Dynamic System Eigenvalue Extraction using a Linear Echo State Network for Small-Signal Stability Analysis a Novel Application

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1 Dynamic System Eigenvalue Extraction using a Linear Echo State Network for Small-Signal Stability Analysis a Novel Alication Jiaqi Liang, Jing Dai, Ganesh K. Venayagamoorthy, and Ronald G. Harley Abstract A large nonlinear dynamic system usually has comlex dynamic modes corresonding to the system s eigenvalues. These eigenvalues govern the system s local behavior and thus are critical information for designing system oeration and control strategies. Without the availability of the system s analytical model, which is often the case for large nonlinear systems, the system s eigenvalues need to be estimated. A linear echo state network (ESN) based method for extracting observable eigenvalues of a dynamic system together with the articiation factors of these eigenvalues in the accessible system states is resented in this aer. A linear ESN is first trained to track the dynamic system s local resonses under injected small erturbation signals. The dynamic system s eigenvalues are then extracted from the ESN s weight matrices. Given the merit of fast training of ESNs, the ESN can be quickly retrained once the system oerating oint changes, and the system eigenvalues can be reestimated. Alication of the roosed eigenvalue extraction method in the ower system small-signal analysis is resented to demonstrate the effectiveness of the roosed method. A I. INTRODUCTION RTIFICIAL neural networks (ANNs) are well-known effective universal function aroximators. Different tyes of ANNs have been roosed in the ast few decades and broadly used in nonlinear system modeling and intelligent neural control [1]. The multi-layer ercetron (MLP) network and the radial basis function (RBF) network have relatively simle structures, but their aroximation abilities are limited by their feed-forward structures. Recurrent neural networks (RNNs) have advantages in modeling dynamic systems due to their internal dynamic memory [2]. However, the training algorithms for RNNs usually have high comutational comlexity and long training cycles. Recently, the echo state network (ESN) [3] has been introduced as a secial tye of RNN, which has the same dynamic modeling caability as a RNN but requires a simler training rocess. ESNs have been successfully alied to the areas of dynamic system modeling and neural adative controls [4]-[7]. However, most of the revious works on dynamic system identification have focused on how to Manuscrit received May 2, 21. This work was suorted in art by the National Science Foundation, USA, under Grants ECCS 8247 and EFRI J. Liang, J. Dai and R. G. Harley are with the School of Electrical and Comuter Engineering, Georgia Institute of Technology, Atlanta, GA 3332, USA. R. G. Harley is also Professor Emeritus and an Honorary Research Associate, University of KwaZulu-Natal, Durban, South Africa. ( jliang@gatech.edu; jingdai@gatech.edu; rharley@ece.gatech. edu) G. K. Venayagamoorthy is the Director of the Real-Time Power and Intelligent Systems Laboratory, Missouri University of Science and Technology, Rolla, MO 6549, USA ( gkumar@ieee.org). train an ANN such that it tracks a certain trajectory or minimizes a certain erformance index, but seldom has anyone investigated the information buried inside an ANN after it has been trained. On the other side, a comlex nonlinear dynamic system (lant) being modeled by an ANN often does not have an accurate analytical model for designers to study its dynamic characteristics and stability margins. This characteristic information, such as system eigenvalues (or modes), however, is often critical for system oerators to maintain the system within a safe stability margin as well as designing system controllers. Existing methods to estimate a dynamic system s observable modes are based on directly analyzing the frequency comonents in the system oututs, for examle, Prony s method [8]. However, if the system has multile oututs and multile observable modes, this method does not rovide information on the couling between different system modes and states, i.e., the articiation factors. This aer rooses a linear ESN (an ESN with linear activation function) based method to extract observable eigenvalues of a dynamic system as well as the articiation factors of these eigenvalues in the accessible system states. For nonlinear system identification, nonlinear activation functions are necessary. However, eigenvalues of a nonlinear dynamic system only reresent its local dynamics around its oerating oint, where the nonlinear system behaves like a linear system. Therefore, a linear ESN is sufficient to extract this information. A brief overview of the ESN structure and its training algorithms, as well as the mathematical simlicity of a linear ESN, is resented in Section II. The roosed method for dynamic system observable eigenvalue extraction is discussed in Section III. Section IV rovides further discussion on finding the articiation factors between the extracted system eigenvalues and the system accessible states. The alication of the roosed method to evaluate the small-signal stability of a ower system is resented in Section V. II. OVERVIEW OF ECHO STATE NETWORKS This section gives a brief overview of ESNs. More detailed descritions can be found in [3],[9]-[12]. A. Structure of ESNs The structure of an ESN is shown in Fig. 1, where the outut of the internal hidden layer (dynamic reservoir) deends on the network inut vector at time k, u(k) (n x 1), the internal state (hidden layer outut) vector at time k, x(k) (q x 1), and the network outut vector at time k, y(k) (m x 1). Note that time k here does not denote the k th execution of the ESN, but reresents the k th samling instant of all signals. W i (q x n)

2 u1( k) u2( k) u( k) = M u ( ) n k Fig. 1. Structure of an ESN. denotes the inut weight matrix, W b (q x m) denotes the outut feedback weight matrix, W s (q x q) denotes the state feedback weight matrix, and W o (m x q) denotes the outut weight matrix. The ESN in Fig. 1 can be mathematically exressed as x(k +1) = f [W i u(k) + W s x(k) + W b y(k)], (1) y(k +1) = W o x(k +1), (2) where the activation function, f( ), is an element-wise oeration that can be chosen as a linear function, or a nonlinear function such as the sigmoid function and the tansig function. W i, W b and, W s are generated randomly beforehand and they are fixed during the training rocess. W o is the only matrix that will be udated during the network training. The random generation of W s follows these secific stes [9]: 1) Randomly generate a sarse matrix, W s_init, with all elements between -1 and 1. 2) Normalize W s_init into a matrix, W s, with unity sectral radius: W ' = W λ, (3) s s _ init x( k + 1) = x1 ( k + 1) M x ( 1) q k + where λ max is the sectral radius of W s_init, i.e., the maximum absolute eigenvalue of W s_init. 3) W s is obtained as max W s = αw s, (4) where α < 1, and usually α is selected between.9 and 1. B. Training Algorithms for ESNs y1 ( k + 1) y2( k + 1) y( k + 1) = M y ( 1) m k + Both offline and online training algorithms have been develoed for training the ESN [9]-[11]. Fig. 2 shows the general scheme for training an ESN to identify the lant dynamics, and the objective of the training is to minimize the error, e(k), between the ESN and the lant oututs. The offline training algorithm is based on the seudo-inverse method [9], [1]. First, a set of lant inut and outut samles are taken, and then a set of over-determined equations are formed with W o unknown (the number of data samles is usually much more than the number of unknown weights in W o ). A solution for W o with minimum mean squared error is obtained by erforming a seudo-inverse oeration. Therefore, the offline training is fast and it leads to an otimal outut weight matrix with resect to the samle data set. Fig. 2. General scheme of lant dynamics identification. The online training algorithm continuously udates the outut weight matrix based on the error signal and the gradient descent algorithm [11]. The online training algorithm offers a method for tracking the variations of the lant dynamics, but the training cycle tyically takes more time than the offline training algorithm. C. ESN with Linear Activation Function If a linear function is used as the activation function in an ESN, then the ESN can be described by x(k +1) = a[w i u(k) + W s x(k) + W b y(k)], (5) y(k +1) = W o x(k +1), (6) where a is a constant. Since a can be absorbed by the outut matrix W o, without loss of generality, a is set to 1 and the resulting ESN is called a linear ESN in this aer. Substitute (6) at time k into (5), to obtain the following equation: x( k + 1) = Wiu( k) + Ws x( k) + WbW ox( k), (7) = W x( k) + W u( k) A where W A =W s +W b W o, which is the state udate matrix of the discrete-time dynamic system (7), i.e. the linear ESN. The modes of the linear ESN are comletely reresented by the eigenvalues of W A. Combine (6) and (7), and the linear ESN is described by i y( k + 1) = W W x( k) + W W u ( k). (8) o A o i When a linear ESN is trained to track the oututs of a nonlinear dynamic system under small erturbation signals, the system s local dynamics around its oerating oint are learned by the linear ESN, i.e., the linear ESN learns the locally linearized dynamic system. Note that the linear ESN is in the discrete-time domain, and thus it indeed learns the time-discretized version of the dynamic system. It will be shown in Section III.B that a time-discretized locally linearized dynamic system can also be reresented in the same analytical form as (8). III. DYNAMIC SYSTEM EIGENVALUE EXTRACTION A. Small-Signal Analysis of a Dynamic System A multile-inut multile-outut (MIMO) time-invariant dynamic system without feed-through oututs can be described in the state equation form, as in & x ( t) = g[ x ( t), u( t)], (9) y ( t) = h[ x ( t)]

3 where u(t) is the inut vector to the MIMO system, x (t) is the system internal state vector, and y (t) is the system outut vector. The overall stability of such a nonlinear system is usually difficult to analyze and rove. However, the smallsignal stability of the linearized system, which is necessary for the overall system to be asymtotically stable at the oerating oint, is often of interest. The locally linearized system of (9) around an oerating oint (u, x, y ) is given by & x ( t) = A x ( t) + B u( t), (1) y ( t) = C x ( t) where u(t)= u(t)+u, x (t)= x (t)+x, y (t)= y (t)+y, g A = x u = u x = x, g u = u B = u x = x, and C = h u x = x The locally linearized system (1) is stable, i.e., the nonlinear system (9) is small-signal stable, if and only if all eigenvalues of matrix A have real arts less than zero. For a small-signal stable nonlinear system, the closer an eigenvalue lies to the imaginary axis, the less stable margin the system would have for the corresonding mode. In the real world, the analytical model of a comlex dynamic system is often unknown. Therefore, the direct eigenvalue analysis on a comlex system is often not ossible. However, the system modes, or art of the modes, can be observed from the system outut, y (t), given that the observability condition is satisfied. Therefore, these observable system modes can be estimated from the system oututs, and the small-signal analysis can then be erformed on these observable modes. B. System Discretization in Time Before introducing the method of system eigenvalue extraction using a linear ESN, it is necessary to first introduce the time-discretized version of (1), because the linear ESN learns in the discrete time domain. The continuous-time system (1) can be discretized [13] to x ( k + 1) = Ad x ( k) + Bd u( k), (11) y ( k + 1) = C x ( k + 1) d where A d = L -1 {(si-a) -1 } t=ts and L -1 { } t=ts is the Lalace inverse transform with the samling time being T s, B d = A -1 (A d -I)B, and C d = C. An aroximation of A d exists based on the bilinear transform which reserves the stability of the continuous-time system [13], given by 1 1 A I AT I AT d = ( + s )( s ). (12) Note that a smaller samling time ste results in a better aroximation by using (12). The modes of the dicretized locally linearized system (11) are comletely reresented by the eigenvalues of A d. System (11) can be further written as. y ( k + 1) = C A x ( k) + C B u ( k), (13) d d d d which has the exact same form as the linear ESN in (8). Note that the linearization and time discretization of the nonlinear system (9) are not assumtions. Under small erturbations and in discrete time domain, the nonlinear system (9) behaves exactly like (11). Therefore, when a linear ESN is trained to model the nonlinear dynamic system (9) under small erturbations, the modeling task is indeed to find out the outut weight matrix, W o, such that (8) and (13) resond as closely as ossible. Note that x(k) in (8) and x (k) in (13) may have different dimensions and different values. C. System Eigenvalue Extraction using Linear ESN The idea of the roosed method is to reconstruct the internal local dynamics of an unknown nonlinear system by training a linear model to follow the system s local resonses and then analyzing the eigenvalues of the trained model. As discussed above, the small-signal stability of a nonlinear system is based on its local dynamics (the linearized system), and thus the reconstruction can be achieved by using a linear dynamic ANN, for examle the linear ESN. Because the dynamics of a linear ESN and the local dynamics of the nonlinear system (lant) are comletely reresented by the eigenvalues of W A in (8) and the eigenvalues of A in (1) resectively, if the linear ESN resonse tracks the lant resonse closely around a local region, then the lant s observable eigenvalues should be shared by the linear ESN. However, because the linear ESN is in the discrete-time domain, the trained linear ESN indeed contains the observable eigenvalues of A d in (11). In order to find the observable eigenvalues of A, it is then necessary to find the continuous-time version of the linear ESN matrix W A. Once the linear ESN is trained, by solving (12), the continuous-time version of W A is given by 2 W ( W I )( W I ) 1 A_ cts = A A +. (14) Ts The eigenvalues of W A_cts then contain the lant s observable eigenvalues. If the lant s oerating oint changes, the ESN can be retrained as necessary, and a new set of eigenvalues can be obtained for the new oerating oint. Because the number of the linear ESN internal states is usually larger than the number of the lant s observable eigenvalues, there are some extra eigenvalues from W A_cts that do not corresond to any of the lant modes. However, since the linear ESN outut dynamics deends mostly on those observable lant modes, the extra eigenvalues are usually well damed and they do not have much effect on the ESN oututs. Therefore, those eigenvalues of the linear ESN that are closer to the imaginary axis are more likely to be the lant modes. Another way of distinguishing the lant modes from the extra eigenvalues is to use the articiation factors, which is

4 introduced in the next section. The eigenvalues of the linear ESN that have high articiation factors in the ESN oututs are more likely to be the lant modes. IV. PARTICIPATION FACTOR EXTRACTION The extraction of the articiation factors between different observable modes and different accessible states is discussed in this section. The accessible states are defined as lant states that are also lant oututs. For the analysis below, art of the lant oututs are assumed to be lant states. This assumtion is often true, because most measured quantities in an actual system cannot change abrutly and thus they can be chosen as state variables. A. Particiation Factor The definition of articiation factor is briefly reviewed below. The lant states x in (1) can be decouled by the lant s modal matrix U [14], where the columns of U are the eigenvectors of matrix A in (1) with unity norms, i.e., AU = U Λ, (15) where Λ = diag (λ 1, λ 2, λ n ) and λ i s are the eigenvalues of A. Define a new set of state variables, z, as z x. (16) 1 ( t) = U ( t) Then the lant dynamics described by (1) can be rewritten as 1 & ( t) ( t) U = Λ + B ( t) z z u. (17) From (17), each state in z is only related to one eigenvalue and thus only related to one system mode. The original state vector is related to z by (16). For examle, the k th state in x is related to the i th element in z by a factor of u ki. Define a matrix V as V = U 1. (18) The rows of V are then the left eigenvectors of A with the same eigenvalue matrix Λ. The articiation factor of the i th mode in the k th state of x is defined as [14] = u v. (19) ki ki ik In the case of comlex eigenvalues and eigenvectors, the normalized articiation factor ki is given by ki = u v q j= 1 ki ik u v kj jk. (2) The articiation factor ki is a measurement of the relative couling between the k th state and the i th mode. It is useful when a daming controller for increasing daming of the i th mode is desired [15]. B. Particiation Factor Extraction using Linear ESN Suose that the first r lant oututs are the lant states and they are distinct (r is less than the total number of the lant states). As a result from the linear ESN modeling, the first r oututs from a trained linear ESN are estimates of the lant states, and the linear ESN has art of the eigenvalues that are corresonding to the observable lant modes. The goal is then to find the articiation factors between the lant observable modes and the r accessible states. In order to find out these articiation factors, the first r oututs from the trained linear ESN is first transformed to be art of the linear ESN states by a linear transformation defined as Wo,1:r T, (21) = I ( r) ( r) where W o,1:r denotes the first to the r th rows of W o, which are linear indeendent due to the assumtion that the r lant states are distinct. If T is full rank, which is true in most cases, a new set of linear ESN states, x new (k), is defined as x new y1( k) M yr ( k). (22) ( k) = T x( k) = xr + 1( k) M xq ( k) With this new set of states, the linear ESN described by (6) and (7) is then rewritten as new 1 new x ( k + 1) = ( TWAT ) x ( k) + TWi u( k). (23) Ir r new y( k + 1) = ( k+1) W x o, r+1:m Define W A new = TW A T -1, which has the same eigenvalues as W A, i.e., they have the same modes. After finding the continuous-time version of W A new from (14) and calculating its right eigenvector matrix U, the articiation factors between the first r states in x new, i.e., y 1, y 2, y r, and all the modes of the linear ESN are obtained by following the same rocedure as described in section IV.A. In other words, the articiation factors between the r accessible lant states and the observable lant modes are obtained. V. APPLICATION IN POWER SYSTEM SMALL-SIGNAL ANALYSIS A electric ower grid is a nonlinear comlex MIMO dynamic system. Most of its internal states are not accessible and cross-couled. The dynamic model for a large ower system is often not available due to lack of accurate system arameters. However, the system dynamic characteristics, such as eigenvalues around different oerating oints, are critical information for oerating the system within a safe

5 Infinite Bus G2 Plant G1 G4 ω 2 ω 4 ω3 V ref3 G3 + + V ref3 PRBS Fig. 3. Benchmark 12 bus ower system. V - ref + P mref v s Seed Gov stability margin, as well as allocating and designing system daming controllers. The generator seed oscillation modes in a 12 bus ower system [15] are studied in this section as an examle alication of the roosed dynamic system eigenvalue extraction technique. A. 12 bus Power System Volt Regulator Generator ω + - ω ω ref = 1 u Fig. 4. Block diagram of controller on each generator. The one-line diagram [15] of the benchmark 12 bus ower system is shown in Fig. 3. It has three detailed modeled generators (G2, G3 and G4) and one voltage source (G1). Two different models of the same 12 bus system are develoed. One detailed dynamic system model, which reresents all three generators by their 8 th order models, is develoed in the PSCAD/EMTDC commercial time domain simulation software. The detailed models for the terminal voltage regulators and generator seed governors are also included, as shown in Fig. 4. This model is used as the lant for generating training data for the linear ESN. Another simlified system analytical model is develoed, which reresents each generator as only a 3 rd order system and reresents each voltage regulator as a 1 st order system. The seed governors are neglected due to their relatively slow resonses, comared to the generator seed oscillations. The analytical modeling and linearizing method of a ower system was discussed in detail in [14] and [15]. In each generator and voltage regulator unit, there are 4 states, including the rotor seed ω, the rotor angle (integral of rotor seed) δ, the internal induced voltage E q, and the voltage regulator outut E f. This simlified analytical model, with a total of 12 states, is linearized around its steady-state oerating oint to obtain the system matrix A and its eigenvalues. These eigenvalues are then used to verify the roosed technique. v s e(k) y (k) + - y(k) Linear ESN Fig. 5. Plant and linear ESN inut-outut signals. TABLE I CASES 1: GENERATOR SET POINTS B. Training of the Linear ESN Model Fig. 5 shows the inut-outut signals of the lant and the linear ESN. The lant has three oututs, which are seed deviation signals from the three generators. The generator seed deviations in a ower system are often couled with modes with long time constants, i.e., their corresonding eigenvalues are close to the imaginary axis. Note that all three lant oututs are also the lant states. A small seudo-random binary signal (PRBS) is fed into the voltage reference signal of generator 3 to erturb the 12 bus system for generating ESN training data. The PRBS is generated by PRBS( k) =.3[ r ( k) + r ( k)] / 2 (u), (24).5.1 where r.5 (k) and r.1 (k) are sequences of i.i.d. random numbers uniformly distributed in [-1, 1]. r.5 (k) udates at the frequency of.5 and r.1 (k) udates at the frequency of.1. The maximum value for the PRBS is set to ±.3 er unit (u), so that the generator terminal voltage stays between.95 u and 1.5 u, which is a requirement in an actual ower system. A linear ESN is trained to track the lant oututs during this series inut PRBS disturbances. The linear ESN thus has one inut (n = 1) and three oututs (m = 3). Thirty internal neurons are used (q = 3), i.e., the linear ESN has 3 eigenvalues. α is set to.95 in the linear ESN W s initialization. Two different cases corresonding to two different oerating oints of the 12 bus system are studied below. For both cases, the offline training based on seudo-inverse is used for fast convergence. C. Case 1: Base Case for Verifying the Proosed Eigenvalue Extraction Technique In this case, the set oints of the three generators, i.e., the reference outut owers and the reference terminal votlages, are listed in Table I. From the simlified analytical model, the u(k) Gen # MVA Rating P mref (MW) V ref (u) Gen Gen Gen

6 TABLE II CASE 1: OSCILLATION MODES FROM ANALYTICAL MODEL Eigenvalues Frequency () Daming Ratio (%) ± j ± j ± j TABLE III CASE 1: PARTICIPATION FACTORS FROM ANALYTICAL MODEL State 1.13 Mode.85 Mode.75 Mode δ G ω G E q G E f G δ G ω G E q G E f G δ G ω G E q G E f G system s eigenvalues at this oerating oint are obtained, among which those corresonding to the oscillation modes are listed in Table II. There are three oscillation modes in this system, the 1.13 mode, the.85 mode, and the.75 mode. Table III shows the articiation factors between these three modes and each of the system states from the analytical model. All of the three oscillation modes are rimarily couled with the generator mechanical states, i.e., the rotor seed and angle. Generator 3 dominates in the 1.13 mode, as indicated by the relatively high articiation factors of δ G3 and ω G3 in the 1.13 mode. Generator 2 dominates in the.85 mode, and Generator 4 dominates in the.75 mode. However, all of the above information is based on the system analytical model, which is often not available for a large comlex real-world system. The linear ESN is now trained with data generated from the detailed dynamic system PSCAD model. 7 s of training data, as shown in Fig. 6, are generated with a samling time of.2 s (T s =.2). The linear ESN is trained offline by seudo-inverse with a final mean squared error (MSE) of 2.19e-16, which is 12 orders of magnitude smaller than the ω data. After the training, all the weights in the linear ESN are fixed and the network is tested under a different set of PRBS disturbance. The testing results with fixed weights are shown in Fig. 7. Under ste change inuts, the trained linear ESN shows good tracking to the dynamic oscillation of the lant, which indicates that the trained linear ESN contains information of the lant oscillation modes. With the mode extraction technique introduced in section III, all the 3 eigenvalues corresonding to the continuous-time version of the trained linear ESN are listed in Table IV. It can be seen from Table IV that the linear ESN contains the lant s three oscillation modes, with accurate dv ref3 (u) dw 2 (u) dw 3 (u) dw 4 (u) 5 x x x Time (s) Fig. 6. Linear ESN offline training data in case 1. To grah: injected PRBS. Bottom three grahs: lant resonses under the injected PRBS. estimation on both the oscillation frequencies and the daming ratios. These three oscillation modes can be searated from the rest of the linear ESN eigenvalues from either the daming ratios or the oscillation frequencies. The other extra eigenvalues of the linear ESN either have a high daming ratio or a high oscillation frequency, which are clearly not the dominant comonents in the ω waveforms. Table V shows the articiation factors between the 3 eigenvalues and the three linear ESN oututs, which are calculated based on the technique introduced in section IV. From the articiation factors, the three modes that corresond to the lant oscillation modes clearly stand out by having relatively high articiation factors. From Table V, it can also be seen that the.75 mode is mostly couled with generator 4, the.84 mode is mostly couled with generator 2, and the 1.13 mode is mostly couled with generator 3. The results are consistent with those derived from the analytical model, i.e., Tables II and III. The roosed technique would be more useful when a certain mode is couled with more than one state, or one state is couled with more than one mode. In such a case, the articiation factor table could show the coulings between multile states and multile modes. D. Case 2: Same System with a Different Oerating Point In this case, the ower generations from all three generators are increased to reduce the energy demand from the voltage

7 dv ref3 (u) dw 2 (u) dw 3 (u) dw 4 (u) x Plant Outut ESN Outut x x 1-4 Plant Outut ESN Outut Plant Outut ESN Outut Time(s) Fig. 7. Testing results of the trained linear ESN (fixed weights) in case 1. To grah: injected PRBS. Bottom three grahs: lant and the linear ESN resonses under the injected PRBS. source (G1). The set oints in this case are listed in Table VI. From the analytical model, the lant oscillation modes are calculated and listed in Table VII for comarison with results extracted from the linear ESN. After changing the system oerating oint, both the oscillation frequency and daming ratio of the mode couled with G3 increase, while the daming ratios of the modes couled with G2 and G4 decrease. 7 s of new data are collected at the new oerating oint under the PRBS erturbation. When using the weights trained in case 1, the MSE increases to 3.41e-14 under this new data set. This is rimarily because the change of the lant oerating oint also changes the eigenvalues of the lant. The linear ESN is then retrained with the new data. After training, the TABLE IV CASE 1: ALL 3 EIGENVALUES OF THE LINEAR ESN Eigenvalues Frequency () Daming Ratio (%) N/A ± j ± j ± j ± j N/A ± j ± j ± j ± j N/A ± j ± j N/A ± j ± j ± j MSE dros back to 2.64e-16. The retrained linear ESN with fixed weights is tested again at the new oerating oint. Fig. 8 shows the testing results, which reveal good tracking erformance of the retrained linear ESN. The system oscillation modes are extracted again from the retrained linear ESN, as shown in Table VIII. Comaring Tables VII and VIII, the modes extracted from the linear ESN are accurate. Note that by monitoring the oscillation modes of the 12 bus system using the linear ESN, it can be seen that generator 2 is close to instability at the new oerating oint. VI. CONCLUSIONS This aer rooses a linear ESN based method for extracting observable eigenvalues of a large nonlinear dynamic system around its oerating oint, as well as extracting the articiation factors between the observable eigenvalues and the system s accessible states. A linear ESN is first trained by using offline training algorithm to learn the nonlinear dynamic system s local dynamics. The system s observable eigenvalues and their articiation factors are then extracted from the linear ESN s weight matrices. Because of the availability of the fast offline training algorithm for ESNs, the roosed method is able to quickly retrain the ESN once the system oerating oint changes, and reestimate the system s eigenvalues. Moreover, for ESNs, it is only necessary to change the ESN outut weights in order to follow the changes of the lant oerating oint. Thus, the ESN demands State TABLE V CASE 1: PARTICIPATION FACTORS (ROUNDED TO THE 3 RD DECIMAL) BETWEEN THE LINEAR ESN OUTPUTS AND THE LINEAR ESN MODES (-2.52) (-49.6) ω G ω G ω G (-131) (-156)

8 dv ref3 (u) dw 2 (u) dw 3 (u) dw 4 (u) x 1-4 Plant Outut ESN Outut x 1-4 Plant Outut ESN Outut x 1-4 Plant Outut ESN Outut Time(s) Fig. 8. Testing results of the retrained linear ESN (fixed weights) at the new oerating oint in case 2. To grah: injected PRBS. Bottom three grahs: lant and the linear ESN resonses under the injected PRBS. less storage if the weights of the linear ESN are to be saved for different lant oerating oints. An examle in estimating the eigenvalues of a 12 bus ower system under different oerating oints is resented in this aer and demonstrates the effectiveness of the roosed method. REFERENCES [1] S. Haykin, Neural networks: a comrehensive foundation, 2nd ed., Prentice Hall, ISBN-13: [2] D. Prokhorov, G. Puskorius, and L. Feldkam, Dynamical neural networks for control, in J. Kolen and S. Kremer (Eds.), A Field Guide to Dynamical Recurrent Networks, IEEE Press, 21. ISBN-13: [3] H. Jaeger, The echo state aroach to analysing and training recurrent neural networks, GMD Reort 148, German National Research Center for Information Technology, 21. [4] J.Mazumdar, R. G. Harley, Utilization of echo state networks for differentiating source and nonlinear load harmonics in the utility network, IEEE Trans. Power Electronics, vol 23, no. 6, : , Nov. 28. [5] J. Mazumdar, G. K. Venayagamoorthy, R. G. Harley and F. C. Lambert, Echo state networks for determining harmonic contributions from nonlinear loads, Proc. IEEE Int. Joint Conf. on Neural Networks (IJCNN 6), Vancouver, Canada, 26, [6] M. Salmen and P.G. Ploger, Echo state networks used for motor control, Proc. IEEE International Conf. on Robotics and Automation, Barcelona, Sain, 25, TABLE VI CASES 2: GENERATOR SET POINTS Gen # MVA Rating P mref (MW) V ref (u) Gen Gen Gen TABLE VII CASE 2: OSCILLATION MODES FROM ANALYTICAL MODEL Eigenvalues Frequency () Daming Ratio (%) Dominant Gen -.41 ± j G ± j G ± j G4 TABLE VIII CASE 2: OSCILLATION MODES ESTIMATED FROM LINEAR ESN Eigenvalues Frequency () Daming Ratio (%) Dominant Gen ± j G ± j G ± j G4 [7] D. Xu, J. Lan and J. C. Princie, Direct adative control: an echo state network and genetic algorithm aroach, Proc. IEEE Int. Joint Conf. on Neural Networks (IJCNN 5), Montréal, Québec, Canada, Jul./Aug. 25, vol. 3, [8] J. F. Hauer, C. J. Demeure, and L. L. Scharf, Initial results in rony analysis of ower system resonse signals, IEEE Trans. Power Systems, vol. 5, no. 1,. 8-89, Feb 199. [9] H. Jaeger, Tutorial on training recurrent neural networks, covering BPTT, RTRL, EKF and the Echo State Network aroach, GMD Reort 159. German National Research Center for Information Technology, Sankt. [1] H. Jaeger, Adative nonlinear system identification with echo state networks, Proc. Advances in Neural Information Processing Systems, , MIT Press, Cambridge, MA, 23. [11] G. Venayagamoorthy, Online design of an echo state network based wide area monitor for a multimachine ower system, Neural Networks, vol. 2, , 27. [12] J. Dai, G. Venayagamoorthy, and R. Harley, An introduction to the echo state network and its alications in ower system, Proc. Intelligent System Alications to Power Systems (ISAP 29), Curitiba, Brazil, Nov. 8-12, 29. [13] R. A. DeCarlo, Linear systems: a state variable aroach with numerical imlementation, Prentice Hall, NJ, ISBN-13: [14] P. Kundur, Power system stability and control, New York: McGraw-Hill, ISBN-13: [15] S. Jiang, U. D. Annakkage, and A. M. Gole, A latform for validation of FACTS models, IEEE Trans. Power Delivery, vol. 21, no. 1, , Jan 26.