Influence of Three Sizes of Sliding Windows on Principle Component Analysis Fault Detection of Air Conditioning Systems

YANG Xuebin(楊學賓), MA Yanyun(馬艷云), HE Ruru(何如如), WANG Ji(王 吉), LUO Wenjun(羅雯軍)

1 College of Environmental Science and Engineering, Donghua University, Shanghai 201620, China2 Shanghai Division, China Ship Development and Design Center, Shanghai 201108, China

Abstract: Principal component analysis(PCA) has been already employed for fault detection of air conditioning systems. The sliding window, which is composed of some parameters satisfying with thermal load balance, can select the target historical fault-free reference data as the template which is similar to the current snapshot data. The size of sliding window is usually given according to empirical values, while the influence of different sizes of sliding windows on fault detection of an air conditioning system is not further studied. The air conditioning system is a dynamic response process, and the operating parameters change with the change of the load, while the response of the controller is delayed. In a variable air volume(VAV) air conditioning system controlled by the total air volume method, in order to ensure sufficient response time, 30 data points are selected first, and then their multiples are selected. Three different sizes of sliding windows with 30, 60 and 90 data points are applied to compare the fault detection effect in this paper. The results show that if the size of the sliding window is 60 data points, the average fault-free detection ratio is 80.17% in fault-free testing days, and the average fault detection ratio is 88.47% in faulty testing days.

Key words: sliding window; principal component analysis(PCA); fault detection; sensitivity analysis; air conditioning system

Introduction

Air handling units(AHUs) are an important part of air conditioning systems. Their operating condition significantly affects the system performance and thermal comfort of heating, ventilating and air conditioning systems. Unfortunately, many faults from sensors, controllers, filters, fans or even coils could affect the normal operation of AHU system and could increase energy consumption.

Many researchers have proposed various fault detection methods to solve the problem[1-2]. Principal component analysis(PCA) method based on multivariate statistics has been widely employed in the field of automated fault detection and diagnosis(AFDD) of AHU[3-4]. Considering the data preprocessing problem of PCA method, Li and Wen[5]proposed a pattern matching method, which adopted sliding windows to search for the historical data windows similar to the current snapshot data. One of the important problems was the selection of training data for PCA method. The training data should be selected accurately or concisely to improve the sensitivity of fault detection. In addition, some measuring errors or system operation uncertainties, might reduce the fault detection performance of PCA method.

The sliding window and sliding speed has been employed in some published literatures. Singhal and Seborg[6-7]used the sliding window with 1 024 data points, and selected 1/10, 1/5, 1/3 and 1/2 of the sliding window size as the sliding speedw. The performance of pattern matching decreased slowly with the increase ofw, and it was suggested thatwcould be 1/10 to 1/5 of the window size, which could match very effectively. Lietal.[8]proposed a method to select the length of time window in dynamic PCA, and found that compared with the traditional PCA method, the dynamic PCA method with appropriate time window length and sliding window width could detect and identify faults better. Ayechetal.[9]selected fixed sliding window sizes with 40, 60, 80 data points, and compared with their PCA fault detection effects based on adaptive sliding windows. Deng and Tian[10]proposed a method of fault pattern recognition, the size of sliding window was 10-60 data points, and pattern recognition was carried out in the selected 300 sample observations. In Ref. [11], the sliding window of 50 data points was employed to simulate and test five kinds of faulty pattern data including 700 samples. Li and Wen[5]adopted the sliding window with 30 data points, and the sliding speedwwas from 1/10 to 1/5 of the sliding window size according to the empirical value. They tested the static pressure sensor bias, outdoor air damper stuck, cooling coil air side fouling, heating coil valve leakage and other faults of the AHU system, and achieved remarkable fault detection effect. Chen and Wen[12]used the sliding window with 30 data points to search for similar historical data. Sheriffetal.[13]proposed a sliding window generalized likelihood ratio detection method based on multi-scale PCA, and compared it with the traditional PCA method. Sukhanovetal.[14]proposed a real-time pattern matching method to support multiple templates. The window size was dynamic, and it showed better performance than other pattern matching methods. Zhuangetal.[15]studied dynamic functional connectivity based on the sliding window method, calculated and verified the window size of single scale time-dependent in dynamic functional connectivity analysis.

It is crucial to evaluate the fault detection effect of pattern matching combined with PCA method. Singhal and Seborg[16]applied the accuracy of candidate pool, searching efficiency and their average to evaluate the effectiveness of pattern matching method. Many researchers[17-20]employed fault detection ratio to express the percent whether the squared prediction error(SPE)Eswas greater than the threshold. Some researchers[21-22]adopted evaluation indexes such as fault detection accuracy, false alarm rate, and compared the fault time to evaluate the sensitivity of fault detection.

From the above research, literatures have seldom investigated on the optimal sliding window size especially for air conditioning systems. Most of them applied the sliding window with a given size, but did not further study the influence of different sliding window sizes on fault detection. In this paper, based on the field measuring data of a variable air volume(VAV) air conditioning system controlled by total air volume method, three different sizes of sliding windows with 30, 60 and 90 data points are selected, and the fault detection ratio is taken as the evaluation index to compare the influence of different sizes of sliding windows on the PCA fault detection of an air conditioning system.

1 Fault Detection Method Based on Different Sliding Window Sizes

The fault detection process based on different sizes of the sliding window is shown in Fig. 1. The method is divided into two stages. The first stage is the threshold training stage, which employs three different sizes of sliding windows to search for the historical fault-free reference data similar to the current snapshot data. The similarity factor values are sorted in a descending order, and the top five data windows with the highest similarity factor value are selected into the candidate pool. The second stage is the online detection stage, which collects the measuring data, selects the current snapshot data window, carries on the standardized processing, calculates the value of SPE, also known asQstatistics, and compares the SPE value with the thresholdδ. If the SPE value is greater thanδ, a fault is flagged.

Fig. 1 Flow chart of the fault detection method based on different sliding window sizes

1.1 Sliding window

The sliding window moves on the historical fault-free reference data, and the sliding speedwis from 1/10 to 1/5 of the sliding window size. One similarity factor for each time point is calculated, and all the similarity factors are sorted in a descending order. The historical fault-free data windows with the largest similarity factor are selected to form the historical fault-free reference data. If duplicate data appear in the reference data, new data from the next top data windows will be selected to replace the duplicate data.

1.2 PCA similarity factor

The similarity factor is an important measuring parameter for pattern matching, which is used to characterize the similarity between current snapshot data and historical fault-free reference data. The current snapshot data are defined asSand the historical fault-free reference data are defined asH. Both are composed ofnvariables andmdata points. Suppose thatk1andk2are the principal components ofSandH, respectively, the firstkprincipal components are selected to form the eigenvector matrices ofLandM, and the PCA similarity factor is calculated by

(1)

whereSprepresents PCA similarity factor, tr is a computer term to calculate the sum of diagonal elements of a two-dimensional square matrix,Ldenotes principal component space ofS,Mmeans principal component space ofH, andkis the number of principal components.

1.3 Fault detection ratio

If SPE value of the data point is greater than thresholdδ, it means that the data point is faulty, otherwise it is fault-free. Fault detection ratioRis the ratio of the number of faulty data points to the total number of data points in the current snapshot data window. Since the field measuring data in this paper is collected per minute, the fault detection ratio can also be expressed by the time proportion of SPE exceeding the thresholdδin a period of time. The fault detection ratioRis calculated by

(2)

whereNfdenotes the number of time points which detects a fault andNmeans total number of time points.

2 Results and Discussion

The field measured data of 5 fault-free testing days and 18 faulty testing days were employed to explore the influence of three sizes of sliding windows. The data were recorded from 9:00 to 17:00 every day. The sizes of the sliding windows are 30, 60 and 90 data points, respectively, which are the same as those of the current snapshot data windows. The sliding speed is 1/6 of the size of a data window according to the empirical value. The sliding window is adopted to accurately reflect the time-variant air conditioning systems.

2.1 Fault-free condition

For the fault-free test, all the fault-free data other than the current snapshot data are taken as the historical fault-free dataset. July 25 and September 7 are taken as the examples of fault-free testing days to study the influence of sliding window sizes on PCA fault detection.

2.1.1Fault-freetest1

For the fault-free testing day on July 25, the fault detection results of different sliding window sizes are shown in Fig. 2. The values of fault detection ratios are labelled during each time period. The average fault detection ratio is 56.04% if the sliding window is 30 data points. The fault detection ratios of 9:00-10:00, 10:30-11:00, 11:30-12:30, 14:00-14:30, 15:00-16:00 and 16:30-17:00 are higher than 83.33%. Some fault is falsely detected in these four and a half hours, and no fault is detected in the rest of the time. The average fault detection ratio is 25.21% if the sliding window is 60 data points, and the fault detection ratios of 9:00-11:00 are all 100.00%. The fault is detected in these two hours, and no fault is detected in the rest of the time. The average fault detection ratio is 20.00% if the sliding window is 90 data points, and the fault detection ratio of 9:00-10:30 is 67.78%. The fault is detected within one and a half hours, and no fault is detected in the rest of the time.

Fig. 2 Fault-free detection results of different sizes of sliding windows on July 25: (a) 30 data points; (b) 60 data points; (c) 90 data points

2.1.2Fault-freetest2

For the fault-free testing on September 7, the fault detection results of different sliding window sizes are shown in Fig. 3. The average fault detection ratio is 15.21% if the sliding window is 30 data points, and the fault detection ratios of 9:00-9:30 and 16:00-16:30 are 43.33% and 50.00%, respectively. The fault detection ratio of 10:30-11:00 is 100.00%, and no fault is detected in the rest of the time. The average fault detection ratio is 15.83% if the sliding window is 60 data points, and the fault detection ratio of 12:00-13:00 is 100.00%. The fault is detected in this hour and no fault is detected in the rest of the time. The average fault detection ratio is 35.33% if the sliding window is 90 data points, and the fault detection ratios of 9:00-10:30 and 12:00-13:30 are 72.22% and 100.00%, respectively. The fault is detected in these three hours, and no fault is detected in the rest of the time.

Fig. 3 Fault-free detection results of different sizes of sliding windows on September 7:(a) 30 data points;(b) 60 data points;(c) 90 data points

2.2 Fault condition

For the fault test, all the fault-free data are taken as the historical fault-free datasets. Two testing days on July 29 and August 25 are taken as the examples to illustrate the influence of sliding window sizes on PCA fault detection.

2.2.1AHUcoolingcoilvalvestuckfaulttest

For the fault testing day on July 29 under the fault of AHU cooling coil valve stuck at 15% opening, the fault detection results are shown in Fig. 4. The average fault detection ratio is 88.54% if the sliding window is 30 data points. The fault detection ratios of 16:00-17:00 are lower than 16.67%. No or less fault is detected in these two half-hour, and the fault can be detected in the rest of the time. The average fault detection ratio is 93.54% if the sliding window is 60 data points, and the fault detection ratio of 16:00-17:00 is 48.33%. The fault can be detected in the rest of the time. The average fault detection ratio is 76.00% if the sliding window is 90 data points. The fault detection ratio of 15:00-16:30 is 24.44%, and no fault is detected in these one and a half hours. The fault detection ratio of 12:00-13:30 is 56.67%. The fault can be detected in the rest of the time.

2.2.2Supplyairtemperaturesensorfaulttest

For the fault testing day on August 25, the system had a fault of supply air temperature sensor. As shown in Fig. 5, the average fault detection ratio is 92.92% if the sliding window is 30 data points. The fault detection ratio of 14:30-15:00 is 33.33%, and no fault is detected in this half an hour. The fault can be detected in the rest of the time. The average fault detection ratio is 98.75% if the sliding window is 60 data points, and the fault can be detected all the time. The average fault detection ratio is 93.00% if the sliding window is 90 data points, and the fault can also be detected all the time.

Fig. 4 Fault detection results of different sizes of sliding windows on July 29:(a) 30 data points;(b) 60 data points;(c) 90 data points

Fig. 5 Fault detection results of different sizes of sliding windows on August 25:(a) 30 data points;(b) 60 data points;(c) 90 data points

3 Performance Evaluation

The size of the sliding window might have significant effect on PCA fault detection. Table 1 and Table 2 show the fault-free detection results and the fault detection results with different sliding window sizes. If the size of the sliding window is 60 data points, the average fault-free detection ratio is 80.17% in fault-free testing days and the average fault detection ratio is 88.47% in faulty testing days. The fault detection performance is the best one among three sizes of sliding windows.

Table 1 Fault-free detection results with different sliding window sizes

Table 2 Fault detection results with different sliding window sizes

4 Conclusions

This study explores the influence of three sizes of sliding windows on PCA fault detection results. The tests are conducted by the measuring data of a VAV air conditioning system. The fault detection ratio is used to evaluate the detection performance.

(1) If the size of the sliding window is defined as 60 data points, the average fault-free detection ratio is 80.17% in fault-free testing days, and the average fault detection ratio is 88.47% in faulty testing days.

(2) For the actual air conditioning system, the size of data window should be selected appropriately, which is beneficial to identify fault-free and fault status more efficiently and accurately, and thus improve the PCA fault detection performance.