Fast Eigenvalue Assessment for Large Interconnected Powers Systems

Transcription

1 1 Fat Eigenvalue Aement or Large Interconnecte Power Sytem S. P. Teeuwen, Stuent Member, IEEE, I. Erlich, Member, IEEE, an M. A. El-Sharkawi, Fellow, IEEE Abtract--Thi paper eal with metho or at eigenvalue preiction in large interconnecte power ytem. The metho can be ue or on-line ocillatory tability aement. Special interet i ocue on the preiction o critical inter-area ocillatory moe. Intea o eigenvalue computation uing the complete power ytem moel, the propoe approach i bae on computational intelligence uch a neural network an eciion tree. Computational intelligence metho nee only a mall et o electe ytem inormation. Thee metho o not require the entire ytem moel an thereore, they are highly applicable in Europe liberalize an competitive energy market. phenomenon wa invetigate in many imulation tuie carrie out in Germany an other European countrie [1]-[5]. For thi purpoe, a ynamic moel o the entire ytem ha been create coniting o hunre o generator, thouan o line an noe. Meaurement were ue or veriication o the moel. Two technique have been applie or the analyi, namely time omain imulation an the application o the moal analyi or the entire European interconnecte power ytem. Inex Term--Ocillatory Stability Aement, Interconnecte Power Sytem, Eigenvalue Computation an Preiction, Computational Intelligence Metho, On-Line Tool I. INTRODUCTION With the eulation o the European electric power market, a competitive ituation or the market participant aroe. Long-itance power traner rom ion with urplu generation are taking place to meet the eman in other ion, wherea economical apect are playing a ominant roll. However, the variation o the loa low over a wie range may lea to angerou ituation, in which the mall-ignal tability i lot an inter-area ocillation, aecting the whole interconnecte power ytem, occur. Such ocillation in the requency range o Hz have been recore in the European interconnecte power ytem many time. Figure 1 an how two example. It i to be een rom Figure 1, that the generator in Spain an France on the one ie an thoe in Polan an Hungary on the other, are winging againt each other, with Germany participating on the requency wing only marginally. However, the ocillating power over the interconnection in Germany i high, becaue Germany i locate between the winging network part. Figure how a low ampe ytem behavior ollowing the lo o active power generation in Spain. A irt analyi o the requency curve reult in the aumption that the behavior i ominate by more than one inter-area moe becaue o the beat eect obervable in the recor. The Fig. 1 Behavior o the European Electric Power Sytem ollowing Lo o 900 MW Generation Power 1) S. P. Teeuwen, Univerity o Duiburg-Een, Germany, (teeuwen@uni-uiburg.e) I. Erlich, Univerity o Duiburg-Een, Germany, (erlich@uni-uiburg.e) M. A. El-Sharkawi, Univerity o Wahington, Seattle, USA, (elharkawi@ee.wahington.eu) Fig. Inter-area Ocillation ater 487 MW Loa Rejection in Spain 1) 1) The meaurement were mae available or preentation by the German electric power company RWE Een

2 Becaue the recore ocillation occurre uring increae power traner in the network, one o the objective wa to invetigate the impact o long itance power traner on the amping o inter-area ocillation. Variou tuie have hown that the limit or traner are etermine by ocillatory tability rather than by the traner capability o line. Thereore, or a ecure ytem operation it i neceary to ae the impact o traner an thu ierent loa low cenario on the ocillatory tability beore the traner are realize. However, or on-line application, the available analyi an imulation metho are not practicable ue to their complexity an the computational requirement. Furthermore, the ipatcher have to make eciion quickly an uually o not have the expertie or etaile mall-ignal tability analyi. Figure 3 how the baic iea. availability an other iue. Thi tak cannot be one bae on engineering knowlege an jugment only. Rather, mathematical eature election metho evelope in the iel o Computational Intelligence (CI) are require. The iue o eature election concerning thi work are icue in [3]. II. INVESTIGATED POWER SYSTEM The power ytem invetigate in thi tuy i the extene UCTE/CENTREL high voltage European network, which alo inclue the Eat-European an Balkan tate. It conit o approximately 500 generator, everal thouan line an noe, hunre o tranormer, etc. The generator are ecribe by 5th orer moel, wherea or controller ierent uer-eine moel are ue. The ytem i prepare or ynamic invetigation in the time range o econ to a ew minute. Figure 4 how comparion between meaurement an imulation reult ue or veriication purpoe. Operation Planning Power Flow Real Sytem Meaurement Lo o 1000 MW generation power in Belgium Meaurement Fat Ocillatory Stability Aement Selecte Variable Computational Intelligence Online Ocillatory Stability Monitoring Frequency an Damping o Dominant Moe Fig. 3 Application o at Eigenvalue Aement In the hort-term operation planning the propoe metho can help to ae the eect o ierent power low cenario on the ocillatory tability. For thi, only electe variable reulting rom power low calculation are require. In on-line moe the ytem can alo be ue or tability monitoring provie that the variable are meaure an exchange by the utilitie. However, the number o variable mut be kept mall becaue o the meauring an tranmiion requirement. Beie, in the competitive electricity market utilitie are uually not interete in exchange o etaile inormation. The election o the input variable i one o the main challenge o the propoe metho. Dierent apect have to be coniere: inormation content, reunancy, robutne, Calculation Fig. 4 Comparion o Meaurement an Simulation Reult Figure 5 contain the ominant eigenvalue or 500 ierent loa-low ituation. It i calculate by varying the

3 3 real power exchange between ierent network area, which correpon to ierent utilitie. By increaing the power generation in one area, the neceary number o generation unit wa taken into operation. At the ame time in the power receiving area, witching o unit reuce the generation. Generally, the loa-low cenario were ajute bae on practical experience. From Figure 5 ollow that all three eigenvalue may experience change up to untable value. Depening on which real power tranaction are realize in the power ytem, the eigenvalue may remain in the table operation ion or they may hit igniicantly to untable location. Furthermore, the impact o tranaction on the eigenvalue i trongly non-linear. Obviouly, it epen on everal actor an cannot be ecribe analytically. Eigenvalue 3 Eigenvalue 1: =0.199 Hz, ξ=13.3% Eigenvalue : =0.9 Hz, ξ=.9% a) b) Eigenvalue Eigenvalue 1 Fig. 5 Calculate Inter-area Eigenvalue or 500 Loa Flow Scenario in the European Power Sytem Eigenvalue 3: =0.368 Hz, ξ=0.9% c) An overview o the ytem inherent ynamic characteritic provie the moe hape epicte in the Figure 6a-c. Thee moe hape repreent the ominant inter-area moe, which will be invetigate next. The arrow how the irection an the relative intenity o the correponing wing, becaue they repreent the eigenvector correponing to the mechanical motion o generator rotor. The poitioning o the arrow irection to the generator geographical location i arbitrary. However, thi kin o viualization provie an excellent overview about the prea o ocillation. The amping o the irt moe (Eigenvalue 1) in the bae cae i beyon expectation. However, it can change with the operating point in a wie range a hown in Figure 5. Eigenvalue an 3 are varying in a maller range. However, or ome loa low cenario they are very cloe to the tability limit. Fig. 6 Moehape to the Dominant Inter-area Moe. Eigenvalue are calculate or one certain Power Flow Situation III. CI-BASED EIGENVALUE CALCULATION CI metho are ue to moel input-output relationhip bae on a mall et o input ata, which correpon to electe variable o the power ytem. The output are irectly relate to the tability problem. Typical CI metho

4 4 are Neural Network (NN), Neuro Fuzzy metho (NF), an Deciion Tree (DT). Thee metho are learne on training pattern an how goo perormance in interpolation. The coorinate o eigenvalue can be calculate with Neural Network irectly. Once a multilayer ee-orwar NN i properly traine, it i able to approximate highly non-linear unction. When ue or OSA, a multilayer ee-orwar NN can be eigne to preict eigenvalue location irectly. In thi cae, the NN output are the coorinate o ominant eigenvalue. The irect eigenvalue preiction by NN i hown in Figure 7 an will be icue in the next ection in etail. Input: Selecte ytem eature Input Layer x 1 x x a x a+1 NN : Non-linear unction W ji Hien Layer 1 b V kj Output Layer out 1 out out c Output: Mappe eigenvalue location Large Power Sytem Stable Intable Output Selecte Input Intable Stable Etimation o Small-Signal Stability Fig. 8 DT-bae Stability Etimation Intable An expert with etaile knowlege about the power ytem perorm the CI training o-line beore the CI metho i implemente a an on-line tool. To enure reliable operation uring changing operating conition, the OSA tool mut be upate requently to learn new ytem tate. b+1 Fig. 7 NN-bae Eigenvalue Preiction Neuro Fuzzy metho are bae on Fuzzy Logic ytem uing rule an memberhip unction to ecribe inputoutput relationhip. However, ince one might have no exact knowlege about the rule in avance, the NF ytem are traine imilarly to NN by a training algorithm in orer to in the rule. Since both the NN an the NF metho are bae on training, they may require long training time epening on the given problem. The mot common NF metho ue i the Aaptive NN bae Fuzzy Inerence Sytem (ANFIS). In OSA, an ANFIS can be ue to etimate minimum-amping coeicient o ominant eigenvalue. In contrat to traine ytem, the Deciion Tree metho i bae on a learning or growing proce, which en once the ull DT i et up. Thereore, the DT metho i much ater to apply to given ata. The DT metho i rule bae an plit the ata into ubet epening on their characteritic. DT can be grown or both claiication an reion. In claiication, the DT can be ue to claiy the power ytem tate into table ituation with uicient amping an untable ituation with inuicient amping. Thi i hown in Figure 8. When implemente a reion tree, the DT metho can be ue imilarly to an ANFIS to etimate minimum-amping coeicient. IV. EIGENVALUE MAPPING A. Direct Eigenvalue Calculation The metho o irect eigenvalue calculation ha been prove in [4] an [5] an i applicable with ome retriction: the locu o ierent eigenvalue may not overlap each other an urthermore, the number o eigenvalue mut remain alway contant. Thee ollow rom the act that eigenvalue are aigne irectly to NN or DT output. The compute eigenvalue o the European interconnecte power ytem UCTE/CENTREL uner variou loa low ituation are hown in Figure 9. The igure how two ominant eigenvalue with amping aroun an below 3%. A thir eigenvalue with amping above 10% become ominant only or certain loa low conition. The compute eigenvalue are marke with x, the eigenvalue preicte by the NN are circle. The irect eigenvalue preiction i highly accurate, but it lack the epenency o a ixe number o eigenvalue to preict ince it i irectly relate to the eigenvalue coorinate. Large power ytem experience more than one (or two) ominant eigenvalue an the number o ominant eigenvalue may vary or ierent loa low ituation. Thereore, the eigenvalue mapping houl be perorme rather by eigenvalue ion preiction, which i highly lexible aring the number o ominant eigenvalue an their poition.

5 10% 3% % 1% 0% Ater the obervation area i ample, the ample point nee to be activate accoring to the poition o the eigenvalue. Thu, the itance between the eigenvalue an the ample i ue to compute activation or the ample point. The eigenvalue are eine by their real part σ an ev their requency ev. The ample point are eine by their location ( σ, ). Then, the itance between a given eigenvalue an a given ample i compute a ollow: 5 = σ σ kσ ev + k ev (1) Fig. 9 Comparion o Eigenvalue o the European Power Sytem UCTE/CENTREL uner variou Loa Flow Conition. The accurate Eigenvalue are marke with x, the Eigenvalue preicte by the NN are circle. B. Eigenvalue Region Preiction The eigenvalue ion preiction metho require that the obervation area in the complex eigenvalue pace be eine irt. The obervation area i locate at the ion o inuicient amping, where typical inter-area eigenvalue can be oun. In thi tuy, the coniere amping range i choen between 4% an 1.0%. Then, thi area i ample along the real axi (σ ) with contant tep with. Thi i one or the given application 5 time or 5 ierent requencie. The ampling tep ue are a ollow: 1 σ = = 0.14 Hz The obervation area o inuicient amping i hown in Figure 10. The ample point are marke by circle. In thi application ierent NN are aigne to the electe 5 requency range. Becaue σ an ue ierent unit an cannot be compare irectly, both are cale. Hence, σ an are ivie by the contant k σ or the real part an k or the requency, repectively. The imum poible itance between an eigenvalue an the cloet ample point occur when the eigenvalue i locate exactly in the geometrical center o 4 neighboring ample point. Thi i hown in Figure 11. ( σ, ) σ Eigenvalue 4% 3% % 1% 0% -1% NN A Fig. 11 Deinition o the Maximum Ditance Accoring to Figure 10 an Equation (1), the imum itance can be compute a NN B NN C = σ k σ + k () Fig. 10 Partitioning an Sampling o Eigenvalue Plain NN D NN E Bae on thi imum itance, the activation value a or a ample i eine a a linear unction epening on the itance o ample point an eigenvalue: a = (3) 0 >

6 6 Thi activation a i compute or a given ample point coniering all eigenvalue reulting rom one pattern. The inal activation value act or the given ample point i the ummation o all activation a b) 3 % % 1 % 0 % act = n i= 1 a whereby n i the number o coniere eigenvalue. The imum itance () an the activation unction (3) lea to the minimum activation or a ample point, when an eigenvalue i nearby: (4) 4 % True eigenvalue poition Preicte eigenvalue ion act 0.5 (5) The ucce o the preiction epen trongly on the choice o the caling parameter. Thee parameter impact both the training proce o the NN an the accuracy o the preicte ion. Thereore, ome experience i neceary to apply the propoe metho to particular power ytem. The ample point activation were compute or the eigenvalue o the 500 calculate loa-low cenario in UCTE power ytem. Then the ata et wa normalize an hule ranomly beore the 5 NN ha been traine inepenently. The training wa carrie out with a ubet o training ata, wherea a maller part o the ata wa retaine or teting. Once the NN are traine properly, the NN output value repreenting the activation o the ampling point, nee to be tranorme into eigenvalue location. For thi purpoe all activation value were ue to etup an activation urace, which i contructe by linear interpolation between all row an column o ample point a hown or a electe pattern in Figure 1a. Then, a plain urace i rawn at the minimum activation level o 0.5, which i calle limit urace. The interection o the activation urace an the limit urace lea to a ion, which i calle preicte ion (Fig. 1b). a) Activation Surace Limit Surace Fig. 1 (a) Activation Surace contructe by Sample Point Interpolation an Limit Surace, (b) View rom above: Preicte Region in the Complex Space I the NN i traine properly, the true eigenvalue poition will be within thi preicte ion (a hown in Figure 1b). For a more oun error evaluation, the preicte ion are compare with the poition o the eigenvalue or all pattern. I an eigenvalue i locate inie a ion, the preiction yiel a correct. I an eigenvalue i locate outie the correponing ion or no range an thereore no ion i contructe or thi eigenvalue, the error i calle ale imial. I a range i contructe, but no correponing eigenvalue exit within the obervation area, the error i calle ale alarm. The error rate are calculate accoring to Equation (6). Table I contain the reult or the invetigate ytem. number o ale imial or ale alarm E [%] = 100 (6) number o pattern TABLE I ERROR RATES FOR NN TRAINING AND TESTING Reult Training Tet Preicte ion matche eigenvalue poition 99. % 97.5 % Fale imial 0.8 %.5 % Fale alarm 0.0 % 0.0 % Fale alarm wa never itere. Fale imial occurre in.5% o all cae, which, however, oe not necearily mean that thee preiction are not utilizable. The eigenvalue were alway locate near to the preicte ion. By uing a lightly moiie activation unction it i poible to reuce the error rate. In oing o, however, the expane o the preicte ion will increae. Thereore, or an evaluation o the propoe metho the with o the ion in requency an amping mut be coniere too. Figure 13 how the einition o ion with ξ an, repectively.

7 eigenvalue, the eigenvalue ion preiction metho i preerre. However, accurate reult can alo be obtaine when ierent CI metho are implemente uch a Neuro Fuzzy Sytem an Deciion Tree. 7 Fig. 13 Deinition o the With o a Preicte Region in Damping ( Frequency ( ) The mean an tanar eviation o ξ ξ an ) an were compute or all pattern, which inclue eigenvalue in the obervation area. Table II contain the reult. Region TABLE II WIDTH OF PREDICTED EIGENVALUE REGIONS Training Teting With Mean STD Mean STD ξ % % % 0.49 % Hz 0.03 Hz Hz 0.04 Hz Auming that the eigenvalue will alway be expecte in the center o the preicte ion, the metho provie an average accuracy or amping o ±0.5% an or requency o ±0.03 Hz. In ummary, by varying the activation unction the accuracy coul be increae at the expene o error rate an vice vera. Thereore a trae-o might be neceary accoring to the actual requirement. V. CONCLUSIONS The recent an uture power ytem expanion, the competitive environment in the liberalize power market, an the continuouly increaing win power generation aect the tability ituation in the European power ytem UCTE/CENTREL. Tranmiion Sytem Operator have to manage more an ierent critical ituation than in the pat. Thereore, they nee at an lexible on-line aement tool. Thi tuy propoe the implementation o Computational Intelligence metho or at on-line OSA. The CI metho are bae on a mall number o input only, which i an important iue in a liberalize power market. Moreover, they are at an accurate. Both the irect eigenvalue preiction an the eigenvalue ion preiction are applicable tool or OSA. However, ince the irect eigenvalue preiction i limite to ew VI. REFERENCES [1] U. Bachmann, I. Erlich an E. Grebe, Analyi o interarea ocillation in the European electric power ytem in ynchronou parallel operation with the Central-European network, IEEE PowerTech, Buapet, 1999 [] H. Breulmann, E. Grebe, et al., Analyi an Damping o Inter-Area Ocillation in the UCTE/CENTREL Power Sytem, CIGRE , Seion 000 [3] S.P. Teeuwen, I. Erlich, U. Bachmann, Small-Signal Stability Aement o the European Power Sytem bae on Avance Neural Network Metho, IFAC 003, Seoul, Korea, September, 003 [4] S.P. Teeuwen, A. Ficher, I. Erlich, M.A. El-Sharkawi, Aement o the Small Signal Stability o the European Interconnecte Electric Power Sytem Uing Neural Network, LESCOPE 001, Haliax, Canaa, June 001 [5] S.P. Teeuwen, I. Erlich, M.A. El-Sharkawi, Feature Reuction or Neural Network bae Small-Signal Stability Aement, PSCC 00, Sevilla, Spain, June 00 VII. BIOGRAPHIES Simon P. Teeuwen (1976) i preently PhD tuent in the Department o Electrical Power Sytem at the Univerity o Duiburg-Een/Germany. He tarte hi tuie at the Univerity o Duiburg in In 000, he went a exchange tuent to the Unverity o Wahington, Seattle, where he perorme hi Diploma Thei. Ater hi return to Germany in 001, he receive hi Dipl.-Ing. egree at the Univerity o Duiburg. He i a member o VDE, VDI, an IEEE. Itvan Erlich (1953) receive hi Dipl.-Ing. egree in electrical engineering rom the Univerity o Dreen/Germany in Ater hi tuie, he worke in Hungary in the iel o electrical itribution network. From 1979 to 1991, he joine the Department o Electrical Power Sytem o the Univerity o Dreen again, where he receive hi PhD egree in In the perio o 1991 to 1998, he worke with the conulting company EAB in Berlin an the Fraunhoer Intitute IITB Dreen repectively. During thi time, he alo ha a teaching aignment at the Univerity o Dreen. Since 1998, he i Proeor an hea o the Intitute o Electrical Power Sytem at the Univerity o Duiburg-Een/Germany. Hi major cientiic interet i ocue on power ytem tability an control, moelling an imulation o power ytem ynamic incluing intelligent ytem application. He i a member o VDE an IEEE. Mohamme A. El-Sharkawi receive the B.Sc. egree in electrical engineering in 1971 rom Cairo High Intitute o Technology, Egypt, an the M.A.Sc. an Ph.D. egree in electrical engineering rom the Univerity o Britih Columbia, Vancouver, B.C., Canaa, in 1977 an 1980, repectively. In 1980, he joine the Univerity o Wahington, Seattle, a a Faculty Member. He erve a the Chairman o Grauate Stuie an Reearch an i preently a Proeor o Electrical Engineering. He i the Vice Preient or Technical Activitie o the Neural Network Society. He organize an taught everal international tutorial on intelligent ytem application, power quality an power ytem, an he organize an chaire numerou eion in IEEE an other international conerence. He i a member o the eitorial boar an Aociate Eitor o everal journal, incluing the IEEE TRANSACTIONS ON NEURAL NETWORKS.