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Determination of PSS location based on singular value decomposition
Electrical Power and Energy Systems 27 (2005) 535–541 www.elsevier.com/locate/ijepes
Determination of PSS location based on singular value decomposition A. Karimpoura,b,1, R. Asgharianb,1, O.P. Malika,* a
Department of Electrical and Computer Engineering, University of Calgary, 2500, University Drive, Calgary, Alta., Canada T2N 1N4 b Department of Electrical Engineering, Mashhad University, Mashhad, Iran Received 7 August 2002; revised 13 July 2004; accepted 13 May 2005
Abstract In a multi-machine power system, it is important to determine the best location for the application of power system stabilizers. A technique based on singular value decomposition to identify generators that contribute in each oscillation mode is introduced. Measures based on singular value decomposition for controllability and observability for every mode are described and the selection of the best location for the installation of the PSSs is done according to these measures. The proposed methodology is designed to use the system transfer function. Studies illustrate the effectiveness of the proposed technique. q 2005 Elsevier Ltd. All rights reserved. Keywords: Power system stability; Singular value decomposition; Controllability
1. Introduction Power systems are known to be subject to low frequency oscillations. If these oscillations persist, they can present limitation on power system performance. Power system stabilizers (PSSs) are used as an effective way to enhance the damping of these electromechanical oscillations [1]. An interesting and important question is to find the generators on which the PSSs be applied for most effective damping of oscillation modes. Several techniques have been proposed in the literature to determine the best location for applying power system stabilizers [2]. Participation factor method is one of the methods in detecting the contribution of various generators in each mode and for suitable location for applying the PSSs [3]. Some authors have proposed sensitivity of PSS effect (SPE) as a measure for determining the suitable location of PSSs [4]. Singular value decomposition (SVD) has been proposed as a measure of the distance of a controllability and * Corresponding author. Tel.: C1 403 220 6178; fax: C1 403 282 6855. E-mail addresses: [email protected] (A. Karimpour), asghrian@um. ac.ir (R. Asgharian), [email protected] (O.P. Malik). 1 Tel.: C98 511 8815100; fax: C98 511 8829536.
0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2005.05.004
observability matrix from singularity in a state space model [5]. This distance is used as a measure to compare the ability of inputs to control an oscillation mode. The main drawback of the above approaches is the determination of linear models from the state space models that usually have large dimensions. Some authors have used transfer function approach to locate the PSSs. In [6] peak of transfer function elements in frequency domain is used as a measure. Transfer function residues are used in [7] to locate PSSs. Singular value is used as a measure to determine the suitable signal for PSSs and its locations in [8–11]. A new method based on the mathematical consideration of SVD through transfer functions is described in this paper. SVD is used as a measure to distinguish the contribution of various generators in each mode. Based on the contribution of each generator to various modes, controllability and observability for each mode is considered and a method for the best position to apply PSSs is introduced. The transfer function of the system is found through identification, so there is no need to construct state space models. The paper is organized as follows: SVD is described in Section 2; a brief explanation of how the transfer function is determined is given in Section 3; a procedure for the selection of the PSS location is described in Section 4; results of simulation studies on non-linear multi-machine power systems are described in Section 5.
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2. Singular value decomposition Consider M is a constant matrix in Cl!m. Then M can be decomposed into its singular value decomposition according to the following theorem [5]. Theorem. Let M2Cl!m Then there exist S2Rl!m and unitary matrices U2Cl!l and V2Cm!m such that M Z USVH " where SZ
0
S
(1)
#
, SZdiag{s1,s2,.,sr} withs1Rs20 0 R.Rsr and r%min(l,m). UZ[u1,u2,.,ul], VZ[v1,v2,.,vm] and VH is complex conjugate transpose of V. The column vector elements of U, identified by ui, are orthogonal and of unit length, and represent the output directions. The column vector elements of V, identified by vi, are orthogonal and of unit length, and represent the input directions [12]. These input and output directions are related through singular values. So one can write: Mvi Z si ui
(2)
It means that if an input is applied in the direction vi, then the output is in the direction ui and has a gain of si. Input direction vi for iOr corresponds to inputs that do not have any influence on outputs and similarly output direction ui for iOr corresponds to the outputs that can not be accessed by any input. Since, the diagonal elements of S are arranged in a descending order, it can be shown that the largest gain for any input direction is equal to the maximum singular value s1 and so one can write s1 Z max ds0
jjMdjj2 jjMv1 jj2 Z jjdjj2 jjv1 jj2
Example: Consider 2 3 1 2 K1 6 7 M Z43 4 1 5 2
0:92
yðsÞ Z GðsÞuðsÞ
8
0:34
K0:17
3H
0
0
0
3
7 07 5 0
0:50 K0:33 K0:80 6 7 7 $6 4 0:35 K0:77 0:53 5 0:79
0:55
3. System transfer function Transfer function identification aims at determining a black box model of the system in the following form
then singular value decomposition of M is: 2 3 2 0:04 K0:53 K0:85 9:77 0 6 7 6 7 6 M Z6 4:53 4 0:38 K0:77 0:51 5$4 0 2
† provides an indication of the relative contribution of different outputs at that frequency, and † shows which generator or group of generators is oscillating against another generator or a group of generators.
(3)
where d is any input signal and k$k2 is the Euclidian norm.
4
0 output is u1 Z 0:04 0:38 0:92 . The gain of 4.53 is
0 for an input in the direction v2 Z K0:33 K0:77 0:55 and the corresponding output direction is
0 u 2 Z K0:53 K0:77 0 0:34 . The input direction v3 Z K0:80 0:53 0:27 has no influence on outputs and
0 the output direction u3 Z K0:85 0:51 K0:17 is not accessible with any input. In this example, input in the direction v1 will excite the largest gain, or, in other words, will be the most effective input. The idea illustrated in the above example can be extended to a matrix having transfer functions, instead of scalars, as its elements. In that case, the SVD of the matrix is determined at any frequency u by substituting s with ju. The peaks of singular values versus frequency plot indicate the frequencies at which the system is likely to exhibit dynamic stability problem. The output direction corresponding to the largest singular value provides an indication of the relative observability of different outputs at that frequency and the input direction corresponding to the largest singular value provides an indication of the relative controllability of different inputs at that frequency. Multiplication of output direction by input direction gives information about residue as well [11]. In a power system the relative magnitude and phase of the elements of the output direction vector (u1), corresponding to the largest singular value, at each frequency of oscillation,
0:27
The largest gain of 9.77 is for an input in the 0 direction v1 Z 0:50 0:35 0:79 and the corresponding
(4)
where y(s) and u(s) are vectors of outputs and inputs of dimension n!1, and G(s) is an n!n transfer function matrix of the system, where n is the total number of generators in the system. The outputs, y(s), of the system are the speeds of various generators and the inputs, u(s), of the system are the excitations of generators. Transfer function gij(s) between input port j and output port i, the ijth element of G(s), can be written as: yi ðsÞ (5) gij ðsÞ Z uj ðsÞ ui Z 0 i sj To find gij(s) inject a small signal to the input port j and record the output at port i. By use of the input and output
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
information and least squares method [13] it is possible to identify gij(s). By performing the same procedure for all elements of G(s) one can directly determine the G(s) of a multi-input/multi-output system.
4. PSS location selection The proposed technique depends upon distinguishing the contribution of various generators to each oscillation mode by finding the minimum number of generators for which the corresponding reduced transfer function has the maximum singular value near to the largest singular value of the real system at that frequency. To do this, the row and column corresponding to each generator is omitted and its effect on the maximum singular value is examined. If the reduction in the maximum singular value is small, one understands that this generator does not contribute in this mode. Otherwise, this generator is contributing in this mode. Repeating this method for all generators and omitting the non-contributing ones, it is possible to find the minimum number of generators whose corresponding reduced transfer function has maximum singular value near to the largest singular value of the real system at that frequency. A procedure to select the appropriate locations for PSSs using SVD is described in the following steps. 1. Substituting ju for s in (4), plot singular values of G(juk) over a range covering the frequencies of oscillation modes. From this plot, find the peaks of maximum singular values that correspond to oscillation modes. Say, these are at frequencies u1,u2,.uk. 2. Determine G(juk) for each mode of oscillation that corresponds to the frequency of u1,u2,.uk. and the corresponding maximum singular values s1max,s2max, .skmax,. 3. Start with one of the oscillation modes, i.e. G(juk). 4. Delete from G(juk) the column and row corresponding to generator i for iZ1,2,3,. one generator at a time and calculate the maximum singular value of the reduced matrix. If this value is much smaller than skmax, generator i contributes to mode uk and should be retained. If instead the maximum singular value is close to skmax, generator i does not contribute to this mode and is deleted. 5. Repeat step 4 until a reduced matrix G(juk) of only those generators that contribute to mode uk is obtained.Note: Since it is possible that uk be the same oscillation mode of more than one group of generators, steps 4 and 5 must be repeated to find all possibilities. 6. For the reduced G(juk) found in step 5 corresponding to the generators contributing to the modeuk, find the SVD of G(juk). The elements of vector u1 are observability measures and the elements of vector v1 are controllability measures for this mode [11]. So one can find the best position to locate PSS and the best signals for applying to PSS.
537
7. For the reduced G(juk) found in step 5, corresponding to the generators contributing to the mode uk, delete one generator at a time and determine the maximum singular value of the remaining system. A generator, the deletion of which leads to the smallest magnitude of the maximum singular value of the remaining system, is the most effective generator for mode uk and is thus the most appropriate location for a PSS to damp this mode. This procedure combines the controllability and observability measures in one step. 8. Repeat steps 3–7 for each mode found in step 1 to determine the appropriate locations for PSSs corresponding to each mode. The above procedure can be easily performed using a computer program. Such a program has been developed to select PSS locations and its application is described in Section 5.
5. Application of SVD in power systems A comparison of the results obtained by the method based on SVD with methods based on participation factor on a four machine system that has closely spaced modes (modal resonance) is given in this Section. Additional studies on a five machine system demonstrate the applicability of the proposed method. 5.1. Four machine system A single line diagram of the 4-machine system is shown in Fig. 1. The parameters of the model are given in [14]. There are three oscillation modes and participation factors (PF) corresponding to the rotor speed for these three modes are given in Table 1. Phasor diagram of the entries corresponding to speed (mode shapes) for the oscillation modes are shown in Fig. 2. As the eigenvalues are practically equal in this case, the eigenvectors are not unique. Thus the mode shapes given by the eigenvectors in Fig. 2 are physically meaningless and participation factors do not give the correct information [14]. The singular value plot of the 4-machine system versus frequency is shown in Fig. 3. Examination of the singular values shows that the gain of the system around 3.36 and 6.63 rad/s is high, so there are oscillation modes near these frequencies. By following the procedure explained in
Fig. 1. Example of a 4-machine power system.
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A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
Table 1 Participation factors for system in Fig. 1 Unit
G1 G2 G3 G4
u12 u12
Mode 1 (K0.39C6.651)
Mode 2 (K0.40C6.631)
PF
PF
PF
0.641 0.8049 0.5641 1
0.6972 1 0.6749 0.9135
0.2503 0.1428 1 0.6749
u14
u13
Mode 3 (0.15C3.291)
u11
u11 (a) 6.63 rad/s
u12 u11
(b) 6.63 rad/s
(c) 3.36 rad/s
Fig. 4. Elements of output direction vector corresponding to maximum singular values.
0.35 Table 2 Controllability measures based on SVD
0.3
Unit
Controllability measures 6.64 rad/s
6.64 rad/s
3.36 rad/s
Exciter of G1 Exciter of G2 Exciter of G3 Exciter of G4
0.711 0.703 *** ***
*** *** 0.693 0.721
0.489 0.502 0.494 0.515
Singular values
0.25 0.2 0.15 1
0.1 0.05
2
0 100
101 Frequency (rad/s) Fig. 2. Mode shapes for system in Fig. 1.
Section 4, it is found that at 6.63 rad/s, the set {G1,G2} has the maximum singular value of 18.1534, the set {G3,G4} has the maximum singular value of 17.5610 and the maximum singular value of the real system is 18.5143. So there are two oscillation modes close to this frequency. Elements of the output direction vector, corresponding to the maximum singular value of each set of generators at 6.63 rad/s, shown in Fig. 4(a) and (b) clearly indicate the two local modes. In one of them G1 oscillates against G2 and in the other G3 oscillates against G4. The same procedure for the frequency of 3.36 rad/s shows that the combination of all generators gives the suitable maximum singular value, and the elements of output direction corresponding to the maximum singular value at this frequency (Fig. 4(c)) show that it is an inter-area G3
G4 G3
G3 G1
G2
G2
Table 3 Observability measures based on SVD Unit
G1
G1
mode such that G1 and G2 combined oscillate against G3 and G4. The next step is to find the best position to apply PSSs. First choose the transfer function corresponding to the generators contributing in each mode. This is done by deleting from the total transfer function the columns and rows corresponding to the generators that do not contribute in this mode. By substituting the frequency of oscillation as ju instead of s and finding SVD of the transfer function, the elements of u1 and v1 (output and input direction corresponding to maximum singular values) show the observability and controllability of the oscillation mode, respectively. Controllability and observability measures based on SVD are shown in Tables 2 and 3. Table 2 shows that the best position for improving the oscillation mode of frequency 3.36 rad/s is exciter of G4 and the best signal for applying PSS is speed of G3, according to Table 3. The same interpretation shows that for local mode between G1 and G2 the best candidate is G1 and the best signal for applying PSS is speed of G2. For local mode between G3 and G4 the best candidate is G4 and the best signal for applying PSS is speed of G4. Now, one can consider both observability and controllability measure through the procedure in Section 4.
G2 G4
G4
(a) Mode 1
(b) Mode 2
(c) Mode 3
Fig. 3. Singular values of the 4-machine system.
Speed of G1 Speed of G2 Speed of G3 Speed of G4
Observability measurs 6.64 rad/s
6.64 rad/s
3.36 rad/s
0.625 0.781 *** ***
*** *** 0.574 0.819
0.155 0.115 0.725 0.661
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541 Table 4 Singular values of 4-machine system
539
Frequency (rad/s)
3.36
6.63
6.63
Convolving generators
#1, #2, #3, #–4
#1, #2
#3, #4
Number of omitted generator
Max singular value
No. omission #1 #2 #3 #4
89.5925 77.17 76.9287 53.0017 57.00463
Velocity (rad/s)
0.8
0.4 0.2 0 –0.2 0
1
2
1
2
3 4 5 6 7 (a) Actual and identified velocity
8
9 10 Time (s)
8
9
0.02
17.561 *** *** 10.5226 7.2949
Input signal
18.1534 10.0695 8.201 *** ***
actual velocity identified velocity
0.6
0.01 0 –0.01 –0.02 –0.03 0
3
4
5
6
7
(b) Signal applied to exciter of generator number 1
By deleting the column and row corresponding to each generator in turn, check the maximum singular value of the remaining system. A generator that leads to the maximum reduction in the largest singular value is the most effective generator for damping of that mode. The largest singular values obtained by deleting each generator in turn for all of modes are shown in Table 4. According to Table 4 omitting G3 reduces the singular values corresponding to 3.36 rad/s the most, so it is the best position to apply PSS for improving the performance of the system, and G2 and G4, respectively, are best positions for improving the local modes. A rank between generators for improving this oscillation mode is also shown in Table 4. 5.2. Five machine system A single line diagram of a 5-machine 8-bus system [15] is shown in Fig. 5. The parameters of the model and operating conditions are given in [15]. In this system, generators 1, 2 and 4 are large and generators 3 and 5 are 1/10 the size of the other three generators. A known signal is applied to the exciter of each generator and both input signal and the speed of each generator, as the output, are recorded. Using this procedure, a twelfth order transfer function for each output/input pair of the system
Fig. 6. Gen #1 to #2 transfer function.
model is determined. Actual speed of generator 2 as obtained by direct simulation of the system and as obtained from the identified transfer function when exciter of generator number 1 is excited is shown in Fig. 6(a) and the input signal used for exciter number 1 is shown in Fig. 6(b). The singular value plot of the 5-machine system versus frequency is shown in Fig. 7. Examination of the singular values shows that the gain of the system around 4 rad/s (0.64 Hz), 6.4 rad/s (1.02 Hz) and 8.5 rad/s (1.40 Hz) is large, so there are oscillation modes near these frequencies. By following the procedure explained in Section 4, it is found that at 8.5 rad/s, the set {G2,G3,G5} has the maximum singular value of 12.727 and the maximum singular value of the real system is 12.734. By referring to the elements of output direction corresponding to maximum singular value at this frequency shown in Fig. 8(a), one can see that this is a local mode between generators 2, 3 and 5. 100 90 80
6
70
1 G1
G3 L2 2
8
G2
Singular values
7
3
60 50 40 σ1
30 L3
4
5
20 10
G5 G4 L1 Fig. 5. Example of a 5-machine power system.
10 Time (s)
σ2
0 100
101 Frequency (rad/s) Fig. 7. Singular values of the 5-machine system.
540
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
The same procedure for the frequency of 6.4 rad/s (Fig. 8(b)) shows that there is a local mode between generators 1 and 4. At 4 rad/s, the combination of all generators gives the suitable maximum singular value and the elements of output direction corresponding to the maximum singular value at this frequency (Fig. 8(c)) show that it is an inter-area mode. The next step is to find the best position to apply PSSs. By the same procedure as before and deleting the column and row corresponding to each generator in turn, and checking the maximum singular value of the remaining system, best position for improving each oscillation mode can be found. The largest singular values obtained by deleting each generator in turn for all modes are shown in Table 5. According to Table 5 for interarea mode (4 rad/s) omitting generator 2 reduces the singular value the most, so it is the best position to apply PSS for improving the performance of the system. A rank between generators for improving this oscillation mode is also shown in Table 5. Similarly best position for improving oscillation mode of frequency 6.4 rad/s is generator 4 as shown in Table 5 and generator 2 is again the most suitable place for improving the 8.5 rad/s local mode. For comparison purposes, a conventional PSS with the following transfer function [16] was installed on each of
Fig. 8. Output direction corresponding to maximum singular values.
Table 5 Singular values of 5-machine system Frequency (rad/s)
4
6.4
8.5
Convolving generators
#1,#2,#3,#–4
#1, #4
#2, #3, #5
Number of omitted generator
Max singular value
No. omission #1 #2 #3 #4 #5
97.444 92.612 45.442 86.038 77.633 85.629
32.003 17.434 *** *** 15.317 ***
17.734 *** 8.893 11.133 *** 10.723
3
3 PSS on g1 PSS on g2 PSS on g4
1 0 –1 –2
0
2
4
PSS on g1 PSS on g2 PSS on g4
2 Velocity (rad/s)
Velocity (rad/s)
2
1 0 –1 –2
6
0
2
4
6
Time (s)
Time (s) (a) Different speed of gen #1 and gen #2
(b) Different speed of gen #2 and gen #4 10
PSS on g1 PSS on g2 PSS on g4
4
Velocity (rad/s)
Velocity (rad/s)
6
2 0 –2
PSS on g1 PSS on g2 PSS on g4
5
0
–5
–4 0
2
4
6
0
2
Time (s) (c) Different speed of gen #1 and gen #2
4
6
Time (s) (d) Different speed of gen #2 and gen #4
Fig. 9. Result for the 5-machine system.
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
the large generators, i.e. 1, 2 and 4 in turn FðsÞ Z 0:5
10s 1 C 0:2s 1 C 0:2s 1 C 10s 1 C 0:04s 1 C 0:04s
541
References (6)
Simulation results in response to a 200 ms 3 phase short circuit on bus 3 are shown in Fig. 9(a) and (b) and for a short circuit on bus 2 are shown in Fig. 9(c) and (d). It can be seen that by applying PSS on generator 2 leads to the best performance of the system for the damping of the inter-area mode as determined in Table 5.
6. Conclusions A new approach based on singular value decomposition is proposed to find the generators contributing in each oscillation mode and also the best location for applying PSSs is determined. This approach only uses the information of transfer function matrix and can be used to find the best location for application of PSS. The advantages of the proposed method are: † No computation involving the right (left) eigenvectors and eigen sensitivity analysis is required. † Information about transfer function that can be determined from system simulation or modeling is only required. † A considerable reduction in the size of involved matrices, since only the transfer function of the system is used and the states of the system are not considered. The proposed method can be used for more complicated power systems that include FACTS devices, since choosing the suitable signal for applying in a controller is an important question with such devices also. Test results on two multi-machine systems show that the proposed method is reliable and straightforward.
[1] Kundur P, Klein M, Rogers GJ, Zywno MS. Application of power system stabilizers for enhancement of overall system stability. IEEE Trans Power Syst 1989;PWRS-4:614–29. [2] Rogers G. Power system oscillations. Boston, MA: Kluwer Academic Publishers; 2000. [3] Hsu YY, Chen CL. Identification of optimum location for stabilizer applications using participation factors. IEE Proc 1987;134(3):238–44. [4] Zhou EZ, Malik OP, Hope GS. Theory and method for selection of power system stabilizer location. IEEE Trans Energy Convers 1991; EC-6:170–6. [5] Hamdan AMA. An investigation of the significance of singular value decomposition in power system dynamics. Int J Electr Power Energy Syst 1999;21:417–27. [6] Castro JC, Araujo CSD. Frequency domain analysis of oscillatory modes in decentralized control systems. Automatica 1998;34(12): 1647–998. [7] Nelson L, Limba Leonardo TG. Determination of suitable locations for power system stabilizers and static VAR compensators for damping electromechanical oscillations in large scale power systems. IEEE Trans Power Syst 1990;PWRS-5(4):1445–63. [8] Heniche A, Kamwa I. Control loops selection to damp Inter-Area Oscillations of electrical networks. IEEE Trans Power Syst 2002;17(2): 378–84. [9] Kamwa I, Gerin-Lajoie L. State-space system identification—toward MIMO models for modal analysis and optimization of bulk power systems. IEEE Trans Power Syst 2000;PWRS-15(1):326–35. [10] Karimpour A, Asgharian R, Malik OP. PSS design in conjunction with scaling effect 17th International power system conference 2002 p. 1–7. [11] Karimpour A, Asgharian R, Malik OP. Singular value decomposition as a measure for control structure design in power systems. Electr Power Components Syst 2004;32(3):295–307. [12] Skogestad S, Postletwaite I. Multivariable feedback control. New York: Wiley; 1997. [13] Eykhof P. System identification. New York: Wiley; 1974. [14] Klein M, Rogers GJ, Moorty S, Kundur P. Analytical investigation of factors influencing power system stabilizers performance. IEEE Trans Ener Convers 1992;7:382–90. [15] Shamsollahi P, Malik OP. Application of neural adaptive power system stabilizer in a multi-machine power system. IEEE Trans Energy Convers 1999;14(3):731–6. [16] IEEE standard 421.5. IEEE recommended practice for excitation system for power system stability studies 1992.
Determination of PSS location based on singular value decomposition A. Karimpoura,b,1, R. Asgharianb,1, O.P. Malika,* a
Department of Electrical and Computer Engineering, University of Calgary, 2500, University Drive, Calgary, Alta., Canada T2N 1N4 b Department of Electrical Engineering, Mashhad University, Mashhad, Iran Received 7 August 2002; revised 13 July 2004; accepted 13 May 2005
Abstract In a multi-machine power system, it is important to determine the best location for the application of power system stabilizers. A technique based on singular value decomposition to identify generators that contribute in each oscillation mode is introduced. Measures based on singular value decomposition for controllability and observability for every mode are described and the selection of the best location for the installation of the PSSs is done according to these measures. The proposed methodology is designed to use the system transfer function. Studies illustrate the effectiveness of the proposed technique. q 2005 Elsevier Ltd. All rights reserved. Keywords: Power system stability; Singular value decomposition; Controllability
1. Introduction Power systems are known to be subject to low frequency oscillations. If these oscillations persist, they can present limitation on power system performance. Power system stabilizers (PSSs) are used as an effective way to enhance the damping of these electromechanical oscillations [1]. An interesting and important question is to find the generators on which the PSSs be applied for most effective damping of oscillation modes. Several techniques have been proposed in the literature to determine the best location for applying power system stabilizers [2]. Participation factor method is one of the methods in detecting the contribution of various generators in each mode and for suitable location for applying the PSSs [3]. Some authors have proposed sensitivity of PSS effect (SPE) as a measure for determining the suitable location of PSSs [4]. Singular value decomposition (SVD) has been proposed as a measure of the distance of a controllability and * Corresponding author. Tel.: C1 403 220 6178; fax: C1 403 282 6855. E-mail addresses: [email protected] (A. Karimpour), asghrian@um. ac.ir (R. Asgharian), [email protected] (O.P. Malik). 1 Tel.: C98 511 8815100; fax: C98 511 8829536.
0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2005.05.004
observability matrix from singularity in a state space model [5]. This distance is used as a measure to compare the ability of inputs to control an oscillation mode. The main drawback of the above approaches is the determination of linear models from the state space models that usually have large dimensions. Some authors have used transfer function approach to locate the PSSs. In [6] peak of transfer function elements in frequency domain is used as a measure. Transfer function residues are used in [7] to locate PSSs. Singular value is used as a measure to determine the suitable signal for PSSs and its locations in [8–11]. A new method based on the mathematical consideration of SVD through transfer functions is described in this paper. SVD is used as a measure to distinguish the contribution of various generators in each mode. Based on the contribution of each generator to various modes, controllability and observability for each mode is considered and a method for the best position to apply PSSs is introduced. The transfer function of the system is found through identification, so there is no need to construct state space models. The paper is organized as follows: SVD is described in Section 2; a brief explanation of how the transfer function is determined is given in Section 3; a procedure for the selection of the PSS location is described in Section 4; results of simulation studies on non-linear multi-machine power systems are described in Section 5.
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A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
2. Singular value decomposition Consider M is a constant matrix in Cl!m. Then M can be decomposed into its singular value decomposition according to the following theorem [5]. Theorem. Let M2Cl!m Then there exist S2Rl!m and unitary matrices U2Cl!l and V2Cm!m such that M Z USVH " where SZ
0
S
(1)
#
, SZdiag{s1,s2,.,sr} withs1Rs20 0 R.Rsr and r%min(l,m). UZ[u1,u2,.,ul], VZ[v1,v2,.,vm] and VH is complex conjugate transpose of V. The column vector elements of U, identified by ui, are orthogonal and of unit length, and represent the output directions. The column vector elements of V, identified by vi, are orthogonal and of unit length, and represent the input directions [12]. These input and output directions are related through singular values. So one can write: Mvi Z si ui
(2)
It means that if an input is applied in the direction vi, then the output is in the direction ui and has a gain of si. Input direction vi for iOr corresponds to inputs that do not have any influence on outputs and similarly output direction ui for iOr corresponds to the outputs that can not be accessed by any input. Since, the diagonal elements of S are arranged in a descending order, it can be shown that the largest gain for any input direction is equal to the maximum singular value s1 and so one can write s1 Z max ds0
jjMdjj2 jjMv1 jj2 Z jjdjj2 jjv1 jj2
Example: Consider 2 3 1 2 K1 6 7 M Z43 4 1 5 2
0:92
yðsÞ Z GðsÞuðsÞ
8
0:34
K0:17
3H
0
0
0
3
7 07 5 0
0:50 K0:33 K0:80 6 7 7 $6 4 0:35 K0:77 0:53 5 0:79
0:55
3. System transfer function Transfer function identification aims at determining a black box model of the system in the following form
then singular value decomposition of M is: 2 3 2 0:04 K0:53 K0:85 9:77 0 6 7 6 7 6 M Z6 4:53 4 0:38 K0:77 0:51 5$4 0 2
† provides an indication of the relative contribution of different outputs at that frequency, and † shows which generator or group of generators is oscillating against another generator or a group of generators.
(3)
where d is any input signal and k$k2 is the Euclidian norm.
4
0 output is u1 Z 0:04 0:38 0:92 . The gain of 4.53 is
0 for an input in the direction v2 Z K0:33 K0:77 0:55 and the corresponding output direction is
0 u 2 Z K0:53 K0:77 0 0:34 . The input direction v3 Z K0:80 0:53 0:27 has no influence on outputs and
0 the output direction u3 Z K0:85 0:51 K0:17 is not accessible with any input. In this example, input in the direction v1 will excite the largest gain, or, in other words, will be the most effective input. The idea illustrated in the above example can be extended to a matrix having transfer functions, instead of scalars, as its elements. In that case, the SVD of the matrix is determined at any frequency u by substituting s with ju. The peaks of singular values versus frequency plot indicate the frequencies at which the system is likely to exhibit dynamic stability problem. The output direction corresponding to the largest singular value provides an indication of the relative observability of different outputs at that frequency and the input direction corresponding to the largest singular value provides an indication of the relative controllability of different inputs at that frequency. Multiplication of output direction by input direction gives information about residue as well [11]. In a power system the relative magnitude and phase of the elements of the output direction vector (u1), corresponding to the largest singular value, at each frequency of oscillation,
0:27
The largest gain of 9.77 is for an input in the 0 direction v1 Z 0:50 0:35 0:79 and the corresponding
(4)
where y(s) and u(s) are vectors of outputs and inputs of dimension n!1, and G(s) is an n!n transfer function matrix of the system, where n is the total number of generators in the system. The outputs, y(s), of the system are the speeds of various generators and the inputs, u(s), of the system are the excitations of generators. Transfer function gij(s) between input port j and output port i, the ijth element of G(s), can be written as: yi ðsÞ (5) gij ðsÞ Z uj ðsÞ ui Z 0 i sj To find gij(s) inject a small signal to the input port j and record the output at port i. By use of the input and output
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
information and least squares method [13] it is possible to identify gij(s). By performing the same procedure for all elements of G(s) one can directly determine the G(s) of a multi-input/multi-output system.
4. PSS location selection The proposed technique depends upon distinguishing the contribution of various generators to each oscillation mode by finding the minimum number of generators for which the corresponding reduced transfer function has the maximum singular value near to the largest singular value of the real system at that frequency. To do this, the row and column corresponding to each generator is omitted and its effect on the maximum singular value is examined. If the reduction in the maximum singular value is small, one understands that this generator does not contribute in this mode. Otherwise, this generator is contributing in this mode. Repeating this method for all generators and omitting the non-contributing ones, it is possible to find the minimum number of generators whose corresponding reduced transfer function has maximum singular value near to the largest singular value of the real system at that frequency. A procedure to select the appropriate locations for PSSs using SVD is described in the following steps. 1. Substituting ju for s in (4), plot singular values of G(juk) over a range covering the frequencies of oscillation modes. From this plot, find the peaks of maximum singular values that correspond to oscillation modes. Say, these are at frequencies u1,u2,.uk. 2. Determine G(juk) for each mode of oscillation that corresponds to the frequency of u1,u2,.uk. and the corresponding maximum singular values s1max,s2max, .skmax,. 3. Start with one of the oscillation modes, i.e. G(juk). 4. Delete from G(juk) the column and row corresponding to generator i for iZ1,2,3,. one generator at a time and calculate the maximum singular value of the reduced matrix. If this value is much smaller than skmax, generator i contributes to mode uk and should be retained. If instead the maximum singular value is close to skmax, generator i does not contribute to this mode and is deleted. 5. Repeat step 4 until a reduced matrix G(juk) of only those generators that contribute to mode uk is obtained.Note: Since it is possible that uk be the same oscillation mode of more than one group of generators, steps 4 and 5 must be repeated to find all possibilities. 6. For the reduced G(juk) found in step 5 corresponding to the generators contributing to the modeuk, find the SVD of G(juk). The elements of vector u1 are observability measures and the elements of vector v1 are controllability measures for this mode [11]. So one can find the best position to locate PSS and the best signals for applying to PSS.
537
7. For the reduced G(juk) found in step 5, corresponding to the generators contributing to the mode uk, delete one generator at a time and determine the maximum singular value of the remaining system. A generator, the deletion of which leads to the smallest magnitude of the maximum singular value of the remaining system, is the most effective generator for mode uk and is thus the most appropriate location for a PSS to damp this mode. This procedure combines the controllability and observability measures in one step. 8. Repeat steps 3–7 for each mode found in step 1 to determine the appropriate locations for PSSs corresponding to each mode. The above procedure can be easily performed using a computer program. Such a program has been developed to select PSS locations and its application is described in Section 5.
5. Application of SVD in power systems A comparison of the results obtained by the method based on SVD with methods based on participation factor on a four machine system that has closely spaced modes (modal resonance) is given in this Section. Additional studies on a five machine system demonstrate the applicability of the proposed method. 5.1. Four machine system A single line diagram of the 4-machine system is shown in Fig. 1. The parameters of the model are given in [14]. There are three oscillation modes and participation factors (PF) corresponding to the rotor speed for these three modes are given in Table 1. Phasor diagram of the entries corresponding to speed (mode shapes) for the oscillation modes are shown in Fig. 2. As the eigenvalues are practically equal in this case, the eigenvectors are not unique. Thus the mode shapes given by the eigenvectors in Fig. 2 are physically meaningless and participation factors do not give the correct information [14]. The singular value plot of the 4-machine system versus frequency is shown in Fig. 3. Examination of the singular values shows that the gain of the system around 3.36 and 6.63 rad/s is high, so there are oscillation modes near these frequencies. By following the procedure explained in
Fig. 1. Example of a 4-machine power system.
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A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
Table 1 Participation factors for system in Fig. 1 Unit
G1 G2 G3 G4
u12 u12
Mode 1 (K0.39C6.651)
Mode 2 (K0.40C6.631)
PF
PF
PF
0.641 0.8049 0.5641 1
0.6972 1 0.6749 0.9135
0.2503 0.1428 1 0.6749
u14
u13
Mode 3 (0.15C3.291)
u11
u11 (a) 6.63 rad/s
u12 u11
(b) 6.63 rad/s
(c) 3.36 rad/s
Fig. 4. Elements of output direction vector corresponding to maximum singular values.
0.35 Table 2 Controllability measures based on SVD
0.3
Unit
Controllability measures 6.64 rad/s
6.64 rad/s
3.36 rad/s
Exciter of G1 Exciter of G2 Exciter of G3 Exciter of G4
0.711 0.703 *** ***
*** *** 0.693 0.721
0.489 0.502 0.494 0.515
Singular values
0.25 0.2 0.15 1
0.1 0.05
2
0 100
101 Frequency (rad/s) Fig. 2. Mode shapes for system in Fig. 1.
Section 4, it is found that at 6.63 rad/s, the set {G1,G2} has the maximum singular value of 18.1534, the set {G3,G4} has the maximum singular value of 17.5610 and the maximum singular value of the real system is 18.5143. So there are two oscillation modes close to this frequency. Elements of the output direction vector, corresponding to the maximum singular value of each set of generators at 6.63 rad/s, shown in Fig. 4(a) and (b) clearly indicate the two local modes. In one of them G1 oscillates against G2 and in the other G3 oscillates against G4. The same procedure for the frequency of 3.36 rad/s shows that the combination of all generators gives the suitable maximum singular value, and the elements of output direction corresponding to the maximum singular value at this frequency (Fig. 4(c)) show that it is an inter-area G3
G4 G3
G3 G1
G2
G2
Table 3 Observability measures based on SVD Unit
G1
G1
mode such that G1 and G2 combined oscillate against G3 and G4. The next step is to find the best position to apply PSSs. First choose the transfer function corresponding to the generators contributing in each mode. This is done by deleting from the total transfer function the columns and rows corresponding to the generators that do not contribute in this mode. By substituting the frequency of oscillation as ju instead of s and finding SVD of the transfer function, the elements of u1 and v1 (output and input direction corresponding to maximum singular values) show the observability and controllability of the oscillation mode, respectively. Controllability and observability measures based on SVD are shown in Tables 2 and 3. Table 2 shows that the best position for improving the oscillation mode of frequency 3.36 rad/s is exciter of G4 and the best signal for applying PSS is speed of G3, according to Table 3. The same interpretation shows that for local mode between G1 and G2 the best candidate is G1 and the best signal for applying PSS is speed of G2. For local mode between G3 and G4 the best candidate is G4 and the best signal for applying PSS is speed of G4. Now, one can consider both observability and controllability measure through the procedure in Section 4.
G2 G4
G4
(a) Mode 1
(b) Mode 2
(c) Mode 3
Fig. 3. Singular values of the 4-machine system.
Speed of G1 Speed of G2 Speed of G3 Speed of G4
Observability measurs 6.64 rad/s
6.64 rad/s
3.36 rad/s
0.625 0.781 *** ***
*** *** 0.574 0.819
0.155 0.115 0.725 0.661
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541 Table 4 Singular values of 4-machine system
539
Frequency (rad/s)
3.36
6.63
6.63
Convolving generators
#1, #2, #3, #–4
#1, #2
#3, #4
Number of omitted generator
Max singular value
No. omission #1 #2 #3 #4
89.5925 77.17 76.9287 53.0017 57.00463
Velocity (rad/s)
0.8
0.4 0.2 0 –0.2 0
1
2
1
2
3 4 5 6 7 (a) Actual and identified velocity
8
9 10 Time (s)
8
9
0.02
17.561 *** *** 10.5226 7.2949
Input signal
18.1534 10.0695 8.201 *** ***
actual velocity identified velocity
0.6
0.01 0 –0.01 –0.02 –0.03 0
3
4
5
6
7
(b) Signal applied to exciter of generator number 1
By deleting the column and row corresponding to each generator in turn, check the maximum singular value of the remaining system. A generator that leads to the maximum reduction in the largest singular value is the most effective generator for damping of that mode. The largest singular values obtained by deleting each generator in turn for all of modes are shown in Table 4. According to Table 4 omitting G3 reduces the singular values corresponding to 3.36 rad/s the most, so it is the best position to apply PSS for improving the performance of the system, and G2 and G4, respectively, are best positions for improving the local modes. A rank between generators for improving this oscillation mode is also shown in Table 4. 5.2. Five machine system A single line diagram of a 5-machine 8-bus system [15] is shown in Fig. 5. The parameters of the model and operating conditions are given in [15]. In this system, generators 1, 2 and 4 are large and generators 3 and 5 are 1/10 the size of the other three generators. A known signal is applied to the exciter of each generator and both input signal and the speed of each generator, as the output, are recorded. Using this procedure, a twelfth order transfer function for each output/input pair of the system
Fig. 6. Gen #1 to #2 transfer function.
model is determined. Actual speed of generator 2 as obtained by direct simulation of the system and as obtained from the identified transfer function when exciter of generator number 1 is excited is shown in Fig. 6(a) and the input signal used for exciter number 1 is shown in Fig. 6(b). The singular value plot of the 5-machine system versus frequency is shown in Fig. 7. Examination of the singular values shows that the gain of the system around 4 rad/s (0.64 Hz), 6.4 rad/s (1.02 Hz) and 8.5 rad/s (1.40 Hz) is large, so there are oscillation modes near these frequencies. By following the procedure explained in Section 4, it is found that at 8.5 rad/s, the set {G2,G3,G5} has the maximum singular value of 12.727 and the maximum singular value of the real system is 12.734. By referring to the elements of output direction corresponding to maximum singular value at this frequency shown in Fig. 8(a), one can see that this is a local mode between generators 2, 3 and 5. 100 90 80
6
70
1 G1
G3 L2 2
8
G2
Singular values
7
3
60 50 40 σ1
30 L3
4
5
20 10
G5 G4 L1 Fig. 5. Example of a 5-machine power system.
10 Time (s)
σ2
0 100
101 Frequency (rad/s) Fig. 7. Singular values of the 5-machine system.
540
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
The same procedure for the frequency of 6.4 rad/s (Fig. 8(b)) shows that there is a local mode between generators 1 and 4. At 4 rad/s, the combination of all generators gives the suitable maximum singular value and the elements of output direction corresponding to the maximum singular value at this frequency (Fig. 8(c)) show that it is an inter-area mode. The next step is to find the best position to apply PSSs. By the same procedure as before and deleting the column and row corresponding to each generator in turn, and checking the maximum singular value of the remaining system, best position for improving each oscillation mode can be found. The largest singular values obtained by deleting each generator in turn for all modes are shown in Table 5. According to Table 5 for interarea mode (4 rad/s) omitting generator 2 reduces the singular value the most, so it is the best position to apply PSS for improving the performance of the system. A rank between generators for improving this oscillation mode is also shown in Table 5. Similarly best position for improving oscillation mode of frequency 6.4 rad/s is generator 4 as shown in Table 5 and generator 2 is again the most suitable place for improving the 8.5 rad/s local mode. For comparison purposes, a conventional PSS with the following transfer function [16] was installed on each of
Fig. 8. Output direction corresponding to maximum singular values.
Table 5 Singular values of 5-machine system Frequency (rad/s)
4
6.4
8.5
Convolving generators
#1,#2,#3,#–4
#1, #4
#2, #3, #5
Number of omitted generator
Max singular value
No. omission #1 #2 #3 #4 #5
97.444 92.612 45.442 86.038 77.633 85.629
32.003 17.434 *** *** 15.317 ***
17.734 *** 8.893 11.133 *** 10.723
3
3 PSS on g1 PSS on g2 PSS on g4
1 0 –1 –2
0
2
4
PSS on g1 PSS on g2 PSS on g4
2 Velocity (rad/s)
Velocity (rad/s)
2
1 0 –1 –2
6
0
2
4
6
Time (s)
Time (s) (a) Different speed of gen #1 and gen #2
(b) Different speed of gen #2 and gen #4 10
PSS on g1 PSS on g2 PSS on g4
4
Velocity (rad/s)
Velocity (rad/s)
6
2 0 –2
PSS on g1 PSS on g2 PSS on g4
5
0
–5
–4 0
2
4
6
0
2
Time (s) (c) Different speed of gen #1 and gen #2
4
6
Time (s) (d) Different speed of gen #2 and gen #4
Fig. 9. Result for the 5-machine system.
A. Karimpour et al. / Electrical Power and Energy Systems 27 (2005) 535–541
the large generators, i.e. 1, 2 and 4 in turn FðsÞ Z 0:5
10s 1 C 0:2s 1 C 0:2s 1 C 10s 1 C 0:04s 1 C 0:04s
541
References (6)
Simulation results in response to a 200 ms 3 phase short circuit on bus 3 are shown in Fig. 9(a) and (b) and for a short circuit on bus 2 are shown in Fig. 9(c) and (d). It can be seen that by applying PSS on generator 2 leads to the best performance of the system for the damping of the inter-area mode as determined in Table 5.
6. Conclusions A new approach based on singular value decomposition is proposed to find the generators contributing in each oscillation mode and also the best location for applying PSSs is determined. This approach only uses the information of transfer function matrix and can be used to find the best location for application of PSS. The advantages of the proposed method are: † No computation involving the right (left) eigenvectors and eigen sensitivity analysis is required. † Information about transfer function that can be determined from system simulation or modeling is only required. † A considerable reduction in the size of involved matrices, since only the transfer function of the system is used and the states of the system are not considered. The proposed method can be used for more complicated power systems that include FACTS devices, since choosing the suitable signal for applying in a controller is an important question with such devices also. Test results on two multi-machine systems show that the proposed method is reliable and straightforward.
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