2013 Fourth International Conference on Intelligent Systems Design and Engineering Applications
Risk based identification of cascading chains based on Generalized Line Outage Distribution Factors Yuanyu Dai1,2, Yusheng Xue2,1, Guo Chen3, Yan Xu4 , Zhao. Xu5, Z.Y. Dong4 1
School of Automation, Nanjing University of Science and Technology, Nanjing, China 2 State Grid Electric Power Research Institute (SGEPRI), Nanjing, China 3School of Electrical and Information Engineering, The University of Sydney, Sydney, Australia 4 Centre for Intelligent Electricity Networks (CIEN), The University of Newcastle, Newcastle, Australia 5 Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong (*E-mail:
[email protected]) N1 criterion. They represent the incremental real power flows on monitored transmission line, which are resulted from a single failed line with a pre-contingency real power flow of one MW [11]. In comparison with dc power flow method, they are faster. Furthermore, the LODFs is lately extended to Generalized LODFs (GLODFs) to estimate possible overloads under multiple-line outages, presenting a potential of rapid power flow computation for cascading analysis [12]. In this paper, a novel risk index based on GLODFs and pre-contingency power flow conditions is proposed to quickly assess the consequences of possible following-up contingencies. Moreover, a fast risk assessment method is developed to properly address the accuracy and efficiency in cascading failure analysis. In the proposed scheme, the computation efficiency can be significantly enhanced due to the post-contingency system states are estimated by using the GLODFs.
Abstract—In the past decades, many countries have suffered from serious blackouts which demonstrate catastrophic consequences caused by rare events in power systems. To address the emerged issues, the contingency screening and ranking should consider the risks of potential failures instead of the associated probabilities of following-up failures alone. However, it is computationally infeasible to evaluate all possible cascading failure sequences beforehand. Thus, fast risk assessment method should be developed to enhance computation efficiency while maintaining acceptable accuracy. In this paper, based on distribution factors and pre-contingency power flow conditions, a novel risk index is proposed to quickly assess the consequences of possible cascading failures. Then creditable cascading failure sequences can be determined and ranked according to the risk index. In addition, case studies on the IEEE-118 bus system show that the computational efficiency of the proposed cascading assessment method can be greatly enhanced with satisfactory accuracy, indicating a high potential of practical implementation.
II. THE GLODFS FOR CASCADING ANALYSIS
Keywords- Risk, Cascading chain, Distribution factors.
A.
Line Outage Distribution Factors Generally LODFl,k represents the percentage of the pre-contingency flow on line k that is to contribute to line l after the outage of line k, and is calculated as follows [13]: PTDFl ,k (1) LODFl ,k 1 PTDFk ,k LODFs can be expressed in pre-contingency network parameters based on Power Transfer Distribution Factors (PTDFs), where the post-contingency network parameters are not needed. As such it can largely improve the computational efficiency.
I. INTRODUCTION The concept of cascading failure is defined in [1] as a sequence of dependent failures of individual components that successively weakens the power system. An initial failure can cause the power flow to be redistributed in a power system due to circuit laws. This may result in some components overloaded. As a consequence, those overloaded components may fail and then trigger the instability of other components. Likewise, a cascading failure may happen. In a power system, there are tremendous amount of components subjected to many different types of faults. Moreover, these failures may interact with each other, resulting in highly complicated cascading procedures [2]. The involving interactions can include deviations in power flows, frequency, voltage, as well as operations or misoperations of protection devices, controls, operator procedures, monitoring and alarm systems. It’s obvious that exhaustive computation of all combinations of cascading failures is infeasible. Thus, the existing cascading failure analysis methods have different compromises [3-7]. For example, analyses focus on cascading phenomena caused by thermal limits, such as steady-state cascading line overloads [8-10]. Traditionally, the Line Outage Distribution Factors (LODFs) are used to perform contingency analysis, mainly as /13 $26.00 © 2013 IEEE 978-1-4799-2791/13 $26.00 © 2013 IEEE DOI 10.1109/ISDEA.2013.531 10.1109/ISDEA.2013.45 10.1109/ISDEA.2013.133
B.
Generalized Line Outage Distribution Factors Generalized line outage distribution factors (GLODFs) extend the LODFs to address multi-line outages [12]. Detailed derivation of GLODFs based on the PTDFs of a pre-contingency network can be found in [11]. The relationship between GLODFs and PTDFs is given by Eq. (2), which actually is an extension of Eq. (1) 1 (2) GLODFl ,o PTDFl ,o I PTDFo,o
where subscript o represents the set of v lines on outage and I is an identity matrix of v×v. PTDFo,o is v×v matrix, which solves the cross effect of multi-line outages. PTDFl,o is 1×v vector, which is obtained from the original PTDF matrix. 169 553
PTDFo1 ,o1 PTDFo,o PTDFo ,o v 1
PTDFo1 ,ov PTDFov ,ov
PTDFl ,o PTDFl ,o1
PTDFl ,ov
probability and consequence [14]. This definition captures the essence of the risk index used for security evaluation in the proposed risk-based approach. Herein, the risk based contingency analysis is applied for all contingencies in the list but not just the highest severities or possibilities.
(3)
(4)
Mathematically, GLODFl,o is a 1 × v vector, where each element respectively represents the percentage of the pre-contingency flow on each outage line that will contribute to line l after the multi-line outages.
A.
Outage probability of transmission lines According to historical data, the tripping probability on a line would rise with the increase of power on the line. The widely adopted technology is to set a tripping probability on each exposed line. This is set as an increasing function of the power flow on each exposed line. The probability is low initially, well below the line security limit and increases linearly to 1 when the line flow reaches the security limit. The mathematical expression is given by p L L0 (6) (1 p )( L L0 ) Pl ( L) p L L0 , Lmax Lmax L0 1 L Lmax where subscripts 0 and max represent base case and line security limit respectively; subscript l represents the monitored line.
C.
Feasibility of using GLODFs in cascading failure The estimation of post-contingency flow on line l is given by (5) Pl c Pl 0 GLODFl ,o Po0 where superscripts 0 and c represent base case and contingency cases; l represents the monitored line, and o represents the set of v lines on outage. Different from traditional linear approximation method, in this paper the pre-contingency power flow is achieved by applying ac power flow model once so as to enhance the computing precision and then the GLODFs is used to make contingency analysis. Take the IEEE-118 bus system for example, a contingency is considered when the system having three transmission lines 44, 45 and 54 outaged simultaneously. Post-contingency line power flow errors of the GLODFs estimation and dc power flow solutions versus full ac power flow solutions are compared in Fig.1. The horizontal ordinate is the ID of transmission lines from 1 to 186. The vertical ordinate describes the line power flow errors versus full ac power flow solutions. The circle dots denote the errors of GLODFs estimation while the cross dots denote the errors of dc power flow solutions.
B.
Outage consequence of transmission lines The cascading risk index proposed in this paper is designed to reflect the possibility and severity of overloaded lines in the cascading process. Admittedly, it is difficult to design an index that can express every aspect of a power system’s security level. This paper only focuses on the consequences of line overloading during cascading failure process. There are several measurements to assess the current security level of a power system, including stability margin, outage cost and control cost. Stability margin reflects the stability after a disturbance. Outage cost can reflect the severity of a happened failure, but could not estimate the unserved buses or duration time for a predictive fault. The minimum control cost can ensure the system security not only in estimating the range and duration time of a failure, but also can meet the actual situation. In this paper, the consequence of a transmission line l after a fault d is supposed by L L0 0 (7) Cl ( L) L L0 ( L L0 ) where represents the price of per MW power, represents the considered insecure ratio of a line power flow to the base case value [15]. The summed consequence of the fault d is given by (8) C C (L )
Power flow error with AC power flow(MW)
50.0 GLODFs estimation DC power flow calculation
40.0 30.0 20.0 10.0 0.0 -10.0 -20.0 -30.0 -40.0 -50.0
0
20
40
60
80
100
120
140
160
180
ID of each branch
Fig. 1. The computation error analysis for outaged lines 44, 45 and 54
The simulation results show that the GLODFs estimation is more accurate than dc power flow solutions for almost all transmission lines. For transmission line 107, the computing error of dc power flow solution is 48.88 MW which is much bigger than 3.07 MW for GLODFs estimation. The worst computing error of GLODFs estimation is only 3.51 MW for transmission line 104, which can be ignored in the cascading failure analysis.
d
C.
il
i
The cascading risk index The risk index should quantitatively capture the factors that determine security level: likelihood and severity of contingency events [16]. This cascading risk index is a good indicator of the actual reliability level, which is decomposable, and can be effectively integrated into economic decision making paradigms. The cascading risk index is defined as the product of probability and consequence
III. RISK ASSESSMENT FOR CASCADING FAILURE The IEEE standard definition for risk is the product of
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R j 1,i Pj Pj 1,i C j 1,i P(d j ) P(d i | d j 1 )C j 1,i
(9)
assumed as key cascading chains which should be paid much more attention to in the system operation, and can be taken as a benchmark to test the proposed algorithm. A simplified cascading tree is obtained according to the proposed algorithm. The computation accuracy can be verified through the coverage level of the top risk ranking of the simplified cascading tree against the benchmark. V. SIMULATION RESULTS In order to verify the proposed algorithm, IEEE-118 bus system is selected for a case study. Simulations are conducted using MATPOWER 4.0b platform.
Fig. 2. The model of risk index
A. Verification of proposed algorithm Over 1,000,000 simulations have been conducted to validate the effectiveness and robustness of the proposed algorithm. In the simulation, is set to 0.9 and is 1 for simplicity sake. Firstly, initial outages are selected by the cascading risk based on contingency screening, as shown in Table I.
By screening the cascading risk index of all the outages in the stage j+1, the most dangerous transmission line faults can be selected. Furthermore, the most important sequences of cascading failure can be identified when the analysis for all stages is finished. IV. THE PROPOSED ALGORITHM FOR CASCADING FAILURE ASSESSMENT BASED ON GLODFS
TABLE I THE INITIAL OUTAGES SELECTED BY RISK INDEX Ranking Line number Risk 1 9 2402.13 2 7 2401.47 3 8 1834.18 4 38 1147.91 5 96 922.53 6 51 904.92 7 36 824.94 8 97 607.24 9 31 469.82 10 33 448.78 11 141 421.61 12 94 364.37
A. Identify the sequences of cascading failure The methodology of identifying the important sequences of cascading failure is displayed in the flowchart as in Fig. 3. The initial operating condition of a power system is assumed to be stable with a standard economic dispatch generated to meet the expected system load scenario. During every outage stage j, the selected amount of outage d can be chosen by the balance of the accuracy and efficiency. However, optimally choosing the accuracy and efficiency remains an open question. A further discussion on a different selected amount p of outage has been done in the simulation part.
Secondly, further discussions are made on the efficient use of the proposed algorithm in the cascading failure analysis. Table II reflects N-3 cascading contingency analyses on IEEE-118 bus system. The parameters D1 and D2 represent the selected amount of outage stage 1 and 2. With a decrease of the parameter D1 and D2, the scale of simplified cascading tree declines obviously and some key cascading chains may be omitted in the algorithm. An appropriate parameter value for the proposed algorithm will greatly enhance the computation efficiency and guarantee the accuracy. TABLE II THE N-3 CASCADING CONTINGENCY SELECTED BY RISK INDEX
Fig. 3. The flowchart of proposed algorithm
B. Validity check scheme for proposed algorithm Through the complete simulation of contingency screening by ac power flow method, the final ranking of all sequences of cascading failure in the cascading tree can be formed by risk assessment method. The top 50 worst sequences are
Testing methodology
The amount of cascading chains
Time(s)
The omitted key chains
AC method Proposed algorithm D1=80, D2=40 D1=50, D2=30 D1=40, D2=30 D1=30, D2=30 D1=30, D2=20 D1=20, D2=20 D1=10, D2=10 D1=10, D2=7
6331440
66825.73
0
595200 279000 223200 167400 116000 74400 18600 13020
1018.26 552.71 438.14 319.58 226.92 152.86 34.45 23.89
2 2 2 2 6 6 6 6
The omitted key chains for case 1 to case 4 are line 8line 9line 96 and line 8line 7line 96. The added four omitting key chains for case 5 to case 8 are line 9line 141line 96 (ranking 22th), line 7line 141line 96
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(ranking 23th), line 9line 33line 96 (ranking 45th) and line 7line 33line 96 (ranking 46th). Table III represents the top 20 risk ranking cascading chains as a reference.
hardly to be fully examined through power flow calculations due to the excessive computing burden. This paper develops a novel risk index to quickly assess the consequences of possible following-up contingencies. Cascading failures of severe consequences can be effectively identified according to the proposed risk index. Furthermore, the GLODFs are introduced to significantly reduce power flow calculations. Simulation results demonstrate that the proposed method is promising for cascading failure analysis with significantly reduced computational burden while ensuring satisfactory precision.
TABLE III THE N-3 CASCADING CONTINGENCY RANKING Ranking Line number Risk Status 1 L8 L36 L51 8197.92 selected 2 6901.72 selected L9 L37 L96 3 6899.38 selected L7 L37 L96 6627.43 selected 4 L9 L96 L108 6625.50 selected 5 L7 L96 L108 6563.90 omitted 6 L8 L9 L96 7 6562.66 omitted L8 L7 L96 6043.87 selected 8 L9 L61 L96 6041.54 selected 9 L7 L61 L96 10 5870.18 selected L9 L36 L37 5868.45 selected 11 L7 L36 L37 5861.56 selected 12 L9 L96 L105 5860.19 selected 13 L7 L96 L105 5741.46 selected 14 L9 L96 L106 5740.08 selected 15 L7 L96 L106 5739.99 selected 16 L9 L96 L59 5738.27 selected 17 L7 L96 L59 5713.58 selected 18 L9 L96 L41 5712.13 selected 19 L7 L96 L41 5693.31 selected 20 L9 L96 L57
Reference [1] [2] [3]
[4]
B. Improvement on proposed algorithm Since the proposed algorithm will omit some key chains, an improvement is designed as follows. 1) Given a supposed operating condition, perform the proposed algorithm; 2) Obtain the top 12 initial outages selected in the proposed algorithm. All the combination of up to three is chosen to form the contingency set; 3) Choose a contingency sequence from contingency set. Check the contingency sequence whether it is included in step 2). If so, go to step 5); otherwise, go to step 4); 4) Recalculate load flow and assessment this contingency sequence by proposed cascading risk index; 5) If all contingency sequences in the contingency set are scanned, screen all the contingency sequences; otherwise, return to step 3). Due to the improvement above, the parameters D1 and D2 can be set small enough to enhance the computation efficiency in order to guarantee the computation accuracy.
[5]
[6] [7]
[8] [9] [10]
[11]
TABLE IV THE N-3 CASCADING CONTINGENCY SELECTED BY RISK INDEX Testing methodology
The amount of cascading chains
Time(s)
The omitted key chains
[12]
Improved algorithm D1=10, D2=7
14340
39.67
0
[13]
With the improvement above, critical cascading failure sequences can be identified accurately and rapidly. The proposed algorithm achieves a good balance between the accuracy and efficiency, indicating a high potential of practical implementation.
[14] [15] [16]
VI. CONCLUSION The prominent role of cascading failure in recent blackouts has highlighted the critical need for a rapid assessment of multiple-line outage in security analysis. The development of a cascading failure can involve many possibilities, which are
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IEEE PES CAMS Task Force, “Initial review of methods for cascading failure analysis in electric power transmission systems,” IEEE Power Engineering Society General Meeting, Pittsburgh, PA, USA, Jul. 2008. IEEE PES CAMS Task Force, “Vulnerability assessment for cascading failures in electric power systems,” Power Systems Conference and Exposition. PES '09. IEEE/PES, vol., no., pp.1-9, 15-18 Mar. 2009. I. Dobson, B.A. Carreras, D.E. Newman, “Branching process models for the exponentially increasing portions of cascading failure blackouts,” 38th Hawaii International Conference on System Sciences, Jan. 2005, Hawaii. I. Dobson, K.R. Wierzbicki, B.A. Carreras, V.E. Lynch, D.E. Newman, “An estimator of propagation of cascading failure,” 39th Hawaii International Conference on System Sciences, Jan. 2006, Kauai, Hawaii. G. Chen, Z. Y. Dong, D. J. Hill, et al., “An improved model for structural vulnerability analysis of power networks,” Physica A: Statistical Mechanics and its Application, Elsevier, vol. 388, pp. 4259-4266. Oct. 2009. I. Dobson, K. R. Wierzbicki, J. Kim, H. Ren, “Towards quantifying cascading blackout risk,” Bulk Power System Dynamics and Control VII, Charleston, South Carolina, USA, Aug. 2007. I. Dobson, B.A. Carreras, D.E. Newman, “A branching process approximation to cascading load-dependent system failure.” 37th Hawaii International Conference on System Sciences, Hawaii, Jan. 2004. C. Davis, “Cascading contingency severity evaluation using linear factors,” M.S. thesis, University of Illinois at Urbana-Champaign, 2005. A. Wang, Y. Luo, G. Tu, P. Liu, “Vulnerability assessment scheme for power system transmission networks based on the fault chain theory,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 442–450, Feb. 2011. G. Chen, Z. Y. Dong, D. J. Hill, et al., “Attack structural vulnerability of complex power grids: a hybrid approach based on complex networks,” Physica A: Statistical Mechanics and its Application, Elsevier, vol. 389, pp. 595-603. Feb. 2010. J. Guo, Y. Fu, Z. Li, and M. Shahidehpour, “Direct calculation of line outage distribution factors,” IEEE Trans. Power syst., vol. 24, no.3, pp. 1633-1634, Aug. 2009. T. Guler, G. Gross, and M. Liu, “Generalized line outage distribution factors,” IEEE Trans. Power syst., vol. 22, no.2, pp. 879-881, May 2007. A. Wood and B. Wollenberg, Power Generation, Operation and Control. New York: Wiley, 1984. The New IEEE Standard Dictionary of Electrical and Electronic Terms, 5th Edition. 1993, IEEE. B.A. Carreras, V.E. Lynch, I. Dobson, and D.E. Newman, “Complex dynamics of blackouts in power transmission systems,” Chaos, vol. 14, no. 3, pp. 643-652, Sep. 2004. J. McCalley, V. Vittal, and N. Abi-Samra, “Overview of risk based security assessment,” in Proc. of the 1999 IEEE PES Summer Meeting, July 18–22, 1999.
