Kasparova
539-492
Application of Fuzzy Sets to Autonomous Robot Vehicle Navigation Application of Fuzzy Sets to Autonomous Robot Vehicle Navigation Using the PIC18F452 Microcontroller
Robert Casey, Michael Hensler
539-219
A Fuzzy Engine Model for Efficient Stock Market Prediction
Shereen Mohamed, Neamat El Gayar,
Hazem Abd
Elazeem
539-672
Application of fuzzy c-means clustering in power system model reduction for controller design
Shu-chen Wang, Pei-hwa Huang, Chi-jui Wu
539-629
報告人的文章「Application of fuzzy c-means clustering in power system model reduction for controller design」安排在「Fuzzy Systems」
的議程口頭宣讀，很多人出席。
三、心得與建議
由上面的分析可知第五屆國際智慧計算與人機介面機制研討會 中，技術分組討論的議程包括電機、機械、經濟等裡領域，大會的訴 求是針對最新主題立刻討論以交流各個領域觀念，統合學習的觀念值 得學習。大會也舉辦Conference Banquet，讓大家交流文化與知識。
隨著資訊科技的發展，先進國家在此領域永遠有大量研究人力。筆者 參加第五屆國際智慧計算與人機介面機制研討會，獲益很多，同時亦 深盼國內能多加強此領域的研究。
WSEAS2006(venice)國科會出國補助報告95-12-06-1
Application of Fuzzy C-Means Clustering in Power System Model Reduction for Controller Design
SHU-CHEN WANG1 PEI-HWA HUANG2 CHI-JUI WU3
Department of Electrical Engineering Department of Electrical Engineering National Taiwan Ocean University National Taiwan University of Science and Technology
No. 2, Peining Rd., Keelung 20224 No. 43, Sec. 4, Keelung Rd., Taipei 10607
TAIWAN TAIWAN
Abstract: -This paper presents the application of fuzzy c-means (FCM) clustering in the order reduction of dynamic models for controller design in a power system. Based on the fuzzy c-means algorithm, a method is proposed for clustering the poles and zeros of the original power system model into new clusters from which a reduced-order model can be obtained. Then the reduced-order model is used to design a proportional-integral type power system stabilizer to improve the damping in system oscillation after a system disturbance. The reduced-order model can contain the critical dynamic characteristics of the original model, but let it easier to design the controller. Results from a sample power system are presented to show the validity of the proposed method. The electromechanical mode of the power system can be improved by the designed power system stabilier from pole assignment.
Key-Words: -Power system dynamics, Model reduction, Fuzzy c-means, Fuzzy Clustering, Pole assignment.
1 Introduction
Model order reduction concerns the transformation of a higher-order model into a lower-order model through some sort of computation [1, 2]. A certain relationship between these two models is preserved and they are similar in the characteristics under consideration. In power system studies, creating a dynamic model is the first step for system stability research, dynamic behaviors analysis, or other system functional tests. As systems become larger, their complexity increases and power system analysis has to tackle high-order model analysis. However, computation on the high-order model is highly complex while the final analysis results may have unnecessary portions. In this case, having a low-order model that maintains the main characteristics of the high-order systemt can replace the original system and significantly simplify the computational problem [3-13].
If the stability performance of a power system is unable to satisfy the specification, the
stabilizing controller can be used to improve the dynamic characteristics. Without stability disturbance compensation, steady-state performance and any other performance index are not possible. Therefore, a stabilizing controller of power system is needed. The most important application of the reduced order model is let it easier in the design of a suitable controller for the original high-order system. Many methods can used to design a power system stabilizier with output feedback scheme. The pole-placement design allows for the power system having the electromechanical mode dynamic to be placed in desired location.
In this paper, the method based on fuzzy c-means clustering analysis [14-18] aims to group poles and zeros of a power system transfer function into some clusters. For each cluster, the original system poles (zeros) can be replaced by each cluster center that becomes the new member representative of the cluster. All new members representing their respective clusters jointly constitute a tentative reduced-order model of the
WSEAS2006(venice)國科會出國補助報告95-12-06-1
original system. The reduced-order model is used to design a proportional-integral power stabilizer to improve the dynamic stability. The results obtained from a sample power system models will be illustrated and the effectiveness of the method is thus confirmed by the example.
2 Fuzzy c-means Cluster Analysis
The method proposed in this paper utilizes fuzzy c-means clustering (FCM) analysis [14-16] to reduce the original high-order model into a low-order model. Cluster analysis [17, 18], of which the task is to classify non-processed data into certain categories depending on various traits, is a basic tool commonly used in several scientific fields. Data in each category have the most resemblance while being very dissimilar with data from other categories.
Suppose there are n data points{ xj }, 1 j n, to be clustered into c data clusters. Let
ij denote the degree of membership that xj belongs to the i th cluster. It is noted that
1
0ij and ci1ij 1 for each j . Define the fuzzy partition matrixU [ij], 1 i c,1 j n. Therefore, the objective of the fuzzy c-means algorithm is to determine all the elements of matrixU . The FCM algorithm is essentially an iterative procedure and can be formulated as the following six steps in which ldenotes the iteration number.
(a) Set the number of clusters c. Initialize U randomly asU( )l [ij], l1,1 i c. (b) Compute the cluster center ci of each
cluster:
Note that the value of m normally falls in the range of 1.5 m 3.
(c) Select the weighting wj of every data point, then the weighted data pointWj as
j j j
W x w (2)
(d) Compute the distance dij between the j th data point and the ith cluster center:
ij i j
otherwise, return to Step (b).
It is worth noting that in the above algorithm, the cluster center ci of each cluster is referred to as the prototype of the cluster and can be considered as the representative of that cluster.
3 Design Method
Given a state space linear model, dynamic characteristics of the system can be best revealed from its poles and zeros. The following steps comprise the proposed model reduction method and controller design.
Step1:
After configuring all the parameters of the power system and linearizing the system state equations, the following system dynamic equations are obtained: matrices of the system; x, u and y denote the state, input and output vectors, respectively.
Step2:
From Equation (5), the transfer function is found to be
Based on the transfer function, the poles and zeros can be computed.
Step3:
Using the fuzzy c-means algorithm, it can cluster separately the poles and zeros in the complex plane to obtain the corresponding cluster centers.
WSEAS2006(venice)國科會出國補助報告95-12-06-1
clustered into distinct groups, and zeros are processed likewise.
Step4:
The calculated cluster centers replace the respective groups of poles and zeros of the original system and collectively constitute the set of poles and zeros for the reduced-order model.
The tentative reduced-order model transfer function is thus set as
Figure 1. Single-machine infinite bus power system
Figure 2. Block diagram of static excitation system
Table 1 The parameters of generator Xd= 2pu Xd' = 0.244pu Tdo' = 4.18sec Xq= 1.91pu Xq'= 0.17pu Tqo' = 0.55sec
Table 2 The parameters of static excitation system KA= 400 TA= 0.05
In order to make the time response of the
reduced-order model compatible with that of the original higher order model, a gain adjustment factor defined by
0
is used to adjust the steady state value of the reduced order model.
Step6:
The parameters of a proportional-integral power system stabilizier (PSS) are to be determined. The power system stabilizier has the transfer funtion as
Then the closed-loop transfer function of the system is
( )
4 Example
Consider the single-machine infinite bus power system shown in Figure 1.
The generator can be represented by the two axis model. The equations are obtained:
' ' '
The parameters of generator are shown in Table 1.
The block diagram of static excitation system is displayed in Figure 2. The parameters of static excitation system are shown in Table 2.
Based on the above-described method, the reduced order model and the controller design for the study system is obtained as follows:
Step1:
Choose the state vector x as:
' '
T
d q FD S
x E E E V
The definitions for each state variable are
'
Ed
direct-axis transient voltage
'
Eq
quadrature-axis transient voltage
WSEAS2006(venice)國科會出國補助報告95-12-06-1
Speed
Rotor angle
EFD exciter output voltage
VS stabilizier transformer output voltage The system matrices are
8.94 0 0 2.79 0 0
Table 3 Poles and zeros of the original model
Poles Zeros
2 2 0 .9 1
0
-8.30 0
-0.21 0
-0.09 j75619000 -0.81 j11.52
Table4. Poles and zeros of the reduced model Clustered
Poles
Clustered Zeros 28.75
j75619000 -0.81 j11.52
Table 5 Electromechanical modes of the power system
Eigenvalue without PSS Eigenvalue with PSS -0.81 j11.52 -3.04j11.50
Figure 3. Comparison the original system with and without power system stabilizer
Step2:
The transfer function of the original model is calculated as
( 1.33)( 0.001)( 0.67 1.15)
The poles and zeros of the original model are displayed in Table 3.
Step3:
Using fuzzy c-means algorithm, the poles and zeros of the original models are processed to obtain some cluster centers to be used for representing the original poles and zeros.
Table 4 shows the poles and zeros after clustering. In Table 4, the poles (28.75 ) are obtained from clustering the poles of the original model, (2 2 0 .9 1), (-8.30), (-0.21), and (-0.09). The poles (-0.81 j11.52 ) of the electromechanical mode are retained. Regarding the zeros, the clustered zeros are (j75619000).
Step4:
The cluster center is obtained after computation and is used to replace the poles and zeros of the original system to become the reduced model. The tentative transfer functions for the reduced model are:
The gain adjustment factor is used to adjust the system response to make the reduced-order model
WSEAS2006(venice)國科會出國補助報告95-12-06-1
system, the gain adjustment factors are calculated as
After the above steps, the transfer function of the reduced order model is R s( )=kR s( ) which is given
The reduced-order model is used to design a proportional-integral power system stabilizer. If the electromechanical mode of the closed-loop system is to be assigned at -2j11, the parameters of power system stabilizier are obtained:
kP kI
8 .8 5 2 0 3 .6
From Table5, the electromechanical mode of of the original model system are obviously improved.
The time responses after a small distrubance is shown in Figure 3.
5 Conclusion
A model reduction method for reducing the order of power system dynamic models in controller design has been proposed in this paper. Based on the fuzzy c-means algorithm, the proposed method performs clustering on the poles and the zeros of the original system model into new clusters from which a reduced-order model can be derived. The reduced-order model that maintains the main characteristics of the high-order system can significantly simplify the design of a power system stabilizer. Results from applying the method to a sample power system have been demonstrated to show the validity of the proposed method.
Acknowledgments
This work was supported in part by the National Taiwan Ocean University and the National Science Council of Taiwan.
References:
[1] A. Bergen, Power Systems Analysis, Prentice Hall, New York, 2000.
[2] D. Trudnowski, Order reduction of large-scale linear oscillatory system models, IEEE Trans. on Power Systems, Vol. 9, No. 1, 1994, pp. 451-458.
[3] F. Saleh and M. Mahmoud, Design of power system stabilizers using reduced-order models, Electric Power System Research, Vol.
33, 1995, pp. 219-226.
[4] A. Feliachi, X. Zhang and C. Sims, Power system stabilizers design using optimal reduced order models, IEEE Trans. on Power Systems, Vol. 3, No. 4, 1988, pp.
1670-1684.
[5] R. Castro and J. de Jesus, A wind park reduced-order model using singular perturbations theory, IEEE Trans. on Energy Conversion, Vol. 11, No. 4, 1996, pp.
735-741.
[6] N. Nihei and T. Oyama, A study on decomposition and model reduction for wide area power system stability assessment, Power Engineering Society 1999 Winter Meeting of the IEEE, Vol. 1, 1999, pp.
651-654.
[7] N. Sinha and J. Pal, Simulation based reduced order modeling using a clustering technique, Computer & Elect. Engineering, Vol. 16, No. 3, 1990, pp. 159-169.
[8] N. Sinha, Reduced-order models for linear systems, Proc. of the IEEE Conference on Systems, Man and Cybernetics, 1992, pp.
537-542.
[9] M. Duric, Z. Radojevic and E. Turkovic, A reduced order multimachine power system model suitable for small signal stability analysis, Electrical Power & Energy Systems, Vol. 20, No. 5, 1998, pp. 369-374.
[10] G. Obinata and B. Anderson, Model Reduction for Control System Design, Springer, 2001.
[11] P. Benner, R. Mayo, E. Quintana-Orti, and G.
Quintana-Orti, A service for remote model reduction of very large linear systems, Proc.
of the International Parallel and Distributed Processing Symposium, 2003, pp. 22-26.
[12] H. Ukai, H. Masubara, M. Kobayashi, and H.
Kandoh, Stabilizing control of series capacitor compensated power system on the basis of reduced order model, Proc. of the 4th International Conference on Advances in Power System Control, Operation and Management, 1997, pp. 603-608.
WSEAS2006(venice)國科會出國補助報告95-12-06-1
[13] F. Saleh and M. Mahmoud, Design of power system stabilizers using reduced-order models, Electric Power System Research, Vol.
33, 1995, pp. 219-226.
[14] F. Hoppner, F. Klawonn, R. Kruse, and T.
Runkler, Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Recognition, John Wiley & Sons, 1999.
[15] J. Bezdek, J. Keller, R. Krisnapuram, and M.
Pal, Fuzzy Models and Algorithms for Pattern Recognition and Image Processing, Kluwer Academic, 1999.
[16] J. Chiang and Y. Chen, Incorporating fuzzy operators in the decision network to improve classification reliability, Computer
& Elect. Engineering, Vol. 28, 2002, pp.
547-560.
[17] M. Aldenderfer and R. Blashfield, Cluster analysis, Sage Publications, Beverly Hills, 1984.
[18] L. Kaufman and P. Rousseeuw, Finding Groups in Data: An Introduction to Cluster Analysis, John Wiley & Sons, 1990.
[19] N. Pal, K. Pal, J. Keller, and J. Bezdek, A possibilistic fuzzy c-means clustering algorithm, IEEE Trans Fuzzy Systems, Vol.
13, No. 4, 2005, pp. 517–530.
[20] S. Nascimento, B. Mirkin and F.
Moura-Pires, A fuzzy clustering model of data and fuzzy c-means, Proc. of the IEEE Conference on Fuzzy Systems, 2000, pp.
302-307.