of the squares of normalised balance residuals A.1

Generalized formula

Equation (A.1) is the generalized formula to calculate the sum of a normalized · · corrected balance residual squares (S ), where, (k,l) are functions of norr S sq XX · malized variables (X[ i](k)) (Equation A.1.1); Nvar is the number of the normalized variables involved. Derivation of this equation and deŽnitions of the normalized · variables X(k) and parameters BX(k) are described in section A.2. · Srsq =

OO

Nv a r Nva r

k=1

· [SXX (k,l)BX(k)BX(l)]

(A.1)

l=1

where · SXX(k,l) =

O

· · X[ i](k)X[ i](l)

(A.1.1)

i

A.2

Derivation of Equation (1) and deŽnition of variables and parameters

A heat balance in the condition of steady state can be represented by Equation (A.2), in a general form, where x[1i] (k) denotes the ow variable, x[2i](k) the temperature, and c[ i] (k) the product (constant) of the ow media density, speciŽc heat capacity, etc.

O

niterm

[c[ i] (k)x[1i] (k)x[2i] (k)] = 0

(A.2)

k=1

Correspondingly, Equations (A.3) and (A.4) represent the raw and corrected heat balance residuals, respectively. Equation (A.5) is the normalized corrected residual according to the deŽnition given by Equation (18) or Equation (19).

O O

niterm

rˆ

[ i]

=

[c[ i](k)xˆ[1i] (k)xˆ[2i] (k)]

(A.3)

k=1

nitem [ i]

r

=

[c[ i](k)(xˆ1[ i](k) - d

k= 1

rˆ r·[ i] =

[ i]

- d r[ i] = N[opi]

OF

nitem

k=1

(k)) (xˆ[2i] (k) - d

(k))]

x1

c[ i](k)(xˆ[1i](k) - d

(A.4)

x2

(k)) (xˆ[2i] (k) - d

x1

(k))]

x2

[ i] op

N

G

(A.5)

By fully expanding Equation (A.5) and considering Equation (A.3), Equation (A.6) can be obtained. Equation· (A.6) can be rewritten as Equation (A.7), where the normalized variables X[ i](j) represent the ratios, e.g., -c[ i] (k)xˆ[1i](k)/N[opi]; the parameters BX(j) represent the associated constants (parameters), e.g., d x2(k). Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

Wang and Wang

259

· Normalized variables and parameters for calculating Srsq.A

Table A.1

Normalized variables

Parameters

· X·[i] (1) X·[i] (2) X·[i] (3) X·[i] (4) X·[i] (4+j) X·[i] (4+j+N) X[i] (4+j+2N)

BX(1) BX(2) BX(3) BX(4) BX(4+j) BX(4+N+j) BX(4+2N+j)

= = = = = = =

[i] rˆ[i] A /N op [i] ˆ - Mb cpw/N[i] op [i] D T[i] b cpw/N op cpw/N[i] op ˆ [i](j)I[i](j)cpw/N[i] M op [i] [i] D Tˆ[i] ch(j)I (j)cpw/Nop [i] [i] -I (j)cpw/Nop

= = = = = = =

1 d

D Tb

d

Mb

d

Mb

d

D Tb

(j) d M(j) d M(j) d D d

D Tch

D Tch(j) = d Tr(j) - d Ts(j); d D Tb = d Trb - d Tsb. Nvar = 3(N+1)+1. N is the total number of chillers in the chilling plant. N[i] op is the number of the chillers simultaneously operating at sampling instant

(j)

Tch

d

OF

nitem

rˆ[ i] r·[ i] = [ i] + Nop r·[ i] =

O

Nvar

k=1

-

c[ i](k)xˆ[1i] (k) d N[opi]

(k) -

x2

c[ i] (k)xˆ[2i] (k) d N[opi]

(k) +

x1

c[ i] (k) d N[opi]

(k)d

x1

.

[i]

(k)

x2

G

(A.6)

· [X[ i] (l)BX(l)]

(A.7)

l=1

Using Equation (A.7), the sum of the squares of the normalized corrected residual is derived as represented by Equation (A.8), which can be rewritten as Equation · (A.1). The normalized variables (X(k)) and the parameters (BX(k)) can be easily deŽned by comparing Equations (A.6) and (A.7). · Srsq =

O

(r·[ i] )2 =

i

OFO

Nva r

i

l=1

· X[ i](l)BX(l)

G O OF 2

Nv ar Nva r

=

k= 1

BX(k)BX(l)

l=1

O

· · X[ i] (k)X[ i] (l)

i

G

(A.8)

Table A.1 and Table A.2 list those variables and parameters for calculating the sum of heat balance residuals of control volume A, · the squares of normalized corrected · , and of control volume B, , Srsq.A SrSq.B respectively. Table A.2

· Normalized variables and parameters for calculating Srsq.B

Normalized variables · X·[i] (1) X·[i] (1+j) X·[i] (1+N+j) X[i] (1+2N+j)

= = = =

[i] ˆr[i] B /N op ˆ [i](j)I[i](j)cpw/N[i] -M op [i] [i] - (T[i] s (j) - Tsb )I (j)cpw I[i](j)cpw/N[i] op

Parameters BX(1) BX(1+j) BX(1+N+j) BX(12N+j)

= = = =

1

(j) - d Tsb (j) d M(j) ( d Ts(j) - d d

Ts

d

M

Nvar = 3N+1

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)

Tsb

260

Fault diagnosis and validation in HVAC systems

Appendix B: Common reference sensor and Estimator 8 This appendix describes the common reference temperature sensor and Estimator 8 used in the FDD&E scheme for the Želd sensor installation condition speciŽed in Figure 6. B.1

Common reference temperature sensor

When using energy (or heat) balance criterion to evaluate the biases of the temperature sensors involved in a control volume, a common reference temperature sensor is necessary. If a bias value was added to all the temperature sensors, the energy balance (residual) would not be inuenced since it is the differential temperatures that count in energy (heat) balances. Therefore, the bias of the reference temperature sensor has to be assumed reliable (its bias is zero), or be determined independently through on-site calibration. The Žnal values of the estimated biases of all other temperature sensors are relative with respect to the reference sensor. Under the sensor installation condition speciŽed in Figure 6, an artiŽcial chiller return temperature sensor (Tref) is constructed and used as the common reference. The artiŽcial temperature sensor is constructed so that its reading is equal to the average of the actual readings from all chiller return temperature sensors (Tr(j)) when all the (N) chillers are supposed in operation (Equation B.1). Use of this temperature sensor as the common reference implies that the sum of the biases of the individual chiller return temperature sensors is zero (Equation B.2), which is proved below.

O

Tˆ[ri](j)/N

(B.1)

d

(j) = 0

(B.2)

N

Tˆ[refi] =

j=1

O N

Tr

j=1

Considering the sensor model (Equation B.3, see Wang and Wang, 1999), Equation (B.1) can be rewritten as Equation (B.4), since the bias of the artiŽcial temperature sensor is assumed zero, where x is the true value of the measured variable; d x is the (Žxed) bias of sensor x; n [xi] is the random noise contained the sensor reading. xˆ[ i] = x[ i] + d

O

x

+n

(B.3)

[ i] x

N

T[refi] + n

[ i] Tref

=

(T[ri](j) + d

(j) + n

Tr

[ i] Tr

(j))/N

(B.4)

j=1

Substituting the physical redundancy relationships represented by Equation (B.5) into Equation (B.4) and manipulating yield Equation (B.6), which can hold only in the condition as represented by Equation (B.7), which is equivalent to Equation (B.2). Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

Wang and Wang

T[ri] (j) = T[refi] , [j = 1, ¼, N]

O O N

n

[ i] Tref

=

O

261

(B.5)

d

N

(j)/N +

Tr

j=1

n

[ i] Tr

(j)/N

(B.6)

j=1

N

d

(j)/N = 0

(B.7)

Tr

j=1

B.2

Estimator 8

Estimator 8 estimates the biases of the individual chiller return temperature sensors, d Tr(j), with the above-described artiŽcial return temperature sensor being used as the common reference. This estimator is developed from solving the minimization problem of Equation (B.8), where the objective function is the total of the sums of the corrected redundancy residual squares. The corresponding raw redundancy residual (rˆ[D i]Tr(j,k)) is deŽned by Equation (B.9), which is the difference of the readings from each pair of return temperature sensors when the corresponding chillers are in operation. The sensor bias effect on the residual is given by Equation (B.10).

O O FO N

Minimizeu d

Tr( j)

N

j=1 k=1,k± 1

(rˆ[D i]Tr(j,k) - d

[ i] rD Tr

(j,k))2

i

G

rˆ[D i]Tr(j,k) = I[ i] (j)I[ i](k) (Tˆ[ri](j) - Tˆ[ri](k)) d

[ i] rD Tr

= I[ i](j)I[ i](k) ( d

(j) - d

Tr

(B.8) (B.9)

(k))

(B.10)

Tr

By applying Equation (10) to Equation (B.8) with respect to each of the return temperature sensor biases ( d Tr(j)) and manipulating properly, Equation (B.11) can be derived. The coefŽcients (a8 ) and constants (b8 ) are given in section B.3.

3

a8 .1,1 ¼ a8.1,j ¼ a8 .1,N ¼

¼ ¼

¼ ¼

a8 .k,1 ¼ a8.k,j ¼ a8 .k,j ¼

¼ ¼

¼ ¼

43 43 4

a8 .N,1 ¼ a8.N,j ¼ a8 .N,N

(1)

d

Tr

¼

¼

(j)

d

Tr

¼ d

b8 .1 b8 .j

(B.11)

¼

(N)

Tr

b8 .N

The coefŽcient matrix of the above equation is intrinsically not full ranked. To settle the problem, Equation (B.2) is used and integrated into Equation (B.11) by adding a constant, a8.max, to each of the cells of one row in the matrix. a8 .max is equal to the cell whose absolute value is the largest among all the cells. The Žnal Downloaded from http://tim.sagepub.com at PENNSYLVANIA STATE UNIV on April 17, 2008 © 2002 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

262

Fault diagnosis and validation in HVAC systems

equation Estimator 8 to deal with is, as an example, represented by Equation (B.12).

3

B.3

43 43 4

a8.1,1 + a8 .m a x ¼ a8 .1,j + a8.m ax ¼ a8.1,N + a8.m a x ¼

¼ ¼

¼ ¼

a8.k,1

¼ a8.k,j

¼ a8 .k,j

¼

¼ ¼

¼ ¼

a8.N,1

¼ a8.N,j

¼ a8.N,N

(1)

d

Tr

¼

¼

(j)

d

Tr

¼ d

O

(N)

Tr

N

Npt(j,k)

k=1

a8. j,k = Npt(j,k); [k ± j]

O N

b8 .j = -

SrD

(j,k)

Tr

k= 1

a8.m a x = max(a8.j,k); [j,k = 1, ¼, N] where SrD

(j,k) =

Tr

O O

(rˆ[D i]Tr(j,k))

i

Npt(j,k) =

b8 .j ¼

CoefŽcients and constants a8.j,j = -

b8.1

I[ i] (j)I[ i] (k).

i

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b8 .N

(B.12)