FEATURE SELECTION FOR SMALL-SIGNAL STABILITY ASSESSMENT

Transcription

1 FEAURE SELECION FOR SMALL-SIGNAL SABILIY ASSESSMEN S.P. eeuwsen Unversty of Dusburg Abstract INRODUCION hs paper ntroduces dfferent feature selecton technques for neural network based small-sgnal stablty assessment. Large-scale power systems lke the European nterconnected network may eperence low frequency oscllatons between remote parts of the system. hese oscllatons are caused by large power transts n the network. In dynamc securty assessment, a fast and accurate artfcal ntellgence technque can be appled. Hereby, the state of the system s predcted by the use of a neural network (NN), whch provdes nformaton about the system egenvalues and therefore the dampng of the oscllatons. Because NN cannot be traned wth the complete power system data, a reducton technque needs to be mplemented. herefore, ths paper ntroduces dfferent feature selecton technques and ther applcatons. he system under study s the European nterconnected power system, also known as UCE/CENREL. he system conssts of the western European Unon for the Coordnaton of ransmsson of Electrcty (UCE) and the central European power system (CENREL), whch ncludes the central European countres Poland, Hungary, the Czech and Slovak Republc. Due to the recent ntegraton of the CENREL power system, the European network has grown rapdly. Further etensons, e.g. n the Balkan area, are under nvestgatons. he ntegraton of the two large power systems (UCE and CENREL) led to a dfferent stablty behavor. Although the European network s strongly mashed, t ncludes parts wth hgh power concentraton, whch could swng aganst each other. Interarea oscllatons are observed partcularly when two or more net groups n the power system (.e. power

2 supply companes) echange energy. hese so-called nterarea oscllatons are slow damped oscllatons wth qute low frequences. In the European system, small-sgnal stablty s largely a problem of nsuffcent dampng of these oscllatons [4], [5]. Wth the deregulaton of the electrcty market n Europe, the utltes are allowed to sell ther generated power outsde ther tradtonal borders and compete drectly for customers. For economcal reasons, the operators are often forced to steer the system closer to the stablty lmts. hus, the operators need dfferent computatonal tools for system stablty. hese tools must be accurate and fast to allow on-lne stablty assessment. he small-sgnal stablty method, the modal analyss, s based on the computaton of egenvalues and egenvectors []. he nter-area modes are assocated wth the swngng of many machnes n one part of the system aganst machnes n other parts. In the European case, three global modes (egenvalues) are of partcular mportance because when they lack dampng the whole system starts to oscllate. For eample, load flow stuatons ncludng large power transts between Span, Portugal, Poland or some Balkan States lead very often to a weakly damped power system. Fgure shows 3 domnant egenvalues n the comple plan. hese egenvalues are nterestng because they show low frequences, whch dentfy them as nterarea egenvalues. he slant lnes n the fgure characterze constant dampng n the range of 0% to 0%. For many dfferent load flow stuatons, these egenvalues reman n the stable regon, but n some cases they shft to the low dampng regon and can cause system nstablty. Fgure : Changes of Domnant Egenvalues under,868 Dfferent Load Flow Stuatons APPROACH he computaton of the small sgnal stablty s a tme consumng process for large networks because t ncludes the load flow computaton, the lnearzaton at the operatng pont, and the egenvalue

3 computaton. Moreover, t requres the knowledge of the complete system data. hus, t s not sutable for on-lne applcatons. An alternatve method s to use a neural network (NN) traned wth off-lne data for dfferent load flow condtons. By usng NN, a fast computaton of the egenvalues s possble, provdng that the network s properly desgned. For on-lne applcatons, the NN predcts the domnant egenvalues based on the current operatng condtons. he off-lne data can be generated by smulatng varous load flow stuatons usng a model of the UCE/CENREL power system. Hereby, the generaton of net groups n the power system s changed to create dverse load flows between the dfferent net groups. Each new load flow stuaton n the network provdes a new pattern for NN tranng and the basc challenge s to smulate load flow cases that are hghly correlated wth the system stablty. Another advantage of the NN s that t can be properly traned wth few nput features. hs s also mportant consderng that due to ncreasng competton utltes may not share essental nformaton. Only very few features are commonly avalable such as the transmtted power or the generaton of complete net groups. Informaton about sngle generators or transmsson lnes s usually not avalable. Once t s traned, the NN can predct the egenvalues wthn mllseconds. However, the key ssue s to fnd the best-nput features that descrbe the system under study. hese nput features have to be measurable and need to contan as much nformaton as possble about the smallsgnal stablty. he prncpal applcablty of NN for stablty predcton has been proven n prevous works about the large-scale dynamc model of the UCE/CENREL power system [6], [7]. 3 FEAURE REDUCION he entre data for the UCE/CENREL system nclude features for power equpment such as the transmsson lnes, transformers, generators, and loads. Hence, there s a large number of features n such an etensve power system. he sze of ths feature set creates the bottleneck problem for NN tranng. herefore, feature etracton or selecton technques are ndspensable for NN based small-sgnal stablty assessment.

4 Frst, a pre-selecton s performed by engneerng udgment, whereby only the avalable and measurable features are used. compared prevously and for ths reason, only more advanced selecton technques are nvestgated n ths study [7]. After pre-selecton, the sze of these feature sets can be reduced usng a reducton technque. 4 APPLIED SELECION ECHNIQUES In ths study, the selected features are: In lterature, one can fnd many dfferent otal generated real and reactve power n each net group Real and reactve power transmtted between neghborng net groups Voltage and voltage angle on generators, loads, and transmsson lnes between neghborng net groups he total power generated n one net group s the sum of all generator power wthn ths net group, and the power flow between two neghborng net groups s the sum of power over all transmsson lnes between them. However, the total number of all preselected features s 4,379, whch s stll too etensve for NN tranng. herefore, the net sectons wll ntroduce some feature selecton technques for further reducton. In contrast to feature selecton methods, feature etracton methods are not appled n ths study. On one sde, they lead to hghly accurate reducton results, but on the other sde the physcal meanng of the features s lost after reducton. Both technques suggested for feature selecton. But as a matter of fact, there are only few methods applcable due to specal constrants for the gven problem. he followng eample usng correlaton as measure of goodness for selecton may help to understand ths statement: If the correlaton between egenvalues and the total generated power n one net group s relatve low, the correlaton between egenvalues and a set of some total generated power features mght be much hgher. hs s reasonable consderng the fact that egenvalues do not depend on the generated power of a sngle net group but on the load flow scenaro n the entre power system ncludng more than one net group. herefore, the selecton usng correlaton as measure of goodness s focused on the canoncal correlaton, whch computes the correlaton between two groups of features. technques have already been appled and

5 Another applcable technque for feature selecton are clusterng methods [7]. Good results can be obtaned by a technque called prncpal feature analyss (PFA), whch s a selecton by clusterng, but on the base of transformed feature vectors reduced n dmenson. he reducton s carred out wth the help of prncpal component analyss (PCA) [8]. However, these technques can be epanded to a multple step selecton (MSS). Hereby, only small homogenous subsets of features are reduced n the frst step usng the PFA method. hen, the results are added to a new set, whch s reduced n a second step. Advantages of ths technque are combnatons of dfferent selecton methods for step and step. 5 DAA PREPROCESSING he frst step n data reducton s the preprocessng of the data. Data preprocessng can be necessary to equal the data set. he data wll be normalzed, whch s a lnear transformaton of each feature n the data set to obtan a zero mean and unt varance. he ntal feature matr X s defned as n X = () p pn whereby p s the number of patterns and n s the number of features. hus, the column vectors are gven by = (,, l, ) ( ) () p n However, the standardzed features f are computed by = σ f (3) Hereby, the standard devaton of gven by s p σ = ( ( ) ) (4) p = and the mean value of = p p = s ( ) (5) herefore, let F be the standardzed feature matr of dmenson p n, whereby n s the number of the orgnal feature vectors and p s the number of patterns. [ f,f, ] F =...,f n (6) [ f f f ] = n f (7) =,,...,,..., p

6 6 CANONICAL CORRELAION In some applcatons t mght be useful to compute the correlaton not between two gven features, but between two groups of features. In ths case, the well-known correlaton coeffcent s not applcable snce t does not regard the correlaton between groups. A soluton to ths problem s provded by the canoncal correlaton method (CCM) []. Hereby, a canoncal correlaton coeffcent s computed, whch determnes the correlaton between two groups of features. Gven are the standardzed feature matrces F and F Y ncludng n and m features, respectvely. he matrces nclude p pattern and the emprcal covarance matrces can be computed by and he matr C C y = = F p F p y F F y (8) (9) C y determnes the covarance between the features n F and matr s gven by F Y, the C y = F Fy (0) p In the net step, the matr Q can be obtaned Q = C C y C y C y () hen, the n egenvalues of Q are gven by λ, whereby the largest egenvalue s denoted λ G. hs egenvalue provdes an estmaton for the mamal canoncal correlaton r. r = () λ G However, n contrast to the correlaton coeffcent, the computaton of canoncal correlaton coeffcents r for dfferent combnatons of nput feature sets allows a much better nvestgaton of the mpact of nput features to the egenvalues. herefore, ths feature selecton method can be appled on the gven problem. Whle the mert of ths technque s proven by the hgh reducton rate and the accurate results after NN tranng, the algorthm can only be mplemented as a Monte-Carlo- Method. he features n the orgnal set are not ordered n any way but added number by number. herefore, a systematc search algorthm or strategy cannot be used to fnd a combnaton of features wth a hgh canoncal correlaton. Consderng that about 60 features need to be selected from a total set of more than 4000 features, only very few combnatons can be computed. herefore, the soluton found by the Monte-Carlo-Algorthm s only a frst appromaton of a set of selected features.

7 One soluton for feature reducton by the canoncal correlaton method s gven n Fgure and Fgure 3. hese fgures show the tranng and testng results of a NN traned wth the selected feature set. he egenvalues marked wth crosses are the ones used as targets. he crcles are the NN outputs. he targets and the NN outputs are connected by lnes. Fgure : Imagnary Part (f/hz) Dampng 0% Dampng % Dampng % Dampng 3% Dampng 0% Ev, argets Ev, Outputs Ev, argets Ev, Outputs Ev 3, argets Ev 3, Outputs ranng Results of the NN after ranng wth Features selected by CCM estng Results of the Neural Network Real Part (σ/s - ) Fgure 3: estng Results of the NN after ranng wth Features selected by CCM 7 PRINCIPAL FEAURE ANALYSIS he PCA technque, a feature etracton method, leads to the best possble reducton results, whch are a good representaton of the orgnal data [8]. hs s obvous, when PCA s analyzed n detal. he proecton onto a smaller number of orthonormal aes leads to a coordnate system wth aes of largest spread. he PCA technque s characterzed by a hgh reducton rate and a mnmal loss of nformaton. Moreover, the technque s fast and can be appled on large data sets. It s only lack s the fact, that the proecton to the lower dmensonal space transforms the orgnal features nto new ones under loss of ther physcal meanng. Because feature selecton technques do not have ths dsadvantage, the combnaton of PCA and feature selecton wll have both benefts. One way of combnng these methods s descrbed n ths secton. Frst, the emprcal covarance matr C of the normalzed feature matr F, gven n equaton (6), s computed C = If s a F p F (3) n n matr ncludng the egenvectors of the covarance matr C,

8 the dagonal varance matr by Σ s gven Σ = C (4) Σ ncludes the varances σ. Notceable s hereby, that the egenvalues λ k of the covarance matr C are equal to the elements of the varance matr Σ, whereby the standard devaton σ k s also called sngular value of F: σ = λ ( k n) (5) k k he n egenvalues of C can be determned and sorted n descendng order λ λ λ n. Whle s a n- dmensonal matr whose columns are the egenvectors of C, q s a n q matr ncludng q egenvectors of C correspondng to the q largest egenvalues of C. he value of q determnes the sze of the new dmenson and s smaller than n. It also determnes the retaned varablty of the features, whch s the rato between the frst q egenvalues and the sum of all n egenvalues. he n rows of vectors q λ = υ = n (6) λ = q can be assumed as v, whch represent the proecton of the -th feature of F onto the lower q- dmensonal space. However, the q elements of v correspond to the weghts of the -th feature of F on the q aes of the subspace. From [3] and [8] follows, that orgnal features, whch are hghly correlated, have smlar absolute value weght vectors v. In other words, two ndependent features wll show a hgh dvergence of ther correspondng weght vectors v. On the other sde, two dentcal features wll lead to dentcal absolute weght vectors. Once the weght vectors they can be clustered to v are computed, p > q groups. he number of cluster needs to be greater than the number of weght vectors v to obtan the same varablty as the PCA. Usually, -5 addtonal dmensons are needed. Because of the smlarty between the features wthn a cluster, one of them can be selected and the others can be treated as redundant nformaton. he feature n one cluster, whch s closest to the centrod of ths cluster, wll be chosen as a prncpal feature. hus, a group of p features wll reman as prncpal features. Hereby, the centrod c of a cluster ncludng n vectors a s defned as n c = a (7) n =

9 and the dstance d between centrod c and vector a s computed by d ( a, c) = c a (8) he PFA method was appled to the gven set of more than 4,000 features. Frst, the set was reduced to a subset usng only the 55 largest prncpal components. Consderng the retaned varablty n the set, whch s 98.3 %, the number of 55 prncpal components s suffcent for the gven problem. hen, the correspondng the prncpal components s not necessary to obtan accurate reducton results [7]. But n case of large feature sets, the k- means cluster algorthm results n ncreasng naccuracy. hs s why the PFA technque s used whch results n transformed and reduced dmensonalty feature vectors. However, these vectors correspond to the orgnal feature vectors. herefore they are more sutable for a fast and accurate clusterng. egenvectors were clustered nto 60 groups. hese groups have been used for NN tranng and the results are shown n Fgure 4 and Fgure 5. Imagnary Part (f/hz) Dampng 0% Dampng % Dampng % Dampng 3% Dampng 0% Ev, argets Ev, Outputs Ev, argets Ev, Outputs Ev 3, argets Ev 3, Outputs estng Results of the Neural Network Real Part (σ/s - ) Fgure 5: estng Results of the NN after ranng wth Features selected by PFA Fgure 4: ranng Results of the NN after ranng wth Features selected by PFA 8 MULIPLE SEP SELECION Multple step selecton (MSS) s the However, t s even possble to skp the PCA computaton n PFA, whch means, that the features are clustered drectly from the begnnng. In fact, the computaton of contnuaton of the PFA technque. In MSS, the PFA method s appled more than one tme or dfferent technques are combned followng each other.

10 hs s obvous comparng to CCM and PFA. hese technques reduce the total set n one selecton step, but ths can be tme consumng (CCM) or problematc regardng the used method (PFA), where the cluster algorthm works fne only wth small sets. However, snce the total set of features s hghly nhomogeneous ncludng features wth dfferent physcal meanng, the splt nto two or even more selecton steps mght be helpful. A frst selecton s done only wthn homogenous groups. hen, a second selecton step bult on the frst one s performed to obtan the fnal set of features. In ths study, the selecton was made n 3 steps. Because the total set of features s etremely nhomogeneous as mentoned before, the set was splt nto 3 homogeneous subsets ncludng power features (total generated power n each net group and power transmtted between net groups), voltage features (voltages on generators, loads, and transmsson lnes between neghborng net groups), and the correspondng voltage angle features. In the frst step, whch s smlar to a preselecton, the correlaton between nput features and outputs s separately for the 3 subsets computed. About 0% of the features n any subset, whch show least nput/output correlaton, are sorted out. he second step of the selecton concentrates on the redundancy n the subsets. herefore, PFA s appled to cluster the subsets to a smaller sze. In ths study, the power features were clustered nto 50 groups, the voltage features nto 0 groups, and the voltage angle features nto 00 groups. he number of clusters s varable and can be defned by the user. Dfferent combnatons lead to dfferent results and an optmum needs to be found. From any group, one feature closest to the centrod s selected and then these features are added to a set of 60 features. In the thrd step, these sets were reduced agan by the PFA method to obtan the desred number of 60 remanng features for NN tranng. Fgure 6: ranng Results of the NN after ranng wth Features selected by MSS

11 he results of the NN tranng and testng are shown n Fgure 6 and Fgure 7, respectvely. Imagnary Part (f/hz) Dampng 0% Dampng % Dampng % Dampng 3% Dampng 0% Ev, argets Ev, Outputs Ev, argets Ev, Outputs Ev 3, argets Ev 3, Outputs estng Results of the Neural Network one pattern. It s defned by the followng equaton: ( σ output σ target ) + ( ωoutput ω target ) E λ = (9) σ + ω target target hs error can be computed for all testng patterns. able shows the mean and the standard devaton for 3 egenvalues and all patterns Real Part (σ/s - ) Fgure 7: estng Results of the NN after ranng wth Features selected by MSS Mean Error Std. Devaton CCM 0.% 0.38% PFA 0. % 0.47 % MSS 0.07 % 0.4 % able : Comparson of the Appled echnques usng Mean Error and Standard Devaton of the Error Functon defned by Equaton (9) 9 CONCLUSIONS Any of the ntroduced technques show good results as shown n the fgures before. However, the data nclude much redundancy and so they are reducble up to a hgh rato. hs fact leads to a reducton rato of 98.6 %, whch results from a set of 60 features selected from a total set of 4,379 features. For more detaled comparson of the dfferent technques, an error functon needs to be defned. In ths study, the error E λ for a gven egenvalue λ = σ + ω determnes the normalzed dstance between targets and outputs for able allows to compare the dfferent technques, but the man crteron for the stablty assessment s the predcton of the dampng coeffcent. herefore, a second error functon E ξ for the dampng coeffcent can be defned as follows by equaton (0) and (): ξ ξ output target E ξ = (0) ξ target σ ξ = () σ + ω able shows the mean and the standard devaton for 3 egenvalues and all patterns usng an error functon regards the dampng coeffcent defned by equaton (0).

12 Mean Error Std. Devaton CCM.0 % 3.99 % PFA.3 % 7.5 % MSS 0.69 %.87 % able : Comparson of the Appled echnques usng Mean Error and Standard Devaton of the Error Functon defned by Equaton (0) able and allow to compare errors of the 3 appled technques. he results of CCM and PFA are accurate and applcable for on-lne stablty assessment and fast egenvalue predcton, but the results of MSS are much more accurate. he reason can be found n the applcaton of more than one selecton step, whch s recommendable for selecton problems wth a large number of nhomogeneous features. Another way of comparson s the tme used for selecton. herefore, able 3 shows the tme comparson of the appled technques. due to the use of a Monte Carlo-Algorthm. But the better the requred soluton, the more tme s needed to compute. For small data sets, the CCM can be used wthn a mantanable duraton. But wth ncreasng number of features, the tme requred to fnd an approprate soluton wll ncrease superproportonal. PFA uses as well much tme for reducton of large feature sets because of the egenvalue computaton of a large feature matr. For small feature sets, the reducton tme wll be very fast. he reducton tme of MSS compared to PFA s almost the same. he egenvalue computaton n MSS s much faster than n PFA because the egenvalues are separately computed for the 3 subsets. In fact, the reducton tme depends mostly on the cluster algorthm, whch clusters much larger feature sets than n PFA. However, even f there s not much dfference n computaton tme between CCM PFA MSS able 3: Reducton me > 4 h 9 mn. 4 mn. Comparson of the Reducton me of the Appled echnques (Pentum 4,.7 GHz) PFA and MSS, the accuracy of MSS s much hgher. Moreover, ths technque s absolutely varable and allows a hgh number of dfferent combnatons and possbltes. herefore, MSS technques are hghly recommended for feature selecton CCM needs much tme to fnd an approprate selecton of features. hs s applcatons for NN based small-sgnal stablty assessment.

13 0 REFERENCES [] P. Kundur, Power System Stablty and Control, McGraw-Hll, New York, 994 [] Hartung, Joachm, Multvarate Statstk : Lehr- und Handbuch der angewandten Statstk, München, 99 [3] I.. Jollffe, Prncpal Component Analyss, Sprnger-Verlag, New York, 986 [4] U. Bachmann, I. Erlch and E. Grebe, Analyss of nterarea oscllatons n the European electrc power system n synchronous parallel operaton wth the Central-European networks, IEEE Powerech, Budapest 999 [5] H. Breulmann, E. Grebe, M. Lösng, W. Wnter, R. Wtzmann, P. Dupus, M.P. Houry,. Margotn, J. Zerény, J. Dudzk, J. Machowsk, L. Martín, J.M. Rodríguez, E. Urretavzcaya, Analyss and Dampng of Inter-Area Oscllatons n the UCE/CENREL Power System, CIGRE 38-3, Sesson 000 [6] S.P. eeuwsen, A. Fscher, I. Erlch, M.A. El-Sharkaw, Assessment of the Small Sgnal Stablty of the European Interconnected Electrc Power System Usng Neural Networks, LESCOPE 00, Halfa, Canada, June 00 [7] S.P. eeuwsen, I. Erlch Feature Reducton for Neural Network based Small-Sgnal Stablty Assessment, PSCC 00, Sevlla, Span, June 00 [8] I. Cohen, Q. an, X. Zhou,. S. Huang, "Feature Selecton and Dmensonalty Reducton Usng Prncpal Feature Analyss", submtted to IEEE Internatonal Conference on Image Processng (ICIP'0), Rochester, New York, Sep. -5, 00