Generator Fault Diagnosis with Bit-Coding Support Vector Regression Algorithm

Abstract

Generator fault diagnosis has a great impact on power networks. With the coupling effects, some uncertain factors, and all the complexities of generator design, fault diagnosis is difficult using any theoretical analysis or mathematical model. This paper proposes a bit-coding support vector regression (BSVR) algorithm for turbine generator fault diagnosis (GFD) based on a support vector machine (SVM) capable of processing multiple classification problems of fault diagnosis. The BSVR can simplify the design architecture and reduce the processing time for detection, where m classifier is needed for m class problems compared to the [m(m − 1)]/2 size of the original multi-class SVM. Compared with conventional methods, numerical test results showed a high accuracy, good robustness, and a faster processing performance.

1. Introduction

Shifting to eco-friendly power generation over the last two decades, gas has replaced 4% of the coal energy and 15% of the oil energy in Taiwan. However, the total energy generation from renewables is only 5.8%, which includes 4.1% of the old hydro plants. This can be seen as a typical example of a country shifting to renewables or clean energy. The power generation from the conventional synchronous turbine-generator (TG) remains high, and the large turbine-generator sets will last for a long time.

The generator’s fault is that it is often accompanied by vibrations, which not only damage the generator but also cause outages and even an economic threat to society. This paper discusses the fault detection for conventional synchronous machines, and the same idea can be extended to machines with a similar shaft and generator.

With the coupling effects, some uncertain factors, and all the complexities of generator design, highly non-linear and sometimes uncertain vibration features can occur, and fault diagnosis is difficult using any theoretical analysis or conventional mathematical model [1,2]. With the recent emphasis on renewables, the fault detection of wind generators has become a popular topic, and there are many publications in this field. In addition to the generator, the unique structure of the windmill needs attention. Common problems encountered are from the drive train [3,4], the gearbox [5], the bearing [6], the imbalance [7,8], the braking system [9], and sometimes the sensor and the actuator [10]. Methods used are conventional current models [3,6], frequency analysis [11], etc. However, with advances in artificial intelligence (AI), an optimized solution becomes possible, such as the use of an artificial neural network (ANN) [12,13,14], the fuzzy logic (FL) theorem [15,16], the fuzzy neural network (FNN) [17], and the expert system [18]. Diagnosis that considers the vibration feature can gain a higher accuracy [19,20]. Combining vibration features with AI could also provide promising results, such as the ANN based method [19], wavelet techniques [21], and the FL-based scheme [22,23].

The ANN technique uses error-back propagation to adjust weighting parameters, achieving the desired diagnosis through a nonlinear mapping relationship. Slower learning processes, weight interferences among fault patterns, and the local minimum are the major drawbacks. The wavelet neural network (WNN) is successful for pattern recognition problems, such as classifying voices and images [24]. The WNN application on TG fault diagnosis appeared in [21]. Initialization of many parameters is important to the success of the complicated WNN network, but the learning time is too long for practical applications. On the other hand, the fuzzy based method strongly depends on experienced experts. The defuzzifier and inference rules must be continuously maintained and revised.

The SVM [25,26,27,28,29,30,31,32,33,34,35] technique is based on a linear machine, nonlinearly related to the input space in a high dimensional feature space, known for fast training even with a large number of input variables and big training sets [32]. The SVM for classification (SVC) is strong for binary classifications. The standard SVC can combine with the one-versus-one (OVO) or one-versus-rest (OVR) approach to solve multi-class problems [36,37]. However, these methods may sometimes suffer from the huge network size, complicated training data preparations, or the heuristic solution scheme.

This paper improves the SVC by the use of the bit-coding approach, combining SVM with a regression technique called bit-coding support vector regression (BSVR), where training becomes an easy and fast regression problem, capable of processing multiple classification problems, including generator fault diagnosis (GFD). The classification problem converges and is coded into a single output bit. BSVR simplifies the design architecture and reduces the processing time. Only m SVR is needed for m classification problems, in comparison with the [m(m − 1)]/2 size of the original SVM. Numerical results show that BSVR can effectively process multiple GFD, which is not easily attainable with other methods.

2. Problem Description

A turbine generator consists of three parts: the turbine, the generator, and the exciter. The turbine can have sections with high, low, and intermediate pressure sections, girdled by the bearings for diagnostic information [38].

Three faults are generally considered:

(1)

The electrical fault.

(2)

Mechanical vibration fault.

(3)

The cooling system fault.

The first two are the common incipient faults in the generator. The electrical fault could be the stator-winding ground fault, the rotor excitation short circuit, and the stator-winding short circuit involving the three-phase fault or line-to-line fault. The rotor fault causes the unbalanced magnetic pull, and vibration could occur. The major mechanical vibration faults studied in the paper are:

(1)

Rotor unbalance: unbalanced rotor weighting or the poor base.

(2)

No orderliness: shaft straight line or insufficient warming.

(3)

Oil membrane oscillation: fault lubricating system or pump.

Physically, (1) and (2) may twist the shaft and deform the turbine, and (3) could cause the bearing to burn. Shaft vibration regulated with a pre-set limit is the restriction to shut down and trip the machine automatically. The displacement meter and accelerometer are generally set at the bearing position. The power spectra data of the vibration signals are important pieces of information to detect the machinery conditions.

Many techniques were used to find the fault symptoms, such as frequency analysis, phase analysis, precession, vibration waveform, and probability density analysis [39]. In addition, FFT is effective at extracting time-domain signals and converting to the frequency-domain for analysis. The frequency spectra of the vibration will continue to increase as incipient faults continue to grow.

Table 1. Limits of Power Spectra in the Frequency Domain [38].

Fault Type	Rotor Vibration Frequency (VF, in Hz)		Afi Amplitude Range (μm) (10−6 m)
			Lower Value	Upper Value
Oil Membrane Oscillation	f1	<0.4 f	2.70	6.50
	f2	1 f	11.00	19.00
	f3	2 f	1.10	4.90
	f4	3 f	0.80	2.40
	f5	>3 f	0.50	3.80
Unbalance (Imbalance)	f1	<0.4 f	0.54	2.70
	f2	1 f	38.00	54.50
	f3	2 f	2.70	6.80
	f4	3 f	0.54	4.10
	f5	>3 f	0.00	2.70
No Orderliness	f1	<0.4 f	0.54	1.90
	f2	1 f	22.00	30.00
	f3	2 f	22.00	26.50
	f4	3 f	14.00	19.50
	f5	>3 f	5.40	16.20
Normal Condition	f1	<0.4 f	0.00	0.54
	f2	1 f	0.00	8.60
	f3	2 f	0.00	3.30
	f4	3 f	0.00	3.30
	f5	>3 f	0.00	2.7

shows the data [38,39] by studying many 50 MW generators, where vibration signals are the high and low limits. Many methods were used for diagnosis [40,41]. In [38,39], typical vibration frequencies (VFs) used are 1 f, 2 f, 3 f, plus three bands <0.4 f, 0.4 f~0.5 f, >3 f, where f is the rotor frequency, f_i is the i-th vibration frequency, and A_fi is the amplitude of f_i. For VF, the high and low values of the power spectrum would indicate the probable fault symptoms, as shown in Table 1 [38]. It took a lot of effort and resources to engage in the field survey. As generator fault records are not easy to come by, extensive field work should be conducted to collect new data. According to the field data, diagnostic information can be generated to monitor mechanical conditions by using the spectrum of the vibration signals. The fault classes chosen were based on the available database, originally built for the study of vibrations.

3. Fundamental Theory

Support Vector Regression (SVR) is a regression technique based on SVM. It can be designed to work in the high dimensional feature space when dealing with highly complicated data by mapping the N-dimensional input vector into a K-dimensional feature space (K > N) through a kernel function. The typical kernel functions used are the radial basis function, and the polynomial or sigmoid functions.

Let ${\Re }^p$ be the p-dimensional real space, and the input and output sets be $X\subseteq {\Re }^n$ and $Y\subseteq \Re$. The example of a training set becomes

$$\mathbf{Tr}:=\{x_i,y_i\}\subseteq X\times Y,\begin{array}{c}\end{array} i=1 ,\dots , l$$

(1)

Let linear regression be

$$f_R^{}(x,w)=w^{\mathit{T}}x+b^{}=\langle w,x\rangle +b^{}$$

(2)

Vapnik introduced an error function, with $\epsilon$-insensitivity. We can isolate a zone for the target output $y$ $\in \mathbf{Y}$ [25,35] by

$$\left|y-f_R{(x,w)}_{\epsilon }\right|=\{\begin{array}{l}0\begin{array}{cc}& if\left|y-f_R(x,w)\right|\end{array}\le \epsilon \\ \left|y-f_R(x,w)\right|-\epsilon \begin{array}{cc}& otherwise\end{array}\end{array}$$

(3)

The empirical risk of training error is

$$R_{Emp}^{\epsilon }(w,b)=\frac{1}{l}{\displaystyle \sum _{i=1}^l{\left|y_i-w^{\mathit{T}}{x}_{\mathit{i}}-b\right|}_{\epsilon }}$$

(4)

For regression, the objective is to minimize R containing $R_{Emp}^{\epsilon }$ with the estimation error ${\Vert w\Vert }^2$ [25,35]. That is,

$$R=2^{-1}{\Vert w\Vert }^2+C{\displaystyle \sum _{i=1}^l{\left|y_i-w^{\mathit{T}}{x}_{\mathit{i}}-b\right|}_{\epsilon }}$$

(5)

From Figure 1, all training data outside $\epsilon$ tube satisfy

$$\left|y-f_R(x,w)\right|-\epsilon =\xi , \mathrm{for} \mathrm{data} “\mathrm{above}” \mathrm{an} \epsilon \mathrm{tube}$$

(6)

$$\left|y-f_R(x,w)\right|-\epsilon =\eta , \mathrm{for} \mathrm{data} “\mathrm{below}” \mathrm{an} \epsilon \mathrm{tube}$$

(7)

Minimizing risk R in (5) is equivalent to minimizing

$$R(w,\xi ,\eta )=2^{-1}\Vert w\Vert +C\left\{{\displaystyle \sum _{i=1}^l{\xi }_i+{\displaystyle \sum _{i=1}^l{\eta }_i}}\right\}$$

(8)

subject to

$$y_i-[\langle w,x\rangle + b]\le \epsilon +{\xi }_i,{\xi }_i\ge 0$$

(9)

$$[\langle w,x\rangle + b]-y_i\le \epsilon +{\eta }_i,{\eta }_i\ge 0,\mathrm{for}i=1, 2, \dots ,l$$

(10)

where C is a user defined trade-off between the risk and the estimation. ${\xi }_i$ and ${\eta }_i$ are slack variables of the measurement of upper and lower bounds of outputs.

Based on the empirical risk, we can derive the regression formula. Applying Karush-Kuhn-Tucker condition (KKT), the dual Lagrangian $L_d(\alpha ,\beta )$ function to maximize is

$$\begin{array}{ll}{L}_d(\alpha ,\beta )=& {\displaystyle \sum _{i=1}^l({\alpha }_i-{\beta }_i)y_i-}\epsilon {\displaystyle \sum _{i=1}^l({\alpha }_i+{\beta }_i)}\\ & -2^{-1}{\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^l}({\alpha }_i-{\beta }_i)({\alpha }_j-{\beta }_j)\langle }{\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}}\rangle \end{array}$$

(11)

This is subject to

$${\displaystyle \sum _{i=1}^l({\alpha }_i-{\beta }_i)}=0, 0\le {\alpha }_i\le C, 0\le {\beta }_i\le C,i=1, 2, \dots ,l$$

(12)

where ${\alpha }_i$ and ${\beta }_i$ will be nonzero for training samples “above” and “below” an $\epsilon$ tube. Let

$$K(x,z)=\langle \varphi (x),\varphi (z)\rangle ,\mathbf{x},\mathbf{z}\in \mathbf{X}$$

(13)

where $\varphi$ is the feature map designed by mapping the input space X to the feature space F equipped with the inner product $\langle \cdot ,\cdot \rangle$. In non-linear cases, we can use $K(x_{\mathit{i}},x_{\mathit{j}})$ to replace $\langle x_{\mathit{i}},x_{\mathit{j}}\rangle$ in (11).

We have

$$\begin{array}{ll}{L}_d(\alpha ,\beta )=& {\displaystyle \sum _{i=1}^l({\alpha }_i-{\beta }_i)y_i-}\epsilon {\displaystyle \sum _{i=1}^l({\alpha }_i+{\beta }_i)}\\ & -2^{-1}{\displaystyle \sum _{i=1}^l{\displaystyle \sum _{j=1}^l}({\alpha }_i-{\beta }_i)({\alpha }_j-{\beta }_j)K(}{\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}})\end{array}$$

(14)

This abides by the same constraints in (12). Now, (14) can be expressed in the matrix notation by [33]

$$L_d(\mathit{A})=\mathbf{0.5}{\mathit{A}}^{\mathit{T}}\mathbf{H}\mathit{A}+{\mathit{c}}^{\mathit{T}}\mathit{A}$$

(15)

where $c={[\epsilon -Y \epsilon +Y]}^T$, $\mathit{A}={[\mathbf{\alpha } \mathbf{\beta }]}^T$, ${(\mathbf{\alpha })}_i={\alpha }_i$, ${(\mathbf{\beta })}_i={\beta }_i$. $\mathbf{H}$ denotes the Hessian matrix with

$$\mathbf{H}=\left[\begin{array}{cc}\mathbf{G}& -\mathbf{G}\\ -\mathbf{G}& \mathbf{G}\end{array}\right]$$

(16)

where ${\mathbf{G}}_{ij}=K({\mathbf{x}}_i,{\mathbf{x}}_j)$.

If ${\alpha }_i{}^o$ and ${\beta }_i{}^o$ ($i=1, 2, \dots , l$) can be found to solve the problem, define ${\gamma }_i{}^o:={\alpha }_i{}^o-{\beta }_i{}^o$, and then the training points with ${\gamma }_i{}^o\ne 0$ are the SVs and (14) depends entirely on the SVs. We have

$$w^o={\displaystyle \sum _{i=1}^l({\alpha }_i^o-{\beta }_i^o})\varphi (x_{\mathit{i}})={\displaystyle \sum _{i\in SVs}^{}{\gamma }_i^o}\varphi (x_{\mathit{i}})$$

(17)

From KKT, choosing any $0<{\alpha }_k^o<C$, we have ${\xi }_k^o=0$

$$0=\epsilon +{\xi }_k^o-y_k+\langle w^{\mathit{o}},x_{\mathit{k}}\rangle +b^o =\epsilon -y_k+\langle w^{\mathit{o}},x_{\mathit{k}}\rangle +b^o$$

(18)

We can solve

$$b^o=y_k-\epsilon -{\displaystyle \sum _{i\in SV{s}_{}}^{}({\alpha }_i^o-{\beta }_i^o})K(x_{\mathit{i}},x_{\mathit{k}})=y_k-\epsilon -{\displaystyle \sum _{i\in SV{s}_{}}^{}{\gamma }_i^o}K(x_{\mathit{i}},x_{\mathit{k}})$$

(19)

We can then find the optimal regression function used in this paper as

$$\begin{array}{ll}{f}_R^o(x)& =\langle w^o,x\rangle +b^o\\ & ={\displaystyle \sum _{i\in SV{s}_{}}^{}({\alpha }_i^o-{\beta }_i^o})K(x_{\mathit{i}},x)+b^o\\ & ={\displaystyle \sum _{i\in SVs}^{}{\gamma }_i^o}K(x_{\mathit{i}},x)+b^o\end{array}$$

(20)

In Figure 2, we can see that a SVR is represented as a feed forward neural network, where the number of SVs can determine the number of hidden units. One SVR structure is built for one fault class with its dataset X_i and its target y_i, as in (1). Training has become an easy and fast optimization problem by computing parameters. When training is finished, parameters 𝑏⁰ and ${\gamma }_i{}^o\ne 0$ are found and frozen, and we can implement the optimal regression function (20) with kernel ${\mathbf{G}}_{ij}=K({\mathit{x}}_i,{\mathit{x}}_j)$, as in Figure 2.

4. The Proposed BSVR Classifier

4.1. Design Architecture and Bit-Coding Approach

According to the field data sorted, diagnostic information can be generated. The major vibration problems concerned are rotor unbalance, rubbing, rotor crack, and oil membrane oscillation, i.e., the purposes of the original field investigation for conventional generators. BSVR contains an input layer, SVR layer, and the output layer. Figure 3 shows the BSVR structure, where one SVR unit is designed for one fault class, i.e., m classifier is needed for m class problems against the original [m(m − 1)]/2 sized SVM problem. This shows a greatly simplified structure of BSVR.

In our example with four faulty classes, only four nonlinear SVRs are needed with the bit-coding approach to solve the nonlinear GFD problem, which shows a greatly simplified classification problem. The network can easily expand with more fault classes without any limitation.

As in

Table 2. The Targets of each SVR in the Training Set.

SVR Type	Targets [O1 O2 O3 O4]
SVR_u	rotor unbalance: 1; otherwise: 0
SVRM_r	rubbing: 1; otherwise: 0
SVR_c	rotor crack: 1; otherwise: 0
SVM_o	oil membrane oscillation: 1; otherwise: 0

, the targets of each SVR are encoded by “1” or “0”, with $y_i$ set to “1” for fault, and “0” otherwise. The GFD problem can be processed as follows:

(1)

Construct the diagnostic system.

(2)

Train the network with known fault patterns.

(3)

Use the diagnostic system to diagnose the fault.

Outputs [O₁ O₂ O₃ O₄] denote the detected fault types, including the rotor unbalance, rubbing, rotor crack, or oil membrane oscillation, respectively. The physical meaning of “1” means “accurate detection,” and 0 means “accurate non-detection.”

4.2. BSVR-Based GFD System (BGFDS)

The process of running the proposed BGFDS are shown below, where BGFDS works with existing equipment without extra measurement devices. We have:

• Data acquisition to collect sampled vibration data periodically at a regular interval.
• Data sent to the Data Processor for interpretation.
• FFT to analyze data in order to acquire the frequency spectrum.
• Data processed by the fault diagnosis processor with two portions of

  (1)

  sampled data from field be constructed in EXCEL Workspace, and

  (2)

  data analysis and storage of BSVR be manipulated on this database.

A fair threshold of 0.5 is preselected to separate “fault” from “no fault”. We have $O_m \ge 0.5$ for when component m is faulty, and $O_m$ < 0.5 for when component m is not faulty.

A flow chart is provided in Figure 4 to show the process. The fault diagnosis processor includes: (1) the sampling data from field measurements in EXCEL Workspace, and (2) the data analysis and storage for BSVR. The diagnostic algorithms can be realized in a PC-based device with the virtual instrument and the hardware device. With the simple structure proposed, the database of EXCEL Workspace is sufficient to support the application. It is simple and easy to use, and it is readily available on most PCs.

5. Simulation and Tests

5.1. Training Patterns Creation

Let the rotor frequency be f, with typical fault features described by swing ratios in nine bands, i.e., (0~0.39) f, (0.40~0.49) f, 0.5 f, (0.51~0.99) f, 1f, 2 f, (3~5) f, odd f, and >5f [23]. The BSVR was trained with 80 samples of faults, 20 samples each category as:

• Rotor unbalance (F₁): [1 0 0 0] with 20 instances.
• Rubbing (F₂): [0 1 0 0] with 20 instances.
• Rotor crack (F₃): [0 0 1 0] with 20 instances.
• Oil membrane osc. (F₄): [0 0 0 1] with 20 instances.

Every fault class has its dataset X_i and target y_i for training. Training will finish with parameters 𝑏⁰ and ${\gamma }_i{}^o\ne 0$ found. We can obtain optimal f^o as in (20) for output detection.

5.2. Simulation Results

The design architecture of BSVR is shown in

Table 3. Architecture of the BSVR.

Network Size			Number of Training Sets	C
I	S	O
9	4	4	80	50

Note: I-S-O: Input Layer–SVR Layer–Output Layer.

. With the normalized input signals [42], $\epsilon$-insensitivity was set in the range (${10}^{-3}-{10}^{-2}$). In this paper, $\epsilon =0.01$ is chosen and the Gaussian Radial Basis Function (GRBF) kernel is used with

$$K(\mathit{x},\mathit{z})=\exp \left[{\displaystyle \sum _{i=1}^9\frac{{(x_i-z_i)}^2}{2{\rho }^2}}\right] ,\rho =0.8, \mathit{x},\mathit{z}\in {\Re }^9$$

(21)

This research also developed a back-propagation neural network (BPNN) for comparison. The popular BPNN was designed to have three layers, trained with the back-propagation learning algorithm [43], using the same number of Input and Output nodes as BSVR. BPNN has one hidden layer, and the number of hidden nodes is from the experience formulas shown in [44,45,46].

Table 4. Architecture of the BPNN.

Network Size			Number of Training Sets	Learning Rate (L)
I	H	O
9	7	4	80	0.2

Note: I-H-O: Input Layer–Hidden Layer–Output Layer.

shows the test configuration of the BPNN. Proper parameters of BSVR and BPNN were determined by using the trial-and-error process. The training parameters are set with C = 50 and L = 0.2. To show the performance and effectiveness of the proposed BGFDS, many tests were conducted and a few cases were chosen for demonstration.

For the accuracy test, the percentage root mean square error (PRMSE) is adopted for each output node by

$$PRMSE=\sqrt{\frac{1}{Q}{\displaystyle \sum _{p=1}^Q{(E(p)-O(p))}^2}} \times 100\%$$

(22)

where

E(p)	Target value of the p-th sample
O(p)	Real output value of the p-th sample
Q	Total number of test samples

5.2.1. Generalization Ability Test

After training finished, this study chose 14 fault samples for tests, as shown in

Table 5. Test Dataset for Simulation.

Sample No.	Input Data									AFT
	(0.01~ 0.39) f	(0.4~ 0.49) f	0.5 f	(0.51~ 0.99) f	1 f	2 f	(3~5) f	Odd f	>5 f
1	0.00256	0.00122	0.00993	0.01826	0.81123	0.07904	0.04958	0.04958	0.00397	F1
2	0.05129	0.00267	0.00227	0.01846	0.7578	0.09388	0.03373	0.03373	0.00596	F1
3	0.00494	0.00162	0.00131	0.01049	0.84174	0.05299	0.01962	0.01962	0.00322	F1
4	0.11805	0.01598	0.00831	0.12527	0.56643	0.01725	0.04653	0.02356	0.07864	F2
5	0.03012	0.01275	0.02175	0.16904	0.61279	0.01977	0.05657	0.02518	0.052	F2
6	0.1167	0.00545	0.00523	0.17401	0.56365	0.02107	0.05358	0.01288	0.05344	F2
7	0.00344	0.00344	0.00553	0.00723	0.54074	0.15488	0.12893	0.12893	0.02687	F3
8	0.00178	0.00178	0.00323	0.00566	0.58058	0.15624	0.11422	0.11422	0.0223	F3
9	0.0132	0.00261	0.00281	0.00642	0.63413	0.14974	0.07667	0.07667	0.0377	F3
10	0.02475	0.18273	0.39201	0.19642	0.05736	0.09657	0.02254	0.02254	0.0051	F4
11	0.00482	0.24	0.50575	0.07214	0.08549	0.03526	0.02535	0.02535	0.0059	F4
12	0.02363	0.14473	0.53938	0.10211	0.05216	0.091177	0.02069	0.02069	0.00548	F4
13	0.00755	0.26129	0.4818	0.0761	0.08415	0.03498	0.02331	0.02331	0.0055	F4
14	0.01321	0.23394	0.488	0.06358	0.09938	0.03841	0.02777	0.02777	0.00791	F4
Noise	−40%	20%	−10%	+10%	+10	−20%	+30%	+20%	−20%

Note: 1. AFT: Actual fault types; 2. NS: Noise; 3. F₁: rotor unbalance; F₂: rubbing; F₃: rotor crack; F₄: oil membrane oscillation; 4. Bold italic type represents the items with the last row noises added in the robustness test.

. These samples do not belong to the group of training data, showing the power spectra of some field data in each frequency band. Data analysis shows the fault type of each sample, e.g., sample No.1~3 are rotor unbalance faults. Bold italic type represents the items with additional noises added in the robustness tests. Noises were added by adjusting the amplitude of A_fi (10⁻³ m) as expained in Table 1 by either increasing or decreasing the amplitude, as indicated in the last row. The data were fed to the trained BSVR and BPNN respectively. The detection and non-detection outputs are shown in Figure 5 and Figure 6. In Figure 5, we can see that both methods can detect the rotor unbalance fault with values above 0.5. For example, sample No.1 has a BSVR value of 0.9247, closer to 1, compared to 0.6895 of BPNN. BSVR provides aggregated data pointed along the expectation line, while BPNN produced scattered points with higher errors.

Figure 6 shows the non-detection with range −0.15~0.3 blown up for more detail. It has the “detection” output of Figure 5 placed on expectation line “0” for reference. Another three outputs of each method are scattered around. For example, sample No.1 of BPNN has the worst value of 0.2958 for rotor unbalance fault non-detection. BSVR provides aggregated data points along the expectation line. The error indices PRMSE are in

Table 6. The PRMSE of Each Output Node.

Method	The PRMSE (%) Value
	O1	O2	O3	O4
BSVR	2.408	2.449	1.897	1.673
BPNN	10.752	6.746	10.817	4.754

to show that BSVR has better accuracy.

5.2.2. Robustness Test

To test the robustness, 10% to 40% of noise was added to the amplitudes of the original field data in Table 5, as noted in bold italic type. The output results with detection and non-detection are shown in Figure 7 and Figure 8. We can see that BSVR has a high noise rejection capability, with output values near the expectation line. Figure 7 shows a range of 0.9~1.06 for detection. Figure 8 blows up the range −0.03~0.06 with increment 0.01 for more detail, since the points tend to cluster with high accuracy.

5.2.3. Consistency Test

To check the consistency of BSVR and BPNN, this test retrains and retests both methods to a specific n number of times. After testing for the given number of times, the mean value $\mu$ and the standard deviation $\sigma$ were recorded for each output O_i.

Part of the results for consistency tests are shown in

Table 7. The Consistency Test of the Two Methods (mean value and standard deviation).

Method	n	SN	Mean Value of Each Output Node O1~O4
			${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$
BSVR	12	1	0.9247	0.0331	0.0466	0.0242
	24	1	0.9247	0.0331	0.0466	0.0242
	12	2	0.9686	0.0190	0.0000	0.0226
	24	2	0.9686	0.0190	0.0000	0.0226
BPNN	12	1	0.7922	−0.0117	0.2260	−0.0079
	24	1	0.8085	−0.0215	0.2212	−0.0089
	12	2	0.8596	−0.0060	0.1491	0.0028
	24	2	0.8598	0.0195	0.1217	0.0023
Method	n	SN	Standard deviation of each output node O1~O4
			${\sigma }_1$	${\sigma }_2$	${\sigma }_3$	${\sigma }_4$
BSVR	12	1	0	0	0	0
	24	1	0	0	0	0
	12	2	0	0	0	0
	24	2	0	0	0	0
BPNN	12	1	0.0831	0.0317	0.0649	0.0186
	24	1	0.0833	0.0466	0.0605	0.0225
	12	2	0.0788	0.0740	0.1031	0.0252
	24	2	0.0932	0.0831	0.1022	0.0417

Note: n number of test times; SN sample no. in Table 5.

. From the tables, we can see that BSVR has the same mean value and “zero” standard deviation for each retrain and retest. The important consistent machine learning character of BSVR shows a more robust nature than BPNN.

5.2.4. Performances Test

The learning performance of BSVR and BPNN are shown in

Table 8. Performance Comparison of BSVR and BPNN.

Method	tr	te	C	L	Tr_ep	Tr_t (sec)	Te_t (sec)
BSVR	80	14	50	-	-	1	1
BPNN	80	14	-	0.2	50 k	13.89	1

Note 1: tr: Number of training data; te: Number of testing data. Note 2: Tr_t: Training time; Te_t: Testing time; Tr_ep: Training epochs. Note 3: L: Learning rate of BPNN. Note 4: Training and testing time of BSVR is set to 1.

. The BPNN was one of the tools developed in house for much research [45,46]. To compare a common base with any hardware, the training and the testing time were normalized with BSVR as the bases. When training finishes, all weights are frozen. From the table, we can see that the training time of BSVR outperformed BPNN substantially, while the testing time of both methods are close to each other. BSVR has a fast learning process, and requires no estimation for the number of layers or for the number of hidden nodes. With similar training data, the proposed BSVR yields a better overall performance than BPNN.

6. Conclusions

In this paper, the BSVR-based Generator Fault Diagnosis System (BGFDS) was proposed. BSVR was designed with a simple network architecture to shorten the processing time. In a turbine generator set, vibration signals are available through data acquisition (DA), and BSVR requires no extra devices. The proposed architecture could effectively detect machine faults with information provided by the frequency spectrum of vibration signals. The BGFDS has many advantages:

• BSVR integrates a bit-coding approach, simple SVR, and a small number of training data for problems that are not linearly separable as GFD.
• SVR trains SVs with standard quadratic optimization technique, which has a unique solution and is globally optimal, requiring much less computation time.
• SVR needs no determination for the size of hidden layers. The number of SVs determines the number of hidden units automatically.
• The training and testing of BSVR are very fast compared with other ANNs.
• A minimum sized network is built with simple learning algorithms.
• Only m SVR is needed for the m classification problem in comparison with [m(m − 1)]/2 size for traditional multi-class SVM (MSVM).
• The design architecture can use existing devices without adding extra measurement devices.
• BSVR has good classification capability, performance, consistency, noise rejection ability, and robustness, i.e., many good characteristics for machine learning.
• The proposed diagnostic algorithms can be realized in a portable device for convenient application.

Test results show that BSVR-based GFDS are precise, very effective, expandable, and easy to work with.