Reduced-order equivalent model to large power networks derived from its spectral dispersion

Electric Power Systems Research 143 (2017) 244–251

Contents lists available at ScienceDirect

Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Reduced-order equivalent model to large power networks derived from its spectral dispersion b,∗ ˜ P. Esquivel a , Carlos E. Castaneda a b

Technological Institute of La Laguna, Torreón, Mexico University Center of Los Lagos of the University of Guadalajara, Mexico

a r t i c l e

i n f o

Article history: Received 15 March 2016 Received in revised form 26 August 2016 Accepted 31 August 2016 Keywords: Empirical orthogonal functions Frequency-domain response MIMO systems Modal distribution Network equivalents State space model Vector fitting

a b s t r a c t This paper describes a general methodology for identification of a reduced-order dynamic equivalent with modal frequency distribution to large power networks, derived from its frequency-varying response. The method is used to define a state space model with modal frequency dispersion established from both, the application of the empirical orthogonal functions (EOFs) analysis and vector fitting (VF) procedure for rational functions approximation from frequency-domain data sets. Initially, our approach uses orthogonal modes of major contributions of spectral dispersion derived from the EOFs analysis to construct a reduced-order approximation with applications to multiple-input, multiple-output (MIMO) linear-time invariant (LTI) systems. This approximation defines an optimal distributed solution to the frequencyvarying data set, where their fundamental properties are based on the interpretation of pre-selected frequencies contained into the eigenvectors of a cross-spectrum matrix. Once the reduced-order empirical modal decomposition is derived, its coefficients are used in the VF procedure in order to generate a rational function approximation into a frequency band with particular level of kinetic energy with applications to MIMO systems. Additionally, the reduced-order equivalent network in a state space model is derived from a VF, which can be efficiently incorporated in a power network simulator to electromechanical studies of multimachine dynamics with modal frequency splitting. Finally, an example for large power networks is examined to both demonstrate the effectiveness for fitting reduced-order dynamic equivalents and to capture its modal coherence and frequency distribution. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Over the last years, many techniques using linear and nonlinear fitting routines have been proposed to fit the frequency response of large power networks. This is in order to ensure accurate equivalent models and decrease remarkably the computational burden in simulators of transient studies where the fitting may be performed either in the s or z domain [1–9]. The analysis of frequency range considering accuracy, the shape of the frequency response, the mathematical model and the possibility for timedomain implementation are examined to decide which one fitting technique is the most appropriated. A recent problem related with these methodologies has been the derivation of a reduced-order equivalent model with applications to multiple-input multipleoutput (MIMO) linear-time invariant (LTI) systems, that considers

∗ Corresponding author. E-mail addresses: [email protected] (P. Esquivel), ˜ [email protected] (C.E. Castaneda). http://dx.doi.org/10.1016/j.epsr.2016.08.039 0378-7796/© 2016 Elsevier B.V. All rights reserved.

the modal-geographical dispersion of large interconnected power networks and its distributed implementation to multimachine systems. This represents a drawback to be efficiently incorporated into a power systems simulator to electromechanical studies [7,10–19] and it will be treated in this work. In [19] is presented an algorithm for identifying a multiphase network equivalent for transient simulations of single-input single-output (SISO) systems, where the method is limited to use the trace of the associated transfer function matrices and to divide the network in two parts: a study zone and a external zone, which the computational efficiency with acceptable degree of accuracy from the results are derived. Additionally, in [2] is given a general methodology to the order reduction of dynamic models by using the singular value decomposition and balanced realization techniques in SISO systems where the issues of sparsity, convergence, and accuracy are examined. Recently, a statistical identification method established on the basis of the empirical orthogonal functions (EOFs) analysis, more commonly called principal components (PCs) analysis, has been widely applied to identify and to extract modal instabilities from a data set. The technique is based on the correlation structure from time-varying fields,

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

which can treat both optimal modal distribution and geographicalspatial dispersion [20–24]. This fact motivates us to extend the EOFs analysis to the studies of frequency-domain responses of large power networks and approaching a reduced-order equivalent model with applications to MIMO systems from a data set. Therefore, in this paper a general methodology for identification of a reduced-order equivalent network with modal frequency distribution to power networks derived from its frequency-domain response, is presented. The method is used to define a state space model with modal frequency splitting established on the application of the EOFs analysis and the vector fitting (VF) procedure for rational functions approximation from frequency-domain data. On the one hand, our approach uses the orthogonal modes of major contributions of modal coherence derived from the EOFs analysis to construct a reduced-order approximation. This approximation defines an optimal distributed solution to the frequency-varying data set. The fundamental properties of this data set are based on the interpretation of pre-selected frequencies contained into the eigenvectors of a cross-spectrum matrix. Once the reduced-order empirical modal decomposition is derived, its spectral coefficients are used in the VF procedure in order to generate a rational function approximation into a particular frequency band with particular level of kinetic energy of applications to MIMO systems. On the other hand, the reduced-order equivalent network given in a state space model form is derived from the VF and efficiently incorporated in a power network simulator to electromechanical studies of multimachine dynamics and inter-area control system design. The procedure incorporates frequency domain responses to study and to characterize coupling frequencies and geographical dispersion into multiple power networks used to the analysis of its oscillatory activity. In addition, the method also incorporates a procedure based in frequency band to effectively define a reduced model from the data used. The objective of this study is to infer the relationship between modal frequencies and the spatial relationship of waveforms present within a particular frequency interval. Some of these difficulties are discussed in the interpretation of results, where a power network model is examined to demonstrate the effectiveness when fitting reduced-order equivalent networks capturing its modal coherence and distribution. This paper is organized as follows: Section 2 introduces some theoretical backgrounds about the EOFs analysis; the method of VF is described in Section 3; next, the proposed method is presented in Section 4; test results are provided in Section 5; finally, discussions and conclusions are given in Sections 6 and 7, respectively.

245

the model is the assumption that X(x, t) is augmented by their imaginary components to form a complex data matrix, XC (x, t), which can be represented as: XC (x, t) = XC [cos( XC t) + i sin( XC t)],

(3)

where XC  and  XC are the magnitude and phase of XC , respectively. Under this assumption, it can be easily verified that: CR = X C XC [cos( X  t)cos( XC t) + sin( X  t)sin( XC t)], C

(4)

C

and CI = X C XC [cos( X  t)sin( XC t) − sin( X  t)cos( XC t)], C

(5)

C

where can be seen that when the time is in phase with the extremum of the cosine or sine, the resulting matrix C = CR + iCI from (2) is a Hermitian matrix, where its real part is a symmetric matrix, i.e., CR = C R , where the superscript  indicates transposed vector, and CI is an asymmetric matrix or hemisymmetric matrix,  i.e., C I = −CI , with det(CI ) = 0 when its size is odd. Since the symmetrical matrix is a particular case of the Hermitian matrix, then its eigenvectors are real with eigenvalue 1 > 2 · · · >0. Due to the fact that all of the elements for the asymmetrical matrix are purely imaginary, then it is a normal matrix, i.e., all of its eigenvectors are in the complex conjugate form. Therefore, the optimal orthogonal basis for the modal decomposition is defined by eigenfunctions ϕR (x) and ϕI (x) for both real and imaginary parts of (2). The orthogonal basis defined in the infinite-dimensional Hilbert space L2 ([0, 1]) are computed from the eigenvalue problem by solving the linear system with the form C␸(x) = ϕ(x), and it is optimal in the sense that maximizes the average projection of the response matrix X(x, t), suitably normalized: max

ϕ ∈ L2 ([1,0])

| (X(x, t), ϕ(x) )|2 subject to  ϕ(x) 2 = 1,

(6)

where |·| denotes the modulus, ||·|| is the L2 -norm and · implies the use of an average operation. Then, the associated approximation to the data set in terms of a truncated sum of dominant empirical modes p and q derived from the kinetic energy contribution contained in the eigenvalue 1 > 2 · · · >0, is given by:

⎡ ⎤ p q   ∗ ∗ X(x, t) ∈ Rm×n = real ⎣ aR(j) (t)ϕR(j) (x) + i aI(j) (t)ϕI(j) (x)⎦ , j=1

j=1

(7) *

where denotes the complex conjugation with temporal coefficients a(t) ∈ C, which are computed as:

2. Empirical orthogonal functions analysis The EOFs analysis is developed to be applied for representations of a data set X(xj , tk ) ∈ Rm×n , with n  m, where xj to j = 1, . . ., n, represents the spatial variables and tk with k = 1, . . ., m, is the time at which the observations are made. A statistical decomposition illustrates the phenomenon of modal distribution, which is derived from the response of large interconnected systems. In [22–24], the model based on the EOFs analysis is given by: X(x, t) ∈ Rm×n = Xswc (x, t) + Xtwc (x, t),

(1)

where Xswc and Xtwc denote standing and traveling waveform components, respectively. The method is established into split a complex autocorrelation matrix computed from the resulting data array defined as: C ∈ Cn×n = √

1 H X (x, t)XC (x, t) = CR + iCI , m C

(2)

with i = −1; where the subscripts C, R and I indicate complex, real and imaginary vectors, respectively, while the superscript H denotes the conjugate transposed of a complex matrix. Implicit in

aj (t) = X(x, t), ϕj (x) /ϕj (x), ϕj (x) .

(8)

From (7), it should be noticed that the statistical representation given from the EOFs analysis is established through the coefficients ϕ(x) and a(t), which represent an optimal modal distribution to the data set. In ϕ(x) are given both, the spatial modal distribution and the variability associated with the mode shape, while a(t) is used to study the modal geographical dispersion. Moreover, the phase variability shows the relative phase fluctuation among various spatial locations where the modal distribution is defined. The modal distribution amplitude and its phase variability are computed, respectively, from (7) by: S=



ϕ∗ (x)ϕ(x),

˚ = tan−1

 ϕ (x) I

ϕR (x)

(9) .

(10)

Moreover, a measurement of the modal variability, in magnitude and phase, for a particular oscillation defined into the spatial modal structure is given by a(t). This information is usually used in

246

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

order to identify waveform components with similar distribution coordinates. Therefore, these waveform components are used to establish coherence groups in vast systems. From (7), the temporal variability and phase fluctuation are estimated respectively by: R=



a∗ (t)a(t),

 = tan−1

(11)

 a (t) I

aR (t)

.

(12)

Thus, the parameters given in (9)–(12) describe distribution properties to the data set, where its waveform envelopes A, angular frequency ω(t), and phase velocity c respectively are defined by: A = R · S, ω(t) = c=

(13)

d , dt

(14)

ω(t) , k(x)

(15)

with k(x) =

d˚ . dx

(16)

Fig. 1. Order reduction scheme to MIMO systems.

identification procedure outlined by VF, can be expressed as the linear time-invariant (LTI) system: x˙ = Ax + Bu

(22)

3. Vector fitting

y = Cx

In [5], it is presented the general methodology for vector fitting (VF), where the objective is to fit a frequency domain response f(s) to s = i · ˝ defined into angular frequency range ˝ = [ωmin ωmax ], with an approximation of N rational functions in the form:

where A is a diagonal matrix with the obtained poles, B is a column vector of ones, and C is a row vector containing the residues [28]. In general, this fitting technique is accurate, robust and efficient on transmission line modeling and to obtain network equivalents [3,5–8,10,11].

f (s) =

N  rk k=1

s − ak

+ d + se,

(17)

where the terms d and e are optional. In this method, the function to be fitted can be either a scalar or a vector. In the latter case, all elements in the vector will be fitted using the same poles. Vector fitting is based on making the approximation in two stages; firstly identifying the poles of f(s) by solving in the least-squares sense, the linear problem: (s)f (s) = p(s),

(18)

with (s) =

N  r˜k k=1

p(s) =

s − hk

(19)

4.1. General theory The methodology here proposed is analogous to the use of EOFs analysis applied to time-varying data series. However, in this work the application of the EOFs analysis is extended to frequencyvarying data series [20,21]. In the method, it is defined that the optimal basis functions are derived from eigenvectors of a crossspectrum matrix, and these are used to study the modal coherence and distribution into the frequency-varying response. Illustratively in Fig. 1 is shown the order reduction scheme for large interconnected power networks, which is used in this work to derive a dynamic equivalent with applications to MIMO systems. 4.2. Data array

N  rk k=1

+ 1,

4. Proposed methodology

s − hk

+ d + se,

(20)

where (s) is a scalar, while p(s) is generally a vector, and hk is a set of initial poles. Then, it can be shown that the poles of f(s) must be equal to the zeros of (s) which can be calculated as the eigenvalues of a matrix: ak = eig(A − b · c  ).

(21)

In (21), A is a diagonal matrix holding the initial poles hk , b is a column vector of ones, and c is a row vector holding the residues r˜k , where the superscript  indicates transposed vector. In addition, an unknown frequency-dependent scaling parameter is introduced, which allows that the scaled function be accurately fitted with the initial poles. A new set of poles is obtained from the fitted function and then used in the second stage in the fitting of the unscaled function. The final approximated function is obtained through an iterative process. Thus, the model obtained from the

In our proposal, it is assumed that the frequency-varying response of the discrete input–output relation of a multiple-input multiple-output (MIMO) linear-time invariant (LTI) system is given in the form of a m × n-dimension data array, which is derived from the transfer function matrix, i.e.:

⎡

g1 (s1 )

g2 (s1 )

···

gn (s1 )

⎤

⎢ g (s ) g (s ) · · · g (s ) ⎥ n 2 ⎥ 2 2 ⎢ 1 2 ⎥, ⎥ .. .. .. ⎣ ... ⎦ . . .

G(, s) ∈ Cm×n = ⎢ ⎢

g1 (sm )

g2 (sm )

···

(23)

gn (sm )

where sk = i · ˝ to k = 1, . . ., m is defined for discrete angular frequencies into the range ˝ = [ωmin(1) · · · ωmax(m) ] subjected to the EOFs analysis. The frequency domain responses or functions gj (s) to j = 1, . . ., n, are derived from the model adopted to study the power networks which defines the frequency-range and the geographical distribution under study. The spatial representation for

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

247

the frequency-varying function in (23) is defined through the variable. 4.3. Method In this section we present the general methodology to identify reduced-order equivalent networks using the EOFs analysis and the VF procedure from frequency-domain responses of large power networks. In our approach, the proposed procedure is based on the analysis of frequency-varying responses from power networks given in a data array in the form (23). In order to analyze the collected measurements, the EOFs analysis is developed in the complex vectors array to detect its modal distribution and a reduced-order equivalent in the form (7). To the case of data series purely imaginary, the Hilbert transform can be used to form its quadrature function in order to get phase information, where the MatLab function used to this end is gc (s) = i · hilbert(g(s)/i). In the method, it is defined that the optimal basis functions are obtained from eigenvectors of a cross-spectrum matrix given in the form,

⎡

Cω () ∈ C

n×n

F11 (ω)

F12 (ω)

···

F1n (ω)

⎤

Fn2 (ω)

···

(24)

gj∗ (ω), gk (ω ) = Fjk (ω)ı(ω − ω ),

(25)

where the angular brackets indicate ensemble average and the asterisk denotes complex conjugation. Since it can be shown directly that the cross spectrum is a Hermitian operator. Then, we may use Fj k (ω) by solving the eigenvalue problem given in the form Cω ()ϕω () = ω ϕω (),

(26)

where ϕω () is the eigenfunction corresponding to the eigenvalue ω , and these are used to study the modal coherence and distribution into large power networks [20,21]. Note that the complex eigenfunctions and the real eigenvalues depend upon frequency ω. Once the empirical orthogonal basis to the spectral decomposition is derived from (26), a reduced-order equivalent model to the frequency-varying data series given in (23) is defined in the form (7) as: p 

∗ aR(ωj ) (, s)ϕR(ω () + i )

q 

j

j=1

∗ aI(ωj ) (, s)ϕI(ω (), ) j

(27)

j=1

its spectral coefficients, aωj (, s) = where G(, s), ϕωj () /ϕωj (), ϕωj (x) , are given by the same convention used with the usual EOFs analysis to time-varying data series. A more detailed discussion can be found in [20,21]. Therefore in the method here presented, the VF procedure is applied to fit the spectral coefficients aω (, s) in a rational function of the form (17), such as: aω (, s) =

N  rk k=1

s − ak

+ d + se

ability of the method to be used in identification of reduced-order dynamic equivalents to MIMO systems. Therefore, the equivalent network in a state space model to a particular frequency band is defined by:

∗ ()x = g (x, u(t), ϕ∗ ()) yj = Cj ϕω j j ωj j

,

(29)

with a transfer function matrix associated to:

Fnn (ω)

where Fj k (ω) denotes the cross-spectrum between the jth and the kth series, which is derived by,

G(, s) =

obtained by multiplication between the estimated residues from the VF procedure and the optimal orthogonal basis derived from the ∗ (). This modification greatly improves the EOF analysis, r k = rk ϕω j

x˙ j = Aj xj + Bj u = fj (x, u(t))

⎢ F (ω) F (ω) · · · F (ω) ⎥ 22 2n ⎢ 21 ⎥ ⎥, =⎢ ⎢ . ⎥ . . .. .. . ⎣ .. ⎦ . . Fn1 (ω)

Fig. 2. Network configuration.

(28)

thus an LTI equivalent model based in (22) is obtained where A contains the estimated poles, B is a column vector of ones, and C is a row vector containing the estimated residues [28]. One of the main steps here is the calculation of the corresponding residue matrices to the proposed empirical equivalent in LTI model, which are

∗ ˆ j (, s) = [Cj ϕω G ()(sI − Aj )−1 Bj ], j

(30)

which considers the modal distribution and a reduced-order model to large power networks. In (30) I is used to define the identity matrix. Then, (29) can be written in real values to the global equivalent network in the form (1) as: x˙ =

n 

f(swc)j (x, u(t)) +

j=1 n

y=



n 

f(twc)j (x, u(t))

j=1 ∗ g(swc)j (x, u(t), ϕω )+ j

j=1

n 

,

(31)

∗ g(twc)j (x, u(t), ϕω ) j

j=1

with ˆ G(, s) =

n 

ˆ (swc)j (, s) + G

j=1

n 

ˆ (twc)j (, s). G

(32)

j=1

The objective in this study has been to infer the relationship between modal coherence and the spatial relationship of waveform present within a particular frequency band. The identification method presented in this section assumes that the frequency-varying responses of power networks is subjected to cross-spectrum analysis. 5. Application 5.1. Test case I: an illustrative case Without loss of generality and, as an illustrative case of study to derive the physical and practical interpretation of results obtained from the proposed methodology in state-space representations to SISO systems, we consider three identical frequencies dependent single-phase overhead transmission lines with aluminum electrical conductor type Ostrich of length l = 100 km each connected, as it is shown in Fig. 2 [25,27,29]. The conductor height is equal to h = 27 m with a radius equal to rc = 1 cm, and dc resistance equals to 8.9954 × 10−5 ohm/m, the ground resistivity is taken to be equal to 100 ohm/m. The voltage source corresponds to u(t) = sin(ω0 (t)), with ω0 = 120 , and its internal resistance and

248

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

Fig. 3. Transmission line (TL) equivalent circuit.

Fig. 5. Current nodal seen from bus 1.

Fig. 4. The admittance seen from the bus 1.

inductance are R0 = 0.001 ohm and L0 = 0.005 H, respectively. At the end of each line, there exists a R − L load connected in parallel with values R = 1 ohm and L = 0.3 H. The line/cable model considering the circuit equivalent shown in Fig. 3, corresponds to a full frequency-dependent model with its voltage/current relation between sending and receiving ends is given by:

is

ir



=

Y1

Y2

Y2

Y1



vs

(33)

,

vr

where A = Yc coth(l), and B =− Yc csch(l), with Yc and  being the line characteristic admittance and propagation function, respectively [27,29]. Let us consider the transfer function (admittance) seen from bus 1 as the function g(s) to be fitted, which is evaluated to 5000 equally spaced points into the range for angular frequencies ˝ = [1 − 1000] · 2 . The current (I)/voltage (V) relation seen from bus 1, which is used to application of the proposed methodology, is given by: I1 −1 = Y11 − Y12 Y22 Y21 − Y0 , V1

(34)

with Y11 = A + Y0 , Y12 = [−B 0 0], Y21 = [−B 0 0] , Y0 =

⎡ ⎢ ⎢ Y22 = ⎢ ⎢ ⎣

R + sL sRL

−B

−B

−B

R + sL A+ sRL

0

−B

0

3A +

A+

R + sL sRL

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

1 , R0 +sL0

and

(35)

Therefore, the identification procedure outlined in Section 4 is applied to the frequency response derived from (33), where the admittance seen from the bus 1 is fitted from two equivalent subnetworks using its spectral distribution. It should be noticed that in the application of EOFs analysis, the data array is defined using the properties of a sparse matrix [11,12]. During the VF procedure, the fitting to the spectral coefficients aω (, s) is carried out with ten pairs of complex poles linearly spaced involving three VF iterations, yielding a root mean square (rms) error belows 10−3 . So far, the number of VF iterations for a given accuracy remain an open topic. Thus, this number has been taken here arbitrarily. Fig. 4 shows a comparative analysis of the admittance selected from

the proposed method and the VF conventional procedure, where it can be seen that the proposed method provides a particular global extraction from measurements allowing to derive a reduced-order equivalent network with multiple spectral distribution, while VF conventional is forced to approximate only an objective function through an iterative process and scale parameters. The main drawback in the application of the EOFs analysis during the process of their modal extraction is related with the setting of an optimal sampling interval ω to be used when acquiring the responses from the given systems. This can be fixed to suitable values considering the signals treated as transient types by ω = (˝min /2)2 . The proposed approach of network equivalent, computationally efficient and derived through this example, provides a practical procedure to the distributed solution to be efficiently incorporated into a power systems simulator to transient studies. In order to illustrate a straightforward comparison between methods about the use of the derived equivalent model in the power networks analysis into the transient simulation package MatLab-Simulink version 8.5 (R2015a), in Fig. 5 is shown the response to the current nodal seen from the bus 1 of the given system in (34), where the model is solved using a sampling interval of t = 1 ×103 s over a t = 0.04 s window. From these results, it can be noticed that the approaches to define a reduced-order equivalent of SISO systems, using the EOF analysis in combination with the VF procedure, provides a manner practical to analyze and define a power equivalent network related with the dispersion and spectral distribution. 5.2. Test case II: 16-generator 68-bus test system In this section, the 16 generator 68 bus New England system is selected to test the application of the proposed method in large power networks, where our attention is focused to examine in detail the relationship between low frequency modes and coherent generator groups in the transmission system network. Fig. 6 shows the single-line diagram to the test system, where linear analysis studies have been conducted in [30] to determine the fundamental modes of oscillation, system’s characteristics and base case operation condition. For the purposes of this analysis, it is assumed that each generator is represented by the classical model, the loads are modeled as constant impedances and the interconnected transmission network is reduced to the generator internal nodes. To stress the system, the circuit of the tie line between bus 1 and bus 27 is taken out of service. In order to assess the proposed method on MIMO systems, the (open-loop) system transfer function matrix G(s) = [C(sI − A)−1 B + D] derived from the linear dynamic model of the proposed power network is used to compute its frequency-varying response, which is evaluated to 5000 equally spaced points into the range for angular frequencies ˝ = [0.01 − 2] · 2 . Using the frequency-varying data set obtained as series in the frequency domain for all system’s generators speeds, G(s) ∈ Cm×n , with n = 256 and m = 5000, the proposed method was applied to derive a reduced-order dynamic equivalent to the power network. In Fig. 7 are presented the locus of the identified

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

249

Fig. 6. The 16 generator 68 bus test system.

Fig. 7. Locus of the identified eigenvalues to the real and equivalent model.

Table 1 Natural frequencies of the electromechanical modes to the 16-generator system. Mode

Eigenvalue

Frequency (Hz)

1 2 3 4

−0.0582 ± 2.4926 i −0.0468 ± 3.0599 i −0.1477 ± 3.8922 i −0.2391 ± 4.9537 i

0.3967 0.487 0.6195 0.7883

eigenvalues to the real and equivalent model of the electromechanical modes of interest. These modes exhibit the exchanges of oscillating energy between the associated areas. Table 1 synthesizes the linear system eigenvalues for the base case condition showing the main electromechanical modes. The data presented in Fig. 7 show that there is good agreement between the locus of the identified eigenvalues and the real model. This is an initial indication that the identified equivalent model is valid, which is used in this test case for simplicity considering the large size of the related coefficients matrix to validate the state space representation for the original and reduced-order equivalent model. Fig. 8 depicts the

Fig. 8. Spectral coefficients showing the captured frequency band.

spectral coefficients computed, which are associated to the two dominant modes that contains the major frequency distribution. This frequency represents 95% of the total energy; furthermore, the obtained modal frequency from linear analysis studies is shown with solid-arrow line to illustrate the applicability of the proposed method. In Fig. 9 is shown the mode shape determined from the spatial base function, which is used to study coherent generator groups. Additionally, in Fig. 9 is shown that the generators of the Set 1 behave coherently in low frequency electromechanical oscillations, and that groups of coherent generators (Set 2 and Set 3) are separated from other groups of coherent generators by weak interconnections in the transmission system network. In order to validate the derived state-space equivalent model, in Fig. 10 are shown the results to the impulse response as the input signal to the global equivalent, where the model is solved using a sampling interval of t = 0.01 s over a t = 15 s window, given a comparison between the speed change to the generator 1 as output signal, with a speed/voltage relation given in Fig. 11. From the numerical results, we observe that the natural response obtained from the state-space equivalent model provides a precise characterization

250

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

Fig. 9. Mode shape showing the coherent generator groups.

system behavior and allows the study of inter-area electromechanical oscillations into large interconnected power systems. Moreover, the proposed approach constitutes a natural extension to choose desirable input–output pairs. Reference [24] presents a numerical algorithm that the reader may be interested to use in order to identify and extract modal components using the EOF analysis, which significantly decreases the computational burden and gives a distributed application in the analysis of large data sets. 6. Discussion In this work, the authors of the method here presented have found a criterion based in level of spectral distribution to define the accuracy of a derived reduced-order equivalent model with applications to MIMO systems. This is achieved by removing orthogonal modes below a threshold, which is directly related to the spectral dispersion of electromechanical instabilities into power networks with a particular frequency band. One of the main steps here is the calculation of the corresponding residue matrices to the proposed empirical model, which are obtained by multiplication between the row vector containing the estimated residues from a VF procedure and the optimal orthogonal basis derived from the

EOF analysis. This modification greatly improves the ability of identification of a reduced-order dynamic equivalent to large power networks. Additionally, a criterion of passivity of network to ordermodel reduction is directly related to enforce that the real part for admittance matrices approximated by rational functions must be positive definite, which represents a drawback in its practical application in modal identification, for instance this was studied in [26]. This criterion is satisfied in a straightforward manner from the proposed methodology in this work by using a Hermitian matrix, i.e., a complex cross-spectrum matrix derived from frequency-domain response of power networks, where its real part is a symmetric matrix, i.e., CR = C R and CI is an asymmetric matrix or hemisym metric matrix, i.e., C I = −CI , with det(CI ) = 0 when its size is odd. Since the symmetrical matrix is a particular case of the Hermitian matrix, then its eigenvectors are real with eigenvalue 1 > 2 · · · >0. Due to the fact that all of the elements for the asymmetrical matrix are purely imaginary, then it is a normal matrix, i.e., all of its eigenvectors are in the complex conjugate form. An interesting result is that the use of the proposed method can shed light on the number of modes and its spatial distribution for predefined oscillations into power networks satisfying with the criterion of passivity. 7. Conclusion

Fig. 10. Speed changes to the generator 1.

In this paper, a frequency-domain identification methodology to reduced-order dynamic equivalents of power networks with modal distribution has been proposed. The method is established on the application of the EOFs analysis and the VF procedure for rational function approximations from frequency-varying data. The application of the method has resulted to identify coupling frequencies and to split waveform disturbances superposed in the same frequency band into electromechanical oscillations in power systems. It makes possible to discriminate between different types of disturbances on the basis of frequency-varying waveforms. The validation of the proposed method has been made through a large power network, where the results have demonstrated its viability for fitting reduced-order equivalent networks and distributed modal responses. The method can be efficiently incorporated in a power network simulator to electromechanical studies of multimachine dynamics with modal frequency splitting. It is important to mention that the propose methodology for identification of a reduced-order network, can be widely applied to MIMO systems. Acknowledgments

Fig. 11. Speed/voltage relation seen from the generator 1.

This paper was supported in part by Consejo Nacional de Ciencia y Tecnología (México) under Cátedras CONACYT, the retention program no. 120489, and by Tecnológico Nacional de México/Instituto Tecnológico de la Laguna under project no. 1982.

P. Esquivel, C.E. Casta˜ neda / Electric Power Systems Research 143 (2017) 244–251

References [1] A. Ramirez, Vector fitting-based calculation of frequency-dependent network equivalents by frequency partitioning and model-order reduction, IEEE Trans. Power Deliv. 24 (1) (2009) 410–415. [2] A. Ramirez, A. Semlyen, R. Iravani, Order reduction of the dynamic model of a linear weakly periodic system – Part I: General methodology, IEEE Trans. Power Syst. 19 (2) (2004) 857–865. [3] A. Ubolli, B. Gustavsen, Comparison of methods for rational approximation of simulated time-domain responses: Arma, zd-vf, and td-vf, IEEE Trans. Power Deliv. 26 (1) (2011) 279–288. [4] L.A. Wedepohl, Frequency Domain Analysis of Transient Phenomena, CEA Transactions, 1989. [5] B. Gustavsen, A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Trans. Power Deliv. 14 (3) (1999) 1052–1061. [6] B. Gustavsen, H.M.J. De Silva, Inclusion of rational models in an electromagnetic transients program: Y-parameters, z-parameters, s-parameters, transfer functions, IEEE Trans. Power Deliv. 28 (2) (2013) 1164–1174. [7] B. Gustavsen, T. Noda, Techniques for the identification of a linear system from its frequency response data, in: J.A. Martinez-Velasco (Ed.), Power System Transients: Parameter Determination, CRC Press, 2009, ISBN 9781420065299. [8] U.D. Annakkage, N.K.C. Nair, Y. Liang, A.M. Gole, V. Dinavahi, B. Gustavsen, T. Noda, H. Ghasemi, A. Monti, M. Matar, R. Iravani, J.A. Martinez, Dynamic system equivalents: a survey of available techniques, IEEE Trans. Power Deliv. 27 (1) (2012) 411–420. [9] T. Noda, A. Ramirez, Z-transform based methods for electromagnetic transient simulation, IEEE Trans. Power Deliv. 22 (3) (2007) 1799–1805. [10] A. Ubolli, B. Gustavsen, Multiport frequency-dependent network equivalencing based on simulated time-domain responses, IEEE Trans. Power Deliv. 27 (2) (2012) 648–657. [11] W.D.C. Boaventura, A. Semlyen, M.R. Iravani, A. Lopes, Robust sparse network equivalent for large systems: Part I – Methodology, IEEE Trans. Power Syst. 19 (1) (2004) 157–163. [12] W.D.C. Boaventura, A. Semlyen, M.R. Iravani, A. Lopes, Robust sparse network equivalent for large systems: Part II – Performance evaluation, IEEE Trans. Power Syst. 19 (1) (2004) 293–299. [13] F. Gao, K. Strunz, Frequency-adaptive power system modeling for multiscale simulation of transients, IEEE Trans. Power Syst. 24 (2) (2009) 561–571. [14] H. Zhou, P. Ju, H. Yang, R. Sun, Dynamic equivalent method of interconnected power systems with consideration of motor loads, Sci. China Technol. Sci. 53 (4) (2010) 902–908.

251

[15] K. Sheshyekani, B. Tabei, Multiport frequency-dependent network equivalent using a modified matrix pencil method, IEEE Trans. Power Deliv. 29 (5) (2014) 2340–2348. [16] M.L. Ourari, L.A. Dessaint, V.Q. Do, Dynamic equivalent modeling of large power systems using structure preservation technique, IEEE Trans. Power Syst. 21 (3) (2006) 1284–1295. [17] M. Matar, R. Iravani, A modified multi-port two-layer network equivalent for the analysis of electromagnetic transients, IEEE Trans. Power Deliv. 25 (1) (2010) 177–186. [18] T. Noda, Application of frequency-partitioning fitting to the phase-domain frequency-dependent modeling of overhead transmission lines, IEEE Trans. Power Deliv. 30 (1) (2015) 174–183. [19] T. Noda, A. Semlyen, R. Iravani, Reduced-order of a nonlinear power network using companion-form state equation with periodic coefficients, IEEE Trans. Power Deliv. 18 (2003) 1478–1488. [20] J.M. Wallace, Empirical orthogonal representation of time series in the frequency domain. Part II: Application to the study of tropical wave disturbances, J. Appl. Meteorol. 11 (6) (1972) 893–900. [21] J.M. Wallace, R.E. Dickinson, Empirical orthogonal representation of time series in the frequency domain. Part I: Theoretical considerations, J. Appl. Meteorol. 11 (6) (1972) 887–892. [22] P. Esquivel, Wave-area wave motion analysis using complex empirical orthogonal functions, in: International Conference on Electrical Engineering Computing Science and Automatic Control, 2009, pp. 1–6. ˜ [23] P. Esquivel, C.E. Castaneda, Spatio-temporal statistical identification methodology applied to wide-area monitoring schemes in power systems, Electr. Power Syst. Res. 111 (1) (2014) 52–61. ˜ [24] P. Esquivel, C.E. Castaneda, F. Jurado, Partitioned modal extraction to power system wide-area measurements using empirical orthogonal functions, Electr. Power Syst. Res. 123 (2015) (2015) 185–191. [25] The Aluminum Association, Aluminum Electrical Conductor Handbook, 3rd ed., The Aluminum Association, New York, 1989. [26] B. Gustavsen, A. Semlyen, Enforcing passivity for admittance matrices approximated by rational functions, IEEE Trans. Power Deliv. 16 (1) (2001) 97–104. [27] R.C. Dugan, M.F. McGranagham, S. Santoso, H.W. Beaty, Electrical Power Systems Quality, McGraw-Hill Companies, 2002. [28] T. Kailath, Linear Systems, Prentice-Hall, 1980. [29] M.E.V. Valkenburg, Introduction to Modern Network Synthesis, Wiley, New York, 1960. [30] R. Rogers, Power System Oscillations, Kluwer Academic Publishers, 2000.