Interruption Modeling in Electrical Power Distribution

Interruption Modeling in Electrical Power Distribution Systems using Weibull-Markov Model R. Medjoudja ∗ , D. Aissanib A. Boubakeurc and K.D. Haimd a

LAMOS Laboratory, Electrical Engineering Department, University of Bejaia, Algeria,

b

LAMOS Loboratory, Operational Research Department, University of Bejaia, Algeria,

c

L.R.E, High voltage Laboratory, Ecole Nationale Polytechnique, Algiers, Algeria

d

Applied Science of Technology, University of Zittau, Germany.

Abstract: This paper develops an application of Markov and Weibull-Markov methods to evaluate the effects of maintenance actions on electrical distribution systems availability and reliability. It is unifying existing models and critically reviewing them on applicability to the electrical components in various configurations. A particular interest is given to availability optimization with regard to a critical reliability value and maximum benefit of maintenance actions. At least, this article provides a comparison of the results obtained using the Markov method and the Weibull-Markov approach.

Keys Words: Electrical distribution system, Preventive maintenance, Reliability and availability, Markov and Weibull-Markov models.

tion [1]. However, the exponential law, usually used to describe failure, is not always suitable for electrical distribution systems. Applications for two Markovian models are considered with the uniform and with the ageing degradation. The results are compared with those obtained using Weibull-Markov approach associated to maintenance effects. This new approach introduced by Van Castaren [2][3] and cited by Pivatolo [4] is applied with success to reliability assessment under preventive maintenance for electrical components taken individually and for electrical distribution systems under various configurations.

Introduction Providing customers with electricity that maintain their operational capability for a long time, and under changing conditions, constitutes one of the most important responsibilities of electrical system manager. Special attention must be paid to the type of maintenance actions, their frequencies and their costs. In many applications, component failure can be defined into two categories, random failures and those arising as a consequence of deterioration. The deterioration can concern a component or the whole system. The purpose of a preventive maintenance (PM) is to reduce a failure rate which gets higher due to age. Models established in order to control the interruptions have been the subject of many publications. Interruptions in distribution systems occur according to exponential law or Poisson distribution according to the network configura∗ Corresponding

1. Electrical distribution system formulation An electric power network can be considered to consist of a generation system, a transmission system, a sub-transmission system and a distribution system. In general, the distribution system is the final mean to transfer the electric power to the ultimate customer. It has been frequently observed that a major part of service interruptions experienced by individual customer , has its ori-

Author, email:[email protected]

1

2 gin in the distribution system failures. 1.1. Distribution system under study analysis The conception of this part of the electrical network is in looped configuration with normallyopen switches. Under normal conditions a normally-open switch (N.O) is open and, therefore, the system operates in radially configuration.

Figure 1. The electrical system in study Where: HV: High Voltage, MV: Medium Voltage, LV: Low Voltage, CB: Circuit Breaker, Tr: Transformer, L1, L2, ...,L5: Load points, N.O: Opening Node. The voltage level of the studied distribution system ( Bejaia city, Algeria) is medium about 30,000V. Each feeder is made up of serialconnected components. The whole system, as shown in Fig.1, is in a bridge structure and the principle components are: two HV/MV transformers T r1 and T r2 with their protections at the both sides( Circuit-Breakers CBHV and CBM V ). The key component is the switchgear Sw connecting the MV barre buses. The MV geographical distribution part is partitioned in ten (10) feeders. The components initiating failures are discussed individually in the following subsection

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim 1.2. Electrical components initiating event frequencies The components of interest are transformers, circuit breakers, underground cables, overhead lines and switchgears. 1. Transformers (T ri ): Transformer failures resulting in a fire or explosion are rare events, but have occurred frequently enough in the past to warrant concern. Fires in mineral oil insulated transformers typically occur due to the breakdown of liquid or solid insulation within the transformer (caused by overloads, switching, lightning surges, or by gradual deterioration), low insulating oil level, moisture intrusion in the insulating oil, or by failure of an insulating bushing. Dry transformers can also experience faults that lead them to burn; the operating experience events are about equal between the both types of transformers. 2. Circuit breakers (CBi ): Experience data from CB reliability studies has shown that an upper bound fire/explosion frequency for 63 kV and larger voltage power CB units is about (6E − 4)/year. However, operating experiences have shown that these faults are often more prevalent in higher voltage CB units, since those units tend to carry more total energy than lower voltage units [5]. 3. Underground cables and overhead lines: It has stated in reference [5] that if electrical cable connections become loose (due to thermal cycling, bolt mechanical relaxation, etc.), the electrical resistance increases. With the resistance increase, the loose connection tends to overheat via Joule (I 2 R) heating. Another important failure mechanism is cable overheating by drawing excessively high current. The high current also results in Joule heating and subsequent cable insulation breakdown from heating. A distinction must be made between underground cables and overhead lines during the studies and data processing. Due to high exposure, most overhead line damage is caused by external factors such as vegetation, animals and severe weather. 4. Switchgear (Sw ): The behavior of this component is developed in Refs. [5][6] by Cadwallader and Hughes respec-

Article submitted to Journal of Risk and Reliability, August 2008 tively for different voltage levels. The range of failure frequency around (1E − 3)/year. In this article, a particular interest is given to the modeling, statistical processing, reliability assessment and maintenance policy for theses components. 2. Statistical treatment Interruptions data for distribution system components are derived from collections that are based on interruption reports obtained from the field in the last ten years or so [7][8]. The expected goal of this study is the evaluation of reliability indices, availability and the determination of the type of probability distribution to model historical data. 2.1. Interruption durations description In the case of forced outage, the commonly steps of restoration actions are organized as follows [9]: - a fault is signaled , - the trouble-shooting crew is activated and travels to the defect feeder,

3

- the repair and/or replacement time - the verification and alinement time. The modeling process obeys to a flowchart shown in Fig.2 and organized as follows: - evaluate the interruption on all the feeders of a source station from the actual operating history of these feeders, - choose the most representative feeder on which further studies will be conducted in order to model the interruptions, - amongst other things, carry out qualitative analysis of the components from the sample, - develop a mathematical model governing the interruptions, - sort out suggestions and recommendations for preventive maintenance of the system. These steps are followed in the application to the studied case and the obtained results are listed in table 1 for components and for a representative feeder (Ville 1).

- the faulted section or component is located, - the faulted section is isolated, - the feeder circuit breaker is closed to power up the first part of the feeder, - the normally opened switch is closed to power up the last part of the feeder. All above defined actions have respectively user durations tk , k = (1, 2, 3, 4, 5, 6). They are summed to define P6 the time to supply restoration, noted tb = k=1 tk . P4 The preventive maintenance time ta = j=1 tj is the sum of the durations tj , j = (1, 2, 3, 4) of the following tasks: - the access time, - the inspection or diagnosis time,

Figure 2. Method flowchart

2.2. Definitions and notations for stochastic model - The reliability definitions given in the literature vary between different practitioners as well as researchers. The generally accepted definition is as follows: More specific, reliability is

4

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim

the probability that an item will operate properly for a specified period of time under designed operating conditions without failure. Mathematically, reliability R(t) is the probability that a system will be successful in the interval time [0, t]: R(t) = P (T > t), t ≥ 0, where T is a random variable denoting the time-to-failure or failure time. If F (t) is the failure distribution function, F (t) = 1 − R(t), t ≥ 0. Its derivative function, if it exists, is the probability density function of T noted, d f (t) = − dt [R(t)]. -The hazard function which denotes the instantaf (t) neous failure rate is, h(t) = R(t) and represents the probability that the item of age t will fail in the small interval of time t to t + dt. As the two parameters Weibull distribution is dominant in this investigation, the above functions are expressed as follow: t R(t) = exp − ( )β , t ≥ 0, η > 0, β > 0 η t F (t) = 1 − exp − ( )β , t ≥ 0, η > 0, β > 0 η f (t) =

h(t) =

β.tβ−1 t .exp − ( )β , t ≥ 0, η > 0, β > 0 ηβ η β.tβ−1 , t ≥ 0, η > 0, β > 0 ηβ

where η ( same unit as t),β (unitless) denote the scale and the shape parameters of Weibull distribution , respectively. When β = 1, the distribution is exponential with a constant parameter λ, however, h(t) = η1 = λ. The Weibull distribution is widely used to analyze the cumulative loss of performance of a complex system in systems engineering. In general, it can be used to describe the data on waiting time until an event occurs. In this manner, it is applied in risk analysis, actuarial science and engineering. -The availability of a system is defined as the probability that the system is successful at time t. It is a measure of success used primarily for repairable systems, A(t) will be equal to or greater than R(t).

-The mean time between failures M T BF is an important measure in repairable systems. This implies that the system has failed and has been repaired. Mathematically, M T BF = M U T + M DT , where the M U T and M DT are the mean up time and the mean down time, respectively. When the system being tested is renewed through maintenance and repair, the expectation E(T ) is known as M T BF . - The Kolmogorov-Smirnov (KS) Test is nonparameters test and assumes only a continuous distribution. Let X1 ≤ X2 ≤ X3 ≤ ... ≤ Xn denote the ordered sample value. The observed distribution function is given by Fn (x) = ni , for xi < x ≤ xi+1 Assume the testing hypothesis, H0 : F (x) = F0 (x), where F0 (x) is a given continuous distribution and F (x) is an unknown distribution. Let dn = sup|Fn (x) − F0 (x)|, −∞ < x < ∞. If dn ≤ dn,α then the hypothesis is not rejected, otherwise, it is rejected (when dn > dn,α . The (KS) test composed of eight steps is widely described in Refs [10][11]. The dn,α is given in critical values (KS) test tables [11], where, n is the sample size and α is the level of significance. In the following application are considered n = 17 and α = 0.05. 2.3. Availability processing Availability is viewed as being the consequence of reliability and maintainability. Operational availability is defined by [12][13] : MUT M U T + M DT Z tm M U T = tm − tb h(t)dt A=

(1) (2)

0

Z

tm

M DT = ta + tb

h(t)dt

(3)

0

Where tm , ta and tb are respectively, the PM interval, the PM and CM (corrective maintenance) times on replacement. Substituting Eqs.2 and 3 into Eq.1, availability can be written as: Rt tm − tb 0 m h(t)dt (4) A= tm + ta

Article submitted to Journal of Risk and Reliability, August 2008

5

Subsequently, the PM interval for maximizing the availability can be derived by differentiating Eq.4 to time tm .

0.7915

0.1762 β=1.992423 , η = 5.474620∗104

λ =0.3203609

Weibull

Exponential

0.318

0.2443

λ =0.649

Weibull

Exponential

17

17

Overhead line

Feeder

0.938

0.9359

0.2246

β=1.128344 , η= 9.595387104

Weibull 17 Underound cable

Exponential

λ =0.066

0.318

0.318

not rejected rejected not rejected rejected not rejected rejected not rejected rejected 0.8775

0.2264

β=1.802051 , η= 3.526378 ∗ 105

β= 2.459579 , η = 5.909754 ∗ 106 Weibull

Exponential

17

(Tr)

MV/LV Transformer

λ =0.007785467

0.318

not rejected rejected 0.9897 λ = 0.01038062

0.185 Weibull

Exponential

17 Internal cable

β= 1.30000 , η = 3.612098 ∗ 106

0.318

0.318 0.8382

0.3136

λ =0.1764706

17

Weibull

Exponential connector (E.C.C)

β=2.874256, η = 4.241950 ∗ 105 External cable

connector (I.C.C)

Decision d(n,0.05) dks Parameters n

Distribution

function

The obtained results show that, dks < d(n,0.05) and the Weibull distribution is not rejected, however with the exponential law, dks > d(n,0.05) , the hypothesis is not accepted. Tatietse et al [15] and Milanovic [16] have stated that the exponential law, usually used to describe failure, is not always 100% suitable for electricity distribution systems. Reliability indices described above are assessed and listed in table 2. To show the contribution of the same type of components to the system failure, is introduced the annual failure frequency (Fi ) processed using Eq.11.

or sub-system

2.4. Reliability Data Analysis It deals with estimation of parameters, selection of distribution functions and many other aspects, such as: modeling and assessment. Failure data are collected from 1999 to 2007, corresponding to 17 years of system operation continuously at the national company of electricity and gas [14]. It concerns forced and planned outages of the components of the system, such as: MV/LV transformers (Tr), internal cable connectors ( I.C.C), external cable connectors (E.C.C), underground cables and overhead lines. (I.C.C), is an electrical component used to assembly an underground cable to the terminations of the MV side of the power transformer in the sub-station. However, (E.C.C) is used to assembly the underground cable to the overhead line at the top of the electric pole. In table 1, are listed the estimated parameters of the components/sub-system and the retained distribution functions.

Component

(6)

The results of availability processing are listed in table 2.

not rejected rejected

(5)

Table 1: Distribution functions and their parameters for working durations

dA =0 dt The differential result is: Z tm ta (tm + ta )h(tm ) − h(t)dt = tb 0

0.3203609 0.9963486 48698.82 177.8183 4.8521 ∗ 104 Feeder

10.79290

0.649 (1/km) 0.981567 9256505 1706.25 90858.8.3 Overhead line

5.678174

0.066 (1/km) 0.9947954 315219.5 313578.9 Underground cable

1640.563

1.667438

0.007785467 0.9998858 5242099 598.7689 MV/LV Transformer (Tr)

5.2415 ∗ 106

0.1002449

0.01038062 0.1575198 0.9999622 3336175 126.1282 Internal cable connector (I.C.C)

3.336049 ∗ 106

1.390368 0.9997107 378219.4 109.3639 External cable connector (E.C.C)

3.7811 ∗ 105

Fi (1/year) A M T B F (mn) M D T (mn) M U T (mn) Component or sub-system

Table 2: Components reliability indices

0.1764706

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim

h (1/year)

6

The results given in Table 2 show that the average annual interruption rate per km is very significant: compared with between 0.298 and 0.015 in the case of well-maintained lines [17]. Despite the interesting value of availability, the frequency of breakdowns is widely spaced from the objective value fixed by the company [7]. By comparison of expected performances of underground cables and overhead lines, depending on the cost/benifit balance, an obvious recommendation could be enumerated: proceed to ageing replacement for underground cables and encourage the use of underground circuits in urban areas, however the number of external cables connectors and their failures will be reduced. 3. Failure and maintenance models In many applications, component failure can be defined into two categories as shown in Fig.3, random failures and those arising as a consequence of deterioration.

Figure 3. a: Two states diagram, b: State diagram with degradation stages Where: W, F: Working and failure states respectively, D1 ,D2 and Dk are degradation states, λ, µ, ρ1 , ρ2 and ρk are transition rates. The deterioration process is represented by a sequence of stages of increasing wear leading to equipment failure. It is assumed that maintenance will bring an improvement to the conditions in the previous stage of deterioration (minimal maintenance) [18]. The reader can find the development of the improvement factors in Ref. [19] and introduced in the fourth section of this paper. In Ref. [20], Chan and Asgrpoor present a model where phasetype distribution can embedded into Markovian model. Endrenyi et al in Ref. [18] and Paneda et al in Ref. [21] state that if deterioration is modeled as occurring in a limited number k of

Article submitted to Journal of Risk and Reliability, August 2008 discrete steps, then minimal preventive maintenance sets back the process by one step. This improves the component from stage i to stage (i−1) of deterioration. The case of three steps is shown in Fig.4, where M1 and M2 are minimal maintenance states and I a possible inspection state which can be omitted.

Figure 4. Degradation stages of a reparable Component/system with minimal maintenance

Two points of view are presented, where the times spent in each stage of deterioration are exponentially distributed: for the first one, the degradation occurs uniformly , with an identi1 cal mean time as ρ11 = ρ12 = ρ13 = 3λ . For the second one, corresponding for an ageing degradation, the transitions between stages occur with decreasing mean times of ρ11 > ρ1 > ρ1 . 2

3

7

with a parameter λm . Times to maintenance are exponentially distributed with a mean of λ1m . Assumption is made that the repair after failure due to deterioration is perfect with a mean of µ1 . 3.1. Markov models development The Markov method has been developed earlier in many publications. In reference [3], Van Casteren et al has defined the homogenous Markov model through out the set of the possible states of a component c, its stochastic history and the set of continuous probability distribution functions. If λc,ij is the transition rate from state i to state j, and with the assumption: N

X 1 1 = λ c,i j=1 λc,ij The expected state duration is given by: 1 E(Dc,i ) = λ c,i The transition probability is given by: λc,i Pc (i, j) = λc,ij

(7)

(8)

(9)

The state probabilities of a component can be calculated as follows: π(i).E(Dc,i ) (10) Pc (i) = PN i=1 π(i).E(Dc,i ) The state frequencies are calculated by: Fc (i) =

Pc (i) E(Dc,i )

(11)

where: π(n) = (P (Xn = 1), P (Xn = 2), ..., P (Xn = N )) : the probabilities vector of the chain states. Figure 5. Degradation stages including perfect maintenance In this paper, as illustrated in Fig.5, is considered that an overall replacement (state M3 ) can occur at any level of deterioration, and always produce as-good-as new condition with a rate of µ0 . Therefore, the time to deterioration failure is represented by an Earlangian distribution. Maintenance actions are modeled as a Poisson process

This theory is applied to the studied system in the above section. This application is more detailed in the next sub-section and the results are discussed. 3.2. Case study application Considering the MV/LV transformer, let [Sij ] the rate transition matrix between different states and [Pi ] the steady state probability vector, (i=1,...,7), e.g P = [P1 P2 P3 P4 P5 P6 P7 ]= [PD1 PD2 PD3 PM1 PM2 PM3 PF ].

8

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim 

[Sij ] =

 

−Σ1 0 0 µm 0 µ0 µ1

ρ1 −Σ2 0 0 µm 0 0

0 ρ2 −Σ3 0 0 0 0

0 λm 0 −µm 0 0 0

0 0 λm 0 −µm 0 0

λ0 λ0 λ0 0 0 −µ0 0



0 0 ρ3 0 0 0 −µ1

4. Weibull-Markov modeling

 

According to Van Castaren in Ref. [3], a component of Weibull-Markov is defined by: 1. the set of the possible states, 2. the stochastic life: 3. the set of continuous probability distribution functions Fc (t) for the conditional state durations Dc,ij . By taking a same shape factor for all conditional state durations in a same state:

where: Σ1 = ρ1 +λ0 , Σ2 = ρ2 +λ0 +λm and Σ3 = ρ3 + λ0 + λm . The set of steady state equation can be solved as: P.S T = 0 7 X

(12)

Pi = 1

(13)

i=1

βc,ij = βc,i

The availability is the sum of the probabilities that the component or system is in service; it is given by: A(λm ) = P1 + P2 + P3

N

(

(14)

Using the method developed in subsection (2.1), the transition probabilities can be gathered in the following matrix, where: Σ4 = ρ2 λm + λ0 λm + ρ2 λ0 and Σ5 = ρ3 λm + λ0 λm + ρ3 λ0 .  0 λ0  ρ1 0 0 0 0   [Pij ] = 

Σ1 0

0

λ0 λm Σ4

ρ2 λ0 Σ4

0

0

0

0

1 0 1 1

0 1 0 0

0 0 0 0

0 0 0 0

0 ρ3 λ0 Σ5 0 0 0 0

Σ1 ρ2 λm Σ4 ρ3 λm Σ5 0 0 0 0

0 λ0 λm Σ5 0 0 0 0

a) first model (Model- I): The degradation is assumed to be uniform, e.g, with identical mean times between states. 1 Let: ρ11 = ρ12 = ρ13 = 3λ =10years. Considering the times defined in sub-sub-section (1.3.1), the matrix of transition probabilities is:  0 0.5882 0  0 0 0.4118 0 [Pij ] =

 

0 0 1 0 1 1

0 0 0 1 0 0

0.50 0 0 0 0 0

0.15 0 0 0 0 0

0 0.15 0 0 0 0

0.35 0.50 0 0 0 0

0 0.35 0 0 0 0

 

b) second model (Model- II): The degradation obeys to an ageing process, e.g, decreasing mean times between states. Let: ρ11 = 15years, ρ12 =10years and ρ13 =5years. Considering the times defined in sub-sub-section (1.3.1), the matrix of transition probabilities is:  0 0.6818 0  0 0 0.3182 0 [Pij ] =

 

0 0 1 0 1 1

0 0 0 1 0 0

0.5 0 0 0 0 0

0.15 0 0 0 0 0

0 0.2 0 0 0 0

0.35 0.4666 0 0 0 0

0 0.3334 0 0 0 0

 

(15)

   

1 βc,i X 1 βc,i ) = ( ) η c,i ηc,ij i=1

(16)

The transition probability matrix can be deduced as follows: ηc,i Pc (i, j) = (17) ηc,ij The state probabilities and the state frequencies can be calculated using Eqs.10 and 11; where the average duration of state i is: E(Dc,i ) = ηi Γ(1 +

1 ) βc,i

(18)

A Weibull-Markov system is a combination of Weibull-Markov components. Considering the studied system presented in section one: the feeder constitutes a sub-system and all the components are presumedly independent and in series and the whole system is in a bridge configuration. 4.1. Application to the case study For the component considered in the above subsection, the probabilities of sojourn in different states, the failure frequency and the availability are processed using equations deduced from the Weibull-Markov approach (noted as Model- III) and the parameters evaluated in section one and illustrated in Fig.6. Where: Si are the different states given in Fig:5 without inspection, and Pij the transition probabilities between these different states. The transition probabilities matrix is given as follow:

Article submitted to Journal of Risk and Reliability, August 2008

9

Table 4 Comparison of state probabilities with maintenance actions P1 P2 P3 P4 P5 P6 P7 A

Figure 6. Weibull-Markov model with transition probabilities

 [Pij ] =

 

0 0 0 1 0 1 1

0.5362 0 0 0 1 0 0

0 0.2167 0 0 0 0 0

0 0.3499 0 0 0 0 0

0 0 0.4343 0 0 0 0

0.6222 0.4333 0.4040 0 0 0 0

0 0 0.1616 0 0 0 0

In tables 3 and 4 are listed the results of processing states probabilities and availability for the three models cited above, considering the effects of maintenance. The maintenance actions are developed in details in section four. Table 3 Comparison between state probabilities with out maintenance actions P1 P2 P3 P7 A

M odel − I 0.3332674 0.3332674 0.3332674 0.0003197 0.9996802

M odel − II 0.4998401 0.3332267 0.1666133 0.0003197 0.9996801

M odel − III 0.5027845 0.3319982 0.1648643 0.0003529 0.9996470

The sojourn probabilities in the degradation states are identical for the first model. For the Weibull-Markov approach, the probability of stay in the initial state is very important comparing with the other states. Looking the intermediary model (Model-II), judged more probable, one can conclude that the results given by WeibullMarkov approach are more realistic than those given by the Markov method. The results given in table 4 confirm what is often reported in theory, that preventive maintenance has no effect when the modeling is done

M odel − I 0.3323 0.3323 0.3323 0.0000910 0.0000910 0.02549773 0.0003188973 0.9969

M odel − II 0.3853 0.3503 0.2627 0.000047988 0.000047988 0.0013, 0.00016808 0.9983

M odel − III 0.6507 0.2464 0.0913 0.00034515 0.00034515 0.0097 0.0012 0.9884

  

throughout exponential distribution, e.g Markov model. However the improvement of maintenance actions is obvious looking the results obtained using Weibull-Markov Approach. The objective of the maintenance is that the component or system may have the state S1 most likely and that state S3 is the least likely as confirmed by the WeibullMarkov model. It would be noted that the models parameters are calculated from data feedback. 5. Maintenance effects modeling In this section we study the preventive maintenance (PM) in considering simultaneously three actions noted (1a), (1b) and (2P) defined by Tsai et al [19], and applied to the cases of mechatronic and mechanical systems, as: mechanical service, repair and replacement respectively. The goal to attempt in this part of this investigation is to apply this theory with success to electrical systems taking into account the WeibullMarkov approach. 5.1. Reliability under preventive maintenance The improvement of maintenance to reliability is developed using two factors and the selection of the action to do for the components on every PM stage is decided by maximizing system benefit in maintenance. Depending on the percent of the survival parts of system when it is maintained, the reliability function giving the probability that the system is

10

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim

always working on the time interval [tj−1 tj ] is: Rj (t) = R0,j .Rν,j (t)

(19)

where R0,j is the initial reliability of the j th stage and Rν,j (t) the reliability degradation of surviving parts on this stage. Considering periodical preventive maintenance which interval is tm , the reliability of surviving parts is defined as: Rν,j (t) = R(

1 (t − (j − 1)tm )) m1

(20)

with:(j − 1)tm ≤ t ≤ jtm and m1 (0 < m1 ≤ 1) is the improvement factor of action (1a). To model the reliability of systems following PM, the effects of various actions on R0,j and Rν,j must be evaluated. R0,j = Rf,j−1 = R0,j−1 .R(tm )

(21)

Where R0,j , Rf,j−1 indicate the initial and final reliability values of the system on the (j −1)th stage. Action (1b) can improve the surviving parts of the system and also recover the failed parts. Generally, the impact of this action on the failed parts can be measured by an improvement factor m2 , which is also set between 0 and 1 representing the restored level except the surviving parts. According to the definition, the initial reliability on the action (1b) can be expressed as: R0,j = Rf,j−1 + m2 (R0 − Rf,j−1 )

(22)

where R0 denotes the initial reliability of the new system. The system reliability is expressed as: Rj (t) = R0,j .exp−[(

1 (t − (j − 1)tm ))/η]β (23) m1

The interest of the Weibull-Markov approach is valorized on the fact that the modeling follows state diagram method as illustrated in Fig.6 and the Weibull distribution is retained for the rest of the applications. 5.2. Optimization The benefit of a component maintenance on the j th stage is defined as [19]: R∞ R∞ Ri,j+1 (t)dt − tj Ri,j (t)dt tj Bi,k = (24) Ci,k

Where: i, k denote the ith sub-system or component and the maintenance action considered respectively and Ci,k , the action cost. The advantageous action will corresponds to the maximum of the benefit, e.g Bi∗ = M ax(Bi,k ). Once the action of maintenance is defined and retained, the availability of the system at any stage is processed as: Pn R tj T − tb,m i=1 tj−1 hi,j (t)dt Pn Asj = (25) T + i=1 ti,k,a Where: n is the number of components or subsystems and ti,k,a is the time of the preventive maintenance ( actions (1a), (1b) and (2P))and T , the cycle time. 5.3. Application to the studied system i) Case of serial sub-system (MV/LV SubStation): Next, are defined the different actions for the components of interest. 1- Transformer: - action (1a): cleaning, lubricating, tightening and oil level verification. - action (1b): oil replacement. - action (2P ): transformer replacement. 2- Internal cable connector: - action (1a): retreading and weeding. - action (2P ): internal cable connector replacement. Parameters are needed to compute the benefits such as: the distribution function parameters (β, η), P M and CM times (ta and tb ), maintenance actions costs ( C1a , C1b , and C2P ) and the critical value of reliability (Rcrit = 0.8) (see table 5).

Table 5 Components parameters Ss

α

β

η

ta (j)

tb (j)

m1

m2

C1a $

C1b $

C

Tr

1

2.45

4103.99

3.5

28

0.80

0.90

600

1500

8

I.C.C

1

1.30

2508.40

0.16

2.83

0.85

0.89

-

150

1

The maintenance interval is processed for each component using Eq.6. The obtained results are:

Article submitted to Journal of Risk and Reliability, August 2008 (tm (I.C.C) =715 days, tm (Tr) = 2415 days), however the maintenance interval for the system is Tm = min{715, 2415}=715days. The determination of the action to do is based on the benefits processing, and the retained one corresponds to the maximum value using Eq.24 ( see table 6).

11

stage, the reliability at the coming stage of maintenance R((j + 1) × Tm ) =R(2 ×Tm ) = 0.618 < Rcrit =0.80 and the benefits of the actions (1b,2p) are (4.900,1.217) respectively. The action (1b) is retained looking the maximum value of the benefit. In table 7, are listed the costs of preventive maintenance and their corresponding values of availability at different stages.

Table 6 The correlation between actions and benefits Stage

Action

proposed

1

2

3

4

5

6

7

8

9

10

1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p 1a 1b 2p

R((j + 1)Tm ) Tr

I.C.C

0.9280

0.618

0.8165

0.559

0.6628

0.468

0.7174

0.369

0.8777

0.272

0.7341

0.618

0.8768

0.560

0.7232

0.468

0.8663

0.370

0.7039

0.920

benef it$

Tr * * * * * * 2.5474 2.1365 0.4713 1.5164 1.7309 0.4284 * * * 0.8158 1.7081 0.4410 * * * 1.0691 1.6781 0.4592 * * * 1.2985 1.6646 0.4807

Action retained

I.C.C

Tr

I.C.C

4.90 1.217

0

2

3.348 1.774

0

2

3.474 2.436

1

2

3.411 2.986

2

2

3.172 3.428

0

3

4.836 1.253

2

2

3.348 1.774

0

2

3.474 1.639

2

2

3.411 2.986

0

2

0.272 3.428

2

3

Notation: ∗:no maintenance is needed; 0:do nothing; 1: action (1a) is carried out; 2: action (1b) is carried out; 3: action (2P ) is carried out; R((j + 1)Tm ) : the instantaneous reliability at (j+1)×Tm . At every stage of maintenance, we verify for each component if its reliability for the coming stage is greater or equal to the Rcrit . If the condition is realized, the decision is do nothing. For the example of transformer, at j = 1, the reliability is R((j + 1) × Tm ) =R(2 ×Tm ) = 0.9280 > Rcrit = 0.80, however no maintenance is needed for the stage (j=1). If no, we compute the benefit for each action proposed and choose the maximum value. For example, for the internal cable connector, at j = 1

Table 7 Availability and costs for maintenance stages Stage tj A Costs(MP)$ j=1 715 0.9816 150 j=2 1430 0.9706 150 j=3 2145 0.9559 750 j=4 2860 0.9399 1650 j=5 3575 0.9410 1050 j=6 4290 0.9728 1650 j=7 5005 0.9724 150 j=8 5720 0.9572 1650 j=9 6435 0.9529 150 j=10 7150 0.9340 2550

An example of reliability changing is illustrated in Fig.7, for a component (case of transformer) and in Fig.8, for a sub-system ( case of transformer in series with the internal cable connector).

Figure 7. Reliability changing of the transformer

12

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim

The reliability changing evolution shown in Fig.7, obeys to the indications given in table 6. The action (1a) is retained at the third stage. At this stage, it slow done the moving velocity of the strength distribution and delay degradation time. However, at the forth stage, the (1b) maintenance action is carried out at tj = 2860days and the reliability improvement is obvious and clearly shown in this figure.

Figure 8. Reliability changing of the serial system

ii) Case of a system with a bridge structure ( see Fig.1): Regarding the studied system, the bridge configuration for reliability modeling is retained. The system is composed with five sub-systems. The maintenance actions concern two feeders (ville1 and zone1) and the both HV/MV transformers. 1- For the feeders: - action (1a): cleaning, lubricating, tightening and oil level verification for MV/LV transformers. - action (1b): oil replacement for MV/LV transformers, weeding and cable connectors replacement. - action (2P ): MV/LV transformers replacement. 2- For HV/MV transformers: - action (1a): cleaning, lubricating, tightening and oil level verification for HV/MV transformers, retreading and weeding. - action (2P ): Oil replacement for transformers. In this section, the replacement of HV/MV transformers is considered unrealistic looking to

the large life duration of the component (about 60 years). This component corresponds to a set of technologies with a specific congestion , subject to multitude changes during this period. In table 8, are gathered the parameters of components and sub-systems under study. A maintenance plant is dressed in table 9 to illustrate the correlation between preventive maintenance actions, theirs costs and the availability of different components/sub-systems.

T r1 T r2 F eeder1 F eeder2

14666 14666 97 39

η 20266 20266 46 8.35

Tm (days) 7 7 3 3

ta (h) 2 2 1 2

tb (days) 0.85 0.85 0.80 0.81

m1 0.85 0.85 0.80 0.81

m2 1400 1400 2250 3750

C1a $

Cost(PM)$ 0 3750 102550 491500 0 3750 16400 43000 2500 3750

0.9375 0.8237 0.6664 0.6386 0.9187 0.7977 0.6203 0.6565 0.8288 0.841

60000 60000 25800 43080

C2p $

A

2960 2960 6150 10250

C1b $

Table 9: Maintenance schedule, availability and costs Maintenance stage tj retained action T r1 T r2 F eeder1 F eeder2 j=1 8.35 0 0 0 0 j=2 16.7 0 0 0 1 j=3 25.05 0 0 0 2 j=4 33.4 0 0 2 3 j=5 42.35 0 0 0 0 j=6 50.7 0 0 0 1 j=7 59.05 0 0 2 2 j=8 67.4 0 0 0 3 j=9 75.75 0 0 3 0 j=10 84.1 0 0 0 1

1.4 1.4 1.58 1.99

β

Table 8: Components and sub-systems parameters

Article submitted to Journal of Risk and Reliability, August 2008 13

14 An example of reliability changing is illustrated in Figs.9 and 10, for feeders (case of components in series). The feeder ville 1 is more long than the feeder Zone 1 and over than 2/3 of its length is corresponds to overhead lines. The investigation by comparison of the reliability changing in these figures, show that the number of maintenances in the aggregate system (feeder 1) is the times greater than in case of the feeder 2. In Fig.11, representing the reliability evolution of the whole studied system (in bridge configuration), the maintenance schedule obeys to the indications given in table 7 and 9. It shows that the reliability level is kept very high. When possible, this kind of configuration is to encourage, it offers more advantages.

R.Medjoudj, D. Aissani, A. Boubakeur and K.D. Haim

Figure 11. Reliability changing of the system in a bridge configuration

GAZ) and supported by a grant from the Algerian ministry of education and scientific research (MESRS)throughout the university of Bejaia. 7. Conclusion

Figure 9. Reliability changing of the representative feeder 1 (ville1)

Figure 10. Reliability changing of the representative feeder 2 (zone1) 6. Acknowledgements The encouragements, the helpful and constructive comments from the referees and editors are very much appreciated. This work has been carried out with the co-operation of the national company of gas and electricity (SONEL-

This paper presented a review of methods used for interruption modeling of electrical system and their applications. Two cases of Markov methods are compared with Weibull-Markov approach and the obtained results confirm the adequacy of this later. A maintenance policy based on a threshold value of reliability and a maximum benefit is developed and applied to the case of electrical system. The applications are carried out for components taken individually and for two types of configurations, as : serial and bridge configurations. In practical operation, the results of the current state of the network can encourage operators and managers to act first on equipment that reduces the performances of the system. This work shows that it is possible to maintain the equipment other than the traditional methods. It is involved in an optimal manner under the constraints known a priori. By following the methodology explained and applications made, this work can be extended to other more complex systems. Some recommendations are necessary, namely: Reduce the very longer departures to reduce the number of consumers affected by an interruption, do lines replacement by underground cables in urban areas and proceed to the renewal of underground cables when the number of joint nodes

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