NIST Technical Note 1831

Detecting Faults in Building Air 

Handling Units – A Background 

Study for Developing Intelligent 

Agents for Optimizing Building 

HVAC Systems

George E. Kelly

Steven T. Bushby

http://dx.doi.org/10.6028/NIST.TN.1831
NIST Technical Note 1831

Detecting Faults in Building Air 

Handling Units – A Background 

Study for Developing Intelligent 

Agents for Optimizing Building 

HVAC Systems

George E. Kelly

Steven T. Bushby

Energy and Environment Division

Engineering Laboratory

http://dx.doi.org/10.6028/NIST.TN.1831

April 2014

U.S. Department of Commerce 

Penny Pritzker, Secretary

National Institute of Standards and Technology 

Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director 
Certain commercial entities, equipment, or materials may be identified in this

document in order to describe an experimental procedure or concept adequately.

Such identification is not intended to imply recommendation or endorsement by the

National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose. 

National Institute of Standards and Technology Technical Note 1831

Natl. Inst. Stand. Technol. Tech. Note 1831, 39 pages (April 2014) 

http://dx.doi.org/10.6028/NIST.TN.1831

CODEN: NTNOEF
iii

Contents

ABSTRACT ............................................................................................................................... 1

INTRODUCTION ....................................................................................................................... 1

FAULT DETECTION BASED ON A SIMPLE MIXING BOX MODEL......................................... 4

FAULT DETECTION BASED ON A SIMPLE COOLING COIL MODEL .................................. 11

MIXING BOX FAULT DETECTION RESULTS....................................................................................................................22

COOLING COIL FAULT DETECTION RESULTS................................................................................................................25

DETECTION OF FAULT IN CHILLED WATER TEMPERATURE SENSOR.....................................................................26

DETECTION OF FAULT IN SUPPLY AIR TEMPERATURE SENSOR..............................................................................27

FAULT IN ENTERING AIR TEMPERATURE SENSOR ......................................................................................................29

FOULING OF COOLING COIL FAULT.................................................................................................................................30

SUMMARY AND RECOMMENDATIONS................................................................................ 31

REFERENCES........................................................................................................................ 32

TERMINOLOGY KEY.............................................................................................................. 34
1

ABSTRACT

One approach to the optimization of building heating, ventilation and air-conditioning (HVAC) systems 

involves the use of intelligent agents. These Agents can be used for minimizing energy consumption, 

maximizing occupant comfort, fault detection, performing diagnostics, and improving system 

commissioning. This paper lays the ground work for using intelligent agents for fault detection in 

HVAC applications involving air handling units (AHUs). Specifically, it looks at which faults in AHU 

mixing boxes and cooling coils can be detected and under what conditions. Such an understanding is 

necessary before intelligent agents can be developed, tested, and implemented in real building HVAC 

systems. A simple mixing box model (Tan 2006) and a simple cooling coil model (Wang 2004) are used 

to represent the performance of a typical AHU, and system identification is performed using piece-wise 

linear approximations. Questions addressed include which variables should be used in the identification 

process, how many linear segments should be used for each variable, how large a fault must be in order 

to be detected given the inherent inaccuracies in the piece-wise linear approximation process, and over 

what range of external variables (e.g., environmental conditions) is fault detection possible. 

INTRODUCTION

Intelligent agents know or can learn the performance and status of the systems and equipment they 

monitor and can communicate and collaborate with other agents to achieve a common goal. In a 

previous investigation (Kelly 2011) (Kelly 2012), a simulation test bed was developed and used to study 

how agents can learn and interact to minimize energy consumption. This learning process, called 

“system identification” (Ljung 1987), makes intelligent agents ideally suited for detecting and 

identifying faults in building HVAC systems. 

Since the proper operation of building systems is important for occupant safety and comfort and for 

minimizing operating and maintenance costs, the detection and diagnosing of operating faults in real 

time is an essential part of ensuring that building systems operate properly. Fault detection and diagnosis 

(FDD) methods have been employed in many applications, but their implementation in real building 

systems has lagged. In the period 1991 through 2001 considerable work on the development and 

implementation of FDD methods for building HVAC equipment and systems was carried out by 

researchers from a number of countries working together in two research projects sponsored by the 

International Energy Agency (IEA). These were IEA Annex 25, which focused on the energy 

performance diagnosis of building and HVAC systems and the fault detection of HVAC components 

and sub processes, and Annex 34, which involved working with control manufacturers, industrial 

partners, and building owners and operators to demonstrate the benefits of computer aided fault 

detection and diagnostic systems in real building applications.

The IEA Annex 25 Building Optimization and Fault Diagnosis Source Book (Hyvärinen 1996) contains 

a detailed list of possible building HVAC faults, including typical faults for variable air volume (VAV) 

air handling units (AHUs). It also discusses in great detail a variety of FDD methods, such as the use of 

physical and black box models, state and parameter estimation, the use of artificial neural networks and 

expert systems, and fuzzy model-based approaches. A more recent review of research in this area can be 

found in two review articles by Katipamula and Brambley (Katipamula 2005 Parts I and II). One of the 

more promising methods for detecting faults in AHUs is called APAR, which stands for “Air Handling 
2

Unit Performance Assessment Rules” (Schein 2006). It uses a set of expert rules derived from mass and 

energy balances to detect faults. A two stage method for the detection and diagnoses of faults in a 

variety of HVAC subsystems (including heat exchangers) in the presence of sensor faults is described in 

a paper by Wang, Zhou, and Xiao (Wang 2010). It uses a sensor fault detection, diagnosis and 

estimation (FDD&E) method to evaluate the health of sensor measurements and recover faulty sensor 

readings. System level faults (e.g., heat exchanger fouling) are then detected and diagnosed using 

performance indices (PIs) (e.g., the coil heat transfer coefficient) chosen to evaluate the performance of 

each HVAC subsystem. This reference paper also gives a good summary of recent research on FDD in 

building HVAC applications.

This paper lays the ground work for using intelligent agents for fault detection in HVAC applications by 

examining the detection of a limited number of faults in a typical VAV AHU, a schematic of which is 

shown in Figure 1. Mixing box faults and cooling coil faults that are normally masked by the operation 

of closed loop controllers are considered. For the mixing box these include stuck recirculation, inlet, and 

exhaust dampers, and for the cooling coil they include faults in the chilled water, supply air, and 

entering air temperature sensors and cooling coil fouling. These are the most difficult types of faults to 

detect because the closed loop controllers normally act to compensate for the fault and the controlled 

variable (in this case the supply air temperature) remains unaffected. 

The paper examines which of the above faults can be detected and under what conditions. A simple 

mixing box model (Tan 2006) and a simple cooling coil model (Wang 2004) are used to represent the 

performance of a typical mixing box and cooling coil in a VAV AHU. Since the purpose of this work 

was to explore how intelligent agents could be used for fault detection in real HVAC applications, 

system identification is performed using results from the simple mixing box and cooling coil models. 

Piece-wise linear approximations are employed to simulate the use of piece-wise linear regression by 

building intelligent agents for system identification. 
3

 Figure 1. A typical variable air volume air handling unit

The models developed by the system identification process are used to represent the performance of the 

mixing box and cooling coil when no faults are present. When a fault is present, the closed loop 

controllers act to maintain the supply air temperature setpoint by adjusting the control signal to the 

mixing box dampers or adjusting the position of the valve controlling the water flow through the cooling 

coil. When this mixing box damper control signal or cooling coil valve position is inserted in the nofault models developed through system identification, it results in a predicted supply air temperature 

different from the supply air temperature setpoint. The difference or residual between this predicted 

supply air temperature and the supply air temperature setpoint can be used to detect the presence of a 

fault. However, errors introduced in the system identification can also cause errors between the 

predicted supply air temperature and the supply air temperature setpoint even when no faults are present. 

These errors, that result from the system identification process, determine how large a fault must be 

before it can be recognized as an actual fault.

The research in this paper looks at which AHU variables should be used in the identification process and 

how many linear segments should be used for each variable. It then addresses how large a mixing box or 

cooling coil fault must be in order to be detected given the inherent inaccuracies in the piece-wise linear 

approximation process, and over what range of external variables (e.g. environmental conditions) is fault 

detection possible for each fault. Recommendations are also made on future research needed for the 

development of intelligent agents capable of performing fault detection in real building AHUs.

Supply

Fan

Exhaust Air 

Damper

Return

Fan

Damper

Actuator

Inlet Air Damper

Damper

Recirculation 

Air Damper

Cooling 

Coil
4

FAULT DETECTION BASED ON A SIMPLE MIXING BOX MODEL

A schematic of a typical mixing box found in building AHUs is shown in Figure 2.

Figure 2. Schematic of a typical AHU mixing box.

The variable x is the control signal to the inlet air and exhaust air dampers and corresponds to the 

position of the dampers (x ranges from 0 = closed to 1 = open) and x ′ = (1 - x) is the control signal to 

the recirculation air damper and corresponds to the position of the damper (0 = closed, 1 = open) and

P0 is the atmospheric pressure

P1(x) is the pressure at the junction of the return air duct and recirculating air duct

P2(x) is pressure at the junction of the recirculating air duct and the supply air duct

Qs is the supply air flow rate

Qr is the return air flow rate

Qe(x) is the exhaust air flow rate

Qx(x) is the recirculating air flow rate

Qi(x) is the inlet air flow rate

Tx(x) is the temperature of the recirculating air

Ts(x) is the temperature of the supply air

Te is the temperature of the exhaust air

Tr is the temperature of the Return Air

Ti is the temperature of the Inlet Air

It will be assumed that there is no Mixed Air Temperature sensor present and the dampers are being 

modulated to maintain the supply air temperature, Ts(x), at a specific setpoint. This is consistent with a 

P0

P0

P1

P2

Qx, Tx

Qe, Te

Qi, Ti

Qr, Tr

Qs, Ts

x

x

x‘
5

typical Dry Bulb Economizer Cycle where modulation of the dampers occurs only after the cooling coil 

valve is completely closed. It is also assumed that the air streams are fully mixed so that a single 

temperature characterizes the air properties.

Performing mass and an energy balances and assuming Tx = Tr and Te = Tr, one finds that:

Qi(x) = Qs - Qx(x)

Qe(x) = Qr - Qx(x)

Qs·Ts(x) = Qx(x)·Tr + Qi(x)·Ti

These three equations can be rearranged to give:

Qx(x)/Qs = (Ts(x) - Ti)/(Tr – Ti)

Qi(x)/Qs = 1 - Qx(x)/Qs

Qe(x)/Qs = Qr/Qs - Qx(x)/Qs

An intelligent agent might perform system identification by dividing x in to several different connected 

regions, observe the operation of the mixing box over time, and record x and the calculated value of 

Qx(x)/Qs as given by the above equation. With closed loop control, Ts(x) should be equal to the supply 

air temperature setpoint, but x (which depends on the characteristics of the dampers) will vary with 

changes in Tr and Ti. Linear regression could then be performed in each region to determine Qx(x)/Qs as 

a function of x. 

An alternative method might involve varying x within each region by moving the dampers and observing 

the resulting value of Ts(x) and calculating Qx(x)/Qs. Presumably, Tr and Ti would be fairly constant 

while this was being done, although this is not a requirement. Linear regression in each region of x 

should yield the same functional relationship between Qx(x)/Qs and x as the first method.

Since in this paper we do not have an actual system to work with, we will assume certain damper 

characteristics and model the performance of the mixing box. Linearization will then be performed in 

different regions of the x variable to simulate system identification. The governing equations are the two 

mass balance equations given above plus the equations:

P1 – P0 = Re(x)·Qe2

P0 – P2 = Ri(x)·Qi2

P1 – P2 = Rx(x)·Qx2

Given values for P0, Qr/Qs, Re(x), Ri(x), and Rx(x), these equations can be solved for the variables P1, 

P2, Qe, Qx/Qs, and Qi/Qs .

Doing this, one finds for the variables of interest (Tan 2006):

ratioQiToQs(x) = 1 – ratioQxToQs(x)

ratioQeToQs(x) = Qr/Qs – ratioQxToQs(x)
6

where:

ratioQxToQs(x) = −b(x) − (b(x)2 − 4 ∙ a(x) ∙ c(x))0.5

2 ∙ a(x)

a(x) = Ri(x) + Re(x) – Rx(x)

b(x) = – 2∙Ri(x) − 2∙Re(x)∙ratioQrToQs

c(x) = Ri(x) + Re(x)∙(ratioQrToQs)2

If we assume

Ri(x) = ((1-Ai)/Ai)·Ri,open + Ri,open·exp(kd(1-x))

Re(x) = ((1-Ae)/Ae)·Re,open + Re,open·exp(kd(1-x))

Rx(x) = ((1-Ax)/Ax)·Rx,open + Rx,open·exp(kd(1-x))

where Ai, Ae, and Ax are the authorities of the inlet, exhaust, and recirculation dampers, then the first 

terms in the above three equations correspond to duct resistance and the second terms represent the 

resistance of the damper as a function of the control signal x. 

Assuming Ri,open = Re,open = Rx,open, it can be easily shown that Ri, Re, and Rx in the equations 

for a(x), b(x), and c(x) can be replaced by Ri′, Re′, and Rx′, where (Tan 2006):

Ri′(x) = ((1-Ai)/Ai) + exp(kd(1-x))

Re′(x) = ((1-Ae)/Ae) + exp(kd(1-x))

Rx′(x) = ((1-Ax)/Ax) + exp(kd(1-x))

Figure 3 shows the results of this mixing box model for values of kd = 7.578 (Legg 1986), Ai = 0.1, Ae 

= 0.5, and Ax = 0.2 (Tan 2006), and assuming a balanced system, i.e., Qr = Qs.
7

Figure 3. Ratios Qi/Qs, Qe/Qs, and Qx/Qs as a function of control signal x, for Qr/Qs = 1.

For values of Qr/Qs < 1, which corresponds to pressurizing the building, one finds that for small values 

of x it is possible to have the flow through the exhaust air damper reversed with some outside air coming 

in through this damper. This phenomenon was discussed in a paper by Seem et. al. (Seem, 2000) and a 

control strategy to prevent it was proposed. This undesirable effect, which can negatively affect indoor 

air quality, can also be avoided by setting an appropriate minimum opening on the inlet and exhaust 

dampers. 

Although the work presented in this paper applies for any value of Qr/Qs and x where there is no reverse 

flow through the exhaust air damper, we will assume for the sake of simplifying the discussion that Qr = 

Qs. For this case, it is seen from Figure 3 that the performance of the dampers as a function of x can be 

divided into three almost linear regions. They are 0 <= x <= 0.3, 0.3 < x <= 0.7, and 0.7 < x <= 1. Using 

actual data from a real mixing box, one could then use linear regression in these three regions to obtain a 

piecewise model of the mixing box performance as a function of x. However, since we are dealing in 

this paper with an idealized mixing box model, we will use straight line interpolation between end points 

to approximate the results of system identification using linear regression. Assuming maximum and 

minimum values of Qx(x)/Qs equal to 1.0 and 0.05 at x =1 and x = 0, respectively, we define:

a1 = 1

a2 = ratioQxToQs(0.3)

a3 = ratioQxToQs(0.7)

b1 = ratioQxToQs (0.3 ) 1

( 0.3 0)

b2 = ratioQxToQs(0.7 ) ratioQxToQ (0.3 )

( 0.7 0.3)
8

where

and calculate estimated ratios (EST) as follows:

ESTratioQxToQs1(x) = a1 + b1(x)

ESTratioQxToQs2(x) = a2 +b2·(x-0.3)

ESTratioQxToQs3(x) = a3 +b3·(x-0.7)

ESTratioQxToQs(x) = ESTratioQxToQs1(x) for x < 0.3

ESTratioQxToQs(x) = ESTratioQxToQs2(x) for 0.3 ≤ x < 0.7

ESTratioQxToQs(x) = ESTratioQxToQs3(x) for x ≥ 0.7

ESTratioQiToQs(x) = 1- ESTratioQxToQs(x)

ESTratioQeToQs(x) =Qr/Qs – ESTratioQxToQs(x)

The result of this linear approximation is shown in Figure 4 and corresponds to what might be obtained 

by an intelligent agent performing system identification.

 Figure 4. Comparison of linear approximations ESTratioQiToQs, ESTratioQeToQs, and

 ESTratioQxToQs with ratioQiToQs, ratioQeToQs, and ratioQxToQs as a function of control signal, x.

b3 = 0.05 ratioQxToQs( 0.7)

( 1 0.7)

ratioQxToQs (x ) = b( x) b ( x)

2 . . 4 a( x) c( x)

0.5

. 2 a( x)
9

. 

System identification introduces some errors in the predicted performance, and these errors will affect

the detection of real faults. Using the mixing box model defined above, the supply air temperature can 

be calculated using:

Similarly, the supply air temperature predicted by the linear approximation of the real model is given by:

ESTTs(x, Ti)= ESTratioQxToQs(x)∙Tr + ESTratioQiToQs(x)∙Ti

ESTratioQxToQs(x) + ESTratioQiToQs(x)

The difference between these two estimates corresponds to the error likely to be introduced by the 

system identification process. Figure 5 shows this error for values of Ti equal to -8.9  ̊C, 0  ̊C, and 

8.9  ̊C, respectively. Also shown in these figures are errors of +1.7  ̊C and -1.7  ̊C.

Figure 5. Difference between ESTTs and Ts corresponding to errors likely to be introduced by the 

system identification process. Figures, from left to right, are for values of Ti = -8.9  ̊C, 0  ̊C, and 8.9  ̊C.

Results for different values of Ti show that faults which result in differences between the actual supply 

air temperature and the predicted supply air temperature (based upon the identified model without faults) 

of less than ±1.7  ̊C cannot be detected because of the error resulting from system identification based 

upon linear approximations. On the other hand, faults resulting in differences equal to or greater than 

-6

-4

-2

0

2

4

6

0 1

ESTTs - Ts (°C)

X

Ti = -8.9 ̊C

1.7°C

-1.7°C

-6

-4

-2

0

2

4

6

0 1

ESTTs - Ts (°C)

X

Ti = 0 ̊C

1.7°C

-1.7°C

-6

-4

-2

0

2

4

6

0 1

ESTTs - Ts (°C)

X

Ti = 8.9 ̊C

1.7°C

-1.7°C

Ts(x,Ti) = . ratioQxToQs(x) Tr . ratioQiToQs( x) Ti

ratioQxToQs(x) ratioQiToQs( x)
10

±1.7  ̊C should be detectable. This is illustrated below for a stuck recirculation air damper. It is assumed 

that the damper is stuck 40% open, corresponding to x = 0.6.

Solving the mixing box model equations with this fault present, yields the performance results shown in 

Figure 6. The variables ratiofQxToQs, ratiofQeToQs, and ratiofQiToQs are the ratios of the 

recirculation, exhaust, and inlet air flow rates to the supply air flow rate with the fault present for 

different values of the control signal, x.

Figure 6. Results from mixing box model when the recirculation air damper is stuck 40% open, 

corresponding to x = 0.6.

Using 

one can find the value of x for which Tsf(x, Ti) is equal to the supply air temperature setpoint, TaSP; 

we will call this value y. If the calculated value of y is greater than 1, it is set equal to 1.

Similarly, the predicted value of the supply air temperature, at x=y, using the linearized model without

faults, is given by:

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

ratio

X

ratiofQiToQs, ratiofQeToQs

ratiofQxToQs

ratiofQxToQs + ratiofQiToQs

Tsf( x,Ti) = . ratiofQxToQs( x) Tr . ratiofQiToQs( x) Ti

ratiofQxToQs( x) ratiofQiToQs( x)

TsPredicted ( y) = ESTratioQxToQs . (y) Tr ESTratioQiToQs . (y) Ti

ESTratioQxToQs(y) ESTratioQiToQs(y)
11

and the observed error in the supply air temperature is:

TsObservedError = Tsf(y,Ti) – TsPredicted(y)

This observed supply air temperature error is shown in Figure 7 for a supply air temperature setpoint, 

TaSP, equal to 12.8  ̊C and values of Ti equal to -8.9  ̊C, -4.4  ̊C, 0  ̊C, 4.4  ̊C, and 8.9  ̊C. Also shown in

these figures are errors of +1.7  ̊C and -1.7  ̊C. One concludes from this graph that this fault (i.e., the 

recirculation air damper stuck 40% open) is likely to be masked by the error caused by system 

identification when Ti is between approximately 0.5  ̊C and 7  ̊C.

 Figure 7. Observed error in supply air temperature for a recirculation air damper stuck at 

 40% open. The supply air temperature setpoint is 12.8  ̊C and values of Ti equal to

 -8.9  ̊C, -4.4  ̊C, 0  ̊C, 4.4  ̊C, and 8.9  ̊C.

The results obtained from the analysis of other recirculation damper faults and exhaust damper and inlet 

damper faults are presented in the section entitled Mixing Box Fault Detection Results.

FAULT DETECTION BASED ON A SIMPLE COOLING COIL MODEL

A simple cooling coil model developed by Wang (Wang 2004) will be used to examine which cooling 

coil faults can be detected and under what conditions. The output (cooling energy, kW) of the cooling 

coil, Q = plf *Qmax, is given by:

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6 8 10

Observed Error in TSA (°C)

Ti (°C)

Observed Error in TSA

1.7°C

-1.7°C
12

where

Qmax = maximum cooling coil capacity

plf = part load factor (Q/Qmax)

C1 = 2.439

C2 = 0.498

e = 0.8

Tai = the entering air wet bulb temperature for a wet coil

Tao = leaving air temperature = Ts in Figure 2

Tchw = entering chilled water temperature

rateSA = supply air flow rate

rateChw = chilled water flow rate

For the work considered in this paper, it is assumed that the cooling coil is dry (i.e., latent loads are not 

considered) and Tai in the above equations is equal to the entering dry bulb temperature.

Substituting the second equation for the rateSA into the first equation we find we have one equation 

with two unknowns, rateChw and Tao. We define the function A equal to:

Then if we know Tao, the chilled water mass flow rate can be calculated using:

MFRchwA(plf,Tai,Tao,Tchw) = root(A(plf, Tai, Tao, Tchw, rateChw), rateChw)

where root(A(_, _, _, _,rateChw), rateChw) gives the value of rateChw for which A = 0.

Similarly if rateChw is known, the temperature of the air leaving the coil (i.e., the supply air 

temperature, Ts) can be calculated using:

TempAirOutA(plf, Tai, Tchw, rateChw) = root(A(plf, Tai, Tao, Tchw, rateChw), Tao)

where root(A(_, _,Tao, _, _), Tao) gives the value of Tao for which A = 0.

Q = . C1.rateSAe

1 C2. rateSA

rateChw

e (Tai Tchw)

rateSA( plf,Tai,Tao ) = plf.Qmax

( ) 1.0 .(Tai Tao )

A(plf,Tai,Tao,Tchw,rateChw) = 2.439. . rateSA(plf,Tai,Tao)

0.8 (Tai Tchw)

1 0.498. rateSA(plf,Tai,Tao)

rateChw

0.8 plf.Qmax
13

In the above equations, a single “A” at the end of MFRchwA and TempAirOutA is used to indicate that 

the chilled water flow rate and leaving air temperature are calculated using the coil model given above 

that was developed by Wang (Wang 2004).

With closed loop control, the controller adjusts the chilled water flow rate, rateChw, through the cooling 

coil to maintain the supply air temperature at the supply air setpoint. As long as there are no faults 

present and the cooling coil valve is not fully open or fully closed, this should result in a chilled water 

flow given by MFRchwA(plf, Tai, Tao, Tchw) with Tao set equal to TaoSP. If this value of MFRchwA 

is inserted for the variable rateChw in the equation for TempAirOutA(plf, Tai, Tchw, rateChw), it will 

result in a value of TempAirOutA equal to the supply air temperature setpoint. This is shown in the 

Figure 8.

Figure 8. When no fault is present, the calculated value of TempAirOutA equals the supply air 

temperature setpoint, TaoSP.

If a fault was present (e.g., an offset error in the supply air temperature sensor), the value of rateChw 

calculated would be different from the value calculated without the fault. If this new (fault present) 

value of rateChw is inserted in the “fault free” equation for TempAirOutA, the resulting value of 

TempAirOutA would diverge from the supply air temperature setpoint. This divergence can be used to 

detect the presence of a fault.

A problem occurs when the supply air temperature setpoint, TaoSP, approaches the temperature of the 

chilled water entering the cooling coil, Tchw. For a fixed cooling coil capacity, this causes the chilled 

water mass flow rate through the cooling coil to rapidly rise and results in flow rates that are unrealistic. 

This is shown in Figure 9 for Tchw = 5.56  ̊C. Also shown in this figure is the calculated design chilled 

water flow rate, mchwDesign, based upon a 4.44  ̊C rise in the temperature of the chilled water as it 

passes through the cooling coil.

8

10

12

14

16

18

8 10 12 14 16 18

TaoSP, TempAirOutA (°C)

TaoSP (°C)

TaoSP

TempAirOutA
14

 Figure 9. Chilled water flow rate, rateChw, and mchwDesign as a function of TaoSP – Tchw.

This problem will be avoided by requiring that TaoSP – Tchw >= 3 ̊ C, which allows for a maximum 

chilled water flow rate only slightly above the design flow rate. This restriction on how close TaoSP can 

approach Tchw will be employed in the remainder of this paper and becomes one of the conditions that 

must be met for the detection of cooling coil faults by the method described in this paper.

The equations given above for the simple cooling coil model present some difficulties if one wants to 

use system identification to come up with an approximate model of a real cooling coil. This problem can 

be overcome by transforming the equations into those for an effectiveness model. The water side 

effectiveness for a cooling coil is defined as:

effectiveness = (Two – Tchw)/(Tai – Tchw)

Using

Two – Tchw = DelTwater(plf,rateChw)

where

and inserting

rateChw = MFRchwA(plf, Tai, Tao, Tchw), 

10

15

20

25

30

35

40

rateChw, mchwDesign (kg/s)

1 2 3 4 5 6 7 8 9 10

TaoSP - Tchw (°C)

rateChw

mchwDesign

DelTwater ( plf,rateChw ) = . plf Qmax

. rateChw 4.2
15

we have:

This gives EFFw as a function of plf, Tai, Tao, and Tchw.

Using the fact that 

(Two – Tchw) / (Tai –Tchw) = plf*Qmax / (rateChw*4.2*(Tai –Tchw)) = EFFw(plf,Tai,Tao,Tchw)

We can define:

Then, MFRchwAA and TempAirOutAA can be calculated using the following equations

MFRchwAA(plf, Tai, Tao, Tchw) = root(AA(plf, Tai, Tao, Tchw, rateChw), rateChw)

and

TempAirOutAA(plf, Tai, Tchw, rateChw) = root(AA(plf, Tai, Tao, Tchw, rateChw), Tao)

where MFRchwAA is the chilled water mass flow rate for a supply air temperature equal to Tao and 

TempAirOutAA is the supply air temperature resulting from a chilled water mass flow rate equal to 

rateChw. Here the “AA” at the end of MFRchwAA and TempAirOutAA is used to indicate that the 

chilled water flow rate and leaving air temperature are calculated using the water side effectiveness 

model.

This form of the cooling coil model is just as accurate as the original form. This is demonstrated by its 

ability to predict a supply air temperature equal to the supply air temperature setpoint when no faults are 

present. This is illustrated in Figure 10 where TempAirOutAA is plotted for various values of TaoSP, 

with rateChw replaced by the function MFRchwA, which is the chilled water mass flow rate given by 

the original cooling coil model without a fault present.

EFFw(plf,Tai,Tao,Tchw) = DelTwater(plf,MFRchwA(plf,Tai,Tao,Tchw))

(Tai Tchw)

AA( plf,Tai,Tao,Tchw,rateChw) = plf. Qmax

4.2. . rateChw (Tai Tchw)

EFFw(plf,Tai,Tao ,Tchw)
16

 Figure 10. When no fault is present, the calculated value of TempAirOutAA equals the 

 supply air temperature setpoint, TaoSP.

If one assumes Tai = 24  ̊C and plots this water side effectiveness, EFFw(plf, Tai, Tao, Tchw), as a 

function of i = (Tao – Tchw) for various values of plf and Tchw and as a function of plf for various 

values of i and Tchw, Figures 11 and 12 are obtained.

 Figure 11. Plot of EFFw(plf, Tai, Tao, Tchw) as a function of i = (Tao – Tchw) for Tai = 24  ̊C

 and values of plf equal to 1, 0.65, and 0.3 and Tchw equal to 3.33  ̊C, 5.56  ̊C, and 8.33  ̊C. The

 bottom, middle, and top group of lines correspond to plf = 1, 0.65, and 0.3, respectively.

10

12

14

16

18

10 12 14 16 18

TaoSP, TempAirOutAA (°C)

TaoSP (°C)

TaoSP

TempAirOutAA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6 7 8 9 10 11

EFFw

i = (Tao - Tchw) (°C)

EFFw(1,Tai,3.33+i,3.33)

EFFw(1,Tai,5.56+i,5.56)

EFFw(1,Tai,8.33+i,8.33)

EFFw(0.65,Tai,3.33+i,3.33)

EFFw(0.65,Tai,5.56+i,5.56)

EFFw(0.65,Tai,8.33+i,8.33)

EFFw(0.3,Tai,3.33+i,3.33)

EFFw(0.3,Tai,5.56+i,5.56)

EFFw(0.3,Tai,8.33+i,8.33)
17

 Figure 12. Plot of EFFw(plf, Tai, Tao, Tchw) as a function of plf for Tai = 24  ̊C and values of

 i = (Tao – Tchw) equal to 2  ̊C, 6  ̊C, and 10  ̊C and Tchw equal to 3.33  ̊C, 5.56  ̊C and 8.33  ̊C.

 The bottom, middle, and top group of lines correspond to i equal to 2  ̊C, 6  ̊C, and 10  ̊C,

respectively.

These two figures show that EFFw is fairly linear over the range of i = (Tao – Tchw) shown and can be 

divided into two approximately linear regions of plf. We will refer to the region (0.3 <= plf <= 0.6) as 

the “A” region, and the region ( 0.6 < plf <=1) as the “B” region. Values of plf below 0.3 will be ignored 

because of convergence problems in calculating EFFw when plf is small, and this becomes a second 

restriction on this fault detection method. Figures 11 and 12 also show that EFFw at Tchw = 5.56  ̊C

provides a very good approximation for EFFw at other values of Tchw.

This demonstrates that if one had real data from a real coil, one should be able to use linear regression to 

arrive at an approximate water side effectiveness model. Since we are dealing in this paper with an ideal 

model, we will approximate the results of linear regression using linear interpolation. 

The linear function EFFwAi3 (where “A” indicates region “A”) is first calculated for Tchw = 5.56  ̊C

and i = 3 by interpolating between values of EFFw at plf = 0.3 and plf = 0.5. EFFwBi3 (where “B” 

indicates region “B”) is then calculated for Tchw = 5.56  ̊C and i = 3 by interpolating between values of 

EFFw at plf = 0.7 and plf = 0.9. This process is then repeated for Tchw = 5.56  ̊C and i =7 to calculate 

EFFwAi7 and EFFwBi7. This gives the following four equations:

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

EFFw

plf

EFFw(plf,Tai,3.33+2,3.33)

EFFw(plf,Tai,5.56+2,5.56)

EFFw(plf,Tai,8.33+2,8.33)

EFFw(plf,Tai,3.33+6,3.33)

EFFw(plf,Tai,5.56+6,5.56)

EFFw(plf,Tai,8.33+6,8.33)

EFFw(plf,Tai,3.33+10,3.33)

EFFw(plf,Tai,5.56+10,5.56)

EFFw(plf,Tai,8.33+10,8.33)

EFFwAi3( plf,Tai) EFFw( 0.3,Tai, 5.56 3, 5.56) . ( plf 0.3) EFFw( 0.5,Tai, 5.56 3, 5.56) EFFw( 0.3,Tai, 5.56 3, 5.56)

( 0.5 0.3)

EFFwBi3( plf,Tai) EFFw( 0.7,Tai, 5.56 3, 5.56) . ( plf 0.7) EFFw( 0.9,Tai, 5.56 3, 5.56) EFFw( 0.7,Tai, 5.56 3, 5.56)

( 0.9 0.7)
18

These equations can be combined to give:

and

EFFwEST(plf,Tai,i) = EFFwA(plf,Tai,i) for plf < 0.6

EFFwEST(plf,Tai,i) = EFFwB(plf,Tai,i) for plf >= 0.6

Figure 13 shows there is good agreement between the full effectiveness model, EFFw, evaluated at Tai 

= 24  ̊C and Tchw equal to 3.33  ̊C, 5.56  ̊C, and 8.33  ̊C and the linearized effectiveness model, 

EFFwEST evaluated at Tai = 24  ̊C and Tchw = 5.56  ̊C. The bottom, middle and top group of lines 

correspond to i = (Tao – Tchw) equal to 3  ̊C, 6  ̊C, and 9  ̊C, respectively.

Figure 13. Comparison of the full effectiveness model with the approximate linear version as a 

function of plf

EFFwAi7( plf,Tai) EFFw( 0.3,Tai, 5.56 7, 5.56) . ( plf 0.3) EFFw( 0.5,Tai, 5.56 7, 5.56) EFFw( 0.3,Tai, 5.56 7, 5.56)

( 0.5 0.3)

EFFwBi7( plf,Tai) EFFw( 0.7,Tai, 5.56 7, 5.56) . ( plf 0.7) EFFw( 0.9,Tai, 5.56 7, 5.56) EFFw( 0.7,Tai, 5.56 7, 5.56)

( 0.9 0.7)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

EFFw, EFFwEST

plf

EFFw(plf,Tai,3.33+3,3.33)

EFFw(plf,Tai,5.56+3,5.56)

EFFw(plf,Tai,8.33+3,8.33)

EFFwEST(plf,Tai,3)

EFFw(plf,Tai,3.33+6,3.33)

EFFw(plf,Tai,5.56+6,5.56)

EFFw(plf,Tai,8.33+6,8.33)

EFFwEST(plf,Tai,6)

EFFw(plf,Tai,3.33+9,3.33)

EFFw(plf,Tai,5.56+9,5.56)

EFFw(plf,Tai,8.33+9,8.33)

EFFwEST(plf,Tai,9)

EFFwA (plf,Tai,i) = EFFwAi3(plf,Tai) . (i 3) EFFwAi7(plf,Tai) EFFwAi3(plf,Tai)

(7 3)

EFFwB(plf,Tai, i) = EFFwBi3(plf,Tai) . (i 3) EFFwBi7( plf,Tai) EFFwBi3(plf,Tai)

(7 3)
19

Figure 14 shows EFFw as a function of i = (Tao – Tchw) for values of Tai equal to 24  ̊C, Tao + 5  ̊C, 

and Tao + 2  ̊C and EFFwEST as a function of i = (Tao – Tchw) for values of Tai equal to 21  ̊C. The 

left column of graphs shows the results when plf = 0.3; the right column when plf = 1. The top, middle, 

and bottom row of graphs correspond to Tchw equal to 3.4  ̊C, 5.56  ̊C, and 10  ̊C, respectively. The 

results show that when Tai is between Tao + 5  ̊C and 24  ̊C, EFFw can be approximated with good 

results by EFFwEST with Tai equal to 21  ̊C. However, when Tao and Tai get closer than 5  ̊C, the 

agreement between EFFw and EFFwEST tends to break down. This is shown by the top line in all six 

graphs where Tai and Tao are only 2  ̊C apart. This places a third and fourth restriction on this fault 

detection approach, namely (Tai –Tao) > 5  ̊C and Tai <= 24  ̊C, respectively.

Figure 14. EFFwEST vs. i with Tai = 21  ̊C and EFFw vs. i when Tai equal to Tao + 

2  ̊C, Tao + 5  ̊C and 24  ̊C.

Thus EFFwEST, with Tai = 21  ̊C, can be used to predict Tao providing the four restrictions discussed 

previously are met. When no faults are present, the difference between the predicted value of Tao and 
20

the supply air temperature setpoint, TaoSP, is due to errors resulting from using an approximate linear 

effectiveness model and a value of Tai = 21  ̊C. 

Assuming (TaoSP - Tchw) >= 3, plf >= 0.3  ̊C, (Tai -TaoSP) > 5, and Tai <= 24  ̊C, one can calculate 

TempAirOutAAA using EFFwEST evaluated at Tai = 21  ̊C. 

Then for a known value of rateChw, we have:

TempAirOutAAA(plf, Tai, Tao, Tchw, rateChw) = root(AAA(plf, Tai, Tao, Tchw, rateChw), Tao)

The “AAA” at the end of TempAirOutAAA is used to indicate that the leaving air temperature is 

calculated using the linearized water side effectiveness model, EFFwEST, with Tai set equal to 21  ̊C.

Setting rateChw equal to the chilled water mass flow rate obtained using the original cooling coil model, 

MFRchwA without a fault present, we can calculate TempAirOutAAA and examine the effect of the 

errors introduced by using EFFwEST with Tai equal to 21  ̊C. Figures 15, 16, and 17 show for plf = 1 

the predicted value of the supply air temperature, TempAirOurAAA, as a function of TaoSP when Tai = 

TaoSP + 5  ̊C, Tai = 24  ̊C, and a midpoint value, TaiMid, halfway between TaoSP + 5  ̊C and 24  ̊C for 

Tchw equal to 3.33  ̊C, 5.56  ̊C, and 10  ̊C, respectively. In each Figure, TaoSP ranges from Tchw + 

3  ̊C to approximately 19  ̊C, which is 5  ̊C below the maximum allowed value of Tai = 24  ̊C. Lines 

showing TaoSP ±1.3  ̊C are also shown in these Figures.

Figure 15. TempAirOurAAA vs. TaoSP for Tchw equal to 3.33  ̊C, Tai = TaoSP + 5  ̊C, Tai 

= 24  ̊C, and a mid-point value, TaiMid; lines TaoSP ± 1.3  ̊C.

5

7

9

11

13

15

17

19

21

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

TempAirOutAAA (°C)

TaoSP (°C)

TaoSP

TempAirOutAAA, Tai = 

TaoSP + 5 ̊C

TempAirOutAAA, Tai =

TaiMid

TempAirOutAAA, Tai = 

24 ̊C

TaoSP -1.3 ̊C

TaoSP + 1.3 ̊C

plf = 1, Tchw = 3.33°C

AAA(plf,Tai,Tao,Tchw,rateChw) = plf. Qmax

4.2. . rateChw (Tai Tchw)

EFFwEST(plf, 21  ̊C ,Tao Tchw)
21

Figure 16. TempAirOurAAA vs. TaoSP for Tchw equal to 5.56  ̊C, when Tai = TaoSP + 

5  ̊C, Tai = 24  ̊C, and a midpoint value, TaiMid; lines TaoSP ± 1.3  ̊C.

Figure 17. TempAirOurAAA vs. TaoSP for Tchw equal to 10  ̊C, when Tai = TaoSP + 5  ̊C, 

Tai = 24  ̊C, and a midpoint value, TaiMid; lines TaoSP ± 1.3  ̊C.

As illustrated by the curve corresponding to Tai = TaiMid, values of TempAirOutAAA for Tai > 

TaoSP+5  ̊C and < 24  ̊C will generally lie between the values of TempAirOutAAA with Tai = 

TaoSP+5  ̊C and TempAirOutAAA with Tai = 24  ̊C. The exception is in regions where the lines for 

these two limiting cases cross, in which case the predicted value of TempAirOutAAA can be said to be 

5

7

9

11

13

15

17

19

21

8 9 10 11 12 13 14 15 16 17 18 19

TempAirOutAAA (°C)

TaoSP (◦C)

TaoSP

TempAirOutAAA, Tai = 

TaoSP + 5 ̊C

TempAirOutAAA, Tai =

TaiMid

TempAirOutAAA, Tai = 

24 ̊C

TaoSP -1.3 ̊C

TaoSP + 1.3 ̊C

plf = 1, Tchw = 5.56°C

5

7

9

11

13

15

17

19

21

13 14 15 16 17 18 19

TempAirOutAAA (°C)

TaoSP (°C)

TaoSP

TempAirOutAAA, Tai = 

TaoSP + 5 ̊C

TempAirOutAAA, Tai =

TaiMid

TempAirOutAAA, Tai = 

24 ̊C

TaoSP -1.3 ̊C

TaoSP + 1.3 ̊C

plf = 1, Tchw = 10°C
22

"close by". This, however, does not appear to cause a significant problem. Values of plf < 1 only bring 

the curves in the above figures closer together.

Based upon these results, we will assume for the remainder of this paper that when TaoSP + 5  ̊C <Tai 

≤ 24  ̊C and the difference between the predicted value of supply air temperature, TempAirOutAAA, 

and the supply air temperature setpoint is equal to or more than ±1.3  ̊C, it is due to the presence of one 

or more faults. The detection of different cooling coil faults is discussed below in the section entitled 

Cooling Coil Fault Detection Results.

MIXING BOX FAULT DETECTION RESULTS

If the Recirculating Air damper is stuck in an open position, the mixing box controller will try to adjust 

the inlet and exhaust dampers to compensate and maintain the supply air temperature setpoint. The 

charts in Figure 18 show the observer error, TsObservedError, in the supply air temperature for an open 

recirculation air damper (Dxf) stuck open at 20%, 40%, 60%, and 80%. The abrupt changes in slope

seen in some of the curves are due to the fact that x cannot be greater than 1, which corresponds to the 

inlet and exhaust dampers being fully open.

Figure 18. TsObservedError vs. inlet temperature for recirculation air damper being stuck open 

at 20%, 40%, 60%, and 80%.
23

From this figure we see that under the following conditions it would be difficult to detect a stuck 

recirculation air damper fault (i.e., where TsObservedError is between ±1.7  ̊C of the supply air 

temperature setpoint) when:

• the recirculating air damper fault is stuck at 20% open and the supply air temperature setpoint is 

10  ̊C, 12.8  ̊C, or 15.6  ̊C and Ti is above (approximately) 6  ̊C, 9.5 ̊, and 12.5  ̊C respectively.

• the recirculating air damper fault is stuck at 40% open and the supply air temperature setpoint is 

10  ̊C, 12.8  ̊C, or 15.6  ̊C and Ti is between (approximately) -4  ̊C and 3  ̊C, 0.5  ̊C and 7  ̊C, 

and 5.5  ̊C and 12  ̊C, respectively.

• the recirculating air damper fault is stuck at 60% open and the supply air temperature setpoint is 

15.6  ̊C and Ti is below (approximately) -5  ̊C,

However, while it may be difficult to detect a stuck recirculation air damper fault under the above 

conditions, it is still likely that this type of fault will sooner or later be detected since Ti will vary over a 

wide range with changes in the outside temperature.

Similar fault detection problems are found for stuck inlet and exhaust damper faults. In this case the 

mixing box controller will try to adjust the recirculating air damper to maintain the supply air 

temperature setpoint.

For TaSP = 10  ̊C, TsObservedError is shown in Figure 19 for an inlet damper (Dif) stuck 20% open and 

an outlet damper (Def) stuck 20% open.

Figure 19. TsObservedError vs. inlet temperature with TaSP = 10  ̊C. Left graph is for an inlet air 

damper stuck open 20%; right graph is for an exhaust air damper stuck open 20%.

We see that stuck inlet and stuck exhaust air dampers have the same effect on TsObservedError and are 

thus indistinguishable from each other without additional information. One can also conclude from this 

figure that for TaSP = 10  ̊C, a fault involving either a 20% stuck open inlet or a 20% stuck open outlet 

damper should be readily detected. TsObservedError is shown in Figure 20 for TaSP = 10  ̊C and the 
24

inlet air damper (Dif) stuck open 40%, 50%, 60%, and 80%. Identical results would be obtained for an 

exhaust damper stuck open.

Figure 20. TsObservedError is shown for TaSP = 10  ̊C and the inlet air damper stuck open 

40%, 50%, 60%, and 80%. 

One finds that for TaSP = 10  ̊C a fault involving an inlet air damper (or an exhaust air damper) stuck 

open 40% is likely to be detected; however, there are serious problems in detecting inlet dampers (or 

exhaust dampers) that are stuck at 50 % or more open. Changing TaSP from 10  ̊C to 15.6  ̊C has only a 

slight effect on TsObservedError. This is shown in Figure 21 for an inlet air damper that is stuck open 

60 %. 

Figure 21. TsObservedError with an inlet air damper stuck open 60% and TaSP equal to 10  ̊C and 

15.6  ̊C.

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

TsObservedError (°C) 

Ti (°C)

Dif = 40%

Dif = 50%

Dif = 60%

Dif = 80%

Error = -1.7 ̊C

Error = +1.7 ̊C

TaSP = 10 ̊C

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6 8 10

TsObservedError °C) 

Ti (°C)

TaSP = 10 ̊C, Dif = 60%

TaSP = 15.6 ̊C, Dif = 60%

Error = -1.7 ̊C

Error = +1.7 ̊C
25

When the inlet and exhaust dampers are both stuck open, the mixing box controller will try to adjust the 

recirculating air damper to maintain the supply air temperature setpoint. Figure 22 shows the results of 

having both the inlet (Dif) and exhaust (Def) dampers stuck open at 20%, 40%, 60%, and 80%. We see 

from this Figure that when both dampers are stuck open 60 %, it will be difficult to detect the fault when 

Ti is greater than -4.5  ̊C. Similarly, when both dampers are stuck open at 80%, it will be difficult to 

detect the fault when Ti is greater than 4.5 ̊C.

 Figure 22. TsObservedError with both the inlet and exhaust dampers are stuck open 20%,

 40%, 60%, and 80%.

From Figures 20, 21, and 22, one may conclude that faults involving either the inlet air damper or 

the exhaust damper stuck open more than 50% are not likely to be detected. However, most faults 

involving both the inlet and exhaust damper stuck open to the same degree are sooner or later 

likely to be detected due to the expected variation in Ti with outside temperature, with the possible 

exception when both dampers are stuck open around 60%.

COOLING COIL FAULT DETECTION RESULTS

Faults involving the valve controlling the flow of chilled water through the cooling coil are easily 

detected because the supply air temperature will not be maintained at the supply air temperature 

setpoint. As discussed in the Introduction, this kind of fault is not the subject of this paper.

Faults involving errors in the chilled water temperature sensor, the supply air temperature sensor, and 

entering air temperature sensor, or fouling of the cooling coil are more difficult to detect. These faults 

result in chilled water mass flow rates that are different from what would occur if no faults were present. 

This error-induced chilled water mass flow rate can be used in the equation for TempAirOutAAA, 

which was discussed above, to predict a supply air temperature that differs from the supply air 

temperature setpoint. Difference between TempAirOurAAA and the supply air temperature setpoint 

-20

-15

-10

-5

0

5

10

15

20

-10 -8 -6 -4 -2 0 2 4 6 8 10

TsObservedError (°C) 

Ti (°C)

Dif, Def = 20%

Dif, Def = 40%

Dif, Def = 60%

Dif, Def = 80%

Error = -1.7 ̊C

Error = 1.7 ̊C
26

equal to or greater than the ± 1.3  ̊C indicate the presence of a fault. The size of the minimum detectable 

sensor error is found by trial and error. 

In a real AHU, the chilled water flow rate resulting from the fault must either be measured directly or 

calculated from the (measured) position of the cooling coil valve. In this paper, the chilled water mass 

flow rate resulting from the existence of a fault will be calculated using the equation for MFRchwA with 

the fault present.

DETECTION OF FAULT IN CHILLED WATER TEMPERATURE SENSOR

For a fault involving a positive error in the chilled water temperature sensor, the actual chilled water

temperature will be lower than the sensor reading. When the calculated chilled water flow rate, 

MFRchwA, based upon this actual (correct) chilled water temperature is inserted into the equation for 

TempAirOutAAA along with the erroneous chilled water temperature, the predicted supply air 

temperature, Tao = TempAirOutAAA, is found to lie above the supply air temperature setpoint. Figure 

23 shows the results for Tchw = 3.4  ̊C, plf = 1, and a positive error in Tchw of 1.5  ̊C for values of Tai 

equal to TaoSP + 5  ̊C, 24  ̊C, and TaiMid, where TaiMid is half way between TaoSP + 5  ̊C and 24  ̊C. 

Since the predicted supply air temperature for all three values of Tai lie on or above the line defined by 

TaoSP + 1.3  ̊C, faults involving positive errors in the chilled water temperature sensor that are greater 

than 1.5  ̊C should be detectable. Additional analysis shows that positive errors greater than 1.5  ̊C

should be detectable independent of changes in the values of Tchw or plf.

Figure 23. Predicted supply air temperature, Tao = TempAirOutAAA, for a positive error 

in Tchw of 1.5  ̊C for values of Tai equal to TaoSP + 5°C, TaoMid, and 24°C.

For a fault involving a negative error in the chilled water temperature sensor, the actual chilled water 

temperature will be higher than the sensor reading. When the calculated chilled water flow rate, 

MFRchwA, based upon this actual (correct) chilled water temperature is inserted into the equation for 

TempAirOutAAA along with the erroneous chilled water temperature, the predicted supply air 

6

8

10

12

14

16

18

20

22

6 8 10 12 14 16 18 20 22

TempAirOutAAA (°C) 

TaoSP (°C)

TaoSP

TaoSP - 1.3 ̊C

TaoSP + 1.3 ̊C

Tai = TaoSP + 5 ̊C

Tai = TaiMid

Tai = 24°C
27

temperature, Tao = TempAirOutAAA, is found to lie below the supply air temperature setpoint. Figure 

24 shows the results for Tchw = 3.4  ̊C, plf = 1, and a negative error in Tchw of 1.5  ̊C for values of Tai 

equal to TaoSP + 5  ̊C, 24  ̊C, and TaiMid, where TaiMid is half way between TaoSP +5  ̊C and 24  ̊C. 

Since the predicted supply air temperature for all three values of Tai lie on or below the line defined by 

TaoSP - 1.3  ̊C, faults involving negative errors in the chilled water temperature sensor that are greater 

than 1.5  ̊C should be detectable. Additional analysis shows that negative errors greater than 1.5  ̊C

should be detectable, independent of changes in the values of Tchw or plf.

Figure 24. Predicted supply air temperature, Tao = TempAirOutAAA, for a negative error 

in Tchw of 1.5  ̊C for values of Tai equal to TaoSP + 5°C, TaoMid, and 24°C.

DETECTION OF FAULT IN SUPPLY AIR TEMPERATURE SENSOR

For a fault involving a positive error in the supply air temperature sensor, inserting the calculated chilled 

water flow rate, MFRchwA (based upon the actual (correct) leaving air temperature (TaoSP - error in 

supply air temperature sensor) into the equation for TempAirOutAAA, results in predicted supply air 

temperatures that lie below the supply air temperature setpoint. Figure 25 shows the results for Tchw = 

3.4  ̊C, plf = 1, and a positive error in the supply air temperature sensor of 2.5  ̊C for values of Tai equal 

to TaoMin, 24  ̊C, and TaiMid, where TaiMin = (TaoSP – Error + 5  ̊C) and TaiMid is half way between 

TaiMin and 24  ̊C. Since the predicted supply air temperature for all three values of Tai lie on or below 

the line defined by TaoSP -1.3  ̊C, faults involving positive errors in the supply air temperature sensor 

that are greater than 2.5  ̊C should be detectable. Additional analysis shows that positive errors greater 

than 2.5  ̊C should be detectable independent of changes in the values of Tchw or plf.

3

5

7

9

11

13

15

17

19

21

3 5 7 9 11 13 15 17 19 21

TempAirOutAAA (°C) 

TaoSP (°C)

TaoSP

TaoSP - 1.3 ̊C

TaoSP + 1.3 ̊C

Tai = TaoSP + 5 ̊C

Tai = TaiMid

Tai = 24°C
28

Figure 25. Predicted supply air temperature, Tao = TempAirOutAAA, for a positive 

error in the supply air temperature sensor of 2.5  ̊C for values of Tai equal to TaoSP + 

5°C, TaoMid, and 24°C.

For a fault involving a negative error in supply air temperature sensor, inserting the calculated chilled 

water flow rate, MFRchwA (based upon the actual (correct) leaving air temperature (TaoSP - error in 

supply air temperature sensor), into the equation for TempAirOutAAA, results in predicted supply air 

temperatures that lie above the supply air temperature setpoint. Figure 26 shows the results for Tchw = 

3.4  ̊C, plf = 1, and a negative error in the supply air temperature sensor of 2.5  ̊C for values of Tai equal 

to TaoMin, 24  ̊C, and TaiMid, where TaiMin = (TaoSP – Error + 5  ̊C) and TaiMid is half way between 

TaiMin and 24  ̊C. Since the predicted supply air temperature for all three values of Tai lie on or above 

the line defined by TaoSP +1.3  ̊C, faults involving negative errors in the supply air temperature sensor 

that are greater than 2.5  ̊C should be detectable. Additional analysis shows that negative errors greater 

than 2.5  ̊C should be detectable independent of changes in the values of Tchw or plf.

6

8

10

12

14

16

18

20

22

24

6 8 10 12 14 16 18 20 22 24

TemoAirOutAAA (°C) 

TaoSP (°C)

TaoSP

TaoSP - 1.3 ̊C

TaoSP + 1.3 ̊C

Tai = TaiMin

Tai = TaiMid

Tai = 24°C
29

Figure 26. Predicted supply air temperature, Tao = TempAirOutAAA, for a negative 

error in the supply air temperature sensor of 2.5  ̊C for values of Tai equal to TaoSP + 

5°C, TaoMid, and 24°C . 

FAULT IN ENTERING AIR TEMPERATURE SENSOR

Figures 27 and 28 show the predicted value of Tao = TempAirOutAAA for an Error of +4  ̊C and a 

-4  ̊C, respectively, in the entering air temperature sensor, Tai, for a chilled water temperature of 3.34  ̊C

and a plf = 1. In both figures, TaiMin = Tao +5  ̊C + Error, TaiMax = 24  ̊C + Error, and TaiMid lies 

half way between TaiMin and TaiMax. These three values (TaiMin, TaiMid, and TaiMax) all 

correspond to the incorrect readings given by the faulty entering air temperature sensor. Larger errors in 

Tai, higher chilled water temperatures, and smaller values of plf all produce curves that lie further away 

from the region bounded by the lines TaoSP +1.3  ̊C and TaoSP -1.3  ̊C. 

From Figure 27, we see that when TaoSP is greater than 9°C, the predicted supply temperature for all 

three values of Tai lie on or below the line defined by TaoSP -1.3°C. Thus faults involving positive 

errors greater than 4°C in the supply air temperature sensor should be detectable when TaoSP ≥ 9°C.

Similarly, from Figure 28, we find that the predicted supply air temperature for all three values of Tai lie 

above the line defined by TaoSP +1.3  ̊C, faults involving negative errors equal to or greater than 4  ̊C in 

the supply air temperature sensor should be detectable for all reasonable values of Tao. Additional 

analysis shows that these conclusions are unaffected by changes in the values of Tchw or plf.

3

5

7

9

11

13

15

17

19

3 5 7 9 11 13 15 17 19

TempAirOutAAA (°C) 

TaoSP (°C)

TaoSP

TaoSP - 1.3 ̊C

TaoSP + 1.3 ̊C

Tai = TaiMin

Tai = TaiMid

Tai = 24°C
30

Figure 27. Predicted value of Tao = TempAirOutAAA for a positive error of 4  ̊C in the 

entering air temperature sensor, Tai, a chilled water temperature of 3.34  ̊C, and plf = 1. 

Figure 28. Predicted value of Tao = TempAirOutAAA for a negative error of 4  ̊C in the 

entering air temperature sensor, Tai, a chilled water temperature of 3.34  ̊C, and plf = 1.

FOULING OF COOLING COIL FAULT

Fouling, whether on the air side or water side of a cooling coil, decreases the cooling capacity of the 

coil. Introducing a fouling factor, FF, (Zhao 2011) that decreases the cooling capacity of the coil gives 

the following equations.

4

6

8

10

12

14

16

18

20

22

24

4 6 8 10 12 14 16 18 20

TempAirOutAAA (°C) 

TaoSP (°C)

TaoSP

TaoSP - 1.3 ̊C

TaoSP + 1.3 ̊C

TaiError = TaiMin

TaiError = TaiMid

TaiError = TaiMax

4

6

8

10

12

14

16

18

20

22

24

4 6 8 10 12 14 16 18 20

TempAirOutAAA (◦C) 

TaoSP (°C)

TaoSP

TaoSP - 1.3 ̊C

TaoSP + 1.3 ̊C

TaiError = TaiMin

TaiError = TaiMid

TaiError = TaiMax
31

MFRchwF(plf, Tai, Tao, Tchw) = root(AF(plf, Tai, Tao, Tchw, rateChw), rateChw)

The results are shown in Figure 29 for FF = 0.85 with Tchw = 3.33  ̊C and plf = 1, where TaiMin = 

TaoSP + 5  ̊C, TaiMax = 24  ̊C, and TaiMid is half way between TaiMin and TaiMax. Since all three 

curves for Tai lie below the line defined by TaoSP - 1.3  ̊C, faults involving either water or air side 

fouling that decrease the coil capacity by more than 15% should be detectable. 

Figure 29. Predicted value of Tao = TempAirOutAAA for water or air side fouling that reduces 

the capacity of the cooling coil by 15%; Tchw and plf equal 3.33  ̊C and 1, respectively. 

SUMMARY AND RECOMMENDATIONS

It has been shown that many faults in building air handling units involving closed loop control could be 

detected if an accurate model of normal operation either exists or can be developed by monitoring the 

system’s performance when no faults are present. The latter approach is called on-line system 

identification and can be readily carried out using intelligent agents. 

AF( plf,Tai,Tao,Tchw,rateChw) 2.439. . . rateSA( plf,Tai,Tao)

0.8 (Tai Tchw) FF

1 0.498. rateSA( plf,Tai,Tao)

rateChw

0.8

plf.Qmax

0

4

8

12

16

20

24

4 6 8 10 12 14 16 18 20

TempAirOutAAA (°C) 

TaoSP (°C)

TaoSP

TaoSP + 1.3C

TaoSP - 1.3C

Tai = TaiMin

Tai = TaiMid

Tai = TaiMax

rateSA( plf,Tai,Tao ) = plf.Qmax

( ) 1.0 .(Tai Tao )
32

The detection of a particular fault also depends upon the nature of the fault (e.g., where the fault is 

located), the limitations of the identified system model, and the size of the fault. Understanding which 

faults are likely to be detectable and under what conditions is important to the successful 

implementation of any fault detection method in a real building HVAC system.

Simple models of an AHU mixing box and a cooling coil were presented. Piece-wise linear 

approximations to represent what might be obtained by on-line system identification using linear 

regression and, in the case of the cooling coil, additional approximations to reduce the number of 

variables involved, were then employed to obtain simplified models representative of those that might 

be developed by intelligent agents using on-line system identification techniques. These simplified 

models were then used to study likely faults in the closed loop control of AHU mixing boxes and 

cooling coils that should be detectable and under what conditions.

For mixing boxes it was found that the detection of a stuck recirculation air damper depended on the 

position of the stuck damper, the supply air temperature, and the entering air temperature. It was 

concluded, however, that because the entering air temperature varies with the outdoor temperature over 

such a wide range, a recirculation air damper stuck at almost any position would likely be detected 

sooner or later.

The detection of a stuck inlet air damper or a stuck exhaust air damper was more difficult. It was 

determined that these faults were only likely to be detected if the damper was stuck at a position less 

than 50% open. However, if both the inlet and exhaust air dampers were both stuck open to the same 

degree, then sooner or later this type of fault would likely be detected, with the exception of when both 

dampers are stuck around 60% open.

For the cooling coil it was found that positive and negative errors greater than 1.5  ̊C in the chilled water 

temperature sensor were likely to be detected. Similarly, positive and negative errors greater than 2.5  ̊C

in the supply air temperature sensor should be detectable. Positive errors greater than 4  ̊C in the 

entering air temperature sensor are detectable when the supply air temperature, Tao, is ≥ 9  ̊C, while 

negative errors ≥ 4  ̊C are detectable for all reasonable values of Tao. An examination of cooling coil 

fouling showed that fouling (either on the water side or air side) that resulted in the cooling capacity 

being reduced by 15% or more was likely to be detected.

While not all the faults likely to occur in the closed loop control of AHU mixing boxes and cooling coils 

can be detected by the method described in this paper, many can. The approach described in this paper is 

worthy of being implemented in real building systems using intelligent agents. However, the authors 

believe that there is little to be gained by studying these fault detecting intelligent agents in a simulation 

environment. Rather, it is recommended that, as a next step, intelligent agents implementing this fault 

detection method be developed and tested in a laboratory setting where faults of various kinds and 

magnitudes can be easily introduced and studied without affecting the comfort of building occupants.

REFERENCES

Hyvärinen, J., Kärki, S., 1996. IEA Annex 25 Building Optimization and Fault Diagnosis Source Book, 

Technical Research Centre of Finland, VTT Building Technology, FIN-02044 VTT (Espoo)
33

Katipamula, S., Brambley, M. R., 2005 Part I. Methods for Fault Detection, Diagnostics, and 

Prognostics for Building Systems – A Review, Part I, HVAC&R Research, Vol. 11, No. 1, January 2005.

Katipamula, S., Brambley, M. R., 2005 Part II. Methods for Fault Detection, Diagnostics, and 

Prognostics for Building Systems – A Review, Part II, HVAC&R Research, Vol. 11, No. 2, April 2005.

Kelly, G. E., Bushby, S. T., 2011. Optimization of Building HVAC Systems Using Intelligent Agents –

A Proof of Concept Study. NIST TN 1707.

Kelly, G. E., Bushby, S. T,. 2012. Are intelligent agents the key to optimizing building HVAC system 

performance? HVAC&R Research, Vol 18, Issue 4, 750-759, 2012.

Legg, R. C., 1986. Characteristics of single and multi-blade dampers for ducted air systems. Building 

Services Engineering Research and Technology, 7(4)

Ljung, L., 1987. System Identification Theory for the User, Prentice-Hall, Inc., Englewood Cliffs, NJ.

Schein, J., Bushby, S. T., Castro, N. S., House, J. M., 2006. A rule-based fault detection method for air 

handling units. Energy and Buildings, Vol 38, Issue 12, 1485-1492, December 2006.

Seem, J. E., House, J. M., Klaassen, C. J., 2000. A damper control system for preventing reverse airflow 

through the exhaust air damper of variable-air-volume air-handling units. HVAC&R Research, Vol. 6, 

No. 2, 135-148, April 2000.

Tan, H., Dexter, A., 2006. Estimating airflow rates in air-handling units from actuator control signals. 

Building and Environment. Vol. 41, Issue 10, 1291-1298, October 2006.

Wang, Y., Cai, W., et al. 2004. A simplified modeling of cooling coils for control and optimization of 

HVAC systems. Energy Conversion and Management 45, p. 2915-2930.

Wang, S. W., Zhou, Q., Xiao, F., 2010. A system-level fault detection and diagnosis strategy for HVAC 

systems involving sensor faults, Energy and Buildings. 42(4), 477-490, April 2010.

Zhao, L., Cai, W., Ding, X., 2011. A Modified Hybrid Model of Heat Exchanger Based on Fouling 

Effect, 6th IEEE Conference on Industrial Electronics and Applications, 2011.
34

TERMINOLOGY KEY

Mixing Box Terminology

x – position of inlet and exhaust dampers (0 = closed, 1 = open)

(1-x) – position of recirculating and exhaust dampers (0 = closed, 1 = open)

P0 – atmospheric pressure

P1(x) – pressure at the junction of the return air duct and recirculating air duct

P2(x) – pressure at the junction of the recirculating air duct and the supply air duct

Qs – supply air flow rate

Qr – return air flow rate

Qe(x) – exhaust air flow rate

Qx(x) – recirculating air flow rate

Qi(x) – inlet air flow rate

Tx(x) – temperature of the recirculating air

Ts(x) – temperature of the supply air

Te – temperature of the exhaust air

Tr – temperature of the Return Air

Ti – temperature of the Inlet Air

Ri(x) – inlet air damper resistance 

Re(x) – exhaust air damper resistance

Rx(x) – recirculating air damper resistance

Ai – authority of the inlet damper

Ae – authority of the exhaust damper

Ax – authority of the recirculation damper

ratioQiToQs(x) – ratio of inlet air flow rate to supply air flow rate without any faults 

present

ratioQxToQs(x) – ratio of recirculating air flow rate to supply air flow rate without any faults 

present

ratioQeToQs(x) – ratio of exhaust air flow rate to supply air flow rate without any faults present

ratiofQiToQs(x) – ratio of inlet air flow rate to supply air flow rate with damper faults present

ratiofQxToQs(x) – ratio of recirculating air flow rate to supply air flow rate with damper fault present

ratiofQeToQs(x) – ratio of exhaust air flow rate to supply air flow rate with damper fault present

ESTratioQiToQs(x) – calculated ratio of inlet air flow rate to supply air flow rate using linearized 

mixing box model without any faults present

ESTratioQxToQs(x) – calculated ratio of recirculating air flow rate to supply air flow rate using

 linearized mixing box model without any faults present

ESTratioQeToQs(x) – calculated ratio of exhaust air flow rate to supply air flow rate using 

 linearized mixing box model without any faults present
35

ESTTs(x, Ti) - calculated supply air temperature using the linear approximation of the real 

model

Tsf(x, Ti) – calculated supply air temperature using full mixing box model with faults present

TaSP – temperature setpoint of air leaving mixing box

y – value of x at which Tsf(x, Ti) equals TaSP

TsPredicted(y) – calculated supply air temperature using linearized mixing box model without any 

faults present

TsObservedError – observed error in supply air temperature equal to the difference between Tsf(x, 

Ti) and TsPredicted(y)

Dxf – recirculating damper fault equal to the percent the damper is stuck open

Dif – inlet damper fault equal to the percent the damper is stuck open

Def – exhaust damper fault equal to the percent the damper is stuck open

Cooling Coil Termonology

Q – cooling coil capacity

Qmax – maximum cooling coil capacity

plf – part load factor (Q/Qmax)

Tai – entering air wet bulb temperature for a wet coil, entering air dry bulb temperature

 for a dry coil

Tao – temperature or air leaving cooling coil, equals supply air temperature, Ts

Tchw – temperature of chilled water entering cooling coil

Two – temperature of water leaving cooling coil

rateSA – supply air flow rate

rateChw – chilled water flow rate

mchwDesign – maximum water flow rate that the cooling coil was designed to handle under

 normal conditions

TaoSP – temperature setpoint of air leaving cooling coil (i.e., the supply air temperature

 setpoint)

MFRchwA – calculated value of the chilled water mass flow rate through the cooling coil using 

the full cooling coil model developed by Wang (Wang 2004)

TempAirOutA – calculated value of the temperature of the air leaving the cooling coil using

 the full cooling coil model developed by Wang (Wang 2004)

MFRchwAA – calculated value of the chilled water mass flow rate through the cooling coil using 

the water side effectiveness model

MFRchwF – calculated value of the chilled water mass flow rate through the cooling coil using 

the fouled cooling coil model

TempAirOutAA – calculated value of the temperature of the air leaving the cooling coil using

 the water side effectiveness model

TempAirOutAAA – calculated value of the temperature of the air leaving the cooling coil using

 the linearized water side effectiveness model
36

EFFw – water side effectiveness model derived from the full cooling coil model developed 

by Wang (Wang 2004)

EFFwEST – linearized version of the water side effectiveness model, EFFw

FF – fouling factor