POWER SYSTEM SMALL SIGNAL STABILITY ANALYSIS BASED ON TEST SIGNAL

Transcription

1 POWE YEM MALL INAL ABILIY ANALYI BAE ON E INAL Zheng Xu, Wei hao, Changchun Zhou Zheang Univerity, Hangzhou, 37 PChina hvdc@ceezueducn Abtract - In thi paper, a method baed on ome tet ignal (et ignal Method) i developed to find the donant ocillating frequencie and damping aociated with any generator in the ytem he method can be realized eaily by time domain imulation uing electromechanical tranient program he theoretical bacground of the tet ignal method i the theory of mechanical adttance and mode identification, which i baed on the frequency canning of the tet ignal and ha already been ued widely in mechanical engineering he tudy cae how that the tet ignal method developed in thi paper can be effectively applied in mall ignal tability analyi and P deign Keyword: Low frequency ocillation, et ignal method, Mechanical adttance, Eigenvalue analyi, P, ime domain imulation ocillating mode are local In the deign and parameter etting of a P aociated with a certain generator, two point are critical he firt point i the ocillating frequencie involved in the generator, the econd point i the damping aociated with thoe ocillating frequencie ecently, ome wor on new mall ignal tability analyi method ha been done [] and [] mainly dicu the ubynchronou ocillation analyi by time domain imulation realization of the complex torque coefficient method or the damping torque coefficient method In thi paper, a new method baed on ome tet ignal for analyi of low frequency ocillation (LFO) i developed, which (et ignal Method) can be effectively ued to find the donant ocillation frequencie and damping aociated with any generator in the ytem INOUCION HE PINCIPLE OF E INAL MEHO Conventionally mall ignal tability analyi of a power ytem i carried out in frequency domain uing the eigenvalue analyi method And the uual tep are: firt developing the linearized mathematical model of the power ytem, then olving for the eigenvalue and eigenvector of the linearized ytem, finally calculating the mode hape, enitivity and participation factor baed on the eigenvalue and eigenvector information Although the method ha nearly become a tandard method, there exit ome difficultie in theory and in practice For intance, it i very hard to find all the eigenvalue of a large power ytem o mall-ignal tability analyi program are not a widely available and ued a electromechanical tranient program or electromagnetic tranient program In a real interconnected power ytem, uually only one or two ocillating mode are inter-regional And to any generator in the ytem, uually only two or three ocillating mode are involved, for example, one ocillating mode i inter-regional and the other Conpectu of Mode Identification Mode identification of a linear ytem i to find the mode parameter, uch a mode frequency, mode damping and mode hape, by tet or theoretical method For any linear ytem, the impoed dynac input will excite ome dynac output repone he input and output of the linear ytem exit ome cauality, which i named a the characteritic of the linear ytem he mot uual method of mode identification i mechanical adttance approach, which i alo called a frequency repone function (FF) approach he general proce of the mechanical adttance approach i: firtly, impoe the input excitation on the ytem and meaure the output repone of the ytem, then calculate the mechanical adttance of the ytem and analyze the mode parameter of the ytem he uual mechanical adttance approach i ingle input multi output (IMO) method, which i to impoe excitation on a ingle point and meaure repone on multiple point

2 he power ytem i a nonlinear ytem, but in mall ignal tability analyi, it i linearized at a certain operating point o the mall ignal tability of a power ytem can be analyzed by uing the mechanical adttance approach efinition of the Mechanical Adttance he mechanical adttance i one of the repreentation of the ytem dynac characteritic And according to the ternology widely ued in mechanical engineering, the baic definition of the mechanical adttance i the ratio of the Laplace tranform of the output to the Laplace tranform of the input [3,4] he output can be diplacement repone, velocity repone or acceleration repone When a power ytem i linearized at a certain operating point, it can be decribed by a tate pace equation a follow: X AX BU () Y CX U If the mechanical torque increment of the generator i elected a a inuoidal excitation, for IMO method, U [,,,,,,] () And if the angular velocity repone of all the machine are elected a the output, then Where [,,, n ] Y (3) i the mechanical torque increment of the i th generator, and ( =,,,n) i the angular velocity increment of the th generator From (), () and (3), we can obtain: N ( (4) Where N and are polynoal of, and i the Laplace operator Here i defined a the mechanical adttance when the exciting point i at the i th generator and the meauring point i at the th generator enerally, if there are n generator in a power ytem, there are n mechanical adttance correponding to an exciting point And the mechanical adttance ii i called a the exciting point mechanical adttance 3 Characteritic of the Mechanical Adttance Firtly, we tudy the mechanical adttance in an one machine infinite bu (OMIB) ytem According to the mall ignal analyi theory, the linearized equation of the motion of the generator in the OMIB ytem can be repreented a follow (the two-order model of the generator i adopted): d ( dt M [ m M m ) ( K e K )] (5) Where M i the inertial time contant of the generator, K and K are the damping and ynchronou torque coefficient aing the Laplace tranform, we can get { m [ K K ( ]} (6) M Becaue ( ) ( /, we have ( ( (7) m M K K By eigenvalue method, the characteritic equation of the OMIB ytem i M K K (8) From (7) and (8), it i hown that the pole of the mechanical adttance are the eigenvalue of the ytem And when the more elaborate model of the generator i adopted, even when the excitation ytem (AV) and power ytem tabilizer (P) are included, (7) and (8) are till tenable, but the order of (7) and (8) are increaed In a multi-machine ytem, through analyi of a erie of mechanical adttance, we can now not only the ocillation mode of LFO but alo the mode hape of the LFO Normally the mechanical adttance ha only a few donant pole repreenting regional or local ocillation mode, o can be approximately repreented a: ( z )( z ) ( z l ) K ( ( p)( p) ( p ) (9) p p p And one couple of the conugated pole of the ytem can be repreented a

3 d p () electromechanical tranient mode, then tae and Where i the mode damping, d i the mode (=,,,n) data in a common period of the added frequency with damping, i the natural mode frequency without damping and i the damping ratio If we can get i ( ), i ( ),,, which are d ni the mechanical adttance of the n generator when the exciting point i at the i th generator, then the mode ocillating torque tep3: o Fourier reolution to and (=,,,n), and obtain the phaor phaor and the (=,,,n) in the different frequencie tep4: Calculate the mechanical adttance in the different frequencie (for all in ): hape correponding to p i [,,,n ] ) ( ) / m ( ), (=,,,n) () ( 3 CALCULAION OF HE MECHANICAL AMIANCE BY IME OMAIN IMULAION In the analyi of LFO problem, becaue the ocillation frequency of the LFO, which i generally in Hz for regional ocillation and 3Hz for local ocillation, i mall compared with the ynchronou frequency, o we can analyze the LFO problem in electromechanical tranient proce And the mathematical model of the HVC and FAC can be the quai-teady tate model without conidering the real witching proce he tep of calculating the mechanical adttance in time domain imulation i a follow: tep: elect ome generator in the ytem and add a erie of mall ocillating torque to the generator rotor co( ) () t Where = 3Hz, and are the amplitude and phae angle of the ocillating torque whoe frequency i he requirement for i that it doe not violate the linearization condition Becaue the ytem i nearly linear in the mall area around the operating point, the different frequency quantitie would not interfere with each other herefore, more than one frequency ocillating torque can be added in one time tep: After adding the ocillating torque to the rotor, imulate the ytem into teady tate in tep5: Identify the approximate expreion of (=,,,n) by curve fitting, and obtain ( ), (=,,,n) p p p (3) tep6: According to the information in equation (3), analyze the characteritic of the LFO, uch a the ocillation frequency, ocillation damping and mode hape In order to apply the tet ignal method efficiently, ome problem mut be clarified he firt problem i the election of the exciting point and the meauring point For a large power ytem, if all the generator in the ytem are choen a exciting point and meauring point, the imulation and calculation wor will be very onerou Becaue there are only a few inter-regional ocillation mode for the LFO, normally electing everal typical exciting point and meauring point i enough to capture all the interregional ocillation mode herefore, we may elect a few typical generator that are located in different region of the power ytem a the exciting point and the meauring point if we are only intereted in the inter-regional ocillating mode he econd important problem i how to deterne the order of If the generator model i m th order (m=, 3, 4, 5, or 6) and the power ytem ha n generator, then the number of the eigenvalue of the power ytem i more than mn, including the eigenvalue aociated with the controller herefore,

4 the order of hould be more than mn hu, the calculated by eigenvalue method r 64rad / and mode identification will be very difficult However, what we focu on i only the mechanical rotor ocillating mode, and to a certain generator, it i nown that there are only two or three donant ocillating mode correponding to the local and inter-regional ocillation herefore, the order of the can be f r Hz [5] he damping ratio and the frequency of the ocillation are 7 and 638rad the mode hape i [ ] rd / And et under ix hu, when we elect ome typical generator a the exciting and the meauring point, we can obtain a et of ocillating mode and their correponding mode hape hen we can deterne which ocillating mode are the inter-regional mode and which ocillating mode are the local mode according to their mode hape o it can be guaranteed that all the inter-regional ocillating mode can be found ( (a) ime omain imulation eult ( (b) Mode Identification eult 4 UY CAE Figure : Amplitude-frequency Characteritic of 4 One Machine Infinite Bu (OMIB) ytem he configuration of an OMIB ytem i hown in Figure, which come from [5] In the power ytem hown in Figure, the contant model) i adopted for the generator L E q ' model (three-order E B Figure : he configuration of an OMIB ytem he amplitude veru frequency characteritic of the mechanical adttance ( ) by time domain imulation and by mode identification i hown in Figure After identifying, we can obtain the mechanical adttance a follow: (4) And by equation (), we can now the natural ocillation frequency of the ytem i ie 638rad, r / f r 6Hz, which i very coincident with the data 4 Multi-machine ytem he configuration of a four-machine ytem i hown in Figure 3, which come from [5] In the power ytem hown in Figure 3, the excitation ytem and the governor ytem are included in the model of the generator Although the four-machine ytem i not large and complex, it i enough to how the characteritic of the multi-machine ytem L7 C7 C9 L9 Figure 3: he configuration of four-machine ytem For example, when generator i elected a the exciting point, the amplitude veru frequency characteritic of the mechanical adttance by time domain imulation and by mode identification i hown in Figure 4 And by equation (), we can now the ocillating frequencie of the ytem aociated with generator are f rd 549Hz and f rd 73Hz he damping 4 3

5 ratio are 7 and 88 he mode hape (without normalization) of them are 3 [33 455, 8 66, 7575, 597] and [97, 3 489,, ] 3 4 (a) ime omain imulation eult 3 4 (b) Mode Identification eult Figure 4: Amplitude-frequency Characteritic of Liewie, we can obtain the ocillating frequencie, damping ratio and mode hape when generator, 3 and 4 are elected to be a exciting point When i elected a the exciting point, the reult are a follow: f rd 548Hz, f rd 74Hz, 6, 78, 3 [546 59, 4, , 7 75], [7 636, 97,, ] When 3 i elected a the exciting point, the reult are a follow: f rd 3 546Hz, f rd 3 Hz, 3 8, 3 8, 3 3 [5476, 34 76, 84 83, 53 ], 3 [,,, 8 49, 57] When 4 i elected a the exciting point, the reult are a follow: f rd 4 545Hz, f rd 4 Hz, 4 7, 4 79, 3 4 [33976, 378, 998, 79 39], 4 [,,, 696, ] From the reult above, we can find that the fourmachine ytem ha three mechanical ocillation mode: f rd 55Hz, f rd 73Hz, and f rd 3 Hz According to the mode hape, it i clear that f rd 55Hz mode i an inter-regional mode, with generator and of area winging againt generator 3 and 4 And f rd 73Hz mode i a local mode of area, with generator winging againt generator And f rd Hz mode i a local mode of area, with generator 3 winging againt generator 4 hee reult are coincident with the reult of the eigenvalue analyi method in [5] 43 eign of P for Multi-machine Power ytem Baed on et ignal Method Power ytem tabilizer (P) have been ued for many year to add damping to electromechanical 3

6 ocillation Although coniderable reearch ha being done in deigning P for multi-machine ytem, the mot widely ued method for P deign i till baed on the theory of OMIB ytem he bai for P deign i the mall ignal tability analyi and phae compenation theory he difficultie in analyzing mall ignal tability of a complex power ytem by the eigenvalue analyi method and the approximation in calculating the generator contant K K 6 by ytem equivalence mae the tet ignal method more advantageou in P deign By adding a mall tet ignal at the reference voltage point of the automatic voltage regulator (AV), we can obtain the generator ternal voltage repone and calculate the phae lag between the generator ternal voltage and the tet ignal at different frequencie he approach i ilar to the calculation of mechanical adttance tated above he phae lag between the generator ternal voltage and the tet ignal at different frequencie i the phae lag we need to compenate in P deign [6] By the tet ignal method, the main tep of P deign are a follow: a) o mall ignal tability analyi uing the tet ignal method b) Chooe ome generator to intall P according to the reult from a) c) Calculate the phae lag of the generator to intall P uing the tet ignal method d) une the P parameter of the generator to compenate the phae lag calculated from c) e) ecalculate the ocillation mode of the ytem uing the tet ignal method, becaue intallation of the P will change ocillation mode and damping of the ytem f) epeat tep c) to e) until all the mechanical ocillation mode are well damped hu, machine by machine, we can deign and tune P for a power ytem uing the tet ignal method Now we till conider the LFO problem of the fourmachine ytem dicued above According to the participation factor and mode hape of the generator to the three ocillation mode, generator and 4 are elected to intall P When peed input ignal i adopted, the bloc diagram of the P i a follow: Input ignal KP Figure 5: W W 4 3 LIM-OWN LIM-UP Bloc iagram of peed Input ignal P Output ignal Where K P i the gain of P, w i the wahout time contant o generator, the phae lag needing to be compenated i calculated by the tet ignal method, which i hown in Figure 6 he tuned P parameter are hown in able he reulted P phae lead i alo hown in Figure 6 From Figure 6, it can be een that the two phae characteritic are well matched in the range of 3Hz to 5Hz, where LFO may occur Phee (eg) Calculated Phae lag for Phae Lead of the P Figure 6 : Phae-frequency Characteritic for Chooe generator 4 a exciting point, do mall ignal tability analyi by the tet ignal method again, we can obtain the new ocillating mode aociated with generator 4 a follow: f rd 4 545Hz and f rd 4 9Hz And liewie, we can deign P for generator 4, the parameter of which i alo lited in able Kp w =3 =4 P P able : Parameter of P and P 4 o evaluate the performance of the two intalled P, we do mall ignal tability analyi by tet ignal method again he reult are a follow: f rd 58Hz, f rd Hz, and f rd 3 3Hz 5, 35, and 3 8 It can be een that all ocillation mode have a

7 damping ratio greater than, which can meet requirement of the operation And the tranient imulation reult hown in Figure 7 and Figure 8 under a three-phae fault alo confirm the good effect of the P he proce of the fault i: when t =, a three-phae fault occur at one of the intertie between the two area, and later, thi intertie i cut off forever Figure 7 how the peed change of the 3 generator (peed of generator 3 i et a reference), and Figure 8 how the active power of the ret intertie between two area Change of peed (pu) Active Power (MW) ime Figure 7: Change of enerator peed With P Without P ime Figure8: Active Power of the Intertie 5 CONCLUION In thi paper, the tet ignal method for mall ignal tability analyi i developed he method i founded on the theory of the mechanical adttance and mode identification he reult of tudy cae how that the method can be efficiently and conveniently applied to analyzing mall ignal tability problem and P deign Compared with the eigenvalue analyi method, the tet ignal method ha the following main advantage: a) he method can be eaily implemented by electromechanical tranient analyi program, which are widely ued and available b) he high-order model of the generator and the excitation ytem and the governor ytem can be included without increaing the complexity of the analyi proce c) he HVC ytem and the FAC can be taen into account facilely d) he very large power ytem can be analyzed by the method conveniently e) It i very handy to deign P baed on the method 6 ACKNOWLEMEN Proect No 9983 upported by National Key Baic eearch pecial Fund of China Proect upported by National cience Foundation of China 7 EFEENCE [] Zheng Xu, Zhouyan Feng, "A novel unified approach for analyi of mall-ignal tability of power ytem", Proceeding of IEEE/PE Winter Meeting,, pp [] Zheng Xu, "he complex torque coefficient approach applicability analyi and it realization by time domain imulation", Proceeding of the CEE, vol, no6,, pp-4 [3] E Newland, "Mechanical vibration analyi and computation", Longman cientific and echnical, 989 [4] V Wow, "Mechanical vibration meaurement and analyi", Mcraw-Hill, Inc, 99 [5] P Kundur, "Power ytem tability and control", Mcraw-Hill, Inc, New yor, 994 [6] EV Laron, A wann, Applying Power ytem tabilizer, Part I: eneral Concept, Part II: Performance Obective and uning Concept, Part III: Practical Conideration, IEEE ranaction on Power Apparatu and ytem, Vol PA-, 98, pp