Electrical Power and Energy Systems 43 (2012) 427–432

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Short-term reliability equivalence algorithm for flexible transmission equipment

Shenghu Li a,⇑, Chaobo Dai b, Ying Zhu a

a School of Electrical Engineering and its Automation, Hefei University of Technology, Hefei 230009, Chinab Smart Grid Research Institute, State Grid Corporation of China, Beijing 102200, China

a r t i c l e i n f o

Article history:Received 5 July 2009Received in revised form 1 April 2012Accepted 24 May 2012Available online 6 July 2012

Keywords:Reliability equivalenceFlexible transmissionInstantaneous state probabilityAverage unavailability

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.05.040

⇑ Corresponding author. Tel./fax: +86 551 290 4435E-mail address: shenghuli@hfut.edu.cn (S. Li).

a b s t r a c t

Flexible power transmission are based on power electronic elements with series/parallel configurationand redundancy design. Because of the limited time span, fast replacement, and not fully reliable redun-dant elements due to unexposed failure, it is better to describe their reliability level with short-term eval-uation instead of long-term evaluation. In short-term evaluation, the reliability of flexible transmissionequipments is time-dependent, and decided by the initial state and the evaluation period, which is dif-ferent from its steady-state value.

In this paper, instantaneous state probability and unavailability for multi-state system within limitedevaluation period are described analytically, and equivalenced with the average values. The algorithm isbased on state transition matrix, and suitable for systems with different configurations. For systems withseries or parallel elements, the steady–state and average unavailability are quantified, and comparedwith the existing method of multiplying the (un) availabilities each element directly. It is found thatdue to simultaneous failure and state transition, the existing method may yield error to the fixed andtime-variant terms of the average unavailability.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Based on power electronics technology, flexible transmissionequipments, such as the high-voltage DC transmission (HVDC)and HVDC-light [1,2], static var compensator (SVC), thyristor con-trolled series compensator (TCSC) [3–6], static synchronous com-pensator (StatCom), and unified power flow controller (UPFC),are applied in power transmission to improve var balance, voltageprofile, transfer capacity, and dynamic performance [7–9].

Flexible HVDC or AC equipments include the hardware and soft-ware components, whose reliability is evaluated mainly by theelectrical hardware components, while the non-electrical part,e.g. the cooling systems, are also important to fulfill the designedfunction [10–14]. The reliability data is the basis for the probabilis-tic adequacy and security evaluation of composite power systems.For multi-element systems with series/parallel configurations,analytical and probabilistic methods, such as contingency enumer-ation, fault tree analysis [15], failure mode and effect analysis [16],frequency and duration analysis, and Monte-Carlo sampling, areapplied to determines the state probability, availability, and capac-ity levels.

Reliability evaluation to power-electronics-based converters orcompensators is performed with different time scales. The long-term evaluation yields yearly indices suitable for power system

ll rights reserved.

.

planning. The reliability data is determined by the equipmentbehavior, and classified by three types [17]: (a) transient forcedoutage, where the unit is undamaged and is restored to serviceautomatically; (b) temporary forced outage, where the unit isundamaged and is restored to service by manual switching withoutrepair, but possibly with on-site inspection; and (c) permanentforced outage, where the unit is damaged and is not restorable toservice until repair or replacement is completed.

However, long term evaluation cannot provide effective coun-termeasures against uncertainties under given scenarios, or in lim-ited time span. For flexible transmission equipments with series/parallel configuration and redundancy design, replacement of faultelement is much shorter than the life span. Therefore short-termreliability evaluation with the period of hours or shorter are moredesirable. Within limited time span, the state probability andunavailability change with time [18], and are dependent on the ini-tial states. Therefore, their instantaneous and average values overthe limited time span should be analyzed, which is easy for two-state elements, but difficult for multi-state systems. Furthermore,the standby unit may be unavailable due to unexposed failures.The average unavailability is different from the steady-state valuefor long-term evaluation. It is also different from the interval anal-ysis, which estimates the zone distribution of the reliability indicesbased on uncertain reliability data, and are not necessarily relevantto short-term reliability evaluation [19–22].

In this paper, the instantaneous state probability and unavail-ability of multi-state systems are analytically quantified, and

t

U (t)

t1 t2

U [t1, t2]

Fig. 1. Average unavailability within a period of time.

428 S. Li et al. / Electrical Power and Energy Systems 43 (2012) 427–432

equivalenced to the average values over limited time span. Theinstantaneous and average reliability data are proved dependenton the initial state and the evaluation period. For series and parallelsystem configurations, the steady-state and average unavailabilityare quantified and compared. It is found that due to simultaneousfailure and insufficient transition within limited time, short-termreliability should be evaluated based on the practical state diagraminstead of directly multiplying the (un)availability of the elements.The proposed algorithm is validated with a converter station ofHVDC transmission system. Influences of the initial state, the eval-uation period, and the redundancy availability on the instanta-neous and average unavailability are quantified.

2. Instantaneous state probability

For a Markov transition process with n states, its long-term reli-ability is described by the steady-state probability P in (1), whereU = [uij] is the transpose of transition rate matrix K, n is the num-ber of the states. It is easy to find that P, the steady-state availabil-ity A, and the steady-state unavailability U, are irrelevant to theinitial states. The unavailability is applied to decide the up/downstates in contingency enumeration, fault tree analysis, and non-sequential Monte-Carlo sampling.

UP ¼ ð0;0; . . . ;0ÞTPPi ¼ 1

(ð1Þ

For short term evaluation, the instantaneous state probabilityP(t) is defined by (2). With given initial state P (t0), P(t) and the cor-responding unavailability U(t) are given by (3) and (4), where D isthe subset of unavailable states.

_PðtÞ ¼ UPðtÞ ð2ÞPðtÞ ¼ eUðt�t0ÞPðt0Þ ð3ÞUðtÞ ¼

Xj2D

PiðtÞ ð4Þ

For a element with two states, i.e. up and down states, P(t) andU(t) are given by (5) and (6), where k and l are the failure and re-pair rates. Subscripts 1 and 2 denote the up and down states. It isclear that P(t) is related to the initial state P(t0). For the power elec-tronics equipments, there will be redundant element, whose avail-ability may be not fixed since its unexposed failure is found onlywhen it is called on to operate.

PðtÞ ¼ 1kþ l

lþ ke�ðkþlÞðt�t0Þ l� le�ðkþlÞðt�t0Þ

k� ke�ðkþlÞðt�t0Þ kþ le�ðkþlÞðt�t0Þ

" #P1ðt0ÞP2ðt0Þ

� �ð5Þ

UðtÞ ¼ P2ðtÞ ¼k

kþ l� kP1ð0Þ � lP2ð0Þ

kþ le�ðkþlÞðt�t0Þ ð6Þ

Since the state probability changes with time, and is differentfrom its steady-state value, the unavailability at the start or theend of evaluation period s = [t1, t2] (t0 6 t1 < t2), i.e. U(t1) or U(t2),may over- or under-estimate the reliability level, as shown inFig. 1. Therefore it is necessary to equivalence the state probabilityand unavailability to the average values in the given period, whichis easy for a two-state system, but a challenge for multi-statesystems.

3. Average unavailability for multi-state systems

In this section, an analytical equivalence method based on thestate transition matrix will be proposed to quantify the averagestate probability and unavailability within the evaluation period,suitable for multi-state systems with different configurations.

The average unavailability U[t1, t2] between [t1, t2] is defined by,

U½t1; t2� ¼1

t2 � t1

Z t2

t1

UðtÞdt ð7Þ

For the two-state element, probability P(0) of the initial statemay be (1, 0)T(up), or (0, 1)T (down). Substituting the P(0) to (6),one can get the average unavailability as following,

Uð1;0Þ½t1; t2� ¼k

kþ lþ k

ðkþ lÞ2ðt2 � t1Þs½e�ðkþlÞðt2�t1Þ � 1� ð8Þ

U 0;1ð Þ½t1; t2� ¼k

kþ l� lðkþ lÞ2ðt2 � t1Þ

½e�ðkþlÞðt2�t1Þ � 1� ð9Þ

For a multi-state system, if the instantaneous probability P (t)between [t1, t2] is known, the average unavailability U[t1, t2] isthe sum of probabilities of the unavailable states divided the timeinterval t2 � t1 (10). Now the problem is to find the analyticalexpression of P(t) with the given initial state P(t0).

U½t1; t2� ¼R t2

t1

Pi2DPiðtÞ

� �dt

t2 � t1¼P

i2D

R t2t1

PiðtÞdth it2 � t1

ð10Þ

The average state probability P[t1, t2] is given by integral to P(t)divided by t2 � t1 (11). Substituting (2) to (11), the analyticalexpression for P[t1, t2] is given by (12). However, U is singular,P[t1, t2] cannot be directly found.

P½t1; t2� ¼1

t2 � t1

Z t2

t1

PðtÞdt ð11Þ

P½t1; t2� ¼U�1½Pðt2Þ � Pðt1Þ�

t2 � t1ð12Þ

A new method is introduced here. Eq. (2) is expanded to (13).Randomly select one state, e.g. n, as the reference. Since sum ofthe state probabilities is 1, Pn(t) may be given by the probabilitiesof other states (14).

_PiðtÞ ¼Xn

j¼1

uijPjðtÞ ð13Þ

PnðtÞ ¼ 1�Xn�1

j¼1

PjðtÞ ð14Þ

Substituting (14) to (13), Pn(t) is deleted. The remaining n � 1states are given by (15), or in matrix form (16), where C and Dare the coefficient matrix. Since C is not singular, the instantaneousprobabilities for the n � 1 states are given analytically by (17).Note if the probabilities of the n � 1 states are found, probabilityof the nth state will also be found.

_PiðtÞ ¼Xn�1

j¼1

uijPjðtÞ þuinPnðtÞ ¼Xn�1

j¼1

ðuij �uinÞPjðtÞ þuin;

i ¼ 1;2; . . . ;n� 1 ð15Þ_PðtÞ ¼ CPðtÞ þ D ð16ÞPðtÞ ¼ eCðt�t0ÞðPðt0Þ þ C�1DÞ � C�1D ð17Þ

UP UP

UP UPDown

element 1

element 2

Time

Time

Down

Simultaneous failure

S. Li et al. / Electrical Power and Energy Systems 43 (2012) 427–432 429

Substituting (16) to (11), the average state probability P[t1, t2] isgiven by (18), where P(t1) and P(t2) are decided by (17), dependenton the initial state P(t0). Then the average unavailability U[t1, t2]will be easily defined. It is actually the average forced outage rate(FOR) of the multi-state equipment between [t1, t2].

P½t1; t2� ¼C�1½Pðt2Þ � Pðt1Þ�

t2 � t1� C�1D ð18Þ

Fig. 4. Simultaneous failure of two components.

4. Equivalence error for series/parallel elements

From the viewpoint of reliability evaluation, the flexible trans-mission system is composed of elements in series and/ or parallelcombination, whose steady-state unavailability is often given bythe multiplication the (un) availabilities (19), where the subscriptss and p denotes the series and parallel elements. In the long-termevaluation, the (un)availability is irrelevant to time.

UsðtÞ ¼ 1�Q

AiðtÞ ¼ 1�Qð1� UiðtÞÞ

UpðtÞ ¼Q

UiðtÞ

�ð19Þ

A two-element system is applied to illustrate the equivalenceerror (16). In Figs. 2 and 3, elements I and II are in series or parallelcombination, where the dashed box indicates the unavailable state,i.e. the system is in down state.

For the series system, the state diagram is checked here. Whenelement I or II is fault, the system stops from work, and failure ofthe other element is impossible, therefore state 4 is not practical,unless the simultaneous failure on both elements is included, asshown in Figs. 4 and 5, where kI,II and lI,II are the simultaneous fail-ure and repair rates between state 1 and 4. In the reliability anal-ysis, simultaneous failure is often ignored due to the assumption ofthe single-step transition, therefore Us in (19) is not suitable forFig. 2b without state (4), nor to Fig. 5.

Now we have two methods to calculate the unavailability of themult-istate systems:

(i) based on Eq. (19), directly multiply the (un)availabilities ofthe series (parallel) elements, and

(ii) based on Eq. (18), analytically solve the state probabilitiesbased on the state transition matrix, then sum the probabil-ities of the failure states to find the unavailability of the totalsystem.

I II

(a) System configuration

Fig. 2. System in seri

I

II

UU1

(a) System configuration

Fig. 3. System in para

In the following, two methods are analyzed for long and shortterm reliability analysis. In the long term evaluation, the steady-state unavailabilities are compared. In the short term evaluation,the average unavailabilities are compared.

4.1. Equivalence error of steady-state unavailabilities

For the parallel systems, the steady-state unavailabilities fromboth methods are the same:

Up ¼kI

kI þ lI

kII

kII þ lIIð20Þ

For the series system, if state 4 is included, the unavailability is gi-ven by (21), otherwise it is given by (22).

Us ¼kIkII þ kIlII þ kIIlI

kIkII þ kIlII þ kIIlI þ lIlIIð21Þ

Us ¼kIlII þ kIIlI

kIlII þ kIIlI þ lIlIIð22Þ

For k identical elements in series configuration, the steady-stateunavailabilities from both methods are given in (23), and the rela-tive error Uerrs(%) is quantified by (24), where g = k/l. As shown inFig. 6, unless g approaches zero, i.e. the system is very reliable, (21)overestimates the unavailability of the series system, yielding pes-simistic judgment.

Us ¼ 1� 1� kkþl

� �k¼ 1� 1� g

gþ1

� �k

Us ¼ kkkkþl ¼

kgkgþ1

8<: ð23Þ

Uerrsð%Þ ¼1� 1� g

gþ1

� �k� kg

kgþ1

kgkgþ1

� 100 ð24Þ

UU1

IλIμ

IIμIIλ

DU2

UD3

DD4

IIμ

Iμ

(b) State-space diagram

es configuration.

DU2

UD3

DD4

IλIμ

IIμIIλ

IIμIIλ

IλIμ

(b) State-space diagram

llel configuration.

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

g

Uer

rs (%

)

k=2

k=4

k=10

Fig. 6. Relative error of steady-state unavailability for series system.

430 S. Li et al. / Electrical Power and Energy Systems 43 (2012) 427–432

4.2. Equivalence error of average unavailabilities

For a system with two elements in either series and parallelconnection, the average unavailability within given time span [0,s] are quantified by two methods.

4.2.1. Equivalence error of average unavailabilities for series systemFor a series system with 2 elements, with the simultaneous fail-

ure ignored, there may be 3 possible initial states, i.e. (1, 0, 0)T, (0,1, 0)T, or (0, 0, 1)T. The corresponding average unavailability are gi-ven by multiplying the availabilities of both elements, as shown in(25), where the 1st term is the steady-state value, while the 2ndand the 3rd terms vary with time, and are dependent on the eval-uation period.

U½0; s� ¼1� ð1� Uð1;0Þ½0; s�Þ2

1� ð1� Uð1;0Þ½0; s�Þð1� Uð0;1Þ½0; s�Þ1� ð1� Uð0;1Þ½0; s�Þ2

8><>:

¼

k2þ2klðkþlÞ2

þ 2lk

ðkþlÞ3se�ðkþlÞs � 1� �

� k2

ðkþlÞ4s2 ½e�ðkþlÞs � 1�2

k2þ2klðkþlÞ2

þ kl�l2

ðkþlÞ3se� kþlð Þs � 1� �

þ klðkþlÞ4s2 ½e�ðkþlÞs � 1�2

k2þ2klðkþlÞ2

þ kl�l2

ðkþlÞ3s½e�ðkþlÞs � 1� þ kl

ðkþlÞ4s2 ½e�ðkþlÞs � 1�2

8>>>><>>>>:

ð25Þ

With the same initial states, the average unavailabilities basedon the analytical solution are given by (26). Compared with (25),both the steady-state term and the time-dependent term aredifferent.

U½0; s� ¼

2k2kþlþ 2k

ð2kþlÞ2s½e�ð2kþlÞs � 1�

2k2kþl�

lð2kþlÞ2s

½e�ð2kþlÞs � 1�2k

2kþl�l

ð2kþlÞ2s½e�ð2kþlÞs � 1�

8>>><>>>:

ð26Þ

4.2.2. Equivalence error of average unavailabilities for parallel systemWith possible initial states of (1, 0, 0, 0)T, (0, 1, 0, 0)T, (0, 0, 1, 0)T,

and (0, 0, 0, 1)T, the average unavailabilities from two methods aregiven by (27) and (28) respectively. It is found that 1st and 2ndterms are the same, but the 3rd terms are different.

U½0;s� ¼

ðUð1;0Þ½0;s�Þ2

ðUð1;0Þ½0;s�ÞðUð0;1Þ½0;s�ÞðUð0;1Þ½0;s�ÞðUð1;0Þ½0;s�ÞðUð0;1Þ½0;s�Þ2

8>>>><>>>>:

¼

k2

ðkþlÞ2þ 2k2

ðkþlÞ3s½e�ðkþlÞs�1�þ k2

ðkþlÞ4s2 ½e�ðkþlÞs�1�2

k2

ðkþlÞ2þ k2�klðkþlÞ3s

½e�ðkþlÞs�1�� klðkþlÞ4s2 ½e�ðkþlÞs�1�2

k2

ðkþlÞ2þ k2�klðkþlÞ3s

½e�ðkþlÞs�1�� klðkþlÞ4s2 ½e�ðkþlÞs�1�2

k2

ðkþlÞ2� 2klðkþlÞ3s

½e�ðkþlÞs�1�þ l2

ðkþlÞ4s2 ½e�ðkþlÞs�1�2

8>>>>>>><>>>>>>>:

ð27Þ

U½0; s� ¼

k2

ðkþlÞ2þ 2k2

ðkþlÞ3s½e�ðkþlÞs � 1� � k2

2ðkþlÞ3s½e�2ðkþlÞs � 1�

k2

ðkþlÞ2þ k2�klðkþlÞ3s

½e�ðkþlÞs � 1� þ kl2ðkþlÞ3s

½e�2ðkþlÞs � 1�k2

ðkþlÞ2þ k2�klðkþlÞ3s

½e�ðkþlÞs � 1� þ kl2ðkþlÞ3s

½e�2ðkþlÞs � 1�k2

ðkþlÞ2� 2klðkþlÞ3s

½e�ðkþlÞs � 1� � l2

2ðkþlÞ3s½e�2ðkþlÞs � 1�

8>>>>>>><>>>>>>>:

ð28Þ

UU1

IλIμ

IIμIIλ

DU2

UD3

DD4

IIμ

Iμ

,I IIλ,I IIμ

(a) Series system

Fig. 5. State transition consid

Based on above analysis, the unavailability by directly multiply-ing the (un)availabilities of the series and/or parallel elementsyields error compared with analytically solving the state equations.The error lies in fixed and time-variant terms of the average unava-ilabilities. In long-term evaluation, the error is caused by thesimultaneous failure incorrectly introduced to the series compo-nents. In short-term evaluation, it is caused by the state transitionbetween the elements.

5. Numerical analysis

The HVDC transmission system is composed of converter sta-tion and DC transmission line. The converter station includes theconverter bridges, transformers, ac filters, dc reactors, controland protection, and auxiliary equipments. As the key part of powertransformation, the bridge is applied to validate the proposedmodel.

5.1. State space for converter bridge

The configuration for a 2-pole, 2-bridge ±800 kV Ultra HVDC(UHVDC) converter station is shown in Fig. 7, with the availablecapacities of 3000–4500 MW, 4500–6400 MW, and 6000–9000 MW respectively[2]. With the failure of one valve, the 12-pulse bridge has to be out of service until the fault valve is repairedand reinstalled. If there are standby valves, the down time of thebridge will be remarkably decreased by fast replacement of thefault valve.

Considering valve outage, there are five possible capacity levels,i.e. 1.0, 0.75, 0.5, 0.25, and 0 for 2-pole double 12 pulse seriesbridge, while there are three capacity levels, i.e. 0.5, 0.25, and 0for single pole bridge. For a single-pole bridge with two redundantvalves, the state space is shown in Fig. 8, where km, lm, cm are thefailure rate, repair rate and installation rate of the valve. The avail-ability of each state, i.e. the converter station satisfies the expectedpower transmission, is decided by the scheduled loading, the over-load capability, and the fast control function. In this paper, the

UU1

DU2

UD3

DD4

IλIμ

IIμIIλ

,I IIμ

IIμIIλ

IλIμ

,I IIλ

(b) Parallel system

ering simultaneous fault.

(a) 12-pulse (b) double 12-pulse in series (c) double 12-pulse in parallel

Fig. 7. Converter station for HVDC transmission.

2Pb 0S

0.51

1Pb 0S

0.254

0Pb 0S

0.08

24 vλ

12 vλ

1Pb 1S

0.255

0Pb 1S

0.09

3 vμ

4 vμ 0Pb 2S

0.010

vγ

vγ 2 vγ

2Pb 1S

0.52

1Pb 2S

0.256

0Pb 3S

0.011

2 vμ

24 vλ

12 vλ 12 vλ3 vγ

2 vγ

2Pb 2S

0.53

1Pb 3S

0.257

0Pb 4S

0.012

vμ

2 vμ

3 vμ

vμ

2 vμ vμ

3 vγ

4 vγ12 vλ

24 vλ

Fig. 8. State space for single converter bridge.

Table 1Steady-state state probability.

State State probability State State probability

1 3.3557 � 10�6 7 4.5599 � 10�6

2 0.002735 8 1.2489 � 10�12

3 0.9971 9 7.4833 � 10�11

4 3.4692 � 10�9 10 8.9646 � 10�9

5 6.7211 � 10�7 11 7.9018 � 10�10

6 1.3680 � 10�4 12 1.9754 � 10�11

0 5 10 15 200

1

2

x 10-4

t/h

U (t

)

(a) From available initial state

4

56

7

1(b) From unavailable initial state

S. Li et al. / Electrical Power and Energy Systems 43 (2012) 427–432 431

unavailable states are defined by states with zero capacity, i.e. 8, 9,10, 11, and 12.

In the following, a single-pole, two-bridge, double 12-pulseconverter is analyzed. The reliability data of the valves are givenby km = 0.6 � 10�5 occ. per hr., lm = 0.05 occ. per hr., cm = 0.5 occ.per hr. The steady-state probability from the long-term reliabilityevaluation is shown in Table 1.

0 5 10 15 200

0.2

0.4

0.6

0.8

t/h

8

9101112

U (t

)

Fig. 9. Instantaneous unavailability with initial states.

5.2. Effect of initial state on unavailability

From different initial states, the state probabilities are shown inFig. 9. They vary remarkably with time, damping to the steady-state values in oscillatory or monotonous manner. From the lessreliable initial state, such as state 4 or 8, the damp time is longer,where the average state probability and unavailability are morenecessary.

The redundant elements are often assumed available whenneeded. However, its is questionable due to the unexposed failure,which may be described by probabilistic initial state. For example,assume the availability is 0.8 for two standby units available, and0.2 for one available, corresponding to the initial state of (0, 0.2,0.8, 0, 0, 0, 0, 0, 0, 0, 0, 0)T. The instantaneous unavailability isshown by the dashed curve in Fig. 10. When the standby unitsare fully available, i.e. the initial state are given by (0, 0, 1, 0, 0,

0, 0, 0, 0, 0, 0, 0)T, the availability is given by the solid curve. Theresults shows that the availability of the standby elements is crit-ical to the overall reliability of flexible transmission equipments.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

x 10-8

t/h

U (t

)

Uncertain initial state

Determined initial state

Fig. 10. Instantaneous unavailability with probabilistic initial state.

Table 2Average unavailability with different evaluation periods.

Initial state s = 0.5 h s = 4 h s = 24 h

1 4.3022 � 10�10 2.3093 � 10�8 1.4625 � 10�7

2 3.8085 � 10�10 1.0240 � 10�8 2.0817 � 10�8

3 3.3817 � 10�10 5.3240 � 10�9 9.0260 � 10�9

4 1.7913 � 10�5 1.1758 � 10�4 1.3186 � 10�4

5 1.5213 � 10�5 3.8453 � 10�5 9.4085 � 10�6

6 1.2966 � 10�5 1.5685 � 10�5 2.8521 � 10�6

7 1.1090 � 10�5 7.8083 � 10�6 1.3314 � 10�6

8 0.9961 0.8516 0.27929 0.8824 0.4025 0.0735

10 0.7856 0.2390 0.040411 0.7030 0.1650 0.027612 0.6321 0.1250 0.0208

432 S. Li et al. / Electrical Power and Energy Systems 43 (2012) 427–432

5.3. Effect of evaluation period on average unavailability

The average unavailabilities with different time spans areshown in Table 2, showing clearly related to the evaluation period.They finally converge to, but are different from the steady-state va-lue. Time dependency of the state probability and unavailability ismore notable at the beginning stage, therefore the instantaneousand average expressions are more valuable with a short time span,such as in hours or shorter.

6. Conclusions

In this paper, the short-term reliability equivalence algorithmfor the flexible transmission equipments is studied. By deletingthe probability of the randomly selected reference state, theinstantaneous and average state probability and unavailabilityare described analytically. The proposed algorithm is applied tothe converter station of the HVDC transmission system. It is foundthat in the short term evaluation, the state probability and unavail-ability is related to the initial state, while their average values arerelated to the initial state and the evaluation period. By multiply-ing the (un)availabilities based on series and parallel configuration,remarkable error exists for the fixed and time-variant terms of theaverage unavailabilities. Uncertainty of the initial states, e.g. possi-ble unexposed outage of standby elements, cannot be ignored inquantifying short-term reliability.

The proposed algorithm is valuable for the reliability evaluationto the electrical equipments with fast transition and redundantelements, such as power converters, relay protections, and windturbine generators.

Acknowledgements

This work was supported by the Program for New Century Excel-lent Talents in University of China (NCET 080765), Fundamental Re-search Funds for the Central Universities (2010HGZY0015), andNational Natural Science Foundation of China (50707006).

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