Research Article Forecasting Models for Hydropower Unit...

Research ArticleForecasting Models for Hydropower UnitStability Using LS-SVM

Liangliang Qiao and Qijuan Chen

College of Power and Mechanical Engineering Wuhan University Wuhan 430072 China

Correspondence should be addressed to Qijuan Chen qjchenwhueducn

Received 14 January 2015 Revised 30 April 2015 Accepted 7 May 2015

Academic Editor Michael Small

Copyright copy 2015 L Qiao and Q ChenThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper discusses a least square support vector machine (LS-SVM) approach for forecasting stability parameters of Francisturbine unit To achieve training and testing data for the models four field tests were presented especially for the vibration in119884-direction of lower generator bearing (LGB) and pressure in draft tube (DT) A heuristic method such as a neural networkusing Backpropagation (NNBP) is introduced as a comparison model to examine the feasibility of forecasting performance Inthe experimental results LS-SVM showed superior forecasting accuracies and performances to the NNBP which is of significantimportance to better monitor the unit safety and potential faults diagnosis

1 Introduction

Hydroelectric powerrsquos low cost near-zero pollution emis-sions and ability to quickly respond to peak loads make it avaluable renewable energy source [1] According to statisticshydropower provides 2245 of the electricity used in Chinaand almost 30 of the nationrsquos electricity from all renewablesources in 2013 [2] By the end of 2013 about 273000MWof hydropower generation capacity exists in China [3] Morethan half of China hydroelectric capacity is in the westernprovinces of Yunnan Tibet and Sichuan with approximately57 of the national total capacity [4 5]

Hydropower generation varies greatly between years withvarying inflows as well as competing water uses such asflood control water supply recreation and in-stream flowrequirements [1] Given hydropowerrsquos economic value and itsrole in complex water systems it is reasonable tomonitor andprotect the hydropower unit from harmful operation modesA unit is often operated through rough zone which will causethe unit vibration and the stability performance will declineThe accident occurred at 813 am on August 17 2009 atturbine number 2 of the Sayano-Shushenskaya Dam Russiarsquoslargest hydropower plant which caused heavy casualties andproperty losses [6] As [7] states the main technologicalcauses are that hydraulic unit number 2 often entered

the nonrecommended band during startup and shutdownoperations and load regulation what is worse the unit wasunder long-term service with inadmissible vibration partic-ularly during the operationwith the temporary turbinewheelto ensure the stability is ultimately connected with the safetyand significant economic efficiency of using hydropowerplants as a source of renewable energy

There are some parameters to describe the unit stabilitysuch as vibration pressure and noise When the parametersexceed a certain value the unit would run in an instabilitycondition The serious vibration of rotating parts will causethe shaftmisalignment Excessive vibration of generator rotorwill increase the abrasion between slip ring andbrush and thebrush would sparkWhat is worse the whole plant house andequipment would be damaged when the resonance occursThe fluctuating pressure in DT will make the flow systemoscillate and the pipe wall crack and even the steel plate willbe lost Abnormal noise generated by unit unstable operationwill be harmful to the workersrsquo physical and mental healthExisting recommendations in Chinese National Standardsregarding stability parameters of hydropower units GBT113485-2008 [8] and GBT 17189-2007 [9] have alarm levelsbased on statistical data and are often used as an aid todetermine and decide if a unit is to be stopped for main-tenance For example the standards GBT113485-2008 and

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 350148 9 pageshttpdxdoiorg1011552015350148

2 Mathematical Problems in Engineering

GBT17189-2007 divide vibration levels into classes withincreasing levels from Class A to Class D where Class A isa good machine that does not need attention while Class D isa machine that should be stopped for immediate correctiveaction The permitted levels for each class vary with theunitrsquos rotational speed a low speed permits higher values ofvibration levels in each class compared to high speed Thestandards are not sufficient as vibrationmonitoring standardssince they do not consider the physical properties of bearingsand brackets as well as specific characteristics of a plant [10]

It is an effective way to understand the stability character-istics of a unit by field test under differentworking conditionsTo determine a machinersquos mechanical condition Nasselqvistet al [10 11] used strained gauges installed inside pivot pinto measure the bearing load in a hydropower unit Talas andToom [12] studied the accurate measurement and analysis ofthe dynamic air gap behavior of large hydroelectric generatorsusing a new fibre-optics instrumentation system and the airgap tests were performed on four 184MVsdotA 156m statorbore diameter generators with 16 radial stator support rodsSun et al [13] made stability tests for the ALSTOM unitson the left bank of the Three Gorge hydropower stationunder low head and gave suggestions for the operationFendin et al [14] gave a black start test of the Swedishpower system which is focused on voltage control andgovernor control as well as on the capability of the individualpower units Khodabakhchian et al [15] performed a morethorough EMTP investigation in which the models and datawere adjusted to reproduce recordings from a field test andproposed a test procedure to determine the parameters of ahydraulic turbine model

For the task of stability parameters identification of ahydropower turbine it is possible to define a regression vectorfrom a set of inputs and nonlinear mapping in order tofinally estimate a model suitable for prediction There aresome typical methods for regression applied in many areas ofengineering research [16ndash18] such as artificial neural network(ANN) and support vector machine (SVM) ANN usuallysuffers from the existence ofmany localminima choosing thenumber of hidden neurons and determining the structure ofthe network the length of the learning cycle and the type ofthe learning process [19] SVM is a relatively novel powerfulmachine learningmethodbased on statistical learning theorywhich was introduced by Shahlaei et al [20] The standardSVM is solved by quadratic programming methods whichare time consuming and finding the final SVM model canbe very difficult because a set of nonlinear equations must besolved [21] As a simplification Rubio et al [22] proposed amodified version of SVM called least square support vectormachine (LS-SVM)which resulted in a set of linear equationsinstead of a quadratic program LS-SVM has been applied toprediction and classification with promising results as can beseen in some works [23ndash26]

In this paper a method based on LS-SVM model ispresented for prediction and regression of hydropower unitstability parametersThe data are obtained from a field test ofa 200MW Francis unit under different working conditionsThe results show good performance of the model which is of

great significance to the unit condition monitoring and faultdetection

The rest of the paper is organized as follows in Section 2a brief description of LS-SVM is given and in Section 3 howto obtain the data based on a field test is shown in detailand the model for prediction and regression of hydropowerunit stability parameters is presented The results usingthe proposed LS-SVM model are discussed in Section 4Finally some conclusions are drawn in Section 5 followed byAcknowledgment and relevant references

2 Methodology

21 Review of LS-SVM LS-SVM is a modification to SVMregression formulation proposed by Rubio et al The mainidea is to transform the problem from quadratic programsto solving a set of linear equations The LS-SVM regressionframework can be formulated as follows [23] Given the dataset 119883

119894 119910119894119897

119894=1 with input vectors 119883119894isin 119877119901 and output values

119910119894isin 119877 consider the regression model 119910

119894= 119891(119883

119894) + 119890119894 where

1199091 1199092 119909119897 are deterministic points 119891 119877119901rarr 119877 is an

unknown real-valued smooth function and 1198901 1198902 119890119897 areuncorrelated random errors with 119864[119890

119894] = 0 119864[1198902

119894] = 120590

2119890lt infin

LS-SVMs have been used to estimate the nonlinear 119891 of theform

119891 (119883119894) = 119882

119879120593 (119883119894) + 119887 (1)

where 120593(119883119894) 119877

119901rarr 119877

119899ℎ denotes the potentially infinite(119899ℎ= infin) dimensional feature mapThe cost function for the

data of the LS-SVMmodel in the primal space is given by

min119882119887119890119897

119875 (119882 119890) =12119882119879119882+120574

12

119897

sum

119894=11198902119894

(2)

st 119910119894= 119882119879120593 (119883119894) + 119887 + 119890

119894 119894 = 1 2 119897 (3)

The formulation includes a bias term as in most standardSVM formulations which is usually not the case in the othermethods The relative importance between the smoothnessof the solution and data fitting is governed by the scalar 120574referred to as the regularization constant The optimizationthat is performed is known as a ridge regression In order tosolve the constrained optimization problem a Lagrangian isconstructed

119871 (119882 119887 119890119894 120572119894) = 119875 (119882 119890

119894)

minus

119897

sum

119894=1120572119894119882119879120593 (119883119894) + 119887 + 119890

119894minus119910119894

(4)

where 120572119894is as the Lagrange multipliers The conditions for

optimality are given by

120597119871

120597119882= 0 997888rarr 119882 =

119897

sum

119894=1120572119894120593 (119883119894)

120597119871

120597119887= 0 997888rarr

119897

sum

119894=1120572119894= 0

Mathematical Problems in Engineering 3

120597119871

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597119871

120597120572119894

= 0 997888rarr 119910119896= 119882119879120593 (119883119894) + 119887 + 119890

119894

(5)

By applying the kernel trick 119870(119883119894 119883119895) = 120593(119883

119894)119879120593(119883119895)

with a positive definite kernel 119870 the dual problem is givenby the following set of linear equations

1119879

119897120572 = 0

(Ω + 119868119873120574minus1) 120572 + 119887 = 119910

997904rArr [

0 1119879

119897

1119897Ω + 120574

minus1119868119897

][

119887

120572] = [

0119910]

(6)

where 119910 = [1199101 1199102 119910119897]119879 1 = [1 1 1]119879 120572 = [1205721 1205722

120572119897]119879 andΩ isin 119877119899times119899 withΩ

119894119895= 119870(119883

119894 119883119895)

The resulting LS-SVM model can be evaluated at newpoint119883lowast by

and

119891 (119883lowast) =

119897

sum

119894=1120572119894119870(119883119894 119883lowast) + 119887 (7)

In (7) 119870(119883119894 119883119895) is defined as the kernel function The

value of the kernel is equal to the inner product of twovectors119883

119894and119883

119895 in the feature spaces120593(119883

119894) and120593(119883

119895) that

is 119870(119883119894 119883119895) = 120593(119883

119894)119879120593(119883119895) This kernel must be positive

definite and must satisfy the Mercer condition

22 Feedforward Neural Network Using Backpropagation(NNBP) The feedforward NNBP is a very popular model inneural networks It does not have feedback connections buterrors are backpropagated duringmodel training Leastmeansquared error is used Many applications can be formulatedwhen using a feedforward NNBP and the methodology isused as the model for most multilayered neural networksErrors in the output determine measures of hidden layer out-put errors which are used as a basis to adjust the connectionweights between the pairs of layers Recalculating the outputsis an iterative process that is carried out until the errors fallbelow a certain tolerance level Learning rate parameters scalethe adjustments to weights Amomentumparameter can alsobe used in scaling the adjustments from a previous iterationand adding to the adjustments in the current iteration [23]

23 Overfitting in LS-SVM and NNBP How well the devel-oping models will make predictions for cases that are notin the training set should be put into consideration LS-SVM and NNBP like other nonlinear parametric modelscan suffer from overfitting problem The models that are toocomplex may fit the noise not just the signal leading tooverfitting Overfitting is dangerous because it can lead topredictions that are far beyond the range of the training datawith LS-SVM and NNBP When the training data includeenough information overfitting can be avoided effectively

Table 1 Specifications

Equipment Type Parameters

Turbine HLK333C-LJ485

Rated power 2041MWRated head 107mRated speed 150 rpmLargest head 127mSmallest head 81mRated discharge 2113m3s

Generator SF2004010800

Rated voltage 138 kVRated excitation voltage38248VRated excitation current13016 AcosΦ 09Frequency 50Hz

In the model applications the data sets applied in LS-SVMand NNBP models are selected from four field tests rangingfrom0MWto 200MWof thewhole load So the training dataof the vibration and pressure have covered all the informationof the unit which can deal with overfitting problem of LS-SVM and NNBP models

LS-SVM is based on the structural risk minimizationprinciple while NNBP is based on the empirical risk min-imization principle LS-SVM includes two structural partsthe error term 120574(12)sum

119897

119894=11198902

119894and the regularization term

(12)119882119879119882 seen as (2) This structure can effectively reduce

the risk of overfitting As for NNBP because the resultsare based on partially neglecting the regularization term(12)119882

119879119882 there is much more danger for overfitting

In addition the selection of the kernel function shouldsatisfy the Mercer condition The radial basis function (RBF)kernel is selected in this paper LS-SVM with RBF kernelyields a good generalization performance And using LS-SVM with an RBF kernel does not risk too much overfittingwhich can be explained by looking to the optimal values ofthe kernel parameter [27]

3 Model Applications

31 Data Sets Based on a Field Test The data sets for theLS-SVM models were selected from field tests of a 200MWFrancis turbine unit in China The test unit located nearthe load center of China Eastern Power Grid is mainlyused to do the peak and frequency regulation It was putinto power generation on August 16 2008 Table 1 givesthe specifications The rated power is of 2041MW and therated speed of 150 revolutions per minute (rpm) Its range ofworking head is between 81m and 127m

The test will mainly measure the following parametersincluding frame vibration guide bearing displacement andpressure fluctuation in DT Figure 1 shows the arrangementof measuring points The capacitance sensor and eddy cur-rent sensor were used for the bearing displacement low-frequency speed sensor was for the vibration measurementpressure transmitter was for the pressure fluctuation mea-surement in DT Figure 2 shows part sensor installation of


Upper bearing bracketUpperlower generator bearing

Stator bearing bracketLower bearing bracket

Turbine head cover Turbine guide bearing

Draft tube

Figure 1 Testing components in a hydropower unit

Figure 2 Part of the sensor installation of LGB

LGB The test working head was 115m 118m 120m and122m In this paper we would select the vibration in 119884-direction of LGB and pressure in DT as the input data of themodels

32 Pressure in DT Forecasting For a Francis turbine it issignificantly meaningful to solve the problem of pressurefluctuation influenced by the low-frequency vortex in DTFrancis turbine works well under the optimal conditionsthat is rated head and wicker gate opening There is lesspressure in DT when the water in runner outlet flows alongthe axial direction while in deviation from the optimaloperating conditions there will be a certain circumferentialvelocity component for the water flowwhich will form vortexphenomenon under the action of centrifugal force

As [28] states Γ2is generally used to describe vortex

intensity of the water flow in runner outlet As Γ2is pro-

portional to 1198811199062

(1198811199062

is absolute velocity component in thecircumferential direction of water flow in runner outlet) itonly needs to carry on the research of 119881

1199062which is shown in

1198811199062 =

120587119903119886

30times 119899minus

cot1205731198872

981198602times119873

119867120578 (8)


050

100150

200

114116

118120

122

050

100150200250300

Power (MW)Head (m)

Peak

-to-p

eak

valu

e (kP

a)

minus50

Figure 3 Pressure of DT changes with power and head

where 119903119886is pitch radius of a certain point in runner blade

edge 1205731198872 is blade angle 119860

2is flow section area of runner

blade outlet 119899 is unit rotation speed rpm119873 is the unit outputpower kW119867 is the working head119898 120578 is unit efficiency

When1198811199062= 0 turbine works under designed conditions

and water flow in DT enters without crashing that is theabsolute velocity is perpendicular to tangential velocity Inthis case there is no circular rector in DT and outlet waterflow is uniformly distributed When 119881

1199062gt 0 turbine works

under small wicket gate openingThe angle between absolutevelocity and tangential velocity is acute and the direction of1198811199062

is consistent with turbine rotation Γ2is positive When

1198811199062

lt 0 turbine discharge is bigger than the rated flowand the unit works under big wicket gate opening The anglebetween absolute velocity and tangential velocity is obtuseThe direction of119881

1199062is opposite to turbine rotationThe water

flow in DT shows reverse rotation In a word when 1198811199062

= 0there will be positive or negative circular rector in DT whichis the direct cause of pressure fluctuation

According to the test results Figure 3 shows that theaverage peak-to-peak pressure in access door of DT changeswith head and power In Figure 3 the values range from10 kPa to 54 kPa in the 0MW to 100MW power sectionthe crest value about 289 kPa appears at 100MW power inhead of 120m between 120MW and 200MWpower sectionthe values are smaller than 70 kPa A trend can be seen inFigure 3 that the pressure will increase with the head Thevalue is 148 kPa at head of 115m 220 kPa at head of 118m and274 kPa at head of 120m Through the amplitude-frequencyanalysis the dominant frequency is 25Hz at both lowerand higher power section which is equal to the rotationfrequency components Between 80MW and 130MWpowerregion there is low-frequency vortex signal and the dominantfrequency is 063Hzwhich is about one-fourth of the rotationfrequency components

Figure 4 gives the time series plot of testing data Underdifferent working head the pressure varies with power Asshown in Figure 4 (a) and (b) give the time series of 80MWand 100MW power in head of 115m and 118m respectivelyIn Figures 4(c) and 4(d) the values are different with 90MWand 130MW power in head of 122m The values show

that nonlinear relationships existed among head power andpressure variables

33 Vibration in 119884-Direction of LGB Forecasting The vibra-tion data related with power and head were collected onAugust 16 2012 September 26 2012 June 6 2013 andOctober 15 2013 respectively The LGB is the main load-bearing part of the whole unit As stated in Chinese NationalStandards GBT113485-2008 and GBT17189-2007 there areallowable values for LGB For example the radial vibration(119883- and 119884-direction) is not allowed to be more than 90120583mand vertical vibration (119885-direction) no more than 70 120583mBased on the data analysis of four times field tests Figure 5shows that the curve of LGB vibration changes with powerand head Figure 6 displays time series plot of testing data

Figure 5 shows that the LGB displacement amplitudevalues change with power and head in 119884-direction Dis-placement amplitude values have no obvious changes withhead variation while the values gradually decrease with theincrease of power In small power region displacement hasits maximum valuesWhen the unit runs in 20MWand if thehead is of 115m displacement amplitude value is 46 120583m in 119884-direction and if the head is of 118m the value is 45120583m Localpeak point appears between 90MW and 140MW In 120mand 122m head the values of local peak point are 40 120583m and41 120583m in power of 130MW and 110MW respectively Whenthe power is close to 200MW displacement amplitude valuesare minimal It is found through spectrum analysis that thedominant frequency of displacement signal is 25Hz (equal tounit rotation frequency) in small and full power region Anddisplacement signal appears as 063Hz of the low-frequencyvortex if the power is between 90MW and 140MW

The vibration of LGB can bemainly affected by hydraulicmechanical and electrical factors Under different workinghead the vibration varies with rotation speed and power Asshown in Figure 6 (a) and (b) give the time series in differentrotation speed of 105 rpm and 165 rpm The values vary fromminus68 120583mto 68120583min 105 rpmandminus56120583mto 56120583min 165 rpmIn Figures 6(c) and 6(d) the vibration values are differentwith power of 80MWand 140MWAlso the ranges are shownfrom minus36 120583m to 36 120583m and minus19 120583m to 19 120583m respectivelyIt is difficult to give the precise mathematical model for therelationship between vibration and working conditions

34 Data Set and Software The data set was divided intotwo groups a training set and a testing set The training andtesting sets were applied for the making of the models and toevaluate the predictive authority of the constructed modelsrespectively The free LS-SVM toolbox (LS-SVM v-18) wasapplied with MATLAB version R2010a to gather all the LS-SVMmodels

35 Model Performance Evaluation The statistical meansof the mean absolute error (MAE) the root mean squareerror (RMSE) and the coefficient of determination (1198772) areused for performance measures of the forecasting models inthis study The magnitude of MAE for forecasting a givenlead time is a measure of the degree of bias The RMSE is


0 1 2 3 4 5 6 7 8

Time (s)

168

minus1

minus170

Pres

sure

(kPa

)

(a) 119899 = 150119867 = 115 and 119875 = 80

0 1 2 3 4 5 6 7 8

Time (s)

326

Pres

sure

(kPa

)

40

minus246

(b) 119899 = 150119867 = 118 and 119875 = 100

0 1 2 3 4 5 6 7 8

Time (s)

353

28

minus297

Pres

sure

(kPa

)

(c) 119899 = 150119867 = 122 and 119875 = 90

0 1 2 3 4 5 6 7 8

Time (s)

134

48

minus38

Pres

sure

(kPa

)(d) 119899 = 150119867 = 122 and 119875 = 130

Figure 4 Time series plot in access door of DT

050

100150

200

114116

118120

12210203040506070

Power (MW)Head (m)

Peak

-to-p

eak

valu

e in

Y-d

irect

ion

(120583m

)

Figure 5 Vibration in 119884-direction of LGB

68

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus68

(a) 119899 = 105119867 = 118 and 119875 = 0

56

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus56

(b) 119899 = 165119867 = 118 and 119875 = 0

36

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus36

(c) 119899 = 150119867 = 118 and 119875 = 80

19

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus19

(d) 119899 = 150119867 = 118 and 119875 = 140

Figure 6 Time series plot of LGB vibration


the average of the forecasting square errors However a fewlarge errors can cause a large RMSE value although most ofthe forecast error magnitudes are within acceptable limitsDespite this disadvantage RMSE is useful as an unbiasedestimate of the variance of the random component And asmaller RMSE indicates better forecasting accuracy betweentwo models These methods can be indicated as follows

MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816

RMSE = radic 1119873

119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)

where 119910test119894 is the predicted value by presented models 119910fore119894is the field test value119873 is the amount of input training dataand 119910

119898is the average value of the field test data set

4 Results and Discussion

41 Vibration Forecasting of LGB Data from the field testson August 16 2012 September 26 2012 and June 6 2013under different working conditions were used for trainingthe LS-SVM model The testing set including 400 pieces ofdata selected from the test on October 15 2013 was usedto validate the performance of the presented model In thisstudy the Gaussian radial basis function was used as thekernel function of LS-SVM The parameters 120574 and 1205902 aredefined as the nonlinear function of the LS-SVMmodel 120574 is aregularization constant and 1205902 is the band width of the radialbasis function (RBF) kernelThe proper selection of these twoparameters is important for the prediction results Since thereare few general guidelines to determine the parameters of LS-SVM this study varied the parameters to select the optimalparameter values for the best forecasting performance Thatis proposed values were chosen over dozens of trial anderror experiments The generalized error was minimum for1205902= 023 and 120574 = 1002 for LS-SVM The parameter values

presented in this paper may be considered the appropriatelevel since the sensitivities of SVM parameters relativelyare not large although the appropriate level of parametersmay differ according to data The activation function of thenetwork was a sigmoid function for NNBP

Figure 7 and Table 2 compare the forecasting perfor-mance among the two models with observed and forecastedvibration value in 119884-direction of LGB LS-SVM showedexcellent performance results for LGB vibration forecastingThe performance of themodels was evaluated by the variableswhich are previously mentionedThe results of the validationtest of the forecasting model as shown in Table 2 clearlyshowed the greater accuracy of the LS-SVM compared to theNNBP model

The testing criteria of MAE RMSE and 1198772 were calcu-lated in order to measure the forecasting performance Theperformance measures of LS-SVM showed lower errors than

Table 2 LS-SVM forecasting performance

Models Performance evaluationMAE RMSE 119877

2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40

Testing dataLS-SVMNNBP

Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)

Figure 7 Vibration forecasting results in 119884-direction of LGB

those of NNBPTheMAE of LS-SVM at 2013 was lower thanthe 2154 of NNBP The RMSE comparisons showed that theerror of NNBP at 3012 was higher than that of LS-SVM at2783 The 1198772 values of the LS-SVM and NNBP were 098and 093 which indicated that LS-SVMhas higher forecastingability

42 Pressure Forecasting of DT Data of pressure in DT fromthe field tests on August 16 2012 June 6 2013 and October15 2013 under different working conditions were used fortraining the LS-SVM model The testing set including 340pieces of data selected from the test on September 26 2012was used to validate the performance of the presentedmodelThe results of forecasting by LS-SVM were compared withthat byNNBPTheoptimized obtained values of1205902 and 120574were037 and 1629 The activation function of the network was asigmoid function for NNBP

Figure 8 displays a plot of observed versus forecast datato compare the performance between the two models withpressure data of DT LS-SVM showed excellent performanceresults for pressure and comparatively good results withrespect to peak value matching The results of the validationtest shown in Table 3 clearly indicated that the LS-SVMforecast wasmore closely aligned to the actual values than theNNBP model because the forecasting errors in the LS-SVMmodel were correspondingly smaller than those in the othermodel

The test criteria parameters achieved for LS-SVM andNNBP in Table 3 show that the coefficient of determination




2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)

Figure 8 Pressure forecasting results of LGB

MAE and RMSE values for LS-SVM model are better thanNNBP model The obtained values of 1198772 for LS-SVM andNNBP models were 095 and 089 respectively The MAE ofLS-SVMwas significantly lower at 3926 than 4261 forNNBPconfirming that the variance forecasting error of LS-SVMwassmaller than that of NNBP The RMSE comparison showedthat the forecasting error of LS-SVM at 7425 was lower thanthat of NNBP at 7920

5 Conclusions

Thispaper has presented an LS-SVMapproach for forecastingstability parameters of a 200MW Francis turbine unit Theobjective of this paper was to examine the feasibility of usingLS-SVM in forecasting the vibration in 119884-direction of LGBand pressure in DT by comparing it with a heuristic methodsuch as NNBP And we would clearly verify predictionperformance of the models by statistical means of MAERMSE and 1198772 The training and testing data for the modelswere selected from four field tests which is an effective wayto understand the unit stability characteristics The field testresults indicate that the stability parameters vary with theunit working conditions such as power rotation speed andworking head For better monitoring of the unit safety andpotential faults diagnosis the evaluation of the models hadshown that prediction performance of LS-SVM is superior toneural networks using backpropagation in prediction of unitstability parameters data Future work will aim at extendingthe methodology developed to deal with more complex

unit working condition models and the LS-SVM and NNBPmodels can be improved tied with optimization algorithmsuch as genetic algorithm (GA)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

Chen Qijuan planned the work and field tests Qiao Lian-gliang drafted the main part of the paper and implementedthe different forecasting methods NNBP and LS-SVM ChenQijuan contributed to the error analysis

Acknowledgment

This research is funded by the National Natural ScienceFoundation of China (no 51379160)

References

[1] KMadani and J R Lund ldquoModelingCaliforniarsquos high-elevationhydropower systems in energy unitsrdquoWater Resources Researchvol 45 no 9 Article IDW09413 2009

[2] China Electric Power Construction Association 2013 AnnualElectric Power Construction Industry China Electric PowerConstruction Association Beijing China 2014

[3] M H Wang Reform and Innovation of Energy China EnergyNews 2014 httppaperpeoplecomcnzgnybhtml2014-0106content 1372716htm

[4] D Han H W Fang B Z Yan and X Y Xu ldquoChinarsquos hydro-power status in 2013rdquo Journal of Hydroelectric Engineering vol33 pp 1ndash5 2014

[5] ldquoChina commissions 139 GW hydropower projectrdquo Power vol158 pp 15ndash16 2014

[6] V S Seleznev A V Liseikin A A Bryksin and P V GromykoldquoWhat caused the accident at the Sayano-Shushenskaya hydro-electric power plant (SSHPP) a seimologistrsquos point of viewrdquoSeismological Research Letters vol 85 pp 817ndash824 2014

[7] V E Fortov M P Fedorov and V V Elistratov ldquoScientificand technological problems of the hydropower industry afterthe accident at the Sayano-Shushenskaya hydropower plantrdquoHerald of the Russian Academy of Sciences vol 81 no 4 pp 333ndash340 2011

[8] Standardization Administration of the Peoplersquos Republic ofChinaMechanical VibrationmdashEvaluation of Machine Vibrationby Measurements on Rotating ShaftsmdashPart 5 Machine Sets inHydraulic Power Generating and Pumping Plants Standardiza-tion Administration of the Peoplersquos Republic of China BeijingChina 2008

[9] Standardization Administration of the Peoplersquos Republic ofChina Code for Field Measurement of Vibrations and Pul-sation in Hydraulic Machines (Turbines Storage Pumps andPump-Turbines) StandardizationAdministration of the PeoplersquosRepublic of China Beijing China 2007

[10] M Nasselqvist R Gustavsson and J-O Aidanpaa ldquoA method-ology for protective vibration monitoring of hydropower unitsbased on the mechanical propertiesrdquo Transactions of the ASME


Journal of Dynamic Systems Measurement and Control vol 135no 4 Article ID 041007 8 pages 2013

[11] M Nasselqvist R Gustavsson and J-O Aidanpaa ldquoBearingload measurement in a hydropower unit using strain gaugesinstalled inside pivot pinrdquo Experimental Mechanics vol 52 no4 pp 361ndash369 2012

[12] P Talas and P Toom ldquoDynamic measurement and analysisof air gap variations in large hydroelectric generatorsrdquo IEEETransactions on Power Apparatus and Systems vol 102 no 9pp 3098ndash3106 1983

[13] J P Sun H Xiong K L Duan and L Y Zheng ldquoFunctionanalysis of ALSTOM units on the left bank of three Gorgehydropower station operated under low headrdquo Journal ofHydroelectric Engineering vol 26 no 3 pp 129ndash133 2007

[14] H Fendin T Hansen M Hemmingsson and D KarlssonldquoBlack start test of the Swedish power systemrdquo in Proceedings ofthe IEEEPESTrondheimPowerTechThePower of Technology fora Sustainable Society (POWERTECH 11) TrondheimNorwayJune 2011

[15] B Khodabakhchian G T Vuong and S Bastien ldquoOn the com-parison between a detailed turbine-generator EMTP simulationand corresponding field test resultsrdquo International Journal ofElectrical Power and Energy Systems vol 19 no 4 pp 263ndash2681997

[16] Y S Diao and H Ren ldquoStructural damage early warning basedon AR model and factor analysisrdquo Journal of Vibration andShock vol 33 pp 115ndash119 2014

[17] D-M Wang L Wang and G-M Zhang ldquoShort-term windspeed forecastmodel for wind farms based on genetic BP neuralnetworkrdquo Journal of Zhejiang University (Engineering Science)vol 46 no 5 pp 837ndash841 2012

[18] G D G Maria C Stefano F Antonio and M C PaololdquoComparison between wind power predction models basedon wavelet decomposition with least squares support vectormachine (LS-SVM) and artificial neural networkrdquo Engergiesvol 7 pp 5251ndash5272 2014

[19] V Dua ldquoAn artificial neural network approximation baseddecomposition approach for parameter estimation of systemof ordinary differential equationsrdquo Computers amp ChemicalEngineering vol 35 no 3 pp 545ndash553 2011

[20] M Shahlaei A Fassihi and L Saghaie ldquoApplication of PC-ANNand PC-LS-SVM in QSAR of CCR1 antagonist compounds acomparative studyrdquo European Journal of Medicinal Chemistryvol 45 no 4 pp 1572ndash1582 2010

[21] M M Adankon M Cheriet and A Biem ldquoSemisupervisedlearning using Bayesian interpretation application to LS-SVMrdquoIEEE Transactions on Neural Networks vol 22 no 4 pp 513ndash524 2011

[22] G Rubio H Pomares I Rojas and L J Herrera ldquoA heuristicmethod for parameter selection in LS-SVM application to timeseries predictionrdquo International Journal of Forecasting vol 27no 3 pp 725ndash739 2011

[23] S H Hwang D H Ham and J H Kim ldquoForecasting perfor-mance of LS-SVM for nonlinear hydrological time seriesrdquoKSCEJournal of Civil Engineering vol 16 no 5 pp 870ndash882 2012

[24] E Comak andAArslan ldquoA biomedical decision support systemusing LS-SVM classifier with an efficient and new parameterregularization procedure for diagnosis of heart valve diseasesrdquoJournal of Medical Systems vol 36 no 2 pp 549ndash556 2012

[25] D Moreno-Salinas D Chaos J Manuel de la Cruz and JAranda ldquoIdentification of a surface marine vessel using LS-SVMrdquo Journal of Applied Mathematics vol 2013 Article ID803548 11 pages 2013

[26] E Yılmaz ldquoAn expert system based on fisher score and LS-SVM for cardiac arrhythmia diagnosisrdquo Computational andMathematicalMethods inMedicine vol 2013 Article ID 8496746 pages 2013

[27] N Pochet F de Smet J A K Suykens and B L R de MoorldquoSystematic benchmarking of microarray data classificationassessing the role of non-linearity and dimensionality reduc-tionrdquo Bioinformatics vol 20 no 17 pp 3185ndash3195 2004

[28] B Yu andZ X Liu ldquoResearch on vibration problemof hydraulicturbines under the high head and low output operation con-ditionsrdquo Journal of Hydroelectric Engineering vol 1 pp 58ndash652001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


GBT17189-2007 divide vibration levels into classes withincreasing levels from Class A to Class D where Class A isa good machine that does not need attention while Class D isa machine that should be stopped for immediate correctiveaction The permitted levels for each class vary with theunitrsquos rotational speed a low speed permits higher values ofvibration levels in each class compared to high speed Thestandards are not sufficient as vibrationmonitoring standardssince they do not consider the physical properties of bearingsand brackets as well as specific characteristics of a plant [10]

It is an effective way to understand the stability character-istics of a unit by field test under differentworking conditionsTo determine a machinersquos mechanical condition Nasselqvistet al [10 11] used strained gauges installed inside pivot pinto measure the bearing load in a hydropower unit Talas andToom [12] studied the accurate measurement and analysis ofthe dynamic air gap behavior of large hydroelectric generatorsusing a new fibre-optics instrumentation system and the airgap tests were performed on four 184MVsdotA 156m statorbore diameter generators with 16 radial stator support rodsSun et al [13] made stability tests for the ALSTOM unitson the left bank of the Three Gorge hydropower stationunder low head and gave suggestions for the operationFendin et al [14] gave a black start test of the Swedishpower system which is focused on voltage control andgovernor control as well as on the capability of the individualpower units Khodabakhchian et al [15] performed a morethorough EMTP investigation in which the models and datawere adjusted to reproduce recordings from a field test andproposed a test procedure to determine the parameters of ahydraulic turbine model

For the task of stability parameters identification of ahydropower turbine it is possible to define a regression vectorfrom a set of inputs and nonlinear mapping in order tofinally estimate a model suitable for prediction There aresome typical methods for regression applied in many areas ofengineering research [16ndash18] such as artificial neural network(ANN) and support vector machine (SVM) ANN usuallysuffers from the existence ofmany localminima choosing thenumber of hidden neurons and determining the structure ofthe network the length of the learning cycle and the type ofthe learning process [19] SVM is a relatively novel powerfulmachine learningmethodbased on statistical learning theorywhich was introduced by Shahlaei et al [20] The standardSVM is solved by quadratic programming methods whichare time consuming and finding the final SVM model canbe very difficult because a set of nonlinear equations must besolved [21] As a simplification Rubio et al [22] proposed amodified version of SVM called least square support vectormachine (LS-SVM)which resulted in a set of linear equationsinstead of a quadratic program LS-SVM has been applied toprediction and classification with promising results as can beseen in some works [23ndash26]

In this paper a method based on LS-SVM model ispresented for prediction and regression of hydropower unitstability parametersThe data are obtained from a field test ofa 200MW Francis unit under different working conditionsThe results show good performance of the model which is of

great significance to the unit condition monitoring and faultdetection

The rest of the paper is organized as follows in Section 2a brief description of LS-SVM is given and in Section 3 howto obtain the data based on a field test is shown in detailand the model for prediction and regression of hydropowerunit stability parameters is presented The results usingthe proposed LS-SVM model are discussed in Section 4Finally some conclusions are drawn in Section 5 followed byAcknowledgment and relevant references

2 Methodology

21 Review of LS-SVM LS-SVM is a modification to SVMregression formulation proposed by Rubio et al The mainidea is to transform the problem from quadratic programsto solving a set of linear equations The LS-SVM regressionframework can be formulated as follows [23] Given the dataset 119883

119894 119910119894119897

119894=1 with input vectors 119883119894isin 119877119901 and output values

119910119894isin 119877 consider the regression model 119910

119894= 119891(119883

119894) + 119890119894 where

1199091 1199092 119909119897 are deterministic points 119891 119877119901rarr 119877 is an

unknown real-valued smooth function and 1198901 1198902 119890119897 areuncorrelated random errors with 119864[119890

119894] = 0 119864[1198902

119894] = 120590

2119890lt infin

LS-SVMs have been used to estimate the nonlinear 119891 of theform

119891 (119883119894) = 119882

119879120593 (119883119894) + 119887 (1)

where 120593(119883119894) 119877

119901rarr 119877

119899ℎ denotes the potentially infinite(119899ℎ= infin) dimensional feature mapThe cost function for the

data of the LS-SVMmodel in the primal space is given by

min119882119887119890119897

119875 (119882 119890) =12119882119879119882+120574

12

119897

sum

119894=11198902119894

(2)

st 119910119894= 119882119879120593 (119883119894) + 119887 + 119890

119894 119894 = 1 2 119897 (3)

The formulation includes a bias term as in most standardSVM formulations which is usually not the case in the othermethods The relative importance between the smoothnessof the solution and data fitting is governed by the scalar 120574referred to as the regularization constant The optimizationthat is performed is known as a ridge regression In order tosolve the constrained optimization problem a Lagrangian isconstructed

119871 (119882 119887 119890119894 120572119894) = 119875 (119882 119890

119894)

minus

119897

sum

119894=1120572119894119882119879120593 (119883119894) + 119887 + 119890

119894minus119910119894

(4)

where 120572119894is as the Lagrange multipliers The conditions for

optimality are given by

120597119871

120597119882= 0 997888rarr 119882 =

119897

sum

119894=1120572119894120593 (119883119894)

120597119871

120597119887= 0 997888rarr

119897

sum

119894=1120572119894= 0


120597119871

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597119871

120597120572119894

= 0 997888rarr 119910119896= 119882119879120593 (119883119894) + 119887 + 119890

119894

(5)


119894)119879120593(119883119895)


1119879

119897120572 = 0

(Ω + 119868119873120574minus1) 120572 + 119887 = 119910

997904rArr [

0 1119879

119897

1119897Ω + 120574

minus1119868119897

][

119887

120572] = [

0119910]

(6)

where 119910 = [1199101 1199102 119910119897]119879 1 = [1 1 1]119879 120572 = [1205721 1205722


119894119895= 119870(119883

119894 119883119895)


and

119891 (119883lowast) =

119897

sum

119894=1120572119894119870(119883119894 119883lowast) + 119887 (7)



119894and119883


119894) and120593(119883

119895) that

is 119870(119883119894 119883119895) = 120593(119883













119897

119894=11198902













Draft tube








(1198811199062



1198811199062 =

120587119903119886

30times 119899minus

cot1205731198872

981198602times119873

119867120578 (8)


050

100150

200

114116

118120

122

050

100150200250300

Power (MW)Head (m)

Peak

-to-p

eak

valu

e (kP

a)

minus50











1198811199062














0 1 2 3 4 5 6 7 8

Time (s)

168

minus1

minus170

Pres

sure

(kPa

)

(a) 119899 = 150119867 = 115 and 119875 = 80

0 1 2 3 4 5 6 7 8

Time (s)

326

Pres

sure

(kPa

)

40

minus246

(b) 119899 = 150119867 = 118 and 119875 = 100

0 1 2 3 4 5 6 7 8

Time (s)

353

28

minus297

Pres

sure

(kPa

)

(c) 119899 = 150119867 = 122 and 119875 = 90

0 1 2 3 4 5 6 7 8

Time (s)

134

48

minus38

Pres

sure

(kPa

)(d) 119899 = 150119867 = 122 and 119875 = 130


050

100150

200

114116

118120

12210203040506070

Power (MW)Head (m)

Peak

-to-p

eak

valu

e in

Y-d

irect

ion

(120583m

)


68

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus68

(a) 119899 = 105119867 = 118 and 119875 = 0

56

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus56

(b) 119899 = 165119867 = 118 and 119875 = 0

36

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus36

(c) 119899 = 150119867 = 118 and 119875 = 80

19

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus19

(d) 119899 = 150119867 = 118 and 119875 = 140




MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816


119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)










2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40


Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)









2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120597119871

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597119871

120597120572119894

= 0 997888rarr 119910119896= 119882119879120593 (119883119894) + 119887 + 119890

119894

(5)


119894)119879120593(119883119895)


1119879

119897120572 = 0

(Ω + 119868119873120574minus1) 120572 + 119887 = 119910

997904rArr [

0 1119879

119897

1119897Ω + 120574

minus1119868119897

][

119887

120572] = [

0119910]

(6)

where 119910 = [1199101 1199102 119910119897]119879 1 = [1 1 1]119879 120572 = [1205721 1205722


119894119895= 119870(119883

119894 119883119895)


and

119891 (119883lowast) =

119897

sum

119894=1120572119894119870(119883119894 119883lowast) + 119887 (7)



119894and119883


119894) and120593(119883

119895) that

is 119870(119883119894 119883119895) = 120593(119883













119897

119894=11198902













Draft tube








(1198811199062



1198811199062 =

120587119903119886

30times 119899minus

cot1205731198872

981198602times119873

119867120578 (8)


050

100150

200

114116

118120

122

050

100150200250300

Power (MW)Head (m)

Peak

-to-p

eak

valu

e (kP

a)

minus50











1198811199062














0 1 2 3 4 5 6 7 8

Time (s)

168

minus1

minus170

Pres

sure

(kPa

)

(a) 119899 = 150119867 = 115 and 119875 = 80

0 1 2 3 4 5 6 7 8

Time (s)

326

Pres

sure

(kPa

)

40

minus246

(b) 119899 = 150119867 = 118 and 119875 = 100

0 1 2 3 4 5 6 7 8

Time (s)

353

28

minus297

Pres

sure

(kPa

)

(c) 119899 = 150119867 = 122 and 119875 = 90

0 1 2 3 4 5 6 7 8

Time (s)

134

48

minus38

Pres

sure

(kPa

)(d) 119899 = 150119867 = 122 and 119875 = 130


050

100150

200

114116

118120

12210203040506070

Power (MW)Head (m)

Peak

-to-p

eak

valu

e in

Y-d

irect

ion

(120583m

)


68

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus68

(a) 119899 = 105119867 = 118 and 119875 = 0

56

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus56

(b) 119899 = 165119867 = 118 and 119875 = 0

36

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus36

(c) 119899 = 150119867 = 118 and 119875 = 80

19

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus19

(d) 119899 = 150119867 = 118 and 119875 = 140




MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816


119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)










2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40


Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)









2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Draft tube








(1198811199062



1198811199062 =

120587119903119886

30times 119899minus

cot1205731198872

981198602times119873

119867120578 (8)


050

100150

200

114116

118120

122

050

100150200250300

Power (MW)Head (m)

Peak

-to-p

eak

valu

e (kP

a)

minus50











1198811199062














0 1 2 3 4 5 6 7 8

Time (s)

168

minus1

minus170

Pres

sure

(kPa

)

(a) 119899 = 150119867 = 115 and 119875 = 80

0 1 2 3 4 5 6 7 8

Time (s)

326

Pres

sure

(kPa

)

40

minus246

(b) 119899 = 150119867 = 118 and 119875 = 100

0 1 2 3 4 5 6 7 8

Time (s)

353

28

minus297

Pres

sure

(kPa

)

(c) 119899 = 150119867 = 122 and 119875 = 90

0 1 2 3 4 5 6 7 8

Time (s)

134

48

minus38

Pres

sure

(kPa

)(d) 119899 = 150119867 = 122 and 119875 = 130


050

100150

200

114116

118120

12210203040506070

Power (MW)Head (m)

Peak

-to-p

eak

valu

e in

Y-d

irect

ion

(120583m

)


68

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus68

(a) 119899 = 105119867 = 118 and 119875 = 0

56

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus56

(b) 119899 = 165119867 = 118 and 119875 = 0

36

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus36

(c) 119899 = 150119867 = 118 and 119875 = 80

19

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus19

(d) 119899 = 150119867 = 118 and 119875 = 140




MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816


119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)










2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40


Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)









2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










050

100150

200

114116

118120

122

050

100150200250300

Power (MW)Head (m)

Peak

-to-p

eak

valu

e (kP

a)

minus50











1198811199062














0 1 2 3 4 5 6 7 8

Time (s)

168

minus1

minus170

Pres

sure

(kPa

)

(a) 119899 = 150119867 = 115 and 119875 = 80

0 1 2 3 4 5 6 7 8

Time (s)

326

Pres

sure

(kPa

)

40

minus246

(b) 119899 = 150119867 = 118 and 119875 = 100

0 1 2 3 4 5 6 7 8

Time (s)

353

28

minus297

Pres

sure

(kPa

)

(c) 119899 = 150119867 = 122 and 119875 = 90

0 1 2 3 4 5 6 7 8

Time (s)

134

48

minus38

Pres

sure

(kPa

)(d) 119899 = 150119867 = 122 and 119875 = 130


050

100150

200

114116

118120

12210203040506070

Power (MW)Head (m)

Peak

-to-p

eak

valu

e in

Y-d

irect

ion

(120583m

)


68

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus68

(a) 119899 = 105119867 = 118 and 119875 = 0

56

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus56

(b) 119899 = 165119867 = 118 and 119875 = 0

36

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus36

(c) 119899 = 150119867 = 118 and 119875 = 80

19

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus19

(d) 119899 = 150119867 = 118 and 119875 = 140




MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816


119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)










2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40


Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)









2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8

Time (s)

168

minus1

minus170

Pres

sure

(kPa

)

(a) 119899 = 150119867 = 115 and 119875 = 80

0 1 2 3 4 5 6 7 8

Time (s)

326

Pres

sure

(kPa

)

40

minus246

(b) 119899 = 150119867 = 118 and 119875 = 100

0 1 2 3 4 5 6 7 8

Time (s)

353

28

minus297

Pres

sure

(kPa

)

(c) 119899 = 150119867 = 122 and 119875 = 90

0 1 2 3 4 5 6 7 8

Time (s)

134

48

minus38

Pres

sure

(kPa

)(d) 119899 = 150119867 = 122 and 119875 = 130


050

100150

200

114116

118120

12210203040506070

Power (MW)Head (m)

Peak

-to-p

eak

valu

e in

Y-d

irect

ion

(120583m

)


68

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus68

(a) 119899 = 105119867 = 118 and 119875 = 0

56

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus56

(b) 119899 = 165119867 = 118 and 119875 = 0

36

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus36

(c) 119899 = 150119867 = 118 and 119875 = 80

19

0

Vibr

atio

n (120583

m)

0 1 2 3 4 5 6 7 8

Time (s)

minus19

(d) 119899 = 150119867 = 118 and 119875 = 140




MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816


119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)










2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40


Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)









2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











MAE = 1119873

119873

sum

119894=1

1003816100381610038161003816119910test119894 minus119910fore1198941003816100381610038161003816


119873

sum

119894=1(1003816100381610038161003816119910test119894 minus 119910fore119894

1003816100381610038161003816)2

1198772= 1minus

sum119873

119894=1 (119910test119894 minus 119910fore119894)

sum119873

119894=1 (119910fore119894 minus 119910119898)

(9)










2

LS-SVM 2013 2783 098NNBP 2154 3012 093

0 50 100 150 200 250 300 350

0

10

20

30

40


Time (s)

minus40

minus30

minus20

minus10

Vibr

atio

n va

lue (

120583m

)









2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2

LS-SVM 3926 7425 095NNBP 4261 7920 089


0 50 100 150 200 250 300 350 400

0

50

100

150

200

250

300

Time (s)

minus100

minus50

Pres

sure

(kPa

)



5 Conclusions







Acknowledgment


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Forecasting Models for Hydropower Unit...

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