Investigating Singularities of the Observability Jacobian for Nonlinear Power Systems

Transcription

1 Investgatng Sngulartes of the Observablty Jacoban for Nonlnear Power Systems C. J. Dafs, Member, IEEE, and C. O. Nwankpa, Member, IEEE Abstract Ths paper focuses on computng and nterpretng the sgnfcance of the observablty Jacoban sngulartes. Prevous efforts have focused on developng an observablty ndex and determnng the effects of model/measurement changes to the observablty formulaton. Usng ths ndex, the sngulartes of the Jacoban are extracted, and prelmnary results ndcate that there s a dualty between the loss of stablty (whether t be phase or voltage stablty), and the loss of observablty. Thus, provdng a sngle dual-use metrc capable of forecastng both stablty and observablty characterstcs of a system. The results are llustrated usng a sample 3-bus system. Index Terms observablty, power systems, system ndex, dfferental algebrac equatons. T I. INTRODUCTION he safe, relable and economc operaton of modern power systems rely on ncreased levels of system automaton, power electronc equpment, and system controllers capable of montorng and controllng the system through varous operatng ponts. One partcular famly of power systems where the effects of system automaton, power converters and rapd system reconfguraton are more profound are naval shpboard power systems. The nherent nature of the shpboard systems, smlar to sland topologes n terrestral systems, has magnfed the need for real-tme system control and therefore the development of dynamc state estmators and nonlnear observers. Shpboard power systems are unque n ther composton, and dffer from terrestral systems n many ways [1]: Sngle dynamc loads consumng a large porton of the generaton capacty Tme constants of the generators are closer to the electrcal tme constants than n terrestral systems System reconfguraton routnes are much faster and a large porton of the system load s senstve to power nterruptons. A small nterrupton of approxmately 1 msec. can cause subsystems to shut-down. Redundancy s ncorporated n all aspects of system desgn to ensure survvablty The power demand on a shp has evolved from klowatts the tens of megawatts, and s constantly ncreasng wth new shp desgns and concepts. One partcular concept affectng drectly the shpboard power system s the vson for an All Electrc Shp [2], whch sgnfes the move of the shpboard power system from an electromagnetc/electromechancal bass to a power electronc bass. Some of the desgn aspects nvolvng the All Electrc Shp concept are: Electrc Drve Integrated Power System Zonal Dstrbuton System Reduced Mannng The enablng technology n all of the above concepts s ntellgent automaton and control. Wth the ncrease n the automaton of system functons wthn the shpboard power system, the dependence on both executve and dstrbuted controllers to mantan the performance of the power system at acceptable levels s becomng more apparent. Tradtonal control approaches based on lnearzed models of the system generaton/dstrbuton capacty wll not be suffcent. The need of ncorporatng the nonlnear dynamcs of the shpboard power system n the control methodology s the soluton to ths problem, and one proposed soluton s the use of nonlnear observer based controllers and dynamc state estmators. Before these solutons are desgned however, the concept of nonlnear observablty s frst nvestgated. Provdng an mplementaton of the observablty formulaton for power systems modeled as Dfferental Algebrac Equatons s the startng pont for the development of nonlnear controllers, dynamc state estmators and performance ndex measures. The authors have nvestgated the problem of system observablty for nonlnear power systems modeled as DAEs [3-6], and ths work emphaszes on the possble dualty between the sngulartes of the observablty Jacoban and the known stablty characterstcs of the system. The prelmnary results ndcate that a system becomes unobservable at the maxmum load pont. Ths work was supported n part by the (NAVSEA-ILIR), under grant number and the Department of Energy under grant number CH C. J. Dafs s wth NAVSEA-Phladelpha, Phladelpha, PA USA (dafscj@nswccd.navy.ml) C. O. Nwankpa s drector of CEPE, Drexel Unversty, Phladelpha, PA 1914 USA (chka@nwankpa.ece.drexel.edu) /5/$2. 25 IEEE. 75 II. PROBLEM FORMULATION The majorty of the system models used n ths work and presented n ths secton are adopted from [4] and repeated here for the sake of contnuty. The Dfferental Algebrac Model of a power system consstng of n generators and m

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 124, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE JUL REPORT TYPE 3. DATES COVERED --25 to TITLE AND SUBTITLE Investgatng Sngulartes of the Observablty Jacoban for Nonlnear Power Systems 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Surface Warfare Center -Carderock Dvson,NAVSEA-Phladelpha,Phladelpha,PA, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 1. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dstrbuton unlmted 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES See also ADM1931. Proceedngs of the 25 IEEE Electrc Shp Technologes Symposum (ESTS 25) Held n Phladelpha, PA on July 25-27, ABSTRACT see report 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed Same as Report (SAR) 18. NUMBER OF PAGES 6 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANSI Std Z39-18

3 statc load buses s gven as: n m x = f( x, u, N) f : 2( + 1) 2( n 1) = gxun (,, ) g: y h x N h 2( n+ m 1) 2( m) 2( n+ m 1) = (, ) : where f(.) s the set of dfferental equatons related to the dynamcs of the generaton equpment, g(.) s the set of nonlnear algebrac equatons related to the statc load buses of the system, h(.) are the measurement equatons, u are the system nputs, N descrbes the characterstcs of the ndvdual power lnes n the system, x s the system vector contanng both the dynamc states and statc varables of the system, and y are the avalable measurements. More nformaton on DAE power system models and ther related characterstcs can be found n [7]. By reformulatng (1) n a more condensed verson: F( x, x, N) = Bu (2) y = h( x, N) the observablty Jacoban (J O ) of the system s gven by [8]: Gx Gx Gw JO = (3) H x Hx H w where FxxN (,, ) (1) F : Fx( x, x, N) x Fx( x, x, N) x G + = ( s) ( s) (4) F :( FxxN (,, )) hxn (, ) y (1) (1) h : hx ( x, N) x y H = = ( r) ( r) ( r) h : h( x, N) y and s s the system dfferentaton ndex, r s the measurement dfferentaton ndex, and w are the hgher dervatves of x (2.w). Based on ths Jacoban, the system s sad to be smoothly observable f [8]: Gx Gw 1. rank( JO ) = n + rank H x H w (5) 2. rank( J ) s constant rank on S A. Power System Model O { = g x u N } S: (,, ) In the adopted structure preservng DAE model of the power system, the generators are modeled usng the rotatonal dynamcs provded by the classcal swng equaton, and the loads are modeled as constant PQ loads (consdered power dstrbuton centers), formng the system equatons: I f : δ ω = II D 1 P M f : ω + ( ω) + PG ( δ, θ, V, N) = (6) M M M III * f : P P ( δθ,, V, N) = j j IV * : j j( δθ,,, ) = f Q Q V N r (1) N : Ycd = Gcd + jbcd = 2 n generators; j= n+ 1 m loads; cd, = 1 n+ m where δ, M, D and P M are the nternal angle, nerta, dampng and constant mechancal power of the th generator respectvely, and θ j and V j are the j th load bus angle and voltage respectvely. The desred power demand on the j th load bus s denoted S * =P * +jq *. The system has the form of (1), wth f(.) n (1) comprsed of f I and f II and g(.) n (1) comprsed of f III and f IV, and wth respect to the form gven n (2): I II III IV T F( x, x, N) = [ f f f f ] (7) PM x= [ δ, ω, θ j, Vj], u = M The avalable measurements are all assumed as real power branch flows, gven as: hxn (, ) = { Pcd ( xn, )} (8) Ths system can be further modfed to ncorporate generator excter dynamcs or dynamcs n the system loads. The observablty formulaton for these varatons was dscussed n [3-6], and the addton of other types of generator and load models (voltage dependent loads for example) wll mpact the constructon of the observablty Jacoban only. The methodology for obtanng the Jacoban, the observablty condtons and the degree of observablty measure s consstent. Also, t should be noted that n (6) generator bus 1 was arbtrarly selected as the reference bus, yeldng δ :δ -δ 1 and θ j :θ j -δ 1. B. Observablty Metrc To quantfy the observablty of the system durng a varaton, a measure of observablty s ntroduced, defned as: l λmax ( JO ) η =, l = 1 L (9) λmn ( JO ) where λ are the sngular values of the observablty Jacoban obtaned va a sngular value decomposton at each pont of nterest l. The maxmum number of L, s dctated by the convergence of the numercal solvers used for ether statc studes (ponts along PV Curve), or dynamc studes (length of tme wndow n a tme doman smulaton). The degree of observablty γ s defned as: l γ = max { η } l = 1 L (1) Based on the formulaton of ths measure, a smaller numercal value of γ s preferred a hgher number ndcates that the observablty Jacoban s becomng sngular. III. APPROACH To determne all the sngulartes of the observablty Jacoban, a symbolc computaton approach s used. Also, tme doman smulatons are used to track the observablty of the system along the system state trajectores, as these unobservable regons are approached. A bref descrpton of both the symbolc and numercal approaches s provded. 76

4 A. Symbolc Approach The symbolc approach examnes the observablty crtera as a functon of the system state space. In partcular, the examnaton s focused on fndng where n the state-space the observablty Jacoban undergoes rank defcences. In essence, by examnng where observablty s lost, all other ponts n the state-space are assumed to be observable. Ths s assumed, rather than guaranteed, because of the performance of the symbolc tools n evaluatng rank defcences. Namely ther performance n locatng hgher order degeneraces n the Jacoban s of queston. For the partcular system examned, the symbolc tools were able to capture hgher order degeneraces. Ths technque s powerful n the sense that t provdes the global solutons where the system s unobservable, however t s lmted by the computatonal capablty of the symbolc solvers, and hence, s lmted to the sze of the system that can be evaluated. The symbolc approach process s outlned n Fgure 1. Statc Power Flow Statc or Dynamc Case Dynamc Tme Doman Solver Obtan System Parameters (Network Parameters and Generator Parameters) Obtan System Measurements Construct System Equatons Intalze s and r Indces, s=r=1 Symbolc Evaluaton of Observablty Jacoban Numercal Evaluaton of Observablty Crtera Is the System Observable Yes No Proceed to Next Parameter Set Update s and r Obtan System Parameters (Network Parameters and Generator Parameters) Obtan System Measurements Fg 2. Flow Dagram of the Numercal Approaches n Determnng Observablty Construct System Equatons NR NRS Inta lze s and r Indces, s =r=1 Symbolc Evaluaton of Observablty Jacoban Symbolc Evaluaton of State Space Solutons for: det(jo)= Fg 1. Investgatng Observablty usng Symbolc Tools B. Numercal Approach The numercal approach conssts of two separate cases: Statc Cases Evaluatng observablty for operatng ponts along system PV Curves Dynamc Cases Evaluatng the observablty crteron along trajectores of the system states The general approach n evaluatng these crtera and determnng the approprate s and r to render the system observable s outlned n Fgure 2. For the statc cases, the soluton of the system PV curves s accomplshed n an teratve fashon usng the Newton- Raphson (NR) and Newton-Raphson-Seydel (NRS) algorthms [9]. When the NR algorthm fals, the NRS algorthm s used to obtan the data ponts near and around the nose pont of the PV curve, as llustrated n Fgure 3. For the numercal cases, the ode15s DAE solver was used n the Matlab/Smulnk envronment to generate the tme doman smulatons of nterest. Fg 3. Typcal Power System PV Curve IV. SIMULATION RESULTS A. Sample System To llustrate the symbolc approach for obtanng the unobservable regons, a 3-bus power system depcted n Fgure 4, wth correspondng system parameters n Table 1, was used. There are two sets of results consderng two dfferent measurements, P 12 and P 32. In both cases, r=s=1, and the sngularty of the Jacoban s nvestgated by solvng the followng relatonshp symbolcally: det ( J O ) = For the selected system, the observablty Jacoban has the general form provded n (11), and the states of the system are gven as: x = θ, ω, θ, V [ ]

5 J O 1 1 j2,1 j2,2 j2,3 j2,4 1 j3,1 j3,3 j3,4 j4,1 j4,3 j4,4 1 1 = j6,1 j6,3 j6,4 j6,5 j6,6 j6,7 j6,8 1 j7,1 j7,3 j7,4 j7,5 j7,7 j7,8 j8,1 j8,3 j8,4 j8,5 j8,7 j8,8 j9,1 j9,3 j9,4 j1,1 j1,3 j1,4 j1,5 j1,7 j1,8 Bus 1 Bus 2 R12 L12 GS GS R13 L23 L13 R23 Bus 3 (11) frst 2 solutons represent ether a fault on bus 3, or a phase nstablty, and the thrd soluton provdes the bus voltage as a functon of the bus phases. TABLE II SYSTEM OBSERVABILITY IS LOST AND THE CORRESPONDING RANK OF THE OBSERVABILITY JACOBIAN P 32 Soluton x 1 x 2 x 3 x 4 Rank 1 x 1 x 2 x x 2 π/2 x x 2 -π/2 x π x 2 π/2 x π x 2 -π/2 x x 1 x 2 x 3 f(x 1, x 3 ) 8 TABLE III SYSTEM OBSERVABILITY IS LOST AND THE CORRESPONDING RANK OF THE OBSERVABILITY JACOBIAN P 12 Soluton x 1 x 2 x 3 x 4 Rank 1 x 1 x 2 x π/2 x 2 x 3 x x 1 x 2 x 3 f(x 1, x 3 ) 8 P3+jQ3 Fg. 4. Sample 3-bus Power System TABLE I SYSTEM PARAMETERS FOR SAMPLE 3-BUS POWER SYSTEM Lne R X D= M= Bus P ntal Q ntal V 1 = V 2 = B. Symbolc Results Loss of Observablty By computng the determnant symbolcally and equatng t to zero, there are 6 solutons for x whch yeld rank defcences (Table II). In analyzng the solutons, and keepng n mnd the sgnfcance of the respectve system state assgnments, the frst fve solutons are not consdered physcally realzable. For Soluton 1, the bus voltage at the load bus would have to go to zero, meanng the generators are ether offlne or the topology of the system has changed (due to a fault on Bus 3, or faults on both lnes connectng Bus 3 to the two generator buses). Smlarly, for Solutons 2-5, the angle dfference between generator 1 and the load bus s 9 degrees, whch would ndcate a phase nstablty n the system. Therefore, for the frst fve solutons, one can assume that the system wll be smoothly observable for all perturbatons that do not cause these specfc system nstabltes, and that one measurement (P 32 ) s suffcent for the system to be smoothly observable. The only soluton of nterest now s Soluton 6, whch represents the load bus voltage as a functon of the two bus phases n the system. Smlarly for the case of a measurement P 12 (Table III), the Examnng Soluton 6 of Table II and Soluton 3 of Table III, and smlar solutons for a lossless system, an unobservable regon can be constructed, gven a range for the bus angles s specfed. To examne the physcal nterpretaton of these solutons, the ntal operatng pont n Table I s selected wth: x = [.869,,.222,1.1] and an arbtrary PV curve s constructed (Fgure 5). Usng the phase quanttes dctated by the PV curve, the relatonshp of the load bus voltage to the phase dfference between bus 2 and 3 s examned and compared to the results for the load bus voltage obtaned usng the PV curve. The phase dfference was used as a parameter to condense the problem nto 2-D. Examnng Fgure 6, the soluton for the load bus voltage magntude ntersects the unobservable solutons obtaned from Tables II and III, and usng the nformaton obtaned by the PV curve, ths ntersecton corresponds to the maxmum load pont of the system (magnfed n Fgure 7). Ths result s consstent wth all the obtaned solutons, and s nvarant to the selected measurement, and to the mplcaton of a lossless or lossy system. Due to the symmetry of the system topology used, the results for P 13 are the same as those for P 32 and are not ncluded. 78 Load Bus Voltage Magntude (pu) A PV Curve for Sample 3-Bus System Real Power Demand - Bus 3 (pu) N N R R S Fg 5. Arbtrary PV Curve of 3-Bus System B C

6 Load Bus Voltage Magntude (pu) Relatonshp Between System PV Curve and Unobservable Regons Losses-P12 Lossless-P12 Losses-P32 Lossless-P32 System PV Curve Theta2-Theta3 (rad) Fg. 6 Relatonshp Between System PV Curve and Unobservable Solutons.682 Relatonshp Between System PV Curve and Unobservable Regons along the trajectores of the system states. It s expected that the observablty ndex for the trajectores assocated wth pont C wll be hgher. By repeatng ths process for the entre set of ponts along the PV curve, a threshold on the observablty ndex was obtaned (γ= 6), and the ndvdual γ s provded n Table V, for the perturbaton level of 4% on the generator phase. TABLE IV ARBITRARY POINTS ALONG PV CURVE CHOSEN FOR TIME DOMAIN SIMULATIONS Pont x 1 x 3 x 4 P 2 P 3 Q 3 A B C Pont A Load Bus Voltage Magntude (pu) Losses-P12 Lossless-P12 Losses-P32 Lossless-P32 System PV Curve Theta2-Theta3 (rad) Pont B 126 Observablty Measure Pont C 3 25 Fg. 7 Relatonshp Between System PV Curve and Unobservable Solutons Intersecton Magnfed 2 15 Tme (sec.) Therefore, for the selected system, a dualty exsts between the loss of observablty and the voltage stablty problem. At the maxmum load pont, the system s unstable and unobservable. Ths s also present wth cases of phase nstablty, represented by the solutons n Table II and III, where large phase dfferences exst between the generator and load bus. C. Dynamc Smulatons Because of the lmtatons wth symbolc tools, analyzng larger systems are not possble wthout the ntroducton of numerc methods. However, unlke the symbolc approach whch yelded the entre famly of unobservable solutons, the numercal approaches are constraned to only approach a pont of nterest. To llustrate that the numercal results are consstent wth the answers from the symbolc computatons, three arbtrary ponts are selected along the PV curve (ponts A, B and C n Fgure 5, gven n Table IV). A 4% perturbaton s appled to the generator phase at each pont, and the resultng tme doman waveforms are extracted (waveforms provded n Appendx). These tme doman waveforms are then used to analyze the observablty ndex 79 Fg. 8. Observablty Measure along State Trajectores for Ponts A, B and C. TABLE V OBSERVABILITY INDEX FOR EACH SET OF TRAJECTORIES INITIATED FROM POINTS A, B AND C Pont γ A B C As expected, the trajectores produced from the ponts closer to the nose pont produced larger ndex values, ndcatng that the system s becomng less observable, and they are consstent wth the results obtaned usng the symbolc approach. However, at no pont does the system become unobservable n a numercal sense. The observablty crtera of Equaton 5 are satsfed for all the ponts along the PV curve, and system trajectores for ths system. A closer examnaton ndcates that a quanttatve change s occurrng as the system s approachng the maxmum load pont. Ths s evdent when examnng the mnmum sngular value of the observablty Jacoban for each pont along the PV curve. As

7 llustrated n Fgure 9, the value approaches zero at the maxmum load pont. From a numercal standpont, the system s stll observable, and ths problem s hence transformed nto a numercal senstvty ssue. The approprate selecton of thresholds on γ provdes a better depcton; however, ths process nvolves extensve system smulatons to set these thresholds. Generator Phase (rad) Generator Speed Trajectores After Perturbaton from Pont B -.1 Mnmum Sngular Value N R N R S Mnmum Sngular Value of Jacoban Along PV Curve Maxmum Load Level Ponts Along PV Curve Fg 9. Mnmum Sngular Value of Jacoban Along PV Curve V. CONCLUSIONS Sngulartes of the observablty Jacoban correspond wth system sngulartes that are typcally assocated wth phase and voltage stablty problems n power systems. Ths dualty can be exploted by usng the observablty ndex as a system metrc capable of forecastng both the stablty and observablty classfcatons of a power system. The dual use of a sngle metrc s very encouragng. In the case of system stablty, snce the common ponts are fewer than the entre famly of unstable ponts for a system, ths metrc would serve as a necessary but not suffcent condton. VI. APPENDIX Tme doman smulatons for the sample system consdered n prevous secton are provded n Fgures Generator Phase (rad) Generator Speed Bus 3 Phase Bus 3 Voltage Magntude Trajectores After Perturbaton from Pont A Tme (sec.) Fg 1. Trajectores After Perturbaton from Pont A Bus 3 Phase Bus 3 Voltage Magntude Tme (sec.) Fg 11. Trajectores After Perturbaton from Pont B Generator Phase (rad) Generator Speed Bus 3 Phase Bus 3 Voltage Magntude Trajectores After Perturbaton from Pont C Tme (sec.) Fg 12. Trajectores After Perturbaton from Pont C VII. REFERENCES [1] K. L. Butler-Perry, N. D. R. Sarma, I. V. Hcks, Servce Restoraton n Naval Shpboard Power Systems, IEE Proceedngs on Generaton, Transmsson and Dstrbuton, Vol. 151, Issue 1, pp [2] Natonal Research Councl, Technology for the Unted States Navy and Marne Corps Becomng a 21st Century Force, Volume 2: Technology, Natonal Academc Press, Washngton, [3] C. J. Dafs, "An Observablty Formulaton for Nonlnear Power Systems Modeled as Dfferental Algebrac Systems," Ph.D. Dssertaton, Dept. Elect. & Comp. Eng., Drexel Unversty, Phladelpha, Feb. 25. [4] C. J. Dafs, C. O. Nwankpa, Characterstcs of Degree of Observablty Measure for Nonlnear Power Systems, n Proc. 25 Hawa Internatonal Conference on System Scence. [5] C. J. Dafs, C. O. Nwankpa, A Nonlnear Observablty Formulaton for Power Systems Incorporatng Generator and Load Dynamcs, n Proc. 22 IEEE Conference on Decson and Control, Vol.3, pp , 22. [6] C. J. Dafs, C. O. Nwankpa, Examnng Characterstcs of an Observablty Formulaton for Nonlnear Power Systems, n Proc. 23 IEEE Internatonal Symposum on Crcuts and Systems, Vol. 3, pp [7] P.W. Sauer, M. A. Pa, Power System Dynamcs and Stablty, Prentce Hall, Upper Saddle Rver, New Jersey, [8] W. J. Terrell, Observablty of Nonlnear Dfferental Algebrac Systems, Crcuts, Systems, and Sgnal Processng, Vol. 16, No.2, 1997, pp [9] R. Seydel, Practcal Bfurcaton and Stablty Analyss From Equlbrum to Chaos, Sprnger-Verlag, New York,