Research Article Optimized Extreme Learning Machine for ...downloads.hindawi.com/journals/mpe/2015/529724.pdf · the PRTSA model mainly includes arti cial neural networks (ANN), decision

Research ArticleOptimized Extreme Learning Machine for Power SystemTransient Stability Prediction Using Synchrophasors

Yanjun Zhang1 Tie Li1 Guangyu Na1 Guoqing Li2 and Yang Li2

1State Grid Liaoning Electric Power Supply Co Ltd Shenyang 110006 China2School of Electrical Engineering Northeast Dianli University Jilin 132012 China

Correspondence should be addressed to Yang Li 876464104qqcom

Received 13 August 2015 Revised 10 September 2015 Accepted 13 September 2015

Academic Editor Mohammed Nouari

Copyright copy 2015 Yanjun Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new optimized extreme learning machine- (ELM-) based method for power system transient stability prediction (TSP) usingsynchrophasors is presented in this paper First the input features symbolizing the transient stability of power systems are extractedfrom synchronizedmeasurementsThen an ELM classifier is employed to build the TSPmodel And finally the optimal parametersof the model are optimized by using the improved particle swarm optimization (IPSO) algorithmThe novelty of the proposal is inthe fact that it improves the prediction performance of the ELM-based TSPmodel by using IPSO to optimize the parameters of themodel with synchrophasors And finally based on the test results on both IEEE 39-bus system and a large-scale real power systemthe correctness and validity of the presented approach are verified

1 Introduction

Monitoring the power system stability status in real-timehas been regarded as an important work to guarantee thepower system safe and stable operation [1 2] Up to nowthe existing transient stability analysis (TSA)methodsmainlycan be divided into 3 classes directmethods [3] time-domainsimulations [4] and the extended equal area criterionmethod[5] Unfortunately these methods cannot work well for real-time stability analysis of modern complex power systems

In recent years pattern-recognition-based TSA (PRTSA)has been attracting the ever-growing attention of researchersall over the world [6 7] This kind of method has proved tobe potential in the area of on-line dynamic security analysisby applying of the techniques of machine learning By farthe PRTSA model mainly includes artificial neural networks(ANN) decision trees (DT) and support vector machines(SVM) [8ndash15] However the reported PRTSA approachesusually suffer from some inherent disadvantages and lackthe ability of big data management and utilization whichrestricts its further application in actual operating scenariosFor example ANN has problems of overfitting local optima

and slow convergence and SVM has difficulty in parameterselectionOn the other hand wide areameasurement systems(WAMS) provide the synchronousmeasurement informationfor the wide area power systems [16] which makes it possibleto explore wide area protection and control schemes to avoidthe system collapse [17ndash19]

In recent years a novelmachine learning algorithm calledextreme learning machine (ELM) is proposed by Huang et al[20] Contrastedwith those conventional PRTSA approachesELM has a lot of significant advantages such as better gen-eralization ability and a much faster learning speed [21ndash23]Inspired by the social behavior of flocks particle swarm opti-mization (PSO) algorithm is proposed in 1995 [24] PSO hasbeen widely used to solve a variety of optimization problemswith many of advantages including good robustness fastconvergence speed and high search efficiency [25]

In this paper a novel ELM-based transient stability pre-diction (TSP) method using synchronized measurements isproposed Moreover to further improve the prediction per-formance the ideal model is obtained by applying theimproved particle swarm optimization (IPSO) algorithm toselect the optimal parameters of the model

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 529724 8 pageshttpdxdoiorg1011552015529724

2 Mathematical Problems in Engineering

The rest of this paper is arranged as follows First of allthe used methodologies including ELM classification PSOare presented briefly Secondly the proposed real-time TSPmethod based on IPSO-ELM is presented in detail Finallythe proposal is tested using the IEEE 39-bus system and a realsystem

2 Related Methodologies

21 ELM Classification Assuming an ELM with 119871 hiddenlayer neurons to model data samples x

119894 y119894119873

119894=1 it can be

mathematically represented as

119871

sum

119894=1

120573119894119866(a119894 119887119894 x119895) = y119895 119895 = 1 119873 (1)

where a119894and120573

119894are respectively the input and output weights

vector 119866(sdot) is activation functions and 119887119894denotes the bias of

the 119894th hidden nodeFor the convenience of expression (1) can be rewritten as

H120573 = Y

H =

[[[[

[

ℎ1(x1) sdot sdot sdot ℎ

119871(x1)


119871(x119873)

]]]]

]

(2)

whereH is the hidden layer output matrixELM is to minimize the training error as well as the norm

of the output weights [20]

Minimize 1003817100381710038171003817H120573 minus Y10038171003817100381710038172 and10038171003817100381710038171205731003817100381710038171003817 (3)

Finally the minimal norm least square method is used inthe original implementation of ELM

= HdaggerY (4)

whereHdagger is the Moore-Penrose generalized inverse ofH

22 PSO The fundamental principle of PSO is to find theoptimal solution in the complex search-space by movingcandidate solutions (called particles) according to the com-petition and collaboration among particles through repetitiveiterations The movement of each particle is determined by amathematical formulae over its position and velocity In the119896 + 1 iteration the velocity and position renewal equation of119894th particle are respectively as follows

V119896+1119894

= 119908V119896119894+ 11988811199031(119875best minus 119904

119896

119894) + 11988821199032(119866best minus 119904

119896

119894)

119904119896+1

119894= 119904119896

119894+ V119896+1119894

(5)

where119875best and119866best are respectively denoted as the local andglobal best known solution 119896 and119908 are respectively denotedas the evolutionary generation and the inertia weight 119888

1and

1198882are the learning factors which represent the self-cognition

and social-cognition in turn 1199031and 1199032are uniform random

numbers obeying the 0-1 distribution

3 Real-Time TSP Based on IPSO-ELM

31 IPSO Algorithm As pointed out by the famous ldquoNo freelunchrdquo theorem the overall performances of different opti-mization algorithms are equivalent [26] which implies thatnone of algorithms can always achieve the optimal for allaspects In this paper a mutation strategy is introduced toavoid the premature convergence to local optimum of PSO

First the optimization process is monitored by dynami-cally monitoring changes in population fitness variance 1205902

1205902=

119873119901

sum

119894=1

(

119891119894minus 119891avg

119891best)

2

(6)

where119873119901denotes the population size119891

119894is the fitness value of

the 119894th individual particle 119891best refers to the best fitness valuein the whole population and 119891avg denotes the average fitnessin the current iteration

Second when premature convergence occurs a mutationstrategy is used to maintain population diversity Specificallythe positions of particles are updated by adding randomperturbations timely as shown as follows

119909(119896+1)

= 119909(119896)+ 119888119898(119909max minus 119909min) sdot (rand minus 05) (7)

where 119888119898is the variation coefficient rand is a real number

randomly generated in the range from 0 to 1 119909(119896+1) and 119909(119896)are respectively the positions of the (119896+1)th and 119896th iterationof particles

The criterion to determine the occurrence of prematureconvergence is given as follows

119898 lt120590119896+1

120590119896

lt 119899 (8)

Here 120590119896+1

and 120590119896are the population fitness variances of the

(119896 + 1)th and 119896th iteration

32 Fitness Function As is known a proper fitness functionplays an important role in optimization problems In thispaper the fitness function is the classification accuracy of 5-fold cross-validation (CV)

119865 (120579) = 1198605-CV (9)

where 120579 is themodel parameter vector to be optimized whichis represented by the position of each particle

33 Coding Scheme A mixed-integer encoding scheme isused in the optimization process [27] Here it is consideredthat the 119896th individual particlestate 120579

119896will be constituted by

120579119896= [a1198791 a119879

119871 1198871 119887

119871 1199041 119904

119899 1198881198911 119888119891

ℎ]119879

(10)

where 119899 is the total count of input features 119904119894(119894 = 1 119899) is

a binary variable and each binary code (ldquo1rdquo or ldquo0rdquo) refers towhether the corresponding feature is selected or not 119888119891

119895=

0 1 2 (119895 = 1 119871) is an integer variable that defines

Mathematical Problems in Engineering 3

Data preprocessing

Parameterinitialization

k = 1

Calculate output weightsand individual fitness

PSO optimization

Prematureconvergence

No

No

Yes

Yes

k = k + 1

Mutation

Termination conditionis satisfied

Output the optimal model

Figure 1 Flowchart of the modeling process

the activation function119891119895of each neuron 119895 of the hidden layer

as follows

119891119895 (120592) =

0 if 119888119891119895= 0

1

(1 + exp (minus120592)) if 119888119891

119895= 1

120592 if 119888119891119895= 2

(11)

The use of parameters 119888119891119895makes possible the adjustment

of the number of neurons (if 119888119891119895

= 0 the neuron isnot considered) and the activation function of each neuron(sigmoid or linear function) To facilitate the optimizationthe decision variables are mapped into real variables withinthe interval [0 1] and all variables need to be converted intotheir true value when computing the fitness value of eachparticle [27]

34 Modeling Process Themodeling process of the proposedmethod can be divided into 8 steps

Step 1 The used data preprocessing approach here is 119911-scorestandardization method [15]

1199111015840=

(119911 minus 119885)

120590119885

(12)

where 119885 is the mean of any feature 119885 in sample data 120590119885is

the standard deviation of the feather 119885 1199111015840 is the normalizedvalue corresponding to 119911 119911 isin 119885

Step 2 Initialization of the parameters the maximum itera-tion number is assigned to 200 the population size119873

119901is set

to 20 and the number of ELM hidden layer neurons is 50

Step 3 Initialization of the population 119873119901solutions are

generated randomly and each solution is corresponding toa particle which is encoded according to (10)

Step 4 According to (4) calculate the output weights 120573 andthe individual fitness values in turn

Step 5 According to the optimization mechanism of PSOalgorithm update the location of individual particle andgenerate the next populations

Step 6 Dynamic monitoring of changes in the populationfitness variance once premature convergence occurs savethe current optimal solution 120579lowast and go on to the mutationoperation If a better solution 1205791015840 (119865(1205791015840) gt 119865(120579lowast)) is found inthe solution space then update the optimal solution 120579lowast = 1205791015840and quit the mutation operation

Step 7 Judgment of termination condition the optimizationprocess will be terminated if the current number of iterations119896 exceeds the prespecified maximum number of iterations orthe value of fitness function is greater than 9900 otherwise119896 = 119896 + 1 and jump to Step 4

Step 8 Acquisition of the ideal model output the optimalsolution 120579lowast and obtain the ideal TSP model

The flowchart of the modeling process is shown inFigure 1

35 Construction of the Initial Feature Set As is known tous all input features play an important role in PRTSA [814 15] However the used features in previous works aremainly prefault static features The reason for this is that


G G

GG

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

10

11

13

14

15

831

126

32

7

Figure 2 IEEE 39-bus test system

the former measurement systems are not able to providewide area dynamic information To take full advantage ofthe synchrophasors information from WAMS [16ndash19] thepresented approach selects input features from both prefaultstatic information and postfault dynamic information

As an extension of related works the selected initialfeatures used in the presented approach are the same as theones in [15] (see [15] for further details)These features are bemade up of prefault static features and the postfault dynamicfeatures Therefore the above selected feathers comprehen-sively indicate the stability of power systems during differentstages of the disturbance process and they are appropriatelyselected to constitute the original feature set of the transientdisturbed pattern space

36 TSP Based on the Trained Model In this section howthe ELM-based TSP model is used after it has been trainedis explained in brief It should be noted that synchrophasorsare simulated through the detailed time-domain simulationsin this work Here all the used input features can be obtainedfrom the following physical quantities comprising the rotorangle angular velocity mechanical power and electromag-netic power of each generator and the generatorsrsquo inertialtime constants and all these physical quantities are availablefrom PMU measurements except for the given inertial timeconstants Therefore the proposed method is able to beapplied to TSP based on PMUmeasurements

In the present approach it is supposed that trippingsignal(s) issued by the local protection is available for trig-gering the TSP system Once a fault is cleared by the actionof relevant relays the trigger allows starting of taking thesamples of the input variables to construct the input vector forthe proposedELM-basedTSPmodel And then the proposedmodel takes 9 consecutive synchronously measured samplesof each generator at the rate of 60 per sec to form the inputvector for the classifier Finally for a specific input vector thetransient stability status of the disturbed power system can beimmediately predicted by using the trained model

4 Results and Discussion

For power system TSA the New England 39-bus test systemis a widely used test system to examine the performance ofvarious assessment methods [9ndash15] The system contains 10generators and 39 buses which is shown in Figure 2

41 Generation of Knowledge Base It is known that forPRTSA the generalization ability of a TSA model largelydepends on the completeness and representativeness of theknowledge base (KB) Hence it is an important work togenerate the used KB reflecting the relationship between theinput features and the stability status In this work KB ismade through extensive time-domain simulations in detailThe employed generator model is four-order model withIEEE DC1 excitation system the load model is the constantimpedance model The fault type is three-phase short-circuitfaults and the fault clearing times are varied from five to tencycles It is assumed that reclosures are successfully appliedand the network topology is not changed when the fault iscleared The load levels used ranged from 80 to 130 of thebasic load level and the active and reactive power outputsof each generator are correspondingly assigned Among thetotal 3300 created samples 2200 ones are randomly chosenas the training samples and the rest as the testing samples

A class label Class Lable of each sample is denoted by atransient stability index which is related to the relative rotorangle deviation during the transient period of a disturbedpower system [13 15] The label Class Lable of a sample isdetermined as

Class Lable = sgn (360∘ minus |Δ120575|max) (13)

where sgn(sdot) is a sign function | sdot | is the absolute valuefunction and Δ120575max is of the maximum relative rotor angledeviation between generators in the period By plotting therotor angle swing curves of the generators an unstable case isillustrated in Figure 3 and a transient stable case is illustratedin Figure 4


0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (cycles)

Roto

r ang

les (

deg

)

BUS30BUS31BUS32BUS33BUS34


Figure 3 A transient unstable case

0 50 100 150 200 250 300minus20minus10

01020304050607080

Time (cycles)

Roto

r ang

les (

deg

)



Figure 4 A transient stable case

42 Training Results of Different Optimization AlgorithmsComparison tests are carried out by using other optimizationalgorithms comprising PSO and genetic algorithm (GA)In order to facilitate comparative analysis all the commonparameters of these algorithms are assigned to the samenumerical values and other parameters are set as followsin GA the mutation probability is set to 001 and thecrossover probability is assigned to 085 in PSO both the twolearning factors 119888

1and 1198882are assigned to 2 and the inertia

weight 120596 is set to be linearly decreased from 09 to 04 Atthe same time taking into account the randomness of theabove-mentioned algorithms all of them are executed 100

Table 1 Comparison results of different algorithms

Algorithm Trainingtimes

Optimal fitnessvalue

Successrate

Number ofhidden nodes

IPSO 1652 9750 92 1835GA 5438 9524 73 2405PSO 2547 9420 61 2900

iterations independentlyThe comparison results of differentoptimization algorithms are listed in Table 1

From Table 1 it can be seen that the proposed IPSOalgorithm has better quality of solution shorter trainingtime and higher search success rate than other optimizationalgorithms such as GA and PSO The reason for this is thatthe dynamic monitoring mechanism and the mutation strat-egy are comprehensively employed during the optimizationprocess in the proposed approach thus it has the best resultand the most stable performance Therefore it can be drawnthat IPSO is able to solve the ELMrsquos parameter optimizationproblem effectively It should be noted that both the trainingtime and the number of hidden nodes are the average valueof 100 times in Table 1

The fitness evolution curves of the optimal individualsfor the three optimization algorithms during the process areshown in Figure 5

Figure 5 shows that all these algorithms have obviouseffects in the parameter optimization process for ELMAmong all these algorithms IPSO has the fast search effi-ciency which reaches the optimal solution only through 106iterations moreover the best fitness value of IPSO is thehighest one among all these optimization algorithms Atthe same time IPSO has a transient pause at 55th iterationbut soon continues to decline This suggests that IPSO canquickly jump out the local optimal solution and overcomepremature phenomenon effectively for its powerful globalsearch capability Therefore the above results demonstratethat IPSO is able to enhance the solution quality and searchefficiency for the proposed TSP approach

43 Test Results First the proposed IPSO-ELM-based TSPmodel is tested It is known that fast prediction of instabilityis crucial for TSA since the transient stability is a veryfast process which demands a control measure within shortperiod of time (lt1 s) [13] The required observation timein the presented approach is 150ms (9 cycles) and thatallows over 400ms for measurement telecommunicationand processing delays Once the data are in the control centerthe prediction time using the trained model is short Forexample with a MATLAB code implemented on a PC with266GHz CPU and 3GB of RAM this calculation requiredonly 42267ms In practical applications the computing timecan be further reduced by means of using the state-of-the-artparallel computing and distributed computing technologyTherefore the conclusion can be drawn on the basis of theevidence that the proposed method is able to predict thetransient stability status of power systems in real-time


0 20 40 60 80 100 120 140 160 180 200

01

02

03

04

05

06

07

08

09

Iteration number

PSOGA

IPSO

Fitn

ess v

alue

Figure 5 Best fitness evolution curves of different algorithms

Table 2 Test results of the proposed approach

TSP model Acc Kap AUC 120578

IPSO-ELM 9709 0969 0979 0973ELM 9573 0919 0953 0943

And then comparison tests by using ELM are carried outwith the results shown in Table 2 Taking into account theoccasionality of test accuracy Acc the classification perfor-mance of the tested TSP model should also be evaluated byusing some statistical indicators such as the Kappa statisticvalue Kap and the area under the ROC curve AUC [15]For this reason a composite indicator 120578 is adopted hereto comprehensively evaluate TSP models as defined by thefollowing equation

120578 =Acc + Kap + AUC

3 (14)

Table 2 illustrates that the proposed algorithmmanages topredict the transient stability with good accuracyThe reasonsfor this are as follows on one hand the classification featuresextracted fromWAMS information fully reflect the dynamicresponse characteristics of the actual system on the otherhand ELM has good generalization ability for it pursuesthe minimization in both the norm of the output weightsand the training error At the same time it can be observedthat the classification ability of IPSO-ELM is much betterthan that of ELM Therefore the conclusions can be safelydrawn that IPSO is an effective way to enhance the predictionperformance for the proposed ELM-based TSP model

44 Results of Comparative Tests In order to further properlytest the proposed scheme comparative tests are carried outby using other TSP models such as DT [7 10] multilayerperception (MLP) [6 9] and SVM [13] which are carried outwith the results shown in Table 3

Table 3 Test results of other TSP models

TSP model Acc Kap AUC 120578 Training timesDT 9482 0908 0950 0935 2834SVM 9564 0925 0964 0949 4715MLP 9409 0896 0946 0928 9758

The parameters of the above-mentioned TSP models areset as follows DT is constructed using the C45 algorithm theradial basis function is used as the kernel function of SVMand its associated parameters are optimized through the gridsearch combined with 5-fold cross validation [13 15] inMLPthe hidden neuron number is set to 25 the learning algorithmis the backpropagation algorithm

Table 3 shows that the proposed method has the superiorpredictive performance and training speed than other TSPmodels not only because of the superior generalizationability and fast training speed of ELM itself but also becauseof application of IPSO to determine the optimal parametersof ELM As a result a conclusion can be safely drawn that theproposed method is effective in real-time transient stabilityprediction for power systems

5 Application to the Power System ofLiaoning Province

In order to further verify the applicability of the proposedapproach to practical large power systems the proposedmethod is examined on the power system of Liaoningprovince China

The modeled system comprises 91 generators 750 majorbuses and some series compensated lines and SVCs It isa highly interconnected grid with an approximate installedcapacity of 396572MW covering an area of 148000 squarekilometersThe systemhas formed 5 connected channels withthe external power network


Table 4 Test results of the power system of Liaoning province


IPSO-ELM 9647 0925 0962 0951

The contingencies considered are three-phase short-circuit faults The stability criterion employed here is exactlythe same as in the former case IEEE 39-bus systemThroughlarge amounts of simulations there are 2000 samples createdtotally Of all the samples 1320 ones are selected as thetraining samples randomly and the rest are used as the testingonesThe results of the power systemof Liaoning province areshown in Table 4

As demonstrated in Table 4 the proposed approach isapplicable to large-scale real power systems as well Further-more the results also show that the proposed approach is ableto determine the transient stability status in the power systemof Liaoning province China Therefore the applicability ofthe proposal to a real power system is verified

6 Conclusions

Machine learning has proved to be promising for solv-ing on-line TSA problems However the existing PRTSAapproaches cannot meet the needs of big data managementand utilization To overcome this problem a new optimizedELM-based approach for real-time power system TSP usingsynchrophasors is presented Based on the test results on thewell-studied IEEE 39-bus system and a large-scale real powersystem the conclusions can be obtained as follows

(1) The method proposed is able to effectively predictthe power system transient stability status using syn-chronized measurements and it has better predictiveperformance and generalization ability than othercommonly used TSP models such as DT MLP andSVM

(2) The predictive performance of the proposal is evi-dently strengthened by using the proposed IPSOto optimize the parameters of the ELM-based TSPmodel with synchronized measurements Further-more by means of the introduction of the dynamicmonitoring mechanism and the mutation strategyIPSO has the better global optimization ability andthe faster searching efficiency than the traditionaloptimization algorithms comprising GA and PSO

(3) For a wide area protection and control system thepresented method may be used as trigger to start therelated emergency control measures Meanwhile theproposed TSP model is able to be applied to othersimilar pattern recognition problems

Conflict of Interests

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

Acknowledgments

This research is supported by the Science and TechnologyProject of State Grid Corporation of China under Grant no2014GW-05 the key technology research of a flexible ringnetwork controller and its demonstration application

References

[1] X P Gu and Y Li ldquoBayesian multiple kernels learning-basedtransient stability assessment of power systems using synchro-nized measurementsrdquo in Proceedings of the IEEE Power andEnergy Society General Meeting (PES rsquo13) pp 1ndash5 VancouverCanada July 2013

[2] P Kundur J Paserba V Ajjarapu et al ldquoDefinition and classi-fication of power system stabilityrdquo IEEE Transactions on PowerSystems vol 19 no 3 pp 1387ndash1401 2004

[3] M A Pai Energy Function Analysis for Power System StabilityKluwer Academic Publishers Boston Mass USA 1989

[4] S Huang Y Chen C Shen W Xue and J Wang ldquoFeasibilitystudy on online DSA through distributed time domain simula-tions inWANrdquo IEEE Transactions on Power Systems vol 27 no3 pp 1214ndash1224 2012

[5] Y Xue L Wehenkel R Belhomme et al ldquoExtended equal areacriterion revisited [EHV power systems]rdquo IEEE Transactions onPower Systems vol 7 no 3 pp 1012ndash1022 1992

[6] D J Sobajic and Y-H Pao ldquoArtificial neural-net based dynamicsecurity assessment for electric power systemsrdquo IEEE Transac-tions on Power Systems vol 4 no 1 pp 220ndash228 1989

[7] S Rovnyak S Kretsinger J Thorp and D Brown ldquoDecisiontrees for real-time transient stability predictionrdquo IEEE Transac-tions on Power Systems vol 9 no 3 pp 1417ndash1426 1994

[8] C A Jensen M A El-Sharkawi and R J Marks II ldquoPowersystem security assessment using neural networks featureselection using fisher discriminationrdquo IEEE Transactions onPower Systems vol 16 no 4 pp 757ndash763 2001

[9] N Amjady and S F Majedi ldquoTransient stability prediction by ahybrid intelligent systemrdquo IEEE Transactions on Power Systemsvol 22 no 3 pp 1275ndash1283 2007

[10] K Sun S Likhate V Vittal V S Kolluri and S Mandal ldquoAnonline dynamic security assessment scheme using phasor mea-surements and decision treesrdquo IEEE Transactions on PowerSystems vol 22 no 4 pp 1935ndash1943 2007

[11] Y Li G Li Z Wang and D Zhang ldquoRule extraction basedon extreme learning machine and an improved ant-mineralgorithm for transient stability assessmentrdquo PLoS ONE vol 10no 6 Article ID e0130814 2015

[12] A D Rajapakse F Gomez K Nanayakkara P A Crossleyand V V Terzija ldquoRotor angle instability prediction using post-disturbance voltage trajectoriesrdquo IEEE Transactions on PowerSystems vol 25 no 2 pp 947ndash956 2010

[13] F R Gomez A D Rajapakse U D Annakkage and I TFernando ldquoSupport vector machine-based algorithm for post-fault transient stability status prediction using synchronizedmeasurementsrdquo IEEE Transactions on Power Systems vol 26no 3 pp 1474ndash1483 2011

[14] Y Xu Z Y Dong K Meng R Zhang and K P Wong ldquoReal-time transient stability assessment model using extreme learn-ingmachinerdquo IET Generation Transmission ampDistribution vol5 no 3 pp 314ndash322 2011


[15] X P Gu Y Li and J H Jia ldquoFeature selection for transient sta-bility assessment based on kernelized fuzzy rough sets andmemetic algorithmrdquo International Journal of Electrical Power ampEnergy Systems vol 64 pp 664ndash670 2015

[16] J De La Ree V Centeno J S Thorp and A G Phadke ldquoSyn-chronized phasor measurement applications in power systemsrdquoIEEE Transactions on Smart Grid vol 1 no 1 pp 20ndash27 2010

[17] C W Taylor D C Erickson K E Martin R E Wilson and VVenkatasubramanian ldquoWACS-wide-area stability and voltagecontrol system RampD and online demonstrationrdquo Proceedings ofthe IEEE vol 93 no 5 pp 892ndash906 2005

[18] V Terzija G Valverde C Deyu et al ldquoWide-area monitoringprotection and control of future electric power networksrdquoProceedings of the IEEE vol 99 no 1 pp 80ndash93 2011

[19] F Aminifar M Fotuhi-Firuzabad M Shahidehpour and ASafdarian ldquoImpact of WAMS malfunction on power systemreliability assessmentrdquo IEEE Transactions on Smart Grid vol 3no 3 pp 1302ndash1309 2012

[20] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine theory and applicationsrdquoNeurocomputing vol 70 no1ndash3 pp 489ndash501 2006

[21] Y Miche A Sorjamaa P Bas O Simula C Jutten and ALendasse ldquoOP-ELM optimally pruned extreme learningmachinerdquo IEEE Transactions on Neural Networks vol 21 no 1pp 158ndash162 2010

[22] X Chen Z Y Dong K Meng Y Xu K P Wong and H WNgan ldquoElectricity price forecasting with extreme learningmachine and bootstrappingrdquo IEEE Transactions on Power Sys-tems vol 27 no 4 pp 2055ndash2062 2012

[23] G-B Huang H Zhou X Ding and R Zhang ldquoExtreme learn-ing machine for regression and multiclass classificationrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 42 no 2 pp 513ndash529 2012

[24] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

[25] R C Eberhart and Y Shi ldquoParticle swarm optimizationdevelopments applications and resourcesrdquo in Proceedings of theCongress on Evolutionary Computation vol 1 pp 81ndash86 IEEESeoul Republic of Korea May 2001

[26] D HWolpert andWGMacready ldquoNo free lunch theorems foroptimizationrdquo IEEE Transactions on Evolutionary Computationvol 1 no 1 pp 67ndash82 1997

[27] T Matias F Souza R Araujo and C H Antunes ldquoLearning ofa single-hidden layer feedforward neural network using anoptimized extreme learning machinerdquo Neurocomputing vol129 pp 428ndash436 2014

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The rest of this paper is arranged as follows First of allthe used methodologies including ELM classification PSOare presented briefly Secondly the proposed real-time TSPmethod based on IPSO-ELM is presented in detail Finallythe proposal is tested using the IEEE 39-bus system and a realsystem

2 Related Methodologies

21 ELM Classification Assuming an ELM with 119871 hiddenlayer neurons to model data samples x

119894 y119894119873

119894=1 it can be

mathematically represented as

119871

sum

119894=1

120573119894119866(a119894 119887119894 x119895) = y119895 119895 = 1 119873 (1)

where a119894and120573

119894are respectively the input and output weights

vector 119866(sdot) is activation functions and 119887119894denotes the bias of

the 119894th hidden nodeFor the convenience of expression (1) can be rewritten as

H120573 = Y

H =

[[[[

[


119871(x1)


119871(x119873)

]]]]

]

(2)

whereH is the hidden layer output matrixELM is to minimize the training error as well as the norm

of the output weights [20]

Minimize 1003817100381710038171003817H120573 minus Y10038171003817100381710038172 and10038171003817100381710038171205731003817100381710038171003817 (3)

Finally the minimal norm least square method is used inthe original implementation of ELM

= HdaggerY (4)

whereHdagger is the Moore-Penrose generalized inverse ofH

22 PSO The fundamental principle of PSO is to find theoptimal solution in the complex search-space by movingcandidate solutions (called particles) according to the com-petition and collaboration among particles through repetitiveiterations The movement of each particle is determined by amathematical formulae over its position and velocity In the119896 + 1 iteration the velocity and position renewal equation of119894th particle are respectively as follows

V119896+1119894

= 119908V119896119894+ 11988811199031(119875best minus 119904

119896

119894) + 11988821199032(119866best minus 119904

119896

119894)

119904119896+1

119894= 119904119896

119894+ V119896+1119894

(5)

where119875best and119866best are respectively denoted as the local andglobal best known solution 119896 and119908 are respectively denotedas the evolutionary generation and the inertia weight 119888

1and

1198882are the learning factors which represent the self-cognition

and social-cognition in turn 1199031and 1199032are uniform random

numbers obeying the 0-1 distribution

3 Real-Time TSP Based on IPSO-ELM

31 IPSO Algorithm As pointed out by the famous ldquoNo freelunchrdquo theorem the overall performances of different opti-mization algorithms are equivalent [26] which implies thatnone of algorithms can always achieve the optimal for allaspects In this paper a mutation strategy is introduced toavoid the premature convergence to local optimum of PSO

First the optimization process is monitored by dynami-cally monitoring changes in population fitness variance 1205902

1205902=

119873119901

sum

119894=1

(

119891119894minus 119891avg

119891best)

2

(6)

where119873119901denotes the population size119891

119894is the fitness value of

the 119894th individual particle 119891best refers to the best fitness valuein the whole population and 119891avg denotes the average fitnessin the current iteration

Second when premature convergence occurs a mutationstrategy is used to maintain population diversity Specificallythe positions of particles are updated by adding randomperturbations timely as shown as follows

119909(119896+1)

= 119909(119896)+ 119888119898(119909max minus 119909min) sdot (rand minus 05) (7)

where 119888119898is the variation coefficient rand is a real number

randomly generated in the range from 0 to 1 119909(119896+1) and 119909(119896)are respectively the positions of the (119896+1)th and 119896th iterationof particles

The criterion to determine the occurrence of prematureconvergence is given as follows

119898 lt120590119896+1

120590119896

lt 119899 (8)

Here 120590119896+1

and 120590119896are the population fitness variances of the

(119896 + 1)th and 119896th iteration

32 Fitness Function As is known a proper fitness functionplays an important role in optimization problems In thispaper the fitness function is the classification accuracy of 5-fold cross-validation (CV)

119865 (120579) = 1198605-CV (9)

where 120579 is themodel parameter vector to be optimized whichis represented by the position of each particle

33 Coding Scheme A mixed-integer encoding scheme isused in the optimization process [27] Here it is consideredthat the 119896th individual particlestate 120579

119896will be constituted by

120579119896= [a1198791 a119879

119871 1198871 119887

119871 1199041 119904

119899 1198881198911 119888119891

ℎ]119879

(10)

where 119899 is the total count of input features 119904119894(119894 = 1 119899) is

a binary variable and each binary code (ldquo1rdquo or ldquo0rdquo) refers towhether the corresponding feature is selected or not 119888119891

119895=

0 1 2 (119895 = 1 119871) is an integer variable that defines


Data preprocessing


k = 1


PSO optimization


No

No

Yes

Yes

k = k + 1

Mutation





as follows

119891119895 (120592) =

0 if 119888119891119895= 0

1

(1 + exp (minus120592)) if 119888119891

119895= 1

120592 if 119888119891119895= 2

(11)






1199111015840=

(119911 minus 119885)

120590119885

(12)




119901is set












G G

GG

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

10

11

13

14

15

831

126

32

7













0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (cycles)

Roto

r ang

les (

deg

)




0 50 100 150 200 250 300minus20minus10

01020304050607080

Time (cycles)

Roto

r ang

les (

deg

)










Successrate


IPSO 1652 9750 92 1835GA 5438 9524 73 2405PSO 2547 9420 61 2900







0 20 40 60 80 100 120 140 160 180 200

01

02

03

04

05

06

07

08

09

Iteration number

PSOGA

IPSO

Fitn

ess v

alue




IPSO-ELM 9709 0969 0979 0973ELM 9573 0919 0953 0943


120578 =Acc + Kap + AUC

3 (14)













IPSO-ELM 9647 0925 0962 0951



6 Conclusions







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Data preprocessing


k = 1


PSO optimization


No

No

Yes

Yes

k = k + 1

Mutation





as follows

119891119895 (120592) =

0 if 119888119891119895= 0

1

(1 + exp (minus120592)) if 119888119891

119895= 1

120592 if 119888119891119895= 2

(11)






1199111015840=

(119911 minus 119885)

120590119885

(12)




119901is set












G G

GG

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

10

11

13

14

15

831

126

32

7













0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (cycles)

Roto

r ang

les (

deg

)




0 50 100 150 200 250 300minus20minus10

01020304050607080

Time (cycles)

Roto

r ang

les (

deg

)










Successrate


IPSO 1652 9750 92 1835GA 5438 9524 73 2405PSO 2547 9420 61 2900







0 20 40 60 80 100 120 140 160 180 200

01

02

03

04

05

06

07

08

09

Iteration number

PSOGA

IPSO

Fitn

ess v

alue




IPSO-ELM 9709 0969 0979 0973ELM 9573 0919 0953 0943


120578 =Acc + Kap + AUC

3 (14)













IPSO-ELM 9647 0925 0962 0951



6 Conclusions







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










G G

GG

G

GG GG G

30

39

1

2

25

37

29

17

26

9

3

38

16

5

4

18

27

28

3624

35

22

21

20

34

23

19

33

10

11

13

14

15

831

126

32

7













0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (cycles)

Roto

r ang

les (

deg

)




0 50 100 150 200 250 300minus20minus10

01020304050607080

Time (cycles)

Roto

r ang

les (

deg

)










Successrate


IPSO 1652 9750 92 1835GA 5438 9524 73 2405PSO 2547 9420 61 2900







0 20 40 60 80 100 120 140 160 180 200

01

02

03

04

05

06

07

08

09

Iteration number

PSOGA

IPSO

Fitn

ess v

alue




IPSO-ELM 9709 0969 0979 0973ELM 9573 0919 0953 0943


120578 =Acc + Kap + AUC

3 (14)













IPSO-ELM 9647 0925 0962 0951



6 Conclusions







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

Time (cycles)

Roto

r ang

les (

deg

)




0 50 100 150 200 250 300minus20minus10

01020304050607080

Time (cycles)

Roto

r ang

les (

deg

)










Successrate


IPSO 1652 9750 92 1835GA 5438 9524 73 2405PSO 2547 9420 61 2900







0 20 40 60 80 100 120 140 160 180 200

01

02

03

04

05

06

07

08

09

Iteration number

PSOGA

IPSO

Fitn

ess v

alue




IPSO-ELM 9709 0969 0979 0973ELM 9573 0919 0953 0943


120578 =Acc + Kap + AUC

3 (14)













IPSO-ELM 9647 0925 0962 0951



6 Conclusions







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 140 160 180 200

01

02

03

04

05

06

07

08

09

Iteration number

PSOGA

IPSO

Fitn

ess v

alue




IPSO-ELM 9709 0969 0979 0973ELM 9573 0919 0953 0943


120578 =Acc + Kap + AUC

3 (14)













IPSO-ELM 9647 0925 0962 0951



6 Conclusions







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












IPSO-ELM 9647 0925 0962 0951



6 Conclusions







Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Optimized Extreme Learning Machine for ...downloads.hindawi.com/journals/mpe/2015/529724.pdf · the PRTSA model mainly includes arti cial neural networks (ANN), decision