Variable Projection Method for Power System Modal Identification

Transcription

1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 6, NOVEMBER Variable Projection Method for Power System Modal Identification Alexander R. Borden, Student Member, IEEE, and Bernard C. Lesieutre, Senior Member, IEEE Abstract Common power system disturbances are characterized by oscillations that either damp out, sustain or grow. This oscillatory response can be measured and analyzed to extract modal information of the system. In this paper we apply a nonlinear least-squares optimization method known as the variable projection method to power system models and data. The variable projection method is compared against several linear methods such as Prony analysis, Matrix-Pencil and ERA, and the results suggest the nonlinear least-squares approach is superior. Index Terms ERA, matrix pencil, modal analysis, power systems, Prony analysis, variable projection. I. INTRODUCTION T HE dominant oscillating modes of a power system characterize its stability properties and reveal important information about the system s behavior. Electromechanical disturbances introduced to the system create the oscillatory responses, and ring-down measurements collected during the disturbance can be analyzed to estimate system modal content and mode shape. Here the modes of a system are characterized by having amplitude, frequency, damping and phase. Accurate mode estimation may help guide control implementations for stabilizing poorly damped modes, and real-time mode estimation permits situational awareness of system stability. In power system studies, Prony analysis is the traditional tool used for estimating modal content of oscillatory disturbance data. Early adoption of Prony s method in power system problems can be seen in [1], where it is applied to several Western Interconnect ringdown events. Since that time, Prony s method has been incorporated into commercial power system analysis software tools [2], [3]. More recently, two other modal analysis techniques have emerged: the Matrix Pencil method and the Eigensystem Realization Algorithm (ERA). Prony analysis, matrix pencil, and ERA all estimate a function consisting of a finite sum of damped or undamped sinusoids and exponentials that approximate a data-set. The Manuscript received April 04, 2013; revised August 09, 2013 and January 10, 2014; accepted February 23, Date of publication March 20, 2014; date of current version October 16, This work was supported under award DE-AC02-05CH11231 through the Consortium for Electric Reliability Technology Solutions (CERTS) under subcontract Paper no. TPWRS The authors are with the Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI USA ( borden@wisc.edu; lesieutre@engr.wisc.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TPWRS estimated solutions of (un)damped sinusoids and exponentials, have the equivalent solution structure of a linear dynamical system. Each method is performed as a two-step procedure: first, modal damping and frequency is estimated, then second, modal amplitude and phase is computed. The solution for modal amplitude and phase is dependent on the previously obtained solution for damping and frequency. It can be shown that these two-step procedures are not optimal in the least-squares sense. In contrast, a different method uses a nonlinear least-squares optimization method to directly match a model to data without directly imposing an underlying linear system assumption. The variable projection method estimates modal content and optimally captures the dynamics present in the data. The variable projection method was originated in 1973 by Golub and Pereyra [4], [5]. They observed that in many nonlinear optimization problems there are optimization variables that appear nonlinearly and optimization variables that appear linearly. It was observed that the linear variables have a dependent relationship corresponding to the optimal solution for the nonlinear variables. Golub and Pereyra identified the separability between the two variable types and called these problems separable least squares problems. They obtained the variable projection functional by eliminating the linear variables from the optimization problem. In addition, they outlined the gradient equation of the variable projection functional; allowing for the use of gradient based line-search optimization methods. In 1977, Bolstad released a FORTRAN implementation of the variable projection method with the varpro.f code. Since then, the code and method have been improved, with a thorough examination of contributions presented in [6] and [7]. A thirdparty MATLAB implementation is available in the varpro.m code [8], which has found its use among a wide area of disciplines [6], [7]. This paper motivates the use of the variable projection method for use in power system problems that have traditionally relied on Prony analysis or other methods for estimating modal content of data. In Section II, a brief overview of traditional modal analysis techniques is given, along with discussion on multi-signal analysis. In Section III, a mathematical overview of the variable projection method is presented. Section IV demonstrates the method on two example disturbance data-sets. In the first example, Section IV-A, analysis is performed on a disturbance simulated in a 16-machine 68-bus test system. In the second example, Section IV-B, Western Interconnection disturbance data is used to demonstrate the variable projection method s ability for analyzing real power system data. Concluding comments and observations are made in Section V IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 2614 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 6, NOVEMBER 2014 II. TRADITIONAL MODAL ANALYSIS METHODS In this section, three traditional modal analysis methods will be discussed: Prony analysis, matrix pencil and ERA. All three methods seek to approximate a function consisting of (un)damped sinusoids and exponentials as in (1), in order to match an oscillatory data-signal : Computationally, the approaches treat the waveform as the response of a linear system whose coefficients are calculated using temporal correlations in the data. The subsequently calculated natural damping and frequencies of the linear system correspond to in (1). The mode shape amplitude and phase correspond to the complex coefficients. In practice, the data comprises equally-spaced discrete-time sample points at,where is the time-sample period. The discrete-time representation of (1) may be represented as shown in (2): The signal is a finite summation of mode pairs, where is the residue coefficient associated with the discrete-time pole.notetheseriessummationto in (1), as opposed to the series summation to in (2). It follows that due to the cancellation from complex conjugate mode pairs and asshownin(3): If there are number of pairs of complex conjugate modes and number of real modes, then and. A. Solving for Linear Model and Discrete-Time Poles The data-signal may require preconditioning before using Prony analysis, matrix pencil or ERA for estimating discretetime poles. The data-signal can be preconditioned so that its dc-offset is subtracted; which by doing so, the methods are alleviated from approximating a zero frequency mode, i.e., a dc-offset mode. A reasonable choice for dc-offset subtraction is the signal s final value, as steady-state is presumed to be approached; however, this conjecture may worsen for unstable or lightly damped data. Additional preconditioning is required if the data-signal is not uniformly spaced by.ifso,then interpolation must be performed to enforce uniformly spaced time-samples; which is required because analysis will be done in the discrete-time domain. Finally if necessary, a noise reduction filter may be applied for overly noisy data. Prony analysis, matrix pencil and ERA are all performed as two-step procedures. In the first-step, the discrete-time poles of the system are estimated, thereby solving for a linear model to match the data. Each of the three methods performs this firststep differently. 1) Prony Analysis: TheuserofProny smethodmustprespecify the number of modes to be estimated in the waveform. (1) (2) (3) The procedure to estimate the discrete-time poles can be reviewed in [9], where the first-step of Prony s method is justified by the linear prediction property of linear systems. 2) Matrix Pencil Method: Literature on the matrix pencil method may be reviewed in [10] and [11]. With this method, a Hankel matrix is formed from the data points in, and singular value decomposition (SVD) is performed on the Hankel matrix. Typically the number of singular values greater than some user-defined threshold are kept, effectively determining the model order. The remaining singular values are discarded. From there, the method goes on to estimate the discrete-time poles. 3) Eigensystem Realization Algorithm: Literature on ERA may be reviewed in [12] and [13]. The ERA method for computing discrete-time poles is also briefly detailed in Section II-C for multi-signal analysis. B. Solving for Continuous-Time Function With the discrete-time poles approximated using one of the three methods described in Section II-A, the system s estimated damping and frequencies can be obtained by. The remaining computations will be performed in the continuous-time domain, where evenly spaced time samples are no longer necessary. Rewrite (1) as shown in (4): A real-valued mode has, and therefore only contributes to the summation in (4). The summation contains basis functions consisting of (un)damped cosines, sines and exponentials. With the basis functions, an matrix is created, where is the number of time-samples in the data-signal : The columns of matrix consist of the basis functions, evaluated at all points in time, and an additional vector of ones to be used as a dc-offset basis function. A 1 vector will be solved for, where the elements in represent the basis function coefficients. The continuous-time approximating signal is. To minimize the least-squares error in, a pseudo-inverse operation is performed in (5): Mode shape amplitudes and phases can be extracted from, and the approximating function in (1) is completed, albeit with an additional dc-off term to improve fit. In testing the three linear methods, it was observed that ERA and matrix pencil performed most reliably, with ERA typically achieving the minimum least-square error most of the time. C. Multi-Signal Analysis All three methods can be extended for simultaneously analyzing multiple data-signals for a particular disturbance event. (4) (5)

3 BORDEN AND LESIEUTRE: VARIABLE PROJECTION METHOD FOR POWER SYSTEM MODAL IDENTIFICATION 2615 As detailed in [14], multi-signal Prony analysis tends to improve modal estimate accuracy. Multi-signal ERA can be performed by the following procedure. 1 Multiple data-signals are collected, for. A Hankel matrix is created from the discrete-data, and a shifted Hankel matrix is created from for. The Hankel matrices are stacked as follows: The SVD of. is computed; an economy size SVD will do:. In (7), was termed the variable projection functional by Golub and Pereyra [4]. For modal analysis problems, the solution to (7) gives the optimal fit to the data-set for a predefined number of modes. The method is not limited to uniformly spaced time-samples, as analysis is done completely in the continuous-time domain. The method is also not limited to basis functions of (un)damped sinusoids, and, and exponentials. Rather the inclusion of any basis function is possible, such as polynomial basis functions for data detrending [15]. Example polynomial trending basis functions could be linear, quadratic or a constant offset. The variable projection functional s gradient equation can also be determined, where the th element equals for : The largest singular values are kept, and the state matrix is computed by. The eigenvalues of are the discrete-time poles discussedinsectionii-a. The gradient can thus be determined once the matrix is calculated, where and its th column is : (8) Jacobian III. VARIABLE PROJECTION METHOD (9) The basis function coefficients in (5),,areentirely dependent on the discrete-time poles solved in the prior step. Therefore is not optimally minimized in the two-step process, as it is biased by the previous solution for damping and frequencies, and. The optimal solution can be obtained by simultaneously solving for, and with a nonlinear optimization method. In relation to the variable projection method, the damping and frequencies, and, are the independent nonlinear parameters and the coefficients are the dependent linear parameters. These variables types are separable as will be shown in Section III-A [16], [17]. A. Variable Projection Functional and Gradient To begin the method, define the set of parameters as. Define the set of basis function vectors, which are nonlinear functions of evaluated at all points in time. The matrix is constructed from the basis function vectors, so that the approximating data-signal is. The residual vector is, which can be expressed as a linear least-squares optimization problem for agiven,asin(6): Golub and Pereyra showed the th column of the Jacobian matrix, shown in (9), can be computed by (10) [4], [5]: (10) For ease of notation, the parameter will be dropped from. The full Moore-Penrose pseudoinverse is not necessary for the calculation; instead the symmetric generalized inverse satisfying and suffices. Denote the projector on the orthogonal complement of the column space as. The conversion between (9) and (10) is proven using the following three observations [4], [5]. 1 (6) Substituting recasts the linear least-squares problem in (6) as a nonlinear least-squares optimization problem, solving for parameters,asin(7): (7) The outlined multi-signal ERA method was not found in literature, but was implemented by the authors with successful results.

4 2616 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 6, NOVEMBER 2014 Therefore TABLE I WEIGHTED ERROR PER SIGNAL TYPE and some of the system s actual eigenvalues are compared to estimated damping and frequencies. In Section IV-B, modal analysis is applied to Western Interconnect disturbance data. With the Jacobian matrix calculated, the gradient of the variable projection functional can be determined by (10) [4], [5]. By having a closed-form expression for the gradient, a line-search method can be employed to solve (7). B. Remarks on Optimization Most nonlinear optimization methods require parameter initial conditions to be specified in order to begin a line-search method [18]. One choice of initial conditions for damping and frequency parameters would use the resultant output from one of the three modal analysis methods detailed in Section II. Basic implementation of the variable projection method is available in the varpro.m code found in [8]. The user must specify initial conditions and must provide equations for both the basis functions and their partial derivatives with respect to. The code can be modified to adjust solver options, and furthermore the authors of this paper modified the code to include multi-signal analysis. For multi-signal variable projection, the objective function in (7) is modified. The summed least-squares error over all signals becomes the objective function as shown in (11). To avoid biasing signals of varying magnitude, each signal is multiplied by a scaling multiplier. One suggested multiplier is,where is the standard deviation of each signal. By weighting each signal by one over its standard deviation, each of the weighted signals will have standard deviation equal to 1. Selecting appropriate scaling multipliers is important, as changes to the cost function in (11) can impact optimization results: (11) The multiple signals belong to the same system, so it is assumed they share the same natural modes in. Consequently, no additional optimization variables are introduced for multi-signal analysis. IV. EXAMPLES In this section, two example data-sets are analyzed. First, a disturbance is simulated in a 16-machine 68-bus test system, A. 16-Machine 68-Bus System The 16-machine 68-bus test system is a simplified New England/New York interconnected system [19]. The system has 283 state-variables. The 16 synchronous machines are represented by sixth-order dynamical models, along with a first-order exciter and third order turbine governor for each machine. There are 29 induction motor loads represented by third-order dynamical models. Lastly there are 12 power system stabilizers represented by third-order dynamical models. A disturbance is simulated in the system using Power System Toolbox [20]. At time, a three-phase fault is applied at bus-1, with bus-2 at the far end of the line. The fault is cleared at bus-1 at, and is cleared at bus-2 at.the simulated response of the synchronous machine angles and speeds and the induction motor slips are recorded as data. Each signal is weighted by the inverse of their standard deviations, and the weighted signals are input to multi-signal ERA with the number of modes specified as plus a zero eigenvalue for a dc-offset. The solution from ERA is used as initial conditions for multi-signal variable projection. The time-frame in the analysis is from to, with a time-step. The total least-squares error obtained by ERA was , and the error obtained by variable projection was Table I breaks down the contributing errors by signal type, where each signal is weighted by one over its standard deviation. In Table I, the error per signal type was calculated by summing each signals contributions to total error, for signals of the same type. For example, there are 16 angles. Each angle is weighted and the summation of the 16 signal contributions to total error equals for ERA and for variable projection. The total error is the squared 2-norm summed across all signals, which is equivalently the objective function in (11). Fig. 1 displays plots of the 16 synchronous machine angles and speeds and the 29 induction motor slips. For ease of display, each signal has its dc-offset subtracted to view oscillatory behavior better. Fig. 1 visually compares results obtained by ERA and variable projection against the actual data. It can be seen both ERA and variable projection obtain a close fit to the data. Fig. 2 displays weighted error versus time for ERA and variable projection for signal types, and slip.theerrorinfig.2 is computed as the absolute value of the difference between the

5 BORDEN AND LESIEUTRE: VARIABLE PROJECTION METHOD FOR POWER SYSTEM MODAL IDENTIFICATION 2617 Fig. 1. Weighted estimated signals from variable projection and ERA versus actual signals. Fig. 2. Weighted error per signal type versus time. estimated signal versus the actual data, summed across signals of the same type. For example, there are 16 angles.thesummation of the absolute value error versus time for all 16 angles is plotted for ERA and variable projection in the left subplot in Fig. 2. In Fig. 2 the overlap between the error from ERA and variable projection is shown in black. Estimated percent damping and frequencies are loosely compared against the actual eigenvalues of the post-fault system in Table II. The actual system has 283 eigenvalues, but only 8 modes were specified for the modal analysis methods. A word of caution is offered at the end of this section on comparing the mode estimations to the actual eigenvalues for nonlinear systems such as in this example. Table II compares percent damping and frequencies estimated by variable projection and ERA against the closest matching of the system s 283 actual eigenvalues. As can be seen the two methods located some different modes than one another. Variable projection seems to have closely converged onto frequencies of the actual eigenvalues. It can be seen TABLE II ESTIMATED %DAMP. AND FREQ. [HZ] VERSUS ACTUAL EIGENVALUES that variable projection converged onto modes that were not estimated by ERA at 1.15 Hz and 1.67 Hz, in order to minimize error in the optimization. ERA also located a mode around 0.93 Hz, where this frequency does not closely match any of the 283 actual eigenvalues. The remaining estimated modes are mostly similar for both methods. This example has demonstrated how multi-signal variable projection can be used for estimating modal content by minimizing the least-squares error to a data-set.

6 2618 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 6, NOVEMBER 2014 Fig. 3. Weighted estimated signals from variable projection and ERA versus actual signals. Fig. 4. Weighted error per signal type versus time. As a cautionary point, all of the estimated modal damping and frequencies should not be expected to exactly match all of the system s actual eigenvalues. For example, in the 16-machine 68-bus system of Section IV-A, not only does the system have 283 states, but the dynamic response of the system is nonlinear. The modal analysis methods approximate this high-order nonlinear response using a subset of just 8 modes. B. Actual Disturbance Data In this section, a large Western Interconnection disturbance is analyzed. A total of 44 signals were collected for analysis, with voltages [kv] and frequencies [Hz] recorded at 14 buses and power flow [MW] recorded along 16 transmission-lines. The disturbance happens around. The mean timestep in the data was. The time-frame used for analysis is to. The 44 signals are weighted by the inverse of their standard deviations and input to multi-signal ERA with number of modes specified as, plus a zero eigenvalue for a dc-offset. The solution from ERA is used as initial conditions for multi-signal variable projection. Table III breaks down the contributing errors by signal type, where each signal is weighted by the inverse of their standard deviations. Again, variable projection achieved a lower leastsquares error of compared to from ERA. Fig. 3 displays plots of the weighted bus voltages and frequencies and the transmission-line power flows.thesignalsare plotted with their dc-offsets subtracted to view oscillatory behavior better. Fig. 4 displays the absolute valued errors in the three signal types versus time. It can be seen in Fig. 3 that the estimated signals for variable projection and ERA closely match the actual signals; however, Fig. 4 reveals larger error with ERA than with variable projection. The 7 estimated modes are compared in Table IV, where there are 3 complex conjugate modes and 1 real mode for both methods. The actual eigenvalues of the system are not known,

7 BORDEN AND LESIEUTRE: VARIABLE PROJECTION METHOD FOR POWER SYSTEM MODAL IDENTIFICATION 2619 TABLE III WEIGHTED ERROR PER SIGNAL TYPE TABLE IV ESTIMATED %DAMPING AND FREQUENCIES [HZ] since the analysis uses real PMU data; thus there is no benchmark to compare the estimated percent damping and frequencies to. Two of the complex modes in Table IV are comparable for both methods. ERA estimated a complex mode around 0.44 Hz that variable projection did not. Variable projection instead located a slow trending mode around 0.04 Hz. This mode does not even complete one cycle within the time-frame of the analysis. In this example, variable projection again achieved a lower least-square error than ERA for an equivalent number of modes. This result should always be expected so long as variable projection is initialized with good initial conditions. If the modal dampingandfrequencyresultsestimatedbyera(oranylinear method) are input to variable projection as initial conditions, then variable projection will never do worse in terms of leastsquares error to the data. If a linear modal analysis method predicts the perfect solution for some number of modes, then variable projection will show the gradient at that solution is zero, i.e., a local minimum was found. If the gradient at that solution is nonzero, then the variable projection method will iterate towards a better solution. The optimization is nonconvex; therefore its ability to find a global rather than local minimizer is dependent on initial conditions provided to the method. In the two examples from this section, it is recognized that increasing the number of modes to fit the data will result in a lower least-squares error. The additional modes typically fitnoiseand negligible deviations in the data, and thus increasing the number of modes does not necessarily improve confidence in the estimations of dominant modes which are of most interest. The number of modes for the two examples in this section were chosen to not only demonstrate variable projection, but also a low enough number of modes were used to allow visual discrepancies in the estimations in Figs. 1 and 3, particularly more so with the ERA estimations. This is not to discredit ERA, as the variable projection optimization relies on quality initial conditions, such as those provided by ERA, to locate a global or local minimum. V. CONCLUSION This paper has outlined a nonlinear least-squares optimization method, called the variable projection method, for estimating modal content of oscillatory data. The authors motivate the use of the variable projection method for use in power system problems where linear modal analysis methods have traditionally been used for modal identification. Two example data-sets were presented. The first example examined a simulated disturbance in a 16-machine 68-bus test system. Estimated damping and frequencies from ERA and the variable projection method were compared to the actual eigenvalues of the system, and it was shown the variable projection method located modes necessary for minimizing error that ERA did not find. The second example examined a Western Interconnection disturbance event, and the data was analyzed by ERA and by variable projection. In both examples, variable projection obtained a lower least-squares error to the data compared to ERA. This result should be expected as long as the variable projection method is provided with quality initial conditions, such as a solution obtained from ERA or the matrix pencil method. The variable projection method offers several advantages over other methods discussed in this paper. The method does not require data for analysis to be evenly spaced in time; and by observation of the authors, the method appears relatively insensitive to time-sampling rate. The method allows for the inclusion of any possible basis function, such as polynomial basis function for data detrending. Solutions with repeated eigenvalues can be enforced with two basis functions having the same damping and frequency, but with one of the basis functions multiplied by time. Further investigation is required for including other nonlinear basis functions such as those having modulated amplitudes and frequencies. The user of the variable projection method has tight control on the type of solution obtained. For example, the user must not only specify the number of modes, but furthermore must specify what number of the modes are complex and real valued. Additional control can be achieved by constrained optimization if desired, where upper and lower bounds are enforced on some of the damping and frequency optimization variables. This type of control allows for specific targeting of modal solutions by an intuitive user. An extension to the variable projection method not discussed in this paper, would enforce high order interaction between natural modes. Linearization is the standard analysis method for approximating solutions to systems of nonlinear differential equations. However the theory of normal form analysis can be used to better approximate solutions. As discussed in [21], a normal form solution structure can be enforced in modal analysis optimization problems. The theory of normal form analysis lends itself well to working with the variable projection method, and this approach could potentially estimate modes of nonlinear disturbance data more accurately. This remains an area of future work. In our analysis we have not directly focused on the impact of noise on our estimates. This is due to the high signal-to-noise ratio observed in ringdown data for large disturbances for which

8 2620 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 6, NOVEMBER 2014 this approach is designed. We note that curve-fitting techniques are not suited for ambient, noisy data analysis. For those applications a correlation based approach must be used. In addition to Prony, the matrix pencil method and ERA. The similar TLS-ES- PRIT [22] method may be useful for that purpose. ACKNOWLEDGMENT The authors would like to thank D. Kosterev and J. Gronquist of Bonneville Power Administration for the valuable discussions and supply of system data for analysis. REFERENCES [1] J. F. Hauer, C. J. Demeure, and L. L. Scharf, Initial results in Prony analysis of power system response signals, IEEE Trans. Power Syst., vol. 5, no. 1, pp , Feb [2] M. A. Johnson, I. P. Zarafonitis, and M. Calligaris, Prony analysis and power system stability-some recent theoretical and applications research, in Proc. IEEE Power Eng. Society Summer Meeting, 2000, 2000, vol. 3, pp [3] J.C.-H.PengandN.-K.C.Nair, Effectsofsamplinginmonitoring power system oscillations using on-line Prony analysis, in Proc. Australasian Universities Power Eng. Conf., 2008 (AUPEC 08), Dec , 2008, pp [4] G.H.GolubandV.Pereyra, Thedifferentiationofpseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal., vol. 10, no. 2, pp , Apr [5] G. H. Golub and V. Pereyra, Separable nonlinear least squares: The variable projection method and its applications, Inverse Probl., vol. 19, no. 2003, pp. R1 R26, Feb [6] D.P.O LearyandB.W.Rust, Variable projection for nonlinear least squares problems, Computat. Optimiz. Applicat., vol. 54, no. 3, pp , [7] L. Kaufman, A variable projection method for solving separable nonlinear least squares problems, BIT Numer. Math., vol. 15, no. 1, pp , [8] Varpro [Online]. Available: varpro.m [9] L.L.ScharfandC.Demeure, Statistical Signal Processing : Detection, Estimation, and Time Series Analysis. Reading, MA, USA: Addison- Wesley, [10] Y. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise, IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 5, pp , May [11] L. L. Grant and M. L. Crow, Comparison of Matrix Pencil and Prony methods for power system modal analysis of noisy signals, in Proc. North Amer. Power Symp. (NAPS), 2011, Aug. 4 6, 2011, pp [12] J.-N. Juang, Applied System Identification. Englewood Cliffs, NJ, USA: Prentice Hall, [13] R. A. De Callafon et al., General realization algorithm for modal identification of linear dynamic systems, ASCE J. Eng. Mech., vol.134, no. 9, pp , Sep [14] D. J. Trudnowski, J. M. Johnson, and J. F. Hauer, Making Prony analysis more accurate using multiple signals, IEEE Trans. Power Syst., vol. 14, no. 1, pp , Feb [15] N. Zhou, D. Trudnowski, J. W. Pierre, S. Sarawgi, and N. Bhatt, An algorithm for removing trends from power-system oscillation data, in Proc IEEE Power and Energy Society General Meeting Conversion and Delivery of Electrical Energy in the 21st Century, Jul , 2008, pp [16] R. Jaramillo and M. Lentini, Stability analysis of a variant of the Prony method, Math. Probl. Eng., vol. 2012, Oct. 2012, Article ID [17] M. R. Osborne and G. K. Smyth, A modified Prony algorithm for exponential function fitting, SIAM J. Sci. Comput., vol. 16, pp , [18] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New York, NY, USA: Springer, [19] G. Rogers, Power System Oscillation. Norwell, MA, USA: Kluwer, [20] J. H. Chow and K. W. Cheung, A toolbox for power system dynamics and control engineering education and research, IEEE Trans. Power Syst., vol. 7, no. 4, pp , Nov [21] A. R. Borden and B. C. Lesieutre, Determining power system modal content of data motivated by normal forms, in Proc. North Amer. Power Symp. (NAPS) 2012, Sep. 9 11, [22] P.Tripathy,S.C.Srivastava,andS.N.Singh, Amodified TLS-ES- PRIT-based method for low-frequency mode identification in power systems utilizing synchrophasor measurements, IEEE Trans. Power Syst., vol. 26, no. 2, pp , May Alexander R. Borden (S 10) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Wisconsin-Madison, Madison WI, USA, in 2008, 2010, and 2014, respectively. Bernard C. Lesieutre (S 86 M 93 SM 06) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign, Urbana, IL, USA. He is currently an Associate Professor of Electrical and Computer Engineering at the University of Wisconsin-Madison, Madison, WI, USA. His research interests include the modeling, monitoring, and analysis of electric power systems and electric energy markets.