Descriptor and nonlinear eigenvalue problems in the analysis of large electrical power systems. Nelson Martins Sergio Gomes Jr. Sergio Luis Varricchio

Transcription

1 Descrptor and nonlnear egenalue problems n the analss o large electrcal power sstems Nelson Martns Sergo Gomes Jr. Sergo Lus Varrccho Workshop on Nonlnear Egenalue Problems March 7 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

2 Presentaton Outlne Electrcal Network Modelng Descrptor Sstems Non-lnear Egenalue ormulaton - Y(s) modelng Y(s) Domnant Pole Algorthm (YDPA) Y(s) Multple Domnant Pole Algorthm (YMDPA) Results Small test sstem Medum-scale test sstem Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

3 Electrcal Network Modelng Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

4 Network Modelng Approaches The dnamc behaor o a lnear sstem can be genercall wrtten n the s-doman as: ( s) x b u Y t c x + where x s the ector contanng the releant sstem arables, u and are the sstem nput and output respectel. Dependng on the technque adopted to model the sstem dnamcs the matrx Y(s) can, or nstance, be equal to: Y( s) ( s I A) ( s T A) Y bus ( s) where I, T and A are constant matrces. d u State-space Descrptor Sstem (DAE) Nodal Admttance 4 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

5 Network Modelng Approaches State space and DAE models eld matrces that are lnear unctons o s. The ecent SADPA and SAMDP algorthms ma be appled to obtan the set o domnant poles or SISO and MIMO transer unctons. The elements o Ybus(s) can be non-lnear unctons o s, allowng the modelng o long transmsson lnes (trbuted and requenc dependent parameters) The releant sstem poles o Y(s) and assocated resdues can be computed usng the Multple Domnant Pole Algorthm. Ths algorthm requres the determnaton o Y(s) and ts derate wth respect to s. Ths derate s bult applng the same rules used to buld Y(s) but usng the admttance derates o the sstem elements. 5 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

6 Electrcal Network Modelng Descrptor Sstem Equatons RLC Seres Branch RLC Parallel Branch R Voltage Source k kj L j L dkj k j R kj + L + dt dc C dt kj Inexstent C d k j R kj + L dt kj C C R + d L dt L L C C + C C d C dt C Inexstent L C dc + C kj R dt k kj j k j R L d dt 6 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

7 Electrcal Network Modelng Descrptor Sstem - Example L R R R L L barra barra C R L L Parallel C connected to node dc C () C () dt barra C C R L L Parallel RLC connected to node dl L C () (5) C dt C d C R L C + dt (4) Parallel RLC connected to node dl L C (6) (8) C dt dc C L (7) C + dt R 7 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

8 Electrcal Network Modelng Descrptor Sstem - Example 8 L R R R L L barra barra C R L L Seres RL between nodes and d L R + (9) dt barra C C R L L Seres RL between nodes and R + Voltage source at node Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7 L d dt d L R + dt Krchho Current Law () () node + () node + + () node + (4)

9 Descrptor Descrptor and and Nonlnear Nonlnear Egenalue Egenalue Problems Problems Appled Appled to to Electrcal Electrcal Power Power Sstems Sstems March March, 7, 7 9 ( ) 7 R dt d C C L C + dt d L L L C L C L C C L C L C () dt d C C + R R R R R C L C L C () () (7) (7) () () () + +

10 Descrptor Descrptor and and Nonlnear Nonlnear Egenalue Egenalue Problems Problems Appled Appled to to Electrcal Electrcal Power Power Sstems Sstems March March, 7, 7 + C L C L C Electrcal Network Modelng Descrptor Sstem - Example ( ) ( ) ( ) t t t u B A x x T + & ( ) ( ) ( ) t t t u D C x + Consderng the nodal oltages as output arables, we hae: Compact orm:

11 Electrcal Network Modelng Y(s) Matrx - Equatons RLC Seres Branch RLC Parallel Branch Voltage Source k k seres R + sl + sc d seres L + s C R + sl + sc + sc parallel R sl + d parallel C s L R k L k n j k kj k j k k + z z R + s L k Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

12 Electrcal Network Modelng Y(s) Matrx - Example L R barra C s C + R z + s L + R + s L R R L L barra barra R + s L Voltage Source Modelng KCL + + C R L L C R L L KVL + z Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

13 Electrcal Network Modelng Y(s) Matrx - Example L R R R L L barra barra C R L L barra C C R L L z + Compact orm: Y ( s) x( s) B u( s) ( s) C x( s) + D u( s) Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

14 Transmsson Lnes Detals n: [] S. Gomes Jr., C. Portela, N. Martns Detaled Model o Long Transmsson Lnes or Modal Analss o ac Networks, Proceedngs o IPST, Ro de Janero, Brazl,. I I s m s m U U s coth ( γ l) m c csch( γ l) c γ Z u ( s) Y ( s) u c Y Z u u ( s) ( s) 4 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

15 Transmsson Lnes - Derates s coth c ( γ l) d s d c coth dγ ( γ l) l csch( γ l) c m csch c ( γ l) d m d c csch dγ ( γ l) l csch( γ l) coth( γ l) c γ Z u ( s) Y ( s) u dγ γ Z u dy u + Y u dz u c Y Z u u ( s) ( s) d Z Y dy c u c u u dz u 5 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

16 Transmsson Lnes Parameters Y u s C u ( e) ( ) ( g) Z Z + Z + Z Z ( e) ( e) ( s L s µ ε C ) Conductor Internal Impedance Z ( ) Bessel unctons Complex depth penetraton Ground Eect (Z ( g ) ) Innte ntegral o Carson Complex depth ground return The expressons or the s-derates, requred b the modal analss algorthms, are presented n the paper 6 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

17 Internal Impedance (Bessel Functons) u ( e) ( ) ( g) Z Z + Z + Z Dagonal matrx wth the ollowng elements: z k ( s) n d ( s) ( s) n d k ( s) s µ σ π ( s) I ( ρ ) K ( ρ ) + I ( ρ ) K ( ) r e ρ ( s) I ( ρ ) K ( ρ ) + I ( ρ ) K ( ) ρ ρ r s µ σ ρ r e s µ σ I, I are the moded Bessel unctons o rst knd whle K and K are the moded Bessel unctons o second knd. Indexes and represent the order o the unctons. The parameter σ s the conductor conductt and µ s the conductor magnetc permeablt. 7 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

18 Ground Impedance (Carson Innte Integral, 96) u ( e) ( ) ( g) Z Z + Z + Z Matrx wth the ollowng elements: ( ) µ e h x g z, s dx π x + x + s µ σ z ( h + ) ( ) µ e h j x cos( x d, j ) g, j s π x + Soled numercall or each alue o s. x + s µ σ dx 8 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

19 Ground Impedance (Complex depth return plane, Semlen, 988) u ( e) ( ) ( g) Z Z + Z + Z Matrx wth the ollowng elements: z z ( e, g ) ( e) ( g ) µ ( h p), z, + z, s + ln π re ( ) ( ) ( ) ( ) µ h + + +, h j p d e g e g, j, j z, j + z, j s ln π ( h ) + h j d, j Where p s: p s µ ( σ + s ε) 9 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

20 Network Modelng Example R Z Voltage Source Modelng Y Y s R L C ( ) Y s dy ( ) Y s ( s) d d d d d d d d ( s) + s C + Y + R + s L Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7 ( R + s L ) Z L d + C + c coth c ( s) coth[ γ( s) l] { ( s) [ γ( s) l] }

21 Y(s) Domnant Pole Algorthm (YDPA) Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

22 Transer Functon The transer uncton can be wrtten as a summaton o partal ractons: The drect term s dened b: G R s λ t ( s) c Y( s) b + d d n Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7 λ t ( s) lm c Y( ) b lm G s s s Sstem pole R Resdue assocated to λ d drect term o G(s) n Number o sstem poles The domnant poles o G(s) are those most releant to tme and requenc responses. The are assocated wth the local maxmum alues (peaks) o the requenc magntude plot. The requences assocated wth these peaks are used as ntal shts n the Y(s) DOMINANT POLE ALGORITHM (YDPA).

23 Y(s) Domnant Pole Algorthm (YDPA) Whle alue o pole sht correcton ( λ (k) ) s hgher than - : Calculate, usng pole estmate: Transer uncton Resdue: Pole Correcton: λ R Y( λ c ( k ) t ( k) Y( λ ) t b ( k+ ) ( k ) ( k ) ( k ) + u R ) t b c ) ( k ( ) k u ) w ( k ( ) k u ( k ) ( k ) t dy( λ ) ( k) [ w ] New pole estmate: ( k ) ( k ) ( k ) λ + λ + λ Next Iteraton: k k + End o Whle Loop Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

24 Reerences on YDPA [] S. Gomes Jr., N. Martns, C. Portela, Modal Analss Appled to s-doman Models o ac Networks, Proceedngs o IEEE Wnter Meetng, Columbus, Oho, USA,. [] S. Gomes Jr., N. Martns, S. L. Varrccho, C. Portela, Modal Analss o Electromagnetc Transents n ac Networks hang Long Transmsson Lnes. IEEE Transactons on Power Deler,., 5. [4] S. L. Varrccho, S. Gomes Jr., N. Martns Modal analss o ndustral sstem harmoncs usng the s-doman approach, IEEE Transactons on Power Deler,.9, 4 4 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

25 Multple Domnant Pole Algorthm (YMDPA) 5 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

26 Y(s) Multple Domnant Poles (YMDPA) The domnant poles o G(s) are those most releant to tme and requenc responses. The modul o the resdues o the domnant poles are larger when compared to those o the remanng poles. The are assocated wth the local maxmum alues o the requenc response magntude plot. These alues can be used as ntal estmates or the Y(s) MULTIPLE DOMINANT POLE ALGORITHM (YMDPA). G(jω) λ G t ( jω) c Y( jω) b ( ) j [ ω ω L ω ω ] 7 8 ω ω ω ω 4 ω 5 ω 6 ω 7 ω 8 Frequenc (ω) Intal estmates or YMDPA 6 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

27 Descrpton o Algorthm (YMDPA) For an estmate λ (k) the matrx Y(λ (k) ) and ts derate are bult. The uncton (s) and ts derate are determned usng the preousl calculated poles and resdues (λ, R ). λ λ ( ( ) λ ) ( k ) k R j j j ( ( ) λ ) k R j ( ( k ) λ λ ) Solng () and (), one obtans the ectors, w and the scalar u. d j j Y ( ( k )) ( k ) λ b c t d ( ) k u ( ) ( ) ( k ) t k Y λ c w () ( ) () t k b d u Usng the alues o u,, w, dy/, and d /, the pole correcton alue s obtaned: ( k ) ( ( k ))( ( k )) ( k ) u λ u λ ( ( k )) ( ( k )) ( k ) t dy λ ( k ) d λ ( k ) 7 ( w ) ( u ) Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

28 Modal Analss MDPA Updated alue o the pole or the next teraton: ( k ) ( k ) ( k) λ + λ + λ Assocated resdue: R ( k+ ) [ ( ) ( ( )) ] k k u λ ( k ) u λ ( k ) G t ( jω) c Y( jω) b G(jω) G ( jω) Ω R jω λ + d Ω set o domnant poles Frequenc (ω) It must be ponted out that the YMDPA s desgned to aod successe conergences to the same pole alue. 8 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

29 Test Sstems 9 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

30 Reerences Small Test Sstem: S. Gomes Jr., N. Martns, S. L. Varrccho, C. Portela, Modal Analss o Electromagnetc Transents n ac Networks hang Long Transmsson Lnes. IEEE Transactons on Power Deler,., 5. Medum Scale Test Sstem (Submtted) S. Gomes Jr., N. Martns, C. Portela, Computng Multple Domnant Poles or Modal Analss o s-doman Models, submted to IEEE Transactons on Power Sstems, 7 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

31 Electromagnetc Transent Problem Small Test Sstem V S µf S Ω.6 H 7 H Transmsson Lne (5 kv, km) TL Parameters: Z u (.8 Ω + s.868 mh) / km Y u s.8 µf / km 4 5 µf 7 H 55 Ω H.4 µf V 5 / V G G R R 8 ( jω) R jω λ t ( jω) c Y( jω) b Frequenc (Hz) Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

32 New England Medum Scale Test Sstem Input: V Output: V Transer Functon : V V Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

33 TL represented b lumped elements (5 π s) Descrptor Sstem or Y(s) model Voltage (pu) Transer Functon : V V Frequenc (Hz) Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

34 TL represented b trbuted elements onl Y(s) model Voltage (pu) Transer Functon : V V Frequenc (Hz) 4 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

35 Frequenc Response Comparson Voltage (pu) π Voltage (pu) π s Frequenc (Hz) Frequenc (Hz) Voltage (pu) π s Voltage (pu) Dstrbuted Parameters ( π s) Frequenc (Hz) Frequenc (Hz) 5 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

36 Frequenc Response Comparson.8 Dstrbuted Parameters 5 Ps Voltage (pu) Frequenc (Hz) 6 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

37 Comparng o Matrx Dmensons Model Lnes Non-zeros States Derental Equatons π poles at nnt 5 π's poles at nnt 5 π's 7, Dstrbuted Parameters 49 5 Innte s-doman 7 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

38 TL represented b lumped elements (P) nz Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

39 TL represented b lumped elements (5P) nz 58 9 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

40 TL represented b lumped elements (5P) nz 988 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

41 TL represented b Dstrbuted Parameters Y(s) model nz 5 4 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

42 Intal Estmates (65 poles) Maxma o Frequenc Response.8 Voltage (pu) Frequenc (Hz) 4 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

43 Conergence Trajectores 4 ). + j j j j j j ). + j j j j j j j ). + j j j j j j ). + j j j j j j j 484. Derges! Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

44 Conergence Trajectores 44 5). + j j j j j j j j j ). + j j j j j j j j 555. Derges! 6). + j j j j j j j ). + j j j j j j j 95.5 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

45 Conergence Report 65 Intal Shts, 6 Conerged Poles Number o Iteratons Number o Pole Estmates Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

46 Choosng Adequate Intal Sht Imagnar part o ntal shts made equal to the requences where the modal error s maxmum (peak) Real Part Estmate o Sht: Zero Search maxmum o model error n s-plane, or a xed magnar part and arng real part o ntal sht. poles were calculated (order 87) 5 real poles 86 complex conjugated pars 46 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

47 Reduced Order Model.8 Complete Model 87th Order Model Voltage (pu) Frequenc (Hz) 47 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

48 Conclusons Deeloped Algorthms: YDPA & YMDPA Full Newton Algorthms (quadratc conergence) Ecent and Robust or large scale sstems Requres arl good ntal shts Allows egenalue analss o nnte sstems Dstrbuted parameters Frequenc Dependent Elements Transport Dela YMDPA (Y(s) Multple Domnant Poles) Sequental Delaton b Full Newton subtracton o combned eects o conerged poles No assocated numercal dcultes Intal shts extracted rom requenc response plots 48 Descrptor and Nonlnear Egenalue Problems Appled to Electrcal Power Sstems March,, 7

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