POZNAN UNIVE RSITY OF TE CHNOLOGY ACADE MIC JOURNALS No 86 Electrcal Engneerng 6 Volodymyr KONOVAL* Roman PRYTULA** PARTICIPATION FACTOR IN MODAL ANALYSIS OF POWER SYSTEMS STABILITY Ths paper provdes a bref overvew on exstng approaches for defnng partcpaton factor n modal analyss whch characterzes the nteracton between modes and state varables of power system. We calculated partcpaton factors usng dfferent methods and compared the results obtaned wth the expressons derved from mode evoluton. For cases of complex egenvalues of lnear dfferental equatons characterstc matrx t was dscovered ncorrect exstng approaches for defnng partcpaton factor of state varable n mode. Modern software applcatons desgned to analyze power system stablty wdely deploy the approaches dscussed that provde ncorrect results of modal analyss and pose rss to the operaton of real power systems. Therefore the problem of calculatng the partcpaton factor remans as mportant as ever. KEYWORDS: modal analyss, partcpaton factor, statc stablty, power system. INTRODUCTION The Modal Analyss s the most modern method currently used to analyse power system stablty. Ths method nvolves decomposton of power system oscllatons to separate components. Consderng complex systems, the process of mergng power systems s the most dffcult to smulate. Frst, the man problem s the sze of such system as t conssts of hundreds of generators connected wth thousands of power lnes, bushes, and hundreds of load centers. Secondly, complex nature of networ physcal processes causes problems due to physcal values wth dfferent tme dynamcs (electrcal changes usually occur faster than mechancal change of generator rotor poston). As s nown the good enough mathematcal model for study of oscllatng statc stablty for power system, s system of lnearzed dfferental equatons that descrbe behavour of system oscllatons caused by mnor dsturbances. The one of man concept of modal analyss s partcpaton factor. Partcpaton factor s a scalar value that determnes the degree of system * Lvv Polytechn Natonal Unversty. ** Ivan Frano Natonal Unversty of Lvv.
98 Volodymyr Konoval, Roman Prytula parameters nfluence n formaton of oscllatons mode for lnear systems. The theory of modal analyss was presented by scentst Perez-Arraga and others n the wors [, ]. Snce then ths technque was developed greatly and appled to problems of power systems stablty. In a seres of papers [3-5] authors (E.H. Abed, W.A. Hashlamoun, and M.A. Hassouneh) revewed the concept of partcpaton factor and showed that t s actually determned usng dfferent formulas for calculatng mode-n-state and state-n-mode partcpaton. In other words, the dfference between the mode-n-state and state-n-mode concepts was ntroduced. And for ths case they used the stochastc nature of the nput state vector. Our goal are comparng dfferent approaches to calculatng the partcpaton factor of lnear tme-nvarant systems and nvestgate ther correctness. The comparson wll be conduct based on numercal examples... Intal equatons. PARTICIPATION FACTOR Today t s nown two dfferent approaches to the partcpaton factor of state varable n mode: the approach of scentsts Perez-Arraga and Verghese, and approach of group of scentsts led by E.H. Abed. Before we consder these approaches let s brefly overvew of the hstory of partcpaton factor problematcs. The power system under study conssts of a number of N synchronous machnes connected by a large number of connectons. In the lnear approxmaton, the descrpton of ths power system can be presented wth the system of dfferental equatons: ẋ = Ax, () where x column vector of state varables, A characterstc matrx of dfferental equatons system by whch power system s descrbed n the lnear approxmaton. In the case of power system descrpton matrx A s real, that s, A * = A. In general, the matrx A has N dfferent egenvalues, some of whch are a complex conjugate: λ = σ ± jω, () where σ real part of egenvalue whch characterzes state stablty margn of power system, ω magnary part of egenvalue whch determnes fluctuaton frequency of power system mode. By stablty margn we shall bascally mean real part module of egenvalue. Left and rght egenvectors that correspond to egenvalue λ are defned by expressons: Ar r, (3) l A l.
Partcpaton factor n modal analyss of power systems stablty 99 The notaton s followng: r rght column egenvector, l left row egenvector. Left and rght egenvectors are normalzed to the symbol Kronecer: l r j = δ j. (4) The soluton of equaton () wth ntal condton x = x() s an expresson: x( t ) R e L x. (5) The notaton of expresson (5) s followng: R rght egenvectors matrx, each column of whch s the rght egenvector; L left egenvectors matrx, each column of whch s the left egenvector; Λ dagonal matrx, the man dagonal of whch contans the egenvalues of matrx A. Condton of normalzaton between the left and rght egenvectors n ths case would loo le ths: L R =. To determne partcpaton factor of -state varable n -mode you need to actually decompose -mode on the bass of the state vector. That s, we need to consder the equaton: z(t) = Lx(t). (6) Components of vector z(t) represent the evoluton of mode assocated wth the correspondng egenvalue. Substtutng the expresson (5) n equaton (6) we get the evoluton of -mode: Λt z ( t) l x e. (7) t In ths approach, the bggest problem s the choce of ntal condtons x. Then we proceed to consder the approach proposed by scentsts Perez- Arraga і Verghese. In the paper [] as ntal condtons the authors chose the rght egenvectors, x = r, whch s not qute correct. In ths case the partcpaton factor of -state varable n the -mode s determned by the formula: r p l (8) Please note that for such defnton of partcpaton factor t s a complex value n cases complex egenvalue. Ths leads to the napplcablty of ths formula as complex numbers cannot be compared wth each other. To avod ths problem we can of course slghtly modfy the expresson as follows: r p l. (9) In ths case the partcpaton factor wll always be postve real value, as t should be. Note that for the convenence partcpaton factor analyss can be normalzed to: N p. ()
Volodymyr Konoval, Roman Prytula Further we wll consder one of the newest approaches (Abed and others) for calculatng the partcpaton factor of -state varable n the formaton of -mode, whch s offered n a seres of papers [3-5]. In the paper [3] the authors use a settheoretc formulaton for calculatng the partcpaton factor of -state varable n the formaton of -mode. In the next paper [4], to avod the problem of choosng the ntal condtons probablstc descrpton of ntal condtons s appled by usng the mathematcal expectaton: l l x p E. () z z Ths formula s proposed for the cases of real and complex egenvalue λ. After calculatng the mathematcal expectaton, the authors obtaned the followng expresson of partcpaton factor of -state varable n the formaton of -mode: p Re( l ). () Re( l ) Re( l ).. Examples showng the nadequacy of expressons for determnng partcpaton factor In general, the matrx A s an arbtrary real matrx of sze N, dependng on the system, whch we descrbe usng dfferental lnear equatons. Let s study the adequacy of expressons for partcpaton factor obtaned by dfferent authors, on specfc examples. Example. Consder the two-dmensonal system, whch state vector s x(t) = [x (t), x (t)]. As a partal case of matrx A, we choose t to be the followng [4]: a b A. (3) d Values a, b, d nonzero real constants, and a d. Equaton () for ths twodmensonal case ths wll loo le: x a b x. x d x The egenvalues of matrx A equal λ = a, λ = d. After calculatng the left and rght egenvectors usng formulas (3) we obtan the followng vectors: r, r b b, l, l. (4) (d a) b a d a d T
Partcpaton factor n modal analyss of power systems stablty It s easy to see that the left and rght egenvectors are normalzed to the symbol Kronecer. Let s explore the mode assocated wth egenvalues λ and determne partcpaton factor of each component of the state vector n the formaton of mode. Usng the approach (9) we obtan: p, p. (5) Hence we see that mode formaton assocated wth egenvalues λ s determned only by the frst component of the state vector x wth a weght of. Then determne the partcpaton factor for the components of the state vector through approach [4] by the formula (6): ( a d ) p, ( a d ) b (6) b p. ( a d ) b Comparng the results for the partcpaton factor (5) and (6) we see that they dffer. A natural queston arses: whch results are correct? To answer ths queston should we should nvestgate the evoluton of mode under study that s assocated wth egenvalue λ usng expresson (7): b t z t x x e ( ). (7) a d From the formula (7) we can see that partcpaton factors are nonzero for the components of the state vector x. Expresson (7) s actually the decomposton of studed mode on the bass of states x, x, then the value that are near x s nothng other than the ampltude of weght. The weght s defned as the square of the ampltude: ( a d ) p, b a d ( a d ) b (8) b a d b p. b a d ( a d ) b Note that partcpaton factors calculated usng (8) are normalzed to. As a result, t can be concluded that the partcpaton factor (5), whch s defned by the formula (9) returns ncorrect results because from the expresson (7) for the evoluton of mode we can see that mode formaton s affected by both components of the state. Comparng expressons (6) and (8) we can conclude that the approach proposed n [4] for determnng factor partcpaton by the formula () provdes correct results. Note that matrx A has real x,
Volodymyr Konoval, Roman Prytula egenvalues, so you need to nvestgate the valdty of the approach [4] for the cases of complex egenvalues of the nput matrx A. Example. Let s study the factor partcpaton usng approach () n the case of complex egenvalues of nput matrx. For smplcty of calculatng egenvalues of matrx A we choose t as follows: c A. (9) b Values b, c constants, and b >, c >. After calculatng egenvalues of matrx A we obtan: j bc, j bc. For each egenvalue usng equaton (3) let s determne the left and rght egenvectors: j j l b c, l b c, j c b j c b r, r. () The left and rght egenvectors satsfy the normalzaton condton. To study partcpaton factor for the mode assocated wth egenvalues λ. Frst, consder the evoluton of mode under study usng the expresson (7): j b c t z t x x e ( ). () As we can see from the expresson (), state varables x and x are ncluded n the expresson () unequally, then t can be concluded that the partcpaton factors of varables x, x n the formaton of nvestgated mode are dfferent and nonzero. Then let s proceed to the calculaton partcpaton factors usng approaches (9) and (). Once agan, usng the formula (9) to calculate the partcpaton factor of state varable n mode, we obtan the followng results: p, p. () Hence we see that the weght of nfluences of state varables x and x on the formaton nvestgated mode assocated wth egenvalue λ, are equal, that contradcts the fndngs obtaned from the expresson for the evoluton of ths mode (). Ths shows the ncorrectness of approach of scentsts Perez-Arraga and Verghese for determnaton of partcpaton factor of state varable n mode for complex egenvalues of the nput matrx.
Partcpaton factor n modal analyss of power systems stablty 3 Our next step s to calculate the partcpaton factor through approach proposed by a group of scentsts led by Abed [4]. Applyng the formula () to calculate partcpaton factors n the formaton of state varables of nvestgated mode we get: p, p. (3) The fact, that ndcatng the ncorrectness of approach () s that the partcpaton factor of component x n the formaton of nvestgated mode assocated wth value λ s not equal to zero, as seen from the expresson for the nvestgated mode evoluton () and got through formula () p (3). Another proof of the ncorrectness of the formula () n case complex egenvalues s the ambguty of left and rght egenvectors as for phase: f we multply the left vector and dvde the rght vector to the same complex number, the result of normalzaton condton (4) between them does not change, but the value of left and rght vector components wll change. Indeed, let s consder the normalzaton condton (4) for -mode: l r =. Suppose z s arbtrary complex number other than zero z. Let us mae the followng changes: l l /z, r r z. Further we wll consder condton of normalzaton between vectors l and r : l r = l / z r z =. Egenvectors l and r satsfy the equaton for the egenvalues and egenfunctons (3) wth the same egenvalue λ, as vectors l and r. The dfference between egenvectors l, r and l, r s the values of ther components: the values of real and magnary parts. Therefore, actually there s an ambguty (nadequacy) of formula () for determnng partcpaton factor for cases of complex egenvalues. Thus, n the case of real egenvalues of ntal matrx the approach proposed n paper [4] (formula ()) provdes correct results for determnng partcpaton factor of state varable n mode, but for the cases of complex egenvalues of ntal matrx, ths approach provdes ncorrect results. 3. CONCLUSION We analyzed dfferent approaches to determnng partcpaton factor of state varable n mode formaton: orgnal approach [, ] and suggested n the paper [4] based on probablstc method of settng the ntal condtons. It was shown the nadequacy of these approaches for determnng the partcpaton factor of state varable n mode formaton assocated wth egenvalue of characterstc matrx of lnear dfferental equatons system. The approach of scentsts led by Abed provdes correct results only for real egenvalues of the nput matrx, but for the cases of complex egenvalues we get ncorrect results.
4 Volodymyr Konoval, Roman Prytula Thus, the problem of determnng the partcpaton factor of state varable n power system mode formaton remans mportant as partcpaton factor plays a ey role n modal analyss. Note that a great number of modern software applcatons desgned to analyze power system stablty deploy the approach of Perez-Arraga and Verghese, causng ncorrectness of modal analyss and as a result rss to the operaton of real power systems. REFERENCES [] Perez-Arraga I. J., Verghese G. C., and Schweppe F. C. Selectve modal analyss wth applcatons to electrc power systems. Part I: Heurstc ntroducton IEEE Transactons on Power Apparatus and Systems, No. 9, Vol., 37 35, 98. [] Verghese G. C., Perez-Arraga I. J. and Schweppe F. C. Selectve modal analyss wth applcatons to electrc power systems. Part II: The dynamc stablty problem IEEE Transactons on Power Apparatus and Systems, No. 9, Vol., 36 334, 98. [3] Abed E. H., Lndsay D. and Hashlamoun W. A. On partcpaton factors for lnear systems, Automatca. Vol. 36, 489 496,. [4] Hashlamoun W. A., Hassouneh M. A. and Abed E. H. New results of modal partcpaton factors: revealng a prevously unnown dchotomy, IEEE Transactons on Automatc Control, No. 7, Vol. 54, 439 449, 9. [5] Sheng L., Abed E. H., Hassouneh M. A., Yang H. and Saad M. S. Mode n Output Partcpaton Factors for Lnear Systems, Amercan Control Conference, Baltmore, 956 96,. (Receved:.. 6, revsed: 9.. 6)