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Description
Existing methods addressing automated fault detection and diagnosis (FDD) for vapor-compression air-conditioning equipment have good performance for faults that occur individually but have difficulty handling multiple simultaneous faults. In addition, these methods either require high-cost measurements or measurements over a wide range of conditions for training reference models, the development of which can be time consuming and cost prohibitive. This paper formulates model-based FDD in a generic way and demonstrates that decoupling is the key to handling multiple simultaneous faults. To eliminate a cost-prohibitive overall system model, an alternative physical decoupling methodology to mathematical decoupling is developed. During the mathematical development, a previously developed FDD method termed the statistical rule-based method is reexamined and cast within the general mathematical framework. The paper also includes an evaluation of the FDD method in terms of both sensitivity and robustness.
INTRODUCTION
Automated fault detection and diagnosis (FDD) has been successfully applied to critical systems, such as space exploration and nuclear power plants, in which early identification of small malfunctions would prevent loss of life and damage to equipment. HVAC systems often do not function as well as expected due to faults introduced during initial installation or developed during routine operation. In the late 1980s, some researchers investigated common faults and methods for FDD in simple vapor-compression cycles, such as household refrigerators (Stallard 1989). With the growing realization of the benefits brought by FDD, many publications related to HVAC FDD have appeared in the last decade (Comstock et al. 1999; Li 2004), and interest is increasing. According to the IEA ANNEX 34 final report (Dexter and Pakanen 2001), 23 prototype FDD performance monitoring tools and 3 validation tools have been developed, 30 demonstrations have taken place in 20 buildings, 26 FDD tools have been tested in real buildings, and 4 performance monitoring schemes have been jointly evaluated on 3 documented data sets from real buildings. Since 2001, 39 more papers have appeared (Li 2004). Overall, the literature can be summarized as follows.
* In terms of focus, the single largest focus of research has been on variable-air-volume (VAV) air-handling units (AHUs), accounting for 45% of the publications. Packaged air-conditioning systems come a distant second with 20% of the publications, and chiller systems have been the third focus at 18%. Rooftop and other packaged air conditioners are used extensively throughout small commercial and institutional buildings, but compared to larger systems, they tend not to be well maintained. In addition, for both packaged and chiller systems, faults occur frequently in vapor-compression cycle equipment. Widespread application of automated FDD for vapor-compression cycle equipment will significantly reduce energy use and peak electrical demand, downtime, and maintenance costs.
* From the methodology point of view, there are three general approaches for FDD, namely, model-based, data-driven, and knowledge-based. Model-based methods use mathematical models often constructed from first principles. They are applicable to information-rich and modeling-manageable systems, where satisfactory models can be built in an affordable way to satisfy FDD application and enough sensors are available. Methods described by Rossi and Braun (1997), Siegel and Wray (2002), Shaw et al. (2002), Yoshida and Kumar (2001a, 2001b), and Dexter and Ngo (2001) fall into this category. The data-driven approach addresses FDD by means of directly processing a large amount of data to capture some meaningful statistics. It mainly applies to large-scale systems, such as the whole-building system, where it is difficult to construct an analytical model but heavy instrumentation is used and an exceptionally large amount of data is produced. Methods used by Riemer et al. (2002) and Reddy et al. (2003) can be classified in this category. The knowledge-based approach uses qualitative models that are based on causal analysis, expert systems, and/or pattern recognition. It is well suited for systems where, like the data-driven approach, detailed mathematical models are not available but, unlike the data-driven approach, a large amount of data is not available. The technique presented by Gerasenko (2002) is an example. Although all three FDD approaches have found their applications in HVAC systems, the model-based approach has been most used (Li 2004). This is because most HVAC systems are relatively small-scale, not heavily instrumented, and modeling-manageable.
* Most of the proposed techniques require either expensive system models or expensive measurements or both. For example, Rossi and Braun (1997) originally proposed the statistical rule-based (SRB) FDD technique and applied it to vapor-compression systems. This technique uses relatively low-cost sensors (nine temperature and one relative humidity) but requires an expensive system model, which entails a wide range of training data. Further reducing the implementation cost is vital for a practical FDD technique.
* Multiple simultaneous faults have barely been addressed in the literature. Most of the publications have only dealt with the presence of single faults. Breuker (1997) investigated the effect of two simultaneous faults on the performance of the SRB technique, which was developed for single faults. He used a simulation model and found that the presence of two simultaneous faults in a rooftop air conditioner did not result in the diagnosis of a third fault that was not present in the system. However, the diagnostic classifier was not capable of making multiple diagnoses, and some combinations of faults increased FDD sensitivity whereas others decreased sensitivity. Further investigations are warranted for testing more simultaneous faults under real operating conditions and for developing a methodology that can diagnose multiple faults.
The goal of the research described in this paper was to develop a low-cost diagnostic methodology for handling multiple simultaneous faults in vapor-compression cycle systems with emphasis on packaged air conditioners. This paper first formulates model-based FDD techniques in a general mathematical way and finds that the methodology of decoupling is the key to handling general multiple-input and multiple-output issues. In order to apply the decoupling methodology to noncritical HVAC & R systems, a physical decoupling methodology is developed that eliminates a cost-prohibitive overall system model. Finally, the method is implemented and evaluated in terms of both sensitivity and robustness.
MATHEMATICAL FORMULATION OF MODEL-BASED FDD
In a broad sense, all FDD approaches involve the use of quantitative or qualitative models. They can be either dynamic or steady-state, either physical or black-box or gray-box, either mathematical or linguistic. Mathematical models relate measured or derived states of the system to external inputs through the use of mathematical equations, while linguistic models, also known as syntactic models, describe the behavior of a system through the use of linguistic expressions such as logic sequences (e.g., fuzzy linguistic models). Since the majority of FDD approaches, especially for HVAC & R applications, are based on mathematical models, the following development is based on mathematical models. However, the development could be extended to linguistic models as well.
The thermodynamic states of a vapor-compression cycle are functions of external driving conditions and various faults, as shown in Figure 1a. It is important for FDD not to misinterpret variations in thermodynamic state-variables caused by changes in the driving conditions for faults. If measurements are classified directly, the classification can be complicated to consider the effect of external driving conditions. In order to simplify classification and improve overall FDD performance, normal operation models are typically used to predict expected values for these measurements under normal operation in terms of measured external driving conditions. For any steady-state measurement, the difference between expected and actual measurement values (residuals) should have a zero mean when there are no faults (see Figure 1b) and a probability distribution that is a weak function of driving conditions but dominantly dependent on faults.
The input-output relationship of the system after being incorporated with a normal operation model (fault-free model) can be described approximately as follows:
Y = F(X) (1)
where X = [[x.sub.1], [x.sub.2],..., [x.sub.n]][.sup.T], Y = [y.sub.1], [y.sub.2],..., [y.sub.m]][.sup.T], and F(X) = [[f.sub.1](X), [f.sub.2](X),..., [f.sub.m](X)][.sup.T]. X is the fault vector, with each entry [x.sub.i] representing a measure of the level for each fault. Y is the state variable residual vector, with each entry [y.sub.i] representing a particular state-variable residual. F(X) is a nonlinear function vector with each individual nonlinear function [f.sub.i]([x.sub.1], [x.sub.2],..., [x.sub.n)] defining the relationship between different faults at different levels, the state-variable residual Y; n is the number of fault types considered; and m is the number of chosen state variables.
After a normal model is incorporated, FDD is simplified to deal with only faults and normal variations in residuals independently. In a broad sense, this is the first occurrence of decoupling, which is between faults and driving conditions. Typically, FDD is achieved by two separate steps--fault detection and fault diagnosis.
[FIGURE 1 OMITTED]
Fault Detection
Fault detection, which is to indicate whether the system is normal or not, can be done essentially by determining whether the resulting Y in Equation 1 is zero or not in a statistical sense. The tool used to achieve fault detection is termed the fault detection classifier. Although some quantitative fault diagnosis techniques can also do fault detection at the same time, implementing fault detection prior to attempting any diagnosis is recommended for the following reasons:
* Fault detection is much easier than fault diagnosis, and the probability for abnormal operation is lower than for normal operation. Therefore, the use of separate FDD steps can reduce computational requirement costs because the diagnosis step can be skipped when there are no faults.
* Fault detection can take statistical analysis into account easily, which makes the fault diagnosis method more flexible.
Original SRB Fault Detection Classifier. Rossi and Braun (1997) proposed a way to evaluate whether Y is zero indirectly by looking into the overlap (see Figure 1b) of the actual distribution and the expected distribution of the residual(s). When the overlap of the actual distribution and the expected distribution of the residual(s) decreases to a preset value (the classification error threshold), a fault is considered to be present.
The direct numerical integration of this overlap for high-dimensional (e.g., seven-dimensional for our case) probability distributions cannot be performed in real time on a microprocessor. Therefore, to obtain the analytical solution of overlap, Rossi and Braun (1997) employed the concept of a Bayes classifier, also known as the Bayes error (Fukunaga 1990).
The other merit of this classifier is that it converted the classification of an individual observation Y among infinite predefined classes [[omega].sub.1], [[omega].sub.2],..., [[omega].sub.n],... inversely... |
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