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Generator Thermal Sensitivity Analysis with Support Vector Regression
Youliang Yang and Qing Zhao*
Abstract— Generator thermal sensitivity issue is studied inthis paper. Currently, thermal sensitivity test is usually adoptedin industries to determine if a generator has been experiencingthermal sensitivity problem. However, this kind of tests has itsown disadvantages. In this paper, Support Vector Regressionis utilized to provide some valuable information regardingthermal sensitivity in a rotating machine based on the normaloperational data of the machine. Experimental results on thesteam turbine generators show that the proposed method canbe used to track the generator condition related to the thermaleffects and make a recommendation to the on-site engineerswhether or not a thermal sensitivity test should be performed.
I. INTRODUCTIONElectrical machine condition monitoring plays an impor-
tant role in modern industries and it has been an active
research topic. Traditionally, electrical machines are allowed
to run until failure then they are either repaired or replaced.
Very limited information regarding the machine condition is
known before the machine is shutdown and hence resulting in
long machine downtime and great economic lost. Recently,
predictive maintenance strategy is adopted in many indus-
tries. In predictive maintenance, machine condition is moni-
tored continuously, and hence valuable information regarding
machine condition can be obtained before the machine is
shutdown and repaired and the machine downtime can be
greatly reduced.Predictive maintenance consists of various aspects. For
example, fault analysis needs to be performed and classi-
fication models can be built with artificial intelligence tech-
niques to classify and identify different machine conditions.
In addition, prognosis models can be built to predict the
future machine conditions and determine if the machine
is experiencing faults related condition degradation. The
machine condition trending based on importance machine
performance index can be carried out to determine when the
machine should be shutdown or a major maintenance/repair
should be scheduled. This way the number of unexpected
shutdown can be greatly reduced and the reliability is greatly
improved.
In this paper, a specific issue, generator rotor thermal
sensitivity, is studied. Thermal sensitivity is a phenomenon
caused by uneven heat distribution around an axis, e.g.
rotor, causing it to bend. Some common causes of generator
thermal sensitivity include shorted turns, blocked ventilation
or unsymmetrical cooling, insulation variation, wedge fit,
Y. Yang and Q. Zhao are with the Dept. of Electrical and ComputerEngineering, University of Alberta, Edmonton, Alberta, T6G2V4 Canada.Email: <youliang, qingz>@ualberta.ca, *: correspondingauthor
distance block fitting, etc. [1]. There are two types of thermal
sensitivity, reversible and irreversible. Normally thermal sen-
sitivity is confirmed and the severity is determined through
a standard test procedure commonly adopted in industries.
However, the thermal sensitivity test is destructive especially
when the machine has irreversible thermal sensitivity prob-
lem.
Support Vector Regression (SVR) has been widely used
in the field of electrical machine condition monitoring. Just
to name a few, in [2], a hybrid model is built with SVR
to predict the future state of a turbo generator. In [3], the
Least-Square Support Vector Machine (LS-SVM) combining
with wavelet decomposition is utilized to predict the futurevibration of a hydro-turbine generating unit. In this paper,
a model for vibration analysis is built with SVR that can
be useful in tracking machine conditions. It is applied to
analyze the thermal sensitivity issue for a type of steam
turbine generators. In this method, only normal machine
operational data are used to build the model. The results
can be used to recommend whether and when a necessary
thermal sensitivity test is needed. The rest of the paper is
organized as follows. The basic theory of SVR is reviewed
in section II. In section III, the background of generator
thermal sensitivity and the procedure of a thermal sensitivity
test are introduced. In section IV, after some discussions
on the industrial practice regarding thermal sensitivity andthe limitations, SVR models are built and the experimental
results are presented. Finally, the conclusion is provided in
section V.
I I . SUPPORT VECTOR REGRESSION
In SVR, the goal is to find a function f (x), which maps the
input to the output, while minimizing the difference between
the predicted value yi and the actual value yi based on
the loss function [4]. Suppose that there are training data
(x1, y1), (x2, y2), ..., (xN , yN ), where xi is the input and yi
is the output. In a linear case, f (x) can be expressed as
y = f (x) = wx + b (1)
where w is the weighted vector and b is a constant. While
trying to minimize the difference between the predicted value
and the actual value, in SVR, it is also desirable to keep
the function f (x) as flat as possible [4], which means wshould be as small as possible. One way to find a small wis to minimize the norm, i.e. ||w||2 =< w, w >. Thus, the
regression problem becomes to
min.1
2
N n=1
(yi − yi)2 +1
2||w||2 (2)
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x
x
x
x
x
x
x
x
x
x
x
xx
xx
x
x
x
x
iξ
iξ
x
x
y
ε + y
ε − y
x
)(ˆ x y
Fig. 1. Linear SVR with slack variables
Quadratic error function is used in Eq. (2) to calculate the
error between the predicted value and the actual value. In
practice, the ǫ-insensitive error function is often used, which
can be mathematically expressed as
E ǫ(yi − yi) =
0, if |yi − yi| < ǫ
|yi − yi| − ǫ, otherwise(3)
Readers can refer to [5] for more details on error functions.
With the ǫ-insensitive error function, the regression problem
becomes to minimize
C N
n=1
E ǫ(yi − yi) +1
2||w||2 (4)
Where C is the trade-off between the flatness of f (x) and the
prediction error. In cases that with the optimal f (x), some
actual values may not lie within the region [y−ǫ, y+ǫ], slack
variables ξ and ξ need to be introduced so that for any given
actual value yi, it lies within the region [yi−ǫ−ξi, yi +ǫ+ξi]
(Please refer to Fig. 1). When yi lies above yi + ǫ, ξi > 0and ξi = 0. On the other hand, when yi lies below yi − ǫ,
ξi = 0 and ξi > 0. Thus, the objective function of the SVR
problem can be rewritten as
min. C N
i=1
(ξi + ξi) +1
2||w||2 (5)
subject to
ξi ≥ 0 (6)
ξi ≥ 0 (7)
yi ≤ yi + ǫ + ξi (8)
yi ≥ yi − ǫ − ξi (9)
The above optimization problem can be transformed into
a Lagrangian dual problem and the final predicted value is
given by [6]:
yi =N
i=1
(αi − αi)k(x, xi) + b (10)
where αi and αi are the Lagrange multipliers, and k is the
kernel function. Common kernel functions include linear,
polynomial, and radial basis functions (RBF) [7].
III. GENERATOR THERMAL SENSITIVITY
A. Causes of thermal sensitivity
As stated in [1], even for a rotor that has thermal sensitivity
issue, it is not affected much when the generator is operating
with low VAR. On the other hand, when the generator is op-
erating with a power factor lower than 0.85 lagging, a thermal
sensitive rotor will be affected and its vibration profile will
change. The rotor vibration may increase, decrease, or its
phase angle may change. Therefore, even with a thermal
sensitive rotor, a generator may not have any issues when
operating with low field current; however, its operation may
be greatly restricted at high field currents or VAR loads as
the rotor vibration excesses the acceptable limit.
Generator rotor thermal sensitivity can be classified into
two types: reversible and irreversible. When the thermal
sensitivity is reversible, rotor vibration changes as field
current varies. That is, when the field current increases, the
rotor vibration increases. Later on, when the field current
decreases, the rotor vibration will decrease as well. This type
of thermal sensitivity usually does not cause major problemsin practice and the rotor can be balanced so that its maximum
vibration will not excess the limit. If the rotor vibration does
not decrease after the field current is reduced, this type of
thermal sensitivity is called irreversible. This type of thermal
sensitivity is troublesome since the rotor vibration will keep
increasing, and the rotor may have to be taken off-line and
repaired in order to reduce the vibration. Readers can refer
to [1] for more details on some common causes of generator
thermal sensitivity and their thermal sensitivity types.
B. Thermal sensitivity test
A standard procedure for determining generator thermal
sensitivity is normally adopted. The purpose of this test is toisolate the machine vibration which is caused by MW (real
power) loading from that caused by VAR loading (reactive
power). Vibration changing with MW loading does not
indicate thermal sensitivity problem. The thermal sensitivity
test consists of 3 parts:
1) The thermal sensitivity test is started by loading the
generator with small MW and MVAR, 10MW and
0MVAR for example, and then MW gradually in-
creases to about 60% of its rated value and MVAR
will be reduced. At each stage, all the important read-
ings, such as the machine vibration, voltage, current,
temperature, etc. are recorded.
2) In the second step of the test, the generator MW is kept
constant while the field current continuously increases
to its rated value, so MVAR increases correspondingly.
It is important that the MVAR is high enough so that
the generator operates with a power factor lower than
0.85 lagging.
3) The last step of the thermal sensitivity test is the
reverse of the first 2 parts. The generator MVAR
decreases while the MW is kept constant, and then
the MW decreases and MVAR increases so that the
final generator MW and MVAR are back to the same
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10
-20
-10
40
30
20
10
706050403020
0 MW
MVAR
Fig. 2. Typical plot of machine output power during a thermal sensitivitytest
Stream
TurbineGenerator
Bearing 1X, 1Y Bearing 4X, 4YBearing 3X, 3YBearing 2X, 2Y
A
X YA
Axial view
Fig. 3. Typical layout of a BPSTG
values as when the test is started. The complete process
of the thermal sensitivity test is illustrated in Figure 2.
If the final machine vibration is similar to the vibration
when the test is started, it can be concluded that the
thermal sensitivity is reversible. Otherwise, if the final
machine vibration remains high, the thermal sensitivity
is irreversible and further maintenance actions may
need to be taken.
IV. SVR MODEL BASED ON MACHINE VIBRATION
TRACKING FOR THERMAL SENSITIVITY ANALYSIS
A. Experimental setup
Two back pressure steam turbine generators (BPSTG) used
in a local oil-sand company are investigated in this paper.
They are labeled as G1 and G2. Figure 3 shows a typical
layout of the BPSTG, which consists of a steam turbine and
a generator. They are 4 bearings in total for each generator
and two vibration sensors are installed on each bearing along
x and y axes of 90 degrees apart. During normal operation,
the machines are running at 3600 RPM and the vibration
waveform for each bearing is captured and updated every 2
hours.
B. Thermal component
The thermal sensitivity test serves 2 purposes. The first one
is to determine if there exists irreversible thermal sensitivity.
The other purpose is to determine the ‘size’ of the thermal
component, i.e. the difference between the vibration when the
generator is operating at the low MW and MVAR loading at
the beginning of the test and the vibration when the generator
is operating at the highest MW and MVAR during the test.
The difference has to be within a certain limit otherwise
the generator will not be able to run with its full capacity.
The method used to calculate the thermal component is as
follows:
0 20 40 60 80 100 120 140−1
0
1
Time (ms)
m i l
(a)
0 20 40 60 80 100 120 140
−1
0
1
Time (ms)
m i l
(b)
Fig. 4. Machine vibration waveform, (a) unfiltered, (b) 1X only
During the thermal sensitivity test, at each stage, the
machine vibration peak-to-peak value and its phase can be
recorded. However, it is believed that the vibration due to
thermal bow is mainly shown on 1X (one times) speed, which
is 60 Hz in this case; therefore, in order to eliminate the other
effects, the 1X vibration peak-to-peak value is used. Figure
4 shows a typical machine vibration waveform together with
1X component only. Thus, every cycle in the 1X vibrationwaveform in the x and y direction can be expressed by:
V x =1
2Axcos(θ − θx)
V y =1
2Aycos(θ − θy)
0 ≤ θ < 2π (11)
where Ax, Ay, θx, and θy are the 1X vibration peak-to-peak
value and phase angle in the x and y direction, respectively,
and they can all be recorded during the thermal sensitivity
test.
θy is subtracted by π/2 (or added by 3π/2 if θy−π/2 < 0)
since the vibration sensor x and y are 90o degree apart. Thus,
θy = θy − π/2
V y =1
2Aycos(θ − θy) (12)
To ensure V x and V y are larger than 0, constant terms, 1
2Ax
and 1
2Ay will be added to V x and V y, respectively. Hence,
V x =1
2Ax +
1
2Axcos(θ − θx)
V y =1
2Ay +
1
2Aycos(θ − θy) (13)
Finally, by iteration, a θ can be found which maximizes
the following equation,
V T =
V x
2+ V y
2(14)
The corresponding phase angle can be denoted as θT . At
this point, the overall maximum machine vibration can be
expressed by a vibration vector with magnitude V T and phase
angle θT .
The maximum vibration vector can be calculated for the
machine vibration at the start of the thermal sensitivity test
and at the point when the machine is operating at the highest
MW and MVAR during the test, and then the vibration
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1 2 3 4 50
-1
-2
-3
1
2
Low
Load
High Load
Thermal Component
mil
mil
-1
Fig. 5. Typical plot of the machine 1X vibration vector during a thermalsensitivity test
difference between those 2 conditions can be calculated.
Figure 5 is a typical plot of the vibration vector during a
thermal sensitivity test indicating the thermal component.
The above analysis is focused on the vibration difference
between the lowest load and the highest load of the machine
operation during the thermal sensitivity test. However, how
the machine vibration changes due to thermal sensitivity ina long term has not been taken into consideration. Also, if
the machine thermal sensitivity is irreversible, the thermal
sensitivity test can be destructive since the machine vibration
may become worse after the test. Moreover, when a machine
is undergone a thermal sensitivity test, it has to be removed
from the production line, so the productivity is reduced. It
is therefore desirable to determine whether or not a machine
has been experiencing thermal sensitivity issue by analyzing
normal machine operational data. In this paper, the SVR
technique is applied to build a model for this purpose.
Many different techniques can be used to build a system
model, including building a physical model. However, this
usually requires an in-depth understanding of the machine
structure. Hence, artificial intelligence techniques are often
preferred. Based on [8] and [9], SVR seems to be a better
choice over Neural Network (NN) and adaptive neuro-fuzzy
inference system (ANFIS) and therefore it is selected in
this paper. The method is tested on the two BPSTG and
some valuable preliminary information about rotor thermal
sensitivity problem is obtained.
C. SVR based vibration model
For tracking the machine vibration, a system model is
needed. In this case, the inputs of the model are the generator
output real power and reactive power, and the output of the
model is the machine 1X vibration. Other than the machine
output power, many other factors, such as the temperature of
the machine operating environment, may also have impacts
on the machine vibration. However, machine output power
can be directly controlled by the on-site engineers, and this
is why they are chosen as the inputs of the model. The model
built to predict the machine vibration based on the generator
output power can be mathematically expressed as
y = f (P, Q) (15)
(a)
(b)
Fig. 6. Plots of (a) V TX and (b) V TY , G1
where P and Q are the machine real and reactive power,
respectively, and y is the output related to machine 1X
vibration amplitude. If the model is properly trained and
the machine thermal sensitivity is irreversible, the difference
between the predicted vibration and the real vibration can be
shown in the thermal component analysis. Instead of using
the magnitude or phase angle of the vibration vector as the
model output, the vibration vector is decomposed into two
components by projecting to X and Y axes:
V T X = V T cosθT , V T Y = V T sinθT (16)
Thus, any changes in the magnitude and phase angle of
the machine 1X vibration are reflected in V T X and V T Y .
D. Case studies and the analysis results
In this section, SVR models are built for both G1 and G2
and used to track their vibrations. Based on previous thermal
sensitivity test results from the plant, it is known that G1
does not have serious problem since the thermal sensitivity isreversible. On the other hand, G2 may have serious thermal
sensitivity issue and it is irreversible. Vibrations for both
generators are analyzed separately in the following sections.
Fig. 6 shows the plots of V T X and V T Y of G1. The
vibration data is obtained during the period from Jan. to Aug.
2003 on bearing 4 and there are 2761 data points in total.
From Fig. 6, no obvious trend can be noticed. V T X and V T Y
do not seem to increase or decrease as time progresses. In
order to confirm that the machine condition did not change
during that period, SVR models can be built to predict the
machine vibration based on the machine output power. If the
machine condition indeed did not change during that period,
the model predicted vibration should be very close to thereal vibration as long as the model is properly trained.
When building the SVR models in this section, different
kernel functions have been tried, including linear and poly-
nomial kernel functions with different degrees. By trial and
error while taking the model training time into consideration
as well, polynomial kernel function with degree 2 is selected.
The first 700 data points are used to train the SVR models
and the prediction error is simply the difference between the
predicted vibration value and the real vibration value:
error = V ibrationT, actual − V ibrationT, predicted (17)
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(a)
(b)
Fig. 7. (a) SVR model prediction results for V TX , predicted values (red),actual values (blue), and (b) prediction error, G1
(a)
(b)
Fig. 8. (a) SVR model prediction results for V TY , predicted values (red),actual values (blue), and (b) prediction error, G1
Fig. 7 and 8 shows the SVR model prediction results along
with the real machine vibration and the prediction error. It
can be seen that, for both V T X and V T Y , the predictionresults are very close to the real values. The mean and
the standard deviation of the prediction error of V T X are
−0.024 and 0.2599, respectively, while they are 0.1202 and
0.5018 for V T Y . Also, from the error plots, there is no
clear trend that the prediction error increases or decreases
as time progresses. Therefore, it can be concluded that the
condition of G1 did not change during the period from Jan. to
Aug. 2003, and a thermal sensitivity test around this period
may not be necessary for G1. From the plots, it can also
be concluded that there is a direct relationship between the
machine output power and the machine 1X vibration. It is
possible to build an accurate model with machine real and
reactive powers as the model inputs to predict the machine1X vibration.
Similar analysis can be applied to generator G2. Fig. 10
shows the plots of V T X and V T Y of G2. The vibration data
is obtained from Jan. to Sep. 2003 on bearing 3 and there are
3123 data points in total. SVR models are built with the same
kernel function and parameter for V T X and V T Y , and again
the first 700 data points are used to trained the models. The
prediction results and prediction errors are shown on Fig.
11 and 12. From Fig. 11, it can be seen that the prediction
results are close to the actual values. The mean and standard
deviation of the prediction errors are 0.0114 and 0.2584,
respectively. On the other hand, on Fig. 12, starting from
data points around 1920, which corresponding to June 23,
2003 in actual date, the actual vibration starts to increase,
which causes the prediction error between the predicted V T Y
and the actual V T Y to increase and finally settles down at
data points around 2200, which corresponding to July 16,
2003 in actual date. The mean and standard deviation of theprediction errors are 0.5502 and 0.613, respectively. Hence,
the mean of the prediction error is much larger than those
in the other 3 cases. The prediction error can be further
analyzed with the moving window method to show how the
mean and standard deviation change more clearly. The results
are shown in Fig. 13, with 500 data points as the window
size and 100 data points as the moving size. From Fig. 13,
it is very clear that the mean of the prediction error starts to
increase rapidly after index 15, which is equivalent to index
1900 in the actual data point. If the vibration vectors are
plotted during the period from June 23 to July 16, the result
would be similar to Fig. 9. V T X did not change too much
during that period and it remained at about 2.5 mil, whileV T Y increased approximately from -1 to 1 mil. Thus, the
vibration vector moves from the forth quadrant to the first
quadrant.
From the Fig. 12, it is noticed that the model prediction
errors are small for the first 1800 data points, it can be
confirmed that the SVR model has been trained properly
and it should generate outputs accordingly with the changing
input power. Therefore the difference shown after the index
number 1920, is mainly due to the reason that the machine
condition has changed. The machine condition may change
due to many mechanical reasons, such as the machine may
have been taken off-line and maintenance work have been
performed to the machine, or some machine componentsare worn out. However, it has been confirmed that G2 was
continuously running for the whole period and there was
not any maintenance work done to the machine. Also, if
the machine condition is changed due to components wear
out, the process should be slow and the vibration should
change slowly instead of increasing abruptly as it is shown
in Fig. 12. Another possible cause for the machine condition
change is irreversible thermal sensitivity. By checking the
generator output powers, it is found out that, from June 23 to
27, the generator was operating with very high MVAR, such
as 25MW and 30MVAR, 45MW and 30MVAR, etc. Also,
G2 was considered running in the normal condition since
its peak-to-peak vibration is under the pre-defined limit and
the change of vibration cannot be noticed if the vibration
data was not processed by the method described previously
in this paper. Thus, considering all the analysis above, it
is believed that the vibration change is due to thermal
sensitivity. Based on the results, one could then recommend
a thermal sensitivity test to be scheduled to confirm this.
Since the valuable information about thermal sensitivity can
be obtained before the severe machine condition degradation
by analyzing past operational data, unexpected shutdowns
can be avoided.
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1 2 3 4 50
-1
1
mil
mil
-1
June 23, 2003
July 16, 2003
Change of vibration vector
Fig. 9. Change of vibration vector, G2
(a)
(b)
Fig. 10. Plots of (a) V TX and (b) V TY , G2
(a)
(b)
Fig. 11. (a) SVR model prediction results for V TX , predicted values (red),actual values (blue), and (b) prediction error, G2
(a)
(b)
Fig. 12. (a) SVR model prediction results for V TY , predicted values (red),actual values (blue), and (b) prediction error, G2
0 5 10 15 20 25 30−0.5
0
0.5
1
1.5
m i l
(a)
0 5 10 15 20 25 300.2
0.3
0.4
0.5
0.6
0.7
m i l
(b)
Fig. 13. SVR model prediction results for V TY , (a) mean, and (b) standarddeviation, G2
V. CONCLUSION
Generator thermal sensitivity is studied in this paper. Cur-
rently, in practice, a thermal sensitivity test can be performed
to determine if a generator has thermal sensitivity issue or
not. However, thermal sensitivity test has some disadvantages
and it is preferred to determine the thermal sensitivity prob-
lem based on the regular machine operational data. In this
paper, system model is built with SVR to predict the machine
vibration based on the machine output power. The proposed
method is applied to analyze the thermal sensitivity in 2
BPSTGs and experimental results show that the proposed
method can be used to keep track of the machine condition
and provide valuable information on whether the generator
has thermal sensitivity issue.
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