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Electric Power Systems Research 77 (2007) 910–916

Incorporating aging failure mode and multiple capacity state model ofHVDC system in power system reliability assessment

Jiaqi Zhou a, Wenyuan Li b,∗, Jiping Lu a, Wei Yan a

a College of Electrical Engineering, Chongqing University, Chongqing 400044, Chinab British Columbia Transmission Corporation, Suite 1100, Four Bentall Center, 1055 Dunsmuir Street, P.O. Box 49260, Vancouver, BC, Canada V7X 1V5

Received 20 May 2006; accepted 8 August 2006Available online 11 September 2006

bstract

The paper presents an approach to incorporating the aging failure mode and multiple capacity states of a HVDC system in power system reliabilityssessment. Subcomponents in HVDC converter stations have a much shorter mean life compared to an ac line or cable. When their age approacheshe mean life, the failure rate due to aging increases greatly and it is necessary to include their aging failure mode in the assessment. The modeling

ethod presented for multiple capacity states includes three reliability block networks. Probabilities of a HVDC system being at the full, derated

nd zero capacities can be easily calculated using the three reliability networks. Once the state probabilities are obtained, the HVDC system cane represented using a three-state model in overall power system reliability evaluation. An actual example in a utility is given to demonstrate thepplication of the presented approach in a power supply system with an aged HVDC link.

2006 Elsevier B.V. All rights reserved.

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eywords: Aging failures; Failure model; HVDC system; Multiple states; Reli

. Introduction

HVDC transmission links are used in power systems for var-ous purposes, for example, interconnecting two neighboringystems of using different frequencies, improving system tran-ient stability as a tie line between two large regions or twoountries, providing a long distance transmission passage fromydrogenation, etc.

Considerable efforts have been devoted to reliability assess-ent of power systems in the past decades [1–7]. There is an

ncreasing need of incorporating a HVDC system model intoower system reliability evaluation. A basic difference in reli-bility modeling between an ac line and an dc link is that anc line is a single component (an overhead circuit or under-round cable) and generally can be modeled by two states of

uccess and failure, whereas a dc link consists of various sub-omponents (valves, transformers, smoothing reactors, filters,verhead lines/cables and auxiliary control equipment) and fail-

∗ Corresponding author. Tel.: +1 604 699 7379.E-mail addresses: zhoujiaqi610@sohu.com (J. Zhou),

en.yuan.li@bctc.com (W. Li), ljp2008@x263.net (J. Lu),quyanwei@21cn.com (W. Yan).

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378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.epsr.2006.08.003

evaluation; Reliability network

res of some subcomponents only result in a derated capacitytate. In other words, multiple states (up, down and derated)ave to be considered in a HVDC reliability model. A fewapers discussed the HVDC reliability model using a mixederies and parallel network that is formed directly from physicalonnecting relationships of subcomponents [8–10]. In general,uch a network can create an equivalent component only withwo states of success and failure but cannot model the multipletates including a derated capacity state. Conceptually, a statenumeration technique or an event tree approach can be usedo capture multiple states [1,11]. However, calculation effortsill exponentially increase with the number of subcomponents

n a HVDC system. For instance, for a system with 20 sub-omponents, an exhaustive enumeration requires to consider20 = 1,048,576 states. An improved enumeration technique wasresented to partially overcome this disadvantage [12].

Another difference between an ac overhead line and a dc links that an overhead line generally does not have an aging concernan ac underground cable may have) whereas the subcompo-

ents in a HVDC link (valves, transformers, reactors, filters andables) age with time and have a much higher failure rate atheir end-of-life stage. Many actual HVDC systems across theorld have been operated for 25–30 years or even longer and are

ystem

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J. Zhou et al. / Electric Power S

pproaching their end-of-life stage although they are still in oper-tion. Therefore, unlike an ac overhead circuit that only needso model repairable failures, both aging and repairable failuresf HVDC subcomponents should be considered in modeling,articularly for an aged HVDC system.

This paper presents a simple method to model multiple capac-ty states of HVDC systems. The method includes three relia-ility networks: (1) a series network; (2) a combined series andarallel network; (3) a k-out-of-n network. The series one isased on the reliability logic and has nothing to do with physi-al connections of HVDC subcomponents. The combined seriesnd parallel network is basically consistent with physical con-ections of HVDC subcomponents except for those with partialedundancy. The k-out-of-n network is used to represent the sub-omponents with partial redundancy for the full capacity (suchs cables) and can be merged into the first two network models.he aging failure mode of HVDC subcomponents are also pro-osed and can be easily incorporated into the model. An actualxample in a utility of Canada is given to demonstrate the appli-ation of the presented approach in reliability evaluation of aower supply system with an aged HVDC link.

. Aging failure model of HVDC subcomponents

As is well known, the unavailability due to repairable failuresf a component is calculated by [11]

r = f × MTTR

8760(1)

here f is the average failure frequency (failures/year) andTTR is the mean time to repair (h/repair).The aging failure mode of a HVDC subcomponent can be

odeled using a conditional probability concept. The unavail-bility due to aging failure is calculated by [2]

a = 1

t

N∑i=1

Pi

[t − (2i − 1)

D

2

](2)

here

i =∫ T+iD

Tf (x)dx − ∫ T+(i−1)D

Tf (x)dx∫ ∞

Tf (x)dx

, i = 1, 2, . . . , N

(3)

here T is the age of a subcomponent, t the subsequent period toe considered, which is often 1 year, N the number of intervalsnto which t is equally divided, D the length of each intervalnd f(x) is the failure density probability distribution functionor the aging failure mode and can be modeled using a normalistribution [2,11].

Eq. (3) gives a conditional failure probability of a subcompo-ent within an interval D given that it has survived for T years.ith the normal distribution model, Eq. (3) can be expressed as

i = Q((T + (i − 1)D − μ)/σ) − Q((T + iD − μ)/σ)

Q((T − μ)/σ),

i = 1, 2, . . . , N (4)

(

s Research 77 (2007) 910–916 911

here μ and σ are the mean and standard deviation of the normalistribution and the function Q is approximated by

Q(y) ={

w(y), if y ≥ 0

1 − w(−y), if y < 0

w(y) = z(y)(b1s + b2s2 + b3s

3 + b4s4 + b5s

5)

z(y) = 1√2π

exp

(−y2

2

)

s = 1

1 + ry

r = 0.2316419, b1 = 0.31938153, b2 = −0.356563782,

b3 = 1.781477937, b4 = −1.821255978,

b5 = 1.330274429

The total unavailability of component considering bothepairable and aging failures can be obtained using a union con-ept:

= Ur + Ua − UrUa. (5)

. Reliability model of a HVDC system

.1. Modeling method

HVDC subcomponents are physically connected in series andarallel to form a HVDC system. However, the physical config-ration in a series and parallel network form cannot be directlysed for its reliability evaluation. First, some subcomponentssuch as cables) may be designed as a connection with partialedundancy for the full capacity. For instance, a HVDC sys-em can contain three cables in parallel while only two of themre needed to transfer the full capacity. In other words, onlyn the case where two or all three cables are out of service,he HVDC system fails. Secondly, if the physical configura-ion of HVDC system is directly used as a reliability blockiagram, the reliability (or availability) obtained from this dia-ram is not the probability of the HVDC system being at theull capacity. It is the probability of the HVDC system beingt the full capacity plus being at its derated capacity sincerobabilities of HVDC system derated states due to failuresf one or more subcomponents in a parallel branch are alsoncluded.

It is proposed to build three reliability block networks toodel a HVDC system. The procedure to evaluate the relia-

ility of a power system containing a HVDC link includes theollowing aspects:

1) Subcomponents with partial redundancy for the full capac-ity are separately modeled using a k-out-of-n network. Theeffects of this network on the HVDC system reliability ismerged after the probabilities of multiple capacity statesof the HVDC system are obtained using the following two

reliability block networks.

2) All other subcomponents except those with partial redun-dancy for the full capacity form a series network. It isobvious from a reliability logic viewpoint that the reliability

9 ystems Research 77 (2007) 910–916

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Fig. 2. Series reliability network for the system.

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12 J. Zhou et al. / Electric Power S

(availability) of this series network is the probability of theHVDC system being at its full capacity excluding the effectof the k-out-of-n network.

3) All other subcomponents except those with partial redun-dancy for the full capacity are connected in the structuresimilar to their physical connections. This is a combinedseries and parallel reliability block network. The reliabil-ity (availability) of this block network is the probability ofthe HVDC system being at the full capacity plus being atits derated capacity excluding the effect of the k-out-of-nnetwork.

4) The probability of the HVDC system being at its deratedcapacity excluding the effect of the k-out-of-n network isthe reliability of the combined series and parallel networkin Step (3) minus the reliability of the series network in Step(2).

5) By applying the intersection concept, the probability of theHVDC system being at the full (or derated) capacity stateincluding the subcomponents with partial redundancy is theprobability of the HVDC system being at the full (or derated)capacity state excluding effect of the k-out-of-n network,which is obtained in Step (2) (or in Step (4)), multiplied bythe reliability of the k-out-of-n network, which is obtainedin Step (1).

6) The HVDC system is modeled using an equivalent com-ponent with multiple capacity states in a power systemreliability evaluation [13].

.2. Explanation of the modeling method

The above concept can be further explained using a simplexample. Consider a six-component system as shown in Fig. 1.he system will be at its zero capacity state if Component 1

ails since it is in a series structure. Components 2 and 3 areonnected in parallel and if one of them fails, the system can beperated at a derated capacity state. Although Components 4, 5nd 6 are also physically in parallel, the system can still operatet its full capacity when any of the three components fails. Inther words, these three components have partial redundancyor the full capacity. This is not a real HVDC system and issed only for the purpose of explaining the concept. To obtainimple analytical results, let us assume that the availability ofach component is A (including both repairable and aging failureodes).

Components 4, 5 and 6 form a 2-out-of-3 network and its reli-

bility is separately evaluated using any method for a k-out-of-network [11,14]. Its reliability (availability) is P(k) = 3A2 − 2A3.

Fig. 1. A six-component system.

tbcrat

4

4

a

Fig. 3. Combined series and parallel reliability network for the system.

The series and combined series and parallel reliability net-orks excluding the Components 4, 5 and 6 are shown inigs. 2 and 3, respectively. Their reliability can be easily assessedsing a simple network modeling method [11]. The reliabil-ty (availability) of the series network is the probability ofhe system being at its full capacity (excluding the effect ofhe Components 4, 5 and 6), which is P(s) = A3. The reliabil-ty (availability) of the combined series and parallel networks the probability of the system being at its full and der-ted capacities (excluding the effect of the Components 4, 5nd 6), which is P(sp) = (A + A − AA)A = 2A2 − A3. Therefore,he probability of the system being at the derated capacitytate (excluding the effect of the Components 4, 5 and 6) is:(sp) − P(s) = 2A2 − 2A3.

The probabilities of the system being at its full, derated andero capacities including the effect of Components 4, 5 and 6re:

The probability at the full capacity = P(s)P(k) = A3(3A2 −2A3).The probability at the derated capacity = [P(sp) − P(s)]P(k)= (2A2 − 2A3)(3A2 − 2A3).The probability at the zero capacity = 1 − P(s)P(k) − [P(sp)− P(s)]P(k) = 1 − P(sp)P(k) = 1 – (2A2 − A3)(3A2 − 2A3).

The above results obtained using the proposed three reliabil-ty networks can be easily verified by enumerating all possibleystem state probabilities for this very simple system.

It can be seen that the probabilities of multiple capacity statesannot be obtained by only using the single combined seriesnd parallel network. Although these probabilities can be calcu-ated using an exhaustive state enumeration technique, utilizinghe proposed three separate reliability networks, which are stillased on the simple series, parallel and k-out-of-n network con-epts, will dramatically reduce computational burdens. For aelatively large system, it is still fast to perform reliability evalu-tion of series and parallel networks whereas a state enumerationechnique often requires considerable calculation efforts.

. Case study

.1. System description

This is an actual power supply system to an island regionnd power sources include two 500 kV ac transmission lines,

J. Zhou et al. / Electric Power System

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Fig. 4. Schematic of the power supply system.

ne HVDC link and local generators. The purpose of the studys to assess the adequacy of power supply to the island regionnd all transmission constraints within the region are omitted.herefore, the system is evaluated as a generation system reli-bility problem since the 500 kV lines and HVDC link can beodeled as power sources as generators. An annual load curve

f the region is considered. Fig. 4 is the schematic of the powerupply system. The focus of the study is placed on modelingf the HVDC link. Both repairable and aging failure modesf HVDC subcomponents and multiple capacity states of the

VDC link are modeled, whereas only repairable forced outagesf the 500 kV lines and generators are considered and repre-ented using up and down two-state models. The skeleton of theVDC system configuration that is consistent with the physical

irvT

Fig. 5. Skeleton of the HVDC system (T, transformer; V, valves; A, auxiliary

Fig. 6. Series network for

Fig. 7. Combined series and parallel

s Research 77 (2007) 910–916 913

onnection of subcomponents is shown in Fig. 5. The transform-rs, valves and auxiliary control/switching equipment at bothides are in parallel branches and a failure of one of them resultsn a derated capacity state. The smoothing reactors, filters andole control at both sides are in series and a failure of any one ofhem leads to a zero capacity state. Two of the three submarineables can deliver the full capacity and therefore they are mod-led using a 2-out-of-3 network while the other subcomponentsre modeled using a series network as shown in Fig. 6 and aombined series and parallel network as shown in Fig. 7.

.2. Data and results

.2.1. Calculating unavailability of HVDC subcomponentsThe data required to model aging failures of HVDC subcom-

onents using the method in Section 2 are the mean life, standardeviation and ages of each subcomponent and are based on fieldssessments and engineering estimation. Their repairable fail-re data are obtained from historical statistics in the past 20ears. The data are given in Table 1. Different subcomponentsre assumed to have different mean lives and/or standard devia-ions. Note that these estimates are specific to the HVDC systemn this utility and have no general implication. The repair time ofmoothing reactor R2 (344 h) is much longer than that of reactor1 (8 h). This is because there is only one spare reactor which

s located at the converter station of R1. The repair time is theeplacement time for R1 but transporting the spare to the con-erter station of R2 needs about 2 weeks (across a sea strait).he repair time of submarine cables is very long (2190 h) due

equipment; R, smoothing reactors; f, filters; P, pole control; C, cables).

the HVDC system.

network for the HVDC system.

914 J. Zhou et al. / Electric Power Systems Research 77 (2007) 910–916

Table 1Data of HVDC subcomponents for aging and repairable failures

Subcomponents Mean life(years)

Standard deviation(years)

Age (years) Reparable failurefrequency (failures/year)

Repair time (h)

Transformer T1, T2, T3, T4 35 10 29 0.028 116Valve V1, V2, V3, V4 33 10 29 0.333 136Auxiliary A1, A2, A3, A4 40 5 29 0.820 3.8Reactor R1 33 5 29 0.0216 8.0Reactor R2 33 5 29 0.0216 344Filter f1, f2 35 5 29 0.2 0.62Pole control P 34 10 29 1.60 2.9Submarine Cable C1, C2, C3 40 10 29 0.118 2190

Table 2Unavailability of HVDC subcomponents due to aging and repairable failures

Subcomponents Unavailability (repairable failure) Unavailability (aging failure) Unavailability (total)

Transformer T1, T2, T3, T4 0.00037 0.02428 0.02464Valve V1, V2, V3, V4 0.00517 0.03859 0.04356Auxiliary A1, A2, A3, A4 0.00036 0.00308 0.00344Reactor R1 0.00002 0.03895 0.03897Reactor R2 0.00085 0.03895 0.03976FPS

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etss

TR

R

2SC

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4t

rdwand represented using the three-capacity-state model given in

ilter f1, f2 0.00001ole control P 0.00053ubmarine Cable C1, C2, C3 0.02950

o the fact that repairing activities under water are extremelyifficult.

The unavailability values of the subcomponents due to agingnd repairable failures were calculated using the method in Sec-ion 2 and are given in Table 2. It can be seen that the totalnavailability of transformers, valves, reactors, filter or poleontrol is dominated by the unavailability due to aging fail-res since the age (29 years) is close to their mean life (33–35ears). The subcomponents with the mean life of 40 years (aux-liary equipment and submarine cable) have much lower agingailure unavailability. The repairable failure unavailability ofhe submarine cables is high because of the long repair time.ote that the total unavailability is not a direct sum of thenavailability due to aging failure and the unavailability dueo repairable failure. It is the union of the two unavailabilityalues.

.2.2. Calculating probabilities of capacity states of theVDC systemThe probabilities of capacity states of the HVDC system were

valuated using the modeling method described in Section 3. Thehree reliability networks are the 2-out-of-3 network for the threeubmarine cables, series network shown in Fig. 6 and combinederies and parallel network shown in Fig. 7.

able 3eliability of three block networks

eliability block network Reliability

-Out-of-3 network 0.99630eries network 0.64042ombined series and parallel network 0.84888

Ttm

TP

S

FDZ

0.02066 0.020670.03080 0.031310.00621 0.03553

The reliabilities for the three block networks are given inable 3. The probability of the HVDC system being at the fullapacity is the reliability of series network. The probability ofhe HVDC system being at the derated capacity (half of the fullapacity) is the reliability of the combined series and paralleletwork minus the reliability of the series network. These tworobabilities are weighted by the reliability of the 2-out-of-3etwork to obtain their probability values including effects ofubmarine cable failures. The probabilities of the HVDC systemt full, derated and zero capacities are shown in Table 4. It cane seen that this HVDC system has a high failure probabilityecause of its aging status.

.2.3. Evaluating reliability of the power supply system tohe island region

A Monte Carlo simulation based method was used to evaluateeliability of the power supply system to the island region. Theata of all components are given in Table 5. The HVDC systemas treated as an equivalent component in the supply system

able 4. The failure data of other components are based on his-orical statistics in the past 20 years. The common cause failure

ode of the two 500 kV lines is also modeled and the relevant

able 4robability of HVDC system at three states

tate Probability(excluding cables)

Probability(including cables)

ull capacity (480 MW) 0.64042 0.63805erated capacity (240 MW) 0.20846 0.20769ero capacity (0 MW) 0.15112 0.15426

J. Zhou et al. / Electric Power System

Table 5Data of components in the island supply system

Component Capacity(MW)

Unavailability Derated stateprobability

Repairtime (h)

Generator 1 24 0.0040 15.35Generator 2 21 0.0795 26.51Generator 3 20 0.0008 2.31Generator 4 20 0.0030 36.32Generator 5 20 0.0026 7.84Generator 6 20 0.0096 28.70Generator 7 21 0.0003 3.77Generator 8 24 0.0010 13.74Generator 9 24 0.0063 19.15Generator 10 24 0.0026 6.60Generator 11 33 0.0027 5.33Generator 12 32 0.0218 28.26Generator 13 165 0.0124 5.99Generator 14 240 0.0650 50.30500 kV line 1 1200 0.0093 137.81500 kV line 2 1200 0.0093 137.81HVDC system 480 0.15426 0.20769 2.10Both 500 kV lines 0.0004 2.98

Table 6Reliability indices of the island supply system

Index Index value

EENS 2238 MWh/yearEA

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FLC 3.26 occurrence/yearDLC 1.88 h/failure

ata is listed in the last line of Table 5. The 8760 hourly loadsn the island region were used to represent an annual load curveith the annual peak of 2194 MW.The reliability indices of expected energy not supplied

EENS), expected frequency of load curtailment (EFLC) andverage duration of load curtailment (ADLC) were calculatednd are shown in Table 6.

. Conclusions

The paper presents an approach to incorporating aging failureode and multiple capacity states of HVDC system in power

ystem reliability assessment. A HVDC system not only con-ists of lines/cables but also converter station subcomponents.ailures of some subcomponents result in the full shutdowntate whereas others lead to a derated capacity state. The sub-omponents in converter stations, such as transformers, valves,moothing reactors and filters, have a much shorter mean lifeompared to an ac line or cable. When their age approacheshe mean life, the failure rate due to aging increases greatlynd it is necessary to include their aging failure mode in thessessment. The modeling method presented for multiple capac-ty states includes three reliability block networks: a k-out-of-n

etwork, a series network, and a combined series and paralleletwork. Probabilities of HVDC system being at the full, der-ted and zero capacities can be easily calculated using the threeeliability networks. Once these state probabilities are obtained,

omWaa

s Research 77 (2007) 910–916 915

he HVDC system can be represented using a three-state modeln power system reliability evaluation.

An actual example in a utility is given to illustrate the appli-ation of the presented approach. It demonstrates the completessessment process including calculations of unavailability ofVDC subcomponents due to both repairable and aging fail-res, reliability evaluation of three block networks, probabilityalculations for the full, derated and zero capacity states of theVDC system, and adequacy assessment of an island power sup-ly system with the aged HVDC link as a major power source.

eferences

[1] R. Billinton, R.N. Allan, Reliability Evaluation of Power Systems, PlenumPress, New York, 1996.

[2] Wenyuan Li, Risk Assessment of Power Systems: Model, Methods, andApplications, IEEE Press/Wiley and Sons, 2005.

[3] R.N. Allan, R. Billinton, A.M. Breipohl, C.H. Grigg, Bibliography on theapplication of probability methods in power system reliability evaluation:1992–1996, IEEE Trans. Power Syst. 14 (1) (1999) 51–57.

[4] R. Billinton, M. Fotuhi-Firuzabad, L. Bertling, Bibliography on the appli-cation of probability methods in power system reliability evaluation1996–1999, IEEE Trans. Power Syst. 16 (November (4)) (2001) 595–602.

[5] IEEE Tutorial Course Text, Electric Delivery System Reliability Evalua-tion, 05TP175, IEEE General Meeting, San Francisco, 2005.

[6] EPRI, Reliability Evaluation for Large Scale Bulk Transmission Systems,Report EL5291, 1988.

[7] CIGRE Task Force 38-03-10 Report, Composite Power System Reli-ability Analysis: Application to the New Brunswick Power Corpora-tion System, CIGRE Symposium on Electric Power System Reliability,September16–18, 1991, Montreal.

[8] R. Billinton, M. Fotuhi-Firuzabad, S.O. Faried, Reliability evaluation ofhybrid multi-terminal HVDC substation systems, in: IEE Proceedings,Generation, Transmission and Distribution, vol. 149, No. 5, September2002, pp. 571–577.

[9] S. Kuruganty, Comparison of reliability performance of group connectedand conventional HVDC transmission systems, IEEE Trans. Power Deliv.10 (October (4)) (1995) 1889–1895.

10] S. Kuruganty, HVDC transmission system models for power systemreliability evaluation, in: IEEE WESCANEX’95 Proceedings, 1995, pp.501–507.

11] R. Billinton, R.N. Allan, Reliability Evaluation of Engineering Systems:Concepts and Techniques, Plenum Press, New York, 1992.

12] W. Li, Probability distribution of HVDC capacity considering repairableand aging failures, IEEE Trans. Power Deliv. 21 (January (1)) (2005)523–525.

13] R. Billinton, W. Li, Reliability Assessment of Electric Power SystemsUsing Monte Carlo Methods, Plenum Press, New York, 1994.

14] E.J. Henley, H. Kumamoto, Probabilistic Risk Assessment: ReliabilityEngineering, Design, and Analysis, IEEE Press, 1992.

iaqi Zhou is currently a full professor at the College of Electrical Engineeringf Chongqing University in China. He has been active in power system planningnd reliability evaluation for about 30 years and published more than 100 papersn this area. He also won several science and technology awards in China due tois contributions in power system reliability methods and applications in electricower utilities.

r. Wenyuan Li (IEEE Fellow) is currently a principal engineer at Britisholumbia Transmission Corporation (BCTC) in Canada and an advisory pro-

essor of Chongqing University in China. He was the co-author of the bookReliability Assessment of Electrical Power Systems Using Monte Carlo Meth-

ds”, Plenum Press, New York, 1994 and the author of the book “Risk Assess-ent of Power Systems: Models, Methods, and Applications”, IEEE Press andiley & Sons, 2005. Dr. Li was the winner of the 1996 “Outstanding Engineer”

warded by the IEEE Canada for the “contributions in power system reliabilitynd probabilistic planning”.

9 ystem

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16 J. Zhou et al. / Electric Power S

r. Jiping Lu is an associate professor at the College of Electrical Engi-eering of Chongqing University in China. His interests include power sys-em relay and protection, automation and probability application in powerystems.

Dnts

s Research 77 (2007) 910–916

r. Wei Yan is an associate professor at the College of Electrical Engi-eering of Chongqing University in China. His interests include power sys-em operation, optimization techniques and probability application in powerystems.