Compound fault diagnosis for cooling dehumidifier based on RBF neural network improved by kernel principle component analysis and adaptive genetic algorithm

Developing fault diagnosis for the cooling dehumidifier is very important for improving the equipment reliability and saving energy consumption. This paper mainly studies and explores the compound fault diagnosis for the cooling dehumidifier. Firstly, the dehumidifier data acquisition system is built, which can be applied to the data acquisition, work status simulation and fault diagnosis. Secondly, a compound fault diagnosis model based on radial basis function neural network (RBFNN) improved by kernel principle component analysis (KPCA) and adaptive genetic algorithm (AGA) is proposed. Aiming at the problems that the selection of RBF width depends on expert experience and knowledge, the network structure scale is large or the training speed is slow in the conventional RBFNN models; on the one hand, AGA and K-means clustering algorithm are employed to automatically optimize the RBF width, the number of hidden layer neurons and the neuron centers, which guarantees the model has small structure and fast running speed on the premise of sufficient output precision, and the local optimization problem of K-means clustering algorithm is well suppressed by AGA group calculation; this also ensures the performance of RBFNN; on the other hand, KPCA is employed to reduce the dimension of the model input data, which not only effectively extract the nonlinear features, but also further simplify the network structure. Finally, the proposed method is validated and compared with the conventional models. The results show that this proposed model can not only be effectively applied to the dehumidifier compound fault diagnosis, but also has prominent application advantages.


Introduction
As a kind of heating, ventilating, air-conditioning and refrigeration (HVAC&R) equipment, the cooling dehumidifier is widely applied to the fields such as building, electronic and precision instrument which need to control the environment temperature and humidity. As the same as other HVAC&R equipment, it's hard to avoid faults occurring in the cooling dehumidifier refrigeration system. When the faults occur in the equipment, there would be some risks such as: (1) the work status can't meet the technical standard so far as to lose effectiveness; (2) the work performance falls down and the energy consumption goes up; (3) the refrigerant leaks so that it destroys environment; (4) the work safety is affected by the fault, e.g., refrigeration compressor suction pressure rise. So developing fault detection and diagnosis (FDD) for cooling dehumidifier is very important for improving equipment work reliability, saving energy consumption, protecting environment, lowering safety risk, reducing equipment maintenance cost and extending equipment life (Lee and Yik 2010). The involved research literatures demonstrate that the HVAC&R system can save 20-30% energy consumption in the operation after adopting FDD technology & Yunguang Gao yung8791@126.com and optimization management (Yoshida and Kumar 1999). Katipamula and Brambley (2005a, b) reviewed the FDD methods for HVAC&R equipment; they consider these methods involve two classes: one is model based and the other is process history dada based. The artificial neural network (ANN) model belongs to a classical black model which is one of methods based on process history data. The cooling dehumidifier fault mainly includes electric fault and mechanical fault, and the latter occurrence frequency is higher than the former. The mechanical fault mainly occurs in the refrigeration system. The cooling dehumidifier refrigeration system fault can be basically divided into two classes, one is the catastrophic fault, and the other is light fault. The surveys indicate that the light fault frequently occurs and isn't easily perceived although it is less serious than the catastrophic fault. The common light fault includes evaporator fouled, condenser fouled, refrigerant insufficient, expansion valve fault, air flow line fault, cooling water line fault and so on. Due to the powerful coupling in the inner components, the nonlinear characteristic of dehumidifier system is obvious; thus, there is some difficulty to accurately diagnose the fault. When the refrigeration system faults occur, the fault symptoms and reasons are often complex; the attributes can be concluded as: (1) the symptoms represented by fault are complex; (2) the fault reasons are complex; (3) the relationship between the fault symptoms and reasons is complex, one reason can arouse multiple symptoms, and multiple reasons can represent the same symptom; (4) it's hard to find the strict logical and quantitative relationship from the system parameters, i.e., there is some fuzziness in the system; (5) there are many inner parameters to be measured. From this, it can be seen that there is some difficulty to build precise mathematic or physical model for cooling dehumidifier fault analysis and diagnosis. At the same time, the process complexity of fault occurring and developing also brings difficulty to determine the fault threshold. For the complex fault compounded of two or more single faults, there is even more difficulty to diagnose due to the parameters coupling and the more complex corresponding relationships between fault symptoms and reasons. It is also hard to find an effective method for diagnosing the cooling dehumidifier compound fault.
As a black-box model, ANN is a kind of method for fitting training data and optimizing performance parameter by input and output data. Due to the merit of high parallel disposing, nonlinear mapping, self-adapting and fault toleration, ANN is suitable for system identification and pattern recognition. The fault diagnosis can be ultimately attributed to pattern recognition problem, so ANN is widely applied to the fault diagnosis fields such as aerospace engineering, chemical industry, pharmacy, petroleum and so on. In the HVAC&R diagnosis field, methods based on ANN were widely studied and applied, e.g., Peitsman and Bakker (1996) employed ANN and autoregressive with exogenous input (ARX) model to fault diagnosis for a reciprocating chiller; Haves et al. (1996) proposed a method based on radial basis function neural network (RBFNN) and gradient regressive estimation to detect coil fouled and valve leakage faults of air handing unite (AHU); House et al. (1999) employed 7 kinds of classification methods for AHU fault diagnosis and these methods performance were analyzed by contrast; Lee et al. (2004) proposed a method based on general regression neural network (GRNN) for AHU FDD; the simulation result demonstrated the GRNN has high precision and reliability for the application to the nonlinear system such as AHU; Dun et al. (2009) Fan et al. (2010) proposed a hybrid FDD strategy for local system of AHU based on ANN and wavelet analysis; furthermore, there are many studies (Kamar et al. 2013;Mohammad et al. 2013;Jani et al. 2016) focused ANN methods on performance prognosis and analysis for airconditioning and the validity is proved by simulation or application. Gao et al. (2016) proposed a nonlinear ARX model for cooling dehumidifier fault detection, but the fault diagnosis needs to be aided by expert knowledge and the fault types study only focused on the single fault.
Reviewing these literatures, in the method research aspect, the black-box model is employed widely, e.g., the ARX model and feed-forward ANN. The ARX model is theoretically suitable for dealing with linear problems, but HVAC&R system has obvious nonlinear characteristics; ANN has strong nonlinear processing ability, so it is more suitable for fault diagnosis of HVAC&R. The most used feed-forward ANN mainly includes back-propagation neural network (BPNN) and RBFNN, the RBFNN has higher generalization ability and learning efficiency than BPNN, and it has no local minimum problem existed in BPNN (Chen et al. 2016;Wei 2017). Although RBFNN has been applied relatively successfully in fault diagnosis, the conventional RBFNN models mainly have two problems in the training. One is the balance between network structure and output precision, in the condition of the RBF width is determined, the output precision will improve with the number of neurons increase, but the structure will become complex; when the number of neurons is reduced, the structure will be simplified, and the output precision will be decreased. The structure size will affect the RBFNN online running speed; the other is the selection of model parameter, RBF width has an important impact on the performance of RBFNN, the conventional artificial selection method needs to be adjusted repeatedly by experiment based on expert experience and knowledge, so the operation is cumbersome; if it is not selected properly, it will lead to poor application effect of RBFNN.
GA is a self-adapting global optimization and probability searching algorithm, which imitates the mechanism of biology evolution such as natural selection and genetic mutation. GA has been applied to many fields such as function optimization and machine learning, e.g.,  proposed a genetic algorithm optimized ANN model for remaining useful life prediction of turbofan engine. In order to optimize the parameter and structure of RBFNN model and reduce the dependence on expert experience and knowledge during the model learning, this paper employs adaptive genetic algorithm (AGA) and K-means clustering algorithm to optimize the RBFNN structure and parameter at the same time, which guarantees that the model has smaller structure and faster running speed on the premise of sufficient output precision.
In the fault research aspect, most studies focus on single fault. When the compound fault occurs on the same time, it is harder to diagnose than the single fault due to the complex equipment parameters coupling. In fact, the compound fault composes of single fault will occur for the real equipment, so research on compound fault diagnosis is necessary despite that this work would be difficult and full of challenges. Employing the powerful nonlinear processing ability of RBFNN, the compound fault diagnosis can be decomposed single fault diagnosis. Compound fault diagnosis usually requires higher sample dimension, which will increase the structure and computational complexity of ANN. In view of this problem, this paper employs the kernel principal component analysis (KPCA) method to reduce the input sample dimension, so as to further reduce the structure of RBFNN, remove the correlation between input variables and better extract the data sample feature.
Based on the two problems of conventional RBFNN models and compound fault research in the HVAC&R fault diagnosis field, this paper proposed an improved RBFNN model based on KPCA and AGA for the cooling dehumidifier compound fault diagnosis. Moody and Darken (1989) developed a novelty RBFNN based on multidimensional space interpolation disposed by radial basis function (RBF). The RBFNN is proposed according to human cerebrum cortical adjustment and overlap receive field, which is also named local receive field neural network. It is a simplified and abstract cerebrum model. The RBFNN is superior to BPNN on the approximation, classification and learning speed, so it is widely applied to the fields such as system identification and pattern recognition, and then, it gained close attentions by many researchers (Chiddarwar and Babu 2010).

RBFNN structure
The RBFNN is a kind of special three-player ANN of feedforward neural networks. The input player unit only transmits input signal to the hidden player; the hidden player unit characteristic function employs nonlinear radial function in order to make local response to the input stimulus signal; the output player unit makes linear combination of the hidden player unit function outputs. The RBFNN structure is shown in Fig. 1.
The radial symmetry Gauss function shown in Fig. 2 is usually employed for the RBFNN nonlinear transmission function, and the transmission function in hidden player is shown as where X ¼ ½x 1 ; x 2 ; . . .; x n T is n-dimensional input vector, C j is the j-th basis function center which has the same dimension with the input vector, r j is the j-th perception variable which determines the basis function width, M is the perception unit number, jj Á jj is Euclidean distance which is defined as follows According to formula (1), when the input vector X is located in the j-th hidden player unit center C j , then jjX À C j jj ¼ 0 and u j ðXÞ ¼ 1. If X is far from C j , u j ðXÞ will be decrescendo. The output of hidden player unit j reflects the distance grade from X to the corresponding RBF center C j which can be looked as a clustering center. The transmission function can produce major output only when the input vector locates in the region nearby the clustering center. So the transmission function works as a signal detector. r j is applied to adjust the clustering center range of action. r j becomes small, u j ðXÞ descends rapidly with jjX À C j jj increasing, and the RBF transmission function output curve becomes very narrow and is sensitive to the input change; contrarily, r j becomes big; the RBF transmission function output curve becomes gentle and is insensitive to the input change.
Because the mapping from hidden player to output player is linear, the RBFNN output is the linear weight sum of hidden player unit output, which is shown as Converting formula (3) into matrix equation, the equation can be obtained as follows: where Y ¼ ½y 1 ; y 2 ; . . .; y l T is l-dimensional output vector, W ¼ ½W 1 ; W 2 ; . . .; W l T is matrix of weight from hidden player to output player, W k ¼ ½w k1 ; w k2 ; . . .; w kM is the k-th unit weight vector of output player, U ¼ ½u 1 ðxÞ; u 2 ðxÞ; . . .u M ðxÞ T is the hidden player output vector.

Learning method
Seeing the RBFNN learning process, there are two methods for training: one is the hybrid mode and the other is the supervised mode.
(1) Hybrid mode Moody and Darken proposed a hybrid method for RBFNN learning. Firstly, the RBF center is determined by unsupervised method after the number of radial basis functions is specified. Secondly, the output player weights are determined by supervised method. According to the hybrid method in unsupervised stage, there are two common methods as follows: (1) Random selecting RBF center.
As a simplest method, the RBF centers in hidden player are selected randomly from input data samples and the centers are changeless. Because the hidden player unit outputs are known, the network connection weights can be solved by linear equations.
If the Guass function is employed for RBF, it can be represented as follows: where M is the number of centers (i.e., hidden player units); d m is the max distance among the selected center. In this case, the mean square deviation (i.e., RBF width) can be specified as follows: (2) Self-organization selecting RBF center. According to the multivariable interpolation RBF methods, the center C j is usually located in the input vector point, so the hidden player units are consistent with the input data. Considering the input data maybe have clustering performance especially to the pattern recognition, there will be so many closed RBF centers that generate redundant points, the learning time becomes long, and the over fitting appears which leads to generalization ability debased. In order to overcome these problems, the input data are usually disposed by clustering analysis technology for selecting representation points as RBF centers, so the number of hidden player units is reduced and the structure complexity is debased. In this method, the RBF centers can move and be located by self-organization learning. The K-mean clustering algorithm which belongs to unsupervised method is usually employed for selecting RBF centers (Pecháč and Sága 2017); the calculation steps are as follows: (a) The clustering center C j ðj ¼ 1; 2; . . .; MÞ is initialized by selecting M samples from input samples X i ði ¼ 1; 2; . . .; NÞ. (b) The input samples are classified according to the nearest neighbor rule. In this way, for that X i is distributed to input sample clustering set h j ðj ¼ 1; 2; . . .; MÞ whose clustering center is C j , i.e., X i 2 h j , and the condition needs to be met where d j is the minimum Euclidean distance.
Calculating the mean of samples in h j , i.e., the clustering center: where M j is the number of input samples in h j . After obtained the center C j and RBF width r j , the RBFNN can be trained to adjust the weight matrix W for minimizing the solution object function. Due to the space mapping from hidden player to output player is linear, i.e., the network output is the weighted sum of hidden player unit output. The relationship between hidden player and output player can be represented the matrix as follows: where U is the object vector. The weight matrix W can be calculated by linear optimization methods, e.g., the least square method (Gonzalez et al. 2002). Solving the pseudomatrix for U in the formula (9), the weight matrix can be obtained: 2. Supervised mode The RBF centers, width and output player weight are determined by supervised learning mode which is the general mode for RBFNN learning. The learning algorithm can be solved by gradient descent algorithm. The weight matrix W can be obtained by iterative algorithm follows gradient descent regulation. The iterative formula is calculated as follows: where g is the learning factor. Because the output is linear unit, the gradient descent algorithm can converge the global solution.
In the learning process, if the central nodes are so excess that the training time will increase and the network will be over fitting which reduces the RBFNN generalization ability; if the central nodes are so insufficient that the training algorithm will not converge, so the nodes need to increase. For the supervised mode, the number of radial basis functions usually needs to be specified before training, the training time is long and the training algorithm maybe not converges, and the initial condition maybe leads to the training algorithm converging in local minimum. In order to dispose the shortages mentioned above, some improved methods are proposed by researchers, e.g., the hybrid method based on gradient descent and least square algorithm, orthogonal least square algorithm (Qian et al. 2017). Yet, there is local minimum problem for gradient descent method; the orthogonal least square is unfit for recursive calculation and the determination for RBF centers needs further study.

Data acquisition system
According to the cooling dehumidifier work analysis, the inherent parameters such as suction temperature, suction pressure, discharge temperature, discharge pressure, evaporation temperature, evaporation pressure, condensation temperature and condensation pressure mostly affect the dehumidifier work performance. In the normal condition, the evaporation temperature and evaporation pressure are corresponding, and this also to condensation temperature and condensation pressure; the evaporation pressure and suction pressure are approximate, and this is also to condensation pressure and discharge pressure. Furthermore, extrinsic parameters also influence the dehumidifier work performance, and they affect the system by causing the inherent parameters changing. In this studied experimental system, the extrinsic parameters include air intake temperature, air intake humidity, air volume, compressor power, cooling water intake temperature and flow. According to these parameters, the dehumidifier data acquisition system is built by employing measurement sensors, transducers, data acquisition card and upper computer. Based on the built system, the dehumidifier work status simulation, data acquisition, fault detection and diagnosis can be implemented. The data acquisition system for the studied tempering cooling dehumidifier is built generally shown in Fig. 3. The illustration of measurement parameters is shown in Table 1. For the cooling dehumidifier, the nominal dehumidification capacity is 21 kg/h, the refrigeration capacity is 32.9 kW, and the input power is 7.7 kW. The requirement environment for dehumidifier normal work is that the temperature is in ranges of 18-32°C and the RH is in ranges of 45-90%. These measurement sensors in the data acquisition system include temperature sensors with accuracy of ± 0.1°C, RH sensors with accuracy of ± 2%, pressure sensors with accuracy of ± 0.02 bar, air velocity sensor with accuracy of ± 0.2 m/s, flow sensor with accuracy of ± 0.15 L/min, compressor power sensor with accuracy of ± 0.02 kW. The measurement uncertainty is as follows: temperature, ± 0.06°C; RH, ± 1.2%; pressure, ± 0.01 bar; air velocity ± 0.12 m/s; flow, ± 0.1 L/min; compressor power, ± 0.01 kW. All these sensors are calibrated before applications according to the requirement environment for dehumidifier work. The data sampling interval is set as 1 s.
After the data acquisition system has been built, the dehumidifier work statuses of that refrigeration system faults commonly occur can be simulated by controlling the work condition and introducing artificial faults. These statuses include normal work, 9 kinds of single faults and 6 kinds of compound faults. The single faults include evaporator fouled, air cooling condenser fouled, air intake volume decrease, air intake filter fouled, air intake temperature lower, water intake volume overfull, refrigerant insufficient, expansion valve open oversize and undersize, and the compound faults include air cooling condenser fouled combined air intake volume decrease, evaporator fouled combined air intake volume decrease, air intake volume decrease combined water intake volume overfull, evaporator fouled combined water intake volume overfull, evaporator fouled combined air cooling condenser fouled and air cooling condenser fouled combined water intake volume overfull.
The dehumidifier work statuses simulation can be operated by introducing artificial faults. In the dehumidifier work statuses simulation process, it needs to determine whether the dehumidifier works in steady by the measurement parameters change and steady detector (Lee et al. 2004) at first. If the simulated status works in steady state, then the measurement data can be acquired and recorded for fault analysis and diagnosis.

Optimization for RBFNN
For model building, the calculation time and storage space will be increased by geometric progression with the number of samples and dimension increasing. So effectively extracting feature becomes an important problem for building system model. By feature extraction, the complexity of learning problem can be debased, the generalization ability can be improved, and the learning model can be simplified. Principle component analysis (PCA) is a multivariate statistical analysis technology for data compression and information extraction (Chen and Lan 2009;Han et al. 2010); it is suitable for analyzing multidimensional process variable and the data set composed of other Fig. 3 The dehumidifier experimental and measurement system elements which affect the process operation. Before building system model, employing PCA method can furthest preserve the useful information, eliminate the relationship from variables and filter redundant information.
Although PCA is an effective feature extraction method, it is virtually a linear mapping. The KPCA method combines PCA and kernel method not only fits for disposing the nonlinear problem but also supplies more information (Schölkopf et al. 1998;Cao et al. 2003). For the HVAC&R system, the nonlinear characteristic is obvious; there are strong coupling among characteristic variables, especially when the compound faults occur. KPCA can better relieve the nonlinear relevancy among variables so that it can better solve the compound fault problem. On the other hand, because the main characteristic of GA is population searching strategy, information interchange and searching independent on gradient information, GA is very suitable for solving nonlinear problem (Palmes et al. 2005;Ashena and Moghadasi 2011;Irani and Nasimi 2011). So it can be employed for ANN-aided design and optimization which will improve the ANN performance (Balaga et al. 2015).

Data dimensionality reduction by KPCA
The main idea of KPCA is that calculate the principle component in the feature space which is mapped from input space by nonlinear mapping. KPCA employs a simple kernel function for finding a calculable and controllable solution, which is essentially to construct a nonlinear mapping from input space to feature space. Therefore, KPCA is a nonlinear PCA accomplished in input space. If the purpose of PCA application is diagonalizing the covariance matrix, weakening the nonlinear relationship in the given data set x k 2 R m ðk ¼ 1; 2; . . .; NÞ, the covariance can be represented in linear feature rather than the nonlinear input space, i.e., Here, supposing P N i¼1 uðx i Þ ¼ 0, and uðÁÞ is a nonlinear mapping function which maps the variable from input space to F feature space. In order to diagonalize the covariance matrix, it needs to resolve the eigenvalue in feature space: The eigenvalue k [ 0, and v 2 F 6 ¼ f0g. C F v can be represented as follows: where x; y h i denotes inner product of x and y. This indicates the solution v corresponding to k must locate in the subspace spanned uðx 1 Þ; uðx 2 Þ. . .; uðx N Þ.
and there will be a factor a j ðj ¼ 1; 2; . . .NÞ to make Synthesizing formula (12), (15) and (16), the following formula can be obtained: The eigenvalue problem only introduces the inner product of mapping vector in feature space demonstrated in formula (17).
Defining a N Â N order kernel matrix K, where matrix element K ij ¼ uðx i Þ T uðx j Þ, 1 i N, 1 j N. Noticing K is a symmetry matrix, formula (17) can be simplified as follows: The feature vector v of C F can be solved by the feature vector a of K, i.e., the principle component direction of mapping space.
In this way, the principle component t of test vector x in the input space is extracted by uðÁÞ mapping to feature vector v k in F space: where k ¼ 1; 2; . . .p, and p is the principle component number which can be calculated according to the following formula: where m is the eigenvalue number and C is a constant spanned 0.85-0.95, here assigned 0.90.
Introducing the inner product kernel function shown as the following formula to feature space can avoid directly calculating nonlinear mapping Selection of kernel function entirely determines the mapping u and feature space F; the common kernel functions (Wu et al. 2007) include: (1) Guass radial basis function (2) q-order polynomial function (3) Two players perceptron function In this paper, the Gauss radial basis function is employed as the KPCA kernel function.

Structure and parameter optimized by AGA
In GA, the possible solution of problem field is considered as one individual or chromosome, and every individual is encoded in the character string form. It repeatedly operates the population based on genetics by imitating the process of genetic selection and nature elimination according to the biology evolution mechanism. Every individual is evaluated depended on the preset object fitness function, and the optimal population is obtained according to the principle that the excellent one exists and the inferior one eliminates. At the same time, the optimal individual in the optimal population is obtained by the global parallel search, which is the optimal solution being satisfied to the requirement. The standard GA flow is shown in Fig. 4. Parameter encoding, fitness function setting, genetic operating, initial population setting and control parameters setting are the basic components of GA, and the selection, cross and mutation are the basic operations of GA.
In GA control parameters, how to select the cross probability p c and the mutation probability p m is very important, because they influence the algorithm performance. Unsuitable selecting for p c and p m will not lead to search the optimal solution. Because control parameters in standard genetic algorithm are fixed, it will not make full use of the three genetic operators, which results in the algorithm being in premature convergence state or getting in local minimum state; especially, the repeating selection for p c and p m by experiment is a fussy work, and it is difficult to find the optimal values for them. Based on this problem, Srinvivas and Patnaik (1994) proposed an AGA which is improved by other researchers. According to the idea of AGA, p c and p m are automatically changing along with the fitness. They are calculated as follows: where f max is the maximum value of all the population's fitness, f avg is the average fitness value of every generation population, f 0 is the bigger fitness value of the two crossover individuals, f is the fitness value of the mutation individual. Here, p c1 takes 0.9, p c2 takes 0.6, p m1 takes 0.2, and p m2 takes 0.006. In view of the limitations in the RBFNN learning methods based on gradient descent algorithm, this paper employs a hybrid learning method based on K-means clustering algorithm for training the RBFNN. But in this conventional K-means clustering algorithm, the initial clustering number M (i.e., the number of radial basis functions in hidden player) and the RBF width r need to be artificially specified according to the experience in advance, which implies that there would be some human subjectivity. When the learning effect is not ideal, these two parameters need to be adjusted repeatedly which leads to the fussy operation and bad practicability. For this problem, this study proposed a hybrid learning algorithm based on AGA which employs AGA for automatically selecting the parameters of M and r, the RBF centers are determined by K-means clustering, and the output player weights are determined by least square algorithm. This proposed method by employing AGA not only overcomes the local minimum problem in the conventional K-means clustering algorithm, but also automatically selects the radial basis functions number and RBF width which overcomes the blindness and subjectivity in the artificial selection method for these parameters. Moreover, this method achieves a perfect balance between the RBFNN structure and output precision due to employing K-means clustering algorithm optimized by AGA, and the overfitting problem is depressed in some degree; thus, the ANN model generalization ability is improved.
In this paper, the genetic resolving procedure is as follows: (1) Encoding.
The parameters of M and r are encoding in real form, the span of former is ½1; N=2, and the span of latter is ½0:01; 5, where N is the total number of samples.
(2) Fitness function designing. The fitness function is designed as where d ij is the j-th expectation output corresponding to the i-th sample, y ij is the real output corresponding to d ij ; N is the number of learning samples and l is the number of output player units.
(3) Genetic operating. The selection operation employs roulette operator, and the optimal preserving strategy is employed. The crossover operation employs arithmetic crossover operator, and the mutation operation employs nonuniform mutation operator. The crossover probability p c and mutation probability p m are adaptively calculated according to the AGA.
(4) Other enactment. The initial population is generated at random, and the scale is 40 based on considering the resolving speed; the algorithm is terminated until the generations achieve to 200.

Application for fault diagnosis
Firstly, the normal and fault work statuses of the dehumidifier are simulated, and the data in each steady-state work status are collected by the data acquisition system as the fault sample database. Secondly, fault samples are selected to train and optimize the RBFNN model. Thirdly, the RBFNN model is validated until it meets the requirement of application precision. Finally, the model is applied to fault diagnosis, the disposed real-time data samples acquired from the dehumidifier are inputted into the RBFNN model for calculation, and the faults are diagnosed by the model output results. It should be pointed out that when the dehumidifier has new faults, the diagnosis result is inconsistent with the actual status, the work environment or the dehumidifier configuration has changed, the sample database needs to be updated, and then, the RBFNN model needs to be retrained and revalidated to ensure the effectiveness and reliability of fault diagnosis. The flow of RBFNN trained and employed for fault diagnosis is shown in Fig. 5.
Before the RBFNN is trained and applied, the input data samples need to be normalized to the [-1, 1] span in order to decrease the computational complexity and model error to prevent the model oscillation. After the model finishes calculation, the output results need to be renormalized.
In order to eliminate the influence from outside interference to data fluctuation, the input data are smoothed by the moving average method for improving modeling precision and fault diagnosis accuracy before the data are normalized.
The normalization is calculated as: where x k0 is original data, max x k0 is the max value of input data, and min x k0 is the minimum value of input data, b is the max value of normalization span, a is the min value of normalization span, and b ¼ 1, a ¼ À1.
The moving average is calculated as: where x kþi is the original measurement data, x i is the weight factor, and P p i¼Àq x i ¼ 1, k ¼ q þ 1; q þ 2; . . .;

Fig. 5 The flow of RBFNN learning and application
Compound fault diagnosis for cooling dehumidifier based on RBF neural network improved by kernel… 1607 N À p, p and q are, respectively, arbitrary positive integer which is less than the smooth span m, and p þ q þ 1 ¼ m, N is the number of measurement data. In order to correctly identify the dehumidifier work statuses mentioned above and improve the RBFNN generalization ability, the training samples are selected according to the rule: 100 groups of samples are selected corresponding to each work status; each group of samples select 18 work parameters, which are compressor suction temperature, suction pressure, discharge temperature, discharge pressure, dehumidifier air intake velocity, air intake temperature, air exhaust temperature, dehumidifier air intake RH, air exhaust RH, condenser water intake flow, water intake temperature, water exhaust temperature, condensing temperature, expansion valve inlet temperature, evaporator inlet temperature, evaporating temperature, evaporator air exhaust temperature and compressor power.
Obviously, if all these parameters are selected as the sample dimension, the RBFNN structure would be large and the training would be complex. So here employs KPCA for extracting input data sample feature in order to reduce the data sample dimension. According to formula (20), when the kernel function width r is assigned 3.1623, the calculation result of principle components number is 7, i.e., the number of input player units corresponding to 7 principle components extracted by KPCA is 7. The output player units are 16 bits which, respectively, corresponds to the dehumidifier work statuses, i.e., if the input sample is corresponding to the i-th (1 B i B 16) work status, then the i-th bit value is 1 and the other bits value are 0; here the whole output bits can be represented as . The relationship of RBFNN outputs corresponding to work statuses is illustrated in Table 2.
Employing the hybrid learning method expounded in Sect. 4 (i.e., Optimization for RBFNN) for RBFNN training and optimizing, the RBFNN structure and parameters can be obtained. In the AGA solving process, the change of fitness function value is shown in Fig. 6.
The AGA converges in the 173 generation, and the offline training time of the algorithm is 190.3342 s. The best individual fitness value is 0.9953, and then, the best solution is obtained, where the number of radial basis functions (i.e., the hidden player neurons) M ¼ 757, and the RBF width r ¼ 0:3441. The mean square error (MSE) of the obtained RBFNN is 1.8359 9 10 -7 .
Corresponding to the 16 kinds of dehumidifier work statuses, 16 groups of test samples besides the training samples are selected to input the trained RBFNN for model validation and verify the model generalization ability, the 16 groups of output results are shown in Fig. 7, and the online running time of the trained model is 3.8150 9 10 -3 s. As demonstrated in Fig. 7, the i-th (1 B i B 16) bit value is 1 (rounded to integer) and other bits value are 0 (rounded to integer) in each output result, which indicates that the output result corresponds to the ith work status illustrated in Table 2 and the fault modes corresponding to the input samples are correctly identified.
In order to verify the RBFNN application effect, corresponding to each work status, 16 groups of fault data samples acquired from the dehumidifier data acquisition system are selected for inputting the RBFNN to calculate; the 16 groups of output results are shown in Fig. 8. As demonstrated in Fig. 8, each output result is consistent with the factual work status (i.e., fault mode) corresponding to the data sample. This indicates the RBFNN model proposed in this paper has favorable fault recognition ability that can correctly recognize not only the single but also the compound fault, and it needs to be referred that the compound fault is more difficult for fault diagnosis due to the close coupling of equipment status parameters, e.g., if the air intake volume decreases fault occurs, then the evaporation pressure drops and the condensation pressure rises; if the compound fault of evaporator fouled ? air cooling condenser fouled occurs, then the evaporation pressure maybe drops and the condensation pressure maybe rises. Due to the coupling relationship between evaporation and condensation pressure, both the evaporation and condensation pressure maybe drops or rises at the same time; even one of them may be changeless. Therefore, it is relatively difficult for accurately discriminating the single fault of air intake volume decrease and the compound fault of evaporator fouled ? air cooling condenser fouled. Moreover, it is also difficult for recognizing the compound fault of evaporator fouled ? air cooling condenser fouled. As can be seen, employing the common methods to diagnose these two single and compound faults is hard, even though these two faults are discriminated; it will need a lot of detailed and trustworthy expert knowledge which is not easily acquired in engineering.
In order to illustrate the advantages of this proposed method in fault diagnosis accuracy, here is a comparative analysis with two conventional RBFNN models. The first model employs the direct creation method (defined as model 1), the number of neurons in the hidden layer is equal to the number of input samples, and the centers are the data sample points; the second model employs the stepby-step creation method (defined as model 2). The number of neurons in the hidden layer gradually increases from zero according to the error, and the increased neuron center is the sample point with the maximum error; the model stops training until it meets the preset accuracy condition. The width of RBF in the two models is selected according to experience, and the parameter value is set to 0.25. It should be pointed out that in order to obtain the ideal RBFNN model, the width of RBF needs to be adjusted repeatedly according to the experiment, and the operation is cumbersome, which brings inconvenience to the application. The same data samples are used to train and validate these two models, and the validated results are, respectively, shown in Figs. 9 and 10.  As demonstrated in Fig. 9, the overall classification precision of model 1 decreases. Although the classification of fault model 3 is correct, the classification precision decreases significantly; the fault model 12 and 15 are both incorrectly classified as fault model 9. As demonstrated in Fig. 10, the overall classification precision of model 2 also decreases. For fault models 12 and 15, the classification precision is decreased significantly, and these two fault models are even incorrectly classified as fault model 4.
From the comparative analysis of fault diagnosis results, it can be seen that the RBFNN model proposed in this paper has prominent advantages in the accuracy of compound fault diagnosis than the conventional RBFNN models.
In order to further illustrate the advantages of this proposed model compared with the other two conventional RBFNN models, the main performances of these models are compared and analyzed. These performances mainly include network structure (number of hidden layer neurons), output precision, fault diagnosis accuracy, online running time and so on. The comparison results are shown in Table 3. In the comparative experiment, based on the same data samples, each model is run for 10 times under the same conditions, and the obtained performance parameters are taken as the average value. For the selection of RBF width parameter value, this proposed model employs automatic mode by employing AGA, and the latter two models employ artificial mode by expert experience. In order to facilitate comparative analysis, the parameter selection value range of these three models is set as [0.01, 5].
As demonstrated in Table 3, except the selection of RBF width without depending on expert experience and knowledge, this proposed method has significant advantages in network structure, online running time, training precision, test precision and fault diagnosis accuracy. Although the training time of model 1 is the shortest, its network structure is the largest, the online running time is the longest, and the fault diagnosis accuracy is the lowest. If the training precision goal of model 2 is higher, its network structure will be larger, the model training time will be longer, and even the training algorithm may not converge. The comparative analysis results in Table 3 further prove that this proposed model can not only be better applied to the cooling dehumidifier compound fault diagnosis, but also has obvious advantages over the other two conventional RBF models.

Conclusions
In this paper, the diagnosis for cooling dehumidifier compound fault is studied and explored. Due to the obvious nonlinear characteristics of the refrigeration system and the strong coupling among various components in the work, it is difficult to establish a model for compound fault diagnosis. In view of the strong nonlinear processing ability of ANN, this paper proposed a compound fault diagnosis method for cooling dehumidifier based on RBFNN improved by KPCA and AGA. On the one hand, AGA and K-means clustering algorithm are used to automatically optimize the RBF width, the number of neurons in the hidden layer and the neuron centers. On the premise of ensuring that the RBFNN model has sufficient output precision, it has a smaller structure and faster online running speed, which solves the problems that the selection of RBF width depends on expert experience and knowledge, large network structure or slow training speed in the conventional RBFNN models; on the other hand, KPCA is employed to reduce the dimension of the model input data, eliminate the correlation between variables and further  Online running time (s) 4.1385 9 10 -3 7.7612 9 10 -3 6.2754 9 10 -3 Offline training time (s) 1.9034 9 10 2 1.0374 5.3135 9 10 2 Training precision (MSE) 2.1429 9 10 -7 7.5832 9 10 -6 9.5958 9 10 -6 Test precision (MSE) 1.5755 9 10 -5 1.9600 9 10 -2 8.8375 9 10 -3 Fault diagnosis accuracy (%) 100% 84.38% 90.63% optimize the network structure while effectively extracting nonlinear features. In order to illustrate the advantages of this proposed method, it is compared with two conventional RBFNN models for performance analysis by employing the same modeling, testing and diagnosis samples. The comparison results show that this proposed method can not only be effectively applied to the compound fault diagnosis for the cooling dehumidifier, but also has obvious application advantages compared with the other RBFNN models.

Discussions
Objectively speaking, the inherent problems of cooling dehumidifier lead to us working hard for the compound fault diagnosis, e.g., the influence of outside interference, the close coupling of inner parameters, the fault artificial simulation difficulty and so on. All these factors will affect the reliability of fault diagnosis. This paper introduces three steps for eliminating these influences for fault diagnosis to sum up: firstly the model built data are coming from the dehumidifier multiple work statuses by artificial simulation in experiment; secondly the model input data are coming from the steady state and smoothed by the moving average method; lastly the strong fault tolerant and generalization ability of RBFNN are applied to restrain the outside interference to the model. These measures ensure the RBFNN model application reliability for dehumidifier fault diagnosis, but the problem of inner mechanism of compound fault, work status simulation, fault profound diagnosis and so on need to deeply research. The proposed RBFNN model employs AGA and K-means clustering algorithm to optimize the RBF width, the number of neurons in the hidden layer and the neuron centers and further employs KPCA to reduce the dimension of the input data, so that the model has a small structure on the premise of sufficient precision, thus guaranteeing the running speed of the online application. By the group calculation of AGA, the obtained best individual is equivalent to performing multiple operations on the K-means clustering algorithm, and the optimal clustering result is selected as the neuron centers in the network hidden layer. At the same time, it also restrains the problem that the K-means clustering algorithm is sensitive to the initialization value and easy to fall into the local minimum.
Although this paper has made a successful exploration in dehumidifier compound fault, there is still some work to be carried out in the further. For the compound fault diagnosis, this paper mainly researches two single faults combination situation because the objective condition such as work status simulation is limited. Theoretically speaking, though there is the relatively lower probability that three or more single faults occur simultaneously, it is still need to research, and the faults of different magnitude also need to research in the further.
This study mainly focuses on fault diagnosis in steady state, but the equipment work state maybe transforms due to control, outside interference and other influence factors; although the equipment work parameters fluctuation and strong coupling bring more difficulties to fault diagnosis in unsteady state, it still needs further research.
In recent years, the application of artificial intelligence technology has promoted the further research and development of machine learning methods (Dahiya and Gupta 2021;Elgendy et al. 2021;Ren et al. 2021;). Among these methods, the deep neural networks have been applied in fault diagnosis with their development (Ye and Yu 2021;Chen et al. 2021;Khorram et al. 2021;Belagoune et al. 2021). The convolutional neural network (CNN) and long short-term memory (LSTM) neural network are the most typical models. CNN can effectively extract the hidden features of high-dimensional data, and its unique network structure can reduce the number of parameters and data complexity while retaining data features; the deep fault characteristics are learned through multi-level nonlinear mapping relationship. However, CNN was originally used to process two-dimensional image data, and its specific application in fault diagnosis for one-dimensional time series needs further research. Recurrent neural network (RNN) can learn history information and is very suitable for dealing with time series problems. As an improved RNN, LSTM neural network makes up for the difficult training problems such as gradient disappearance and explosion of ordinary RNN, and its practicability is further improved. In the further work, these two ANN models can be studied for applications in fault diagnosis and prognosis according to their respective characteristics. Like other network models, the optimization of structure and parameters of these two ANN models also needs to be studied, e.g., the literature (Gupta and Vasant 2021) employs nondominated sorting genetic algorithm II (NSGA-II)) for optimizing the weights to combine RNN and LSTM neural network.