Mode in Output Participation Factors for Linear Systems

Transcription

1 2010 American ontro onference Marriott Waterfront, Batimore, MD, USA June 30-Juy 02, 2010 WeB05.5 Mode in Output Participation Factors for Linear Systems Li Sheng, yad H. Abed, Munther A. Hassouneh, Huizhong Yang and Mohamed S. Saad Abstract The common definition of moda participation factors was intended to provide a measure of participation of modes in states and of states in modes for inear timeinvariant (LTI) systems. In this paper, recent wor by the authors that revisited this common definition is extended to yied definitions and cacuations for quantifying the reative contribution of system modes in system outputs. When the system outputs are simpy the system states, the mode-in-output participation factors are found to reduce to the origina modein-state participation factors. A numerica exampe is given to iustrate the issues addressed and the resuts obtained. I. INTRODUTION The concept of moda participation factors is an important component of the Seective Moda Anaysis (SMA) framewor introduced in [7] and [9]. Participation factors provide a mechanism for assessing the eve of interaction between system modes and system state variabes for inear systems. They have been used for stabiity anaysis, sensor and actuator pacement, order reduction, and coherency and custering [7], [9], [8], [2], [3]. In their study of moda participation factors, the authors of [7] and [9] seected particuar initia conditions and introduced definitions of mode in state and state in mode participation factors using those initia conditions. Based on their seected initia conditions, the authors of [7] and [9] gave a singe formua for both mode in state and state in mode participation factors. Unti recenty, it has been routiney taen for granted that the measure of participation of mode in state is identica to that of participation of states in modes. However, the recent wor [4] presented a new fundamenta approach to moda participation anaysis by using averaging over an uncertain set of system initia conditions. Based on this new approach, it has been shown in [4] that participation of modes in states and participation of states in modes shoud not be viewed as interchangeabe and a new formua for measuring the participation of states in modes This wor was supported by the Nationa Natura Science Foundation of hina (No ), and by the U.S. Nationa Science Foundation and the U.S. Office of Nava Research. L. Sheng is with Schoo of ommunication and ontro ngineering, Jiangnan University, 1800 Lihu Rd, Wuxi, , P.R.hina (emai: victory8209@sina.com).. H. Abed is with the Department of ectrica and omputer ngineering and the Institute for Systems Research, University of Maryand, oege Par, MD USA (emai: abed@umd.edu). M. A. Hassouneh is with the Institute for Systems Research and the Department of Mechanica ngineering, University of Maryand, oege Par, MD USA (emai: munther@umd.edu). H. Yang is with Schoo of ommunication and ontro ngineering, Jiangnan University, 1800 Lihu Rd, Wuxi, , P.R.hina (emai: yhz jn@163.com). M. S. Saad is with the ectrica Power and Machines Department, airo University, airo, gypt (emai: mssaad@eng.cu.edu.eg). was proposed. Furthermore, the formua for participation factors measuring participation of modes in states based on the averaging approach taen in [4] was found to coincide with the commony used formua originay introduced in [7] and [9]. Previous wor on participation factors [7], [9], [1] and [4] has focused mainy on participations associated with the state variabes of inear time-invariant systems. In many practica situations, not a system states are avaiabe for measurement and ony system outputs can be measured. The goa of this paper is to deveop a notion of participation factors measuring the participation of system modes in system outputs. Such a notion can be usefu in appications such as system monitoring, actuator pacement, custering studies, detection of coseness to instabiity and oca and coordinated contro design. Other reated concepts have been considered in the iterature. For exampe, the concept of mode observabiity factor and transfer function residues introduced by [5], were used to determine the most suitabe ocation for instaing power system stabiizers. In [6], a concept of extended eigenvector was introduced and used to identify the dominant output variabe associated with a critica mode. However, a drawbac of using these definitions is that they do not reduce to the conventiona state participation factors when the output is simpy a state variabe. Motivated by the discussion above, we focus on defining a notion of output participation factors for continuous-time LTI systems ẋ(t) = Ax(t), (1) y(t) = x(t), (2) where x R n is a vector of states, y R m is the vector of outputs, A is a rea n n matrix and is an m n matrix. This paper continues our previous wor [4], and the approach taen here foows that in our previous wor [1], [4]. In this approach, participation factors are formuated as the average reative contribution of modes in outputs over an uncertain set of system initia conditions. The new definition proposed here has the desirabe property that it reduces to the origina state participation factors definition when the outputs are simpy the system states. The paper proceeds as foows. In Section II, the origina definition of participation factors [7], [9] and the new definitions of participation factors [1], [4] are recaed. In Section III, the definition and cacuation of participation of modes in outputs are given. In Section IV, a mechanica system exampe is provided to iustrate the effectiveness of /10/$ AA 956

2 the deveoped theory. oncusions and suggestions for future wor are coected in Section V. II. BAKGROUND AND MOTIVATION In this section, the definition of participation factors for inear systems are recaed from [7], [9], [1] and [4]. onsider the inear time-invariant continuous-time system ẋ = Ax(t), (3) where x R n, and A is a rea n n matrix. In the previous studies [1], [4], [7], [9] and in this paper, the banet assumption is made that A has n distinct eigenvaues (λ 1, λ 2,..., λ n ). Let (r 1, r 2,...,r n ) be right (coumn) eigenvectors of the matrix A associated with the eigenvaues (λ 1, λ 2,...,λ n ), respectivey. Let ( 1, 2,..., n ) denote eft (row) eigenvectors of the matrix A associated with the eigenvaues (λ 1, λ 2,..., λ n ), respectivey. The right and eft eigenvectors are taen to satisfy the normaization, where δ ij is the Kronecer deta: 1, i = j δ ij =. 0, i j i r j = δ ij, (4) The soution of the dynamic system (3) satisfying the initia condition x(0) = is x(t) = e At. (5) Since the eigenvaues of A are assumed distinct, A is simiar to a diagona matrix. Using this, (5) can be rewritten as x(t) = ( i )e λit r i. (6) Therefore, the -th state x (t) is given by x (t) = ( i )e λit r i. (7) A. Origina definition of participation factors In order to determine the reative participation of the i-th mode in the -th state, the authors of [7] and [9] seected an initia condition = e, the unit vetor aong the -th coordinate axis. With this choice of, the evoution of the -th sate becomes x (t) = (r i )e i λit =: p i e λit. (8) The quantities p i := i ri (9) are taen in [7] and [9] as measures of reative participation of the i-th mode in the -th state; p i is defined in [7] and [9] as the participation factor for the i-th mode in the -th state. On the other hand, the reative participation of the -th state in the i-th mode is investigated in [7] and [9] by first appying the simiarity transformation z := V 1 x, (10) to system (3), where V is the matrix of right eigenvectors of A : V = [r 1 r 2 r n ] (11) and V 1 is the matrix of eft eigenvectors of A : 1 V 1 2 =.. (12) n Then z foows the dynamics ż(t) = V 1 AV z(t) = Λz(t), (13) where Λ := diag(λ 1, λ 2,..., λ n ), with initia condition z 0 := V 1. Hence, the evoution of the new state vector components z i, i = 1, 2,..., n is given by z i (t) = z 0 i eλit = i e λit. (14) Obviousy, z i (t) represents the evoution of the i-th mode. To determine the reative participation of the -th state in the i-th mode, the authors of [7] and [9] seected an initia condition = r i, the right eigenvector associated with λ i. With this choice of initia condition, the evoution of the i-th mode becomes [ n ] [ n ] z i (t) = i r i e λit = e λit = p i e λit. (15) =1 i ri =1 Based on (15), the authors of [7] and [9] proposed the formua p i = i ri (16) as a measure of the reative participation of the -th state in the i-th mode. Note that this is the same formua as (9) which was proposed as a measure of participation of modes in states. B. Recenty proposed new definition of participation factors The authors of [1] and [4] presented a new approach to defining moda participation factors for the inear system (3). This approach invoves taing an average or a probabiistic expectation of a quantitative measure of reative moda participation. The average is performed with respect to a possibe vaues of the initia state vector and is assumed to be unnown but to ie in a nown set or satisfy a nown probabiity distribution. Since (9) and (16) provide identica formuas for participation of modes in states and participation of states in modes, respectivey, the same notation p i was used for both types of participation factors and the participation of modes in states and the participation of states in modes have been viewed as interchangeabe. However, the recent wor of [4] demonstrated through simpe exampes that the participation of modes in states and the participation of states in modes shoud not in fact be viewed as interchangeabe and the origina state-in-mode participation factors formua is inadequate. Moreover, the authors of [4] proposed a new definition and 957

3 cacuation to repace the existing state-in-mode participation factors formua, whie the previousy existing participation factors formua shoud be retained but viewed ony in the sense of mode-in-state participation factors. Next, we reca from the previous wor in [1] and [4] the new definitions of mode-in-state and state-in-mode participation factors. Denote by the expectation operator. The genera definition for the participation factor p i measuring participation of mode i in state x is p i := ( i )r i, (17) where the expectation is evauated using some assumed joint probabiity density function f( ) for the initia condition uncertainty. Under the assumption that the initia condition components 1, x0 2,...,x0 n are independent with zero mean, definition (17) is shown to reduce to [1], [4] p i = i ri (18) which is the same expression originay introduced by [7], [9] for the participation of the i-th mode in the -th state. For a compex eigenvaue λ i, the mode can be viewed as consisting of the combined contributions from λ i and its compex conjugate eigenvaue λ i [4]. In this case, an aternative expression for the participation factor of a compex mode associated with λ i in state x is given by [4] p i = 2Re i r i. (19) The participation factor measuring contribution of the -th state in the i-th mode is defined in [4] as i x0, if λ zi π i := 0 i is rea ( i +i )x 0 (20), if λ i is compex z 0 i +z0 i whenever this expectation exists. Note that in (20), the notation z 0 i means z i (t = 0) = i and and the asteris denotes compex conjugation. In order to obtain a simpe formua from definition (20), the authors of [4] assumed that the probabiity density function f( ) is such that the components 1, 2,..., n are jointy uniformy distributed over the unit sphere in R n centered at the origin. Under this assumption, a new formua (21), was given based on definition (20) for the participation factor for the -th state in the i-th mode π i = i ri + n j=1,j i i j ( i ) T rj i ( i ) T. (21) Under the initia condition uncertainty assumption based on which (21) was obtained, the participation factor for the i-th mode in the -th state is equa to the participation factor for the -th state in the i-th mode (i.e., π i = p i ) if the eigenvectors of the system matrix A are mutuay orthogona, which is a very restrictive case. III. OUTPUT PARTIIPATION FATORS In this section, definitions and cacuations are given for participation factors measuring the reative contribution of modes in outputs for the system (1)-(2). The soution to (1) which is given in (6) is repeated here for convenience: x(t) = ( i )e λit r i. (22) Using (22), (2) can be rewritten in the form y(t) = x(t) = ( i )e λit r i. (23) From (23), the -th output y (t) is given by y (t) = n ( i )e λit r i, (24) where is the -th row of. The participation factor measuring participation of the i-th mode in the -th output y is defined as foows. Definition 1. For the continuous LTI system (1)-(2), the participation factor for the i-th mode in the -th output is p y i := ( i ) r i y 0 ( i ) r i +( i ) r i y 0, if λ i is rea, if λ i is compex (25) whenever this expectation exists. Note that in (25), the notation y 0 means. To obtain a simpe cosed-form expression for the mode in output participation factors p y i using (25), we need to find an assumption on the probabiity density function f( ) governing the uncertainty in the initia condition that is intuitivey appeaing and aows us to expicity evauate the integras inherent in the definition. In the reminder of this section, we mae the foowing assumption. Assumption 1. Assume that the units of the state variabes have been scaed to ensure that the probabiity density function f( ) is such that the components 1, x0 2,...,x0 n are jointy uniformy distributed over the unit sphere in R n centered at the origin: f(, x ) = 0 1, (26) 0, otherwise. The constant in Assumption 1 is chosen to ensure the normaization f( )d = 1. (27) 1 The vaue of the constant can be determined by evauating the integra in (27) using f( ) in (26): f( )d = d 1d 2...d n = V n = 1, 1 1 (28) 958

4 where V n is the voume of the unit sphere in R n. The constant is then given by = 1 V n. (29) The foowing Lemma wi be used beow. Lemma 1. [4] For vectors a n, b R n with b 0 we have a T x b T x d nx = at b b T b V n, x 1 where d n x denotes the differentia voume eement dx 1 dx 2 dx n, and V n is the voume of a unit sphere in R n which is given by 2, n = 1, V n = π, n = 2, 2π n V n 2, n 3. A. Rea-mode-in-output participation factors To determine the participation of a mode associated with a rea eigenvaue λ i in the -th output, substitute y 0 = in the first case of (25) and write i = i + i, where i is the projection of i in the direction and i is the projection of i in the direction orthogona to. Then ( p y i = r i i + i ) = r i i + i. (30) Moreover, i can be written in terms of i and as foows: i = i i ( ) T i = i ( ) T 2. (31) Substituting (31) into (30) yieds p y i = r i i ( ) T 2 + i = r i i ( ) T 2 + r i i. (32) In genera, the second term in (32) does not vanish. This is true even in case the components 1, x0 2,...,x0 n representing the initia conditions of the state are assumed to be independent. This is due to the fact that the second term invoves the components which need not to be independent even under the assumption that the are independent, due to the transformations i and. We now use Lemma 1 to simpify the expression (32) for the participation factor ( for ) the i-th (rea) mode in the -th T output. Denote a = i and b = ( ) T and note that a is orthogona to b. Using Lemma 1, the expectation term in (32) reduces to i i = 1 f(x0 )d a T = b T dx0 1 = at b b T b V n = 0. (33) Therefore, the second term in (32) vanishes under Assumption 1 and the rea-mode-in-output participation factors are given by p y i = r i i ( ) T 2. (34) Another way to obtain formua (34) under Assumption 1 is as foows. Reca from Definition 1, the genera averagingbased definition for mode-in-output participation factors for the case of a rea eigenvaue: p y i = ( i ) r i = r i i. (35) Denote a = ( i ) T and b = ( ) T. Using Lemma 1 and the normaization V n = 1, (35) reduces to p y i = r i at b b T b = r i i ( ) T 2, (36) which is exacty (34). B. ompex-mode-in-output participation factors To determine the participation factor for a compex mode, i.e., a mode associated with a compex conjugate pair of nonrea eigenvaues λ i and λ i, in an output, we use the second case of (25): ( i ) r i + ( i ) r i = p y i x 0 = r i i + r i i. (37) Write i = i + i and i = i + i, where i, i are the projections of i and i in the direction, respectivey, and i, i are the projections of i and i in the direction orthogona to, respectivey. Substituting i = i + i and i = i + i in (37) gives p y i = r i i + i + r i i + i. (38) Furthermore, i and i can be written in terms of i, i and as foows: i = i ( ) T 2, (39) i = i ( ) T 2. (40) 959

5 Substituting (39) and (40) into (38) yieds p y i = r i i ( ) T 2 + i + r i = r i i ( ) T 2 i ( ) T + r i i ( ) T i i + r i + r i i = r i i ( ) T 2 + r i i ( ) T 2 + r i i + r i i.(41) It is easy to show that the ast two terms in (41) vanish as was done in (33). Hence, (41) reduces to p y i = r i i ( ) T ( ) T = + r i i ( ) T ( ) T 2 2Re ( r i i ( ) T). (42) Next, we give another way to obtain formua (42) under Assumption 1. Reca from Definition 1, the genera averaging-based definition for mode-in-output participation factors for the case of compex eigenvaues: ( i ) r i + ( i ) r i = p y i x 0 = r i i + r i i Using Lemma 1, it easy to show that i = i ( ) T i ( ) T, = i ( ) T ( ) T. Thus, (43) can be rewritten as p y i = r i i ( ) T ( ) T = + r i i ( ) T ( ) T. (43) 2 2Re ( r i i ( ) T), (44) which is exacty (42). Note that in formua (44), the participation factors for the -th output depend on the matrix ony through its -th row. An immediate impication of this is that when an output coincides with a state variabe, the moda participations in the output equa the traditiona moda participations in the state. IV. NUMRIAL XAMPL The foowing exampe is borrowed from [10]. onsider the transation mechanica system depicted in Fig. 1, where d 1 (t) and d 2 (t) denote the dispacements of mass 1 and mass 2, respectivey, from the static equiibrium. The system Fig. 1 Mechanica system. parameters are the masses m 1 and m 2, viscous damping coefficients c 1 and c 2, and the spring constants 1 and 2. A state space representation is obtained by defining the system states as x 1 (t) = d 1 (t), x 2 (t) = d 1 = ẋ 1, x 3 (t) = d 2 (t), x 4 (t) = d 2 = ẋ 3. According to [10], the system dynamics can be described by a inear time-invariant differentia equation where A = ẋ(t) = Ax(t), (45) m 1 c 1+c 2 2 c 2 m 1 m 1 m c 2 m 2 m 2 2 m 2 c 2 m 2. With the system parameters seected as m 1 = 39g, m 2 = 17g, c 1 = 19Ns/m, c 2 = 33Ns/m, 1 = 374N/m, 2 = 196N/m, the system state dynamics matrix A becomes A = The eigenvaues of A are λ 1,2 = ± j2.315 and λ 3,4 = ±j We denote the modes associated with the eigenvaues as z 1 (t) and z 2 (t), respectivey. The dominant modes are the ones associated with the compex conjugate pair of eigenvaues cosest to the imaginary axis, i.e., λ 1,2. Modes associated with eigenvaues cose to the imaginary axis are of considerabe interest since they can be used as an indication of coseness to system instabiity. Hence, our emphasis wi be on z 1 (t). The right (coumn) eigenvectors of the system matrix A associated with λ 1 and λ 3 are r 1 = j j j , r 3 = j j j , respectivey. Since λ 2 = λ 1 and λ 4 = λ 3, we have that r 2 = r 1 and r 4 = r 3, where asteris represents compex conjugation. The eft (row) eigenvectors of the system matrix A associated with λ 1 and λ 3 are 1 = j j j j T, 3 = respectivey, and 2 = 1 and 4 = j j j j T, 960

6 TABL I MOD IN OUTPUT PARTIIPATION FATORS FOR SYSTM (45), (46) BASD ON FORMULA (42) y 1 p y 11 = py 12 = y 2 p y 21 = py 22 = y 3 p y 31 = py 32 = TABL II MOD IN OUTPUT PARTIIPATION FATORS FOR SYSTM (45), (46) BASD ON AVRAGING INITIAL ONDITIONS y 1 p y 11 = py 12 = y 2 p y 21 = py 22 = y 3 p y 31 = py 32 = TABL III MOD IN OUTPUT PARTIIPATION FATORS FOR SYSTM (47), (48) ỹ 1 ỹ 2 ỹ 3 BASD ON FORMULA (42) pỹ 11 = pỹ 12 = pỹ 21 = pỹ 22 = pỹ 31 = pỹ 32 = TABL IV MOD IN STAT PARTIIPATION FATORS FOR TH SYSTM (45) BASD ON FORMULA (19) x 1 p 11 = p 12 = x 2 p 21 = p 22 = x 3 p 31 = p 32 = x 4 p 41 = p 42 = Next, the output of the system (45) is chosen as where y(t) = x(t), (46) = [ The mode in output participation factors p y i evauated using formua (42) are given in Tabe I. To vaidate this formua numericay, a set of different initia conditions of the state vector that ie in the unit sphere is randomy generated and the reative contribution of the mode in the output for each of these initia conditions is evauated. The average of a mode in output participation from a initia conditions is computed as shown in Tabe II. The vaues of the mode in output participation factors of Tabe II are in good agreement with the vaues of Tabe I obtained based on formua (42). To verify that the formua of mode in output participation factors (42) reduces to the formua of mode in state participation factors (19), we consider the same system with outputs taen as system states ] ẋ(t) = Ax(t), (47) ỹ(t) = x(t), (48) where A is the same matrix defined in (45) and [ ] 0 = For this output matrix, the mode in output participation factors are given in Tabe III whereas the mode in state participation factors (19) are shown in Tabe IV. eary, the output participation factors reduce to the state participation factors when the output is simpy a state variabe. V. ONLUSION In this paper, we have presented an approach to quantifying the participation of system modes in system outputs for. inear time-invariant systems. We have proposed a definition and a formua for these output participation factors by using averaging over an uncertain set of system initia conditions. An exampe was given to demonstrate the usefuness of the resuts obtained. The case of participation of outputs in modes requires a different anaysis and wi be considered in future wor. RFRNS [1].H. Abed, D. Lindsay, and W.A. Hashamoun, On participation factors for inear systems, Automatica, Vo. 36, pp , [2] L..P. da Siva, Y. Wang, V.F. da osta, and W. Xu, Assessment of generator impact on system power transfer capabiity using moda participation factors, I Proceedings on Generation, Transmission and Distribution, Vo. 149, No.5, pp , [3] I. Genc, H. Schatter, and J. Zaborszy, ustering the bu power system with appications towards Hopf bifurcation reated osciatory instabiity, ectric Power omponents and Systems, Vo. 33, No.2, pp , [4] W.A. Hashamoun, M.A. Hassouneh, and.h. Abed, New resuts of moda participation factors: reveaing a previousy unnown dichotomy, I Transactions on Automatic ontro, Vo. 54, No. 7, pp , [5] N. Martins, and L.T.G. Lima, Determination of suitabe ocations for power system stabizers and static VAR compensators for damping eectromechanica osciations in arge scae power systems, I Transactions on Power Systems, Vo. 5, No. 4, pp , [6] N. Mithuananthan,.A. anizares, J. Reeve, and G.J. Rogers, omparison of PSS, SV, and STATOM controers for damping power system osciations, I Transactions on Power Systems, Vo. 18, No. 2, pp , [7] I.J. Pérez-Arriaga, G.. Verghese, and F.. Schweppe, Seective moda anaysis with appications to eectric power systems, Part I: Heuristic introduction, I Transactions on Power Apparatus and Systems, Vo. 101, No. 9, pp , [8] I.J. Pérez-Arriaga, G.. Verghese, F.L. Sancha, and F.. Schweppe, Deveopments in seective moda anaysis of sma-signa stabiity in eectric power systems, Automatica, Vo. 26, No. 2, pp , [9] G.. Verghese, I.J. Pérez-Arriaga, and F.. Schweppe, Seective moda anaysis with appications to eectric power systems, Part II: The dynamic stabiity probem, I Transactions on Power Apparatus and Systems, Vo. 101, No. 9, pp , [10] R.L. Wiiams II, and D. A. Lawrence, Linear State-Space ontro Systems. Hoboen, NJ: Wiey,