Voltage Stability Assessment Index for Recognition of Proper Bus for Load Shedding Mir Sayed Shah Danish Student Member, IEEE
Atsushi Yona Member, IEEE
Tomonobu Senjyu Senior Member, IEEE
Electrical & Electronics Dept. University of the R yukyus Nishihara, Okinawa, Japan mirsayedshah@yahoo.com
Electrical & Electronics Dept. University of the R yukyus Nishihara, Okinawa, Japan yona@tec.u-ryukyu.ac.jp
Electrical & Electronics Dept. University of the Ryukyus Nishihara, Okinawa, Japan b985542@tec.u-ryukyu.ac.jp b985542@tec.u-ryuk yu.ac.jp
Abstract — Nowadays, voltage stability of large interconnected power power system systemss due due to the multip multipli licit city y of the power power system system operation and control components, complicated configuration and overall performance of the systems have been recognized as a comple complex x probl problem em for identi identific ficati ation on of the fun fundam dament ental al mechanism of voltage stability. Contributory factors to voltage instability in power systems are heavy loading and faults occur in the system. Treating to alleviate heavy loading at the stressed bus requires selection of the proper bus, and it became a challenge to find and predict it easily especially in an interconnected system. In this paper, an assessment index of voltage stability is derived with supports supports of linear linear algebra algebra through the Singular Singular Value Decomposit Decomposition ion (SVD) and Pseudo-inverse Pseudo-inverse of the power flow Jacobian Jacobian matrix. matrix. Singular Singular values and right right eigenvec eigenvector tor are applied as indicators of load shedding recognition. Meanwhile, VSM (Voltage (Voltage Stability Stability Margin) Margin) enhanceme enhancement nt is pursued pursued through through load shedding with ranking of sensitivity for assessment index. Finally, the study is tested on the IEEE 14-bus and 23-bus case studies as a point of reference were carried out to illustrate the effectiveness and accuracy of the proposed assessment index. Index Terms--Lo Terms--Load ad shedding; shedding; Sensitivi Sensitivity; ty; Voltage Voltage stability stability assessment index; Weak bus recognition.
I.
I NTRODUCTION
area, affect the voltages at the load busses. Frequently, in the literature voltage stability indices and load shedding has been focused through researches. Reference [3], introduced multicriteria criteria integrate integrated d voltage voltage stability stability index index was proposed proposed for weak buses identification with application of P-V and P-Q curves. The relationship of reactive power reserves and VSM (Voltage Stability Margin) is quantitatively analyzed in [4]. Sinha [5] has proposed comparative study of voltage stability using index-L and modal analysis and summed with three indices up. Reactive power compensation method using shunt capacitors and reactive power planning are conducted in [6], [7]. Modal analysis and Jacobian matrix are used to evaluate VSM in [9], [10]. Whereas, in [8], introduced a method with using shunt capacitors to improve VSM and applied the modal analysis methodology as well. In [11] presented a method for determining the location and quantity of the load to be shed, through mathematical calculation of voltage stability indicator. L Index, study of voltage collapse index are bade based on voltage collapse point theory, which in this point, power flow becomes unsolvable unsolvable [5], [11]. The others indices are also counted as voltage stability limits, such as sensitivity index and singular value index [12], load proximity index [13], [14], and line line stab stabil ilit ity y inde index x [15] [15].. Thes Thesee inde indexe xess have have vari variou ouss consideration during there’s foundation.
The increase in power demand and limited sources for electr electric ic power power have have resul resulted ted in an increa increasin singly gly comple complex x interconnected system, forced the system to operate closer to the limits of stability. Voltage stability is considered to be one of the keen interest of industry and research sectors around the world since the power system is being operated closed to the limit whereas the network expansion is restricted due to many reasons reasons such as lack of investme investment nt or serious serious concerns concerns on environmental problems [1]. A system enters a stat of voltage instability when a disturbance, increase in load demand, or chan change ge in syst system em cond conditi ition on caus caused ed a prog progre ress ssiv ivee and and uncontrollable decline in voltage [2]. Voltage stability depends on how how variat variation ionss in reacti reactive, ve, as well well as activ activee power power in the load load
In this paper paper predomin predominantly antly is used Jacobian Jacobian eigenva eigenvalue lue and eigenvectors indexes via support of linear algebra with ranking of sensitivit sensitivity y for assessme assessment nt index index of voltage voltage stability and improving VSM. The present method is leant on load shedding at appropriate worse bus/area. The test systems show that this assessment index method can be obtained to distinguish the weak and sensitive buses in the power system for the sake of load load shed sheddi ding ng;; simu simulta ltane neou ousl sly, y, the the crit criter erio ion n volt voltag ages es magn magnitu itude de can be acquir acquiree on weak weak buses buses throug through h load load shedding. At different conditions, critical and normal operation the case studies studies are pursued pursued to indicate indicate the accuracy accuracy and effectiveness of the proposed assessment index.
This work was supported supported in part by the SDP (Strategic (Strategic Developme Development nt Plan) Plan) projec projectt under under admini administr strati ation on of MUDA MUDA (Ministry (Ministry of Urban Urban Development Affairs) in 2012, in Afghanistan, and PEACE (Promotion and Enhancement of the Afghan Capacity for Effective Development) project of JICA (Japan International Cooperation Agency) Agency) in 2013.
____________________________ ________ ________ /$31.00 .00 ©2014 ©2014 IEEE 978-1-4799-3197-2/14 /$31
II.
ASSESSMENT I NDEX FORMULATION
At normal operation of the electric power system voltage stability of the system can be analyzed using computing eigenvalue, and right and left eigenvectors of Jacobian matrix [8]. The premise is that, at the point of voltage stability limit (collapse point) in a power system, the determinant of Jacobian matrix of load flow solution becomes zero. Jacobian matrix is defined using linearized voltage-power equation at operation point as follows [16], [17]:
=
=
If
=
=
(3)
(13)
(14)
According singular-value decomposition and Pseudoinverse [18] under the condition of and are orthonormal columns respectively.
=
(5)
and are nonzero vectors.
0 0
(6)
Whereas, is a singular value of ( is a × matrix) and, and are corresponding right and left singular vectors, respectively. Where, = = , then = ( is × diagonal, nonnegative real values called singular values).
= (, , … , ) ordered so that . In this study, is a singular value of , its square is an eigenvalue of . Therefore, = . From Pseudo-inverse (linear algebra method) [19]:
(7) (8) (9)
). Bus with maximum
eigenvalue and minimum right eigenvectors (the left eigenvectors are not more effective in sensitivity) can be the most sensitive or worse bus in the system. So, the assessment index can be used as a firs decision making reference for next procedures of VSM improvement in the system, based on identification of the proper weak bus. A bus caused the voltage instability easily is called the weak bus [20], and this behavior identifies often by bus sensitivity. The sensitivity index of bus voltage with respect to
(12)
and with sensitivity index (
(4)
Where, is the reduced Jacobian matrix of the system. Decomposition may be applied directly to assess voltage stability in a power system. In this study, the Singular Value Decomposition (SVD) and Pseudo-inverse method (linear algebra) is applied. At the point of voltage stability limit in a power system, the determinant of Jacobian matrix of load flow solution becomes zero.
Form this index easily can be observe the relations of ,
=
= = 0 0 0 = =
then:
With the substitution of we can obtain the voltage stability assessment index:
(2)
=
0
, , ,
(11)
By using (3) we can obtain:
By setting = 0, of the power flow Jacobian matrix given by (1). This yields;
=
This summation can be calculated from 1 to , where is the rank of .
Where, , , , and are Jacobian sub-matrices representing the sensitivities of active and reactive power [ ] respect to voltage angles and magnitudes [ ] .
is a nonnegative scalar and
=
(1)
=
= ( , ,, ) = , ,, and = ,
active power load
[5].
Furthermore, the present voltage stability assessment index identifying voltage-weak bus/area susceptible to voltage instability for the purpose of load shedding. Assess of weak bus via this index can be more precise, where the case studies were carried out to demonstrate the accuracy and effectiveness of the proposed assessment index. III.
CASE STUDIES
The study has been conducted on test cases with IEEE 14 and 23 bus. The practical application of the voltage stability assessment index (14) derived in section II. For the purpose of analysis, Jacobian matrix, eigenvalue, eigenvectors, and sensitivity are calculated by voltage stability function of NEPLAN® software. The voltage stability and load shedding were analyzed for two cases. Case 1 present validity of assessment index with analysis of weak bus and bus sensitivity. Case 2 presents load shedding and VSM improvement through numerical representation. Today different stability margins are considered for power system %5 or %6. I n this study minimum stability margin is set %5 [8].
Case I: Weak Bus Recognition and Bus Sensitivity The IEEE 14 Bus test system is used in this case. Bus 14, 10 and 9 were recognized as weak buses for analysis which is depicted in appendix A. Sensitivity analysis is effective in weak bus identification, however, sensitivity index alone will not be sufficient to identify weak buses especially in an interconnected system. To obtain an accurate result from the assessment index (14), the least eigenvalue and greatest right eigenvector should be
(10)
considered. Bus 14, 10 and 9 having the greatest right eigenvectors at the minimum eigenvalue. Therefore, these buses were known the weakest buses in the system. Meanwhile, bus 7 is the fourth weakest bus in the system while its sensitivity is greater than the sensitivity of bus 9.Table. 1, demonstrates the indexes with the minimum eigenvalue 2.079206 and maximum eigenvectors 0.48619, 0.48392, 0.476716 corresponding to bus 14, 10 and 9 respectively. TABLE I.
IEEE 14-BUS TEST SYSTEM ( = 2.079206)
/
Bus
14
0.486190
0.223312
10
0.483920
0.162144
09
0.476716
0.137696
07
0.401572
0.141678
11
0.302497
0.135261
13
0.138994
0.087230
04
0.119854
0.043989
12
0.096489
0.137652
05
0.080149
0.042725
At the first scenario, the eigenvalue is changed from 2.076186 to 2.086527 and at the second scenario the eigenvalue increase from 2.086527 to 2.21654 and also there is declined in eigenvectors, as well. Bus 26 can be identified as the weakest bus with the highest voltage collapse point, the lowest reactive power margin and the highest percentage change in voltage. All these can be evidence of assessment index validity and the load shedding effectiveness. Voltage stability steady-state analyses can be assessed by obtaining voltage profiles or shortly P-V curves of critical buses as a function of their loading conditions [21]. Figure. 1, shows an increase of voltage magnitude at the operating point, likewise improvement of voltage stability of the entire system with 47 MW load decrease at bus 26 as a function of the parameter value. 23 BUS SYSTEM
TABLE II.
Bus
Critical Operation
Normal Operation
= .
= .
/
/
26
0.417374
0.107948
0.408093
0.099490
27
0.411640
0.096384
0.406467
0.089750
12
0.393176
0.091369
0.390434
0.085775
28
0.377982
0.134475
0.379584
0.128950
13
0.365807
0.096258
0.365063
0.091385
In this case, 23 bus system (Appendix B) is simulated using load shedding method to enhance voltage stability and change the eigenvalue to its maximum possible value and eigenvectors to minimum values. In this case, bus 26 (weakest bus) was selected for load shedding.
24
0.262055
0.116076
0.262258
0.112248
11
0.222030
0.090070
0.227594
0.087888
10
0.181515
0.098856
0.191070
0.097655
09
0.173424
0.162855
0.186474
0.161465
At the first scenario, %6 load and in the second scenario completely connected load (55.758 MVA) are shed at bus 26. The bus voltage is changed from stressed condition 0.948 p.u to 0.95 p.u and 0.985p.u at bus 26 respectively. These scenarios represent bus 26 transition to a stable region via increasing in eigenvalue and decreasing eigenvectors, Table. 2.
30
0.157115
0.072784
0.160910
0.071747
14
0.122397
0.050822
0.124604
0.050089
08
0.065302
0.074600
0.070781
0.074373
15
0.009206
0.003155
0.009551
0.003152
22
0.007082
0.004064
0.007237
0.004061
The results obtained in Table. 2, it can be seen that by using load shedding matching with the principle of assessment index.
20
0.004291
0.090098
0.004374
0.090063
16
0.001144
0.001117
0.001174
0.001117
18
0.000316
0.000150
0.000327
0.000150
0.000017
0.000020
0.000019
0.000020
This case is simulated for 5 eigenvalues,the is the minimum value.
Case II: Load Shedding Performance Evaluation and Voltage Stability
Critical Operation Normal Operation
0.985 p.u
105 100 95 90 85 e 80 g 75 a t l 70 o 65 V 60 55 50 45 40
07
0.948 p.u
Loading 0 30 60 90 120150180210240270300330360390420450
.
Figure 1.
= 0.140354 1
IV.
= =
= = 47 The aim of load shedding at bus 26
CONCLUSION
Voltage stability problem has been a widespread concern in power systems as a result of heavier loadings. One procedure of avoiding voltage instability is to shed load of the critical buses/areas in the system. This paper is conducted on separate aspect of voltage stability; proposed assessment index load shedding via weak bus identification. The effectiveness of the proposed assessment index is tested on two relative test systems in normal and critical operations. The Jacobian matrix is simplified through singular value decomposition and Pseudo-
inverse (linear algebra). The coordination of all procedures of the study can give a quick overview on the system steady state voltage stability. V.
[2]
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APPENDIXES
Case A: IEEE-14 Bus test system single line diagram
Single line diagram of IEEE-14 Bus test system
[10] R.B. Prada and J.O.R. dos Santos, “Fast nodal assessment of static voltage stability including contingency analysis,” ELSEVIER. Electrical Power and Energy Systems, vol. 51, pp. 55-59, 1999.
Case B: 23-Bus system In this system which is selected from NEPLAN ® [22], the voltage stability of a 220 kV transmission network with five power stations is analyzed. The module contains 23 buses.
[11] P Ajay-D-Vimal Raj and M. Sudhakaran. (2010, Jan.). Optimum Load Shedding in Power System Strategies with Voltage Stability Indicators. SciRes., Atlanta, Georgia. [Online]. Available: http://www.scirp.org/jounal/eng/ [12] M.M. Begovic and A.G. Phadke, "Control of voltage stability using sensitivity analysis," IEEE Trans. Power System, vol. 7, pp. 114-123, Feb. 1992. [13] T. Nagao, K. Tanaka and K. Takenaka, "Development of static and simulation programs for voltage stability st udies of bulk power system," IEEE Trans. Power System, vol. 12, pp. 273-281, Feb. 1997. [14] K.Iba, H. Suzuki, M. Egawa and T. Watanabe, "Calculation of critical loading condition with nose curve using homotopy continuation method," IEEE Trans. Power System, vol. 6, pp. 584-593, May. 1991. [15] C. Reis, a. Andrade and F.P. Maciel, “Line stability indices for voltage collapse prediction,” Lisbon. 2009 International Conference on Power Engineering, Energy and Electrical Drives., pp. 239-243. [16] J. Duncan Glover, S. Mulukutla Sarma and J. Thomas Overbye, “Power system analysis and design,” 5th ed. Stamford: Cengage Learning, 2012, pp. 311-352. [17] L. Mariesa Crow, “Computational method for electric power systems,” 2nd ed. Boca Raton, Florida: Tylor and Francis Group, LLC, 2010, pp. 72-98. [18] Germund Dahlquist and Ake Bjorck, “Numerical Methods,” 1st ed. Stamford: Prentice-Hall, Inc, 1974, pp. 142-144. [19] N. Philip Sabes, "Linear Algebraic Equations, SVD, and the PseudoInvers," San Francisco, Oct. 2001.
[1]
Single line diagram of 23 bus
[20] Yao Zhang and Wennan Song, “Power system static voltage stability limit and the identification of weak bus,” Beijing. 1993 IEEE Region 10 Conference on Computer, Communication, Control and Power Engineering., vol. 5, pp. 157-160.
R EFERENCES
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