NIST Special Publication 1087 

COOLING MODE FAULT DETECTION AND DIAGNOSIS 

METHOD FOR A RESIDENTIAL HEAT PUMP 

Minsung Kim 

Seok Ho Yoon 

W. Vance Payne 

Piotr A. Domanski 
NIST Special Publication 1087 

COOLING MODE FAULT DETECTION AND DIAGNOSIS METHOD FOR A 

RESIDENTIAL HEAT PUMP 

Minsung Kim 

Korea Institute of Energy Research 

Geothermal Energy Research Center 

71-2 Jang-dong, Yuseong-gu, Daejeon 305-343, Korea 

Seok Ho Yoon 

Korea Institute of Machinery and Materials 

Energy Systems Research Division 

171 Jang-dong, Yuseong-gu, Daejeon 305-343, Korea 

W. Vance Payne 

Piotr A. Domanski 

U.S. DEPARTMENT OF COMMERCE 

National Institute of Standards and Technology 

Building Environment Division 

Building and Fire Research Laboratory 

Gaithersburg, Maryland 20899-8631, USA 

October 2008 

U.S. DEPARTMENT OF COMMERCE 

Carlos M. Gutierrez, Secretary 

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY 

James M. Turner, Deputy 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 Special Publication 1087 

Natl. Inst. Stand. Technol. Spec. Publ. 1087, 98 pages (October 2008) 

CODEN: NSPUE2 
i

BRIEF SUMMARY OF THE RESEARCH 

This research addresses the need for fault detection and diagnosis (FDD) in residential-style, air 

conditioner, and heat pump systems in an attempt to make these systems more trouble free and energy 

efficient over their entire lifetime. This work is one of the first to apply FDD techniques to a residential 

system with the added control element of a thermostatic expansion valve (TXV). Any control element 

actively seeks to perform its duties and thus obscures any faults occurring by making adjustments. This 

research work takes this into account and shows how FDD techniques may be applied to this type of 

system operating in the cooling mode. 

Performance characteristics of an R410A residential unitary split heat pump equipped with a TXV were 

investigated in the cooling mode under no-fault and faulty conditions. Six artificial faults were imposed: 

compressor/reversing valve leakage, improper outdoor air flow, improper indoor air flow, liquid-line 

restriction, refrigerant undercharge/overcharge, and presence of non-condensable gas. 

An automated method of steady-state detection was developed to produce consistent collection of data for 

all tests. The no-fault test measurements were used to develop a multivariate polynomial reference model 

for those system features (temperatures) that varied the most when a single fault was imposed. Outdoor 

air dry-bulb temperature, indoor air dry-bulb temperature, and indoor air dew-point temperature were 

used as the independent variables. From the no-fault reference model, feature residuals (differences 

between model predictions and measured values) were determined. Since the system was controlled by a 

TXV, the system could adapt itself to external variation much easier than a system with a fixed area 

expansion device. This added measure of refrigerant flow control provided by the TXV meant that the 

system compensated for faulty behavior more easily than a fixed area expansion device system. The 

distinctiveness of a fault depended on the TXV control status (fully open or fully closed), and thus the 

TXV affected the fault response of the selected features. 

The rule-based chart method of fault detection and diagnosis presented in this work requires knowledge 

of the variation of system features at steady-state and during transient operation. Knowledge of the 

transient variation of the various features is necessary to establish the size of the moving window used by 

the steady-state detector, which is a key part of our FDD method. The goodness of fit of the no-fault 

reference model along with the lack of measurement repeatability from one day to the next was included 

in developing the FDD algorithm. The steady-state detector is the foundation upon which the FDD 

algorithm rests; only when all system features are within their strictly defined steady-state limits does the 

FDD algorithm begin applying the rule sets as defined in its rule-based chart. These steady-state limits 

are based upon the standard deviation of the feature values, within a fixed-time-interval moving window, 

being less than three times their no-fault steady-state standard deviation values. 

Once the steady-state detector indicates that the important FDD features are steady, the difference in the 

moving window mean and the no-fault reference model values, feature residuals, are calculated using the 

no-fault reference model. A feature residual may have one of three values; positive or up-arrow Ç, 

negative or down-arrow È, or neutral [no change, NC (−)]. The NC case is defined by a positive and 

negative threshold about the moving window mean value. Each case has an associated probability of 

occurrence, P(C | X), as shown in Figure i. 
ii

( | ) P C X ↑ i P(C X − | i) ( | ) P C X ↓ i

μi μ + ε μ i i − ε i i xi

PDF , (μi i σ )

( ) 2

,NF ,NF , N μ σ i i

μi,NF

Moving Window Mean Value

No-Fault Reference Model Prediction

Figure i. Threshold for a moving window mean value 

The calculation of the neutral threshold value, ε, involves selecting a confidence level (for example 99 %) 

to avoid a false alarm. For a given confidence level, the calculation of the appropriate confidence interval 

involves determining the appropriate variances (uncertainty) associated with steady-state measurement 

variations, modeling, and lack of measurement repeatability. These three sources of variance have 

individual confidence intervals associated with them, and their square rooted sum value gives the final 

value of ε at the given confidence level. With ε defined, the probability for each of the three cases may be 

calculated as shown in Figure i. 

Looking at an example of a rule-based chart as shown in Table i, each positive, negative, or neutral case 

index (rows of the table) for the important features (columns of the table) has an associated probability 

(0< P(C | X)<1) including the probability of no-fault. The probability of a given fault is the product of all 

probabilities in a given row. When the probability of a given fault is higher than the no-fault probability 

(all calculated at the previously stated confidence level), the system most likely has this fault occurring. 

Table i. Example rule-based chart 

Fault Type TE Tsh TD TC Tsc ∆TCA ∆TEA

Refrigerant undercharge È Ç Ç È È È È

Refrigerant overcharge – – Ç Ç Ç – – 

No-fault – – – – – – – 

The techniques discussed above are applied to a residential heat pump in the cooling mode with our 

results discussed herein. The addition of a complete definition of the no-fault case and the method of 

handling the TXV are unique to this work. The methods discussed in this work may be applied to any 

vapor compression system that can experience similar faults. 
iii

TABLE OF CONTENTS 

BRIEF SUMMARY OF THE RESEARCH.................................................................................................i

TABLE OF CONTENTS............................................................................................................................iii 

LIST OF TABLES....................................................................................................................................... v 

LIST OF FIGURES ....................................................................................................................................vi 

NOMENCLATURE .................................................................................................................................viii

CHAPTER 1. BACKGROUND ................................................................................................................. 1

CHAPTER 2. OVERVIEW OF FAULT DETECTION AND DIAGNOSIS 

2.1 Concept of Fault Detection and Diagnosis .............................................................................. 3 

2.2 Previous Research.................................................................................................................... 4 

CHAPTER 3. STEADY-STATE DETECTOR 

3.1 A Moving Window for Steady-State Detection ....................................................................... 8 

3.2 Development of the Steady-State Detector ............................................................................ 10 

3.2.1 Experimental setup and conditions ........................................................................ 10

3.2.2 Selection of measurement features......................................................................... 12 

3.2.3 Setting feature thresholds based upon steady-state data ........................................ 12 

3.2.4 Selection of moving window size using startup transient tests .............................. 15 

3.3 Verification of the Steady-State Detector Using an Indoor Load Change Test ..................... 19 

3.3.1 Tests with no faults imposed.................................................................................. 19 

3.3.2 Test with a 20 % refrigerant undercharge fault imposed ....................................... 21 

3.4 Verification of the Steady-State Detector During a Startup 

 Transient Test with a Fault Imposed...................................................................................... 21 

3.4.1 Test with condenser fault imposed......................................................................... 21 

3.4.2 Test with an evaporator air flow fault imposed...................................................... 24 

3.5 Verification of the Steady-State Detector During Continuous and Cyclic 

Operations........................................................................................................................... 25 

3.6 Steady-State Detector Summary ............................................................................................ 27 

CHAPTER 4. NO-FAULT REFERENCE MODEL 

4.1 Data Collection for the Reference Model .............................................................................. 28 

4.2 Multivariate Polynomial Regression (MPR) Reference Model ............................................. 29 

4.3 Artificial Neural Network (ANN) Reference Model ............................................................. 30 

4.4 Sensitivity of the Reference Model to the Steady-State Detector.......................................... 35 

4.4.1 Effect of the steady-state detector threshold values on the reference 

model................................................................................................................... 35 

4.4.2 Example of the steady-state detector applied to the experimental heat 

pump.................................................................................................................... 36 

4.5 Conclusions on System Reference Model.............................................................................. 38 
iv

CHAPTER 5. STATISTICAL METHODOLOGY FOR FAULT DIAGNOSIS USING A 

RULE-BASED APPROACH 

5.1 Fault Influences on System Parameters ................................................................................. 39 

5.2 Classification of Feature Rules .............................................................................................. 41 

5.2.1 Statistical rule-based classification with two case indices; positive or 

negative residuals ................................................................................................ 41 

5.2.2 Statistical rule-based classification with three case indices; positive, 

negative, or neutral.............................................................................................. 43 

5.2.3 Decision rule index and building a rule-based chart .............................................. 46 

5.2.3.1 Fault classification error in a three case classification ........................... 49 

5.2.3.2 Probability of a fault classification error................................................ 50 

5.3 Setting the Neutral Threshold by Calculating Total Residual Uncertainty............................ 51 

5.3.1 Uncertainties due to steady-state variation and lack of measurement 

repeatability......................................................................................................... 51 

5.3.2 Uncertainties due to the reference models.............................................................. 52 

5.3.3 Combining uncertainties to determine the neutral case, NC, threshold ................. 54 

5.3.3.1 Confidence interval, k1, for the steady-state uncertainty .......................... 54 

5.3.3.2 Confidence interval, k2, for model uncertainty......................................... 55 

5.3.3.3 Confidence interval, k3, for lack of measurement repeatability................ 57 

5.3.4 Neutral case, NC, threshold value for the required credibility............................... 57 

5.4 Application of the FDD Algorithm to a Residential Heat Pump System Operating 

in the Cooling Mode .............................................................................................................. 58 

5.4.1 Implementing single faults ..................................................................................... 58 

5.4.2 Quantifying the control limits of the TXV............................................................. 58 

5.4.3 Building a rule-based chart .................................................................................... 61 

5.4.4 Example FDD module............................................................................................ 63 

5.5 Performance of the FDD System ........................................................................................... 65

5.5.1 Comparative evaluation of fault effects on cooling capacity and EER.................. 65 

5.5.2 Example implementation of the FDD system ........................................................ 68 

CHAPTER 6. CONCLUDING REMARKS............................................................................................. 77 

REFERENCES .......................................................................................................................................... 79 

APPENDIX A. FDD MODULES............................................................................................................. 82 

A.1 Main Module......................................................................................................................... 82 

A.2 Moving Window Module...................................................................................................... 83 

A.3 Steady-State Detector Module .............................................................................................. 83 

A.4 Preprocessor Module............................................................................................................. 84 

A.5 Reference Model Correlation Interface................................................................................. 85 

A.6 No-Fault MPR Reference Module ........................................................................................ 86 

A.7 Statistical Rule-Based FDD Module ..................................................................................... 86 
v

LIST OF TABLES 

Table i. Example rule-based chart.........................................................................................................ii 

Table 2.1. Summary of system types and FDD methodology for selected previous 

investigations ........................................................................................................................... 7 

Table 3.1. Measurement uncertainties..................................................................................................... 11 

Table 3.2. System features used in fault detection.................................................................................. 12 

Table 3.3. Variations of selected features during steady state ................................................................ 15 

Table 4.1. Operating conditions for fault-free tests ................................................................................ 28 

Table 4.2. MSR of the models fit to a reduced dataset for the seven selected features .......................... 31 

Table 4.3. MSR of the models fit to the full dataset for the seven selected features .............................. 31 

Table 4.4. MSR of the reduced dataset model applied to the full dataset of seven selected 

features................................................................................................................................... 32 

Table 4.5. F-statistic and percent change in MSR with respect to the full models for seven 

features................................................................................................................................... 33 

Table 5.1. Comparison of rule-based patterns with regard to three different configurations of 

vapor compression systems.................................................................................................... 40

Table 5.2. Rule-based fault pattern chart of seven features .................................................................... 41 

Table 5.3. Standard deviation of the selected features............................................................................ 52 

Table 5.4. Net model uncertainties of the features using the 1st, 2nd, and 3rd order MPR 

models.................................................................................................................................... 52 

Table 5.5. Two-sided confidence intervals with degrees of freedom of four and nine........................... 55 

Table 5.6. Two-sided confidence interval of the seven features for the 3rd order MPR model............... 57 

Table 5.7. Two-sided confidence intervals of Gaussian distribution...................................................... 57 

Table 5.8. Feature thresholds at different confidence levels for two sample sizes ................................. 58 

Table 5.9. Description of studied faults .................................................................................................. 58 

Table 5.10. Operating conditions for FDD tests from Kim et al. (2006) .................................................. 64 

Table 5.11. FDD results for no fault (NF) cases ....................................................................................... 68 

Table 5.12. FDD results for compressor leakage fault (CMF).................................................................. 70 

Table 5.13. FDD results for lowered outdoor air flow fault (CF)............................................................. 71 

Table 5.14. FDD results for lowered indoor air flow fault (EF) ............................................................... 72

Table 5.15. FDD results for liquid-line restriction fault (LL)................................................................... 73 

Table 5.16. FDD results for refrigerant undercharge fault (UC)............................................................... 74

Table 5.17. FDD results for refrigerant overcharge fault (OC)................................................................. 75 

Table 5.18. Misdiagnosed faults ............................................................................................................... 76 
vi

LIST OF FIGURES 

Figure i. Threshold for a moving window mean value..........................................................................i 

Figure 2.1. Supervision loop for fault detection and diagnosis (Isermann, 1984).................................... 3 

Figure 2.2. Equipment supervision for fault detection and diagnosis of a vapor compression 

system .................................................................................................................................... 4 

Figure 3.1. Moving windows of n data points at k

th time ......................................................................... 8 

Figure 3.2. Schematic diagram of the experimental setup...................................................................... 11 

Figure 3.3. The variation of seven features during steady-state operation (TID = 26.7±0.5 °C; 

TOD = 27.8±0.5 °C)............................................................................................................... 15 

Figure 3.4. Variation of seven features during a startup transient, TID = (26.7±0.5)°C, TOD = 

(27.8±0.5)°C ........................................................................................................................ 16 

Figure 3.5. Onset of steady state during a no-fault startup transient test, TID = (26.7 ± 0.5)°C, 

TOD = (27.8 ± 0.5)°C; (a) Decision on steady state; (b) σMW(Tsh); (c) σMW(Tsc)................... 17 

Figure 3.6. The startup transient of compressor discharge temperature and cooling and 

heating capacities, TID = (26.7±0.5)°C, TOD = (27.8±0.5)°C ............................................... 19 

Figure 3.7. Identification of steady state during a no-fault indoor load change test, TOD = 

(35 ± 0.5)°C, (a) TID; (b) Tsh and Tsc; (c) σMW(Tsh); (d) σMW(TE); (e) σMW(ΔTEA)................. 20 

Figure 3.8. Identification of steady state during an indoor load change test with a 20 % 

refrigerant undercharge fault, TOD = (35 ± 0.5)°C, (a) TID; (b) Tsh and ΔTEA; (c) 

σMW(Tsh); (d) σMW(TE); (e) σMW(ΔTEA) ................................................................................ 22 

Figure 3.9. Variation of Tsc, Tsh, TE and TD during startup transient with a 30 % condenser air 

flow fault level with standard deviations calculated for a 144 s (9 sample) 

moving window ................................................................................................................... 23 

Figure 3.10. Variation of Tsc, Tsh, TE and TD during startup transient with a 30 % evaporator 

air flow fault level with standard deviations calculated for a 144 s (9 sample) 

moving window ................................................................................................................... 24 

Figure 3.11. Sample operation of a steady-state detector during continuous system operation 

(TID = varied, TOD = (27.8±0.5) °C, moving window size = 150 s); (a) Variation 

of indoor temperature; (b) Temperature differences; ΔTEA, Tsh, and Tsc; (c) 

Decision from the steady-state detector: 0-unsteady, 1-steady ........................................... 26 

Figure 3.12. Operation of a steady-state detector during system on-off transient operation (TID

= 26.7±0.5 °C; TOD = 27.8±0.5 °C); (a) Variation of TC, TE, and Tsh; (b) Decision 

from the steady-state detector: 0-unsteady, 1-steady .......................................................... 27 

Figure 4.1. Indoor test conditions on a psychrometric chart for fault-free model experiments 

at a fixed outdoor temperature; (a) Indoor condition change as electric heaters 

activate; (b) Sampled indoor air conditions at four TOD (27.8, 32.2, 35.0, and 

37.8) °C................................................................................................................................ 29 

Figure 4.2. Artificial neural network structure....................................................................................... 30 

Figure 4.3. Performance of MPR models and ANN model to predict features during a 

sample operation period; (a) TID; (b) Tsh; (c) Tsc; (d) TD; (e) steady-state ............................ 34 

Figure 4.4. Comparison between measured data and predicted data by 3rd order MPR model 

at all test conditions; (a) TE; (b) Tsh; (c) TC; (d) Tsc .............................................................. 36 
vii

Figure 4.5. Results filtered by the steady-state detector on the plots with feature residuals 

and independent variables (zone A: startup transient, zone B: TXV hunting) .................... 37 

Figure 5.1. Probability density functions (PDF) of i

th feature, xi, for the current measurement 

and the no-fault model prediction ........................................................................................ 42 

Figure 5.2. Example of a three case classification using a Boolean and a smoothed belief 

function with regard to the residual ri and its probability distribution P(Ri) for a 

given threshold on ri of ±εi (Kramer 1987) .......................................................................... 44 

Figure 5.3. Estimation of conditional probability of the ith feature with regard to the current 

measurement complying with individual rules .................................................................... 45 

Figure 5.4. Classification probability functions for positive change, negative change, and 

neutral change with regard to the measurement threshold standard deviation 

multiplier, s=1 and s=2 ........................................................................................................ 47 

Figure 5.5. Relation between εi(s) and s ................................................................................................ 48 

Figure 5.6. Classification error, α, versus measurement threshold standard deviation 

multiplier, s, with regard to the number of features............................................................. 49 

Figure 5.7. Classification error with regard to the constant s for three different MPR models 

with seven features............................................................................................................... 51 

Figure 5.8. Comparison of standard deviations of the seven features due to measurement 

noise at steady state, due to the 3rd order MPR model, and due to the lack of 

measurement repeatability ................................................................................................... 54 

Figure 5.9 Comparison of the distributions of the Tsh residuals from NFSS data and 3rd

order MPR model based on the model RMS error; Gaussian distribution, student 

t-distribution, and the calculated distribution from the test data.......................................... 56 

Figure 5.10. Estimation of refrigerant mass flow rate at ΔTsc_TXV.up > 0.5 °C using the 

compressor map with superheat correction by Dabiri and Rice (1981)............................... 60 

Figure 5.11. CdA versus Tsh for all the test cases....................................................................................... 61 

Figure 5.12. Residuals of seven parameters for different faults. (Test #5: TOD = 27.8 °C 

(82.0 °F), TID = 26.7 °C (80.0 °F), and RHID = 50 %): Residual of (a) TE, (b) Tsh, 

(c) TD, (d) TC, (e) Tsc, (f) ∆TCA, and (g) ∆TEA ...................................................................... 62 

Figure 5.13. Main Module of the FDD system running at 15 % of improper indoor air flow 

fault. ..................................................................................................................................... 63 

Figure 5.14. Sequential algorithm of the sample FDD system ................................................................. 64 

Figure 5.15. Variation of system performance under fault-free conditions (NISTIR 7350)..................... 66 

Figure 5.16. Estimated fault levels for each fault at a 5 % degradation in EER at four different 

operating conditions............................................................................................................. 67 

Figure 5.17. Estimated EER degradation for a 10% compressor fault level and a 20 % fault 

level for the remaining faults ............................................................................................... 67
viii

NOMENCLATURE 

Symbols 

a coefficient of multivariate polynomial or a constant 

A air side or area 

C mass flow correction coefficient used with a TXV, or a general case-variable 

F fault case 

g number of coefficients used in a reduced coefficient correlation 

h specific enthalpy (kJ kg-1) 

i feature index 

k index of time 

m mass flow rate (kg/h) 

m mean or number of coefficients 

n number of data samples in a moving window 

N compressor rotational speed, or number of data points, or normal distribution 

P pressure (kPa) or probability 

Q capacity (W) 

r residual 

s positive constant multiplier of the measurement threshold 

t time

T temperature (oC) 

u mean value 

v variance

W work (W) 

x variable representing measured data 

x moving window average of measured data 

z feature residual 

Z set of feature residuals 

Greek Symbols 

α probability of a classification error (%) 

Δ difference

φ feature or performance parameter 

ε threshold value 

σ standard deviation 

ρ density 

Subscripts 

avg average 

comp compressor 

c coefficients in the TXV mass flow rate equation 

C condenser 

CA condenser air 

CW condenser water in water-to-water heat pump 

d TXV discharge 

D compressor discharge 

DB dry bulb 

exp experimental 

E evaporator 

EA evaporator air 

EB evaporator brine in water-to-water heat pump 

full full model with all coefficients included 

ID indoor 
ix

IDP indoor dew point 

m number of features 

map compressor performance map 

NF no fault 

R refrigerant 

reduced model with some coefficients removed as compared to full model 

S saturated refrigerant property 

sc subcooling 

sh superheat 

ss steady-state 

Acronyms/Abbreviations 

AC air conditioner 

ANN artificial neural network 

CF condenser air flow fault 

CFM volumetric flow rate in cubic feet per minute (ft3

 min-1) 

CMF compressor fault 

COP coefficient of performance 

DOE United States Department of Energy 

EER energy efficiency ratio 

EF evaporator air flow fault 

EPA Environmental Protection Agency 

FDD fault detection and diagnosis 

HP heat pump 

HVAC(&R) heating, ventilating and air-conditioning (and refrigeration) 

LL liquid line or liquid line fault 

MPR multivariate polynomial regression 

MS mean squared 

MSR mean squared residual 

MW moving window 

OC refrigerant overcharged fault 

OD outdoor 

PDF probability density function 

RH relative humidity 

scfm standard cubic feet per minute (70 o

F dry air with a density of 0.075 lbm ft3) 

SSR sum of squared residuals 

TXV thermostatic expansion valve 

UC refrigerant undercharge fault 

VAV variable air volume 
1

CHAPTER 1. BACKGROUND 

This work examines cooling mode fault detection and diagnosis of a residential heat pump system with 

the intent of making these systems more trouble free and energy efficient over their entire lifetime. An 

increasing emphasis on energy saving and environmental conservation requires that air conditioners and 

heat pumps be highly efficient. To this end, a few government initiatives have been undertaken. For 

example, the EPA’s Global Programs Division is responsible for the assessment of alternative 

refrigerant’s performance and enforcement of the Clean Air Act. Another prime example is the ENERGY 

STAR initiative, a program formulated by the EPA/Climate Protection Partnerships Division and the 

Department of Energy (DOE), which promotes products that offer energy efficiency gains and pollution 

reduction. Directly affecting residential equipment, a recent DOE regulation imposed a 30 % increase in 

the minimum seasonal energy efficiency ratio (SEER) for central air conditioners, from 10.0 to 13.0, 

beginning January 23, 2006. 

To ensure that heating, ventilating, and air-conditioning (HVAC) equipment operates in the field at its 

design efficiency, the efforts exerted by equipment manufacturers to improve equipment’s SEER must be 

paralleled in the field by good equipment installation and maintenance practices. However, a survey of 

over 55 000 residential and commercial units found the refrigerant charge to be incorrect in more than 

60 % of the systems (Proctor, 2004). Another independent survey of 1500 rooftop units showed that the 

average efficiency was only 80 % of the expected value, primarily due to improper refrigerant charge 

(Rossi, 2004). A low refrigerant charge in the system may be due to a refrigerant leak or improper 

charging during system installation. While the most common concern about a refrigerant leak is that a 

greenhouse gas has been released to the atmosphere, the greater impact is caused by the additional CO2

emissions from a fossil fuel power plant due to a lowered air-conditioner (AC) efficiency. 

Proctor’s survey shows a correlation between the quality of installation and a technician’s training and 

supervision. As is no surprise, proper training of the technician is the condition sine qua non for proper 

installation. But the survey also showed clearly that the number of return calls to correct improper 

installation was lowest when routine oversight of the installation work was provided, and that the number 

of faulty installations markedly increased when post-installation inspection visits were eliminated. At 

present, the homeowner has no quality assurance method for equipment installation as long as his/her 

comfort is not compromised. 

The goal of this project is to study and develop fault detection and diagnostic (FDD) methods which 

could provide a technician with a fault diagnosis and could alert a homeowner when performance of his 

AC unit falls below the expected range, either during commissioning or post-commissioning operation. 

For the homeowner, this FDD capability could be incorporated into a future smart thermostat in the form 

of a readout to provide basic oversight of the service done on the unit and of its performance. 

The benefits of fault detection and diagnosis (FDD) methods for ACs and HPs are numerous, and they 

include; 

- reduction of energy use 

- reduction in peak demand of electricity 

- reduction in CO2 emissions from fossil fuel power plants 

- reduction in refrigerant emissions from AC and HP systems 

- improvement of thermal comfort 

- reduction in down time and maintenance cost 

Fault detection and diagnosis is accomplished by comparing a system’s current performance or features 

with those expected based on the measurements taken from the system when it was known to operate 

fault-free. FDD method development includes a laboratory phase during which fault-free and faulty 
2

operations are mapped, and an analytical phase during which FDD algorithms are formulated. Kim et al. 

(2006) documented performance of the cooling mode fault-free and faulty heat pump operation, while this 

report describes the development of FDD concepts and their analytical implementation. 
3

CHAPTER 2. OVERVIEW OF FAULT DETECTION AND DIAGNOSIS 

2.1 Concept of Fault Detection and Diagnosis 

Generally, a fault is a defect or condition that prevents the system from performing as intended. For a 

heat pump, a fault will result in degraded system capacity or efficiency, or both. The use of computers 

and microprocessors permits the application of FDD methods, which results in an earlier detection of 

faults than is possible by limit and trend checks that can be implemented by a conventional controller. 

Isermann (1984) outlined a supervision loop for model-based FDD adopted in this study, shown in Figure 

2.1. Once a fault is observed, the fault detection module produces a fault message and activates a fault 

diagnosis module, which identifies the cause of the fault and its location. Then, the identified fault is 

evaluated in a fault evaluation module by the class of its hazardous effect. Eventually, the decision 

module will decide the proper maintenance needed based upon economic considerations. After all the 

processes have been executed, the supervision process is reinitialized. The two modules of fault detection 

and diagnosis are typically combined and referred to as an FDD module. 

FAULT

DETECTION

FAULT

DIAGNOSIS

FAULT

EVALUATION DECISION MAINSYSTEM

FAULT

MESSAGES

CAUSE OF FAULT

AND LOCATION

HAZARD CLASS

MAINTENANCE

MEASUREMENTS

Figure 2.1. Supervision loop for fault detection and diagnosis (Isermann, 1984) 

Figure 2.2 shows a detailed FDD procedure for a vapor compression system. System parameter 

measurements are first filtered through a steady-state detector to remove transient variation and external 

noise. The filtered data are converted into a variety of useful parameters such as EER, cooling capacity, 

flow rate, etc., using semi-physical derivations. Here, energy balance evaluations or manufacturer’s 

formulations are utilized. Also from the filtered data, external parameters such as indoor and outdoor 

temperatures - called independent variables - are measured to determine the status of the system. These 

are used by the preprocessor to estimate the expected performance parameters - called features - using a 

no-fault, steady-state system model. The fault classifier analyzes and identifies the fault. The diagnostic 

result is evaluated by a decision module, which recommends a corrective reaction and communicates it to 

another authority. 
4

INPUT

Field Measurements

Decision

Module

FDD

Module

FDD System

OUTPUT

Reactions

FDD Module

FAULT

DIAGNOSIS

INPUT PREPROCESSOR Classifier FAULT DIAGNOSIS

Refrigerant leakage

Condenser fouling

Non-condensables

FEATURES

PREPROCESSOR

PARAMETER

ESTIMATION

STEADY-STATE

DETECTOR FEATURES

EER

Reference values

Estimated parameters 

FILTERED

SIGNAL

INPUT

Figure 2.2. Equipment supervision for fault detection and diagnosis of a vapor compression system 

Isermann’s supervision loop differentiates fault detection from fault diagnosis, although both concepts 

originated from pattern recognition theories. Fault detection classifies the system status into two 

categories; no-fault and fault. When the FDD module diagnoses the system as being fault-free, the 

supervision process starts over. When the system is diagnosed as being faulty, the fault diagnostic 

classifier analyzes the system status in more detail to identify any on-going faults. 

Sometimes the procedures of fault detection and fault diagnosis are unified in one step. Stylianou (1997) 

classified the status of a 5-ton commercial reciprocating chiller into one of five classes; a normal class and 

four fault classes. Stylianou applied a Bayesian decision rule and statistical training data to derive a 

family of classification functions. 

2.2 Previous Research 

A variety of FDD studies have been performed for HVAC systems. Since the automation of HVAC 

systems is strongly connected with building management systems, initial FDD studies were performed for 

whole HVAC systems and not discrete components (Anderson et al., 1989; Pape et al. 1991; Norford and 

Little, 1993; Lee et al., 1996a; Lee et al., 1996b; Peitsman and Bakker, 1996). Many of the studies were 

related to variable-air-volume (VAV) air-handling units. During the development of the first FDD 

techniques for large HVAC systems, energy savings was a secondary consideration to preventing 

equipment malfunction. 

Even though there is a large body of literature on FDD applications for HVAC systems, relatively few 

publications exist for monitoring performance of residential vapor compression heat pumps. Grimmelius 

(1995) developed an on-line failure diagnosis method for a vapor compression refrigeration system. He 

established a symptom matrix based on the combination of casual analysis, expert knowledge, and 

simulated failure modes. He suggested using fuzzy logic for a real-time recognition of the failure mode. 

The author built a reference model based on a simple regression analysis and commented on the need to 

develop more general techniques for reference state estimation. 
5

Stylianou and Nikanpour (1996) presented an FDD methodology for a reciprocating chiller using 

thermodynamic modeling, pattern recognition, and expert knowledge. The authors suggested three fault 

detection modules for off-cycle, start-up, and steady-state operations. In the off-cycle module, the 

authors estimated a temperature decay period, which activated FDD for sensor failure or drift. In the 

steady-state module, they estimated COP and capacity using a semi-empirical chiller model, which was 

then applied to fault detection. In order to isolate a fault, a rule-based fault pattern table was applied. In a 

successive investigation, Stylianou (1997) presented a fault diagnostic methodology using a statistical 

pattern recognition algorithm (SPRA) based on a Bayesian decision rule. The SPRA assigned different 

faults and a no-fault status to a set of five classes. The five classes were allocated equal probability, 

which biased the fault results more often than no-fault results. By shifting the scores of the classification 

functions, the bias impact was reduced. 

Rossi and Braun (1997) developed a statistical FDD method for a roof-top air conditioner. The FDD 

system operated with seven representative temperature measurements. The residual values were used as 

performance indices for both fault detection and diagnosis. The residuals of seven FDD features were 

assumed to obey a Gaussian distribution. Statistical properties of the residuals for current and normal 

operation were used to classify the current operation as faulty or normal. The authors created a fault 

detection classifier module based on a Bayesian decision rule, which was optimized by minimizing the 

classification error. Once the system was suspected to have a fault, a fault diagnostic classifier was 

activated. A rule-based pattern chart which consists of the direction of residual change was used as an 

indication of fault symptoms. The fault diagnostic classifier module was devised assuming individual 

features as a series of independent probabilistic accidents. Since fault detection and fault diagnosis were 

running as independent modules, the fault diagnostic classifier functioned isolated from the detection 

module. Five types of faults could be distinguished from the diagnosis. 

Breuker and Braun (1998a) surveyed frequently occurring faults for a packaged air conditioner. Based on 

field data, they sorted field faults into three different categories according to the cause of the fault, service 

frequency, and service cost. With respect to the cause of the fault, system shutdown failures were caused 

by electrical or control problems approximately 40 % of the time and mechanical problems approximately 

60 % of the time. When sorted by service frequency, refrigerant leakage occurred most frequently, 

followed by condenser, air handler, evaporator, and compressor faults. When sorted by service cost, 

compressor failure contributed 24 % of total service costs. Control related faults contributed 10 % of 

total service costs. 

Breuker and Braun (1998b) performed steady-state and transient tests on a packaged air conditioner and 

described an overall procedure to develop an FDD system. The authors developed no-fault steady-state 

models using polynomial representations. They introduced five types of faults with different levels. The 

impacts of steady-state detector threshold, detection error safety factor, and fault decision threshold were 

evaluated. 

Chen and Braun (2001) developed a simplified FDD method for a 5-ton rooftop air conditioner with a 

thermostatic expansion valve (TXV). They modified an FDD technique by simplifying Rossi and 

Braun’s method (1997). They used measurements and model predictions of temperatures for fault-free 

system operation to compute ratios sensitive to individual faults. The authors selected the two most 

sensitive measurements and developed a simple rule-based FDD algorithm. The FDD algorithm was 

based on sequential rules comparing the sensitivity of residuals organized within a fault characteristic 

chart. 
6

Castro (2002) applied a k-nearest neighbor and k-nearest prototype method for fault detection of a chiller. 

The author calculated Euclidean distances for the current state based on the selected two largest residuals, 

and estimated the possibility of a fault from the distance information. In this research, the software 

MATCH was developed as a tool for the controls package to combine monitoring, fault detection, and 

diagnostic features. After detecting faults, data were input to the rule-based fault diagnosis algorithm. 

Castro preferred the nearest prototype classifier since the nearest neighbor classifier is more 

computationally intensive. 

Comstock and Braun (2001) tested eight common faults in a 316 kW (90 ton) centrifugal chiller to 

identify the sensitivity of different measurements to the faults. The eight faults considered in the study 

were selected through a fault survey among major American chiller manufacturers. The fault testing led 

to a set of generic rules for the impacts of faults on measurements used for FDD. The impact of faults on 

cooling capacity and coefficient of performance was also evaluated. 

Smith and Braun (2003) performed field-site tests of more than 20 units to identify local installation and 

operation problems. Using an 11 kW rooftop unit with a fixed orifice expansion device and a 18 kW unit 

with a TXV, they proposed a decoupling-based unified FDD technique to handle multiple simultaneous 

faults. Li (2004) re-examined the statistical rule-based method initially formulated by Rossi and Braun 

(1997) and presented two additional FDD schemes, which improved the sensitivity of the FDD module. 

He also developed virtual sensors to estimate characteristic parameters from indirect component modeling. 

For a reference model, Li combined a multivariate polynomial model and generalized regressive neural 

network (GRNN). 

Kim and Kim (2005) tested a water-to-water heat pump system with a variable-speed compressor and an 

electrical expansion valve (EEV). They found the system parameters to be less sensitive to faults 

compared to a constant-speed compressor system. They reported that the control of compressor speed 

reduced the fault sensitivity of the system. They also developed an FDD algorithm along with two 

different rule-based charts that depended on the compressor status. The authors suggested that COP 

degradation due to a fault is much more severe with a variable-speed compressor than with a constantspeed compressor. 

Navarro-Esbri et al. (2007) developed an adaptive Kalman filter/forgetting-factor based algorithm applied 

as a refrigerant charge leak detector for a breadboarded water-to-water chiller. Compressor rotation speed, 

suction refrigerant vapor superheat and suction pressure were the most sensitive variables to refrigerant 

charge. Proper experimental tuning of the forgetting-factor allowed this algorithm to immediately detect 

refrigerant charge leakage. This is very promising considering other investigators have noted difficulty in 

reliably detecting an undercharged condition until a fault level of at least 40 % lost refrigerant charge 

(Breuker and Braun 1998b, Stylianou and Nikanpour 1996). 

The works cited above are summarized in Table 2.1. The investigators have extended previously known 

techniques and added unique FDD components to these varying system installations. The rule-based 

chart method based upon application of statistical methods was shown by several investigations to be very 

effective and adaptable. This is one reason this method has been adapted for use in the present work. 
7

Table 2.1. Summary of system types and FDD methodology for selected previous investigations 

System types FDD methodology 

VAV air-handling units and whole building system Component FDD and black box model FDD 

Vapor compression refrigeration system Symptom matrix with expert system decision 

making algorithm (IF..THEN..ELSE). 

Reciprocating chiller Off, Start, and Steady FDD 

Rooftop AC w/ orifice Feature residual, rule-chart FDD based upon 

statistical method 

Packaged AC Five faults, FDD with steady-state detector 

Rooftop with TXV Ratios sensitive to certain faults were identified and 

a rule-based chart FDD scheme was used 

Centrifugal chiller Residuals calculated and k-nearest neighbor 

method used for FDD 

Centrifugal chiller Eight faults and impacts of faults on efficiency 

were studied 

Rooftop unit w/ TXV FDD with virtual sensors, polynomial and neural 

network reference model 

Variable speed, mini split AC Rule-based chart FDD 

Small breadboard chiller Fast detection of charge loss 
8

CHAPTER 3. STEADY-STATE DETECTOR 

In this study, we envision the fault detection and diagnosis (FDD) process to be performed every time the 

system is in steady state. For this reason, identification of the steady state is an important task in our 

FDD analysis. Some of the first investigations of system steady-state identification came from process 

control field studies including model development, optimization, fault detection, and control (Mahuli et 

al., 1992, Cao and Rhinehart, 1995, Jiang et al. 2003). Steady state can be detected by observing global 

system characteristics, e.g., capacity, or – more simply – by monitoring selected parameters. If the only 

goal was to check system performance, providing enough time to reach steady-state capacity and power 

input could be a sufficient approach. However, reaching steady capacity would not guarantee the actual 

steady-state of all parameters used in a particular FDD scheme. 

In previous investigations within the HVAC field, several investigators suggested steady-state 

identification techniques for FDD purposes (Glass et al., 1995, Rossi, 1995, Breuker and Braun, 1998a, Li, 

2004). Since the goal of these investigations was mainly fault detection, the evaluation of the steady-state 

detectors was not performed in detail. In this investigation, a moving window steady-state detector was 

evaluated and optimized. A systematic methodology to design a steady-state detector is provided below. 

3.1 A Moving Window for Steady-State Detection 

The concept of the steady-state detector originates from noise filter theory. When a system is not steady, 

thermodynamic system parameters are highly unstable. The variance, or standard deviation, of important 

parameters is typically utilized to indicate the statistical spread within the data distribution and can be 

used to characterize random variation of the measured signals. 

The most common and simple steady-state detectors analyze data over a predefined moving window, as 

illustrated in Figure 3.1. A predefined time interval is established over which important parameters are 

sampled at regular intervals. This produces an array of system parameters that are continuously updated 

and held in memory. Since a moving window replaces each data point within the timespan, the moving 

window average is equivalent to a low-pass filter. 

t, time

n

n

n

k− n− 1 k− n k− n+1 k− 2 k− 1 k

Figure 3.1. Moving windows of n data points at k

th time 
9

Several methods of steady state detection were investigated by other researchers. Glass et al. (1995) 

formulated a steady-state detector using a geometrically weighted running method. In this method, the 

old data are exponentially and automatically attenuated through multiplication with a “forgetting factor”, 

which is based upon a system transition time constant. Some FDD investigations of vapor compression 

systems applied this method (Rossi 1995, Breuker and Braun 1998b). Equations 3.1 and 3.2 denote 

geometrically weighted average and geometrically weighted variance, respectively. Geometrically 

weighted average and variance may have different weights. 

∑

∑

=

−

=

−

= n

k

n k

n

k

k

n k

n

x

x

0

0 ( )

α

α

α (3.1) 

∑

∑

=

−

=

− −

= n

k

n k

k n

n

k

n k

n

x x

v

0

2

0

( ( ))

α

α α

 (3.2) 

The geometric weighting factor, α is calculated as in Equation 3.3 and has a value between 0 and 1. In 

this equation, τss is a time constant and Δt is a time increment between measurements. 

t ss

ss

+ Δ = τ

τ

α (3.3) 

Li (2004) utilized moving window slopes and variances of evaporator exit superheat and liquid line 

subcooling as the key parameters of the steady-state detector in his FDD research on roof-top air 

conditioners. 

In our study the moving window standard deviation calculated for an optimized moving window size with 

defined feature thresholds indicates the steady state of the heat pump. The steady-state detector calculates 

standard deviation of parameters in a recursive fashion. Suppose that at any instant k, the average of the 

latest n samples of a data sequence, xi, is given by, 

1

1

=−+

= ∑

k

k i

ikn

x x

n

 (3.4) 

A difference between two averages of the latest n samples at the current time, k, and at the previous time 

instant, k −1, is, 

[ ]

1

1

1

1 1 −

− −

=−+ =−

⎡ ⎤ −= − = − ⎢ ⎥ ⎣ ⎦ ∑ ∑

k k

k k i i k kn

ikn ikn

xx x x xx

n n

 (3.5) 

Rearranged, the current average is calculated by, 

1 ( ) 1k k k kn =+ − − − xx xx

n

 (3.6) 
10

This approach is known as a moving window average because the average at each k

th instant is based on 

the most recent set of n values. In other words, at any instant, a moving window of n values is used to 

calculate the average of the next data sequence. A moving window variance can be defined similarly. 

( )2 2 2

1 1

1 1

=−+ =−+

= −= − ∑ ∑

k k

k ik i k

ikn ikn

v xx x x

n n

. (3.7) 

( ) ( ) 22 2 2

1 1

1

k k k kn k k =+ − −− − −− vv xx xx

n

 (3.8) 

The moving window standard deviation is then given as 

σ k k = v (3.9) 

The steady-state detector identifies steady operation if the standard deviations for the selected features 

representing the status of the system fall below a predefined threshold. 

3.2 Development of the Steady-State Detector 

3.2.1 Experimental setup and conditions 

The studied system was an R410A, 8.8 kW (2.5 ton) split residential heat pump with Seasonal Energy 

Efficiency Ratio (SEER) of 13 (ARI, 2006). The unit consisted of an indoor fan-coil section, outdoor 

section with a scroll compressor, cooling mode and heating mode thermostatic expansion valves (TXV), 

and connecting tubing. Both the indoor and outdoor air-to-refrigerant heat exchangers were of the finnedtube type. The unit was installed in environmental chambers and charged with refrigerant according to 

the manufacturer’s specifications. Figure 3.2 shows a schematic of the experimental setup indicating the 

measurement locations of temperature, pressure, and mass flow rate. On the refrigerant side, pressure 

transducers and T-type thermocouple probes were attached at the inlet and exit of every component of the 

system. The refrigerant mass flow rate was measured using a Coriolis flow meter. The air enthalpy 

method served as the primary measurement of the system capacity, and the refrigerant enthalpy method 

served as the secondary measurement. These two measurements always agreed within 5 %. Table 3.1 

lists uncertainties of the major quantities measured during this work. Detailed specification of the test rig 

including indoor ductwork, dimensions, data acquisition, measurement uncertainty, and instrumentation is 

described in Kim et al. (2006). 
11

INDOOR

CHAMBER

OUTDOOR

CHAMBER

Evaporator

Condenser

Compressor Scroll

Mass flow meter

4-way

Valve

ΔP ΔP

ΔP

MFR

TDB TDP

SCFM

W

Thermostatic

Expansion Valve

T

P, T

P, T, Tsat

P, T

P, T

P, T

TDB

TDB

P, T, Tsat

P, T, Tsat

TDB TDP Valves 

for LL fault

Valve 

for CMF fault

Figure 3.2. Schematic diagram of the experimental setup 

Table 3.1. Measurement uncertainties 

Measurement Range Uncertainty at the 95 % 

Confidence Level 

Individual Temperature -18 °C to 93 °C ±0.3 °C 

Temperature Difference 0 °C to 28 °C ±0.3 °C 

Air Nozzle Pressure 0 Pa to 1245 Pa ±1.0 Pa 

Refrigerant Mass Flow Rate 0 kg/h to 544 kg/h ±1.0 % 

Dew point Temperature -18 °C to 38 °C ±0.4 °C 

Dry-Bulb Temperature -18 °C to 40 °C ±0.4 °C 

Total Cooling Capacity 3 kW to 11 kW ±4.0 % 

COP 2.5 to 6.0 ±5.5 % 
12

3.2.2 Selection of measurement features 

Table 3.2 shows system features used in fault detection. In this laboratory study, we measured pressure at 

the exits of the evaporator and the condenser to obtain TE and TC, respectively. The obtained pressures 

were converted into saturation temperatures using REFPROP 7 (Lemmon et al., 1998). For the other five 

features, T-type thermocouples were used. 

Table 3.2. System features used in fault detection 

Independent Features Dependent Features 

Outdoor dry-bulb temperature TOD

Evaporator exit refrigerant saturation 

temperature 

TE

Indoor dry-bulb temperature TID Evaporator exit refrigerant superheat Tsh

Indoor dew point temperature TIDP

Condenser inlet refrigerant saturation 

temperature 

TC

 Compressor discharge refrigerant 

temperature 

TD

 Condenser exit liquid line refrigerant 

subcooled temperature 

Tsc

 Evaporator air temperature change ΔTEA

 Condenser air temperature change ΔTCA

3.2.3 Setting feature thresholds based upon steady-state data 

The threshold of each feature that bounds the steady-state signal is an important parameter in determining 

the performance of a steady-state detector. The smaller a threshold is, the more sensitive an FDD module 

is because steady state is more conservatively identified. However, if a threshold is too small, it takes 

more time for the given feature to settle within its threshold range. Also, an excessively small threshold 

value may inhibit the functioning of the steady-state detector when operating in the field. Large 

thresholds allow faster data collection but carry a risk of including some transient data, which decreases 

the performance of an FDD system. Therefore, the thresholds must be selected to both minimize the 

inclusion of non-steady-state data and maximize the recognition of steady state. 

To determine the thresholds, we collected steady-state data over 60 minutes commencing 3 hours after 

startup at fixed chamber conditions and calculated the standard deviations, σ, for all seven features. As an 

example, Figure 3.3 shows the variation and standard deviation of the steady-state data for all features, 

and Table 3.3 summarizes the ranges and standard deviations of all seven features. Since the system 

operated under stable conditions, the standard deviations and ranges in Table 3.3 are small. As expected, 

superheat shows the largest fluctuation. 
13

7.5

8.0

8.5

9.0

0 10 20 30 40 50 60

Time (min)

Temperature (oC)

+ σ = 0.12 

+ 3σ = 0.37

Average Tsh

Tsh (oC)

Figure 3.3(a). Variation of superheat (TID = 26.7±0.5 °C; TOD = 27.8±0.5 °C) 

4.0

4.5

5.0

5.5

0 10 20 30 40 50 60

Time (min)

Temperature (oC)

+ σ = 0.05

+ 3σ = 0.16

Average Tsc Tsc (oC)

Figure 3.3(b). Variation of subcooling (TID = 26.7±0.5 °C; TOD = 27.8±0.5 °C) 

10

10.2

10.4

10.6

10.8

0 10 20 30 40 50 60

Time (min)

TE (oC)

Figure 3.3(c). Variation of evaporator saturation temperature (TID = 26.7±0.5 °C; TOD = 27.8±0.5 °C) 
14

65.6

65.8

66

66.2

66.4

66.6

0 10 20 30 40 50 60

Time (min)

T D (oC)

Figure 3.3(d). Variation of compressor discharge temperature (TID = 26.7±0.5 °C; TOD = 27.8±0.5 °C) 

39

39.2

39.4

39.6

39.8

40

0 10 20 30 40 50 60

Time (min)

T C (oC)

Figure 3.3(e). Variation of condenser saturation temperature (TID = 26.7±0.5 °C; TOD = 27.8±0.5 °C) 

9

9.2

9.4

9.6

9.8

10

0 10 20 30 40 50 60

Time (min)

ΔT CA (oC)

Figure 3.3(f). Variation of air temperature increase through condenser (TID = 26.7±0.5 °C; TOD = 

27.8±0.5 °C) 
15

11

11.2

11.4

11.6

11.8

12

0 10 20 30 40 50 60

Time (min)

ΔTEA (oC)

Figure 3.3(g). Variation of air temperature drop through evaporator (TID = 26.7±0.5 °C; 

TOD = 27.8±0.5 °C) 

Table 3.3. Variations of selected features during steady state 

Features Tsh Tsc TE TD TC ΔTCA ΔTEA

Range* (°C) 0.49 0.22 0.14 0.25 0.17 0.27 0.25 

Standard deviation, σ (°C) 0.124 0.052 0.024 0.058 0.035 0.063 0.058 

Calculated thresholds, 3σ (°C) 0.37 0.16 0.07 0.17 0.11 0.19 0.17

 * Difference between the maximum and minimum value 

3.2.4 Selection of moving window size using startup transient tests 

Figures 3.4, 3.5, and 3.6, presented in this section, were derived from the same startup transient test. 

Figure 3.4 displays variation of the seven features in the post-startup period. The figure demonstrates that 

Tsh and Tsc fluctuate the most and are the dominant indicators of system instability during startup. Figure 

3.5(a) further examines fluctuations of Tsh and Tsc showing them with ±3σ thresholds superimposed (these 

thresholds were presented in Figure 3.3). The vertical dashed line extending to Figures 3.5(b) and 3.5(c) 

indicates the onset of steady state at approximately 6 minutes and 30 seconds. From this point in time 

and on, the values of both features are retained within their respective ±3σ thresholds. 

While Figure 3.5(a) convincingly shows us, based on individual measurements, that steady state was 

attained at 6 minutes and 30 seconds after the startup, we must realize that we could make this 

determination only after extending the data collection much further in time beyond 6 minutes and 30 

seconds to be able to calculate Tsh and Tsc mean values for steady-state operation. For this reason, using 

individual feature measurements for steady state indication proves not to be a robust approach, and it is 

rather attractive to base a steady-state detector on some statistical quantity based upon measurements 

taken within a predefined moving time window. In this study we applied the ±3σ steady-state thresholds 

and standard deviations of moving-window-measured Tsh and Tsc values, σMW(Tsh) and σMW(Tsc), for 

steady-state detection during the startup transient. 
16

Temperature difference (oC) Temperature (oC)

Time (min)

ΔTCA

TD

TC

TE

ΔTEA

Tsc

Tsh

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14

5

10

15

2

4

6

8

10

12

Figure 3.4. Variation of seven features during a startup transient, TID = (26.7±0.5)°C, TOD = (27.8±0.5)°C 

Figures 3.5(b) and 3.5(c) explain the procedure we used to determine the size of the moving window by 

showing Tsh and Tsc standard deviations, σMW(Tsh) and σMW(Tsc), respectively, for three windows sizes with 

a sample period of 14 seconds/sample: 70 seconds (MW70s), 140 seconds (MW140s), and 210 seconds 

(MW210s). It is our interest to establish the minimum window size for which the calculated σMW(Tsh) and 

σMW(Tsc) are below their respective threshold values past the onset of steady state, which is shown in 

Figure 3.5 to occur at 6 minutes and 30 seconds after startup. 
17

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14

Time (min)

σ MW (Δ

Tsc) (oC)

MW 70 s

MW 140 s

MW 210 s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

σ MW (Δ

Tsh) (oC)

MW 70 s

MW 140 s

MW 210 s

2

4

6

8

10

12

±0.37 oC line

±0.16 oC line

steady

threshold 0.37

Temperature (oC)

Tsc

Tsh

(b)

threshold 0.16

(a)

(c)

Moving Window Standard Deviation (oC)

σMW (Tsh) (oC)

σMW (Tsc) (oC)

MW70s

MW140s

MW210s

MW70s

MW140s

MW210s

Figure 3.5. Onset of steady state during a no-fault startup transient test, TID = (26.7 ± 0.5)°C, 

TOD = (27.8 ± 0.5)°C; (a) Decision on steady state; (b) σMW(Tsh); (c) σMW(Tsc) 

The general procedure for determining the minimum moving window size, illustrated in Figure 3.5, is as 

follows: 

1. Collect selected feature data during the start-up period for at least 30 samples into the 

steady-state region at a sampling rate equal to the sampling rate used for steady-state 

sampling. The steady-state region is defined here to occur when the instantaneous values 

of the selected features fluctuate within ±3σ of their steady-state mean values, as 

determined in Figure 3.3. The dashed vertical line in Figure 3.5(a) illustrates this onset of 

steady-state using the two most fluctuating FDD features during the startup, Tsh and Tsc. 

2. For all features, calculate the moving window standard deviation versus time for a range 

of moving window sample sizes, as presented in Figures 3.5(b) and 3.5(c) for Tsh and Tsc. 

3. The moving window size that results in all features’ standard deviations crossing and 

remaining within the ±3σ threshold after steady-state is attained (as defined in Step 1), is 

the minimum acceptable moving window size. 
18

The minimum acceptable moving window size (and thus sample size for our sampling rate) is determined 

by plotting the moving window standard deviations as a function of time for all features used in the FDD 

algorithm. In our case the last two features to vary within their ±3σ thresholds, of all the features listed in 

Table 3.3, were Tsc and Tsh. The moving window standard deviations of Tsh and Tsc (σMW(Tsh) and 

σMW(Tsc)) are plotted as a function of time in Figures 3.5(b) and 3.5(c) with the vertical steady-state line 

determined in Figure 3.5(a) extending down to indicate the onset time of steady-state. 

Selecting the 70 second long moving window, MW70s, would not be appropriate because both σMW(Tsh) 

and σMW(Tsc) values calculated for this window size drop below the respective thresholds well before the 

onset of steady state at 6 minutes and 30 seconds. The MW140s appears to be a good selection because it 

produces standard deviations that remain below the steady-state threshold after the vertical steady-state 

line; with a few seconds past this instance for σMW(Tsh) and 90 seconds later for σMW(Tsc) at 8 minutes. 

Hence, MW140s would indicate the onset of steady state 90 seconds after it actually has occurred, but it 

would not provide a false indication of steady state at any time earlier because of the relatively smooth 

and oscillation-free character of σMW(Tsh) and σMW(Tsc) lines. For MW210s, steady-state detection was 

indicated at approximately 9 minutes with σMW(Tsh) being the defining factor; this moving window could 

be acceptable, but it is not the minimum moving window size to satisfy our steady-state criteria that all 

features remain within their ±3σ thresholds. 

In this startup test, we took measurements every 14 seconds. We should note that the data sampling rate 

(samples/s or Hz) should be based upon a Nyquist frequency (greater than twice the frequency of the most 

varying feature) (Franklin et al. 1991), as determined by the frequency of variation in the features 

important to the FDD algorithm. For this investigation, the heat pump features during steady-state varied 

at a maximum frequency well below 0.03 Hz. Sampling at greater than twice this frequency, or 1 sample 

every 16 seconds, would capture all the feature variations. The sample rate, fixed at 1 sample per 14 

seconds, was sufficient to eliminate or greatly reduce the likelihood of signal measurement bias. A 

greater sampling rate could be desirable, but sampling at much greater than the Nyquist frequency gains 

no new information. 

As defined by the ±3σ steady-state threshold, the system was stable approximately 6 minutes and 30 

seconds after startup; however, the discharge line wall temperature, TD, continued to increase. Figure 3.6 

shows that TD increased by 2.1 °C for the remainder of the time shown in the figure while the evaporator 

capacity, QEA, increased by 1.3 % with no change in the condenser capacity, QCA. The unchanging value 

of QCA tends to indicate a thermal inertia effect for TD. The slow increase, or drift, in this temperature 

does not preclude its use as a fault detection feature, however, it will result in a higher uncertainty for TD

within the system’s steady-state no-fault reference model. 
19

0

2000

4000

6000

8000

10000

12000

14000

16000

0 2 4 6 8 10 12 14

Time (min)

Heat capacity (W)

30

35

40

45

50

55

60

65

70

T

D (oC)

QEA

QCA

TD

TD increases by 2.1oC

The onset of steady state

QEA increases by 1.2%

Compressor

preheating

Capacity (W)

Figure 3.6. The startup transients of compressor discharge temperature and evaporator/condenser 

capacities, TID = (26.7±0.5)°C, TOD = (27.8±0.5)°C 

3.3 Verification of the Steady-State Detector Using an Indoor Load Change 

Test 

3.3.1 Tests with no faults imposed 

Figure 3.7 presents verification of the developed steady-state detection procedure during transients driven 

by changes in the indoor air temperature with no faults imposed. Stepped increases in TID in Figure 3.7(a) 

reflect the energizing of additional indoor chamber electric duct heaters. In addition to TID, the figure 

shows Tsh, Tsc, and standard deviations of Tsh, TE and ΔTEA for an 18 second sample interval and moving 

window sizes of 72 seconds (4 samples), 144 seconds (8 samples), and 216 seconds (12 samples). Tsh, Tsc

and σMW(Tsh) are included because they were important during the startup transient. While Tsh shows 

some variation during an indoor load change test, Tsc is much more stable than during the startup transient. 

The figures show two transient regions, which were identified using a 144 second moving window. The 

gray areas in the plots indicate transient regions for individual features as determined by the ±3σ

threshold steady-state detector algorithm within a 144 seconds moving window. Figure 3.7(e) shows that 

ΔTEA consistently indicates the beginning of the transient while Figure 3.7(d) shows that TE indicates the 

end of the transient. The variation of Tsh in Figure 3.7(c) cannot filter out the whole transient region for 

all of these parameters. Instead, the variation of TE and ΔTEA in Figures 3.7(d) and 3.7(e) control the 

detection of steady state. Since the transients in Figure 3.7(a) are due to changes in TID, the features that 

characterize the evaporator (indoor coil), TE and ΔTEA, proved to be necessary for steady state detection. 
20

0

0.1

0.2

0.3

0.4

0.5

σ MW (TE) (oC)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

σ MW (Δ

Tsh) (oC)

27

29

31

33

4

5

6

7

8

9

10

Temperature (oC)

threshold 0.37

Tsh

TID

Identified as transient region

0

0.1

0.2

0.3

0.4

0.5

0.6

145 155 165 175 185 195

Time (min)

σ MW (Δ

TEA) (oC)

MW 72 s

MW 144 s

MW 216 s

threshold 0.07

threshold 0.17

Tsc

Standard Deviation (oC) Standard Deviation (oC) Moving Window Standard Deviation (oC)

σMW (Tsh)

σMW (TE)

σMW (ΔTEA)

(b)

(c)

(a)

(d)

(e)

Moving Window Standard Deviation (oC)

σMW (Tsh) (oC)

σMW (TE) (oC)

σMW (Δ

TEA) (oC)

MW72s

MW144s

MW216s

MW72s

MW144s

MW216s

MW72s

MW144s

MW216s

Figure 3.7. Identification of steady state during a no-fault indoor load change test, TOD = (35 ± 0.5)°C, (a) 

TID; (b) Tsh and Tsc; (c) σMW(Tsh); (d) σMW(TE); (e) σMW(ΔTEA) 
21

3.3.2 Test with a 20 % refrigerant undercharge fault 

Since the goal of an FDD scheme is to detect a system fault, the steady-state detector must be able to 

identify steady state during a system’s faulty operation. For this reason, we applied the developed steadystate detector during a test with changing indoor temperature with a 20 % refrigerant undercharge fault. 

Figure 3.8 shows the selected system parameters and features during the 35 °C outdoor temperature test. 

The moving window sizes are 66 seconds (3 samples), 132 seconds (6 samples), and 198 seconds (8 

samples) with a data sampling rate of 22 seconds. Figure 3.8(a) shows two rapid drops of TID due to 

turning off chamber electric duct heaters and a continuous decrease of TID due to chamber cooling by the 

tested system. 

As during the fault-free indoor temperature change transient test, the change of σMW(Tsh) in Figure 3.8(c) 

is not large enough to filter out the transient state, but changes in σMW(TE) and σMW(ΔTEA), taken together, 

correctly identify steady state, as shown in Figures 3.8(d) and 3.8(e). Hence, the same features identified 

steady state during the undercharge fault and no-fault operations. Further, our review of transient data of 

the system operating under different faults indicated that the developed steady-state detector would work 

reliably with other faults as well. 

3.4 Verification of the Steady-State Detector During a Startup Transient Test 

with a Fault Imposed 

To verify the steady-state detector developed in this study, several startup transient tests were performed 

with evaporator and condenser faults imposed on the system. Operating conditions for all tests were 

(27.8±0.5) °C outdoor dry-bulb, (26.7±0.5) °C indoor dry-bulb, and indoor relative humidity of 

(50.0±1.0) %. 

 

3.4.1 Test with condenser fault imposed 

A condenser fault was imposed by blocking the condenser heat exchanger face area with paper from the 

bottom up. The fault level for this transient test was set at 30 % face area having no air flow. Figure 3.9 

shows the variation of four features during the startup of the heat pump. Superheat and subcooling are 

shown in Figure 3.9(a). The standard deviation of Tsh, Tsc, TE, and TD in a 144 seconds (9 sample) moving 

window are shown in Figure 3.9 in sequence. Tsh and Tsc reach steady state 4 and 5 minutes after the 

system starts, but the compressor discharge temperature, TD, continues to change thus delaying the 

beginning of steady state to 10 minutes. With the condenser air flow fault, the discharge pressure of this 

system is higher than that of the no-fault system; hence this causes the discharge temperature to be higher. 

The thermal inertia time constant for the discharge wall temperature results in TD attaining steady state 

later than other features. This test shows that the steady-state detector still safely determines steady state 

even if the system has a condenser fouling fault. 
22

22

24

26

28

0

0.1

0.2

0.3

0.4

0.5

0.6

Standard deviation of Superheat

MW 66s

MW 132s

MW 198s

9

10

11

12

13

0

0.1

0.2

0.3

0.4

0.5

75 80 85 90 95 100 105 110

Time (min)

Standard deviation of DTea

MW 66s

MW 132s

MW 198s

0

0.1

0.2

0.3

0.4

0.5

Standard deviation of DTea

Temperature (oC)

Temperature (oC) Moving Window Standard Deviation (oC)

threshold 0.37

threshold 0.17

Identified as 

transient region Tsh

TID

ΔTEA

σMW (Tsh)

σMW (ΔTEA)

(b)

(c)

(a)

(d)

Standard Deviation (oC) Standard Deviation (oC) Moving Window Standard Deviation (oC) Moving Window Standard Deviation (oC)

σMW (Tsh) (oC)

σMW (TE) (oC)

σMW (Δ

TEA) (oC)

(e)

threshold 0.07

σMW (TE)

MW66s

MW132s

MW198s

MW66s

MW132s

MW198s

MW66s

MW132s

MW198s

Figure 3.8. Identification of steady state during an indoor load change test with a 20 % refrigerant 

undercharge fault, TOD = (35 ± 0.5)°C, (a) TID; (b) Tsh and ΔTEA; (c) σMW(Tsh); (d) σMW(TE); 

(e) σMW(ΔTEA) 
23

+3σ=0.37

σMW (Tsh) (oC)

σMW (Tsc) (oC)

+3σ=0.16

MW144s

σMW (Tsh)

σMW (Tsc)

σMW (TE) (oC)

+3σ=0.07

σMW (TE)

σMW (TD) (oC)

+3σ=0.17

σMW (TD)

0

0.1

0.2

0.3

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0

0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14

time (min)

Figure 3.9. Variation of Tsc, Tsh, TE and TD during startup transient with a 30 % condenser air flow fault 

level with standard deviations calculated for a 144 s (9 sample) moving window 
24

3.4.2 Test with an evaporator air flow fault imposed 

An evaporator fault was imposed by reducing the air flow rate through the evaporator heat exchanger. 

The fault level was a 30 % reduction in normal air flow. 

Figure 3.10. Variation of Tsc, Tsh, TE and TD during startup transient with a 30 % evaporator air flow fault 

level with standard deviations calculated for a 144 s (9 sample) moving window 
25

Figure 3.10 shows evaporator exit refrigerant superheat and inlet refrigerant liquid subcooling in addition 

to the standard deviation of Tsh, Tsc, TE, and TD within a 144 second (9 sample) moving window. The 

standard deviation of superheat remains below its threshold indicating steady state at about 15 minutes 

after the system starts. Because this system has an evaporator air flow fault, the evaporator saturation 

temperature and superheat became unstable. On the contrary, the subcooling and the discharge 

temperature reach steady state well before Tsh. The steady-state detector can still indicate the initiation of 

steady state for all of the features, but this indication occurred much later than the no-fault steady-state 

startup time interval of 6 min 30 s. 

3.5 Verification of the Steady-State Detector During Continuous and Cyclic 

Operations 

Figure 3.11 shows an example of the operation of the steady-state detector based on the thresholds listed 

in Table 3.3. The data in Figure 3.11 were sampled while the system was continuously operating. Figure 

3.11(a) shows rises and falls in indoor temperature as electric air heaters were turned on and off. 

Based on the Boolean chart in Figure 3.11(c), the features TD and Tsh have the largest number of 

transitions from 0 (unsteady) to 1 (steady); however, the majority of the time was identified as steady 

state. Even though indoor temperature varied as the indoor electric heaters were energized, the system 

was still regarded as steady when the feature variations were small enough to satisfy the steady–state 

detector. From Figures 3.11(a) and 3.11(c), several minutes of uniform temperature do not necessarily 

produce a steady state. Since TOD was fixed during these tests, the condenser saw little change as 

indicated by the steady Boolean values of TC and Tsc. 

Figure 3.12 presents performance of the steady-state detector for on-off transient operation at fixed 

operating conditions. At approximately 3 minutes the system turned off and remained off for 27 minutes. 

It turned off again at 33 minutes, on at 40 minutes, off at 47 minutes, on at 57 minutes, off at 63 minutes 

and finally on again at 88 minutes. The figure includes five of seven features. Based on the value (0 or 

1) of the ‘SS detector’ in Figure 3.12(b), it takes 5 minutes or more of system runtime to reach a steady 

state. 
26

TE

ΔTsh

TD

TC

ΔTsc

ΔTCA

ΔTEA

SS Detector

Steady-state

System status

Non steady-state

Time (min)

0 100 200 300

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

Feature signal

Time (min)

0 100 200 300

20

25

30

35

0

5

10

15

ΔTEA

ΔTsh

ΔTsc

TID

Temperature

(oC)

Temperature

(oC)

(a)

(c)

(b)

Figure 3.11. Sample operation of a steady-state detector during continuous system operation 

(TID = varied, TOD = (27.8±0.5) °C, moving window size = 150 s); (a) Variation of indoor 

temperature; (b) Temperature differences; ΔTEA, Tsh, and Tsc; (c) Decision from the steadystate detector: 0-unsteady, 1-steady 
27

TE

ΔTsh

TD

TC

ΔTsc

SS Detector

Steady-state

System status

Non steady-state

Time (min)

0 30 60 90 120

1

0

1

0

1

0

1

0

1

0

1

0

1

0

Feature signal

(b)

Temperature

(oC)

(a)

0

10

20

30

40

0 30 60 90 120

Time (min)

TE

ΔTsh

TC

Figure 3.12. Operation of a steady-state detector during system on-off transient operation 

(TID = (26.7±0.5) °C; TOD = (27.8±0.5) °C); (a) Variation of TC, TE, and Tsh; (b) Decision 

from the steady-state detector: 0-unsteady, 1-steady 

3.6 Steady-State Detector Summary 

The presented methodology for developing a steady-state detector for a vapor compression system is 

based on a moving window and using standard deviations of seven parameters selected as features. When 

the threshold band of the features was set at ±3σ, reflecting their fluctuations during steady-state 

operation, the optimal moving window size was approximately 140 seconds for a 14 second sampling rate. 

Of the seven monitored features, Tsh and Tsc measurements were sufficient for determining the onset of 

steady state during the startup transient. However, they were not the dominant steady-state indicators 

during indoor temperature change tests, where TE and ΔTEA were needed for proper steady-state 

identification. Consequently, we recommend including all FDD features in the steady-state detector to 

ensure the robustness of the detector because different features may play key roles with different 

transients. While the proposed steady-state detector was developed from no-fault data, we verified that it 

can perform correctly with a faulty system. 

A practical steady-state detector must be defined based upon the heat pump and system controller that 

will perform the fault detection and diagnosis. For any given system, the designer may choose FDD 

features other than those identified here. The system controller and instrumentation used in a commercial 

product most likely will have a different resolution or noise immunity than a dedicated lab quality data 

acquisition system. This would translate into larger standard deviations and threshold values with 

different sampling rates and moving window sizes. 
28

CHAPTER 4. NO-FAULT REFERENCE MODEL 

FDD systems recognize a set of key system performance parameters and function by comparing predicted 

fault-free parameter values to the current values, and analyzing their residuals. Thus, a reference model is 

required to estimate the fault-free system parameters at any operating condition. Since an FDD system 

model requires precise estimation of system parameters, generalized conventional analytical modeling 

techniques were replaced with empirical correlations in several studies (Gordon and Ng, 1995; Rossi, 

1995). Lee et al. (1996b) used an artificial neural network to relate the dominant symptoms and faults of 

an air-handling unit. To improve the modeling capability of an FDD system, Li and Braun (2003) 

implemented a polynomial/generalized neural network regression in their reference model, and they 

produced improved interpolation and extrapolation results for a roof top unit. Navarro-Esbri et al. (2007) 

developed a low-data-requirement model based on neural networks for a water-to-water vapor 

compression system focused on refrigerant leak detection. In this study we collected extensive data for 

cooling mode, and evaluated multivariable polynomial and artificial neural network reference models for 

their ability to predict system features selected for the FDD scheme. 

4.1 Data Collection for the Reference Model 

We systematically varied three independent variables, TOD, TID, and TIDP and monitored the seven features 

as listed in Table 3.2. To implement the most efficient test procedure, outdoor temperature was fixed at 

one of four constant values, the addition of steam to the indoor chamber was set at one of several discrete 

levels by modulating a steam valve, and the indoor dry-bulb temperature was changed over the desired 

operating range by energizing 10 fixed heaters sequentially. For example, as the number of indoor 

electric heaters increased, the test conditions moved from A to B in Figure 4.1(a) with indoor temperature 

increasing and relative humidity decreasing. The data were recorded continuously and filtered through 

the steady-state detector, which qualified steady-state data for use in development of the reference model. 

In this process, instability of the system due to on-off transients and rapid load changes was filtered out 

by the steady-state detector. 

Table 4.1 shows operating conditions for the fault-free tests. The four outdoor temperatures were 

maintained within ±0.3 °C. For the indoor conditions, the amount of steam introduced to the indoor 

chamber was fixed such that the humidity ratio varied from 0.0037 to 0.0168. Data were recorded, every 

18 s, as indoor dry-bulb temperature varied from 15.3 °C to 33.9 °C. The range of operating conditions 

for which data were collected defines the applicable limits for the FDD scheme. 

Table 4.1. Operating conditions for fault-free tests 

Outdoor dry-bulb temp. (°C) 27.8, 32.2, 35.0, 37.8 

Indoor dry-bulb temp. (°C) 15.3 to 33.9 

Indoor humidity ratio 0.0037 to 0.0168 

From the total number of 10409 recorded data sets, 5830 data sets passed through the steady-state 

detector. Among these steady-state sets, 2176 sets were collected at 27.8 °C outdoor temperature, 1732 

sets at 32.2 °C, 633 sets at 35.0 °C, and 1289 sets at 37.8 °C. Figure 4.1(b) shows a sample of data (every 

fifth data point) taken at four fixed outdoor temperatures. In addition, we performed ARI standard rating 

tests at 27.8 °C indoor and 35.0 °C outdoor dry-bulb temperatures (ARI, 2006) and included these data to 

develop the fault-free steady-state reference models. 
29

(a)

(b)

A B

10 15 20 25 30 35 40

0

0.005

0.01

0.015

0.02

0.025

0.03

Humidity ratio

Dry bulb temperature ( oC)

10

20

30

RH 80%

60%

40%

20%

Dew point temperature (oC)

27.8 oC

32.2 oC

35.0 oC

37.8 oC

TOD

Figure 4.1. Indoor test conditions on a psychrometric chart for fault-free model experiments at a fixed 

outdoor temperature; (a) Indoor condition change as electric heaters activate; (b) Sampled 

indoor air conditions at four TOD (27.8, 32.2, 35.0, and 37.8) °C 

4.2 Multivariate Polynomial Regression (MPR) Reference Model 

The MPR model belongs to the “black-box” category of models, which do not consider the physics of the 

system and require a large data set to accurately predict a system’s performance. In our study, we 

evaluated 1st, 2nd, 3rd, and 4th order MPR models representing the seven key features of the heat pump. 

The higher order MPR models offer better accuracy of prediction; however, excessive polynomial order 

for a relatively small database may worsen data interpolation. The MPR models presented in this work 

have an advantage in that they have a simple structure and can be programmed easily. In addition, they 

can be implemented for any other experimental database with little modification, because they have no 

physical basis. 

We used outdoor dry-bulb temperature (TOD), indoor dry-bulb temperature (TID), and indoor dew-point 

temperature (TIDP) as independent variables. These variables were regressed upon the database generated 

from the fault-free tests. Equations (4.1a), (4.1b), (4.1c), and (4.1d) show the general form of the 

regressed equations for the i

th feature (or ith dependent variable) as 1st, 2nd, 3rd, and 4th order MPR models, 

respectively. 

(1) φi =+ + + a aT aT aT 0 1 OD 2 ID 3 IDP (1st order) (4.1a) 

(2) (1) 22 2

i i 4 OD 5 ID 6 IDP 7 OD ID 8 ID IDP 9 IDP OD φ φ =+ + + + + + aT aT aT aT T aT T aT T (2nd order) (4.1b) 
30

(3) (2) 33 3

10 OD 11 ID 12 IDP 13 OD ID IDP

22 22 2 2

14 OD ID 15 OD IDP 16 ID OD 17 ID IDP 18 IDP OD 19 IDP ID

i i aT aT aT aT TT

aT T aT T aT T aT T aT T aT T

φ φ =+ + + +

++ + + + +

 (3rd order) (4.1c) 

(4) (3) 4 4 4 22 2 2 22

20 OD 21 ID 22 IDP 23 ID OD 24 ID IDP 25 IDP OD

3 3 33 3 3

26 ID OD 27 ID IDP 28 IDP OD 29 ID OD 30 ID IDP 31 IDP OD

22 2

32 ID OD IDP 33 ID OD IDP 34 ID OD IDP

i i aT aT aT aT T aT T aT T

aTT aTT aT T aT T aT T aT T

aT T T aTT T aTT T

φ φ =+ + + + + +

++ + + + +

++ +

 (4th order) (4.1d) 

4.3 Artificial Neural Network (ANN) Reference Model 

An Artificial Neural Network (ANN) model was developed for the seven features. The relationship of 

independent variables and features is learned by an artificial neural network using a back propagation 

algorithm (Wasserman, 1989; Hassoun, 1995). Figure 4.2 shows the structure of the ANN used in this 

study. It has three input variables (TOD, TID, and TIDP) and one output. This neural network has three 

layers consisting of an input, hidden, and output layer with the input and hidden layers having three 

nodes. The sigmoid function is used as the activation function of the hidden layer. The weight 

coefficients and offsets are “learned” using a momentum back propagation algorithm through more than 

10,000 iterations. 

The input layer acts only as input nodes; no processing of TOD, TID, or TIDP occurs within the input layer. 

The various arrows between the input layer and the hidden layer indicate weights, or multipliers, applied 

to each input variable before passing to the sigmoid function within each hidden layer node. Equation 2 

illustrates the output of a neuron, f(s), and how the sigmoid function is applied within the layers. 

f ( )s ( ) s s x w c

k

k k ⎟ +

⎠

⎞ ⎜

⎝

⎛ = + = ∑=

3

1

1 tanh(2 ) 2

1 (2) 

The three-node hidden layer uses 9 adjustable weighting coefficients and 3 offset coefficients. The single 

output node has its own 3 adjustable weighting coefficients and a single offset to produce the final 

predicted value of the feature, Fmodel. Each heat pump feature to be represented by the ANN is calculated 

in this manner using TOD, TID, or TIDP as inputs. The predicted value is compared to the measured value to 

produce an error value (residual) for each feature. The back propagation algorithm is then used to adjust 

the weights and offsets to minimize the error, or “train” the ANN. 

Figure 4.2. Artificial neural network structure 
31

In validating the model, we used a dataset of 111 points, a subset of the full 5830 point dataset, to 

improve the performance of the ANN learning process. The 111 point dataset was created by selecting 

the data points that were spaced by a minimum predetermined distance in the independent variables space. 

First, TID , TOD , and TIDP were mapped onto xyz-coordinates. Each (x, y, z) point was then compared to 

all of the other points to determine if any of the other points fell within the predetermined distance of the 

point. This is equivalent to drawing a sphere of radius r at a particular (x, y, z) location and examining 

this sphere volume to determine if any other points fall within. If another (x, y, z) point fell within this 

sphere, it was removed from the data set; thus all of the independent variable (x, y, z) points were 

examined and winnowed in this manner. In this study, r was selected to be 0.96 °C. 

Table 4.2 shows the mean squared residual (MSR), as calculated by Eq. 4.3, for the multivariate 

polynomial regression models and ANN model when fit to the reduced dataset consisting of 111 points. 

The mean squared residual is the sum of the squared residuals divided by the degrees of freedom for the 

regression and is an estimate of the model variance (Graybill and Iyer, 1994a). 

( ) { }

( )

2

MSR

N m1

n

i i

i

x −φ

= − +

∑ (4.3) 

Table 4.2. MSR of the models fit to a reduced dataset for the seven selected features 

Feature 1st order 2nd order 3rd order 4th order ANN 

TE (°C) 1.298 0.095 0.052 0.016 0.122 

Tsh (°C) 0.980 0.442 0.222 0.140 0.263 

TC (°C) 0.136 0.014 0.007 0.005 0.072 

TD (°C) 2.215 0.373 0.246 0.110 1.341 

Tsc (°C) 0.198 0.139 0.081 0.036 0.395 

ΔTEA (°C) 1.431 0.087 0.030 0.012 0.159 

ΔTCA (°C) 0.106 0.019 0.012 0.009 0.024 

N 111 

m+1 4 10 20 35 16 

Table 4.3 shows the MSRs for the models fit to the full dataset of 5830 points while Table 4.4 shows the 

MSRs for the reduced dataset models applied to the full dataset. The reduced models’ MSRs of Table 4.4 

and the full dataset MSRs of Table 4.3 differ by an average of 26 %. Thus the reduced dataset is a good 

representation of the system features, but the full dataset model produces smaller MSRs. 

Table 4.3. MSR of the models fit to the full dataset for the seven selected features 

Feature 1st order 2nd order 3rd order 4th order ANN 

TE (°C) 0.979 0.072 0.051 0.015 0.129 

Tsh (°C) 1.055 0.397 0.176 0.121 0.273 

TC (°C) 0.108 0.011 0.007 0.005 0.068 

TD (°C) 2.078 0.342 0.213 0.088 1.403 

Tsc (°C) 0.202 0.135 0.066 0.028 0.480 

TEA (°C) 1.103 0.066 0.027 0.011 0.159 

TCA (°C) 0.085 0.018 0.014 0.012 0.030 

N 5830 

m+1 4 10 20 35 16 
32

Table 4.4. MSR of the reduced dataset model applied to the full dataset of seven selected features 

Feature 1st order 2nd order 3rd order 4th order ANN 

TE (°C) 0.992 0.077 0.061 0.023 0.129 

Tsh (°C) 1.160 0.465 0.258 0.213 0.273 

TC (°C) 0.110 0.012 0.0080 0.0078 0.068 

TD (°C) 2.403 0.409 0.285 0.161 1.407 

Tsc (°C) 0.213 0.144 0.083 0.055 0.482 

TEA (°C) 1.116 0.072 0.033 0.017 0.160 

TCA (°C) 0.087 0.019 0.017 0.019 0.030 

N 5830 

m+1 4 10 20 35 16 

As expected, a higher order MPR model produces a smaller mean squared residual. However, the 

number of model coefficients increases exponentially due to the addition of the crossterm coefficients. 

The number of coefficients used to model each feature may be reduced by applying an F-Test to each 

coefficient of the respective models (Graybill and Iyer, 1994b). The F-statistic is calculated using the 

following equations: 

( )

m - g

SSR SSR MS reduced full

drop

− = (4.4) 

N - ( ) m 1

SSR MSR full

full

+ = (4.5) 

full

drop

MSR

MS

F = (4.6) 

where g+1 is the number of coefficients in the reduced model. The F-statistic follows an F distribution 

with m-g and N-(m+1) degrees of freedom. Large values of F indicate that the terms removed from the 

reduced model were significant. One may use the F-statistic as a means of ranking the contribution of a 

particular coefficient to the fit of the regressed model. By dropping one term at a time and sorting the 

reduced models in terms of their F-statistics, the effect of removal of a particular term may be assessed by 

comparing the MSR of the reduced model to that of the full model. Table 4.5 shows the results of this 

technique when applied to a backward elimination on the full 3rd order polynomial model. 
33

Table 4.5. F-statistic and percent change in MSR with respect to the full models for seven features 

3rd Order MPR Model Backward Elimination, x = TOD, y = TID, z = TIDP

Feature Less than 1 % 

higher MSR 

Less than 5 % 

higher MSR 

Less than 10 % 

higher MSR 

Term(s) removed 

TE z, xy, z

2

 z, xy, z

2

, x

2

, y, y3 N/A 

Tsh y

2

, z, z

2

, y

2

z, x y

2

, z, z

2

, y

2

z, x, y, z3 y

2

, z, z

2

, y

2

z, x, y, 

z

3

, y

2

x, y3

TC y y, y

2

z, y3 y, y

2

z, y3, yz, z3

TD x, y3 x, y3

, yz, x

2

y, z

2

y, 

xyz 

x, y3, yz, x

2

y, z

2

y, 

xyz, z

2

Tsc y, z

2

, z

2

y, y3 y, z

2

, z

2

y, y3, y

2

x, 

yz, z

2

x 

N/A 

ΔTEA y N/A y, z 

ΔTCA x

2

z, y, z3 x

2

z, y, z3, y3, z, x

2

y, 

zx 

N/A 

N/A: The % change in MSR does not fall within the 

bounds with the removal of a single coefficient. 

Table 4.5 shows that one to five terms may be removed from the full 3rd order polynomial while the 

models’ MSRs remain less than 1 % from the full models’ MSRs. If the criteria is raised to within 5 % of 

the full models’ MSRs, up to seven terms may be removed from several of the 3rd order polynomials. 

Raising the percentage change in MSR to within 10 % allows the removal of up to nine coefficients for 

Tsh. The decision as to how many terms to remove is at the discretion of the model developer, but as with 

any regression equation, the confidence interval on the mean value of any of the features determined at a 

particular value of the independent parameters will be larger for larger MSRs. The confidence interval is 

a function of the MPR model standard residual ( MSR ), the Student’s t-value for the particular 

confidence level, and degrees of freedom (Ott 1984). 

Figure 4.3 shows the performance of the full MPR models and ANN model during operation at a TOD of 

27.8±0.3 °C. If an air conditioning system is installed in the field, TID may change continuously 

according to indoor cooling load or thermostat settings. The three features, Tsh, Tsc, and TD, are shown in 

Figure 4.3 because they varied the most as indoor temperature changed. In Figure 4.3(a), where there is 

an abrupt change in TID, the steady state of the system is broken, as indicated by the steady-state detector 

in Figure 4.3(e). 

Figures 4.3(b), 4.3(c), and 4.3(d) show Tsh, Tsc and TD as predicted by the 1st, 2nd, and 3rd order MPR 

models and the ANN model. The 3rd order MPR model shows the best fit to the measured data during 

steady-state operation. As the order of the polynomial model decreases, the fit to the experimental data 

set degrades. Predictions by the ANN model are worse than those by the 3rd order MPR model. The Tsh

predicted by the ANN model is comparable to that of the 3rd order MPR model (Figure 4.3(b)), but Tsc

was not predicted well by the ANN model, as shown in Figure 4.3(c). The predicted values of TD for the 

ANN model are between the 1st and 2nd order MPR models, as shown in Figure 4.3(d). 
34

Figure 4.3. Performance of MPR models and ANN model to predict features during a sample operation 

period; (a) TID; (b) Tsh; (c) Tsc; (d) TD; (e) steady state 
35

4.4 Sensitivity of the Reference Model to the Steady-State Detector 

4.4.1 Effect of the steady-state detector threshold values on the reference model 

Data used to generate the reference model was taken by changing indoor or outdoor temperatures in a 

stepwise manner and holding the new conditions for a fixed time interval. If we had selected a very strict 

definition of steady state (small threshold values such as a fraction of one standard deviation of the 

important features), the time required to reach steady state would have increased. During the stepwise 

changes in conditions, a given temperature state was only maintained for 30 to 45 minutes before moving 

on to the next condition. A stricter more sensitive steady-state detector would have indicated steady state 

for a shorter period of time, thus limiting the number of data points collected for the given time period. 

The number of data points collected along with their variance from their respective means would be 

greatly affected by this change in threshold values and thus influence the reference model’s dataset. A 

strict, small threshold steady-state detector produces fewer data points with lower variance, while a less 

strict steady-state detector would produce more data points with higher variance. 

The threshold values of standard deviation for the monitored system features define the performance of 

the steady-state detector. The smaller a threshold is the more stable the system must be before steady 

state is declared. On the other hand, if a threshold is too small more time is required to collect reference 

data, and this small threshold value may inhibit data collection. Large thresholds increase the variability 

of the reference model and the effectiveness of the FDD system. Therefore, it is necessary to find 

thresholds that minimize the uncertainty and variability of the reference model while maximizing the use 

of experimental data. 

To investigate the relation between reference model and threshold values, we compared the various 

reference models as a function of various threshold values. To do this, the system reference model was 

re-calculated after all data were filtered through the steady-state detector using different thresholds. 

Specifically, we developed the 2nd order polynomial reference model from the steady-state data 

determined by the steady-state detector with (0.06, 0.17, 0.28, 0.56, and 1.11) °C ((0.1, 0.3, 0.5, 1.0, and 

2.0) o

F) thresholds for all features. Then, we calculated the standard deviation between the steady-state 

data and predicted data for all features. 

( )

N

Predicted value - Measured value

Standard Deviation 

2

= ∑ (4.3) 

Figure 4.4 shows the polynomial reference model standard deviation and data extraction percentage for 

each system feature. The x-axis represents the threshold for the seven features. As the threshold of the 

steady-state detector increases, the standard deviations of TD, Tsh, TE, and Tsc pass through an optimal 

point. Discharge temperature and superheat show this tendency clearly. In this figure, we can see that the 

optimal threshold exists around 0.17 °C (0.3 °F) for TD, Tsh, TE, and Tsc. Below 0.17 °C (0.3 °F), the 

uncertainty worsens due to the increase of model residuals caused by excessive filtration by the steadystate detector. Considering both data extraction rate and uncertainties of the thermocouple sensors, the 

thresholds should be set between 0.11°C to 0.28 °C (0.2 °F to 0.5 °F). This figure compares well to the 

±3σ thresholds listed in Table 3.3 which indicates that the selection of thresholds that at three times larger 

than the steady-state standard deviation produces a near minimum model standard deviation (as calculated 

by Equation 4.3) while at the same time yielding acceptable use of the available data. 
36

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Steady-state detection threshold (o

C)

Standard deviation (oC)

0

10

20

30

40

50

60

70

80

90

Data extraction rate (%)

DTca DTea

Te dsh

Td Tc

dsc Data No

ΔTCA

TE

TD

Tsc

ΔTEA

Tsh

TC

Data extraction rate

Figure 4.4. Standard deviation for the seven features of the no-fault system reference model and data 

extraction rate as a function of the steady-state detector thresholds (sampling interval: 18 

seconds, moving window size: 144 seconds) 

4.4.2 Example of the steady-state detector applied to the experimental heat pump 

Figures 4.5(a) and 4.5(b) show sample feature residuals as a function of indoor temperature and relative 

humidity. In Figure 4.5, all of the non steady-state data having large deviation from the reference were 

filtered out. For both figures, zone A represents the non steady-state data filtered out during the startup 

transient. In the low humidity area of Figure 4.5(b), marked as zone B or ‘TXV hunting,’ highly 

fluctuating data were filtered out by the steady-state detector. Since the evaporator capacity is reduced 

for dry inlet air conditions, refrigerant exiting the evaporator oscillates between saturated and superheated. 

These abrupt condition changes at the evaporator exit cause TXV hunting. When the TXV hunting 

occurs, other temperatures, especially on the evaporator side, also oscillate within a large range. The nofault model would thus estimate the parameters inaccurately due to a lack of steady-state data in that 

range. Once the model estimations are acceptable for other conditions, there is no reason to set the 

steady-state thresholds so low as to filter all the oscillating data out. By including some data in this 

oscillating zone within a no-fault model, we can obtain a model that covers a wider range of operating 

conditions. 
37

-10

-5

0

5

10

5 10 15 20 25 30 35

Steady-state

Non steady-state

Δ

T

sh,exp

 - 

Δ

T

sh,model (oC)

T

ID (o

C)

(A)

(a) Residual of evaporator exit superheat versus indoor temperature 

(b) Residual of evaporator exit saturation temperature versus indoor relative humidity 

Figure 4.5. Results filtered by the steady-state detector on the plots with feature residuals and 

independent variables (zone A: startup transient, zone B: TXV hunting) 
38

4.5 Conclusions on System Reference Model 

Seven features of a residential heat pump system were modeled using the 1st, 2nd, 3rd, and 4th order MPR 

models and an ANN model. The laboratory data were filtered by a steady-state detector, which 

automatically examined and processed data to improve data collection consistency and the resulting 

reference model for steady state. When we consider the accuracy of a reference model and usefulness of 

field data, the 3rd order polynomial model provides a reasonable fit uncertainty. The F-Test is one method 

of reducing the number of coefficients in a general linear model, and this technique may aid a developer 

in creating a more compact representation of the system features. The ANN model predicted all features 

with less accuracy than the 3rd order MPR model. Because of insignificant non-linearity between the 

independent variables and the features in the cooling mode, the ANN model did not produce lower MSRs 

than the best MPR models. 

The reference model should not be over-specified by increasing the order of the MPR model. The model 

variance (MSR) should be lower than that associated with the measurement uncertainty of the 

independent variables. The smaller the models’ variances, the smaller their contributions to the overall 

uncertainty of the predicted features. The model developer must decide on an acceptable level of model 

variance (associated with model order) based upon his FDD requirements and his ability to measure the 

variables. 

In general, any order MPR model is a “black box model” and may be applied to any unknown system. 

When the system is changed slightly due to aging, a rigid non-physical model cannot accurately predict 

performance, which will cripple the FDD system. Although the model may be updated by ongoing 

commissioning, a model with semi-physical concepts can be easily adjusted to compensate for gradual 

system changes. Models that can adapt to changing system conditions must be developed. It would be 

advantageous to revise an existing model using up-to-date information, which would eventually evolve 

into a model capable of learning and self-tuning during normal operation, yet still ensure that the system 

remained within the manufacturer’s performance limits. 
39

CHAPTER 5. STATISTICAL METHODOLOGY FOR FAULT DIAGNOSIS 

USING A RULE-BASED APPROACH 

The concept of a rule-based FDD system originated from expert system theory. Typically, knowledgebased expert systems are designed to solve problems that would normally be tackled by a human expert. 

Though the application domain of expert systems might be narrow and restricted, it provides experienced 

aid to developers (Tzafestas, 1989). 

Since expert systems originated from human experts’ knowledge, most of the related schemes can be 

interpreted by rules such as IF, THEN, ELSE algorithms. Based on this interpretation, there are a number 

of different methods for rule-based algorithms. Fault tree methods (Lee et al., 1985), fuzzy theory-based 

techniques (Krandel, 1992), and signed directed graph (SDG) methods (Kramer and Palowitch, 1987) 

were examined by several investigators. 

A rule-based approach has various advantages, including the transparency, flexibility, and adaptability 

obtained by expressing the knowledge base as a set of rules. Commissioning engineers prefer using the 

rule-based algorithm because it helps with intuitive judgment of the system status. The rule-based 

algorithm is simple, so it is convenient to develop new rules for an unknown object system. Furthermore, 

the rule-based module is suitable to run on an embedded device like a microprocessor since it requires a 

relatively small computational capacity and memory requirement (Schein and Bushby, 2005). If FDD for 

HVAC systems are to be affordable, rule-based approaches are of great benefit because they require low 

computing overhead and are relatively inexpensive to implement. 

5.1 Fault Influences on System Parameters 

A handful of researchers have attempted rule-based algorithms for identifying vapor compression system 

faults. Since the pattern of system response changes with the system configuration, the researchers 

provided their unique rule-based charts for every corresponding system. 

Rossi and Braun (1997) provided a rule-based chart of seven feature residuals with regard to five different 

faults of a rooftop air conditioner. The rooftop system was equipped with a constant speed compressor 

and a short tube expansion device. The authors merged the rule-based knowledge into a statistical pattern 

recognition technique in order to develop a fault detection classifier. Assuming the individual 

temperature measurements being independent, the type of each fault was classified. Breuker and Braun 

(1998b) modified the rule-based chart of the same system based on their experimental verification. 

Castro (2001) tested a 41 kW (12 ton) air-cooled liquid chiller with a constant speed, two-stage 

reciprocating compressor. Twelve performance parameters were originally monitored to track system 

performance. Eight representative parameters were selected to build a rule-based chart. The author 

calculated Euclidean distances of the current state based on the selected two largest residuals, and 

estimated the possibility of a fault from the distance information. After detecting a fault, the data were 

input to the rule-based fault diagnosis algorithm. 

Kim and Kim (2005) tested a water-to-water heat pump system with an electronic expansion valve (EEV) 

driven by an open-type variable-speed compressor. Depending on the cooling load, the compressor was 

operated in two modes, variable speed or constant speed. The authors provided two separate rule-based 

charts depending on the operating mode. 

Table 5.1 lists rule-based charts from previous investigations on: (a) a rooftop air conditioner with a short 

tube orifice and a constant-speed compressor; (b) a water-to-water heat pump with an Electronic 
40

Expansion Valve (EEV) and a constant-speed compressor; and (c) a heat pump with an EEV and a 

variable-speed compressor. The indices of features represent three rules; up-arrows (increased residual), 

down-arrows (decreased residual), and no-change (neutral residual). 

Table 5.1. Comparison of rule-based patterns with regard to three different configurations of vapor 

compression systems 

(a) Rooftop air conditioner with a short tube orifice and constant-speed compressor1

 (no controller) 

Fault Type TE Tsh TD TC Tsc ΔTCA ΔTEA

Compressor valve leakage Ç È Ç È È È È

Refrigerant leakage È Ç Ç È È È È

Liquid-line restriction È Ç Ç È Ç È È

Condenser fault Ç È Ç Ç È Ç È

Evaporator fault È È È È È È Ç

1

 Breuker and Braun (1998b) 

(b) Water-to-water heat pump with an EEV and constant-speed compressor2

 (one controller) 

Fault Type TE Tsh TD TC Tsc ΔTCW ΔTEB N

Compressor valve leakage Ç – Ç È È È È – 

Refrigerant leakage < 15% – – – – È È Ç – 

> 15% È Ç Ç È È È – – 

Condenser fault È – Ç Ç – È È – 

Evaporator fault È – – È Ç È È – 2

 Kim and Kim (2005) 

(c) Water-to-water heat pump with an EEV and variable-speed compressor3

 (two controllers) 

Fault Type TE Tsh TD TC Tsc ΔTCW ΔTEB N

Compressor valve leakage – – Ç – – – – Ç

Refrigerant leakage < 15% – – – – È – – Ç

> 15% Ç È Ç Ç È È È – 

Condenser fault – – Ç Ç È – – Ç

Evaporator fault È – Ç – Ç – – Ç

3

 Kim and Kim (2005) 

Table 5.1(a) does not contain a neutral (NC (-)) index, which implies that a system response is directly 

associated with a fault regardless of the fault type. However, for the fixed orifice system (Table 5.1(a)) 

compared to an EEV-equipped system (Table 5.1(b)), many of the positive or negative indices become 

neutral indices. When compressor speed control is added, additional rules become neutral indices (see 

Table 5.1(b) and (c)). 

If we pair the EEV and the variable-speed compressor as a single-controller unit, Table 5.1(a) can be 

regarded as a rule-based chart of a system with no controller, Table 5.1(b) for a single-controller system, 

and Table 5.1(c) for a double-controller system. As the number of controllers increases, the variability of 

the system features decreases. When a system changes due to an external influence, faults, or controller 

noise, controllers attempt to adjust the system to maintain the original status. Thus, controllers reduce 

system variability creating a much more fault-tolerant system (more neutral indices) (Kim and Kim, 2005). 
41

The work done during this investigation shows that the addition of the extra control provided by a TXV 

does in fact create a more fault-tolerant system, and thus produces more neutral indices in the feature set. 

Table 5.2 shows the feature set and their changes with faulty operation for the faults imposed for the 

current cooling mode investigation. This added variety within the feature set can be directly compared to 

that of the fixed orifice expansion device system of Table 5.1(a). 

Table 5.2 Rule-based fault pattern chart of seven features 

Fault Type zone1 TE Tsh TD TC Tsc ∆TCA ∆TEA

Compressor valve leakage2 A Ç – Ç È È È È

A Ç – Ç Ç È – È Improper outdoor air flow B È Ç Ç Ç È È È

Improper indoor air flow2 A È – – È – È Ç

A – – – – – – – Liquid-line restriction3

B È Ç Ç È Ç È È

A – – – È È È È Refrigerant undercharge B È Ç Ç È È È È

Refrigerant overcharge2 A – – Ç Ç Ç – – 

No-fault – – – – – – – – 

1

 Zone A: Tsh < 9 oC, Zone B: Tsh > 9 oC 

2

 The case of Tsh > 9 oC was not observed within the fault levels of this study. 

3

 The rules when Tsh < 9 oC were identical with no-fault cases. 

Compared to the rule-based chart in Table 5.1(a), the rule-patterns for a liquid-line restriction and 

refrigerant undercharge fault were the same as Breuker and Braun (1998b) at a high fault level in zone B. 

Considering that they tested a rooftop air conditioner with a short tube orifice, this tends to indicate that 

the TXV is working like a fixed area expansion device; the TXV is saturated or at its control limits. 

However, the improper outdoor air flow fault rule-pattern in zone B is different from that of a condenser 

fault in Table 5.1(a), both of which were simulated by blocking the flow passage through the condenser. 

This is because the air inflow area of the outdoor unit was blocked from the bottom in this research, but 

Breuker and Braun (1998b) blocked the condenser randomly. 

5.2 Classification of Feature Rules 

5.2.1 Statistical rule-based classification for two cases; positive or negative 

residuals 

Figure 5.1 shows two probability distributions corresponding to the current measurement and the no-fault 

model’s estimate. The current mean, μi, and standard deviation, σi, are calculated in the moving window 

of on-going measurements. The estimated mean, μi,NF, and standard deviation, σi,NF, of the i

th feature are 

determined from the no-fault model and associated uncertainty, respectively. The difference in the two 

means, μi – μi,NF, is defined as the residual of the feature, ri, in Equation 5.1. 

i i,NFi r = μ − μ (5.1) 

Assuming that the features are random variables, the estimated features from the no-fault model can be 

assumed to obey a Gaussian probability distribution, N (μi.NF, σi.NF

2

). Based on the probability 

distributions and the sign of the residual (positive Ç or negative È), the probability of the feature being 
42

higher or lower than the reference model prediction can be estimated in a statistical fashion. Looking at 

Table 5.1(a), a set of seven features establishes a 7-dimensional multivariate Gaussian probability 

distribution. With the assumption that each dimension is independent, the probability corresponding to a 

given fault type can be calculated from a single variable Gaussian distribution (Rossi and Braun, 1997). 

Though the independent assumption was reported to be less accurate compared to other schemes that do 

not force independence (Li and Braun, 2003), the assumption of independence is still applicable in this 

type of FDD analysis, or in other words we may ignore any cross-correlation, which simplifies this 

analysis. 

( ) 2

,NF ,NF , N μ σ i i

Measured feature No-fault model

Residual, ri

PDF , (μi i σ )

μi μi,NF xi

Figure 5.1. Probability density functions (PDF) of i

th feature, xi, for the current measurement and the nofault model prediction 

Equation 5.2 defines the probability of seeing the j

th fault, P(Fj | X), corresponding to the set of m

independent variables, X, for the current measurements. 

( ) ( )( ) ( ) ( ) 11 2 2

1

||| | |

m

j j j ij i ij i

i

PF PC X PC X PC X PC X =X =⋅ = L ∏ (5.2) 

P(Cij | Xi) denotes the individual probability of seeing case variable C (positive Ç or negative È) for the 

i

th feature (Tsh, Tsc, ΔTEA, etc.) with the jth fault type. Since the features are assumed to be independent, 

the total conditional probability, P(Fj | X), can be obtained by the multiplication of individual P(Cij | Xi)’s. 

Rossi and Braun (1997) assumed that the current distribution follows a Gaussian pattern and suggested 

P(Cij | Xi) be represented by the form of Equation 5.3. 

( )

( )

,NF

,NF ,NF

1 1 | 1 erf 1 erf 2 2 2 2

1 if 0, where

1 if 0.

i i i

ij i i i

i i

i

i i

i

r PC X a a

r

a r

r

μ μ

σ σ

⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛⎞ − =+ =+ ⎢ ⎥⎢ ⎥ ⎜ ⎟ ⎜⎟

⎣ ⎦⎣ ⎦ ⎝ ⎠ ⎝⎠

⎧+ > = ⎨

⎩− ≤

 (5.3) 

Standard deviation σi,NF is the square root of the i

th diagonal component in the covariance matrix of the 

no-fault model. The set of rules in Table 5.1(a) consists of positive Ç or negative È discrete values. 

Noting that an up-arrow rule implies positive ri and a down-arrow rule implies negative ri, the constant ai
43

is assigned a value of 1 for positive ri or –1 otherwise. Since Equation 5.3 can be applied only to rules 

with two classes (positive Ç or negative È), it is necessary to evaluate the neutral (– or NC) class in a 

different way. 

5.2.2 Statistical rule-based classification with three case indices; positive, negative, 

or neutral 

A set of measurement features can be regarded as a multi-dimensional Gaussian probability distribution. 

In contrast to the rules of Rossi and Braun in Section 5.2.1, the neutral index (– or NC) is required for the 

system features shown in Table 5.1(b) and (c) and Table 5.2. When we use the assumption that each 

dimension is independent, we can extend the statistical rule-based classification to a three-class problem 

(positive Ç, negative È, or neutral –) with each class being represented by a simple Gaussian distribution 

rather than a more complicated distribution with multiple degrees of freedom and dependencies. 

When the operation is controlled by feedback controllers, a system will adjust itself based upon the 

control variable. Since the active control reduces system variation, many features will not change clearly 

enough to apply a two-rule diagnosis. Therefore, researchers using rule-based charts, in several 

investigations for more complicated systems, added a neutral rule resulting in a three-class variation 

(positive, negative, neutral). Applying different rules for different systems as seen in the previous 

researches, allowed us to more easily find the appropriate FDD rules for the system. 

A conventional way of classifying residuals as positive, negative, or neutral is by enumerating the 

probable faults based on belief functions. For example, Figure 5.2 illustrates a Boolean and a smoothed 

belief function, b(Cij | Xi), of the i

th feature with regard to residual ri. P(Ri) is the Gaussian distribution of 

residual ri. Boolean classification results in a case of 0 or 1 depending on the value of residual ri and the 

given threshold εi. When measurement noise produces random fluctuation in the signal around the 

thresholds, +εi or −εi, a Boolean classification returns highly unstable belief values rapidly fluctuation 

between 0 and 1. To solve this problem, later investigators implemented smoothed belief functions to 

alleviate the diagnostic instability of Boolean classification. The smoothed belief function in Figure 5.2 

represents a sigmoidal type suggested by Kramer (1987). The factor 1 − ui (one minus the uncertainty in 

ri) permits the introduction of uncertainty into the belief function. Although the Boolean belief function, 

b(Cij | Xi), is similar conceptually to the probability function P(Cij | Xi) in Equation 5.2, they should be 

kept separate since belief functions were derived empirically and have only a weak background in 

probability theory. 
44

( ) | bC X ↓ i

Boolean belief 

function

Smoothed belief 

function

−ε i +ε i

( | ) bC X ↑ i

ri

Belief

1.0

1.0 − i u

( ) P Ri

0

Figure 5.2. Example of a three case classification using a Boolean and a smoothed belief function with 

regard to the residual ri and its Gaussian probability distribution P(Ri) for a given threshold 

on ri of ±εi (Kramer 1987) 

When we assume that the measured signal is a normally distributed random variable with zero mean and a 

given variance, σi

2

, the three cases (È,Ç, and −) can be estimated using standard statistics. The chisquared (χ2

) probability distribution gives the probability of the squared sum of the standardized multidimensional variables that have normal independent distributions with zero-mean and unity variance. The 

number of variables is the degree of freedom. When the degree of freedom is one, the probability 

determined by a chi-square distribution is identical to a normal distribution. For the current research, the 

seven features, with the assumption of independence, provide seven individual chi-square distributions 

with 1 degree of freedom. Thus, statistical inferences can be based on a Gaussian distribution. 

Figure 5.3 represents the probability for three cases of (a) negative (È), (b) positive (Ç), and (c) neutral 

(−). The conditional probability, P(Cij | Xi), is shown as the cross-hatched area depending on the 

corresponding rule. The probability for each rule was described in Equation 5.3. 

( )

( )

( )

|

|

|

↓

↑

−

i

i

i

PC X

PC X

PC X

 (5.4) 

If P(x) is a Gaussian distribution, P(xi < ai) can be expressed in an explicit form by Equation 5.5. 

( ) ,NF

,NF

1 1 erf

2 2

i i

i i

i

a

Px a μ

σ

⎡ ⎛ ⎞ − ⎤

<= + ⎢ ⎜ ⎟⎥ ⎢ ⎥ ⎣ ⎝ ⎠⎦

 (5.5) 

where the error function, erf, is defined as follows: 

( ) 2

0

2 erf

π

− = ∫

z t z e dt (5.6) 
45

( | ) PC X ↓ i

μi μ + ε μ i i − ε i i xi

PDF , (μi i σ )

( ) 2

,NF ,NF , N μ σ i i

μi,NF

(a) Probability of negative (È) case 

( | ) PC X ↑ i

μi μ + ε μ i i − ε i i xi

PDF , (μi i σ )

( ) 2

,NF ,NF , N μ σ i i

μi,NF

(b) Probability of positive (Ç) case 

( ) 2

,NF ,NF , N μ σ i i

PDF , (μi i σ )

PC X ( − | i)

μi i − ε μi μi i + ε μi,NF xi

(c) Probability of neutral (–) case 

Figure 5.3. Estimation of conditional probability of the i

th feature with regard to the current measurement 

complying with individual rules. The subscript of the case variable C corresponds to the 

rules in Table 5.2 and represents a (a) negative change, (b) positive change, or (c) no change 
46

From Equation 5.5 and Figure 5.3, the three probability terms in Equation 5.4 can be evaluated. Note that 

the cross-hatched area on the right side of the current distribution represents the negative change 

conditional probability since the current distribution produces a negative residual. The probabilities are 

expressed as functions of the threshold εi. 

( ) ( )

,NF

1 | 1 erf 2 2

i i

i i ii

i

r PC X Px ε

μ ε

σ

↓

⎡ ⎛ ⎞⎤ +

= ≥+ = − ⎢ ⎜ ⎟⎥ ⎜ ⎟ ⎢ ⎥ ⎣ ⎝ ⎠⎦

 (5.7a) 

( ) ( )

,NF

1 | 1 erf 2 2

i i

i i ii

i

r PC X Px ε

μ ε

σ

↑

⎡ ⎛ ⎞⎤ − = ≤− = + ⎢ ⎜ ⎟⎥ ⎜ ⎟ ⎢ ⎥ ⎣ ⎝ ⎠⎦

 (5.7b) 

( )( )

,NF ,NF

1 | erf erf 2 2 2

ii ii

i ii i ii

i i

r r PC X P x ε ε

με με

σ σ −

⎡ ⎛ ⎞⎛ ⎞⎤ + −

= −<< + = − ⎢ ⎜ ⎟⎜ ⎟⎥ ⎢ ⎥ ⎣ ⎝ ⎠⎝ ⎠⎦

 (5.7c) 

The summation of Equations 5.7a, 5.7b, and 5.7c equals unity. 

( ) | | |1 ( ) ( − ) ↓ ↑ PC X PC X PC X iii + + = (5.8) 

5.2.3 Decision rule index and building a rule-based chart 

In our analysis a non-faulty, current measurement is assumed to show the same variation as the no-fault 

steady-state reference experiments. 

2 2 σ σ i i NF ≈ , (5.9) 

Let the measurement threshold, εi, be the multiplication of the measurement standard deviation, σi, by a 

positive constant s.

() , 0 i i ε sss = σ > (5.10) 

Let the standardized measurement residual be represented by zi. 

i i

i

i

x

z μ

σ

− = (5.11) 

Substituting Equations 5.9 to 5.11 into Equations 5.7 yields more compact definitions of the Probability 

Distribution Functions (PDF) for the standardized measurement residual of the i

th feature. 

( ) ( ) 1 1 | 1 erf 2 2

i

i i

i

z PC Z Pz s s

σ ↓

⎡ ⎧ ⎛ ⎞⎫⎤ ⎪ ⎪ = ≥= − + ⎢ ⎨ ⎜ ⎟⎬⎥ ⎢⎣ ⎪⎩ ⎭ ⎝ ⎠⎪⎥⎦

 (5.12a) 

( ) ( ) 1 1 | 1 erf 2 2

i

i i

i

z PC Z Pz s s

σ ↑

⎡ ⎧ ⎛ ⎞⎫⎤ ⎪ ⎪ = ≤= + − ⎢ ⎨ ⎜ ⎟⎬⎥ ⎢⎣ ⎪⎩ ⎭ ⎝ ⎠⎪⎥⎦

 (5.12b) 

( )( ) 11 1 | erf erf 2 2 2

i i

i i

i i

z z PC Z P s z s s s

σ σ −

⎡ ⎧ ⎛⎞ ⎛⎞ ⎫⎧ ⎫⎤ ⎪ ⎪⎪ ⎪ = −< < = + − − ⎢ ⎨ ⎜⎟ ⎜⎟ ⎬⎨ ⎬⎥ ⎢⎣ ⎩ ⎭⎩ ⎭ ⎪ ⎝⎠ ⎝⎠ ⎪⎪ ⎪⎥⎦

 (5.12c) 
47

Figure 5.4 shows probability distributions of the above three cases when the measurement threshold 

standard deviation multiplier, s, has values of 1 and 2. With a zero residual (zi = 0), the no-change 

probability has a maximum value but the probabilities for up-change and down-change are non-zero. 

Compared with Figure 5.2 where there was no representation of a no-change case, the no-change case can 

be evaluated based on the value of zi and its accompanying threshold εi ( = sσi,NF). Depending on the 

value of the residual zi and threshold εi, the probability of a feature’s three possible states can be classified 

into one of the three categories as seen in Figure 5.4. 

0

0.2

0.4

0.6

0.8

1

1.2

-6 -4 -2 0 2 4 6

Normalized residual, zi

Conditional probability, 

P(Cij | Zi )

P(CÈ| Zi)

P(C− | Zi

)

P(CÇ| Zi

)

− ɛs= 1 + ɛs= 1

(a) P(Cij | Zi) at s = 1 

0

0.2

0.4

0.6

0.8

1

1.2

-6 -4 -2 0 2 4 6

Normalized residual, zi

Conditional probability, 

P(Cij | Zi )

P(CÈ| Zi) P(C− | Zi

)

P(CÇ| Zi

)

− ɛs= 2 + ɛs= 2

(b) P(Cij | Zi) at s = 2 

Figure 5.4. Classification probability functions for positive change, negative change, and neutral change 

with regard to the measurement threshold standard deviation multiplier, s=1 and s=2. 
48

When we build a rule-based chart, each feature is associated with a case (or status) group (C↓, C−, C↑) to 

which its residual, z, belongs. To clarify, Equation 5.13 shows the regimes for determining the rule index 

for the i

th feature with the jth fault. 

, ( )

() ()

( )

ε

ε ε

ε

↓

−

↑

≤ − ∈

− < <+ ∈

≥+ ∈

ij i s ij

i ij i ij

ij i ij

z s ZC

s z s ZC

z s ZC

 (5.13) 

Equation 5.13 presents general cases of a case group for the threshold ±εi(s) (εi > 0) regardless of the 

assumption in Equation 5.10. For logical consistency, the probabilities of no-change and either of up-ordown change must be identical when the residual, zij, equals the threshold, εi, as in Equation 5.14. 

 In case ( ) | | ( − ) ↓ PC Z PC Z i i = , then = −ε ( ) ij i z s (5.14a) 

 In case ( ) | | ( − ) ↑ PC Z PC Z i i = , then = +ε ( ) ij i z s (5.14b) 

From Equations 5.14, the relation between the constant s and threshold εi(s) can be derived by setting 

Equations 5.12a and 5.12c or 5.12b and 5.12c to be equal. In either case, the equality can be formulated 

in the same form as in Equation 5.15. 

( ) 1 1 , 1 2erf erf 0

2 2

i i

i

i i

fs s s ε ε

ε

σ σ

⎧ ⎫⎧ ⎫ ⎪ ⎪⎪ ⎪ ⎛⎞ ⎛⎞

=− + + − = ⎨ ⎬⎨ ⎬ ⎜⎟ ⎜⎟ ⎪ ⎪⎪ ⎪ ⎩ ⎭⎩ ⎭ ⎝⎠ ⎝⎠ (5.15) 

Figure 5.5 represents the relation between the constant s and the threshold εi(s)/σi. As s increases, the 

function asymptotically approaches a linear function, εi/σi = s. When s > 1, the threshold is within 0.4 % 

of εi/σi = s. Therefore, we can simply assume εi(s) ≈ sσi as in Equation 5.10, in most cases, when s > 1. 

0.431

-2

-1

0

1

2

3

4

01234

Constant, s

εi

/σi in Equation 5.15

Figure 5.5. Relation between εi(s) and s
49

5.2.3.1 Fault classification error in a three case classification 

In order to estimate the probability of making a mistake and misinterpreting no-fault as being faulty, the 

no-fault probability was evaluated. If all the measurements are identical to the model expectations (z = 0), 

the system is performing exactly as predicted by the reference model, and the NC or neutral case for each 

feature will have a maximum probability as seen in Figure 5.4. Equation 5.16 is the classification error, α, 

based upon the overall probability, P(C− | Z)|z=0, calculated when all the features have zero residuals. For 

example, an α =1 % probability would occur when the probability of NC (P(C− | Z)|z=0) equals 99 %. 

( ) 0 α 1 | P C− = = − z Z (5.16) 

From the previous definitions and assumption of the measurement of m features being independent, the 

classification error for a particular fault may be simplified. 

( ) ( ) ( ) ( ) 1 2 1 2 00 0 0 1| | | 1 | m

m

m zz z z α PC Z PC Z PC Z PC −− − − == = = =− =− ⎡ ⎤ ⎣ ⎦ L Z (5.17) 

From Equations 5.12c and 5.17, the relation between the threshold standard deviation multiplier s and 

classification error α can be obtained. Figure 5.6 shows the plot of s versus α with regard to the number 

of features. In most cases, the classification error is below 5 % when s > 2. 

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

Constant, s

Classfication error, 

α

1

2

3

4

5

6

7

8

9

10

··

50

Number of features

α = 5 %

Figure 5.6. Classification error, α, versus measurement threshold standard deviation multiplier, s, with 

regard to the number of features 

For a 5 % value of α, the independent assumption forces s to be larger as the number of features increases. 

When k cases are to be classified using m features, the combination will manipulate k

m classes. For 

example, a three case classification using two features combines 32

 = 9 classes, but 10 features combines 
50

310 = 59049 classes. Since no-fault is one class among all of the classes, the increase of independent 

features will reduce the no-fault probability exponentially. Thus, when the classification error, α, is fixed 

at a constant value, the threshold standard deviation multiplier s must increase as seen with 50 features in 

Figure 5.6. If the FDD system operates with a large number of features, the independent assumption may 

prevent the FDD system from detecting concurrent faults due to less sensitivity caused by a high s value. 

Therefore, the threshold standard deviation multiplier s needs to be set below a reasonable value that may 

require some engineering judgment to determine. The sensitivity of the no-fault reference model, the 

maintenance cost, and the uncertainty of the sensors should be accounted for when optimizing the 

threshold standard deviation multiplier s. 

Since this evaluation is biased toward minimizing the risk of false alarm, the opposite case which 

interprets a fault state as being no-fault, is underestimated. Excessive credibility (large s value) degrades 

the sensitivity of an FDD system; however, false alarms decrease its usefulness and dependability. Since 

FDD systems for HVAC applications are mainly intended to reduce operating costs, false alarms are more 

detrimental than low sensitivity in terms of cost effectiveness. Therefore, it is reasonable to set a large 

threshold standard deviation multiplier (large s value) to minimize false alarms. 

5.2.3.2 Probability of a fault classification error 

Figure 5.7 presents the classification error using the polynomial reference models discussed in Chapter 4. 

The 1st, 2nd, and 3rd order MPR models were compared with regard to the constant s. Classification error, 

α, for the case with seven independent features is shown by a dashed line. In contrast to the curves seen 

in Figure 5.6, classification error decreases rapidly for smaller values of s than it did for the independent 

assumption cases of Figure 5.6. Unless the seven features are perfectly independent, the classification 

error would be less for this more realistic situation of some dependence between features. The three MPR 

models show similar trends; when s < 0.5, most of the no-fault cases are incorrectly categorized as being 

faulty; when s = 2.0, the classification error for the 3rd order MPR model drops to 0.58 % from 3.2 % as 

compared to the independent assumption. 

0

0.2

0.4

0.6

0.8

1

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Constant, s

Classification error, 

α

1-D model

2-D model

3-D model

Independent assumption

1st order MPR model

2nd order MPR model

3rd order MPR model

Independent assumption

(7 feature case)
51

Figure 5.7. Classification error with regard to the constant s for three different MPR models with seven 

features 

5.3 Setting the Neutral Threshold by Calculating Total Residual Uncertainty 

5.3.1 Uncertainties due to steady-state variation and lack of measurement 

repeatability 

To determine a realistic value of the neutral threshold, εi, as discussed in Section 5.2, validation of the 

measurements is mandatory. Naturally, the system measurements have uncertainties due to sensors – 

mostly thermocouples – and due to lack of measurement repeatability. The uncertainty of a thermocouple 

may come from measurement noise and drift. Considering the measurement noise behaves like zeromean white noise, its natural variation can be characterized closely by the steady-state standard deviation, 

σi,SS, defined in Chapter 3. Thermocouple drift is the measurement bias that varies over longer time 

periods than noise. However, the thermocouple drift can be regarded as negligible in this research since 

the same built-in sensors are used for model development and application to the tested system for FDD, 

thus their bias has been considered in the reference model measurements. 

To examine the repeatability of the system measurements, we analyzed repetitive tests. Referencing 

Table 3.3 in Kim et al., 2006, the tests were performed at 6 month intervals with 9 tests at condition 4 

(TOD = 27.8 °C (82.0 °F), TID = 21.1 °C (70.0 °F), and RHID = 50 %), 11 tests at condition 5 

(TOD = 27.8 °C (82.0 °F), TID = 26.7 °C (80.0 °F), and RHID = 50 %), 8 tests at condition 8 (TOD = 37.7 °C 

(100.0 °F), TID = 21.1 °C (70.0 °F), and RHID = 50 %), and 10 tests at condition 9 (TOD = 37.7 °C 

(100.0 °F), TID = 26.7 °C (80.0 °F), and RHID = 50 %). The feature standard deviations from repeatability 

tests, σi,Repeat, are listed in Table 5.3. 
52

Table 5.3. Standard deviation of the selected features 

(°C) Tsh Tsc TE TD TC ΔTCA ΔTEA

Steady-state standard deviation σi,SS 0.124 0.052 0.024 0.058 0.035 0.063 0.058 

Standard deviation from 

repeatability tests σi,Repeat 0.101 0.156 0.084 0.280 0.166 0.088 0.111 

Table 5.3 gives us the measurement uncertainty due to steady-state variation, σi,SS, and due to variation 

from test-to-test (measurement repeatability), σi,Repeat, for similar test conditions. These two values will be 

used to calculate total residual threshold uncertainty for each feature. 

5.3.2 Uncertainties due to the reference models 

In section 5.2 we studied the variation of probabilities for the three feature cases (C↓, C−, C↑) with the 

definition of Equation 5.8 and assumption specified by Equation 5.9. Since measurements are used with 

the reference model predictions to determine residuals, the square-root of the sum of residuals presents a 

RMS error, which is not Gaussian. In this section, we analyze the no-fault measurements distribution in 

detail to provide the methodology for determining a proper value of the threshold εi, as defined in Section 

5.2.3. 

In most cases, it is hard to obtain a reference model covering all operating conditions. To train a 

reference model after installation, a real-time decision of fault-free or faulty status is mandatory. In 

contrast to the steady-state and repetition uncertainty, the model uncertainty comes from the 

imperfections associated with any mathematical model. Here, we define average bias of the model 

estimation as the averaged residual between the model and the current measurement with no zero-mean 

noise. The model standard deviation, σi,Model, characterizes the model uncertainty. Since the zero-mean 

noise uncertainty (σi,SS) and model uncertainties (σi,Model) amplify the variability of residuals 

independently, it is reasonable to assume that no joint effect exists between the two uncertainties. 

Therefore, the covariance between the two uncertainties (σi,SS·σi,Model) is zero, and σi,NF will be a squared 

sum of σi,SS and σi,Model as shown in Equation 5.18. By combining Equation 5.18 and Table 5.3, σi,Model

can be estimated as shown in Table 5.4. Comparing Table 5.3 to Table 5.4, we can see that the reference 

model is the same order of magnitude as the steady-state and repeatability uncertainties. 

2 2 σ σσ i ii ,NF ,SS ,Model = + (5.18) 

Table 5.4. Net model uncertainties of the features using the 1st, 2nd, and 3rd order MPR models 

Model uncertainties, σi,Model (°C) Tsh Tsc TE TD TC ΔTCA ΔTEA

1st order MPR model 0.557 0.244 0.549 0.799 0.179 0.150 0.581 

2nd order MPR model 0.328 0.197 0.147 0.319 0.047 0.040 0.131 

3rd order MPR model 0.197 0.133 0.123 0.250 0.029 0.019 0.071 

Equation 5.18 defines the overall uncertainties of the no-fault residual for one measurement. When a 

moving window average of n samples is used, the steady-state standard deviation is reduced by the square 

root of n as shown in Equation 5.19. 
53

,SS

,SS MW

i

i

n

σ

σ = (5.19) 

Equation 5.19 will be applied throughout our calculations since σi,SS in Table 5.4 is constant as 

determined in Section 3.2.3. 

MW

,SS MW 1

i

i

n

σ

σ = − (5.20) 

Since the model uncertainty, σi,Model, remains the same without regard to the moving window size, the nofault total uncertainty for the moving window average will be, 

2

,SS 2

,NF ,Model MW

i

i i

n

σ

σ σ

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

 (5.21) 

where n is the number of measurements in the moving window. Figure 5.8 compares the standard 

deviations of the three characteristics contributing to the total standard deviation: steady-state, 3rd order 

MPR model, and repeatability. As expected, steady-state uncertainty, σi,SS, shows the highest value for 

Tsh. The σi,Model of TD shows the highest value among the features, but the σi,SS shows moderate variation. 

This means that the polynomial model has a higher uncertainty when estimating TD, compared to the 

steady-state standard deviation. Some of this higher modeling standard deviation is due to the heat inertia 

of the compressor causing a higher time constant for TD. This higher variation of TD also appears in the 

σi,Repeat having the largest value in Figure 5.8. 
54

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Tsh

Tsc

TE

TD

TC

DTCA

DTEA

Standard deviations (oC)

계열1

계열2

계열3

r(Tsh) r(Tsc) r(TE) r(TD ) r(TC ) r(ΔTCA ) r(ΔTEA )

Feature residuals

Steady-state measurement

Third order MPR model

Repeatability

Figure 5.8. Standard deviations for seven feature residuals due to measurement noise at steady state, the 

3rd order MPR model, and the lack of measurement repeatability 

5.3.3 Combining uncertainties to determine the neutral case, NC, threshold 

From sections 5.3.1 and 5.3.2, we observed uncertainties from the no-fault measurements. Considering 

that total uncertainty is due to steady-state uncertainty, modeling uncertainty, and lack of measurement 

repeatability uncertainty, the threshold is proportional to the total uncertainty as shown by Equation 5.22, 

where the constants k1, k2, and k3 define the confidence intervals for each uncertainty. 

( ) ( ) 2

2 2 ,SS

1 2 ,Model 3 ,Repeat

i

i ii kk k

n

σ

ε σσ

⎛ ⎞

= ++ ⎜ ⎟ ⎝ ⎠

 (5.22) 

Depending on the confidence level, the constants k1, k2, and k3 can be determined based on the 

distribution for each uncertainty term. 

5.3.3.1 Confidence interval, k1, for the steady-state uncertainty 

Since we use measurements and standard deviations in a preset moving window, their distribution 

depends on the characteristics of the moving window. When n no-fault data are sampled from a Gaussian 

distribution with standard deviation, σi, the quantity t can be defined by Equation 5.23. 

/

i i

i

x

t

n

μ

σ

− = (5.23) 
55

where μi is the current mean of the moving window of n samples. In such a case, t in Equation 5.23 

follows a Student’s t-distribution with n – 1 degrees of freedom. When we set a 1 – α probability that the 

two values are equal (xi = μi), the confidence interval, k1= tα/2, n – 1, is described by Equation 5.24. 

/2, 1 1 i Px t ii n

n α

σ

μ − α

⎛ ⎞ − < =− ⎜ ⎟ ⎝ ⎠ (5.24) 

where tα/2, n – 1 is a two-sided confidence interval with n – 1 degrees of freedom. In Chapter 3, we 

determined the appropriate moving window size to be 140 seconds, or 10 samples in the moving window 

with a 14 second sampling interval. With a 1 – α = 99 % confidence, the t-value, t0.005,9, is 3.25, which is 

larger than a Gaussian distribution of 2.58. Table 5.5 shows the values of k1= tα/2, 9. Thus, constant k1 in 

Equation 5.22 can be defined by Equation 5.25. For 99 % confidence (or credibility) level, k1 is 3.25. 

1 /2, 1 n k t = α − (5.25) 

Table 5.5. Two-sided confidence intervals with degrees of freedom of four and nine 

1 – α (%) 80.0 90.0 95.0 99.0 

α/2 (%) 10.0 5.0 2.5 0.5 

tα/2,4

1 1.533 2.132 2.776 4.604

k1=tα/2,9

2 1.383 1.833 2.262 3.250

1

 5 sample moving window 

2

 10 sample moving window

5.3.3.2 Confidence interval, k2 , for model uncertainty 

Figure 5.9 shows the distribution of the Tsh residual using the 3rd order MPR model applied to the no-fault 

steady-state data discussed in Chapter 4. As seen in the plot, a Gaussian assumption tracks the residual 

distribution of the feature to a certain extent. However, residuals near zero are distributed narrower than a 

Gaussian. At high residual values where | r(Tsh) | > 1.7 °C (3 °F), a Gaussian assumption underestimates 

the probability. From the Gaussian approach, 99 % of data fall within the range of ±0.6 °C (1.08 °F), but 

no-fault test data have a wider range of ±0.78 °C (1.41 °F) to cover 99 % of all data. 

Equation 5.26 shows the normalized 99 % confidence intervals for a Gaussian assumption. In Equation 

5.27, normalized 99 % confidence intervals of each feature residual are shown based upon the 

experimental data. Any bounds greater than an absolute value of 2.58 in the experimental results imply 

that the measurements have a wider 99 % distribution than the infinitely sampled Gaussian assumption. 
56

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-3 -2 -1 0 1 2 3

r (Tsh) (o

C)

Probability density function

Experiment

N(0, ssh2)

Student t

Experiment

Gaussian distribution, N(0, σi,NF

2)

Student t-distribution, Degree of freedom = 9

3rd order MPR model

σNF(Tsh) = 0.233 oC

Figure 5.9. Comparison of the distributions of the Tsh residuals from no-fault steady-state data and 3rd

order MPR model based on the model RMS error; Gaussian distribution, student tdistribution, and the calculated distribution from the test data. 

 Gaussian distribution: 

,NF

2.58 2.58 i

i

r

σ

−< < (5.26) 

 Experiment: ( )

( )

sh

NF sh

3.37 3.37

r T

σ T

−< < (5.27a) 

( )

( )

sc

NF sc

3.03 3.03

r T

σ T

−< < (5.27b) 

( )

( )

E

NF E

2.96 2.96

r T

σ T

−< < (5.27c) 

( )

( )

D

NF D

2.65 2.65

r T

σ T

−< < (5.27d) 

( )

( )

C

NF C

2.63 2.63

r T

σ T

−< < (5.27e) 

( )

( )

CA

NF CA

2.64 2.64

r T

σ T

Δ

−< <

Δ (5.27f) 

( )

( )

EA

NF EA

3.22 3.22

r T

σ T

Δ

−< <

Δ (5.27g) 
57

As shown by comparing the confidence intervals of Equation 5.26 and Equations 5.27, applying the 

infinite degree of freedom Gaussian confidence interval would underestimate the threshold εi. Table 5.6 

lists the confidence intervals for three confidence levels. 

Table 5.6. Two-sided confidence interval of the seven features for the 3rd order MPR model 

1 – α (%) 75.0 97.5 99.5 

Tsh 1.00 2.26 2.96 

Tsc 0.93 2.22 3.37 

TE 1.10 2.06 2.65 

TD 1.15 1.96 2.63 

TC 1.03 2.03 3.03 

ΔTCA 1.14 1.95 2.64 

k2= 

tα/2, n-1 

Δ TEA 0.95 2.16 3.22 

5.3.3.3 Confidence interval, k3 , for lack of measurement repeatability 

Repetitive measurements of a random variable will follow a Gaussian distribution, thus, under similar 

measurement conditions, repetitively measured feature residuals will also follow a Gaussian distribution. 

Table 5.7 shows the confidence interval with regard to the confidence level for a Gaussian distribution at 

various confidence levels. Thus, constant k3 in Equation 5.22 is defined by the Gaussian confidence 

interval in Equation 5.28. For example, with 99 % credibility, k3 equals 2.576 as shown below. 

3 /2 k g = α (5.28) 

Table 5.7. Two-sided confidence intervals of Gaussian distribution 

1 – α (%) 75.0 80.0 85.0 90.0 95.0 97.5 99.0 99.5 99.75 99.9 99.95

k3 = gα/2 1.156 1.282 1.445 1.645 1.960 2.241 2.576 2.807 3.024 3.291 3.481

5.3.4 Neutral case, NC, threshold value for the required credibility 

In the previous sections we defined the confidence intervals k1, k2, and k3 for each uncertainty term as 

defined by Equations 5.24, 5.27, and 5.28. With this, the thresholds, εi, at any given level of confidence 

may be calculated using Equation 5.21. Table 5.8(a) and 5.8(b) list the feature thresholds with 50 %, 

95 % and 99 % credibility for the moving window size of 5 and 10 samples. 
58

Table 5.8. Feature thresholds at different confidence levels for two sample sizes 

(a) Moving window of 5 samples 

Threshold of the features Tsh Tsc TE TD TC ∆TCA ∆TEA

50 % credibility, εi,0.50 (°C) 0.131 0.134 0.092 0.243 0.082 0.064 0.086 

95 % credibility, εi,0.95 (°C) 0.504 0.413 0.324 0.756 0.239 0.192 0.274 

99 % credibility, εi,0.99 (°C) 0.758 0.579 0.426 0.987 0.317 0.265 0.383 

(b) Moving window of 10 samples 

Threshold of the features Tsh Tsc TE TD TC ∆TCA ∆TEA

50 % credibility, εi,0.50 (°C) 0.130 0.134 0.092 0.243 0.082 0.064 0.086 

95 % credibility, εi,0.95 (°C) 0.496 0.411 0.323 0.755 0.237 0.187 0.271 

99 % credibility, εi,0.99 (°C) 0.735 0.574 0.424 0.983 0.313 0.248 0.373 

5.4 Application of the FDD Algorithm to a Residential Heat Pump Operating 

in the Cooling Mode 

5.4.1 Implementing single faults 

Table 5.9 lists the six types of faults investigated. The online classification of the presence of noncondensable gases is not included in this study, since it can be diagnosed by checking its partial pressure 

while the system is off (Stylianou and Nikanpour, 1996; Li, 2004). Implementation of the seven artificial 

faults was detailed in Chapter 3. 

Table 5.9. Description of studied faults 

Fault Abbr. Determination of level of fault during tests 

Compressor leakage (4-way valve leakage) CMF % of refrigerant flow rate 

Improper outdoor air flow rate CF % of coil area blocked 

Improper indoor air flow rate EF % of correct air flow rate 

Liquid line restriction LL % of normal pressure drop through TXV 

Refrigerant overcharge OC % overcharge from the correct charge 

Refrigerant undercharge UC % undercharge from the correct charge 

5.4.2 Quantifying the control limits of the TXV 

When the TXV is fully open, it behaves like a fixed area expansion device; thus, any faults that occur 

during this time will affect system features differently than they would if the TXV were within its normal 

control range. For this reason, the cooling mode TXV’s control limits must be assessed so that 

appropriate feature rule-sets may be developed for the various realms of TXV operation. The delineation 

of these TXV control realms is aided by correlation of the refrigerant mass flow rate to measured system 

features. 
59

The cooling mode TXV experiences a large range of inlet refrigerant conditions and must control 

evaporator exit superheat over a broad range of refrigerant mass flow rates; thus selection of the 

appropriate TXV is somewhat of an art aided by extensive experimental knowledge. The TXV may reach 

its control limits when two-phase refrigerant is present or more/less refrigerant flow is required by its 

superheat setting. Since the TXV is working as a superheat controller, its characteristic behavior should 

be quantified as a means of assessing its control limits. Equation 5.29 is a simple formulation of mass 

flow rate via isenthalpic expansion 

m CA P R TXV.up TXV = Δ d ρ (5.29) 

where Cd is a characteristic flow coefficient; A is the area, which is a function of valve stem displacement; 

ρTXV.up is TXV upstream refrigerant density; and ΔPTXV is pressure drop across the TXV. For the case of 

two-phase refrigerant at the inlet of the TXV, an accurate mass flow measurement is required in order to 

determine the appropriate value of CdA. 

Since two-phase refrigerant flow degrades or completely prevents accurate measurement of refrigerant 

mass flow through the Coriolis type meter used in this study, mass flow is estimated indirectly using a 

compressor map when TXV upstream subcooling (Tsc_TXV.up) is below 0.5 °C. Equation 5.30 presents the 

form of the mass flow equation provided by manufacturers according to ANSI/ARI Standard 540 (ARI 

2004). 

2

R 1 2 S.sat 3 D.sat 4 S.sat 5 S.sat D.sat

2 32 2 3

6 D.sat 7 S.sat 8 S.sat D.sat 9 S.sat D.sat 10 D.sat

m c cT cT cT cT T

cT cT cT T cT T c T

=+ + + + +

++ + +

 (5.30) 

To increase the usefulness of Equation 5.30, Dabiri and Rice (1981) suggested Equation 5.31 to adjust 

mass flow rate when the compressor suction superheat is different from the standard conditions. 

R S

R.map S.map

1 0.75 1 m

m

ρ

ρ

⎛ ⎞ = + − ⎜ ⎟

⎝ ⎠

 (5.31) 

The compressor map was created at a standard suction superheat (TS.sh) of 11.1 °C ( 20.0 °F). Figure 5.10 

shows calculated and measured refrigerant mass flow rate with Tsc_TXV.up > 0.5 °C for suction superheats 

of 10.2 °C to 13.7 °C (18.4 °F to 24.6 °F). All the measurements are well estimated within a ±5 % 

deviation. This figure indicates that these two equations are accurate estimators of mass flow rate and 

that they should be applicable even when two-phase conditions exist at the TXV inlet. 

In order to characterize the control limits of the TXV, refrigerant flow rate is calculated with Equations 

5.30 and 5.31 when the upstream is two-phase as defined by a Tsc_TXV.up < 0.5 °C. In this case, 

thermodynamic properties including refrigerant enthalpy and mass quality upstream of the TXV are 

estimated by an energy balance calculation on the condenser air-side. Two-phase density is calculated 

using mass quality and a harmonic weight of saturated liquid and vapor densities. 
60

+ 5 % line

–5 % line

140

160

180

200

220

140 160 180 200 220

Measured mR (kg/h)

Estimated 

mR (kg/h)

Figure 5.10. Estimation of refrigerant mass flow rate at ΔTsc_TXV.up > 0.5 °C using the compressor map 

with superheat correction by Dabiri and Rice (1981) 

Figure 5.11 shows CdA with respect to Tsh at the evaporator exit. Note that the level of Tsh in Figure 5.11 

measured at the evaporator exit is smaller than the compressor suction superheat due to heat transfer 

within the TXV. When Tsh is below 9 oC (zone A), CdA and Tsh show a linear trend where the TXV 

successfully controls Tsh. But in some cases of single-phase and two-phase conditions, the TXV cannot 

change the superheat because it is fully open (zone B). This condition is known as actuator saturation. 

In most cases, TXV saturation occurs when there is two-phase refrigerant upstream; however, actuator 

saturation also occurs at subcooled upstream conditions due to a non-condensable gas fault. Noncondensable gas collects on the high pressure side of the system and raises the condensing pressure. If 

non-condensable gas passes through the TXV or so much non-condensable is present that the discharge 

pressure is forced abnormally high, the TXV can reach its control limits even though the refrigerant is in a 

subcooled state. In zone B, CdA does not increase with Tsh. Applying Bernoulli’s orifice equation, 

Equation 5.29, to two-phase flow, underestimates mass flow rate through an expansion valve (Aaron and 

Domanski, 1989). Considering that CdA is estimated by Equation 5.29 with the homogeneous two-phase 

assumption, CdA values in zone B are located below the horizontal line which represents the maximum 

CdA of zone A. 

From Figure 5.11, when the TXV is saturated, CdA is near constant, and the TXV is working like a fixed 

expansion device. Therefore, the fault characteristics of a system with a TXV should be observed in zone 

A and B, separately. 
61

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20

ΔT sh (

o

C)

C

Ad (x 10-6 m2

)

1-phase upstream

2-phase upstream

A: In-control zone of TXV B: Full open zone (TXV saturation)

ΔTsc_TXV.up > 0.5 oC (1φ zone)

ΔTsc_TXV.up < 0.5 oC (2φ zone)

Figure 5.11. CdA from Equation 5.29 versus Tsh for the selected tests 

5.4.3 Building a rule-based chart 

Figure 5.12 shows a comparative presentation of the effects of single faults on the seven parameters at test 

condition #5 (TOD = 27.8 °C (82.0 °F), TID = 26.7 °C (80.0 °F), and RHID = 50 %) (as in Table 3.3 of Kim 

et al. 2006). The standard deviations, σ’s, of the various features are listed in Table 3.3. Some features 

show clearly increasing or decreasing trends, but some show little change within a standard deviation. 

ΔTLL can be a good indicator that shows whether the condenser exit is in two-phase or single-phase. 

However, ΔTLL is not a rapidly changing feature when subcooled refrigerant at the condenser exit flashes 

into two-phase during liquid line migration. In addition, the degree of ΔTLL highly depends upon ΔTsc

and ΔPLL. Thus, a two-phase condition upstream does not always guarantee the TXV being saturated or 

fully open. This is observed also in Figure 5.11, where Tsh’s in a number of conditions are well controlled 

at Tsc_TXV.up < 0.5 °C. Thus, ΔTLL is not selected as a feature of the FDD system in this study, though it is 

useful for some typical faults like liquid line restriction. 

Based on the bounds defined by the model standard deviations (dashed lines on the figure), rules for each 

feature can be determined as in Table 5.2. The rule-based chart in Table 5.2 uses three residual indices of 

no-change (–), positive-change (Ç), negative-change (È). By using the no-change index, the no-fault 

case can also be included in the rule-based chart thus creating a complete FDD algorithm that deals with 

all possible residual states. Even if a fault case is within the no-fault threshold and is assigned a nochange value for a particular fault level, as the fault level increases, the residual may move out of the nofault threshold bounds. In such a case, the changed case should be properly adjusted to positive/negativechange. 

This re-assignment of case indices within the rule set for a particular fault occurs during TXV saturation 

as mentioned in Figure 5.8; the rule set was separated into zone A and B. Note that the residual of Tsh has 

no change in zone A but increases in zone B. TXV saturation is not observed within the test range for 

compressor valve leakage, improper indoor air flow, or refrigerant overcharge faults. The rule set for the 

liquid-line restriction fault were all NC indices for zone A. 
62

0 10 20 30 40

-6

-4

-2

0

2

Fault Level [%]

R[

T

E] (oC)

0 10 20 30 40

-2

0

2

4

6

8

10

Fault Level [%]

R[

Tsh] (oC)

0 10 20 30 40

-2

0

2

4

6

8

10

Fault Level [%]

R[

T

D] (oC)

0 10 20 30 40

-3

-2

-1

0

1

2

3

Fault Level [%]

R[

T

C] (oC)

+ ε

− ε

+ ε

− ε

+ ε

+ ε − ε

− ε

ε(TE) = 0.424 oC

ε(TD) = 0.983 oC

ε(TC) = 0.313 oC

ε(Tsh) = 0.735 oC

(d)

(b)

(c)

(a)

0 10 20 30 40

-5

0

5

Fault Level [%]

R[

T

sc] (oC)

0 10 20 30 40

-2

-1.5

-1

-0.5

0

0.5

Fault Level [%]

R[

Δ

TCA] (oC)

0 10 20 30 40

-2

-1

0

1

2

3

Fault Level [%]

R[

Δ

TEA] (oC)

Compressor leakage

Improper OD air flow rate

Improper ID air flow rate

Liquid line restriction

Refrigerant undercharge

Refrigerant overcharge

Non-condensable gases

+ ε

− ε

+ ε

− ε

+ ε

− ε

ε(Tsc) = 0.574 oC

ε(ΔTEA) = 0.373 oC

ε(ΔTCA) = 0.248 oC

(f)

(g)

(e)

Figure 5.12. Residuals of seven parameters for different faults. (Test #5: TOD = 27.8 °C (82.0 °F), 

TID = 26.7 °C (80.0 °F), and RHID = 50 %): Residual of (a) TE, (b) Tsh, (c) TD, (d) TC, (e) Tsc, 

(f) ∆TCA, and (g) ∆TEA (Kim et al. 2006) 
63

5.4.4 Example FDD module 

Appendix A presents screenshots of the FDD modules comprising the entire FDD system. Figure 5.13 

shows the main interface window of the FDD module developed under this study. Moving Window 

Module (A.2), Steady-state Detector Module (A.3), Preprocessor Module (A.4), No-fault MPR Reference 

Module (A.5 and A.6), and Statistical Rule-based FDD Module (A.7) are running in the background 

while the main module is running. 

In Figure 5.13, sample operation at a 20 % reduced indoor air flow fault is demonstrated. The moving 

window average is manipulated through the moving window module at the left hand side. Based on the 

MPR model, feature residuals is calculated and fed into Rule-based FDD Module, which indicates ongoing fault. In the meanwhile, Preprocessor Module calculates real-time EER, from which EER 

degradation is evaluated. From the FDD Module, we can see the EER degradation warning is ON and 

improper indoor air flow fault (EF) is properly diagnosed. 

EER degradation warning

Reference model 

estimation

Indication of 

evaporator fouling

Figure 5.13. Main interface window of the FDD system indicating an improper indoor air flow fault. 

The flow diagram in Fig. 5.14 shows a working algorithm for the FDD system in Figure 5.13. After a 

steady state is identified, NFSS parameters are calculated using the operating condition as input 

parameters. When the performance degradation parameter, in this case EER, is over the high limit in the 

fault detection module, the system is regarded to be faulty and the detailed fault status is analyzed in the 

fault diagnostic modules. In this research, the fault detection alarm is activated when EER is degraded 

more than 3.0 % below the NFSS reference EER value. Then the appropriate rule-based chart is 

determined based upon whether or not the TXV is in control or out of control. The TXV status is 

evaluated, as in Figure 5.11, from the degree of subcooling at the TXV upstream (greater than or less than 

0.5 °C). Based on the rule-based chart and the fault calculation algorithm in sections 5.2 and 5.3, the 

current fault is identified in the fault diagnostic module. 
64

Starting FDD

Operating condition: TOD, TID, TID,dew

Field measurements

No-fault Reference Model

Rule-based FDD Analysis

Fault Diagnosis

Rule-based chart 

in zone B

End of FDD

Steady-state?

Reference Module

Steady-state Detector

FDD Modules

yes

yes EERmodel – EERsystem < ε

No

ΔTsc_TXV.up > 0.5 oC

(TXV in control)

Calculate feature residuals

Rule-based chart 

in zone A

yes

No

Figure 5.14. Sequential algorithm of the sample FDD system 
65

5.5 Performance of the FDD System 

5.5.1 Comparative evaluation of fault effects on cooling capacity and EER 

For the validation of the FDD algorithm, the experimental data in Kim et al. (2006) were utilized. The six 

single-fault cases–compressor leakage, improper OD air flow rate, improper ID air flow rate, liquid-line 

restriction, refrigerant undercharge, and refrigerant overcharge–were tested. Seventeen combinations of 

indoor and outdoor conditions in Table 5.10 were selected as experimental conditions. 

Table 5.10. Operating conditions for FDD tests from Kim et al. (2006) 

Indoor Outdoor Test 

number Dry-bulb temp. 

°C (°F) 

Relative humidity

% 

Dry-bulb temp. 

°C (°F) 

Relative humidity

% 

1* 26.7 (80.0) 51 35.0 (95.0) 40 to 60 

2 21.1 (70.0) 50 21.1 (70.0) 40 to 60 

3 26.7 (80.0) 50 21.1 (70.0) 40 to 60 

4** 21.1 (70.0) 50 27.8 (82.0) 40 to 60 

5** 26.7 (80.0) 50 27.8 (82.0) 40 to 60 

6 21.1 (70.0) 50 32.2 (90.0) 40 to 60 

7 26.7 (80.0) 50 32.2 (90.0) 40 to 60 

8** 21.1 (70.0) 50 37.8 (100.0) 40 to 60 

9** 26.7 (80.0) 50 37.8 (100.0) 40 to 60 

10 21.1 (70.0) dry coil 21.1 (70.0) 40 to 60 

11 26.7 (80.0) dry coil 21.1 (70.0) 40 to 60 

12 21.1 (70.0) dry coil 27.8 (82.0) 40 to 60 

13 26.7 (80.0) dry coil 27.8 (82.0) 40 to 60 

14 21.1 (70.0) dry coil 32.2 (90.0) 40 to 60 

15 26.7 (80.0) dry coil 32.2 (90.0) 40 to 60 

16 21.1 (70.0) dry coil 37.8 (100.0) 40 to 60 

17 26.7 (80.0) dry coil 37.8 (100.0) 40 to 60 

* ARI Standard 210/240 (2006) rating point 

** Combination of test conditions selected for fault tests except improper ID air flow rate fault

The test data show to what extent faults effect cooling capacity and EER and at what ambient temperature 

conditions particular faults have the greatest effect. Cooling loads are proportional to the indoor and 

outdoor temperature difference or temperature lift. The highest to lowest temperature lift occurs with test 

conditions 8 > 9 > 4 > 5. Cooling capacity, QER in Figure 5.15(a), is highest to lowest with test condition 

5 > 9 > 4 > 8. This illustrates that the system produces the highest capacity at the minimum temperature 

lift. The EER in Figure 5.15(b) is inversely proportional to the temperature lift. We have seen from our 

testing that the feature residuals will show the most distinctive impact at the high capacity, test #5, 

conditions where the temperature lift is the lowest, and EER is the highest. 
66

20 25 30 35 40

0.6

0.7

0.8

0.9

1

1.1

Outdoor Temperature [C]

QER [kW]

ID 21.1 °C DB/50% RH

ID 26.7 °C DB/50% RH

ID 21.1 °C DB/Dry-coil

ID 26.7 °C DB/Dry-coil 5

9

8

4

(a) Cooling capacity 

20 25 30 35 40

10

12

14

16

18

20

22

Outdoor Temperature [C]

EER

5

9

8

4

ID 21.1 °C DB/50% RH

ID 26.7 °C DB/50% RH

ID 21.1 °C DB/Dry-coil

ID 26.7 °C DB/Dry-coil

(b) EER 

Figure 5.15. Variation of system performance under fault-free conditions 

Figure 5.16 presents a histogram of estimated fault levels that would cause a 5 % reduction in EER 

relative to the fault-free EER. The bars marked by an asterisk represent linear extrapolation of fault levels 

to those which would cause a 5 % degradation in EER. EER is relatively robust with evaporator fouling 

or refrigerant overcharge faults. For a refrigerant overcharge fault, the fault level is more sensitive at the 

highest temperature lift seen with test #8. This is because higher operating temperature raises optimum 

refrigerant charge. Since a high temperature lift adversely effects capacity, EER degradation is more 

sensitive to fault level at a higher capacity (low lift test #5) but less sensitive at a lower capacity (higher 

temperature lift test #8) for all faults except evaporator fouling or refrigerant overcharge. All faults, 

except the compressor leakage fault, must have a fault level greater than 10 % to produce a 5 % decrease 

in EER. 

Figure 5.17 presents estimated EER degradation relative to the fault-free EER at a 10 % level compressor 

fault and a 20 % fault level for the remaining five faults. With a 20 % improper refrigerant charge, 

undercharge causes more EER degradation than overcharge, especially at the lowest temperature lift 

occurring for test #5. EER degradation is greater when TID is at 26.7 °C than at 21.1 °C. 
67

0

5

10

15

20

25

30

35

40

45

CMF CF EF LL UC OC

Fault type

Fault level (%)

 # 8 (ID 21.1 oC/OD 37.7 oC)

 # 4 (ID 21.1 oC/OD 27.8 oC)

 # 9 (ID 26.7 oC/OD 37.7 oC)

 # 5 (ID 26.7 oC/OD 27.8 oC)

* *** **

Figure 5.16. Estimated single-fault levels for a 5 % degradation in EER at four different operating 

conditions. (‘*’ below the x-axis bars represents linear extrapolation beyond the test 

condition bounds) 

0

2

4

6

8

10

12

14

CMF CF EF LL UC OC

Fault type

EER degradation (%)

 # 8 (ID 21.1 oC/OD 37.7 oC)

 # 4 (ID 21.1 oC/OD 27.8 oC)

 # 9 (ID 26.7 oC/OD 37.7 oC)

 # 5 (ID 26.7 oC/OD 27.8 oC)

*

Figure 5.17. Estimated EER degradation for a 10 % compressor fault level and a 20 % fault level for the 

remaining faults 
68

5.5.2 Example implementation of the FDD system 

Performance of the FDD system was verified using the steady-state data in Kim et al. (2006). The NFSS 

reference model trained through the random operations in Chapter 4 were applied as a base to the FDD 

system. The considerations of the FDD system in this section are listed as follows. 

- EER warning alarm : 3 % EER degradation from the NFSS reference value. 

- NFSS reference model: The 3rd order multivariate polynomial model. 

- Isolation of diagnosed system status: The case of the maximum probability more than 3 times of the 

second high probability. Otherwise, the case regarded as undetermined. 

- Constant s in Equation 5.10: s = 2.0 

Table 5.11 presents diagnostic results of no fault cases using the FDD system built in section 5.4.4. From 

Table 5.11, no fault cases were well diagnosed by the FDD system. EER degradations in Table 5.11 is 

not zero even though the cases were set to zero fault level. This is because the reference EER is 

generated by the NFSS model which had non-zero mean square error for the regression. 

Table 5.11. FDD results for no fault (NF) cases 

Probability for each case 

Test condition 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation (%) 

Compressor Leakage 

(CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air flow 

rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge (UC) 

Refrigerant 

overcharge (OC) 

No Fault (NF) 

2 NF A 0 -0.26 2.88E-21 8.45E-19 9.73E-14 7.68E-36 1.86E-14 8.34E-14 0.27146 

3 NF A 0 0.05 4.68E-23 3.32E-14 1.15E-10 1.23E-10 8.29E-16 1.07E-04 0.00320 

4 NF A 0 0.19 3.37E-16 4.24E-13 3.81E-12 5.42E-16 8.22E-11 1.32E-07 0.91963 

5 NF A 0 -0.06 1.80E-16 1.55E-14 1.09E-11 2.02E-22 3.61E-10 4.60E-11 0.91433 

6 NF A 0 -1.35 1.28E-16 3.99E-17 2.31E-07 8.93E-16 1.21E-11 8.36E-08 0.69309 

7 NF A 0 0.10 1.90E-29 2.11E-16 2.83E-23 1.63E-40 9.74E-27 4.04E-07 0.08289 

8 NF A 0 -0.06 6.24E-17 4.02E-14 2.22E-11 1.46E-20 1.24E-11 5.36E-09 0.96387 

9 NF A 0 0.04 4.17E-16 6.68E-14 8.62E-11 4.63E-17 7.81E-11 4.09E-08 0.95847 

10 NF A 0 -0.17 8.00E-18 4.16E-14 3.52E-14 1.15E-25 1.40E-10 8.59E-13 0.77679 

11 NF A 0 0.38 3.99E-16 1.94E-13 1.23E-13 5.72E-23 1.51E-10 4.41E-10 0.88323 

12 NF A 0 -0.20 8.08E-18 1.01E-15 3.23E-09 2.00E-17 3.00E-13 1.03E-06 0.89601 

13 NF A 0 -0.28 6.08E-17 1.76E-15 5.61E-11 1.97E-24 1.08E-09 4.98E-13 0.81954 

14 NF A 0 0.10 3.26E-17 6.46E-15 1.96E-10 1.94E-20 3.47E-13 3.81E-08 0.88826 

15 NF A 0 1.06 9.05E-16 7.05E-14 1.96E-15 3.56E-28 1.09E-09 6.52E-13 0.56744 

16 NF A 0 0.42 8.41E-16 1.05E-12 1.63E-13 1.14E-18 6.47E-10 3.15E-09 0.90353 

17 NF A 0 -0.61 2.48E-18 1.77E-15 9.79E-11 3.65E-22 9.61E-13 3.98E-09 0.93935 
69

Table 5.12 thru Table 5.17 present detection and diagnostic results for six single faults. The no-fault 

threshold, ε, confidence level was set at 99 % for all results shown below. The probability of each fault 

was calculated according to the method outlined in Section 5.2 with Equations 5.12 and 5.17 

implemented according to the rule-chart of Table 5.2 (repeated here for ease of reference). The 

probability of a given fault was the product of the probabilities of each case as represented in a row of the 

rule-chart, which includes the no-fault or neutral (NC) case. 

Table 5.2 (repeated here). Rule-based fault pattern chart of seven features 

Fault Type zone1 TE Tsh TD TC Tsc ∆TCA ∆TEA

Compressor valve leakage2

(CMF) A Ç – Ç È È È È

Improper outdoor air flow A Ç – Ç Ç È – È

(CF) B È Ç Ç Ç È È È

Improper indoor air flow2

(EF) A È – – È – È Ç

A – – – – – – – Liquid-line restriction3

 (LL) B È Ç Ç È Ç È È

A – – – È È È È Refrigerant undercharge (UC) B È Ç Ç È È È È

Refrigerant overcharge2 (OC) A – – Ç Ç Ç – – 

No-fault – – – – – – – – 

1

 Zone A: Tsh < 9 oC, Zone B: Tsh > 9 oC 

2

 The case of Tsh > 9 oC was not observed within the fault levels of this study. 

3

 The rules when Tsh < 9 oC were identical with no-fault cases. 
70

Table 5.12 shows the performance of the FDD system with a compressor valve leakage fault (CMF) at 

four different fault levels and four different test conditions (#4, #5, #8, and #9) with the TXV in superheat 

zone A as shown in Figure 5.10. At fault levels greater than 5 %, the FDD system made a correct 

detection and diagnosis. At fault levels less than 5 %, no-fault (NF) and undercharged fault (UC) were 

incorrectly indicated. For test condition #4 at a fault level of 2.5 %, EER was degraded by 3.4 % while 

the FDD system indicated NF. As fault level increased to 5 %, the compressor fault was correctly 

detected and diagnosed while EER degraded by 5.5 %. For test condition #8, the 2.5 % CMF was not 

detected with an EER degradation of 2.5 %. For test condition #9, the 2.4 % CMF was incorrectly 

detected and diagnosed as an UC fault while EER was degraded by 1.3 %. 

Table 5.12. FDD results for compressor leakage fault (CMF) 

Probability for each case 

Test condition 

Fault level 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation 

(%) 

Compressor 

Leakage (CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air 

flow rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge 

(UC) 

Refrigerant 

overcharge 

(OC) 

No Fault (NF) 

4 2.5 NF A 1 -3.40 2.52E-08 3.33E-19 1.26E-15 4.15E-19 1.99E-06 1.01E-16 0.0133 

4 5.0 CMF A 1 -5.51 2.05E-05 0 0 0 1.54E-08 0 3.53E-20

4 9.3 CMF A 1 -8.88 0.015091 0 0 0 0 0 0 

4 11.4 CMF A 1 -10.84 0.209672 0 0 0 0 0 0 

 

5 4.0 CMF A 1 -3.20 9.35E-13 0 0 0 3.11E-13 0 1.73E-26

5 6.7 CMF A 1 -5.53 6.12E-12 0 0 0 0 0 0 

5 9.5 CMF A 1 -7.39 3.45E-11 0 0 0 0 0 0 

5 27.2 CMF A 1 -23.03 0.385201 0 0 0 0 0 0 

5 38.2 CMF A 1 -33.50 0.378976 0 0 0 0 0 0 

 

8 2.5 NF A 0 -2.53 1.24E-07 4.38E-21 1.93E-21 2.89E-26 7.21E-05 2.80E-23 0.000226

8 5.5 CMF A 1 -4.95 0.000691 0 0 0 2.35E-11 0 3.76E-33

8 9.0 CMF A 1 -7.92 0.096084 0 0 0 0 0 0 

9 2.4 UC A 0 -1.28 3.53E-07 0 1.31E-18 1.08E-23 5.07E-05 0 1.41E-07

9 5.6 CMF A 1 -3.89 1.61E-05 0 0 0 8.79E-07 0 0 

9 8.9 CMF A 1 -7.06 0.071916 0 0 0 0 0 0 

9 11.5 CMF A 1 -9.25 0.113017 0 0 0 0 0 0 
71

Table 5.13 shows the performance of the FDD system with an improper outdoor air flow fault (CF) with 

the TXV in superheat zones A and B. At fault levels near 20 %, the correct detection and diagnosis was 

made. At fault levels less than 20 %, no-fault (NF) and compressor leakage fault (CMF) were incorrectly 

indicated, but the EER degradation was less than 3 % for these cases. For test condition #4 at a fault level 

of 10 %, EER was degraded by 1.9 % while the FDD system indicated NF. As fault level increased to 

20 %, the compressor fault was correctly detected and diagnosed while EER degraded by 5.7 %. For test 

condition #5, the CF was not diagnosed at fault levels of 5 % and 10 % with an EER degradation of 0.4 % 

and 1.6 %, respectively. For test condition #8, the CF was incorrectly diagnosed as a CMF fault while 

EER was degraded by 0.9 %. 

 

Table 5.13 FDD results for lowered outdoor air flow fault (CF) 

Probability for each case 

Test condition 

Fault level 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation 

(%) 

Compressor 

Leakage (CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air 

flow rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge 

(UC) 

Refrigerant 

overcharge 

(OC) 

No Fault (NF) 

4 10 NF A 0 -1.86 4.35E-24 6.75E-10 9.80E-20 3.08E-24 1.23E-19 0.000165 0.07567 

4 20 CF A 1 -5.68 0 0.288118 0 0 0 0 0 

4 35 CF A 1 -9.82 0 3.35E-07 0 0 0 0 0 

 

5 5 NF A 0 -0.43 2.14E-23 9.73E-13 6.64E-18 1.26E-27 5.00E-19 6.99E-06 0.659723

5 10 CMF A 0 -1.61 0 3.70E-09 0 0 0 9.90E-05 0.000196

5 20 CF B 1 -5.95 0 2.72E-05 0 0 0 0 0 

5 35 CF B 1 -16.54 0 0.999999 0 0 0 0 0 

5 50 CF B 1 -36.88 0 1 0 0 0 0 0 

 

8 10 CMF B 0 -0.93 1.26E-28 1.85E-07 1.24E-28 6.85E-37 2.30E-26 1.75E-06 0.013527

8 20 CF B 1 -5.23 0 2.82E-01 0 0 0 0 0 

8 35 CF B 1 -12.19 0 0.000877 0 0 0 0 0 

9 5 CMF B 0 -0.71 1.41E-33 3.32E-19 2.85E-19 3.54E-24 9.69E-28 0.062688 0.063006

9 10 NF B 0 -1.51 0 1.24E-10 0 0 0 0.006799 4.12E-06

9 20 CF B 1 -5.66 0 0.264652 0 0 0 0 0 

9 35 CF B 1 -11.91 0 0.005673 0 0 0 0 0 
72

Table 5.14 shows the fault detection and diagnosis performance of the FDD algorithm for a lowered 

indoor air flow rate. Note that even with a more than 30 % reduction in indoor air flow, the EER 

decreased by less than 10 %. The dry-coil test #13 showed the largest decrease in EER. For test 

conditions #5 and #9, a NF was indicated with an approximately 6 % decrease in indoor air flow rate, 

while EER was negligibly effected. Otherwise, the FDD algorithm performed very. 

Table 5.14. FDD results for lowered indoor air flow fault (EF) 

Probability for each case 

Test condition 

Fault level 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation 

(%) 

Compressor 

Leakage (CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air 

flow rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge 

(UC) 

Refrigerant 

overcharge 

(OC) 

No Fault (NF) 

4 13.0 EF A 0 -1.79 0 0 0.779818 0 0 2.59E-48 5.42E-35

4 22.5 EF A 0 -2.45 0 0 0.751554 0 0 0 0 

4 31.4 EF A 1 -3.77 0 0 0.305285 0 0 0 0 

 

5 5.9 NF A 0 -0.18 0 0 1.27E-07 0 0 3.67E-11 0.000678

5 11.0 EF A 0 -0.67 0 0 0.000713 0 0 9.98E-24 6.04E-16

5 20.7 EF A 0 -1.39 0 0 0.710153 0 0 0 0 

5 30.8 EF A 0 -2.73 0 0 0.210412 0 0 0 0 

 

8 11.8 EF A 0 -0.99 0 0 1.99E-05 0 0 7.79E-23 2.32E-19

8 21.1 EF A 0 -1.59 0 0 0.010518 0 0 0 0 

8 33.1 EF A 1 -3.50 0 0 2.88E-06 0 0 0 0 

9 5.7 NF A 0 -0.42 4.76E-35 1.08E-34 0.000371 1.22E-19 7.64E-22 7.61E-08 0.003588

9 10.6 EF A 0 -0.65 0 0 0.026696 0 0 4.75E-20 8.03E-14

9 20.9 EF A 0 -0.62 0 0 0.957496 0 0 0 0 

9 30.5 EF A 0 -1.73 0 0 0.079909 0 0 0 0 

 

12 14.3 EF A 0 -2.49 0 0 0.674034 0 0 0 0 

12 24.1 EF A 1 -4.43 0 0 0.039897 0 0 0 0 

12 34.6 EF A 1 -6.70 0 0 1.56E-07 0 0 0 0 

13 6.1 EF A 0 -0.80 0 0 0.034282 0 0 2.80E-38 6.88E-20

13 11.0 EF A 0 -1.57 0 0 0.208393 0 0 0 0 

13 21.1 EF A 1 -4.08 0 0 0.147573 0 0 0 0 

13 30.9 EF A 1 -7.63 0 0 8.77E-05 0 0 0 0 
73

Table 5.15 shows the performance of the FDD algorithm with a liquid line restriction fault imposed. A 

large LL fault level was needed before EER degradation exceeded 3 %. At fault levels where the EER 

degradation exceeded 3 %, the fault was correctly diagnosed. At EER degradations lower than 3 %, the 

LL fault tended to be diagnosed as NF. 

Table 5.15. FDD results for liquid-line restriction fault (LL) 

Probability for each case 

Test condition 

Fault level 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation 

(%) 

Compressor 

Leakage (CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air 

flow rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge 

(UC) 

Refrigerant 

overcharge 

(OC) 

No Fault (NF) 

4 3.7 NF A 0 -1.48 9.18E-21 2.44E-16 1.24E-10 2.36E-14 2.01E-14 4.62E-05 0.665639

4 7.0 NF A 0 -1.08 5.15E-16 2.28E-13 2.17E-11 3.17E-14 9.68E-11 2.23E-07 0.777571

4 12.5 NF A 0 -0.94 4.24E-16 9.84E-15 1.78E-15 2.29E-12 4.38E-10 3.68E-10 0.123811

4 19.0 LL B 0 -1.24 0 3.47E-16 5.20E-49 2.96E-08 4.08E-18 1.23E-21 1.26E-36

 

5 5.3 NF A 0 -0.22 1.69E-19 6.68E-14 4.72E-14 1.06E-25 4.79E-16 1.28E-07 0.793808

5 10.4 NF B 0 -0.21 5.81E-16 9.50E-12 2.13E-09 7.25E-09 2.29E-07 8.36E-11 0.055037

5 20.2 LL B 1 -5.89 0 0 0 0.996187 0 0 0 

5 31.9 LL B 1 -24.43 0 0 0 1 0 0 0 

 

8 4.4 NF A 0 -0.22 1.33E-24 3.14E-15 1.35E-14 1.25E-20 1.32E-18 0.000209 0.607407

8 8.8 NF A 0 0.34 1.98E-22 2.77E-13 4.07E-15 5.14E-24 3.18E-18 3.83E-06 0.717091

8 13.3 NF A 0 0.22 1.05E-28 2.16E-15 5.06E-18 5.77E-24 1.86E-25 0.017534 0.175031

 

9 7.1 NF A 0 0.35 2.65E-11 2.32E-15 1.49E-08 9.25E-15 1.24E-05 3.57E-14 0.557706

9 13.3 NF A 0 0.21 5.64E-20 2.36E-15 1.57E-15 5.67E-13 6.70E-16 4.61E-08 4.63E-05

9 20.0 LL B 0 -0.44 0 0 0 0.002216 0 0 0 
74

Table 5.16 shows that the FDD algorithm correctly diagnosed the undercharged refrigerant fault at all 

fault levels presented for both TXV zones A and B as seen in Figure 5.10. This is very promising due to 

the common occurrence of this fault. 

Table 5.16. FDD results for refrigerant undercharge fault (UC) 

Probability for each case 

Test condition 

Fault level 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation 

(%) 

Compressor 

Leakage (CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air 

flow rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge 

(UC) 

Refrigerant 

overcharge 

(OC) 

No Fault (NF) 

4 10 UC A 0 -2.87 7.03E-08 0 0 0 0.099239 0 0 

4 20 UC B 1 -3.56 0 0 0 0 0.999967 0 0 

4 30 UC B 1 -68.81 0 0 0 0 1 0 0 

 

5 10 UC B 0 -2.95 0 0 0 0 0.999208 0 0 

5 20 UC B 1 -7.25 0 0 0 0 1 0 0 

5 30 UC B 1 -20.14 0 0 0 0 1 0 0 

8 10 UC A 0 -2.36 6.75E-15 0 0 0 0.011534 0 0 

8 20 UC A 1 -4.09 8.54E-17 0 0 0 0.001896 0 0 

8 30 UC B 1 -9.45 0 0 0 0 1 0 0 

9 10 UC B 0 -0.87 3.61E-17 0 0 0 0.005703 0 0 

9 20 UC B 1 -6.06 0 0 0 0 1 0 0 

9 30 UC B 1 -15.79 0 0 0 0 1 0 0 
75

Table 5.17 shows the performance of the FDD algorithm for an overcharged refrigerant fault. In all cases 

the fault was correctly diagnosed, even when the EER degradation was below 3 %. 

Table 5.17. FDD results for refrigerant overcharge fault (OC) 

Probability for each case 

Test condition 

Fault level 

Diagnosed system 

status 

TXV zone 

EER warning 

EER degradation 

(%) 

Compressor 

Leakage (CMF) 

Improper OD air 

flow rate (CF) 

Improper ID air 

flow rate (EF) 

Liquid-line 

restriction (LL) 

Refrigerant 

undercharge 

(UC) 

Refrigerant 

overcharge 

(OC) 

No Fault (NF) 

4 10 OC A 0 -0.95 0 0 0 0 0 0.176407 0 

4 20 OC A 0 -1.89 0 0 0 0 0 0.854211 0 

4 30 OC A 1 -3.41 0 0 0 0 0 0.696819 0 

 

5 10 OC A 0 1.14 0 0 0 0 0 0.002974 1.30E-19

5 20 OC A 0 -0.10 0 0 0 0 0 0.646483 0 

5 30 OC A 0 -2.84 0 0 0 0 0 0.015035 0 

8 10 OC A 0 -0.92 0 0 0 0 0 0.214176 0 

8 20 OC A 0 -2.21 0 0 0 0 0 0.051813 0 

8 30 OC A 1 -3.71 0 0 0 0 0 0.002555 0 

9 10 OC A 0 0.90 0 0 0 0 0 0.00027 1.11E-20

9 20 OC A 0 -0.21 0 0 0 0 0 0.142298 0 

9 30 OC A 0 -2.32 0 0 0 0 0 0.029946 0 

The FDD algorithm correctly diagnosed all faults causing an EER degradation of more than 3.5 %. 

Misdiagnosed faults occurred under the conditions noted in Table 5.17 at a maximum EER degradation of 

3.4 %. 

The liquid line restriction fault produced the highest number of misdiagnosed faults, but these faults 

occurred with a 1.5 % or less degradation in EER. Liquid line restriction was consistently misdiagnosed 

as a NF (no-fault) even at fault levels as high as 13 %. Even at this fault level, the liquid line fault was 

not degrading system EER. EER was relatively insensitive to this fault at the fault levels and test 

conditions of this study. 
76

Table 5.18. Misdiagnosed faults 

Single-Fault 

Imposed 

Indoor to Outdoor 

Temperature Lift (oC) Fault Level (%) Fault Indicated EER degradation (%) 

CMF 6.7 2.5 NF 3.4 

CMF 16.7 2.5 NF 2.5 

CMF 11.1 2.4 UC 1.3 

CF 6.7 10 NF 1.9 

CF 1.1 5 NF 0.4 

CF 1.1 10 CMF 1.6 

CF 16.7 10 CMF 0.9 

CF 11.1 5 CMF 0.7 

CF 11.1 10 NF 1.5 

EF 1.1 5.9 NF 0.2 

EF 11.1 5.7 NF 0.4 

LL 6.7 3.7 NF 1.5 

LL 6.7 7.0 NF 1.1 

LL 6.7 12.5 NF 0.9 

LL 1.1 5.3 NF 0.2 

LL 1.1 10.4 NF 0.2 

LL 16.7 4.4 NF 0.2 

LL 16.7 8.8 NF 0.0 

LL 11.1 7.1 NF 0.0 

LL 11.1 13.3 NF 0.0 
77

CHAPTER 6. CONCLUDING REMARKS 

Single-fault test data for an R410A residential unitary split heat pump tested by Kim et al. (2006) in the 

cooling mode was used to develop an FDD algorithm. This heat pump was equipped with a TXV for the 

cooling mode expansion device; the TXV tried to maintain evaporator exit superheat even though a fault 

was occurring. This added level of control meant that faults occurring when the TXV was within its 

operational range (Zone A in Figure 5.10) or outside of its operational range (Zone B) had to be treated 

differently. The response of system temperatures to these single-faults was monitored, and their changes 

during faulty operation formed the basis for the FDD algorithm. 

The steady-state detector, SSD, acted as a filter of the dependent variables (features). The SSD was based 

upon a moving window in the which standard deviation of every feature used by the FDD algorithm must 

fall below a threshold value before the system is defined as steady-state. The feature standard deviation 

thresholds were based upon no-fault steady state standard deviations multiplied by a factor of three. The 

sampling rate and size of the moving window were determined by transient (start-up) tests of the no-fault 

system. 

Once a SSD was developed, a steady-state reference model was formulated. Two different types of 

feature-correlating models were considered; a multi-variate polynomial regression (MPR) and an artificial 

neural network (ANN). The 3rd order MPR model produced a better fit than the ANN and was further 

employed in the FDD system. The 3rd order MPR model was exposed to an F-test to illustrate how the 

number of coefficients could be reduced to produce a more compact correlation. 

Once the no-fault reference model was developed, the cooling mode heat pump’s features (temperatures) 

could be compared to their reference model counterparts and residual values calculated. These residuals 

were positive, negative or neutral with respect to their no-fault values. A feature’s neutrality was defined 

by how far its residual was from zero. A feature’s neutrality threshold was defined by three sources of 

uncertainty; measurement uncertainty, lack of measurement repeatability, and modeling uncertainty. 

Each of these uncertainty terms was multiplied by a coverage factor to ensure the desired confidence level, 

and then a square rooted sum of squares was taken to represent the total uncertainty or neutral threshold 

value for each feature. A rule-based chart was then generated by noting that certain faults produced a 

certain arrangement of positive, negative, or neutral residuals within the FDD feature set. Also, the 

probability of a residual being positive, negative, or neutral was calculated allowing for the determination 

of a fault probability relative to a no-fault probability. When the fault probability was greater than the nofault probability, a fault was diagnosed in the system. 

The person implementing this FDD method should be aware of the following points: 

1. When this FDD algorithm is applied to a real system, some method of determining when 

to notify the homeowner or responsible authority must be implemented. A fixed 

percentage degradation in EER can be selected as the notification impetus. Calculating 

EER using compressor maps and measured temperatures proved to be very effective due 

to the accurate compressor maps for the tested system. 

2. Using only temperatures is not the preferred method for determining the suction and 

discharge refrigerant saturation temperatures needed by the power and mass flow 

compressor map equations. A more accurate FDD system would include suction and 

discharge pressure transducers to provide input to the compressor map equations and 

allow more accurate superheat and subcooling calculations. 
78

3. Other implementations could choose some parameter other than EER as a notification 

impetus, or the fault level itself could be correlated and used. 

Future work should focus on developing a complete FDD module for a residential heat pump/AC system 

that is “self training.” Adaptable FDD algorithms are necessary for split, field-installed heat pump 

systems because each installation is unique. A rigid FDD algorithm will not perform correctly under all 

installation conditions; therefore, a method for evolving the reference model for newly installed systems 

must be developed. 
79

REFERENCES 

Aaron, A. A. and Domanski, P A., 1989, “An experimental investigation and modeling of the flow rate of 

refrigerant 22 through the short tube restrictor,” NISTIR 89-4120, National Institute of Standards 

and Technology, Washington, DC. 

Anderson, D., Graves, L., Reinert, W., Kreider, J.F., Dow, J., and Wubbena, H., 1989, “A Quasi-RealTime Expert System for Commercial Building HVAC Diagnostics,” ASHRAE Transactions, Vol. 

95, Part 2, pp. 25-28. 

ARI, 2004, “Performance rating of positive displacement refrigerant compressors and compressor units,” 

ARI Standard 540, Air Conditioning and Refrigeration Institute, Arlington, VA, USA 

ARI, 2006, “Performance rating of unitary airconditioning and airsource heat pump equipment,” ARI 

Standard 210/240, Air Conditioning and Refrigeration Institute, Arlington, VA, USA 

Breuker, M.S. and Braun, J.E., 1998a, “Common faults and their impacts for rooftop air conditioners,” 

International Journal of Heating, Ventilating, Air-Conditioning and Refrigerating Research, Vol. 

4, No. 3, pp. 303-318. 

Breuker, M.S. and Braun, J.E., 1998b, “Evaluating the performance of a fault detection and diagnostic 

system for vapor compression equipment,” HVAC&R Research, Vol. 4, No. 4, pp. 401-425. 

Cao S, Rhinehart R. An efficient method for on-line identification of steady-state. J Process Control 

1995;5(6);363-374. 

Castro, N.S., 2002, “Performance Evaluation of a Reciprocating Chiller Using Experimental Data and 

Model Predictions for Fault Detection and Diagnosis,” ASHARE Transactions, Vol. 108, Pt. 1, pp. 

889-903. 

Chen, B. and Braun, J.E., 2001, “Simple Rule-Based Methods for Fault Detection and Diagnostics 

Applied to Packaged Air Conditioners,” ASHRAE Transactions, Vol. 107, Pt. 1, pp. 847-857. 

Comstock, M.C., Braun, J.E., and Groll, E.A., 2001, “The Sensitivity of Chiller Performance to Common 

Faults,” HVAC&R Research, Vol. 7, No. 3, pp. 263-279. 

Dabiri, A. E. and Rice, C. K., 1981, “A compressor simulation model with corrections for the level of 

suction gas superheat,” ASHRAE Transactions, Vol. 87, Part 2, pp. 771-782. 

Franklin GF, Powell JD, Emami-Naeini A, 1991, Feedback control of dynamic systems (2nd Edition), 

Addison-Wesley Publishing Company, Reading, MA USA, 1991;617. 

Glass AS, Gruber P, Roos M, Tödtli J. Qualitative model-based fault detection in air-handling units. 

IEEE Control Systems Magazine, 1995;15(4);11-22. 

Gordon, J. M. and Ng, K. C., 1995, “Predictive and diagnostic aspects of a universal thermodynamic 

model for chillers,” International Journal of Heat and Mass Transfer, Vol. 38, No. 5, pp. 807-818. 

Graybill, F. A. and Iyer, H. K., 1994a, Regression analysis: concepts and applications, 2nd edition, 

Duxbury Press, Belmont California, pp. 239-241. 

Graybill, F. A. and Iyer, H. K., 1994b, Regression analysis: concepts and applications, 2nd edition, 

Duxbury Press, Belmont California, pp. 431-437. 

Grimmelius, H.T., Woud, J.K., and Been, G., 1995, “On-line Failure Diagnosis for Compression 

Refrigeration Plants,” International Journal of Refrigeration, Vol. 18, No. 1, pp. 31-41. 

Hassoun, M.H., 1995, “Fundamentals of Artificial Neural Networks”, The MIT Press, Cambridge, MA. 

Isermann, R. 1984, “Process fault detection based on modeling and estimation methods—A survey,” 

Automatica, Vol. 20-4, 387-404. 

Jiang T, Chen B, He X, Stuart P. Application of steady-state detection method based on wavelet 

transform. Computers and Chemical Engineering 2003;27(4);569-578 

Kim, M. and Kim, M.S., 2005, “Performance Investigation of a Variable Speed Vapor Compression 

System for Fault Detection and Diagnosis,” International Journal of Refrigeration, Vol. 28, No. 4, 

pp. 481-488. 
80

Kim, M., Payne, W.V., Hermes, C.J.L., and Domanski, P.A., 2006, “Performance of a Residential Air 

Conditioner at Single-Fault and Multiple-Fault Conditions,” NISTIR 7350, National Institute of 

Standards and Technology, Gaithersburg, MD. 

Kim, M., Yoon, S. H., Payne, W. V., and Domanski, P. A., 2008, “Design of a steady-state detector for 

fault detection and diagnosis of a residential air conditioner,” International Journal of 

Refrigeration 31(5), 790-99. 

Kramer, M.A., 1987, “Malfunction Diagnosis Using Quantitative Models with Non-Boolean Reasoning 

in Expert Systems,” AIChE Journal, Vol. 33, No. 1, pp. 130-140. 

Kramer, M.A. and Palowitch, B.L., 1987, “Rule-Based Approach to Fault Diagnosis Using the Signed 

Directed Graph,” AIChE Journal, Vol. 33, No. 7, pp. 1067-1078. 

Krandel, A., 1992, Fuzzy Techniques in Pattern Recognition, Wiley, New York. 

Lee, W.S., Grosh, D.L., Tillman, F.A., and Lie, C.H., 1985, “Fault Tree Analysis, Methods, and 

Applications – A Review,” IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 194-203. 

Lee, W.Y., Park, C., and Kelly, G.E., 1996a, “Fault Detection in an Air-Handling Unit Using Residual 

and Recursive Parameter Identification Methods,” ASHRAE Transactions, Vol. 102, Part 1, pp. 

528-539. 

Lee, W.Y., House, J.M., Park, C., and Kelly, G.E., 1996b, “Fault Diagnosis of Air-Handling Unit Using 

Artificial Neural Networks,” ASHRAE Transactions, Vol. 102, Part. 1, pp. 540-549. 

Lemmon EW, McLinden MO, Huber ML., 1998, NIST Reference fluid thermodynamic and transport 

properties – REFPROP, Version 7.0. National Institute of Standards and Technology, 

Gaithersburg, MD USA. 

Li, H. and Braun, J.E., 2003, “An Improved Method for Fault Detection and Diagnosis Applied to 

Packaged Air Conditioners,” ASHRAE Transactions, Vol. 109, Pt. 2, pp. 683-692. 

Li H. A decoupling-based unified fault detection and diagnosis approach for packaged air conditioners. 

Ph.D. Thesis, West Lafayette, IN: Purdue University, 2004. 

Mahuli SK, Rhinehart R, Riggs JB. Experimental demonstration of non-linear model-based in-line 

control of pH. J Process Control 1992;2(3);145-153. 

Navarro-Esbri J, Torrella E, Cabello R., 2007, “A vapour compression chiller fault detection technique 

based on adaptive algorithms. Application to on-line refrigerant leakage detection,” International 

Journal of Refrigeration, Vol. 29, pp. 716-723. 

Norford, L.K. and Little, R.D., 1993, “Fault Detection and Load Monitoring in Ventilation System,” 

ASHRAE Transactions, Vol. 99, Part 1, pp. 590-602. 

Ott, L., 1984, An introduction to statistical methods and data analysis, 2nd edition, PWS Publishers, 

Duxbury Press, Boston, Massachusetts, ISBN 0-87150-473-1, pp. 438-443. 

Pape, F.L.F., Mitchell, J.W., and Beckman, W.A., 1991, “Optimal Control and Fault Detection in 

Heating, Ventilating, and Air-Conditioning System,” ASHRAE Transactions, Vol. 97, Part 1, pp. 

729-745. 

Peitsman, H.C. and Bakker, V.E., 1996, “Application of Black-Box Models to HVAC Systems for Fault 

Detection,” ASHRAE Transactions, Vol. 102, Part. 1, pp. 628-640. 

Proctor, J., 2004, “Residential and Small Commercial Central Air Conditioning; Rated Efficiency isn’t 

Automatic,” Presentation at the Public Session. ASHRAE Winter Meeting, January 26, Anaheim, 

CA. 

Rossi, T.M., 1995, Detection, diagnosis, and evaluation of faults in vapor compression cycle equipment, 

Ph.D. Thesis, Purdue University, West Lafayette, IN, USA 

Rossi, T.M. and Braun, J.E., 1997, “A statistical, rule-based fault detection and diagnostic method for 

vapor compression air conditioners, International Journal of Heating, Ventilating, AirConditioning and Refrigerating Research,” Vol. 3, No. 1, pp. 19-37. 

Rossi, T.M., 2004, “Unitary Air Conditioner Field Performance,” International Refrigeration and Air 

Conditioning Conference at Purdue, Paper No. R146, July 12-15, West Lafayette, IN. 
81

Schein, J. and Bushby, S., 2005, “A Simulation Study of a Hierarchical, Rule-Based Method for SystemLevel Fault Detection and Diagnostics in HVAC Systems,” NISTIR 7216, National Institute of 

Standards and Technology, Gaithersburg, MD. 

Smith, V.A. and Braun, J.E., 2003, “Fault Detection and Diagnostics for Rooftop Air Conditioners,” Final 

Report Compilation for Project 2.1, Publication #P500-03-096, California Energy Commission, 

http://www.archenergy.com/cec-eeb/reports.htm 

Stylianou, M. and Nikanpour, D., 1996, “Performance Monitoring, Fault Detection and Diagnosis of 

Reciprocating Chillers,” ASHRAE Transactions, Vol. 102, Pt. 1, pp. 615-627. 

Stylianou, M., 1997, “Application of Classification Functions to Chiller Fault Detection and Diagnosis,” 

ASHRAE Transactions, Vol. 103, Pt. 1, pp. 645-656. 

Tzafestas, S.G., 1989, “System Fault Diagnosis Using the Knowledge-Based Methodology,” Fault 

Diagnosis in Dynamic Systems Theory and Applications, Prentice Hall, New York, pp. 514-543. 

Wasserman, P.D., 1989, “Neural Computing Theory and Practice”, Van Nostrand Reinhold, New York, 

NY. 
82

APPENDIX A. FDD MODULES 

This appendix presents individual windows of the FDD system developed and implemented on the split 

residential system used in this study. 

A.1 Main Module 
83

A.2 Moving Window Module 

A.3 Steady-State Detector Module 
84

A.4 Preprocessor Module 
85

A.5 Reference Model Correlation Interface 
86

A.6 No-Fault MPR Reference Module 

A.7 Statistical Rule-Based FDD Module